Advances in Neural Networks - ISNN 2006: Third International Symposium on Neural Networks, ISNN 2006, Chengdu, China, May 28 - June 1, 2006, Proceedings,

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

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Jun Wang Zhang Yi Jacek M. Zurada Bao-Liang Lu Hujun Yin (Eds.)

Advances in Neural Networks – ISNN 2006 Third International Symposium on Neural Networks Chengdu, China, May 28 – June 1, 2006 Proceedings, Part I

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Volume Editors Jun Wang The Chinese University of Hong Kong Dept. of Automation and Computer-Aided Engineering Shatin, New Territories, Hong Kong E-mail: [email protected] Zhang Yi University of Electronic Science and Technology of China School of Computer Science and Engineering Chengdu, Sichuan, China E-mail: [email protected] Jacek M. Zurada University of Louisville, Dept. of Electrical and Computer Engineering Louisville, Kentucky, USA E-mail: [email protected] Bao-Liang Lu Shanghai Jiao Tong University, Dept. of Computer Science and Engineering Shanghai, China E-mail: [email protected] Hujun Yin University of Manchester, School of Electrical and Electronic Engineering Manchester M60 IQD, UK E-mail: [email protected] Library of Congress Control Number: 2006925897 CR Subject Classification (1998): F.1, F.2, D.1, G.2, I.2, C.2, I.4-5, J.1-4 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-540-34439-X Springer Berlin Heidelberg New York 978-3-540-34439-1 Springer Berlin Heidelberg New York

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Preface

This book and its sister volumes constitute the Proceedings of the Third International Symposium on Neural Networks (ISNN 2006) held in Chengdu in southwestern China during May 28–31, 2006. After a successful ISNN 2004 in Dalian and ISNN 2005 in Chongqing, ISNN became a well-established series of conferences on neural computation in the region with growing popularity and improving quality. ISNN 2006 received 2472 submissions from authors in 43 countries and regions (mainland China, Hong Kong, Macao, Taiwan, South Korea, Japan, Singapore, Thailand, Malaysia, India, Pakistan, Iran, Qatar, Turkey, Greece, Romania, Lithuania, Slovakia, Poland, Finland, Norway, Sweden, Demark, Germany, France, Spain, Portugal, Belgium, Netherlands, UK, Ireland, Canada, USA, Mexico, Cuba, Venezuela, Brazil, Chile, Australia, New Zealand, South Africa, Nigeria, and Tunisia) across six continents (Asia, Europe, North America, South America, Africa, and Oceania). Based on rigorous reviews, 616 high-quality papers were selected for publication in the proceedings with the acceptance rate being less than 25%. The papers are organized in 27 cohesive sections covering all major topics of neural network research and development. In addition to the numerous contributed papers, ten distinguished scholars gave plenary speeches (Robert J. Marks II, Erkki Oja, Marios M. Polycarpou, Donald C. Wunsch II, Zongben Xu, and Bo Zhang) and tutorials (Walter J. Freeman, Derong Liu, Paul J. Werbos, and Jacek M. Zurada). ISNN 2006 provided an academic forum for the participants to disseminate their new research findings and discuss emerging areas of research. It also created a stimulating environment for the participants to interact and exchange information on future challenges and opportunities of neural network research. Many volunteers and organizations made great contributions to ISNN 2006. The organizers are grateful to the University of Electronic Science and Technology of China and the Chinese University of Hong Kong for their sponsorship; to the National Natural Science Foundation of China and K.C. Wong Education Foundation of Hong Kong for their financial supports; and to the Asia Pacific Neural Network Assembly, European Neural Network Society, IEEE Computational Intelligence Society, IEEE Circuits and Systems Society, and International Neural Network Society for their technical cosponsorship. The organizers would like to thank the members of the Advisory Committee for their supports, the members of the International Program Committee for reviewing the papers and members of the Publications Committee for checking the accepted papers in a short period of time. Particularly, the organizers would like to thank the publisher, Springer, for publishing the proceedings in the prestigious series of

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Preface

Lecture Notes in Computer Science. Last but not least, the organizers would like to thank all the speakers and authors for their active participation at ISNN 2006, which is essential for the success of the symposium. May 2006

Jun Wang Zhang Yi Jacek M. Zurada Bao-Liang Lu Hujun Yin

ISNN 2006 Organization

ISNN 2006 was organized and sponsored by the University of Electronic Science and Technology of China and the Chinese University of Hong Kong. It was technically cosponsored by the Asia Pacific Neural Network Assembly, European Neural Network Society, IEEE Circuits and Systems Society, IEEE Computational Intelligence Society, and International Neural Network Society. It was financially supported by the National Natural Science Foundation of China and K.C. Wong Education Foundation of Hong Kong. Jun Wang, Hong Kong (General Chair) Zhang Yi, Chengdu, China (General Co-chair) Jacek M. Zurada, Louisville, USA (General Co-chair) Advisory Committee Shun-ichi Amari, Tokyo, Japan (Chair) Hojjat Adeli, Columbus, USA Guoliang Chen, Hefei, China Chunbo Feng, Nanjing, China Kunihiko Fukushima, Tokyo, Japan Okyay Kaynak, Istanbul, Turkey Yanda Li, Beijing, China Erkki Oja, Helsinki, Finland Marios M. Polycarpou, Nicosia, Cyprus Shoujue Wang, Beijing, China Youlun Xiong, Wuhan, China Shuzi Yang, Wuhan, China Siying Zhang, Qingdao, China Shoubin Zou, Chengdu, China

Walter J. Freeman, Berkeley, USA (Co-chair) Zheng Bao, X’ian, China Ruwei Dai, Beijing, China Toshio Fukuda, Nagoya, Japan Zhenya He, Nanjing, China Frank L. Lewis, Fort Worth, USA Ruqian Lu, Beijing, China Nikhil R. Pal, Calcutta, India Tzyh-Jong Tarn, St. Louis, USA Paul J. Werbos, Washington, USA Lei Xu, Hong Kong Bo Zhang, Beijing, China Nanning Zheng, Xi’an, China

Steering Committee Zongben Xu, Xi'an, China (Chair) Tianping Chen, Shanghai, China Wlodzislaw Duch, Torun, Poland Xiaoxin Liao, Wuhan, China Zhiyong Liu, Beijing, China Zhengqi Sun, Beijing, China Donald C. Wunsch II, Rolla, USA Fuliang Yin, Dalian, China Liming Zhang, Shanghai, Chinac Mingtian Zhou, Chengdu, China

Houjun Wang, Chengdu, China (Co-chair) Andrzej Cichocki, Tokyo, Japan Anthony Kuh, Honolulu, USA Derong Liu, Chicago, USA Leszek Rutkowski, Czestochowa, Poland DeLiang Wang, Columbus, USA Gary G. Yen, Stillwater, USA Juebang Yu, Chengdu, China Chunguang Zhou, Changchun, China

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Organization

Program Committee Bao-Liang Lu, Shanghai, China (Chair) Shigeo Abe, Kobe, Japan Khurshid Ahmad, Surrey, UK A. Bouzerdoum, Wollongong, Australia Jinde Cao, Nanjing, China Matthew Casey, Surrey, UK Luonan Chen, Osaka, Japan Yen-Wei Chen, Kyoto, Japan Yuehui Chen, Jinan, China Yiu Ming Cheung, Hong Kong Sungzoon Cho, Seoul, Korea Emilio Corchado, Burgos, Spain Shuxue Ding, Fukushima, Japan Meng Joo Er, Singapore Mauro Forti, Siena, Italy Marcus Gallagher, Brisbane, Australia Chengling Gou, Beijing, China Lei Guo, Nanjing, China Min Han, Dalian, China Zhifeng Hao, Guangzhou, China Zengguang Hou, Beijing, China Jinglu Hu, Fukuoka, Japan Guangbin Huang, Singapore Marc van Hulle, Leuven, Belgium Danchi Jiang, Hobart, Australia Hoon Kang, Seoul, Korea Samuel Kaski, Helsinki, Finland Tai-hoon Kim, Seoul, Korea Yean-Der Kuan, Taipei, Taiwan James Lam, Hong Kong Xiaoli Li, Birmingham, UK Yuanqing Li, Singapore Xun Liang, Beijing, China Lizhi Liao, Hong Kong Fei Liu, Wuxi, China Ju Liu, Jinan, China Hongtao Lu, Shanghai, China Fa-Long Luo, San Jose, USA Jinwen Ma, Beijing, China Stanislaw Osowski, Warsaw, Poland Ikuko Nishkawa, Kyoto, Japan Paul S. Pang, Auckland, New Zealand Yi Shen, Wuhan, China Michael Small, Hong Kong Ponnuthurai N. Suganthan, Singapore Fuchun Sun, Beijing, China

Hujun Yin, Manchester, UK (Co-chair) Ajith Abraham, Seoul, South Korea Sabri Arik, Istanbul, Turkey Jianting Cao, Saitama, Japan Wenming Cao, Hangzhou, China Liang Chen, Prince George, Canada Songcan Chen, Nanjing, China Xue-wen Chen, Kansas, USA Xiaochun Cheng, Berkshire, UK Zheru Chi, Hong Kong Jin-Young Choi, Seoul, Korea Chuanyin Dang, Hong Kong Tom Downs, Brisbane, Australia Shumin Fei, Nanjing, China Wai Keung Fung, Winnipeg, Canada John Qiang Gan, Essex, UK Chengan Guo, Dalian, China Ping Guo, Beijing, China Qing-Long Han, Rockhampton, Australia Daniel W.C. Ho, Hong Kong Dewen Hu, Changsha, China Sanqing Hu, Chicago, USA Shunan Huang, Singapore Malik Magdon Ismail, Troy, USA Joarder Kamruzzaman, Melbourne, Australia Nikola Kasabov, Auckland, New Zealand Tae Seon Kim, Seoul, Korea Hon Keung Kwan, Windsor, Canada James Kwok, Hong Kong Shaowen Li, Chengdu, China Yangmin Li, Macao Hualou Liang, Houston, USA Yanchun Liang, Changchun, China Meng-Hiot Lim, Singapore Guoping Liu, Treforest, UK Meiqin Liu, Hangzhou, China Wenlian Lu, Leipzig, Germany Zhiwei Luo, Nagoya, Japan Qing Ma, Kyoto, Japan Zhiqing Meng, Hangzhou, China Seiichi Ozawa, Kobe, Japan Jagath C. Rajapakse, Singapore Daming Shi, Singapore Jochen J. Steil, Bielefeld, Germany Changyin Sun, Nanjing, China

Organization

Norikazu Takahashi, Fukuoka, Japan Yu Tang, Mexico City, Mexico Christos Tjortjis, Manchester, UK Michel Verleysen, Louwain, Belgium Dan Wang, Singapore Si Wu, Brighton, UK Cheng Xiang, Singapore Simon X. Yang, Guelph, Canada Yingjie Yang, Leicester, UK Dingli Yu, Liverpool, UK Gerson Zaverucha, Rio de Janeiro, Brazil Huaguang Zhang, Shenyang, China Liqing Zhang, Shanghai, China Tao Zhang, Tianjin, China Yanqing Zhang, Atlanta, USA Jin Zhou, Shanghai, China Organizing Committee Yue Wu (Chair), Chengdu, China

Pu Sun, Ann Arbor, USA Ying Tan, Hefei, China Peter Tino, Birmingham, UK Dan Ventura, Provo, USA Bing Wang, Hull, UK Kesheng Wang, Trondheim, Norway Wei Wu, Dalian, China Daoyi Xu, Chengdu, China Xiaosong Yang, Wuhan, China Zhengrong Yang, Exeter, UK Wen Yu, Mexico City, Mexico Zhigang Zeng, Hefei, China Jie Zhang, Newcastle, UK Qingfu Zhang, Essex, UK Ya Zhang, Kansas, USA Yunong Zhang, Maynooth, Ireland

Xiaofeng Liao (Co-chair), Chongqing, China

Publications Committee Chuandong Li (Chair), Chongqing, China Mao Ye (Co-chair), Chengdu, China Jianwei Zhang (Co-chair), Hamburg, Germany Publicity Committee Bin Jiang (Chair), Chengdu, China Zeng-Guang Hou (Co-chair), Beijing, China

Jennie Si (Co-chair), Tempe, USA

Registration Committee Xiaorong Pu (Chair), Chengdu, China Local Arrangements Committee Hongli Zhang (Chair), Chengdu, China Secretariats Jiancheng Lv, Chengdu, China

Tao Xiang, Chongqing, China

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Table of Contents – Part I

Neurobiological Analysis The Ideal Noisy Environment for Fast Neural Computation Si Wu, Jianfeng Feng, Shun-ichi Amari . . . . . . . . . . . . . . . . . . . . . . . . . .

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How Does a Neuron Perform Subtraction? – Arithmetic Rules of Synaptic Integration of Excitation and Inhibition Xu-Dong Wang, Jiang Hao, Mu-Ming Poo, Xiao-Hui Zhang . . . . . . . . .

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Stochastic Resonance Enhancing Detectability of Weak Signal by Neuronal Networks Model for Receiver Jun Liu, Jian Wu, Zhengguo Lou, Guang Li . . . . . . . . . . . . . . . . . . . . . .

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A Gaussian Dynamic Convolution Models of the FMRI BOLD Response Huafu Chen, Ling Zeng, Dezhong Yao, Qing Gao . . . . . . . . . . . . . . . . . .

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Cooperative Motor Learning Model for Cerebellar Control of Balance and Locomotion Mingxiao Ding, Naigong Yu, Xiaogang Ruan . . . . . . . . . . . . . . . . . . . . . .

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A Model of Category Learning with Attention Augmented Simplistic Prototype Representation Toshihiko Matsuka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On the Learning Algorithms of Descriptive Models of High-Order Human Cognition Toshihiko Matsuka, Arieta Chouchourelou . . . . . . . . . . . . . . . . . . . . . . . . .

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A Neural Model on Cognitive Process Ru-bin Wang, Jing Yu, Zhi-kang Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Theoretical Analysis Approximation Bound of Mixture Networks in Lpω Spaces Zongben Xu, Jianjun Wang, Deyu Meng . . . . . . . . . . . . . . . . . . . . . . . . . .

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Integral Transform and Its Application to Neural Network Approximation Feng-jun Li, Zongben Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Essential Approximation Order for Neural Networks with Trigonometric Hidden Layer Units Chunmei Ding, Feilong Cao, Zongben Xu . . . . . . . . . . . . . . . . . . . . . . . . .

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Wavelets Based Neural Network for Function Approximation Yong Fang, Tommy W.S. Chow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Passivity Analysis of Dynamic Neural Networks with Different Time-Scales Alejandro Cruz Sandoval, Wen Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Exponential Dissipativity of Non-autonomous Neural Networks with Distributed Delays and Reaction-Diffusion Terms Zhiguo Yang, Daoyi Xu, Yumei Huang . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Convergence Analysis of Continuous-Time Neural Networks Min-Jae Kang, Ho-Chan Kim, Farrukh A. Khan, Wang-Cheol Song, Jacek M. Zurada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Global Convergence of Continuous-Time Recurrent Neural Networks with Delays Weirui Zhao, Huanshui Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Global Exponential Stability in Lagrange Sense of Continuous-Time Recurrent Neural Networks Xiaoxin Liao, Zhigang Zeng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Global Exponential Stability of Recurrent Neural Networks with Time-Varying Delay Yi Shen, Meiqin Liu, Xiaodong Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 New Criteria of Global Exponential Stability for a Class of Generalized Neural Networks with Time-Varying Delays Gang Wang, Hua-Guang Zhang, Chong-Hui Song . . . . . . . . . . . . . . . . . . 129 Dynamics of General Neural Networks with Distributed Delays Changyin Sun, Linfeng Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 On Equilibrium and Stability of a Class of Neural Networks with Mixed Delays Shuyong Li, Yumei Huang, Daoyi Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Stability Analysis of Neutral Neural Networks with Time Delay Hanlin He, Xiaoxin Liao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Table of Contents – Part I

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Global Asymptotical Stability in Neutral-Type Delayed Neural Networks with Reaction-Diffusion Terms Jianlong Qiu, Jinde Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Almost Sure Exponential Stability on Interval Stochastic Neural Networks with Time-Varying Delays Wudai Liao, Zhongsheng Wang, Xiaoxin Liao . . . . . . . . . . . . . . . . . . . . . 159 Stochastic Robust Stability of Markovian Jump Nonlinear Uncertain Neural Networks with Wiener Process Xuyang Lou, Baotong Cui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Stochastic Robust Stability Analysis for Markovian Jump Discrete-Time Delayed Neural Networks with Multiplicative Nonlinear Perturbations Li Xie, Tianming Liu, Guodong Lu, Jilin Liu, Stephen T.C. Wong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Global Robust Stability of General Recurrent Neural Networks with Time-Varying Delays Jun Xu, Daoying Pi, Yong-Yan Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Robust Periodicity in Recurrent Neural Network with Time Delays and Impulses Yongqing Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Global Asymptotical Stability of Cohen-Grossberg Neural Networks with Time-Varying and Distributed Delays Tianping Chen, Wenlian Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 LMI Approach to Robust Stability Analysis of Cohen-Grossberg Neural Networks with Multiple Delays Ce Ji, Hua-Guang Zhang, Chong-Hui Song . . . . . . . . . . . . . . . . . . . . . . . . 198 Existence and Global Stability Analysis of Almost Periodic Solutions for Cohen-Grossberg Neural Networks Tianping Chen, Lili Wang, Changlei Ren . . . . . . . . . . . . . . . . . . . . . . . . . 204 A New Sufficient Condition on the Complete Stability of a Class Cellular Neural Networks Li-qun Zhou, Guang-da Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Stability Analysis of Reaction-Diffusion Recurrent Cellular Neural Networks with Variable Time Delays Weifan Zheng, Jiye Zhang, Weihua Zhang . . . . . . . . . . . . . . . . . . . . . . . . 217

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Table of Contents – Part I

Exponential Stability of Delayed Stochastic Cellular Neural Networks Wudai Liao, Yulin Xu, Xiaoxin Liao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Global Exponential Stability of Cellular Neural Networks with Time-Varying Delays and Impulses Chaojin Fu, Boshan Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Global Exponential Stability of Fuzzy Cellular Neural Networks with Variable Delays Jiye Zhang, Dianbo Ren, Weihua Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Stability of Fuzzy Cellular Neural Networks with Impulses Tingwen Huang, Marco Roque-Sol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Absolute Stability of Hopfield Neural Network Xiaoxin Liao, Fei Xu, Pei Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Robust Stability Analysis of Uncertain Hopfield Neural Networks with Markov Switching Bingji Xu, Qun Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Asymptotic Stability of Second-Order Discrete-Time Hopfield Neural Networks with Variable Delays Wei Zhu, Daoyi Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Convergence Analysis of Discrete Delayed Hopfield Neural Networks Sheng-Rui Zhang, Run-Nian Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 An LMI-Based Approach to the Global Stability of Bidirectional Associative Memory Neural Networks with Variable Delay Minghui Jiang, Yi Shen, Xiaoxin Liao . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Existence of Periodic Solution of BAM Neural Network with Delay and Impulse Hui Wang, Xiaofeng Liao, Chuandong Li, Degang Yang . . . . . . . . . . . . 279 On Control of Hopf Bifurcation in BAM Neural Network with Delayed Self-feedback Min Xiao, Jinde Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Convergence and Periodicity of Solutions for a Class of Discrete-Time Recurrent Neural Network with Two Neurons Hong Qu, Zhang Yi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Table of Contents – Part I

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Existence and Global Attractability of Almost Periodic Solution for Competitive Neural Networks with Time-Varying Delays and Different Time Scales Wentong Liao, Linshan Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Global Synchronization of Impulsive Coupled Delayed Neural Networks Jin Zhou, Tianping Chen, Lan Xiang, Meichun Liu . . . . . . . . . . . . . . . . 303 Synchronization of a Class of Coupled Discrete Recurrent Neural Networks with Time Delay Ping Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Chaos and Bifurcation in a New Class of Simple Hopfield Neural Network Yan Huang, Xiao-Song Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Synchronization of Chaotic System with the Perturbation Via Orthogonal Function Neural Network Hongwei Wang, Hong Gu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Numerical Analysis of a Chaotic Delay Recurrent Neural Network with Four Neurons Haigeng Luo, Xiaodong Xu, Xiaoxin Liao . . . . . . . . . . . . . . . . . . . . . . . . . 328 Autapse Modulated Bursting Guang-Hong Wang, Ping Jiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Neurodynamic Optimization A Neural Network Model for Non-smooth Optimization over a Compact Convex Subset Guocheng Li, Shiji Song, Cheng Wu, Zifang Du . . . . . . . . . . . . . . . . . . . 344 Differential Inclusions-Based Neural Networks for Nonsmooth Convex Optimization on a Closed Convex Subset Shiji Song, Guocheng Li, Xiaohong Guan . . . . . . . . . . . . . . . . . . . . . . . . . 350 A Recurrent Neural Network for Linear Fractional Programming with Bound Constraints Fuye Feng, Yong Xia, Quanju Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 A Delayed Lagrangian Network for Solving Quadratic Programming Problems with Equality Constraints Qingshan Liu, Jun Wang, Jinde Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

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Wavelet Chaotic Neural Networks and Their Application to Optimization Problems Yao-qun Xu, Ming Sun, Guangren Duan . . . . . . . . . . . . . . . . . . . . . . . . . . 379 A New Optimization Algorithm Based on Ant Colony System with Density Control Strategy Ling Qin, Yixin Chen, Ling Chen, Yuan Yao . . . . . . . . . . . . . . . . . . . . . . 385 A New Neural Network Approach to the Traveling Salesman Problem Paulo Henrique Siqueira, S´ergio Scheer, Maria Teresinha Arns Steiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Dynamical System for Computing Largest Generalized Eigenvalue Lijun Liu, Wei Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 A Concise Functional Neural Network for Computing the Extremum Eigenpairs of Real Symmetric Matrices Yiguang Liu, Zhisheng You . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Learning Algorithms A Novel Stochastic Learning Rule for Neural Networks Frank Emmert-Streib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Learning with Single Quadratic Integrate-and-Fire Neuron Deepak Mishra, Abhishek Yadav, Prem K. Kalra . . . . . . . . . . . . . . . . . . . 424 Manifold Learning of Vector Fields Hongyu Li, I-Fan Shen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Similarity Measure for Vector Field Learning Hongyu Li, I-Fan Shen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 The Mahalanobis Distance Based Rival Penalized Competitive Learning Algorithm Jinwen Ma, Bin Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Dynamic Competitive Learning Seongwon Cho, Jaemin Kim, Sun-Tae Chung . . . . . . . . . . . . . . . . . . . . . . 448 Hyperbolic Quotient Feature Map for Competitive Learning Neural Networks Jinwuk Seok, Seongwon Cho, Jaemin Kim . . . . . . . . . . . . . . . . . . . . . . . . 456

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A Gradient Entropy Regularized Likelihood Learning Algorithm on Gaussian Mixture with Automatic Model Selection Zhiwu Lu, Jinwen Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Self-organizing Neural Architecture for Reinforcement Learning Ah-Hwee Tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 On the Efficient Implementation Biologic Reinforcement Learning Using Eligibility Traces SeungGwan Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Combining Label Information and Neighborhood Graph for Semi-supervised Learning Lianwei Zhao, Siwei Luo, Mei Tian, Chao Shao, Hongliang Ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 A Cerebellar Feedback Error Learning Scheme Based on Kalman Estimator for Tracing in Dynamic System Liang Liu, Naigong Yu, Mingxiao Ding, Xiaogang Ruan . . . . . . . . . . . . 489 An Optimal Iterative Learning Scheme for Dynamic Neural Network Modelling Lei Guo, Hong Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 Delayed Learning on Internal Memory Network and Organizing Internal States Toshinori Deguchi, Naohiro Ishii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 A Novel Learning Algorithm for Feedforward Neural Networks Huawei Chen, Fan Jin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 On H∞ Filtering in Feedforward Neural Networks Training and Pruning He-Sheng Tang, Song-Tao Xue, Rong Chen . . . . . . . . . . . . . . . . . . . . . . . 515 A Node Pruning Algorithm Based on Optimal Brain Surgeon for Feedforward Neural Networks Jinhua Xu, Daniel W.C. Ho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 A Fast Learning Algorithm Based on Layered Hessian Approximations and the Pseudoinverse E.J. Teoh, C. Xiang, K. Cheng Tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

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Table of Contents – Part I

A Modular Reduction Method for k-NN Algorithm with Self-recombination Learning Hai Zhao, Bao-Liang Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Selective Neural Network Ensemble Based on Clustering Haixia Chen, Senmiao Yuan, Kai Jiang . . . . . . . . . . . . . . . . . . . . . . . . . . 545 An Individual Adaptive Gain Parameter Backpropagation Algorithm for Complex-Valued Neural Networks Songsong Li, Toshimi Okada, Xiaoming Chen, Zheng Tang . . . . . . . . . . 551 Training Cellular Neural Networks with Stable Learning Algorithm Marco A. Moreno-Armendariz, Giovanni Egidio Pazienza, Wen Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 A New Stochastic PSO Technique for Neural Network Training Yangmin Li, Xin Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 A Multi-population Cooperative Particle Swarm Optimizer for Neural Network Training Ben Niu, Yun-Long Zhu, Xiao-Xian He . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 Training RBF Neural Network with Hybrid Particle Swarm Optimization Haichang Gao, Boqin Feng, Yun Hou, Li Zhu . . . . . . . . . . . . . . . . . . . . . 577 Robust Learning by Self-organization of Nonlinear Lines of Attractions Ming-Jung Seow, Vijayan K. Asari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Improved Learning Algorithm Based on Generalized SOM for Dynamic Non-linear System Kai Zhang, Gen-Zhi Guan, Fang-Fang Chen, Lin Zhang, Zhi-Ye Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 Q-Learning with FCMAC in Multi-agent Cooperation Kao-Shing Hwang, Yu-Jen Chen, Tzung-Feng Lin . . . . . . . . . . . . . . . . . . 599 Q Learning Based on Self-organizing Fuzzy Radial Basis Function Network Xuesong Wang, Yuhu Cheng, Wei Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 A Fuzzy Neural Networks with Structure Learning Haisheng Lin, Xiao Zhi Gao, Xianlin Huang, Zhuoyue Song . . . . . . . . . 616

Table of Contents – Part I

XIX

Reinforcement Learning-Based Tuning Algorithm Applied to Fuzzy Identification Mariela Cerrada, Jose Aguilar, Andr´e Titli . . . . . . . . . . . . . . . . . . . . . . . 623 A New Learning Algorithm for Function Approximation Incorporating A Priori Information into Extreme Learning Machine Fei Han, Tat-Ming Lok, Michael R. Lyu . . . . . . . . . . . . . . . . . . . . . . . . . . 631 Robust Recursive Complex Extreme Learning Machine Algorithm for Finite Numerical Precision Junseok Lim, Koeng Mo Sung, Joonil Song . . . . . . . . . . . . . . . . . . . . . . . . 637 Evolutionary Extreme Learning Machine – Based on Particle Swarm Optimization You Xu, Yang Shu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 A Gradient-Based ELM Algorithm in Regressing Multi-variable Functions You Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 A New Genetic Approach to Structure Learning of Bayesian Networks Jaehun Lee, Wooyong Chung, Euntai Kim . . . . . . . . . . . . . . . . . . . . . . . . 659

Model Design Research on Multi-Degree-of-Freedom Neurons with Weighted Graphs Shoujue Wang, Singsing Liu, Wenming Cao . . . . . . . . . . . . . . . . . . . . . . . 669 Output PDF Shaping of Singular Weights System: Monotonical Performance Design Hong Yue, Aurelie J.A. Leprand, Hong Wang . . . . . . . . . . . . . . . . . . . . . 676 Stochastic Time-Varying Competitive Neural Network Systems Yi Shen, Meiqin Liu, Xiaodong Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 Heterogeneous Centroid Neural Networks Dong-Chul Park, Duc-Hoai Nguyen, Song-Jae Lee, Yunsik Lee . . . . . . . 689 Building Multi-layer Small World Neural Network Shuzhong Yang, Siwei Luo, Jianyu Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Growing Hierarchical Principal Components Analysis Self-Organizing Map Stones Lei Zhang, Zhang Yi, Jian Cheng Lv . . . . . . . . . . . . . . . . . . . . . . . 701

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Table of Contents – Part I

Hybrid Neural Network Model Based on Multi-layer Perceptron and Adaptive Resonance Theory Andrey Gavrilov, Young-Koo Lee, Sungyoung Lee . . . . . . . . . . . . . . . . . . 707 Evolving Neural Networks Using the Hybrid of Ant Colony Optimization and BP Algorithms Yan-Peng Liu, Ming-Guang Wu, Ji-Xin Qian . . . . . . . . . . . . . . . . . . . . . 714 A Genetic Algorithm with Modified Tournament Selection and Efficient Deterministic Mutation for Evolving Neural Network Dong-Sun Kim, Hyun-Sik Kim, Duck-Jin Chung . . . . . . . . . . . . . . . . . . . 723 A Neural Network Structure Evolution Algorithm Based on e, m Projections and Model Selection Criterion Yunhui Liu, Siwei Luo, Ziang Lv, Hua Huang . . . . . . . . . . . . . . . . . . . . . 732 A Parallel Coevolutionary Immune Neural Network and Its Application to Signal Simulation Zhu-Hong Zhang, Xin Tu, Chang-Gen Peng . . . . . . . . . . . . . . . . . . . . . . . 739 A Novel Elliptical Basis Function Neural Networks Optimized by Particle Swarm Optimization Ji-Xiang Du, Chuan-Min Zhai, Zeng-Fu Wang, Guo-Jun Zhang . . . . . 747 Fuzzy Neural Network Optimization by a Particle Swarm Optimization Algorithm Ming Ma, Li-Biao Zhang, Jie Ma, Chun-Guang Zhou . . . . . . . . . . . . . . 752 Fuzzy Rule Extraction Using Robust Particle Swarm Optimization Sumitra Mukhopadhyay, Ajit K. Mandal . . . . . . . . . . . . . . . . . . . . . . . . . . 762 A New Design Methodology of Fuzzy Set-Based Polynomial Neural Networks with Symbolic Gene Type Genetic Algorithms Seok-Beom Roh, Sung-Kwun Oh, Tae-Chon Ahn . . . . . . . . . . . . . . . . . . . 768 Design of Fuzzy Polynomial Neural Networks with the Aid of Genetic Fuzzy Granulation and Its Application to Multi-variable Process System Sung-Kwun Oh, In-Tae Lee, Jeoung-Nae Choi . . . . . . . . . . . . . . . . . . . . . 774 A Novel Self-Organizing Fuzzy Polynomial Neural Networks with Evolutionary FPNs: Design and Analysis Ho-Sung Park, Sung-Kwun Oh, Tae-Chon Ahn . . . . . . . . . . . . . . . . . . . . 780 Design of Fuzzy Neural Networks Based on Genetic Fuzzy Granulation and Regression Polynomial Fuzzy Inference Sung-Kwun Oh, Byoung-Jun Park, Witold Pedrycz . . . . . . . . . . . . . . . . . 786

Table of Contents – Part I

XXI

A New Fuzzy ART Neural Network Based on Dual Competition and Resonance Technique Lei Zhang, Guoyou Wang, Wentao Wang . . . . . . . . . . . . . . . . . . . . . . . . . 792 Simulated Annealing Based Learning Approach for the Design of Cascade Architectures of Fuzzy Neural Networks Chang-Wook Han, Jung-Il Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798 A New Fuzzy Identification Method Based on Adaptive Critic Designs Huaguang Zhang, Yanhong Luo, Derong Liu . . . . . . . . . . . . . . . . . . . . . . 804 Impacts of Perturbations of Training Patterns on Two Fuzzy Associative Memories Based on T-Norms Wei-Hong Xu, Guo-Ping Chen, Zhong-Ke Xie . . . . . . . . . . . . . . . . . . . . . 810 Alpha-Beta Associative Memories for Gray Level Patterns Cornelio Y´ an ˜ez-M´ arquez, Luis P. S´ anchez-Fern´ andez, Itzam´ a L´ opez-Y´ an ˜ez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 Associative Memories Based on Discrete-Time Cellular Neural Networks with One-Dimensional Space-Invariant Templates Zhigang Zeng, Jun Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 Autonomous and Deterministic Probabilistic Neural Network Using Global k-Means Roy Kwang Yang Chang, Chu Kiong Loo, Machavaram V.C. Rao . . . . 830 Selecting Variables for Neural Network Committees Marija Bacauskiene, Vladas Cibulskis, Antanas Verikas . . . . . . . . . . . . . 837 An Adaptive Network Topology for Classification Qingyu Xiong, Jian Huang, Xiaodong Xian, Qian Xiao . . . . . . . . . . . . . 843 A Quantitative Comparison of Different MLP Activation Functions in Classification Emad A.M. Andrews Shenouda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 Estimating the Number of Hidden Neurons in a Feedforward Network Using the Singular Value Decomposition Eu Jin Teoh, Cheng Xiang, Kay Chen Tan . . . . . . . . . . . . . . . . . . . . . . . . 858 Neuron Selection for RBF Neural Network Classifier Based on Multiple Granularities Immune Network Jiang Zhong, Chun Xiao Ye, Yong Feng, Ying Zhou, Zhong Fu Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866

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Table of Contents – Part I

Hierarchical Radial Basis Function Neural Networks for Classification Problems Yuehui Chen, Lizhi Peng, Ajith Abraham . . . . . . . . . . . . . . . . . . . . . . . . . 873 Biased Wavelet Neural Network and Its Application to Streamflow Forecast Fang Liu, Jian-Zhong Zhou, Fang-Peng Qiu, Jun-Jie Yang . . . . . . . . . . 880 A Goal Programming Based Approach for Hidden Targets in Layer-by-Layer Algorithm of Multilayer Perceptron Classifiers Yanlai Li, Kuanquan Wang, Tao Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 889 SLIT: Designing Complexity Penalty for Classification and Regression Trees Using the SRM Principle Zhou Yang, Wenjie Zhu, Liang Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 Flexible Neural Tree for Pattern Recognition Hai-Jun Li, Zheng-Xuan Wang, Li-Min Wang, Sen-Miao Yuan . . . . . . 903 A Novel Model of Artificial Immune Network and Simulations on Its Dynamics Lei Wang, Yinling Nie, Weike Nie, Licheng Jiao . . . . . . . . . . . . . . . . . . 909

Kernel Methods A Kernel Optimization Method Based on the Localized Kernel Fisher Criterion Bo Chen, Hongwei Liu, Zheng Bao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915 Genetic Granular Kernel Methods for Cyclooxygenase-2 Inhibitor Activity Comparison Bo Jin, Yan-Qing Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922 Support Vector Machines with Beta-Mixing Input Sequences Luoqing Li, Chenggao Wan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928 Least Squares Support Vector Machine on Gaussian Wavelet Kernel Function Set Fangfang Wu, Yinliang Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936 A Smoothing Multiple Support Vector Machine Model Huihong Jin, Zhiqing Meng, Xuanxi Ning . . . . . . . . . . . . . . . . . . . . . . . . . 942 Fuzzy Support Vector Machines Based on Spherical Regions Hong-Bing Liu, Sheng-Wu Xiong, Xiao-Xiao Niu . . . . . . . . . . . . . . . . . . 949

Table of Contents – Part I

XXIII

Building Support Vector Machine Alternative Using Algorithms of Computational Geometry Marek Bundzel, Tom´ aˇs Kasanick´y, Baltaz´ ar Frankoviˇc . . . . . . . . . . . . . . 955 Cooperative Clustering for Training SVMs Shengfeng Tian, Shaomin Mu, Chuanhuan Yin . . . . . . . . . . . . . . . . . . . . 962 SVMV - A Novel Algorithm for the Visualization of SVM Classification Results Xiaohong Wang, Sitao Wu, Xiaoru Wang, Qunzhan Li . . . . . . . . . . . . . 968 Support Vector Machines Ensemble Based on Fuzzy Integral for Classification Genting Yan, Guangfu Ma, Liangkuan Zhu . . . . . . . . . . . . . . . . . . . . . . . . 974 An Adaptive Support Vector Machine Learning Algorithm for Large Classification Problem Shu Yu, Xiaowei Yang, Zhifeng Hao, Yanchun Liang . . . . . . . . . . . . . . . 981 SVDD-Based Method for Fast Training of Multi-class Support Vector Classifier Woo-Sung Kang, Ki Hong Im, Jin Young Choi . . . . . . . . . . . . . . . . . . . . 991 Binary Tree Support Vector Machine Based on Kernel Fisher Discriminant for Multi-classification Bo Liu, Xiaowei Yang, Zhifeng Hao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997 A Fast and Sparse Implementation of Multiclass Kernel Perceptron Algorithm Jianhua Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004 Mutual Conversion of Regression and Classification Based on Least Squares Support Vector Machines Jing-Qing Jiang, Chu-Yi Song, Chun-Guo Wu, Yang-Chun Liang, Xiao-Wei Yang, Zhi-Feng Hao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1010 Sparse Least Squares Support Vector Machine for Function Estimation Liang-zhi Gan, Hai-kuan Liu, You-xian Sun . . . . . . . . . . . . . . . . . . . . . . . 1016 A Multiresolution Wavelet Kernel for Support Vector Regression Feng-Qing Han, Da-Cheng Wang, Chuan-Dong Li, Xiao-Feng Liao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022

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Table of Contents – Part I

Multi-scale Support Vector Machine for Regression Estimation Zhen Yang, Jun Guo, Weiran Xu, Xiangfei Nie, Jian Wang, Jianjun Lei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1030 Gradient Based Fuzzy C-Means Algorithm with a Mercer Kernel Dong-Chul Park, Chung Nguyen Tran, Sancho Park . . . . . . . . . . . . . . . . 1038 An Efficient Similarity-Based Validity Index for Kernel Clustering Algorithm Yun-Wei Pu, Ming Zhu, Wei-Dong Jin, Lai-Zhao Hu . . . . . . . . . . . . . . . 1044 Fuzzy Support Vector Clustering En-Hui Zheng, Min Yang, Ping Li, Zhi-Huan Song . . . . . . . . . . . . . . . . . 1050 An SVM Classification Algorithm with Error Correction Ability Applied to Face Recognition Chengbo Wang, Chengan Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 A Boosting SVM Chain Learning for Visual Information Retrieval Zejian Yuan, Lei Yang, Yanyun Qu, Yuehu Liu, Xinchun Jia . . . . . . . . 1063 Nonlinear Estimation of Hyperspectral Mixture Pixel Proportion Based on Kernel Orthogonal Subspace Projection Bo Wu, Liangpei Zhang, Pingxiang Li, Jinmu Zhang . . . . . . . . . . . . . . . 1070 A New Proximal Support Vector Machine for Semi-supervised Classification Li Sun, Ling Jing, Xiaodong Xia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 Sparse Gaussian Processes Using Backward Elimination Liefeng Bo, Ling Wang, Licheng Jiao . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 Comparative Study of Extreme Learning Machine and Support Vector Machine Xun-Kai Wei, Ying-Hong Li, Yue Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

ICA and BSS Multi-level Independent Component Analysis Woong Myung Kim, Chan Ho Park, Hyon Soo Lee . . . . . . . . . . . . . . . . . 1096 An ICA Learning Algorithm Utilizing Geodesic Approach Tao Yu, Huai-Zong Shao, Qi-Cong Peng . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

Table of Contents – Part I

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An Extended Online Fast-ICA Algorithm Gang Wang, Ni-ni Rao, Zhi-lin Zhang, Quanyi Mo, Pu Wang . . . . . . . 1109 Gradient Algorithm for Nonnegative Independent Component Analysis Shangming Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115 Unified Parametric and Non-parametric ICA Algorithm for Arbitrary Sources Fasong Wang, Hongwei Li, Rui Li, Shaoquan Yu . . . . . . . . . . . . . . . . . . 1121 A Novel Kurtosis-Dependent Parameterized Independent Component Analysis Algorithm Xiao-fei Shi, Ji-dong Suo, Chang Liu, Li Li . . . . . . . . . . . . . . . . . . . . . . . 1127 Local Stability Analysis of Maximum Nongaussianity Estimation in Independent Component Analysis Gang Wang, Xin Xu, Dewen Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 Convergence Analysis of a Discrete-Time Single-Unit Gradient ICA Algorithm Mao Ye, Xue Li, Chengfu Yang, Zengan Gao . . . . . . . . . . . . . . . . . . . . . . 1140 A Novel Algorithm for Blind Source Separation with Unknown Sources Number Ji-Min Ye, Shun-Tian Lou, Hai-Hong Jin, Xian-Da Zhang . . . . . . . . . . 1147 Blind Source Separation Based on Generalized Variance Gaoming Huang, Luxi Yang, Zhenya He . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 Blind Source Separation with Pattern Expression NMF Junying Zhang, Zhang Hongyi, Le Wei, Yue Joseph Wang . . . . . . . . . . 1159 Nonlinear Blind Source Separation Using Hybrid Neural Networks Chun-Hou Zheng, Zhi-Kai Huang, Michael R. Lyu, Tat-Ming Lok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 Identification of Mixing Matrix in Blind Source Separation Xiaolu Li, Zhaoshui He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171 Identification of Independent Components Based on Borel Measure for Under-Determined Mixtures Wenqiang Guo, Tianshuang Qiu, Yuzhang Zhao, Daifeng Zha . . . . . . . 1177

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Estimation of Delays and Attenuations for Underdetermined BSS in Frequency Domain Ronghua Li, Ming Xiao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 Application of Blind Source Separation to Five-Element Cross Array Passive Location Gaoming Huang, Yang Gao, Luxi Yang, Zhenya He . . . . . . . . . . . . . . . . 1189 Convolutive Blind Separation of Non-white Broadband Signals Based on a Double-Iteration Method Hua Zhang, Dazhang Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195 Multichannel Blind Deconvolution Using a Novel Filter Decomposition Method Bin Xia, Liqing Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 Two-Stage Blind Deconvolution for V-BLAST OFDM System Feng Jiang, Liqing Zhang, Bin Xia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208

Data Preprocessing A Comparative Study on Selection of Cluster Number and Local Subspace Dimension in the Mixture PCA Models Xuelei Hu, Lei Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214 Adaptive Support Vector Clustering for Multi-relational Data Mining Ping Ling, Chun-Guang Zhou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222 Robust Data Clustering in Mercer Kernel-Induced Feature Space Xulei Yang, Qing Song, Meng-Joo Er . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 Pseudo-density Estimation for Clustering with Gaussian Processes Hyun-Chul Kim, Jaewook Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238 Clustering Analysis of Competitive Learning Network for Molecular Data Lin Wang, Minghu Jiang, Yinghua Lu, Frank Noe, Jeremy C. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 Self-Organizing Map Clustering Analysis for Molecular Data Lin Wang, Minghu Jiang, Yinghua Lu, Frank Noe, Jeremy C. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1250

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A Conscientious Rival Penalized Competitive Learning Text Clustering Algorithm Mao-ting Gao, Zheng-ou Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256 Self-Organizing-Map-Based Metamodeling for Massive Text Data Exploration Kin Keung Lai, Lean Yu, Ligang Zhou, Shouyang Wang . . . . . . . . . . . . 1261 Ensemble Learning for Keyphrases Extraction from Scientific Document Jiabing Wang, Hong Peng, Jing-song Hu, Jun Zhang . . . . . . . . . . . . . . . 1267 Grid-Based Fuzzy Support Vector Data Description Yugang Fan, Ping Li, Zhihuan Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273 Development of the Hopfield Neural Scheme for Data Association in Multi-target Tracking Yang Weon Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280 Determine Discounting Coefficient in Data Fusion Based on Fuzzy ART Neural Network Dong Sun, Yong Deng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286 Scientific Data Lossless Compression Using Fast Neural Network Jun-Lin Zhou, Yan Fu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 HyperSurface Classifiers Ensemble for High Dimensional Data Sets Xiu-Rong Zhao, Qing He, Zhong-Zhi Shi . . . . . . . . . . . . . . . . . . . . . . . . . . 1299 Designing a Decompositional Rule Extraction Algorithm for Neural Networks Jen-Cheng Chen, Jia-Sheng Heh, Maiga Chang . . . . . . . . . . . . . . . . . . . . 1305 Estimating Fractal Intrinsic Dimension from the Neighborhood Qutang Cai, Changshui Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Dimensionality Reduction for Evolving RBF Networks with Particle Swarms Junying Chen, Zheng Qin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 Improved Locally Linear Embedding Through New Distance Computing Heyong Wang, Jie Zheng, Zhengan Yao, Lei Li . . . . . . . . . . . . . . . . . . . . 1326 An Incremental Linear Discriminant Analysis Using Fixed Point Method Dongyue Chen, Liming Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334

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Table of Contents – Part I

A Prewhitening RLS Projection Alternated Subspace Tracking (PAST) Algorithm Junseok Lim, Joonil Song, Yonggook Pyeon . . . . . . . . . . . . . . . . . . . . . . . 1340 Classification with the Hybrid of Manifold Learning and Gabor Wavelet Junping Zhang, Chao Shen, Jufu Feng . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346 A Novel Input Stochastic Sensitivity Definition of Radial Basis Function Neural Networks and Its Application to Feature Selection Xi-Zhao Wang, Hui Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352 Using Ensemble Feature Selection Approach in Selecting Subset with Relevant Features Mohammed Attik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 A New Method for Feature Selection Yan Wu, Yang Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367 Improved Feature Selection Algorithm Based on SVM and Correlation Zong-Xia Xie, Qing-Hua Hu, Da-Ren Yu . . . . . . . . . . . . . . . . . . . . . . . . . 1373 Feature Selection in Text Classification Via SVM and LSI Ziqiang Wang, Dexian Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1381 Parsimonious Feature Extraction Based on Genetic Algorithms and Support Vector Machines Qijun Zhao, Hongtao Lu, David Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387 Feature Extraction for Time Series Classification Using Discriminating Wavelet Coefficients Hui Zhang, Tu Bao Ho, Mao-Song Lin, Xuefeng Liang . . . . . . . . . . . . . 1394 Feature Extraction of Underground Nuclear Explosions Based on NMF and KNMF Gang Liu, Xi-Hai Li, Dai-Zhi Liu, Wei-Gang Zhai . . . . . . . . . . . . . . . . . 1400 Hidden Markov Model Networks for Multiaspect Discriminative Features Extraction from Radar Targets Feng Zhu, Yafeng Hu, Xianda Zhang, Deguang Xie . . . . . . . . . . . . . . . . . 1406 Application of Self-organizing Feature Neural Network for Target Feature Extraction Dong-hong Liu, Zhi-jie Chen, Wen-long Hu, Yong-shun Zhang . . . . . . . 1412

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Divergence-Based Supervised Information Feature Compression Algorithm Shi-Fei Ding, Zhong-Zhi Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427

Table of Contents – Part II

Pattern Classification Design an Effective Pattern Classification Model Do-Hyeon Kim, Eui-Young Cha, Kwang-Baek Kim . . . . . . . . . . . . . . . . .

1

Classifying Unbalanced Pattern Groups by Training Neural Network Bo-Yu Li, Jing Peng, Yan-Qiu Chen, Ya-Qiu Jin . . . . . . . . . . . . . . . . . .

8

A Modified Constructive Fuzzy Neural Networks for Classification of Large-Scale and Complicated Data Lunwen Wang, Yanhua Wu, Ying Tan, Ling Zhang . . . . . . . . . . . . . . . .

14

A Hierarchical FloatBoost and MLP Classifier for Mobile Phone Embedded Eye Location System Dan Chen, Xusheng Tang, Zongying Ou, Ning Xi . . . . . . . . . . . . . . . . . .

20

Iris Recognition Using LVQ Neural Network Seongwon Cho, Jaemin Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Minimax Probability Machine for Iris Recognition Yong Wang, Jiu-qiang Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Detecting Facial Features by Heteroassociative Memory Neural Network Utilizing Facial Statistics Kyeong-Seop Kim, Tae-Ho Yoon, Seung-Won Shin . . . . . . . . . . . . . . . . .

40

Recognizing Partially Damaged Facial Images by Subspace Auto-associative Memories Xiaorong Pu, Zhang Yi, Yue Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

A Facial Expression Classification Algorithm Based on Principle Component Analysis Qingzhang Chen, Weiyi Zhang, Xiaoying Chen, Jianghong Han . . . . . .

55

Automatic Facial Expression Recognition Huchuan Lu, Pei Wu, Hui Lin, Deli Yang . . . . . . . . . . . . . . . . . . . . . . . .

63

Facial Expression Recognition Using Active Appearance Model Taehwa Hong, Yang-Bok Lee, Yong-Guk Kim, Hagbae Kim . . . . . . . . . .

69

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Facial Expression Recognition Based on BoostingTree Ning Sun, Wenming Zheng, Changyin Sun, Cairong Zou, Li Zhao . . . .

77

KDA Plus KPCA for Face Recognition Wenming Zheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Face Recognition Using a Neural Network Simulating Olfactory Systems Guang Li, Jin Zhang, You Wang, Walter J. Freeman . . . . . . . . . . . . . . .

93

Face Recognition Using Neural Networks and Pattern Averaging Adnan Khashman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Semi-supervised Support Vector Learning for Face Recognition Ke Lu, Xiaofei He, Jidong Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Parts-Based Holistic Face Recognition with RBF Neural Networks Wei Zhou, Xiaorong Pu, Ziming Zheng . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Combining Classifiers for Robust Face Detection Lin-Lin Huang, Akinobu Shimizu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Face Detection Method Based on Kernel Independent Component Analysis and Boosting Chain Algorithm Yan Wu, Yin-Fang Zhuang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Recognition from a Single Sample per Person with Multiple SOM Fusion Xiaoyang Tan, Jun Liu, Songcan Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Investigating LLE Eigenface on Pose and Face Identification Shaoning Pang, Nikola Kasabov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Multimodal Priority Verification of Face and Speech Using Momentum Back-Propagation Neural Network Changhan Park, Myungseok Ki, Jaechan Namkung, Joonki Paik . . . . . 140 The Clustering Solution of Speech Recognition Models with SOM Xiu-Ping Du, Pi-Lian He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Study on Text-Dependent Speaker Recognition Based on Biomimetic Pattern Recognition Shoujue Wang, Yi Huang, Yu Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 A New Text-Independent Speaker Identification Using Vector Quantization and Multi-layer Perceptron Ji-Soo Keum, Chan-Ho Park, Hyon-Soo Lee . . . . . . . . . . . . . . . . . . . . . . . 165

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XXXIII

Neural Net Pattern Recognition Equations with Self-organization for Phoneme Recognition Sung-Ill Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Music Genre Classification Using a Time-Delay Neural Network Jae-Won Lee, Soo-Beom Park, Sang-Kyoon Kim . . . . . . . . . . . . . . . . . . . 178 Audio Signal Classification Using Support Vector Machines Lei-Ting Chen, Ming-Jen Wang, Chia-Jiu Wang, Heng-Ming Tai . . . . 188 Gender Classification Based on Boosting Local Binary Pattern Ning Sun, Wenming Zheng, Changyin Sun, Cairong Zou, Li Zhao . . . . 194 Multi-view Gender Classification Using Local Binary Patterns and Support Vector Machines Hui-Cheng Lian, Bao-Liang Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Gender Recognition Using a Min-Max Modular Support Vector Machine with Equal Clustering Jun Luo, Bao-Liang Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Palmprint Recognition Using ICA Based on Winner-Take-All Network and Radial Basis Probabilistic Neural Network Li Shang, De-Shuang Huang, Ji-Xiang Du, Zhi-Kai Huang . . . . . . . . . . 216 An Implementation of the Korean Sign Language Recognizer Using Neural Network Based on the Post PC Jung-Hyun Kim, Kwang-Seok Hong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Gait Recognition Using Wavelet Descriptors and Independent Component Analysis Jiwen Lu, Erhu Zhang, Cuining Jing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Gait Recognition Using Principal Curves and Neural Networks Han Su, Fenggang Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 An Adjacent Multiple Pedestrians Detection Based on ART2 Neural Network Jong-Seok Lim, Woo-Beom Lee, Wook-Hyun Kim . . . . . . . . . . . . . . . . . . 244 Recognition Method of Throwing Force of Athlete Based on Multi-class SVM Jinghua Ma, Yunjian Ge, Jianhe Lei, Quanjun Song, Yu Ge, Yong Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

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Table of Contents – Part II

A Constructive Learning Algorithm for Text Categorization Weijun Chen, Bo Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Short-Text Classification Based on ICA and LSA Qiang Pu, Guo-Wei Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Writer Identification Using Modular MLP Classifier and Genetic Algorithm for Optimal Features Selection Sami Gazzah, Najoua Essoukri Ben Amara . . . . . . . . . . . . . . . . . . . . . . . 271 Self-generation ART Neural Network for Character Recognition Taekyung Kim, Seongwon Lee, Joonki Paik . . . . . . . . . . . . . . . . . . . . . . . . 277 Handwritten Digit Recognition Using Low Rank Approximation Based Competitive Neural Network Yafeng Hu, Feng Zhu, Hairong Lv, Xianda Zhang . . . . . . . . . . . . . . . . . . 287 Multifont Arabic Characters Recognition Using HoughTransform and Neural Networks Nadia Ben Amor, Najoua Essoukri Ben Amara . . . . . . . . . . . . . . . . . . . . 293 Recognition of English Calling Card by Using Multiresolution Images and Enhanced ART1-Based RBF Neural Networks Kwang-Baek Kim, Sungshin Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 A Method of Chinese Fax Recipient’s Name Recognition Based on Hybrid Neural Networks Zhou-Jing Wang, Kai-Biao Lin, Wen-Lei Sun . . . . . . . . . . . . . . . . . . . . . 306 Fast Photo Time-Stamp Recognition Based on SGNN Aiguo Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Hierarchical Classification of Object Images Using Neural Networks Jong-Ho Kim, Jae-Won Lee, Byoung-Doo Kang, O-Hwa Kwon, Chi-Young Seong, Sang-Kyoon Kim, Se-Myung Park . . . . . . . . . . . . . . . 322 Structured-Based Neural Network Classification of Images Using Wavelet Coefficients Weibao Zou, King Chuen Lo, Zheru Chi . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Remote Sensing Image Classification Algorithm Based on Hopfield Neural Network Guang-jun Dong, Yong-sheng Zhang, Chao-jie Zhu . . . . . . . . . . . . . . . . . 337

Table of Contents – Part II

XXXV

Tea Classification Based on Artificial Olfaction Using Bionic Olfactory Neural Network Xinling Yang, Jun Fu, Zhengguo Lou, Liyu Wang, Guang Li, Walter J. Freeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Distinguishing Onion Leaves from Weed Leaves Based on Segmentation of Color Images and a BP Neural Network Jun-Wei Lu, Pierre Gouton, Yun-An Hu . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Bark Classification Based on Textural Features Using Artificial Neural Networks Zhi-Kai Huang, Chun-Hou Zheng, Ji-Xiang Du, Yuan-yuan Wan . . . . 355 Automated Spectral Classification of QSOs and Galaxies by Radial Basis Function Network with Dynamic Decay Adjustment Mei-fang Zhao, Jin-fu Yang, Yue Wu, Fu-chao Wu, Ali Luo . . . . . . . . . 361 Feed-Forward Neural Network Using SARPROP Algorithm and Its Application in Radar Target Recognition Zun-Hua Guo, Shao-Hong Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Computer Vision Camera Calibration and 3D Reconstruction Using RBF Network in Stereovision System Hai-feng Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 A Versatile Method for Omnidirectional Stereo Camera Calibration Based on BP Algorithm Chuanjiang Luo, Liancheng Su, Feng Zhu, Zelin Shi . . . . . . . . . . . . . . . 383 Evolutionary Cellular Automata Based Neural Systems for Visual Servoing Dong-Wook Lee, Chang-Hyun Park, Kwee-Bo Sim . . . . . . . . . . . . . . . . . 390 Robust Visual Tracking Via Incremental Maximum Margin Criterion Lu Wang, Ming Wen, Chong Wang, Wenyuan Wang . . . . . . . . . . . . . . . 398 An Attention Selection System Based on Neural Network and Its Application in Tracking Objects Chenlei Guo, Liming Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Human Motion Tracking Based on Markov Random Field and Hopfield Neural Network Zhihui Li, Fenggang Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

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Table of Contents – Part II

Skin-Color Based Human Tracking Using a Probabilistic Noise Model Combined with Neural Network Jin Young Kim, Min-Gyu Song, Seung You Na, Seong-Joon Baek, Seung Ho Choi, Joohun Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Object Detection Via Fusion of Global Classifier and Part-Based Classifier Zhi Zeng, Shengjin Wang, Xiaoqing Ding . . . . . . . . . . . . . . . . . . . . . . . . . 429 A Cartoon Video Detection Method Based on Active Relevance Feedback and SVM Xinbo Gao, Jie Li, Na Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Morphological Neural Networks of Background Clutter Adaptive Prediction for Detection of Small Targets in Image Data Honggang Wu, Xiaofeng Li, Zaiming Li, Yuebin Chen . . . . . . . . . . . . . . 442 Two Important Action Scenes Detection Based on Probability Neural Networks Yu-Liang Geng, De Xu, Jia-Zheng Yuan, Song-He Feng . . . . . . . . . . . . 448 Local Independent Factorization of Natural Scenes Libo Ma, Liqing Zhang, Wenlu Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Search Region Prediction for Motion Estimation Based on Neural Network Vector Quantization DaeHyun Ryu, HyungJun Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 Hierarchical Extraction of Remote Sensing Data Based on Support Vector Machines and Knowledge Processing Chao-feng Li, Lei Xu, Shi-tong Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Eyes Location Using a Neural Network Xiao-yi Feng, Li-ping Yang, Zhi Dang, Matti Pietik¨ ainen . . . . . . . . . . . 474

Image Processing Gabor Neural Network for Endoscopic Image Registration Vladimir Spinko, Daming Shi, Wan Sing Ng, Jern-Lin Leong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 Isomap and Neural Networks Based Image Registration Scheme Anbang Xu, Ping Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Table of Contents – Part II

XXXVII

Unsupervised Image Segmentation Using an Iterative Entropy Regularized Likelihood Learning Algorithm Zhiwu Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 An Improvement on Competitive Neural Networks Applied to Image Segmentation Rui Yan, Meng Joo Er, Huajin Tang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 Image Segmentation by Deterministic Annealing Algorithm with Adaptive Spatial Constraints Xulei Yang, Aize Cao, Qing Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 A Multi-scale Scheme for Image Segmentation Using Neuro-fuzzy Classification and Curve Evolution Da Yuan, Hui Fan, Fu-guo Dong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 A Robust MR Image Segmentation Technique Using Spatial Information and Principle Component Analysis Yen-Wei Chen, Yuuta Iwasaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Adaptive Segmentation of Color Image for Vision Navigation of Mobile Robots Zeng-Shun Zhao, Zeng-Guang Hou, Min Tan, Yong-Qian Zhang . . . . . 523 Image Filtering Using Support Vector Machine Huaping Liu, Fuchun Sun, Zengqi Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 The Application of Wavelet Neural Network with Orthonormal Bases in Digital Image Denoising Deng-Chao Feng, Zhao-Xuan Yang, Xiao-Jun Qiao . . . . . . . . . . . . . . . . 539 A Region-Based Image Enhancement Algorithm with the Grossberg Network Bo Mi, Pengcheng Wei, Yong Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Contrast Enhancement for Image Based on Wavelet Neural Network and Stationary Wavelet Transform Changjiang Zhang, Xiaodong Wang, Haoran Zhang . . . . . . . . . . . . . . . . 551 Learning Image Distortion Using a GMDH Network Yongtae Do, Myounghwan Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 An Edge Preserving Regularization Model for Image Restoration Based on Hopfield Neural Network Jian Sun, Zongben Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

XXXVIII Table of Contents – Part II

High-Dimensional Space Geometrical Informatics and Its Applications to Image Restoration Shoujue Wang, Yu Cao, Yi Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Improved Variance-Based Fractal Image Compression Using Neural Networks Yiming Zhou, Chao Zhang, Zengke Zhang . . . . . . . . . . . . . . . . . . . . . . . . . 575 Associative Cubes in Unsupervised Learning for Robust Gray-Scale Image Recognition Hoon Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 A Novel Graph Kernel Based SVM Algorithm for Image Semantic Retrieval Songhe Feng, De Xu, Xu Yang, Yuliang Geng . . . . . . . . . . . . . . . . . . . . . 589 Content Based Image Retrieval Using a Bootstrapped SOM Network Apostolos Georgakis, Haibo Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Unsupervised Approach for Extracting the Textural Region of Interest from Real Image Woo-Beom Lee, Jong-Seok Lim, Wook-Hyun Kim . . . . . . . . . . . . . . . . . . 602 Image Fakery and Neural Network Based Detection Wei Lu, Fu-Lai Chung, Hongtao Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Object Detection Using Unit-Linking PCNN Image Icons Xiaodong Gu, Yuanyuan Wang, Liming Zhang . . . . . . . . . . . . . . . . . . . . 616 Robust Image Watermarking Using RBF Neural Network Wei Lu, Hongtao Lu, Fu-Lai Chung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 An Interactive Image Inpainting Method Based on RBF Networks Peizhi Wen, Xiaojun Wu, Chengke Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 No-Reference Perceptual Quality Assessment of JPEG Images Using General Regression Neural Network Yanwei Yu, Zhengding Lu, Hefei Ling, Fuhao Zou . . . . . . . . . . . . . . . . . 638 Minimum Description Length Shape Model Based on Elliptic Fourier Descriptors Shaoyu Wang, Feihu Qi, Huaqing Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Neural Network Based Texture Segmentation Using a Markov Random Field Model Tae Hyung Kim, Hyun Min Kang, Il Kyu Eom, Yoo Shin Kim . . . . . . 652

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Texture Segmentation Using SOM and Multi-scale Bayesian Estimation Tae Hyung Kim, Il Kyu Eom, Yoo Shin Kim . . . . . . . . . . . . . . . . . . . . . . 661 Recognition of Concrete Surface Cracks Using the ART1-Based RBF Network Kwang-Baek Kim, Kwee-Bo Sim, Sang-Ho Ahn . . . . . . . . . . . . . . . . . . . . 669

Signal Processing SVM-Enabled Voice Activity Detection Javier Ram´ırez, Pablo Y´elamos, Juan Manuel G´ orriz, Carlos G. Puntonet, Jos´e C. Segura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 A Robust VAD Method for Array Signals Xiaohong Ma, Jin Liu, Fuliang Yin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 A Flexible Algorithm for Extracting Periodic Signals Zhi-Lin Zhang, Haitao Meng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688 A Neural Network Method for Blind Signature Waveform Estimation of Synchronous CDMA Signals Tianqi Zhang, Zengshan Tian, Zhengzhong Zhou, Yujun Kuang . . . . . . 694 A Signal-Dependent Quadratic Time Frequency Distribution for Neural Source Estimation Pu Wang, Jianyu Yang, Zhi-Lin Zhang, Gang Wang, Quanyi Mo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700 Neural Network Channel Estimation Based on Least Mean Error Algorithm in the OFDM Systems Jun Sun, Dong-Feng Yuan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 Higher-Order Feature Extraction of Non-Gaussian Acoustic Signals Using GGM-Based ICA Wei Kong, Bin Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 Automatic Removal of Artifacts from EEG Data Using ICA and Exponential Analysis Ning-Yan Bian, Bin Wang, Yang Cao, Liming Zhang . . . . . . . . . . . . . . 719 Identification of Vibrating Noise Signals of Electromotor Using Adaptive Wavelet Neural Network Xue-Zhi Zhao, Bang-Yan Ye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727

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Table of Contents – Part II

Fractional Order Digital Differentiators Design Using Exponential Basis Function Neural Network Ke Liao, Xiao Yuan, Yi-Fei Pu, Ji-Liu Zhou . . . . . . . . . . . . . . . . . . . . . . 735 Multivariate Chaotic Time Series Prediction Based on Radial Basis Function Neural Network Min Han, Wei Guo, Mingming Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 Time Series Prediction Using LS-SVM with Particle Swarm Optimization Xiaodong Wang, Haoran Zhang, Changjiang Zhang, Xiushan Cai, Jinshan Wang, Meiying Ye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 A Regularized Minimum Cross-Entropy Algorithm on Mixtures of Experts for Time Series Prediction Zhiwu Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Prediction for Chaotic Time Series Based on Discrete Volterra Neural Networks Li-Sheng Yin, Xi-Yue Huang, Zu-Yuan Yang, Chang-Cheng Xiang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759

System Modeling A New Pre-processing Method for Regression Wen-Feng Jing, De-Yu Meng, Ming-Wei Dai, Zongben Xu . . . . . . . . . . 765 A New On-Line Modeling Approach to Nonlinear Dynamic Systems Shirong Liu, Qijiang Yu, Jinshou Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 Online Modeling of Nonlinear Systems Using Improved Adaptive Kernel Methods Xiaodong Wang, Haoran Zhang, Changjiang Zhang, Xiushan Cai, Jinshan Wang, Meiying Ye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 A Novel Multiple Neural Networks Modeling Method Based on FCM Jian Cheng, Yi-Nan Guo, Jian-Sheng Qian . . . . . . . . . . . . . . . . . . . . . . . 783 Nonlinear System Identification Using Multi-resolution Reproducing Kernel Based Support Vector Regression Hong Peng, Jun Wang, Min Tang, Lichun Wan . . . . . . . . . . . . . . . . . . . 790 A New Recurrent Neurofuzzy Network for Identification of Dynamic Systems Marcos A. Gonzalez-Olvera, Yu Tang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796

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XLI

Identification of Dynamic Systems Using Recurrent Fuzzy Wavelet Network Jun Wang, Hong Peng, Jian Xiao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 Simulation Studies of On-Line Identification of Complex Processes with Neural Networks Francisco Cubillos, Gonzalo Acu˜ na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808 Consecutive Identification of ANFIS-Based Fuzzy Systems with the Aid of Genetic Data Granulation Sung-Kwun Oh, Keon-Jun Park, Witold Pedrycz . . . . . . . . . . . . . . . . . . . 815 Two-Phase Identification of ANFIS-Based Fuzzy Systems with Fuzzy Set by Means of Information Granulation and Genetic Optimization Sung-Kwun Oh, Keon-Jun Park, Hyun-Ki Kim . . . . . . . . . . . . . . . . . . . . 821 A New Modeling Approach of STLF with Integrated Dynamics Mechanism and Based on the Fusion of Dynamic Optimal Neighbor Phase Points and ICNN Zhisheng Zhang, Yaming Sun, Shiying Zhang . . . . . . . . . . . . . . . . . . . . . . 827

Control Systems Adaptive Neural Network Control for Nonlinear Systems Based on Approximation Errors Yan-Jun Liu, Wei Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 Adaptive Neural Network Control for Switched System with Unknown Nonlinear Part by Using Backstepping Approach: SISO Case Fei Long, Shumin Fei, Zhumu Fu, Shiyou Zheng . . . . . . . . . . . . . . . . . . . 842 Adaptive Neural Control for a Class of MIMO Non-linear Systems with Guaranteed Transient Performance Tingliang Hu, Jihong Zhu, Zengqi Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849 Adaptive Neural Compensation Control for Input-Delay Nonlinear Systems by Passive Approach Zhandong Yu, Xiren Zhao, Xiuyan Peng . . . . . . . . . . . . . . . . . . . . . . . . . . 859 Nonlinear System Adaptive Control by Using Multiple Neural Network Models Xiao-Li Li, Yun-Feng Kang, Wei Wang . . . . . . . . . . . . . . . . . . . . . . . . . . 867

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Implementable Adaptive Backstepping Neural Control of Uncertain Strict-Feedback Nonlinear Systems Dingguo Chen, Jiaben Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 A Discrete-Time System Adaptive Control Using Multiple Models and RBF Neural Networks Jun-Yong Zhai, Shu-Min Fei, Kan-Jian Zhang . . . . . . . . . . . . . . . . . . . . . 881 Robust Adaptive Neural Network Control for Strict-Feedback Nonlinear Systems Via Small-Gain Approaches Yansheng Yang, Tieshan Li, Xiaofeng Wang . . . . . . . . . . . . . . . . . . . . . . 888 Neural Network Based Robust Adaptive Control for a Class of Nonlinear Systems Dan Wang, Jin Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898 Robust H∞ Control for Delayed Nonlinear Systems Based on Standard Neural Network Models Mei-Qin Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904 SVM Based Nonlinear Self-tuning Control Weimin Zhong, Daoying Pi, Chi Xu, Sizhen Chu . . . . . . . . . . . . . . . . . . 911 SVM Based Internal Model Control for Nonlinear Systems Weimin Zhong, Daoying Pi, Youxian Sun, Chi Xu, Sizhen Chu . . . . . . 916 Fast Online SVR Algorithm Based Adaptive Internal Model Control Hui Wang, Daoying Pi, Youxian Sun, Chi Xu, Sizhen Chu . . . . . . . . . . 922 A VSC Method for MIMO Systems Based on SVM Yi-Bo Zhang, Dao-Ying Pi, Youxian Sun, Chi Xu, Si-Zhen Chu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928 Identification and Control of Dynamic Systems Based on Least Squares Wavelet Vector Machines Jun Li, Jun-Hua Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934 A Nonlinear Model Predictive Control Strategy Using Multiple Neural Network Models Zainal Ahmad, Jie Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943 Predictive Control Method of Improved Double-Controller Scheme Based on Neural Networks Bing Han, Min Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

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Discrete-Time Sliding-Mode Control Based on Neural Networks Jos´e de Jes´ us Rubio, Wen Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956 Statistic Tracking Control: A Multi-objective Optimization Algorithm Lei Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962 Minimum Entropy Control for Stochastic Systems Based on the Wavelet Neural Networks Chengzhi Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968 Stochastic Optimal Control of Nonlinear Jump Systems Using Neural Networks Fei Liu, Xiao-Li Luan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975 Performance Estimation of a Neural Network-Based Controller Johann Schumann, Yan Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 Some Key Issues in the Design of Self-Organizing Fuzzy Control Systems Xue-Feng Dai, Shu-Dong Liu, Deng-Zhi Cui . . . . . . . . . . . . . . . . . . . . . . 991 Nonlinear System Stabilisation by an Evolutionary Neural Network Wasan Srikasam, Nachol Chaiyaratana, Suwat Kuntanapreeda . . . . . . . 998 Neural Network Control Design for Large-Scale Systems with Higher-Order Interconnections Cong Ming, Sunan Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 Adaptive Pseudo Linear RBF Model for Process Control Ding-Wen Yu, Ding-Li Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 An Improved BP Algorithm Based on Global Revision Factor and Its Application to PID Control Lin Lei, Houjun Wang, Yufang Cheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019 Neuro-fuzzy Generalized Predictive Control of Boiler Steam Temperature Xiang-Jie Liu, Ji-Zhen Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 Model-Free Control of a Nonlinear ANC System with a SPSA-Based Neural Network Controller Yali Zhou, Qizhi Zhang, Xiaodong Li, Woonseng Gan . . . . . . . . . . . . . . 1033 Robust Control for AC-Excited Hydrogenators System Using Adaptive Fuzzy-Neural Network Hui Li, Li Han, Bei He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1039

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Adaptive Fuzzy Neural Network Control for Transient Dynamics of Magneto-rheological Suspension with Time-Delay Xiaomin Dong, Miao Yu, Changrong Liao, Weimin Chen, Honghui Zhang, Shanglian Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046 Adaptive Fuzzy Basis Function Network Based Fault-Tolerant Stable Control of Multi-machine Power Systems Youping Fan, Yunping Chen, Shangsheng Li, Qingwu Gong, Yi Chai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052 Simulation Research on Applying Fault Tolerant Control to Marine Diesel Engine in Abnormal Operation Xiao-Yan Xu, Min He, Wan-Neng Yu, Hua-Yao Zheng . . . . . . . . . . . . . 1062 Hybrid Neural Network and Genetic Algorithms for Self-tuning of PI Controller in DSPM Motor Drive System Rui-Ming Fang, Qian Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068 An Efficient DC Servo Motor Control Based on Neural Noncausal Inverse Modeling of the Plant ¨ calık . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 H. Rıza Oz¸ A Dynamic Time Delay Neural Network for Ultrasonic Motor Identification and Control Yanchun Liang, Jie Zhang, Xu Xu, Xiaowei Yang, Zhifeng Hao . . . . . . 1084 Application of PSO-Optimized Generalized CMAC Control on Linear Motor Qiang Zhao, Shaoze Yan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090 PID Control of Nonlinear Motor-Mechanism Coupling System Using Artificial Neural Network Yi Zhang, Chun Feng, Bailin Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 Design and Simulation of a Neural-PD Controller for Automatic Balancing of Rotor Yuan Kang, Tsu-Wei Lin, Ming-Hui Chu, Yeon-Pun Chang, Yea-Ping Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104 PD Control of Overhead Crane Systems with Neural Compensation Rigoberto Toxqui Toxqui, Wen Yu, Xiaoou Li . . . . . . . . . . . . . . . . . . . . . 1110 A Study on Intelligent Control for Hybrid Actuator Ke Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116

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Double Inverted Pendulum Control Based on Support Vector Machines and Fuzzy Inference Han Liu, Haiyan Wu, Fucai Qian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124 Adaptive Wavelet Neural Network Friction Compensation of Mechanical Systems Shen-min Song, Zhuo-yi Song, Xing-lin Chen, Guangren Duan . . . . . . 1131

Robotic Systems Application of Collective Robotic Search Using Neural Network Based Dual Heuristic Programming (DHP) Nian Zhang, Donald C. Wunsch II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 RBF Neural Network Based Shape Control of Hyper-redundant Manipulator with Constrained End-Effector Jinguo Liu, Yuechao Wang, Shugen Ma, Bin Li . . . . . . . . . . . . . . . . . . . 1146 Robust Adaptive Neural Networks with an Online Learning Technique for Robot Control Zhi-gang Yu, Shen-min Song, Guang-ren Duan, Run Pei . . . . . . . . . . . . 1153 A Particle Swarm Optimized Fuzzy Neural Network Control for Acrobot Dong-bin Zhao, Jian-qiang Yi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1160 Adaptive Control Based on Recurrent Fuzzy Wavelet Neural Network and Its Application on Robotic Tracking Control Wei Sun, Yaonan Wang, Xiaohua Zhai . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 Dynamic Tracking Control of Mobile Robots Using an Improved Radial Basis Function Neural Network Shirong Liu, Qijiang Yu, Jinshou Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 Grasping Control of Robot Hand Using Fuzzy Neural Network Peng Chen, Yoshizo Hasegawa, Mitushi Yamashita . . . . . . . . . . . . . . . . . 1178 Position Control Based on Static Neural Networks of Anthropomorphic Robotic Fingers Juan Ignacio Mulero-Mart´ınez, Francisco Garc´ıa-C´ ordova, Juan L´ opez-Coronado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188 Control of Voluntary Movements in an Anthropomorphic Robot Finger by Using a Cortical Level Neural Controller Francisco Garc´ıa-C´ ordova, Juan Ignacio Mulero-Mart´ınez, Juan L´ opez-Coronado . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198

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Learning Control for Space Robotic Operation Using Support Vector Machines Panfeng Huang, Wenfu Xu, Yangsheng Xu, Bin Liang . . . . . . . . . . . . . . 1208 Neural Networks for Mobile Robot Navigation: A Survey An-Min Zou, Zeng-Guang Hou, Si-Yao Fu, Min Tan . . . . . . . . . . . . . . . 1218 Fault Diagnosis for Mobile Robots with Imperfect Models Based on Particle Filter and Neural Network Zhuohua Duan, Zixing Cai, Jinxia Yu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 Adaptive Neural Network Path Tracking of Unmanned Ground Vehicle Xiaohong Liao, Zhao Sun, Liguo Weng, Bin Li, Yongduan Song, Yao Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233

Power Systems A Nuclear Power Plant Expert System Using Artificial Neural Networks Mal rey Lee, Hye-Jin Jeong, Young Joon Choi, Thomas M. Gatton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239 Short-Term Load Forecasting Based on Mutual Information and Artificial Neural Network Zhiyong Wang, Yijia Cao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246 Short Term Load Forecasting by Using Neural Networks with Variable Activation Functions and Embedded Chaos Algorithm Qiyun Cheng, Xuelian Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252 Short Term Load Forecasting Using Neural Network with Rough Set Zhi Xiao, Shi-Jie Ye, Bo Zhong, Cai-Xin Sun . . . . . . . . . . . . . . . . . . . . . 1259 Application of Neural Network Based on Particle Swarm Optimization in Short-Term Load Forecasting Dong-Xiao Niu, Bo Zhang, Mian Xing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269 Study of Neural Networks for Electric Power Load Forecasting Hui Wang, Bao-Sen Li, Xin-Yang Han, Dan-Li Wang, Hong Jin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277 A Neural Network Approach to m-Daily-Ahead Electricity Price Prediction Hsiao-Tien Pao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284

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Next-Day Power Market Clearing Price Forecasting Using Artificial Fish-Swarm Based Neural Network Chuan Li, Shilong Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 Application of Evolutionary Neural Network to Power System Unit Commitment Po-Hung Chen, Hung-Cheng Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 Application of BP Neural Network in Power Load Simulator Bing-Da Zhang, Ke Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304 Feeder Load Balancing Using Neural Network Abhisek Ukil, Willy Siti, Jaco Jordaan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1311 A Neural Network Based Particle Swarm Optimization for the Transformers Connections of a Primary Feeder Considering Multi-objective Programming Cheng Chien Kuo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317 3-D Partial Discharge Patterns Recognition of Power Transformers Using Neural Networks Hung-Cheng Chen, Po-Hung Chen, Chien-Ming Chou . . . . . . . . . . . . . . 1324 Design of Self-adaptive Single Neuron Facts Controllers Based on Genetic Algorithm Quan-Yuan Jiang, Chuang-Xin Guo, Yi-Jia Cao . . . . . . . . . . . . . . . . . . . 1332 Generalized Minimum Variance Neuro Controller for Power System Stabilization Hee-Sang Ko, Kwang Y. Lee, Min-Jae Kang, Ho-Chan Kim . . . . . . . . . 1338 Adaptive Control for Synchronous Generator Based on Pseudolinear Neural Networks Youping Fan, Yunping Chen, Shangsheng Li, Dong Liu, Yi Chai . . . . . 1348 A Research and Application of Chaotic Neural Network for Marine Generator Modeling Wei-Feng Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354 Ship Synchronous Generator Modeling Based on RST and RBF Neural Networks Xihuai Wang, Tengfei Zhang, Jianmei Xiao . . . . . . . . . . . . . . . . . . . . . . . 1363 A New Control Strategy of a Wind Power Generation and Flywheel Energy Storage Combined System Jian Wang, Long-yun Kang, Bing-gang Cao . . . . . . . . . . . . . . . . . . . . . . . 1370

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Wavelet-Based Intelligent System for Recognition of Power Quality Disturbance Signals Suriya Kaewarsa, Kitti Attakitmongcol, Wichai Krongkitsiri . . . . . . . . . 1378 Recognition and Classification of Power Quality Disturbances Based on Self-adaptive Wavelet Neural Network Wei-Ming Tong, Xue-Lei Song, Dong-Zhong Zhang . . . . . . . . . . . . . . . . 1386 Vibration Fault Diagnosis of Large Generator Sets Using Extension Neural Network-Type 1 Meng-hui Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395 Fault Data Compression of Power System with Wavelet Neural Network Based on Wavelet Entropy Zhigang Liu, Dabo Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1402 Intelligent Built-in Test (BIT) for More-Electric Aircraft Power System Based on Hybrid Generalized LVQ Neural Network Zhen Liu, Hui Lin, Xin Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 Low Voltage Risk Assessment in Power System Using Neural Network Ensemble Wei-Hua Chen, Quan-Yuan Jiang, Yi-Jia Cao . . . . . . . . . . . . . . . . . . . . . 1416 Risk Assessment of Cascading Outages in Power Systems Using Fuzzy Neural Network Wei-Hua Chen, Quan-Yuan Jiang, Zhi-Yong Wang, Yi-Jia Cao . . . . . 1422 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1429

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Transportation Systems Traffic Volume Forecasting Based on Wavelet Transform and Neural Networks Shuyan Chen, Wei Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Prediction of Railway Passenger Traffic Volume by means of LS-SVM Zhen-Rui Peng, Fu Wu, Zhao-Yuan Jiang . . . . . . . . . . . . . . . . . . . . . . . . .

8

Traffic Flow Modeling of Urban Expressway Using Artificial Neural Networks Guo-Jiang Shen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Radial Basis Function Network for Traffic Scene Classification in Single Image Mode Qiao Huang, Jianming Hu, Jingyan Song, Tianliang Gao . . . . . . . . . . .

23

A New Method for Traffic Signs Classification Using Probabilistic Neural Networks Hang Zhang, Dayong Luo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Detection for Triangle Traffic Sign Based on Neural Network Shuang-dong Zhu, Yi Zhang, Xiao-feng Lu . . . . . . . . . . . . . . . . . . . . . . . .

40

Vanishing Point and Gabor Feature Based Multi-resolution On-Road Vehicle Detection Hong Cheng, Nanning Zheng, Chong Sun, Huub van de Wetering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Recognition of Road Signs with Mixture of Neural Networks and Arbitration Modules Boguslaw Cyganek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Vehicle Classification in Wireless Sensor Networks Based on Rough Neural Network Qi Huang, Tao Xing, Hai Tao Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Neural Network Approach to Identify Model of Vehicles Hyo Jong Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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An Intelligent Vehicle Security System Based on Modeling Human Driving Behaviors Xiaoning Meng, Yongsheng Ou, Ka Keung Lee, Yangsheng Xu . . . . . .

73

Adaptive Neural Network Control of Helicopters Shuzhi Sam Ge, Keng-Peng Tee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Communication Networks Modified Hopfield Neural Network for CDMA Multiuser Detection Xiangdong Liu, Xuexia Wang, Zhilu Wu, Xuemai Gu . . . . . . . . . . . . . .

88

Blind Multiuser Detection Based on Kernel Approximation Tao Yang, Bo Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

A Novel Blind Multiuser Detection Model over Flat Fast Fading Channels Hongbo Tian, Qinye Yin, Ke Deng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Robust Multiuser Detection Method Based on Neural-net Preprocessing in Impulsive Noise Environment Ying Guo, Tianshuang Qiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Channel Equalization Using Complex Extreme Learning Machine with RBF Kernels Ming-Bin Li, Guang-Bin Huang, Paramasivan Saratchandran, Narasimhan Sundararajan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Nonlinear Channel Equalization Using Concurrent Support Vector Machine Processor Jae Woo Wee, Tae Seon Kim, Sung Soo Dong, Chong Ho Lee . . . . . . . 120 Recursive Complex Extreme Learning Machine with Widely Linear Processing for Nonlinear Channel Equalizer Junseok Lim, Jaejin Jeon, Sangwook Lee . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A Study on the Detection Algorithm of QPSK Signal Using TDNN Sun-Kuk Noh, Jae-Young Pyun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The BP Network for Carrier Frequency Offset Estimation in OFDM-Based WLANs Feng Zhu, Yafeng Hu, Saiyi Wang, Peng Wei . . . . . . . . . . . . . . . . . . . . . 144 The LD-CELP Gain Filter Based on BP Neural Network Gang Zhang, Keming Xie, Zhefeng Zhao, Chunyu Xue . . . . . . . . . . . . . . 150

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A Neural Network Based Application Layer Multicast Routing Protocol Peng Cheng, Qiong-Hai Dai, Qiu-Feng Wu . . . . . . . . . . . . . . . . . . . . . . . 156 A Neural Network Decision-Making Mechanism for Robust Video Transmission over 3G Wireless Network Jianwei Wen, Qionghai Dai, Yihui Jin . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A Multilevel Quantifying Spread Spectrum PN Sequence Based on Chaos of Cellular Neural Network Yaqin Zhao, Nan Zhao, Zhilu Wu, Guanghui Ren . . . . . . . . . . . . . . . . . . 171 An Experimental Hyper-Chaos Spread Spectrum Communication System Based on CNN Jianye Zhao, Quansheng Ren, Daoheng Yu . . . . . . . . . . . . . . . . . . . . . . . . 178 A Resource Allocating Neural Network Based Approach for Detecting End-to-End Network Performance Anomaly Wenwei Li, Dafang Zhang, Jinmin Yang, Gaogang Xie, Lei Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Recurrent Neural Network Inference of Internal Delays in Nonstationary Data Network Feng Qian, Guang-min Hu, Xing-miao Yao, Le-min Li . . . . . . . . . . . . . 190 Multiscale BiLinear Recurrent Neural Networks and Their Application to the Long-Term Prediction of Network Traffic Dong-Chul Park, Chung Nguyen Tran, Yunsik Lee . . . . . . . . . . . . . . . . . 196 Bandwidth Prediction and Congestion Control for ABR Traffic Based on Neural Networks Zhixin Liu, Xinping Guan, Huihua Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Information Security Application of Neural Networks in Network Control and Information Security ´ Angel Grediaga, Francisco Ibarra, Federico Garc´ıa, Bernardo Ledesma, Francisco Brot´ ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Enhancing the Transmission Security of Content-Based Hidden Biometric Data Muhammad Khurram Khan, Jiashu Zhang . . . . . . . . . . . . . . . . . . . . . . . . 214

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Building Lightweight Intrusion Detection System Based on Random Forest Dong Seong Kim, Sang Min Lee, Jong Sou Park . . . . . . . . . . . . . . . . . . . 224 Intrusion Detection Based on Fuzzy Neural Networks Ji-yao An, Guangxue Yue, Fei Yu, Ren-fa Li . . . . . . . . . . . . . . . . . . . . . . 231 Intrusion Detection Using PCASOM Neural Networks Guisong Liu, Zhang Yi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 A Mutated Intrusion Detection System Using Principal Component Analysis and Time Delay Neural Network Byoung-Doo Kang, Jae-Won Lee, Jong-Ho Kim, O-Hwa Kwon, Chi-Young Seong, Se-Myung Park, Sang-Kyoon Kim . . . . . . . . . . . . . . . 246 A Novel Intrusion Detection Model Based on Multi-layer Self-Organizing Maps and Principal Component Analysis Jie Bai, Yu Wu, Guoyin Wang, Simon X. Yang, Wenbin Qiu . . . . . . . 255 A Modified RBF Neural Network for Network Anomaly Detection Xiaotao Wei, Houkuan Huang, Shengfeng Tian . . . . . . . . . . . . . . . . . . . . 261 Anti-worm Immunization of Web System Based on Normal Model and BP Neural Network Tao Gong, Zixing Cai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Data Hiding in Neural Network Prediction Errors Guangjie Liu, Jinwei Wang, Shiguo Lian, Yuewei Dai, Zhiquan Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 The Minimum Detectable Capacity of Digital Image Information Hiding Fan Zhang, Ruixin Liu, Xinhong Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Robust Digital Image Watermarking Algorithm Using BPN Neural Networks Cheng-Ri Piao, Wei-zhong Fan, Dong-Min Woo, Seung-Soo Han . . . . . 285 A Novel Watermarking Method with Image Signature Xiao-Li Niu, Ju Liu, Jian-De Sun, Jian-Ping Qiao . . . . . . . . . . . . . . . . . 293 Robust Halftone Image Watermarking Scheme Based on Neural Networks Xiang-yang Wang, Jun Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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A Blind Source Separation Based Multi-bit Digital Audio Watermarking Scheme Xiaohong Ma, Xiaoyan Ding, Chong Wang, Fuliang Yin . . . . . . . . . . . . 306 A 2DPCA-Based Video Watermarking Scheme for Resistance to Temporal Desynchronization Jiande Sun, Ju Liu, Hua Yan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 A Fast Decryption Algorithm for BSS-Based Image Encryption Qiu-Hua Lin, Fu-Liang Yin, Hua-Lou Liang . . . . . . . . . . . . . . . . . . . . . . 318 A Novel Cryptographic Scheme Based on Wavelet Neural Networks Guo Chen, Feng Tan, Degang Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Combining RBF Neural Network and Chaotic Map to Construct Hash Function Pengcheng Wei, Wei Zhang, Huaqian Yang, Jun Chen . . . . . . . . . . . . . 332 Multiple-Point Bit Mutation Method of Detector Generation for SNSD Model Ying Tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 An Erotic Image Recognition Algorithm Based on Trunk Model and SVM Classification Qindong Sun, Xinbo Huang, Xiaohong Guan, Peng Gao . . . . . . . . . . . . 346

Fault Detection Sensor Validation Using Nonlinear Minor Component Analysis Roger Xu, Guangfan Zhang, Xiaodong Zhang, Leonard Haynes, Chiman Kwan, Kenneth Semega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Fault Diagnosis with Enhanced Neural Network Modelling Ding-Li Yu, Thoon-Khin Chang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 Fault Detection and Diagnosis Using Neural Network Design Kok Kiong Tan, Sunan Huang, Tong Heng Lee . . . . . . . . . . . . . . . . . . . . 364 Certainty Improvement in Diagnosis of Multiple Faults by Using Versatile Membership Functions for Fuzzy Neural Networks Yuan Kang, Chun-Chieh Wang, Yeon-Pun Chang, Chien-Ching Hsueh, Ming-Chang Chang . . . . . . . . . . . . . . . . . . . . . . . . . . 370

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Fault Detection of Reactive Ion Etching Using Time Series Neural Networks Kyung-Han Ryu, Song-Jae Lee, Jaehyun Park, Dong-Chul Park, Sang J. Hong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Intelligent Diagnostics for Sound Reproduction System by the Use of PEAQ Byung Doo Jun, Nakjin Choi, Hyun-Woo Ko, KoengMo Sung . . . . . . . 382 Active Learning of Support Vector Machine for Fault Diagnosis of Bearings Zhousuo Zhang, Wenzhi Lv, Minghui Shen . . . . . . . . . . . . . . . . . . . . . . . . 390 Growing Structure Multiple Model System Based Anomaly Detection for Crankshaft Monitoring Jianbo Liu, Pu Sun, Dragan Djurdjanovic, Kenneth Marko, Jun Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Fault Diagnosis for Induction Machines Using Kernel Principal Component Analysis Jang-Hwan Park, Dae-Jong Lee, Myung-Geun Chun . . . . . . . . . . . . . . . . 406 Application of RBF and SOFM Neural Networks on Vibration Fault Diagnosis for Aero-engines Kai Li, Dongxiang Jiang, Kai Xiong, Yongshan Ding . . . . . . . . . . . . . . . 414 Predictive Fault Detection and Diagnosis of Nuclear Power Plant Using the Two-Step Neural Network Models Hyeon Bae, Seung-Pyo Chun, Sungshin Kim . . . . . . . . . . . . . . . . . . . . . . 420 Kernel PCA Based Faults Diagnosis for Wastewater Treatment System Byong-Hee Jun, Jang-Hwan Park, Sang-Ill Lee, Myung-Geun Chun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Financial Analysis On the Symbolic Analysis of Market Indicators with the Dynamic Programming Approach Luk´ aˇs Pichl, Takuya Yamano, Taisei Kaizoji . . . . . . . . . . . . . . . . . . . . . . 432 Neural Network Method to Predict Stock Price Movement Based on Stock Information Entropy Xun Liang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

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Stock Time Series Forecasting Using Support Vector Machines Employing Analyst Recommendations Zhi-yong Zhang, Chuan Shi, Su-lan Zhang, Zhong-zhi Shi . . . . . . . . . . . 452 Stock Index Prediction Based on the Analytical Center of Version Space Fanzi Zeng, Yonghua Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Comparison of Forecasting Performance of AR, STAR and ANN Models on the Chinese Stock Market Index Qi-an Chen, Chuan-Dong Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Index Prediction of KOSPI 200 Based on Data Models and Knowledge Rules for Qualitative and Quantitative Approach Hyeon Bae, Sungshin Kim, Joing-Il Bae . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Modular Neural Network Rule Extraction Technique in Application to Country Stock Cooperate Governance Structure Dang-Yong Du, Hai-Lin Lan, Wei-Xin Ling . . . . . . . . . . . . . . . . . . . . . . . 477 A Hybrid Support Vector Machines and Discrete Wavelet Transform Model in Futures Price Forecasting Fan-yong Liu, Min Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 A Novel Learning Network for Option Pricing with Confidence Interval Information Kyu-Hwan Jung, Hyun-Chul Kim, Jaewook Lee . . . . . . . . . . . . . . . . . . . . 491 An Adaptive BP Algorithm with Optimal Learning Rates and Directional Error Correction for Foreign Exchange Market Trend Prediction Lean Yu, Shouyang Wang, Kin Keung Lai . . . . . . . . . . . . . . . . . . . . . . . . 498 Recurrent Self-Organising Maps and Local Support Vector Machine Models for Exchange Rate Prediction He Ni, Hujun Yin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Selection of the Appropriate Lag Structure of Foreign Exchange Rates Forecasting Based on Autocorrelation Coefficient Wei Huang, Shouyang Wang, Hui Zhang, Renbin Xiao . . . . . . . . . . . . . 512 Exchange Rate Forecasting Using Flexible Neural Trees Yuehui Chen, Lizhi Peng, Ajith Abraham . . . . . . . . . . . . . . . . . . . . . . . . . 518 Local Volatility Function Approximation Using Reconstructed Radial Basis Function Networks Bo-Hyun Kim, Daewon Lee, Jaewook Lee . . . . . . . . . . . . . . . . . . . . . . . . . 524

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Neuroinformatics Visualization of Dynamic Brain Activities Based on the Single-Trial MEG and EEG Data Analysis Jianting Cao, Liangyu Zhao, Andrzej Cichocki . . . . . . . . . . . . . . . . . . . . 531 Multichannel Classification of Single EEG Trials with Independent Component Analysis Dik Kin Wong, Marcos Perreau Guimaraes, E. Timothy Uy, Logan Grosenick, Patrick Suppes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Application of SVM Framework for Classification of Single Trial EEG Xiang Liao, Yu Yin, Chaoyi Li, Dezhong Yao . . . . . . . . . . . . . . . . . . . . . 548 Normal and Hypoxia EEG Recognition Based on a Chaotic Olfactory Model Meng Hu, Jiaojie Li, Guang Li, Xiaowei Tang, Walter J. Freeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Nonlinear Dynamics of EEG Signal Based on Coupled Network Lattice Model Minfen Shen, Guoliang Chang, Shuwang Wang, Patch J. Beadle . . . . . 560 ICA-Based EEG Spatio-temporal Dipole Source Localization: A Model Study Ling Zou, Shan-An Zhu, Bin He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 Networking Property During Epileptic Seizure with Multi-channel EEG Recordings Huihua Wu, Xiaoli Li, Xinping Guan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Time-Frequency Analysis of EEG Based on Event Related Cognitive Task Xiao-Tong Wen, Xiao-Jie Zhao, Li Yao . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Adaptable Noise Reduction of ECG Signals for Feature Extraction Hyun Dong Kim, Chul Hong Min, Tae Seon Kim . . . . . . . . . . . . . . . . . . 586 Mining the Independent Source of ERP Components with ICA Decomposition Jia-Cai Zhang, Xiao-Jie Zhao, Yi-Jun Liu, Li Yao . . . . . . . . . . . . . . . . . 592 Multiple Signal Classification Based on Chaos Optimization Algorithm for MEG Sources Localization Jie-Ming Ma, Bin Wang, Yang Cao, Li-Ming Zhang . . . . . . . . . . . . . . . 600

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Automatic Segmentation of Putamen from Brain MRI Yihui Liu, Bai Li, Dave Elliman, Paul Simon Morgan, Dorothee Auer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 A Neural Network Model for the Estimation of Time-to-Collision Ling Wang, Hongjin Sun, Dezhong Yao . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Classification of Movement-Related Potentials for Brain-Computer Interface: A Reinforcement Training Approach Zongtan Zhou, Yang Liu, Dewen Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Bioinformatics Two-Class SVM Trees (2-SVMT) for Biomarker Data Analysis Shaoning Pang, Ilkka Havukkala, Nikola Kasabov . . . . . . . . . . . . . . . . . . 629 Interpreting Gene Profiles from Biomedical Literature Mining with Self Organizing Maps Shi Yu, Steven Van Vooren, Bert Coessens, Bart De Moor . . . . . . . . . . 635 Mining Protein Interaction from Biomedical Literature with Relation Kernel Method Jae-Hong Eom, Byoung Tak Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642 A Study of Particle Swarm Optimization in Gene Regulatory Networks Inference Rui Xu, Ganesh Venayagamoorthy, Donald C. Wunsch II . . . . . . . . . . . 648 Support Vector Machine Approach for Retained Introns Prediction Using Sequence Features Huiyu Xia, Jianning Bi, Yanda Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 Prediction of Protein-Protein Interface Residues Using Sequence Neighborhood and Surface Properties Yasir Arafat, Joarder Kamruzzaman, Gour Karmakar . . . . . . . . . . . . . . 660 Prediction of Protein Subcellular Multi-locations with a Min-Max Modular Support Vector Machine Yang Yang, Bao-Liang Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 Prediction of Protein Domains from Sequence Information Using Support Vector Machines Shuxue Zou, Yanxin Huang, Yan Wang, Chunguang Zhou . . . . . . . . . . . 674

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Using a Neural Networking Method to Predict the Protein Phosphorylation Sites with Specific Kinase Kunpeng Zhang, Yun Xu, Yifei Shen, Guoliang Chen . . . . . . . . . . . . . . . 682 Neural Feature Association Rule Mining for Protein Interaction Prediction Jae-Hong Eom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 Prediction of Contact Maps Using Modified Transiently Chaotic Neural Network Guixia Liu, Yuanxian Zhu, Wengang Zhou, Chunguang Zhou, Rongxing Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 Identification of Cell-Cycle Phases Using Neural Network and Steerable Filter Features Xiaodong Yang, Houqiang Li, Xiaobo Zhou, Stephen T.C. Wong . . . . . 702 Prediction of the Human Papillomavirus Risk Types Using Gap-Spectrum Kernels Sun Kim, Jae-Hong Eom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710 Extreme Learning Machine for Predicting HLA-Peptide Binding Stephanus Daniel Handoko, Kwoh Chee Keong, Ong Yew Soon, Guang Lan Zhang, Vladimir Brusic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 Identifying Transcription Factor Binding Sites Based on a Neural Network Xianhua Dai, Zhiming Dai, Jiang Wang . . . . . . . . . . . . . . . . . . . . . . . . . . 722 TSFSOM: Transmembrane Segments Prediction by Fuzzy Self-Organizing Map Yong Deng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728

Biomedical Applications Analysis of Multifibre Renal Sympathetic Nerve Recordings Dong Li, Yingxiong Jin, Zhuo Yang, Tao Zhang . . . . . . . . . . . . . . . . . . . 734 A New Color Blindness Cure Model Based on BP Neural Network Yu Ma, Xiao-Dong Gu, Yuan-Yuan Wang . . . . . . . . . . . . . . . . . . . . . . . . 740 Design of RBF Network Based on Fuzzy Clustering Method for Modeling of Respiratory System Kouji Maeda, Shunshoku Kanae, Zi-Jiang Yang, Kiyoshi Wada . . . . . . 746

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Recognition of Fatty Liver Using Hybrid Neural Network Jiangli Lin, XianHua Shen, Tianfu Wang, Deyu Li, Yan Luo, Ling Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 A Novel Fast Fuzzy Neural Network Backpropagation Algorithm for Colon Cancer Cell Image Discrimination Ephram Nwoye, Li C. Khor, Satnam S. Dlay, Wai L. Woo . . . . . . . . . . 760 Poultry Skin Tumor Detection in Hyperspectral Images Using Radial Basis Probabilistic Neural Network Intaek Kim, Chengzhe Xu, Moon S. Kim . . . . . . . . . . . . . . . . . . . . . . . . . 770 Combination of Network Construction and Cluster Analysis and Its Application to Traditional Chinese Medicine Mingfeng Wang, Zhi Geng, Miqu Wang, Feng Chen, Weijun Ding, Ming Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Differentiation of Syndromes with SVM Zhanquan Sun, Guangcheng Xi, Jianqiang Yi . . . . . . . . . . . . . . . . . . . . . 786 Neural Network Based Posture Control of a Human Arm Model in the Sagittal Plane Shan Liu, Yongji Wang, Jian Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

Industrial Applications Neural Network Models for Transforming Consumer Perception into Product Form Design Chung-Hsing Yeh, Yang-Cheng Lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 2D Pattern Design of Upper Outer from 3D Specification Based on Neural Networks Dongyun Wang, Yulin Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 Design of a Broadband Microwave Amplifier Using Neural Performance Data Sheets and Very Fast Simulated Reannealing Yavuz Cengiz, H¨ useyin G¨ oksu, Filiz G¨ une¸s . . . . . . . . . . . . . . . . . . . . . . . . 815 An Intelligent System for the Heatsink Design Yao-Wen Hsueh, Hsin-Chung Lien, Ming-Hsien Hsueh . . . . . . . . . . . . . 821 Learning Shape for Jet Engine Novelty Detection David A. Clifton, Peter R. Bannister, Lionel Tarassenko . . . . . . . . . . . . 828

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Support Vector Machine in Novelty Detection for Multi-channel Combustion Data Lei A. Clifton, Hujun Yin, Yang Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 A Levinson Predictor Based Compensatory Fuzzy Neural Network and Its Application in Crude Oil Distillation Process Modeling Yongfeng He, Quanyi Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844 Laminar Cooling Process Model Development Using RBF Networks Minghao Tan, Xuejun Zong, Heng Yue, Jinxiang Pian, Tianyou Chai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 Hybrid Intelligent Control Strategy of the Laminar Cooling Process Minghao Tan, Shujiang Li, Tianyou Chai . . . . . . . . . . . . . . . . . . . . . . . . . 858 Application of Adaptable Neural Networks for Rolling Force Set-Up in Optimization of Rolling Schedules Jingming Yang, Haijun Che, Yajie Xu, Fuping Dou . . . . . . . . . . . . . . . . 864 Multiple Neural Network Modeling Method for Carbon and Temperature Estimation in Basic Oxygen Furnace Xin Wang, Zhong-Jie Wang, Jun Tao . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870 Air-Fuel-Ratio Optimal Control of a Gas Heating Furnace Based on Fuzzy Neural Networks Heng Cao, Ding Du, Yunhua Peng, Yuhai Yin . . . . . . . . . . . . . . . . . . . . . 876 An Evolutionary Hybrid Model for the Prediction of Flow Stress of Steel Ai-ling Chen, Gen-ke Yang, Zhi-ming Wu . . . . . . . . . . . . . . . . . . . . . . . . . 885 Meta-Learning Evolutionary Artificial Neural Network for Selecting Flexible Manufacturing Systems Arijit Bhattacharya, Ajith Abraham, Crina Grosan, Pandian Vasant, Sangyong Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 A Multi-Criteria Decision Making Procedure Based on Neural Networks for Kanban Allocation ¨ ¨ ur Eski, Ceyhun Araz . . . . . . . . . . . . . . . . . . . . . . 898 Ozlem Uzun Araz, Ozg¨ On-Line Measurement of Production Plan Track Based on Extension Matter-Element Theory Zhi-Lin Sheng, Song-Zheng Zhao, Xin-Zheng Qi, Chen-Xi Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906

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Modeling and Optimization of High-Technology Manufacturing Productivity Sheng Xu, Hui-Fang Zhao, Zhao-Hua Sun, Xiao-Hua Bao . . . . . . . . . . . 914 Scheduling of Re-entrant Lines with Neuro-Dynamic Programming Based on a New Evaluating Criterion Ying Wang, Huiyu Jin, Shunzhi Zhu, Maoqing Li . . . . . . . . . . . . . . . . . . 921 A Constraint Satisfaction Adaptive Neural Network with Dynamic Model for Job-Shop Scheduling Problem Li-Ning Xing, Ying-Wu Chen, Xue-Shi Shen . . . . . . . . . . . . . . . . . . . . . . 927 Neural Network Based Industrial Processes Monitoring Luis P. S´ anchez-Fern´ andez, Cornelio Y´ an ˜ez-M´ arquez, Oleksiy Pogrebnyak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933 A New Method for Process Monitoring Based on Mixture Probabilistic Principal Component Analysis Models Zhong-Gai Zhao, Fei Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939 On-Line Nonlinear Process Monitoring Using Kernel Principal Component Analysis and Neural Network Zhong-Gai Zhao, Fei Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945 On-Line Batch Process Monitoring Using Multiway Kernel Independent Component Analysis Fei Liu, Zhong-Gai Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 Tool Wear Monitoring Using FNN with Compact Support Gaussian Function Hongli Gao, Mingheng Xu, Jun Li, Chunjun Chen . . . . . . . . . . . . . . . . . 957 Intelligent Classification of Cutting Tool Wear States Pan Fu, Anthony D. Hope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 Neural Networks-Based In-Process Surface Roughness Adaptive Control System in Turning Operations Julie Z. Zhang, Joseph C. Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970 Modeling of Micro Spring Tension Force for Vertical Type Probe Card Fabrication Chul Hong Min, Tae Seon Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976 Identification of Crack Location and Depth in Rotating Machinery Based on Artificial Neural Network Tao Yu, Qing-Kai Han, Zhao-Ye Qin, Bang-Chun Wen . . . . . . . . . . . . . 982

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Natural Color Recognition Using Fuzzification and a Neural Network for Industrial Applications Yountae Kim, Hyeon Bae, Sungshin Kim, Kwang-Baek Kim, Hoon Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991 Design of a High Precision Temperature Measurement System Kenan Danisman, Ilker Dalkiran, Fatih V. Celebi . . . . . . . . . . . . . . . . . . 997 Integrating Computational Fluid Dynamics and Neural Networks to Predict Temperature Distribution of the Semiconductor Chip with Multi-heat Sources Yean-Der Kuan, Yao-Wen Hsueh, Hsin-Chung Lien, Wen-Ping Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 Modeling and Characterization of Plasma Processes Using Modular Neural Network Seung Soo Han, Dong Sun Seo, Sang Jeen Hong . . . . . . . . . . . . . . . . . . . 1014 Prediction of Plasma Enhanced Deposition Process Using GA-Optimized GRNN Byungwhan Kim, Dukwoo Lee, Seung Soo Han . . . . . . . . . . . . . . . . . . . . . 1020 Prediction of Radio Frequency Impedance Matching in Plasma Equipment Using Neural Network Byungwhan Kim, Donghwan Kim, Seung Soo Han . . . . . . . . . . . . . . . . . 1028 Recognition of Plasma-Induced X-Ray Photoelectron Spectroscopy Fault Pattern Using Wavelet and Neural Network Byungwhan Kim, Sooyoun Kim, Sang Jeen Hong . . . . . . . . . . . . . . . . . . 1036 Polynomial Neural Network Modeling of Reactive Ion Etching Process Using GMDH Method Seung-Soo Han, Sang Jeen Hong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 Wafer Yield Estimation Using Support Vector Machines Lei-Ting Chen, David Lin, Dan Muuniz, Chia-Jiu Wang . . . . . . . . . . . . 1053 Dynamic Soft-Sensing Model by Combining Diagonal Recurrent Neural Network with Levinson Predictor Hui Geng, Zhihua Xiong, Shuai Mao, Yongmao Xu . . . . . . . . . . . . . . . . 1059 Thermal Properties Reduced Models by ANN in Process Simulation Xia Yang, Rongshan Bi, Yugang Li, Shiqing Zheng . . . . . . . . . . . . . . . . . 1065

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Nonlinear Identification of a PEM Fuel Cell Humidifier Using Wavelet Networks Xian-Rui Deng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 Application of RBF Neural Networks Based on a New Hybrid Optimization Algorithm in Flotation Process Yong Zhang, Jie-Sheng Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Estimation of Some Crucial Variables in Erythromycin Fermentation Process Based on ANN Left-Inversion Xianzhong Dai, Wancheng Wang, Yuhan Ding . . . . . . . . . . . . . . . . . . . . 1085 The Control of Membrane Thickness in PECVD Process Utilizing a Rule Extraction Technique of Neural Networks Ming Chang, Jen-Cheng Chen, Jia-Sheng Heh . . . . . . . . . . . . . . . . . . . . . 1091 PCA-Based Neural Network Modeling Using the Photoluminescence Data for Growth Rate of ZnO Thin Films Fabricated by Pulsed Laser Deposition Jung Hwan Lee, Young-Don Ko, Min-Chang Jeong, Jae-Min Myoung, Ilgu Yun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 Wood Defects Classification Using a SOM/FFP Approach with Minimum Dimension Feature Vector Mario I. Chacon, Graciela Ramirez Alonso . . . . . . . . . . . . . . . . . . . . . . . . 1105 A Kernel Based Multi-resolution Time Series Analysis for Screening Deficiencies in Paper Production Marcus Ejnarsson, Carl Magnus Nilsson, Antanas Verikas . . . . . . . . . . 1111 Using Directed Acyclic Graph Support Vector Machines with Tabu Search for Classifying Faulty Product Types Ping-Feng Pai, Yu-Ying Huang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117 Product Quality Prediction with Support Vector Machines Xinggao Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126 Hierarchical Neural Network Based Product Quality Prediction of Industrial Ethylene Pyrolysis Process Qiang Zhou, Zhihua Xiong, Jie Zhang, Yongmao Xu . . . . . . . . . . . . . . . 1132 A Sub-stage Moving Window GRNN Quality Prediction Method for Injection Molding Processes Xiao-Ping Guo, Fu-Li Wang, Ming-Xing Jia . . . . . . . . . . . . . . . . . . . . . . 1138

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Joint Time-Frequency and Kernel Principal Component Based SOM for Machine Maintenance Qianjin Guo, Haibin Yu, Yiyong Nie, Aidong Xu . . . . . . . . . . . . . . . . . . 1144

Other Applications Automatic Recognition and Evaluation of Natural Language Commands Maciej Majewski, Wojciech Kacalak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 Natural Language Human-Machine Interface Using Artificial Neural Networks Maciej Majewski, Wojciech Kacalak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161 Implementing a Chinese Character Browser Using a Topography-Preserving Map James S. Kirk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167 A Soft Computing Method of Economic Contribution Rate of Education: A Case of China Hai-xiang Guo, Ke-jun Zhu, Jin-ling Li, Yan-min Xing . . . . . . . . . . . . . 1173 Improving Population Estimation with Neural Network Models Zaiyong Tang, Caroline W. Leung, Kallol Bagchi . . . . . . . . . . . . . . . . . . 1181 Application of Fuzzy Neural Network for Real Estate Prediction Jian-Guo Liu, Xiao-Li Zhang, Wei-Ping Wu . . . . . . . . . . . . . . . . . . . . . . 1187 Local Neural Networks of Space-Time Predicting Modeling for Lattice Data in GIS Haiqi Wang, Jinfeng Wang, Xuhua Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192 Modeling Meteorological Prediction Using Particle Swarm Optimization and Neural Network Ensemble Jiansheng Wu, Long Jin, Mingzhe Liu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 A Fast Cloud Detection Approach by Integration of Image Segmentation and Support Vector Machine Bo Han, Lishan Kang, Huazhu Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 Application of Support Vector Machines to Vapor Detection and Classification for Environmental Monitoring of Spacecraft Tao Qian, Xiaokun Li, Bulent Ayhan, Roger Xu, Chiman Kwan, Tim Griffin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216

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Artificial Neural Network Methodology for Three-Dimensional Seismic Parameters Attenuation Analysis Ben-yu Liu, Liao-yuan Ye, Mei-ling Xiao, Sheng Miao, Jing-yu Su . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 Estimation of the Future Earthquake Situation by Using Neural Networks Ensemble Tian-Yu Liu, Guo-Zheng Li, Yue Liu, Geng-Feng Wu, Wei Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 An Expert System Based on BP Neural Networks for Pre-splitting Blasting Design Xiaohong Li, Xinfei Wang, Yongkang Dong, Qiang Ge, Li Qian . . . . . 1237 Surface Reconstruction Based on Radial Basis Functions Network Han-bo Liu, Xin Wang, Xiao-jun Wu, Wen-yi Qiang . . . . . . . . . . . . . . . 1242 Determination of Gas and Water Volume Fraction in Oil Water Gas Pipe Flow Using Neural Networks Based on Dual Modality Densitometry Chunguo Jing, Guangzhong Xing, Bin Liu, Qiuguo Bai . . . . . . . . . . . . . 1248 Application of Neural Network in Metal Loss Evaluation for Gas Conducting Pipelines Wei Zhang, Jing-Tao Guo, Song-Ling Huang . . . . . . . . . . . . . . . . . . . . . . 1254 Soft Sensor Using PNN Model and Rule Base for Wastewater Treatment Plant Yejin Kim, Hyeon Bae, Kyungmin Poo, Jongrack Kim, Taesup Moon, Sungshin Kim, Changwon Kim . . . . . . . . . . . . . . . . . . . . . 1261 Knowledge Acquisition Based on Neural Networks for Performance Evaluation of Sugarcane Harvester Fang-Lan Ma, Shang-Ping Li, Yu-Lin He, Shi Liang, Shan-Shan Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1270 Application of Artificial Neural Network in Countercurrent Spray Saturator Yixing Li, Yuzhang Wang, Shilie Weng, Yonghong Wang . . . . . . . . . . . 1277 Wavelet Neural Networks Approach for Dynamic Measuring Error Decomposition Yan Shen, Bing Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283

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Maneuvering Target Tracking Based on Unscented Particle Filter Aided by Neutral Network Feng Xue, Zhong Liu, Zhang-Song Shi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 Application of Principal Component-Artificial Neural Networks in Near Infrared Spectroscopy Quantitative Analysis Hai-Yan Ji, Zhen-Hong Rao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296 Application of Neural Networks for Integrated Circuit Modeling Xi Chen, Gao-Feng Wang, Wei Zhou, Qing-Lin Zhang, Jiang-Feng Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304 Power Estimation of CMOS Circuits by Neural Network Macromodel Wei Qiang, Yang Cao, Yuan-yuan Yan, Xun Gao . . . . . . . . . . . . . . . . . . 1313

Hardware Implementation An Efficient Hardware Architecture for a Neural Network Activation Function Generator Daniel Larkin, Andrew Kinane, Valentin Muresan, Noel O’Connor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 A Design and Implementation of Reconfigurable Architecture for Neural Networks Based on Systolic Arrays Qin Wang, Ang Li, Zhancai Li, Yong Wan . . . . . . . . . . . . . . . . . . . . . . . . 1328 Hardware In-the-Loop Training of Analogue Neural Network Chip Liang Zhang, Joaquin Sitte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334 Implementation of a Neural Network Processor Based on RISC Architecture for Various Signal Processing Applications Dong-Sun Kim, Hyun-Sik Kim, Duck-Jin Chung . . . . . . . . . . . . . . . . . . . 1340 Fully-Pipelining Hardware Implementation of Neural Network for Text-Based Images Retrieval Dongwuk Kyoung, Keechul Jung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1350 FPGA Implementation of a Neural Network for Character Recognition Farrukh A. Khan, Momin Uppal, Wang-Cheol Song, Min-Jae Kang, Anwar M. Mirza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357 A Silicon Synapse Based on a Charge Transfer Device for Spiking Neural Network Application Yajie Chen, Steve Hall, Liam McDaid, Octavian Buiu, Peter Kelly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366

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Effect of Steady and Relaxation Oscillations in Brillouin-Active Fiber Structural Sensor Based Neural Network in Smart Structures Yong-Kab Kim, Soonja Lim, ChangKug Kim . . . . . . . . . . . . . . . . . . . . . . 1374 A Novel All-Optical Neural Network Based on Coupled Ring Lasers Ying Chen, Qi-guang Zhu, Zhi-quan Li . . . . . . . . . . . . . . . . . . . . . . . . . . . 1380 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387

The Ideal Noisy Environment for Fast Neural Computation Si Wu1 , Jianfeng Feng2 , and Shun-ichi Amari3 1

Department of Informatics, University of Sussex, UK 2 Department of Computer Science, Warwick University, UK 3 Lab. for Mathematical Neuroscience, RIKEN Brain Science Institute, Japan

Abstract. A central issue in computational neuroscience is to answer why neural systems can process information extremely fast. Here we investigate the effect of noise and the collaborative activity of a neural population on speeding up computation. We find that 1) when input noise is Poissonian, i.e., its variance is proportional to the mean, and 2) when the neural ensemble is initially at its stochastic equilibrium state, noise has the ‘best’ effect of accelerating computation, in the sense that the input strength is linearly encoded by the number of neurons firing in a short-time window, and that the neural system can use a simple strategy to read out the stimulus rapidly and accurately.

1

Introduction

Neural systems can process information extremely fast. Taking the primate visual systems as an example, the event-related potential study has revealed that human subjects are able to carry out some complex scenes analysis in less than 150 ms [1]. Recoding also shows that the latency of neural responses in V1 can be as short as 40 ms [2], and in the temporal cortex 80 − 110 ms [3]. Understanding why neural systems can carry out information processing at such a rapid speed is of critical importance in understanding computational mechanisms underlying brain functions. Recent studies on the properties of neural dynamics have suggested that stochastic noise, which is ubiquitously observed in biological systems and is often thought to degrade information processing, may actually play a critical role in speeding up neural computation [4-5]. To see this point clearly, let us first look at the limitation on the speed of neural response in a noiseless environment. Without loss of generality, we consider the neural dynamics is simply written as τ dv/dt = I, with v representing the membrane potential of the neuron, I the strength of external input and τ the membrane time constant. Consider I is a constant and the initial state of the neuron is fixed to be v0 . The firing moment of the neuron is then calculated to be T = τ (θ − v0 )/I, where θ is the firing threshold. Thus, in general the response speed of a neuron J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 1–6, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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in a noiseless environment is in the order of τ , which has a value of 10 − 20 ms. This cannot account for the fast computational behaviours as observed in data [1-3]. To increase the speed of neural computation, one idea is to take into account the effect of noise and the collaborative activities of a neural ensemble (see, e.g. [4, 5]). The underlying picture is intuitively understandable. Noise randomizes the initial state of the neural population, i.e., the distribution of membrane potentials of neurons. As a result, those neurons whose potentials are close to the threshold will fire rapidly after the onset of a stimulus. Therefore, we can expect that if the noisy environment (which includes both the form of input noise and the initial state of the neural ensemble) is appropriate, then even for small inputs a non-zero percentage of the neural population will fire quickly to report the presence of the stimulus. Ideally, if the joint activity of the neural population exhibits a simple statistical feature which encodes the input strength faithfully, then the neural system can utilize a simple strategy to read out the stimulus rapidly and accurately. Although the idea seems straightforward, the detailed mechanism by which noise accelerates computation and its exact achievement remain largely unclear. In particular, these issues include: The encoding scheme. While noise speeds up the reaction time of the neural system, it also complicates the way of encoding external stimuli. Simply thinking, because of noise the spike timing of individual neurons can no longer encode inputs reliably in a short-time window, and the activities of the neural ensemble need to be considered. Furthermore, to simplify the decoding process (presuming this is associated with low computational cost, and hence also affect the speed of computation), it is critical that external information is encoded by a simple feature of neural population activity. The suitable noisy environment. Fast computation relies on neural activity in the very initial moment of presenting stimuli. Thus, its performance is sensitive to the structure of input noise and the initial state of the system. A critical question is: what kind of noisy environment is most suitable for fast computation, so that the neural system can use a simple strategy to read out stimuli rapidly and accurately. The goal of this study is to explore the above two issues.

2

The Model

We consider a simple population coding paradigm, in which a one-dimensional continuous variable x is encoded by N neurons [8]. Neurons are clustered according to their preferred stimuli, denoted as c. The dynamics of a single neuron is specified by the I & F operation, i.e, τ

dvci = −vci + Ici (t), dt

(1)

The Ideal Noisy Environment for Fast Neural Computation

3

where vci denotes the potential of ith neuron in the cluster c and Ici (t) the input. The fluctuations of Ici (t) are assume to be Gaussian white noise, that is, Ici (t) = μc + σc ξci (t), μc ξci (t) ξci (t1 )ξci (t2 )

= Ae−(c−x) = 0,

2

(2) 2

/2a

= δ(t1 − t2 ),

,

(3) (4) (5)

where the variable μc denotes the mean drift of the external inputs, which is a Gaussian function of the difference between the true stimulus x and the neuronal preferred stimulus c. A is a positive constant controlling the magnitude of the input. The variable ξci (t) represents Gaussian white noise of zero mean and unit variance, and σc the noise strength. We consider fluctuations of external inputs between different neurons are independent of each other. Notably, for neurons in the same cluster, their inputs have the same mean drift. The form of noise is the key that determines the performance of fast computation. It turns out that the following noise structure, σc2 = αμc ,

α > 0,

(6)

has the best effect of accelerating neural computation. This noise structure is stimulus-dependent. For convenience, we call it Poissonian, taking into account that when α = 1, noise is Poisson.

3

The Poissonian Equilibrium Theory

To achieve fast computation, the key is that the neural system can use a simple statistical feature of its population activity measured in a short-time window to encode external stimuli. For the model we consider, since neural activities within a cluster are homogeneous and independent of each other, a natural choice for such a simple feature is the number of neurons firing in a short-time window, which we denote as Nc (t), i.e. the number of neurons fired in the cluster c within the time interval t. The detailed firing pattern of the cluster within the timewindow is irrelevant. But, can Nc represent the input information faithfully in a short-time window? Clearly, this depends on the noisy environment. Ideally, we may expect that there exists a noisy environment, in which the mean drift of external inputs, which is correlated with the stimulus in our case, is linearly and instantly encoded by Nc (t) (more exactly, the mean value of Nc ), that is, Nc (t) ∼ μc t.

(7)

If this encoding scheme holds, then the neural estimator can use a simple strategy (NE) to read out the stimulus, i.e.,  Nc (t)c xˆN E (t) = c . (8) c Nc (t)

4

3.1

S. Wu, J. Feng, and S.-i. Amari

The Ideal Noisy Environment

Our finding of the ideal noisy environment is inspired by the following observations. We compute the equilibrium state of the neural ensemble given a fixed stimulus. Since neural clusters are independent of each other, it is adequate for us to consider only one of them. For simplicity, let us consider first there is no decay term (−vci ) in the neural dynamics of eq. (1). Denote pc (v, t) as the probability of finding the potential of a randomly chosen neuron in cluster c at v at time t. The Fokker-Plank equation for pc (v, t) is given by [6, 7] 1 ∂pc (v, t) ∂pc (v, t) σc2 ∂ 2 pc (v, t) = −μc + . (9) τ ∂t ∂v 2τ ∂v 2 The equilibrium distribution, pc (v), of the cluster, is determined by ∂pc (v, t)/∂t = 0, i.e., ∂pc (v) σc2 ∂ 2 pc (v) − μc + = 0. (10) ∂v 2τ ∂v 2 Assume noise is Poissionian, i.e., σc2 = αμc , we have −

∂pc (v) α ∂ 2 pc (v) + = 0. ∂v 2τ ∂v 2

(11)

Thus, from the above equation, we see that for Poissonian noise, pc (v) is independent of μc , i.e., independent of the value of stimulus. At the equilibrium state, pc (v) satisfies a set of boundary conditions [7], which are pc (θ) = 0, ∂pc (θ) 2τ 2 γc =− 2 , ∂v σc + − ∂pc (vr ) ∂pc (vr ) 2τ 2 γc − =− 2 , ∂v ∂v σc

(12) (13) (14)

where vr denotes the resting potential and γc the instant firing rate of the cluster c. θ By further utilizing the normalization condition, i.e., −∞ pc (v)dv = 1, pc (v) can be exactly solved, which is given by ⎧1 ⎨ θ (1 − e−2τ θ/α )e2τ v/α v < 0, pc (v) = 1θ (1 − e2τ (v−θ)/α ) (15) 0 ≤ v ≤ θ, ⎩ 0 v > θ, where we have set vr = 0. The profile of pc (v) is shown in Fig.1A. The instant firing rate γc in the equilibrium state is calculated to be, γc = − =

σc2 ∂pc (v) |θ , 2τ 2 ∂v

μc . τθ

(16)

The Ideal Noisy Environment for Fast Neural Computation

5

Thus, the mean of Nc (t), i.e., Nc (t) = M μc t/(τ θ), is proportional to μc t. The above study reveals two important features of the network dynamics when the noise structure is Poissonian, which are: 1. In the equilibrium state, the linear encoding scheme eq.(7) holds. 2. The equilibrium state of a neural cluster is insensitive to the stimulus value. It may not be straightforward to see the importance of this second property which guarantees that the equilibrium state of the neural system serves to be the optimal initial condition for fast computation. To understand this point, let us imagine the opposite situation that pc (v) is sensitive to μc . We then have difficulty identifying a unique optimal initial condition, albeit at each pc (v) eq.(7) holds. Interestingly, it turns out that when the decay term (−vci ) is added in the neural dynamics, the above properties still hold roughly (see Fig.1B). A

B

0.9

μc=0.5

0.8

μc=1

1 0.7 0.8

pc(v)

pc(v)

0.6 0.5 0.4

μ =1.5 c

0.6

0.4

0.3 0.2

0.2 0.1 0 −1

−0.5

0

0.5

1

0 −1

−0.5

0

0.5

1

v

v

Fig. 1. The equilibrium state of a neural cluster under Poissonian noise. (A) Without the decay term. (B) With the decay term.

Drawing the above observations together, we come to the conclusion that the ideal noisy environment for fast computation consists of two elements, namely, – The structure of input noises satisfies σc2 = αμc ; – The neural ensemble is initially at its equilibrium state. We call this encoding scheme the Poissonian equilibrium theory. In the ideal noisy environment, consider N t is sufficiently large, the error of neural estimator (NE) is calculated to be (ˆ xN E − x)2  ≈ √

κτ a . 2πAρt

(17)

where κ is the variance-mean ratio of Nc and ρ is the density of neurons. Fig.2 shows the simulation results, which confirms that the error of NE decreases with the length of decoding time t (Fig.2A) and the number of neurons (Fig.2B). The error of NE can be arbitrarily small at any decoding time, provided the number of neurons is sufficiently large.

6

S. Wu, J. Feng, and S.-i. Amari A

−3

1.5

B

x 10

0.06 α=1 α=0.8 α=0.6

0.05

Decoding Error

Decoding Erroe

α=0.6 α=0.8 α=1 1

0.5

0.04

0.03

0.02

0.01

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Decoding Time

0.8

0.9

1

0 10

20

30

40

50

60

70

80

90

100

M

Fig. 2. Illustrating the performance of NE. (a) For different decoding time. (b) For different number of neurons.

4

Conclusions

The present study investigates the ideal noisy environment for fast neural computation. We observe that the stimulus-dependent Poissonian noise, rather than stimulus-independent ones, has the best effect of accelerating neural information processing. This property is somehow intuitively understandable. For the strong diffusive noise, in a short time-window, it is fluctuations, rather than the mean √ drift, √ that dominate the value of external inputs (that is, W (t) ≈ μt + ση t ≈ ση t, for t  1, where η is a Gaussian random number of zero mean and unit variance). The signal-noise correlation is the key that enables the stimulus information to be adequately propagated to the neural system quickly. As elucidated in this study, with a proper structure, the stimulus-dependent Poissonian noise not only increases the response speed of the neural ensemble, but also provides a simple linear scheme to encode the stimulus.

References 1. Thorpe, S., Fize, D. & Marlot, C.: Speed of Processing in the Human Visual System. Nature 381 (1996) 520-522. 2. Celebrini, S., Thorpe, S., Trotter, Y. & Imbert, M.: Dynamics of Orientation Coding in Area V1 of the Awake Primate. Vis. Neurosci. 10 (1993) 811-825 3. Sugase, Y., Yamane, S., Ueno, S. & Kawano, K.: Global and Fine Information Coded by Single Neurons in the Temporal Visual Cortex. Nature 400 (1999) 869-873 4. Gerstner, W.: Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous State and Locking. Neural Computation 12 (1999) 43-89 5. van Rossum, M., Turrigiano, G. & Nelson, S.: Fast Propagation of Firing Rates through Layered Networks of Neurons. J. Neurosci. 22 (2002) 1956-1966 6. Tuckwell, H.: Introduction to Theoretical Neurobiology. Cambridge University Press, Cambridge (1996) 7. Brunel, N. & Hakim, V.: Fast Global Oscillations in Networks of Integarte-and-Fire Neurons with Low Firing Rates. Neural Computation 11 (1999) 1621-1671 8. Wu, S., Amari, S. & Nakahara, H.: Population Coding and Decoding in a Neural Field: a Computational Study. Neural Computation 14 (2002) 999-1026

How Does a Neuron Perform Subtraction? ─ Arithmetic Rules of Synaptic Integration of Excitation and Inhibition Xu-Dong Wang1,2, Jiang Hao1,2, Mu-Ming Poo1,3, and Xiao-Hui Zhang1 1

Institute of Neuroscience Graduate School of Chinese Academy of Sciences, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, Shanghai, 200031, China {wxd, acchj, xhzhang}@ion.ac.cn 3 Division of Neurobiology, Department of Molecular and Cell Biology, University of California, Berkeley, California 94720, USA [email protected] 2

Abstract. Numerous learning rules have been devised to carry out computational tasks in various neural network models. However, the rules for determining how a neuron integrates thousands of synaptic inputs on the dendritic arbors of a realistic neuronal model are still largely unknown. In this study, we investigated the properties of integration of excitatory and inhibitory postsynaptic potentials in a reconstructed pyramidal neuron in the CA1 region of the hippocampus. We found that the integration followed a nonlinear subtraction rule (the Cross-Shunting Rule, or CS rule). Furthermore, the shunting effect is dependent on the spatial location of inhibitory synapses, but not that of excitatory synapses. The shunting effect of inhibitory inputs was also found to promote the synchronization of neuronal firing when the CS rule was applied to a small scale neural network.

1 Introduction One of fundamental issues in neuroscience is to understand how a neuron integrates excitatory and inhibitory synaptic inputs of a variety of spatial and temporal characteristics at the dendrite [1]. Recent neurophysiology studies have revealed a linear [2] or super-linear [3, 4] summation of excitatory postsynaptic potentials (EPSPs) at the dendrite under different experimental conditions. Furthermore, though the neuron-like unit usually used in artificial neural networks is the classical “point” neuron [5， 6， 7], it is reported that the single pyramidal neuron may function as a two-layer neural network in response to excitatory synaptic inputs [8， 9， 10， 11， 12]. In the present study, we examined the computational rules for summation of EPSPs and inhibitory postsynaptic potentials (IPSPs) in a realistic neuron model. In this model, the neuronal morphology is derived from a reconstructed pyramidal neuron of rat hippocampus CA1 region, and the distribution and density of ion channels are largely based on those physiological results reported in the literature. We also examined the functional J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 7 – 14, 2006. © Springer-Verlag Berlin Heidelberg 2006

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significance of the computational rule obtained in this study on the firing properties of a population of connected neurons.

2 Methods The morphology (see Fig. 1A) of the model neuron was obtained from DukeSouthampton archive of neuronal morphology [13]. The model was run within the NEURON simulation environment [14]. The ion channel properties in this model cell include: (1) Hodgkin-Huxley-type sodium and potassium channels, distributed uniformly along the soma and dendrites; (2) A-type potassium channels KA and hyperpolarization-activated channels Ih, with a progressively incremental channel density from the soma to distal dendrites [15]; (3) Four types of synaptic conductance AMPA, NMDA, GABA-A and GABA-B conductances at the sites of the synaptic input. (A)

(B)

Fig. 1. (A) The reconstructed morphology of the rat hippocampus CA1 pyramidal neuron. (B) Typical traces of the EPSP (dotted trace), IPSP (dashed trace) and the integrated response (solid trace, the membrane potential change when both excitatory and inhibitory synapses are activated simultaneously).

The parameters in our simulation are adjusted based on the previous studies [10， 16]. In our simulation paradigm, we placed two synaptic inputs along the main trunk of the apical dendrite, in a configuration of inhibitory inputs more proximal to excitatory inputs. We recorded the EPSP, IPSP, and integrated responses at the soma (see Fig. 1B). We defined the peak amplitude of EPSP as x, the amplitudes of IPSP and integrated potential as y and z at the time point when EPSP reached the peak.

How Does a Neuron Perform Subtraction?

9

The neural network consisted of fifteen artificial neurons fully connected with one another. Each neuron possessed an intrinsic inter-spike interval, which varied among different neurons [17].

3 Results 3.1 Cross-Shunting Rule (CS Rule) First, we systematically studied the model cell with following two synaptic sites: the inhibitory and excitatory synapses were located at 46.68 and 118.05 from the soma, respectively. We found that the simulation results can be best fitted by the function (1), as illustrated by Fig. 2.

μm

μm

Fig. 2. 3-D map of EPSP, IPSP and SUM from the formula (1). A prominent feature in this map is that a straight line is obtained when a section parallel to x-z or y-z plane intersects the function S. The magnitude of synaptic potentials is also coded by the graded grey intensity.

S ( x, y ) = 1.00595 x − 1.0165 y − 0.07891xy + 0.00698

(1)

In the second simulation, we used a different set of locations - 46.68 and for the inhibitory and excitatory inputs, respectively. The best fitting func268.27 tion showed a similar form:

μm

S ( x, y ) = 1.00403x − 0.99415 y − 0.08212 xy + 0.01295

(2)

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The above two sample results strongly suggested that the relationship among the three quantities (S, x and y) takes the following form, with the constant term abandoned (due to the constraint that S should be zero when both x and y equal zero): S ( x, y ) = x − y − kxy

(3)

We refer this nonlinear subtraction equation as the Cross-Shunting rule (CS rule) and k as the shunting factor. (A)

(B)

Fig. 3. Comparison between the results predicted by the CS rule and that obtained by simulation. (A) The comparison of the summation value. (B) The relative error between the CS rule and simulation.

We further tested the CS rule by randomly taking 11 sets of the amplitude of EPSP and excitatory input at and IPSP, with the inhibitory input placed at 46.68 268.27 from the soma, and compared the results of summation from the simulation with that calculated from the CS rule. As shown in Fig. 3, the two have an excellent agreement.

μm

μm

3.2 Dependence of Shunting Factor k on Synapse Location The CS rule indicates that the shunting effect (kxy) is dependent on both excitatory and inhibitory inputs. We thus examined the spatial dependence of k on the location of excitatory synapse, when that of the inhibitory synapse was kept constant, and vice versa. The result of simulation is shown in the Fig. 4. Our results indicated that k value is independent on the spatial location of excitatory synapse, but increases with the distance of inhibitory synapse from the soma. This suggests that the inhibitory effect of GABAergic synapses on more distal afferent excitatory input is independent of the relative spatial location of the excitatory synapse along the dendrite. More distal GABAergic synapses exhibit a relatively larger shunting effect.

How Does a Neuron Perform Subtraction?

(A)

11

(B)

(C)

Fig. 4. Dependence of the shunting factor k on the synapse location. (A) & (B) The inhibitory synapse was located at 46.68 and 158.62 from the soma and the location of the excitatory synapse was varied. The shunting factor k was found to remain constant. (C) The excitatory synapse was at 268.27 , while the location of inhibitory synapse was varied. The value of k increased with the distance between the inhibitory synapse and the soma. The relation between the two quantities can be fitted by a first-order exponential function (black line): k = 0.069 + 0.005exp(x/83).

μm μm

μm

3.3 CS Rule in Neural Networks How does the CS rule work in neural networks? To address this question, we further investigated the effect of this nonlinear subtraction rule in the framework of neural network described by Carnevale and Hines (2005) [17]. In this neural network, each neuron fires action potentials spontaneously with a particular inter-spike interval and receives excitatory or inhibitory inputs from all other neurons. We additionally added the shunting term kxy when the cell in this network received inhibitory inputs. The network consisted of 15 neurons, whose activities were described by a membrane state variable m, which satisfied the following equation:

τ

dm + m = m∞ dt

m∞ >1

(4)

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(A)

(B)

(C)

Fig. 5. Synchronous activity of 15 artificial cells in the neural network. (A) No inhibition was introduced and each cell fired action potentials with its intrinsic inter-spike-interval. (B) Only inhibition in a manner of S = x - y was introduced. (C) Strong synchrony in the presence of both the inhibition and the shunting terms (S = x - y - kxy).

How Does a Neuron Perform Subtraction?

13

The values for the parameters used in this simulation were: synaptic weight: -0.12; synaptic delay: 2 ms; cell time constant: 10 ms; shunting factor k: -0.08; inter-spiking interval: 10-11 ms. In this neural network, the neurons showed a progressive asynchrony of neuronal activity over time in the absence of the inhibition, due to the accumulative difference of inter-spiking intervals (Fig. 5A). When the inhibition y was introduced to the network without the shunting term kxy, the neurons only exhibited a partial synchrony (Fig. 5B). However, if both the inhibition and the shunting effect were added, the synchronization of the neuronal activity was immensely enhanced (Fig. 5C). An intuitive explanation of this finding is as follows: For the neurons that begin with shorter inter-spike-intervals, they have a higher possibility in staying a higher membrane state (x) close to the firing threshold, and the shunting effect is bigger, given the same inhibitory input (y). Hence, their firing activity tends to slow down and becomes synchronous to those neurons that begin with longer inter-spike intervals. This result suggests that the shunting term contributes to the synchronizing activity of neurons in the neural network.

4 Conclusion Taken together, our results have revealed a simple computational rule - the CrossShunting rule that underlies the integration of EPSP and IPSP in a realistic neuron model. The form of the nonlinear shunting term (kxy) suggests that the shunting effect results from the interaction between excitation and inhibition and this provides a means for a neuron to implement multiplication, a very important nonlinear operation. The shunting factor (k) increases with the increasing distance of the inhibitory synapse from the soma but is independent of the location of excitatory synapses. Our preliminary data of physiological experiments, in which the dendritic integration of IPSP and EPSP has been studied in a single CA1 pyramidal neuron in the rat hippocampus slice, support the above conclusion (Hao et al., unpublished data). Moreover, our results further suggest that the shunting term may contribute to the synchronization of the neuronal activity in a neural network.

Acknowledgments We thank the members in our lab: Ning-long Xu, Chang-quan Ye, Dong Wang, Jiang-teng Lv, Hai-tian Zhang for useful comments and discussions. This project was supported by a grant from Major State Basic Research Program of China.

References 1. Yuste, R., Tank, D.W.: Dendritic Integration in Mammalian Neurons, A Century After Cajal. Neuron 16 (1996) 701–716 2. Cash, S., Yuste, R.: Linear Summation of Excitatory Inputs by CA1 Pyramidal Neurons. Neuron 22 (1999) 383–394 3. Margulis, M., Tang, C.M.: Temporal Integration Can Readily Switch Between Sublinear and Supralinear Summation. J. Neurophysiol. 79 (1998) 2809–2813

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4. Nettleton, J.S., Spain, W.J.: Linear to Supralinear Summation of AMPA-mediated EPSPs in Neocortical Pyramidal Neurons. J. Neurophysiol. 83 (2000) 3310–3322 5. McCulloch, W.S., Pitts, W.: A Logical Calculus of the Ideas Immanent in Nervous Activity. Bull Math Biophys. 5 (1943) 115–133 6. Rosenblatt, F.: Principles of Neurodynamics. New York, Spartan (1962) 7. Rumelhart, D., Hinton, G., and McClelland, J.: A General Framework for Parallel Distributed Processing. in: D. Rumelhart and J. McClelland, J. (Eds): Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol.1. Cambridge, MA, Bradford (1986) 45–76 8. Mel, B.W., Ruderman, D.L., Archie, K.A.: Translationinvariant Orientation Tuning in Visual ‘Complex’ Cells Could Derive from Intradendritic Computations. J. Neurosci. 17 (1998) 4325–4334 9. Archie, K.A., Mel, B.W.: An Intradendritic Model for Computation of Binocular Disparity. Nat. Neurosci. 3 (2000) 54–63 10. Poirazi, P., Brannon, T.M., Mel, B.W.: Arithmetic of Subthreshold Synaptic Summation in A Model CA1 Pyramidal Cell. Neuron 37 (2003) 977–987 11. Poirazi, P., Brannon, T.M., Mel, B.W.: Pyramidal Neuron as Two-Layer Neural Network. Neuron 37 (2003) 989–999 12. Polsky, A., Mel, B.W., Schiller: Computational Subunits in Thin Dendrites of Pyramidal Cells. J. Nat. Neurosci. 7 (2004) 621−627 13. Cannon, R., Turner, D., Pyapali, G.K., Wheal, H.: An On-line Archive of Reconstructed Hippocampal Neurons. J. Neurosci. Methods 84 (1998) 49–54 14. Hines, M.L., Carnevale, N.T.: The NEURON Simulation Environment. Neural Comput. 9 (1997) 1179–1209 15. Migliore, M., Shepherd, G. M.: Emerging Rules for the Distributions of Active Dendritic Conductances. Nature Rev. Neurosci. 3 (2002) 362–370 16. Migliore, M.: On the Integration of Subthreshold Inputs from Perforant Path and Schaffer Collaterals in Hippocampal CA1 Pyramidal Neurons. J. Comput. Neurosci. 14 (2003) 185–192 17. Carnevale, N.T., Hines, M.L.: The NEURON Book. Cambridge Univ. Press (2005)

Stochastic Resonance Enhancing Detectability of Weak Signal by Neuronal Networks Model for Receiver Jun Liu, Jian Wu, Zhengguo Lou, and Guang Li Department of Biomedical Engineering, Key Laboratory of Biomedical Engineering of Ministry of Education, Zhejiang University, 310027 Hangzhou, P.R. China [email protected]

Abstract. Stochastic resonance phenomenon in a biological sensory system has been studied through signal detection theories and psychophysical experiments. There is a conflict between the real experiments and the traditional signal detection theory for stochastic resonance because the latter treats the receiver as linear model. This paper presents a two-layer summing network of Hodgkin-Huxley (HH) neurons and a summing network of threshold devices to model the receiver, respectively. The simulation results indicate that the relevant index of signal detectability exhibit the stochastic resonance characteristics.

1 Introduction Stochastic resonance (SR) is a phenomenon that a nonzero noise level optimizes system performance in a non-linear field and especially contributes to weak signal detection and transfer in an intensive noisy background [1,2]. Some psychophysical experiments have indicated that the capacity of sensory systems to detect a weak stimulus can be enhanced by SR [3-9]. Accordingly, the signal detection theory has been used to analyze and model the psychophysical experiment by the detector model that separates detection process into reception and classification [10-13]. The detector model based on signal detection theory is generally consisted of two parts, defined as a receiver and a classifier respectively. The former transforms observation signal into decision information and is evaluated by the index of detectability (d’). The latter discriminates the decision information according to a binary classification criterion and is evaluated by the index of percent-correct classification. This model is consistent with two basic functions of sensory systems, namely reception and classification so that it may be an appropriate model to simulate sensory systems including peripheral receptors and cortical cells. Some research results suggest that SR is closely linked to the non-linear classifier with the binary classification criterion [10,11], but the receiver of a detector (sensory system) is considered as linear system according to the initial description of signal detection theory and this part does not exhibit any SR phenomenon. It is worth noting that detectability (d’) has a close relationship with the signal-noise-ratio (SNR) and the two-interval forced choice (2IFC) rule of psychophysical experiment through which SR phenomenon can be observed [7,15]. For explaining this conflict, some new models and general indices of detectbability are used to describe the non-linear J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 15 – 20, 2006. © Springer-Verlag Berlin Heidelberg 2006

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characteristic of a receiver [13]. The simulation results indicate that SR can occur in either parts of the detector as long as they are non-linear. In this paper, we will further discuss the above issue using neuronal networks a two-layer summing network of Hodgkin-Huxley (HH) neurons [16] and a summing network of threshold devices to model the receiver. Based on two kinds of network model and signal detection theory, the detectability will be used to investigate SR phenomena in the receiver.

2 The Signal Detection Theory and Detectability In terms of signal detection theory [18], the corresponding decision model is written as two hypotheses: H1 and H2, representing the input conditions of noise-alone and signal-plus-noise, respectively. After the receiver transforms inputs into spikes and preprocesses the spikes into the information used for classification, the classifier discriminates whether the inputs belong to H1 or H2. A straightforward and useful measure tool is the index of detectability that implies the characteristics of receiver. The detecbability of a signal depends both on the separation and the spread of the noise-alone and signal-plus-noise. The most widely used measure is called d-prime (d’), and its formula is simply:

d ' = separation spread

(1)

Where the separation corresponds to the difference between the response from noise-alone and signal-plus-noise, and the spread corresponds to the deviation induced by noise.

3 Detectability Based on the Two-Layer HH Neuronal Network The left figure in Fig.1 shows the structure of the receiver based on two-layer HH neuronal network model. The details of a HH neuronal model are seen in the paper [16]. The first layer network is composed of N parallel neurons represented by n11， n12， …n1N . The second layer has one neuron represented by n2 and act as the output part of network. The total network is an analogical structure of the spikes generation system, in which the first layer can be considered as the part of the reception and transmission of external stimulus and converges on the neuron of the second layer. Especially, the input signal S(t) of the first layer is considered as the periodic stimulation I1 sin(2 fst) and constant stimulation I0 in order to study the spectrum characteristic of the systems. The I1 sin(2 fst) denotes the external stimulation information and the I0 is regarded as the average effect of the internal environment in sensory systems. Each neuron in the first layer is subjected to the external input noises represented by η1， η2,……, ηN , which are simply assumed as an independent or uncorrelated Gaussian white noise, satisfying <η(t)> = 0， <η(ti) η(tj)> = 2Dδ(ti-tj), D is intensity of noises. The neuron of the second layer receives all outputs of the neurons in the first layer and the same internal environment stimulation I0. The neurons in the first layer are parallel connected with the second layer through a synapse. The synaptic current of the neuron n1i is described in reference [19].

π

π

Stochastic Resonance Enhancing Detectability of Weak Signal

17

Fig. 1. (Left) The structure of the receiver based on the two-layer network. The inputs (the periodic signal s(t) plus noise or only noises ) are transformed into spikes. The spikes are converted into power spectrum by spectral analyzer. (Right) The d’ varying with the intensity of noise D for N 1, 50 and 100, respectively. The rest parameters: I1 =1μA/cm2, I0 =1μA/cm2, fs=50Hz.

＝

The corresponding spikes sequences occur when the inputs are applied to the receiver. It is widely acknowledged that each of the auditory, visual, and tactile sensory systems can be modeled as spectral frequency analyzers [17], thus a spectrum analysis is used to preprocess the information for classification. We simplify firstly the spikes sequences into the standard rectangle pulse sequences (the amplitude of each pulse is 1mV and the width is 2ms), and then obtain the power spectrum density (PSD) by summing the results from the Fast Fourier Transform (FFT) of the pulse sequences. The PSD in a small bandwidth f and centered on the signal frequency fs is used to define the detectability of the detection process. Referring to the paper [17], the corresponding expression can be written as:

Δ

d ' = [ PsΔ+fn ( f s ) − PnΔ f ( f s )] PnΔ f ( f s )

(2)

In this equation, PsΔ+fn ( f s ) − PnΔ f ( f s ) represents the power separation between the conditions of noise-alone and signal-plus-noise and PnΔf ( f s ) is the power from the noise. The result represents the total power contained in the signal feature of the spectrum compared to the total power in the noise in the same bandwidth centered on the signal frequency. Obviously, the d’ have a monotonic relationship with the SNR which usually represent the ratio of the signal peak and the mean amplitude of background noise at the input signal frequency f s in the power spectrum. The right figure of fig. 1 shows the d’ versus the intensity of noises D when N=1,N=50 and N=100. The amplitude of input signal is taken as the subthreshold value ( I1 =1μA/cm2). Three curves exhibit the typical characteristic of SR: first a rise and then a drop. Differently, the optimal intensity of noise in the case of N=50 and N=100 varies from 1 to10 and has wider range than that in the case of N=1. This means that the range and the optimal d’ increase with the number of neurons in first layer. The reason may be the collective behavior of many neurons can restrain the noises and decrease the power of noise so as to improve the d’.

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4 Detectability Based on the Summing Network of Threshold Devices Because the HH model is only numerically analyzed, the network of threshold devices is introduced to mimic the threshold characteristic of reception. In view of its simplicity in form, the corresponding probability model can be used to obtain the analytic results.

Fig. 2. (Left) The structure of the receiver based on the summing network of threshold devices. The inputs (the signal s(t) plus noise or only noises ) are transformed into spikes. The spikes are summed and analyzed statistically. (Right) The d’ varying with the intensity of noise D for N 1, 50 and 100, respectively. The subthreshold signal s=5 and the threshold value θ=10.

＝

As shown in the left figure of fig.2, a receiver model can also consist the network of threshold devices. The basic elements of the receiver are MacCulloch-Pitts neurons that are considered as simple threshold devices described by Heaviside functions. Each output, yi (t), is equal to 1, if si(t)+ηi(t)>θi or si(t)+ηi(t)=0. Each threshold device is subject to the same input signal, s(t). And all noise variables, η1(t), … , ηi(t), … ,ηN(t), are assumed to be independent and identically distributed (IID) with mean zero and distribution function F. The θi is the threshold where i=1, 2, …, N. Let thresholds of all devices equal to θ, independent Bernoulli random variables with probabilities are observed at time, t, instantly:

p(t) = pyi (t) = P[yi (t) =1] = Pyi (t ) (θ, ∞) =1− F[θ − s(t)]

(3)

The receiver is considered as a summing network of N threshold devices. The output, y(t), represents the number of the devices which input cross the threshold at any time. According to Eq.(3), when the value of N is large enough, the probability density of the outputs, y(t), approximate the Gaussian distribution with the mean μ(t)=Np(t) and the deviation σ2(t)=Np(t)[1-p(t)] at time t. The corresponding decision model is written as two hypotheses: N N H : y (t ) = g[η (t )] and H : y (t ) = g[ s (t ) + η (t )] . The function, g(.), is Heaviside 0

∑ i =1

i

1

∑ i =1

i

i

function. H0 represents the condition with only noise input and H1 is the condition with signal plus noise input. For a detector, y(t) corresponds to the decision variable which is the output of receiver. The corresponding detectability [20] at time t is defined as:

Stochastic Resonance Enhancing Detectability of Weak Signal

d ' = 2[ μ1 (t ) − μ0 (t )]

σ 02 (t ) + σ 12 (t )

19

(4)

Here this d’ is a complete characterization of the detectability of the signal assuming that the noise follows a normal (Gaussian) distribution with a fixed variance, independent of the signal strength, satisfying <η(t)> = 0， <η(ti) η(tj)> = 2Dδ(ti-tj), D is intensity of noises. For a stationary signal input, the output of receiver, y(t), is stationary random variable, so the population distribution can be replaced by the distribution at time, t. The mean and the deviation become constant values: μi=N [1-F (θ-si)] and σi2=N F (θ-si) [1-F (θ-si)], where, i=0 or 1, s0=0, s1=s (amplitude of the input signal). Numeric simulation results are shown in the right figure of fig.2. It can be found that the detectability has the characteristics of SR when the signal amplitude is lower than the threshold. Similarly, we investigate that how the number of threshold devices affects the d’. The results indicate that the d’ increase with the number of devices, but the range of optimal d’ does not exhibit wider and wider with the number increase like the two-layer HH network. The possible reason is that the neurons in the first layer are parallel connected with the second layer through a synapse with the nonlinear summing function.

5 Conclusion Many human psychophysical and animal behavioral experiments have shown SR in a d’-like measure of performance but the traditional signal detection theory cannot explain these phenomena. The reason is that the receiver of a detector (sensory system) is considered as linear. In this paper we present two models to replace the linear receiver part in a detector from different angles. It is noting that when the stimulus is transformed and processed from the peripheral neural system to the cortical of brain, there are the gathers of many neurons or neuron groups. These gathering processes may be considered as the different levels of classification. The models in this paper are constructed according as this idea. The HH network is biologically realistic, but it is only numerically analyzed. The network of threshold devices is simple in construction and the corresponding probability model can be obtained. According to two network model, two kinds of definition of d’ for different models both exhibit the SR. This may be a complemental explanation to the corresponding experiment.

Acknowledgment This work was supported by the National Natural Science Foundation of China (projects No. 30470470) and the Project of Zhejiang Province Education Department of China (20050907).

References 1. Benzi, R., Parisi, G., Sutera, A., Vulpiani, A.: Stochastic Resonance in Climatic Change. Tellus 34(1982) 10-16 2. Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.: Stochastic Resonance. Rev. Mod. Phys. 70(1998) 223-287

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3. Gingl, Z., Kiss, L. B., Moss, F.: Non-dynamical Stochastic Resonance—Theory and Experiment with White and Arbitrarily Colored Noise. Europhys Lett.. 29 (1995) 191-196 4. Pei, X., Bachmann, K., Moss, F.: The Detection Threshold, Noise and Stochastic Resonance in the Fitzhugh-Nagumo Neuron Model. Phys. Lett. (1995) 61–65 5. Collins, J. J., Imhoff, T. T., Grigg, P.: Noise-enhanced Tactile Sensation. Nature 383(1996) 770-772 6. Simonotto, E., Riani, M., Seife, C.: Visual Perception of Stochastic Resonance. Phys. Rev. Lett. 78(1997) 1186-1189 7. Zeng, F.G., Fu, Q.J., Morse. R.: Human Hearing Enhanced by Noise. Brain Researcher 869 (2000) 251–255 8. Russell, D. F., Tucker, A., Wettring, B.: Noise Effects on the Electrosense- mediated Feeding Behavior of Small Paddlefish. Fluct. Noise Lett. 1 (2001) L71–L86 9. Freund, J., Schimansky-Geier, L., Beisner, B.: Behavioral Stochastic Resonance: How the Noise from a Daphnia Swarm Enhances Individual Prey Capture by Juvenile Paddlefish. J. Theor. Biol. 214 (2002) 71–83 10. Tougaard, J.: Stochastic Resonance and Signal Detection in an Energy Detector–Implications for Biological Receptor Systems. Biol. Cybern. 83 (2000) 471–480 11. Gong, Y. F., Marrhews, N., Qian, N.: Model for Stochastic-resonance-type Behavior in Sensory Perception. Phy. Rew. 65 (2000) 0319041-5 12. Tougaard, J.: Signal Detection Theory, Detectability and Stochastic Resonance Effects. Biol. Cybern. 87 (2002) 79-90 13. Ward, L. M., Neiman, A., Moss, F.: Stochastic Resonance in Psychophysics and in Animal Behavior. Biol.Cybern. 87 (2002) 91-101 14. Macmillan, N. A., Creelman, C. D.: Detection Theory: a User’s Guide. Cambridge University Press, Cambridge (1991) 15. Manjarrez, E.,Rojas-Piloni, G., Mẻndez, I.: Stochastic Resonance with the Somatosensory System: Effects of Noise on Evoked Field Potentials Elicited by Tactile Stimuli. Journal of Neuroscience 23 (2003) 1997-2001 16. Hodgkin, A., Huxley, A.: A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve. J. Physiol. 117 (1952) 500-544 17. Coren, S., Ward, L. M., Enns, J.: Sensation and perception, 5th edn. Harcourt, San Diego (1999) 18. Green, D. M., Swets, J. A.: Signal Detection Theory and Psychophysics. Krieger, Huntington, N, Y. (1966) 19. Koch, C., Segev, I.: Methods in Neuronal Modeling: From Ions to Networks (second edition). Cambridge (Massachusetts, USA): MIT Press (1998) 98~100 20. Simpson A. J., Fitter, J.: What is the Best Index of Detectability? Psychol. Bull. 80 (1973) 481–488

A Gaussian Dynamic Convolution Models of the FMRI BOLD Response Huafu Chen, Ling Zeng, Dezhong Yao, and Qing Gao School of Applied Mathematics, School of Life Science & Technology, University of Electronic Science and Technology of China, Chengdu 610054, China [email protected]

Abstract. Blood oxygenation level dependent (BOLD) contrast based functional magnetic resonance imaging (fMRI) has been widely utilized to detect brain neural activities and great efforts are now stressed on the hemodynamic processes of different brain regions activated by a stimulus. The focus of this paper is Gaussian dynamic convolution models of the fMRI BOLD response. The convolutions are between the perfusion function of the neural response to a stimulus and a Gaussian function. The parameters of the models are estimated by a nonlinear least-squares optimal algorithm for the fMRI data of eight subjects collected in a visual stimulus experiment. The results show that the Gaussian model is better in fitting the data.

1 Introduction A primary goal of functional MRI (fMRI) is to accurately characterize the underlying neural activities by the changes of the measured signals. The most common technique in fMRI is the gradient-echo imaging with blood oxygenation level dependent (BOLD) contrast to measure the hemodynamic changes accompanying neural activation [1-5]. In block design experiments the subject is instructed to perform a task that activates a certain region of the brain during a specific time interval. The resultant hemodynamic response (HDR) of the cortex under investigation changes with the stimulus. Early fMRI experiments [6-7] suggested a linear relationship between the parameters in the stimulus and the resultant HDR, but later experiments [8-12] showed an increasing amount of nonlinearity in the response. To analyze the nonlinear relationship, Miller et al proposed a fMRI dynamic model of convolution between the perfusion function of the neural response to a stimulus with a Gamma response function of the Cerebral Blood Flow (CBF) hemodynamics of neuron [8,4]. Recently, we proposed an extended Gamma convolution model where a baseline is introduced into the original Gamma convolution model to represent the background activities [13]. In general, it is still a frontier problem to construct a better BOLD dynamic model to reveal the neural mechanism of cerebral activities and to investigate the dynamic pattern of various cerebral regions in brain function exploration and cognitive neuroscience [14-16]. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 21 – 26, 2006. © Springer-Verlag Berlin Heidelberg 2006

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In this paper, the effectiveness of the new Gaussian model is evaluated. As a preliminary application, the new model is applied to an fMRI data of a visual stimulus where we concern the phenomena of hemispheric lateralization. Understanding the specific nature of how spatial abilities are lateralized in the brain is important for many reasons and is still a topic without common viewpoint [17-18]. In this paper, the parameters of the two fMRI models were estimated by a nonlinear least-square optimal algorithm, and according to the parameter changes in the left and right cerebral visual region, the difference of the hemodynamic processes are discussed.

2 Gaussian Dynamic Convolution Models 2.1 The Perfusion Response Many experiments indicate that there are neural adaptation between neural responses and stimulus, and it is the general tendency of the neural firing rate to decay from a high initial value to a lower steady-state value during a sustaining stimulus [6,8,16]. If the parameterization of the neural response to a block stimulus begins at t=0 and ends at t=T, then we have: ⎧⎪1 + ae −t / tn n(t ) = ⎨ ⎪⎩0

0≤t ≤T otherwise

(1)

where parameter t n is a decay time constant, a is the amount that the initial response overshoots the steady-state response and n(t ) is assumed being the perfusion function of neural response[8]. 2.2 Gaussian Response of a Neural Cluster It has been shown that the regional response of collectively activated neural cluster to a stimulus is a normal distribution, and it is consistent to the law of large numbers in probability [18], so the response of a neural cluster may be expressed by a Gaussian function: g (t ) =

θ0 (t − θ 2 ) 2 exp(− ) θ1 2θ 1 2

(2)

where θ 0 and θ1 are two constants, θ 2 is the time interval from the presentation of a stimulus to the maximum neural response, namely it is a response delay parameter. 2.3 Gaussian Convolution Model The BOLD signal of the cerebral activation is a collective response of an activated region and it can be explained as a mutual interaction process between the perfusion of the neural response to a stimulus and the Hemodynamic change due to the activation of a neural cluster. Therefore, the BOLD signal x(t ) of a cerebral activation can be expressed by convolution of this perfusion response of the neural response with a Gaussian function of the CBF hemodynamic response of a neural cluster[17]:

A Gaussian Dynamic Convolution Models of the FMRI BOLD Response

23

Model I:

x(t ) = n(t ) ⊗ g (t ) + (1 + α × t ) × η + e(t )

(3)

where (1 + α × t ) × η is introduced to represent the background fMRI signal. The η is a constant, and the time dependence is assumed being a linear function α × t × η with a baseline shift factor α , e(t ) represents various additive noises. Fig. 1 is the simulation result of model I without e(t ) . 2.4 Nonlinear Least-Squares Estimation

Gaussian convolution model I is composed of two nonlinear functions with parameters a, t n ,θ 0 ,θ1 ,θ 2 , η , e(t ) and α .These parameters are estimated with a nonlinear least-squares algorithm as follow. 1) Equations (3) and (6) are reformulated as x(t ) = f (t , β ) + e(t )

(4) r

where x(t ) is the observed signal value at time t, β is an unknown parameter vector including all the above noted unknown parameters except the e(t ) . f is the nonlinear r function with respect to parameter t and β . r 2) Using a nonlinear least-squares estimator, the parameter β is estimated as following r

[

][

]

r T r ⎤ ⎡ 1 x(t ) − f (t , β ) x (t ) − f (t , β ) ⎥ ⎦ ⎣ n

β * = arg max ⎢−

(5)

In our model, the detail cost function to be minimized is n

∑ [ x(t ) − (1 + ae

−t / t n

) ⊗ ϕ (t ) − (1 + α × t ) × η ] 2

(6)

t =1

where x (t ) is the measured BOLD signal at time t, t = 1,2,L , n . ϕ (t ) is the Gaussian function.

3 Application in fMRI Data 3.1 In Vivo fMRI Experiment

For testing the Gaussian model in fMRI experiments in vivo, normal subjects were involved in a visual task with a Block paradigm. Experiment conducted at Beijing MRI center， Chinese Academy of Sciences, with SIEMENS 3 TESLA MAGNETOM TRIO. EPI: repetition time TR=1.0s, matrix size=64×64, voxel o

size=3.00×3.00×2mm.TE=30ms, flip angle 90 . The first 4 scans of each run were discarded to allow for magnetic saturation effects. Then 320 volume images comprised time-series were analyzed.

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3.2 fMRI Image Result

According to F-test statistical method (P value <0.05) and the prior response information, the data is analyzed with SPM principle [www.fil.ion.ucl.ac.uk/spm;]. Shown in Fig.1 is the result of the fMRI activations of the subjects in Talairach brain corresponding to left and right VFs. The black points in Fig. 1 represent the activated voxels in a brain slice, where regions of interest (ROIs) comprise those in the left and right occipital lobes of the primary visual cortex, respectively.

Fig. 1. fMRI result

3.3 Modelling Results of fMRI Signals

First, we choose two measured activated fMRI signals, marked them as A and B (Fig.1), from the selected ROIs in the left and the right primary visual cortexes, and minimize the cost function in Eq.(6) for the Gaussian model with optimal algorithm offered by MATLAB to get the model parameter estimations.

(a) voxel A Fig. 2. BOLD signal in Gaussian model

(b) voxel B

A Gaussian Dynamic Convolution Models of the FMRI BOLD Response

25

Fig.2 are the signal curves of the measured BOLD signals and the estimated BOLD signals of the model. The horizontal axis is the sample point, and the vertical axis is the signal value. Fig. 1(a) is the result of the signal at voxel A in the left occipital lobe, where the solid lines note the measured BOLD signal, the dotted solid lines note the signals estimated by the Gaussian model. These results show the new Gaussian model not only of a better curve trend adapted to the measured BOLD signals. The model parameters are different in the left and right occipital regions, which indicate that the dynamic processes seem different in various cerebral functional regions.

4 Conclusion In this paper, Gaussian convolution model is presented. The parameters was presented to estimate a vivo fMRI data evoked by a visual stimulation The results show that the Gaussian model, which is based on the law of large numbers in probability thus may reflects the statistical characteristics of neural cluster activation, can simulate the BOLD response. fMRI is a new functional imaging technology and the BOLD signal is the key to investigate the relation between the neuronal activation and fMRI signals [4,8,10]. It is still a challenging topic to combine characteristics of various experimental paradigms to develop and improve a BOLD model [14].

Acknowledgment This work is supported by the 973 Project No. 2003CB716100, NSFC#90208003, 30525030, 30570507, Fok Ying Tong Education Foundation (91041), New Century Excellent Talents in University.

References 1. Kwong, K., Belliveau, J., Chesler, D.: Dynamic Magnetic Resonance Imaging of Human Brain Activity During Primary Sensory Stimulation. Proc Natl Acad Sci USA 89 (1992) 5675–5679 2. Ogawa, S, Tank, D. W., Menon, R., Ellermann, J.M., Kim, S.G., Merkle, H., Ugurbil, K: Intrinsic Signal Changes Accompanying Sensory Stimulation: Functional Brain Mapping with Magnetic Resonance Imaging. Proc. Natl. Acad Sci. USA 89 (1992) 5951-5955 3. Bandettini, P.A., Jesmanowicz, A., Wong, E.C., Hyde, J.S.: Processing Strategies for Time-Course Data Sets in Functional MRI of the Human Brain. Magnetic Resonance in Medicine 30(2) (1993) 161-173 4. Rasmus, M.B., Ziad, S.S., Peter, A.N.: Spatial Heterogeneity of the Nonlinear Dynamics in the FMRI BOLD Response. NeuroImage 14(5) (2001) 817-826 5. Yao, D.: Highresoltion EEG Mappings: A Spherical Harmonic Spectra Theory and Simul ation Results. Clinical Neurophysiology 111(1) (2000) 81-92 6. Boynton, G.M.,. Engel, S.A, Glover, G.H., Heeger, D.J.: Linear Systems Analysis of Functional Magnetic Resonance Imaging in Human V1. J. of Neuroscience 16(13) (1996) 4207-4221

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7. Rao, S.M., Bandettini, P.A., Binder, J.R., Bobholz, J.A., Hammeke, T.A., Stein, v, Hyde, v.: Relationship Between Finger Movement Rate and Functional Magnetic Resonance Signal Change in Human Primary Motor Cortex. J. of Cerebral Blood Flow and Metabolis 16(6) (1996) 1250-1254 8. Miller. V., Luh, W.M., Liu, T.T., Martinez, A., Obata, T., Wong, E.C., Frank, L.R., Buxton, R.B.: Nonlinear Temporal Dynamics of the Cerebral Blood Flow Response. Human Brain Mapping 13(1) (2001) 1-12 9. Buxton, R.B., Liu, T.T.,Wong, E.C.: Nonlinearity of the Hemodynamic Response: Modeling the Neural and BOLD Contributions. Proc. 9th ISMRM (2001) 1164 10. Liu, H.L., Gao, J.H.: An investigation of the Impulse Functions for the Nonlinear BOLD Response in functional MRI. Magnetic Resonance Imaging 18(8) (2000) 931-938 11. Vazquez, A.L., Noll, D.C.: Nonlinear Aspects of the BOLD Response in Functional MRI. Neuroimage 7(2) (1998) 108-118 12. Aguirre, G. K., Zarahn, E., Esposito, M.D.: The Variability of Human BOLD Hemodynamic Responses. NeuroImage 8 (4) (1998) 360-369 13. Chen, H. and Yao, D.: An Extended Gamma Dynamic Model of fMRI BOLD Response. Neurocomputing 61 (2004) 395-400 14. Logothetis, N.K., Pauls, J., Augath, M., Trinath, T., Oeltermann, A.: Neurophysiological Investigation of the Basis of the FMRI Signal. Nature 412 (2001) 150-157 15. Rajapakse, J.C., Kruggel, F., Maisog, J.M. and VonCramon, D.Y.: Modelling Hemodynamic Response for Analysis of Functional MRI Time-series. Human Brain Mapping 6(4) (1998) 283-300 16. Rosen, B.R., Buckner, R.L., Dale, A.: Event-related Functional MRI: Past, Present, and Future. Proc. Natl. Acad. Sci. USA 95 (1998) 773-780 17. Chen, H., Yao, D., Liu, Z.: A Study on Asymmetry of Spatial Visual Field by Analysis of the FMRI BOLD Response, Brain topography 17(1) (2004) 39-46 18. Chen, H., Yao, D., Zhuo, Y, and Chen, L.: Analysis of FMRI Data by Blind Separation into Independent Temporal Component. Brain Topography 15(4) (2003) 223-232

Cooperative Motor Learning Model for Cerebellar Control of Balance and Locomotion* Mingxiao Ding, Naigong Yu, and Xiaogang Ruan** Beijing University of Technology, Beijing 100022, China [email protected] {yunaigong, adrxg}@bjut.edu.cn

Abstract. A computational model in the framework of reinforcement learning, called Cooperative Motor Learning (CML) model, is proposed to realize cerebellar control of balance and locomotion. In the CML model, cerebellum is a parallel pathway that the vermis and the flocculonodular lobe play a role of reflex actions executor, and that the intermediate zone of the cerebellum participates in initiating voluntary actions. During the training phase, the cerebral cortex provides the predictive error through climbing fiber to modulate the concurrently activated synapses between parallel fibers and purkinje cells. Meanwhile, the intermediate zone of the cerebellum computes the temporal difference (TD) error as a training signal for the cerebral cortex. In the simulation experiment for the balance of double inverted pendulum on a cart, a well-trained CML model can smoothly push the pendulum into the equilibrium position.

1 Introduction Cerebellum has long been seen as a learning machine for motor control and motor skill [1], [2], [3], [4]. In all of these classical theories, an instructive signal via the inferior olive-climbing fiber system is the critical basis of successful motor learning. Therefore, how to obtain the instructive signal for the neural network of cerebellum is the key to understand the physiological functions of cerebellum. To address this problem, the idea that anticipations influence and guide motor control has been increasingly appreciated over the last decades. Learning by prediction and reward has been used to understand the learning and memory function in the basal ganglia [5], hippocampus [6], and cerebral cortex [7]. Evidences show that the anticipatory mechanism in the framework of reinforcement learning is suitable for identifying the cerebellum for one aspect of basic associative learning and memory: classical conditioning [8]. Inspired by recent progress in neurophysiology, we propose a hypothesis that the cerebral cortex is responsible to predict future instructive signal for the cerebellar control, and the cerebellum and the cerebral cortex compose an overall actor-critic version of reinforcement learning agent. Our current work is focused on the cooperative learning scheme of the cerebellum and the cerebral cortex compose an overall actor-critic version of reinforcement learning agent. Our current * **

This work is supported by National Natural Science Foundation of China (60375017). Corresponding author.

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 27 – 33, 2006. © Springer-Verlag Berlin Heidelberg 2006

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M. Ding, N. Yu, and X. Ruan

work is focused on the cooperative learning scheme of the cerebellum and the cerebral cortex during the control and adaptation of dynamic motor balance, which is a fundamental issue for humanoid robot. In this paper, we attempt to model this Cooperative Motor Learning (CML) scheme in the framework of reinforcement learning. Because the recurrent neural activity exists across both cortical modules [9], [10] and subcortical areas [11], both the cerebellum and the cerebral cortex are realized using the recurrent network and real-time recurrent learning (RTRL) algorithm [12]. The effectiveness of the CML model is demonstrated by the balance task of double inverted pendulum on a cart.

2 The Cooperative Motor Learning Model The CML model is in the framework of temporal difference (TD) learning which aims to deal with reinforcement learning problem with delayed reward. Since the neural circuits of cerebellum and cerebral cortex are taken as an integral agent in this paper, everything outside the integral agent is labeled as the “environment”. The sensory inputs from the environment stimulate both cerebellum and cerebral cortex, and the action output to the environment is the response of the integral agent. The cerebellum is responsible for action decision, and the cerebral cortex is responsible to predict the future instructive signal for action network based on the current sensory inputs and current action output. The learning aim of the cerebellum is to minimize the square of the predictive error derived from the cerebral cortex:

Ecerebellum = (eˆ(t ))2

(1)

The reward from environment is computed according to:

⎧0 r(t ) = ⎨ ⎩ −1

if the action is successful if the action is

failed

(2)

Thus the expected temporal difference (TD) error for the cerebral cortex is computed [6] according to:

δ (t ) = r (t ) + γ eˆ(t ) − eˆ(t − 1)

(3)

The learning aim of the cerebral cortex is to minimize the goal: Ecerebrum = (δ (t ))2

(4)

In the CML model, the reward signal guarantees that the behavior of an animal is generally goal directed, thus an interesting question is where the reward signal come. 2.1 Cerebellum

Based on afferent and efferent connectivity, the cerebellum is functionally divided into the medial zone (vermis), the intermediate zone, the lateral zone and the flocculonodular lobe. It is believed that the vermis and the flocculonodular lobe may play a

CML Model for Cerebellar Control of Balance and Locomotion

29

strong role in the control of balance and locomotion by modulating the limb flexor and extensor muscle activity [13]. The intermediate zone receives abundant inputs not only from receptors in muscles, joints and skin, but also from the cerebral cortex via the corticopontocerebellar system. The intermediate zone projects out to the red nucleus and to the cerebral cortex via the thalamus. Therefore, the intermediate zone has been proposed to serve as a “comparator” which indicates the mismatch between the expected and actual outcome of the action [14], [15]. These facts enable us to speculate that the function of the cerebellum is both an action executor and a sensory comparator, as depicted in Fig. 1. In the CML model, the cerebellum can output an action response corresponding to sensory inputs, and the cerebral cortex provides the predictive error through climbing fiber to modulate the concurrently activated synapses between parallel fibers and purkinje cells. Meanwhile, the intermediate zone of the cerebellum computes the TD error as a training signal for the cerebral cortex. When the cerebral cortex has built up a forward predictive model for the outer environment, the cerebellum can be trained as an inverse model for the environment. If the inverse model is accurate, the cerebellum can reflex action responses to shift the body into the equilibrium position. This whole process is a “trial and error” learning frame, thus the cooperative motor learning scheme can be depicted in Fig. 2. Based on the sound anatomy knowledge, the formula description of the vermis is given by equation (5)-(7): I action ( i ) (t ) = X sensor ( i ) (t )

i = 1, 2

TD error

(5)

,N

predictive error

action feedback to cerebral cortex

intermediate zone comparator

sensory input

reward signal

-

Z-1

vermis action output

Z-1

Cerebellum

Fig. 1. Subdivided function zones of the cerebellum and its signal relation with the cerebral cortex. In the intermediate zone, it is assumed that there is build-in knowledge about the equilibrium position in the cerebellum, such as a small region around the vertical position. The movement with a variable expected position may be in relation to motion plan and is out of the research scope of this paper.

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M. Ding, N. Yu, and X. Ruan

sensory input

Cerebral Cortex

predictive error

TD error action output

Cerebellum

Fig. 2. The architecture of the Cooperative Motor Learning model. The instructive signals for the cerebral cortex and the cerebellum are set in dashed line.

Oaction (t ) = F ( s(t ))

(6)

s(t ) = ∑ (Winput ( i ) i I action ( i ) (t )) + Wrecurrent iOaction (t − 1)

(7)

i

where X sensor (t ) is the sensory input from the eyes, the muscles and joints of the body, and the vestibular system. This afferent information is delivered to the granule cells via mossy fiber and parallel fibers carry the outputs of granule cells to the output neurons of the cerebellar cortex, the Purkinje cells. The motor impulses, denoted by Oaction (t ) , are sent to the muscles of the body to maintain the balance during standing, walking and running. During the learning phase, the climbing fibers derived from the inferior olive carry an instructive signal to adjust the synaptic strength between the Purkinje cells and the parallel fibers. 2.2 Cerebral Cortex

Admittedly, the real neural circuits in the cerebral cortex are perfectly complex. On the other hand, the excitation of the neurons in the cerebral cortex is problem depended, thus a four-layer recurrent network is employed here to simulate the anticipative function in humanoid motor balance task. This anticipative network is active when the action output of the cerebellum is unfavorable to the motor balance of the body. When the cerebellum can balance the body precisely, the activity in the anticipative network should be inhibited. The formula description of the anticipative network in the cerebral cortex is given by equation (8)-(15): I critc ( i ) (t ) = X sensor ( i ) (t )

i = 1, 2

, M −1

I critc ( M ) (t ) = Oaction (t )

H 1j (t ) = F j ( rj1 (t ))

j = 1, 2

(8) (9)

, P1

(10)

CML Model for Cerebellar Control of Balance and Locomotion 1 1 rj1 (t ) = ∑ (Vinput ( ji ) i I critic ( i ) (t )) + ∑ (Vrecurrent ( jk ) i H k ( t − 1))

i

k = 1, 2

, P1

k

H s2 (t ) = Fs ( rs2 (t ))

s = 1, 2

, P2

2 2 rs2 (t ) = ∑ (Vhidden ( sj ) i H 1j (t )) + ∑ (Vrecurrent ( su ) i H u ( t − 1))

j

u

31

(11) (12)

u = 1, 2

, P2

(13)

Ocritic (t ) = ∑ (Voutput ( s ) i H s2 (t ))

(14)

eˆ(t ) = Ocritic (t )

(15)

s

where X sensor (t ) is a copy version of the sensory inputs into the cerebellum, and eˆ(t ) is the predictive instructive signal for the cerebellum. The distributed recurrent loop in every hidden layer is essential to capture the nonlinear dynamic by form a short term memory. During the learning phase, the weights in the cerebral cortex are updated using the real-time recurrent learning (RTRL) algorithm [12].

3 Simulation Experiments and Results The double inverted pendulum on a cart (DIPC) is adopted as the control object because the nonlinear dynamics of human legs can be well simulated by a double inverted pendulum model. Here the dynamic equation of the DIPC system is obtained from Lagrange’s equation of motion as used in [16]. The training phase consisted of a number of trials, setting the DIPC system to a given state chosen at random between ±5° both for the bottom pendulum and the top pendulum. Each trial run until a failure signal occurs or the plant reaches 12,000 steps successfully. The sample-time is chosen as 0.02 seconds. The learning performance is showed by Fig. 3 where the number of steps per trial is plotted versus the number of trials.

Fig. 3. Learning curve for the CML model

In Fig. 3, the stand-up duration raises up suddenly at around 220 trials. This experiment result is very possible because similar individual learning curves can be seen in [17]. Once the action network in the cerebellum has been trained, the model is

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tested using the initial state: θ1 = 9° , θ 2 = −9° , where θ1 defines the angle of bottom pendulum and θ 2 defines the angle of top pendulum. The controlling performance is showed in Fig. 4 where the time course of the bottom and top pendulum is depicted.

Fig. 4. Controlling performance of the CML model

A 3-D phase plane of the bottom pendulum is represented in Fig. 5 to illustrate the non-linear dynamics during the controlling process. This figure suggests that the CML model has learnt to control the balancing task. In the presence of a steady-state input, the weights in the CML model successfully approach the fixed point attractor.

Fig. 5. 3-D phase plane of the bottom pendulum

These experiment results show that the CML model can acquire the technique of keeping the DIPC upright and finally focus to the equilibrium position without any prior knowledge and additional instructive signal. Although the controlling performance of the trained CML model is less efficient than the fuzzy controller proposed in [16], these preliminary results reveal the CML model’s potential to explain or represent the neural dynamics in the cerebellum. In future, more delicate learning principles can be developed to improve the controlling performance of the trained CML model.

4 Conclusions Both the anticipatory and feedback mechanisms are used in the cerebellar control of balance and locomotion. The CML model proposed here represents the parallel path-

CML Model for Cerebellar Control of Balance and Locomotion

33

way in the cerebellum and the correlation between the cerebellum and the cerebral cortex. The cerebral cortex plays a crucial role during the adaptation of cerebellar control pattern. Once a new control pattern is learned by cerebellum, the cerebellum could automatically response to the sensory stimulus without the involvement of the cerebral cortex. The experiment results showed that the CML model can learn the new balancing task from scratch, and a well-trained CML model can smoothly push the pendulum into the equilibrium position. It remains to test the model on further tasks, and to compare it to other cerebellar learning and control models. To cope with various complexities of controlled objects, an evolving neural architecture for the model is also to be considered.

References 1. Albus, J.S.: A Theory of Cerebellar Function. Mathematical Bioscience 10 (1971) 25-61 2. Marr, D.: A Theory of Cerebellar Cortex. Journal of Physiology 202 (1969) 437-470 3. Ito, M.: Neural Design of the Cerebellar Motor Control System. Brain Research 40 (1972) 81-84 4. Kawato, M. and Gomi, H.: A Computational Model of Four Regions of the Cerebellum Based on Feedback Error Learning. Biological Cybernetics 69 (1992) 95-103 5. Schultz, W., Dayan, P., and Montague, P.R.: A Neural Substrate of Prediction and Reward. Science 275 (1997) 1593-1599 6. Schmajuk, N.A. and Dicarlo, J.J.: A Neural Network Approach to Hippocampal Function in Classical Conditioning. Behavioral Neuroscience 105 (1991) 82-110 7. Bao, S., Chan, V.T., and Merzenich, M.M.: Cortical Remodelling Induced by Activity of Ventral Tegmental Dopamine Neurons. Nature 412 (2001) 79-83 8. Thompson, R.F., Thompson, J.K., Kim, J.J., Krupa, D.J., and Shinkman, P.G.: The Nature of Reinforcement in Cerebellar Learning. Neurobiology of learning and memory 70 (1998) 150-176 9. Fuster, J.M.: Executive Frontal Functions. Experimental Brain Research 133 (2000) 66-70 10. Suri, R.E. and Sejnowski, T.J.: Spike Propagation Synchronized by Temporally Asymmetric Hebbian Learning. Biological Cybernetics 87 (2002) 440-445 11. Houk, J.C., Buckingham, J.T., and Barto, A.G.: Models of the Cerebellum and Motor Learning. Behavioral and Brain Sciences 19 (1996) 368-383 12. Williams, R.J., and Zipser, D.: A Learning Algorithm for Continually Running Fully Recurrent Neural Networks. Neural Computation 1 (1989) 270-280 13. Morton, S.M. and Bastian, A.J.: Cerebellar Control of Balance and Locomotion. The Neuroscientist 10 (2004) 247-259 14. Stein, J.F. and Glickstein, M.: Role of The Cerebellum in Visual Guidance of Movement. Physiological Reviews 74 (1992) 967-1017 15. Matsumura, M., Sadato, N., Kochiyama, T., Nakamura, S., Naito, E., Matsunami, K., Kawashima, R., Fukuda, H., and Yonekura, Y.: Role of the Cerebellum in Implicit Motor Skill Learning: A PET Study. Brain Research Bulletin 63 (2004) 471-483 16. Yi, J.Q., Yubazaki, N., and Hirota, K.: Stabilization Control of Series-type Double Inverted Pendulum Systems Using the SIRMs Dynamically Connected Fuzzy Inference Model. Artificial Intelligence in Engineering 15 (2001) 297-308 17. Barto, A.G., Sutton, R.S., and Anderson, C.: Neuron-like Adaptive Elements That Can Solve Difficult Learning Control Problems. IEEE Transactions on Systems, Man, and Cybernetics 13 (1983) 834-846

A Model of Category Learning with Attention Augmented Simplistic Prototype Representation Toshihiko Matsuka Center for Decision Technologies, Stevens Institute of Technology, Hoboken, NJ 07030, USA [email protected]

Abstract. Cognitive models has been a main tool for quantitatively testing theories on human cognition. The results of previous cognitive modeling research collectively suggest the comparative advantage of exemplar over prototype accounts in human cognition. However, we hypothesized that unsuccessful outcomes by traditional prototype models may be the unforeseen consequences of the algorithmic constraints imposed on the models, but not of the implausibility of the theory itself. To test this hypothesis, a new cognitive model based on prototype theory with a more complex and realistic attention system is introduced and evaluated in the present study. A simulation study shows that a new model termed CASPRE resulted in a substantial improvement as compared with the traditional prototype model in replicating empirical findings and that it performed marginally better than an exemplar model, thus confirming our hypothesis.

1 Introduction There are two main competing theories on how humans internally represent categorical knowledge. Exemplar theory suggests that humans categorize an objects by utilizing its similarities to many if not all memorized exemplars they previously encountered, while Prototype theory assumes that humans utilize similarities between the input object and a smaller number of summarized reference points, i.e., prototypes. The results of previous cognitive modeling research collectively suggest the comparative advantage of exemplar over prototype accounts of internal representation in human cognition. For example, for one of the most extensively tested stimulus sets, namely Medin and Shaffers 5-4 stimulus set shown in Table 1 (referred to as MS78 hereafter), exemplar models have been more successful than prototype models in replicating empirical observations [1]. Moreover, although it is not always observed, a psychological phenomenon called A2 advantage, which is a tendency that people often categorize the less ”prototypical” stimulus A2 (see Table 1) more accurately than more ”prototypical” stimulus A1, have been pointed as supporting evidence for exemplar theory and challenging evidence against prototype theory. Another simple yet robust evidence against traditional prototype models is a category structure with exclusive-or (XOR) logic. Consider a two-dimension XOR stimulus set (i.e., [0 0, 1 1] for Category A, [0 1, 1 0] for Category B) whose prototypes for Categories A and B would be theoretically identical, i.e., [0.5, 0.5]. By internally representing categorical knowledge with these identical prototypes, previous prototype models with traditional selective attention mechanisms J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 34–40, 2006. c Springer-Verlag Berlin Heidelberg 2006 

A Model of Category Learning

35

have not been able to categorize or learn to categorize these stimuli, because the models mathematical formulations yield identical psychological similarity measures for any input stimulus to both prototypes, providing no constructive information regarding a category to which the input stimulus should belong. These previous prototype models are comprised of two key assumptions on their selective attention mechanisms that result in the invariant similarities for the simple XOR stimulus set and possibly other unsuccessful replications of some empirical findings. The first key assumption is that, like many other similarity-based models of categorization, the models assume that the selective attention mechanism is mediated by the dimension-by-dimension basis (i.e., selective attention is applied orthogonally). Secondly, the models also assume that the dimensional selective attention process is applied to the entire psychological stimulus space universally. These two assumptions have been widely accepted and incorporated in virtually all computational models of human categorization and category learning, and many of these models have been shown to be successful in replicating several empirical phenomena (e.g., [2]). However, in spite of their prevalent acceptance and recognized successes in some applications, the present research casts doubt on the two key assumptions about the selective attention mechanisms incorporated in the models. What if attention processes are uniquely applied to individual prototypes and what if humans are capable of attending correlations among feature dimensions, then is the prototype theory still psychologically less plausible than exemplar theory? A recent simulation study suggested the possibilities of interdependencies between models internal representation systems and selective attention mechanisms on effectiveness in replicating human category learning and that the traditional selective attention mechanisms, described above, incorporated into prototype models might not be the most sensible ones [3]. The main objective of the present works is not to reiterate the debate over the validity of prototype and exemplar accounts in human cognition per se, but to raise a concern that unsuccessful outcomes by some previous prototype models may be the unforeseen consequences of the algorithmic constraints imposed on the models, but not of the implausibility of the theory itself. In particular, the present study introduces a new model of category learning on the basis of prototype theory with a new feedforward algorithm consisting of more complex yet plausible selective attention mechanisms and a backward (learning) algorithm that emphasizes human tendency in knowledge acquisition.

2 A New Model of Category Learning: CASPRE Our new model is termed CASPRE for Category learning with Attention augmented Simplistic Prototype REpresentation. It is a cognitive model based on prototype theory and thus it assumes that categorical knowledge is organized by small numbers of prototypes and that humans utilize psychological similarity between input stimulus and prototypes to categorize the input. As its name suggests, CASPRE is comprised of two important components. First, it assumes a somewhat complex attention process, namely a local attention process (i.e., each prototype having unique selective attention process) and sensitivity to correlations among features (paying attention to correlations

36

T. Matsuka

among feature dimensions) to form attention augmented prototype representations. The selective attention may be interpreted as processes of mental rotation and psychological scaling of proximity or similarity between input stimuli and prototypes. Thus, in CASPRE each prototype is augmented with a unique selective attention process to form a uniquely shaped and oriented prototype conceptual field. Therefore, unlike traditional prototype models, characteristics of prototypes cannot be explained only by centroids, but by both conteroids and within-prototype psychological scaling processes. Some empirical studies provide supporting evidence for CASPRE’s attention mechanism (e.g., local attention process [4] and correlation sensitivity [5]). Another important component is the principle of simplicity in high-order human cognition. One plausible theoretical justification of prototype theory is its compact representation of knowledge, allowing a limited-processing-capacity human brain to handle rich information. CASPRE follows the simplicity principle, but the incorporation of complex attention mechanisms can confound the principle. To avoid the possible interference, CASPRE incorporates a simple multi-objective learning algorithm, trying to acquire manageably simple yet sufficiently accurate concepts. 2.1 Forward Algorithm In CASPRE, psychological similarity or distance, dj (x), between an input stimulus, x, and prototype j, πj , are defined by Mahalanobis distance (in quadratic form) between them, allowing it to be sensitive to correlations among features dimensions. Thus, dj (x) =

I I i

m

αjim (πji − xi )(πjm − xm )

(1)

where αj defines directions and strengths of attention field for πj , subscripts i and m indicate feature dimensions, and I is the number of feature dimensions. Note that it is assumed that αim = αmi , α2im ≤ |αii · αmm |, and , αii ≥ 0, ∀i. For off-diagonal entries (i.e., i  = m), an attention weight can be a negative value, where its signum indicates direction of attention field while its magnitude indicates the strength of attention. Psychological distance measures activate prototype units using the following function hj = exp(−c · dj (x)) (2) where c controls overall sensitivity. Prototype units activations are then fed forwarded to category output nodes, or  Ok (x) = wkj hj (3) j

where wkj is an association weights between πj and category node k. The output activations will be used to obtain response probability by the following function: P (k) = exp(φOk )/ l exp(φOl ), where φ scales decisiveness of response. In short, CASPRE assumes that humans utilize psychological similarity between input object and prototypes, psychologically scaled by correlation sensitive local selective attention processes, as evidence for categorizing the input instance into the most probable category.

A Model of Category Learning

37

2.2 Learning Algorithm CASPRE incorporates the gradient descent version of Stochastic Context-Dependent (multi-objective) Learning algorithms [6] [7]. In particular, CASPRE assumes that in ordinary situations humans would prefer and try to acquire compact yet accurate knowledge. Thus, its objective functional consists of at least two sub-objectives; accuracy and simplicity, or Υ (wkj , αim ) =

1 k

2

(tk − Ok )2 +



2 wkj +

j

k

I  I−1   j

i

m=i+1

(αjim )2 /Z 1 + (αjim )2 /Z

(4)

I−1 I j 2 where tk is a target output for category node k, and Z = i m=i+1 (αim ) . The first term is a function defining categorization accuracy, the second term is weight decay function, regularizing association weights, and the third term is an attention elimination function [7], reducing the number of dimensions and correlations among dimensions attended on the basis of the relative attention strengths. A framework [6] [7] that allows more general multi-objective context dependent learning can also be applied to CASPRE. Learnable coefficient w and α are updated by following functions, Ω Δwkj = λE w ek hj − 2λw wkj + νkj

Δαjim

=

−λE α



ek wkj hj δji δjm c −

λA α

k

2αjim ·



2(αjim )2 +

(5)

(αjl )2 l∈im /



(αjl )2 l∈im /

2 + νim

(6)

where ek = (tk −Ok ), δji = (πji −xi ), hj is as given Eq. 2, λs are learning rates, and ν are independent Gaussian noise in learning with means equal to 0 and time-decreasing standard deviations. The noise in learning is introduced in CASPRE, because, recent cognitive modeling studies indicate the importance of stochasticity or noise in human learning for qualitative fits and quantitative interpretations [6] [7]. The centroid of prototype j is updated with a simple competitive learning algorithm [8] [9], thus  (t) (t) π j + η (t) (x − π j ) if x and πj belong the same category (t+1) πj = (7) (t) πj otherwise. where η is a time-decreasing learning rate.

3 Simulations In order to investigate the capability of CASPRE to replicate observed empirical data, a simulation study was conducted. In particular, we simulated a classical study in human category learning [1] where prototype models previously had trouble replicating observed classification profile. Table 1 shows the schematic representation of stimulus set used in the present simulation study. In the empirical study, subjects were asked to learn to classify nine unique training exemplars (A1 - B4) to either Category A or

38

T. Matsuka Table 1. Schematic representation of stimulus set used in simulations (MS78)

ID A1 A2 A3 A4 A5 B1 B2 B3 B4

Training Cat D1 D2 D3 A 1 1 1 A 1 0 1 A 1 0 1 A 1 1 0 A 0 1 1 B 1 1 0 B 0 1 1 B 0 0 0 B 0 0 0

D4 0 0 1 1 1 0 0 1 0

ID T1 T2 T3 T4 T5 T6 T7

Transfer D1 D2 D3 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0

D4 1 1 1 1 0 0 0

B with corrective feedback. The training session was followed by a transfer session in which subjects were asked to categorize the nine training exemplars and seven novel exemplars. There are two empirically observed criterion response profiles involved in the present study, namely the original MS78 [1] and the average classification response profile from 30 independent replications (but also includes the original study) of MS78 summarized in Smith and Minda [11] (referred to as SM00). The two criterion profiles were employed to facilitate fair comparisons as Smith and Minda [11] claim the original MS78 data dubiously favor the exemplar models. The first and fourth rows in Table 2 show the observed classification profile in the transfer session in MS78 and SM00 respectively. There were three models involved in the present simulation study, namely CASPRE, a prototype model (PRO) with traditional selective attention mechanism (i.e., αjim = αlim , ∀j, l and αjim = 0, ∀i  = m), and a typical successful exemplar model (EXE) that has the same selective attention mechanism with PRO (e.g., [10], see also [3]). The only difference between EXE and PRO was their reference point. In EXE all unique nine training exemplars were used as the reference points and the locations of those nine reference exemplars were static. [10]. PRO and EXE were included to evaluate descriptive validity of attention augmented prototype representation. Methods: The models were run in a simulated training procedure with 50 trial blocks, where each block consisted of a random presentation of the nine unique training exemplars exactly once, in order to learn the correct classification responses for the stimulus set. The model parameters were optimized so that the sum of squared errors (SSE) between observed and predicted classification profiles were minimized. The selected paΩ rameters were as follows: For MS78 replication,c = 1.17, φ = 1.96, λE w = 0.30, λw = E A E E 0.0001, λα = 0.36, λα = 0.002 for CASPRE; c = 1.38, φ = 2.11, λw = 0.43, λα = E 0.11 for PRO; and c = 2.01, φ = 1.82, λE w = 0.12, λα = 0.006 for EXE. For SM00, E Ω E c = 4.13, φ = 2.48, λw = 0.30, λw = 0.0003, λα = 0.10, λA α = 0.002 for CASPRE; E c = 1.379, φ = 2.113, λE = 0.434, λ = 0.106 for PRO; and c = 1.01, φ = w α E Ω A 1.56, λE = 0.17, λ = 0.04 for EXE. λ andλ were held at 0 for PRO and EXE w α w α to simulate traditional prototype and exemplar models. Each model was replicated 100 times.

A Model of Category Learning

39

Table 2. Results of Simulation Study (MS78 & SM00 : observed human data [1] [11]; PRO: prototype model; CAS: CASPRE; EXE: exemplar model) Model SSE MS78 PRO CAS EXE SM00 PRO CAS EXE

N/A 0.197 0.023 0.020 N/A 0.086 0.015 0.034

A1 0.78 0.84 0.83 0.80 0.83 0.81 0.84 0.86

A2 0.88 0.75 0.89 0.88 0.82 0.73 0.84 0.79

A3 0.81 0.94 0.87 0.87 0.89 0.91 0.86 0.93

A4 0.88 0.76 0.83 0.85 0.79 0.71 0.74 0.70

A5 0.81 0.76 0.83 0.85 0.74 0.69 0.71 0.66

Probability of ’A’ Response B1 B2 B3 B4 T1 T2 0.16 0.16 0.12 0.03 0.59 0.94 0.32 0.32 0.25 0.03 0.67 0.97 0.19 0.14 0.14 0.10 0.58 0.90 0.17 0.17 0.16 0.11 0.62 0.93 0.30 0.28 0.15 0.11 0.62 0.88 0.36 0.34 0.30 0.06 0.65 0.95 0.25 0.30 0.19 0.11 0.60 0.84 0.39 0.34 0.28 0.04 0.64 0.97

T3 0.50 0.30 0.53 0.51 0.40 0.34 0.41 0.30

T4 0.62 0.67 0.60 0.62 0.55 0.63 0.58 0.60

T5 0.31 0.24 0.29 0.33 0.40 0.32 0.35 0.34

T6 0.34 0.24 0.32 0.33 0.34 0.30 0.35 0.30

T7 0.16 0.04 0.10 0.09 0.17 0.07 0.18 0.05

Results and Discussion: Table 2 shows the results of Simulation study. There were dramatic decreases in SSEs for CASPRE as compared with those in the traditional prototype model (PRO). The SSEs decreased approximately 90 % for MS78 and 85% for SM00. CASPRE performed almost as good as EXE for MS78 and its SSE was twice as small than EXE in SM00. Overall, CASPRE was the best performing model among three models tested. To the best of our knowledge, CASPRE is the first prototype model that performed better than EXE for this particular stimulus set. Furthermore, although CASPRE is in fact a prototype model, it successfully replicated A2 advantage in MS78 replication (Table 1, third row), more correctly categorizing less ”prototypical” stimuli, A2, than more ”prototypical” stimulus, A1. This is because CASPRE perceived A2 to be more similar to the prototype of Category A (Prototype A) than A1 to Prototype A. This was caused by CASPRE attention process, elongating concept coverage area of Category A (or Prototype A) to the direction of the least prototypical stimulus characteristics, namely 0s in both Dimensions 2 and 4 in order to reduce classification error. This in turn resulted in an increased perceived similarity between Prototypes A and Stimulus A2 above and beyond the fact that the two most diagnostic dimensions (i.e., Dimensions 1 and 3) provide supporting evidence that A2 may belong to Category A. We also conducted a simulation study to test if CASPRE is capable of learning to categorize simple XOR logic (i.e., [0 0; 1 1] for Category A and [0 1; 1 0] for Category B), which was also pointed out as evidence against the previous prototype models. CASPRE successfully learned the XOR logic. In sum, CASPRE’s performance in replicating observed data was on average better than that of a typical successful exemplar model. In addition, the present simulation studies showed CASPRE’s attention augmented representation system significantly improved a prototype model of category learning in replicating empirical psychological phenomena.

4 Conclusion Cognitive modeling research has been the main vehicle for quantitatively testing theories on human cognition. Prototype theory has been criticized or receiving less fa-

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vorable review, because models based on prototype theory used in these studies often resulted in less accurate replication of empirical data. Simulation studies on MS78 were one of the most frequently utilized for criticizing the theory. However, with proper attention system, a prototype model, CASPRE, performed as good as or better than an exemplar model, showing no comparative advantage of exemplar theory over prototype theory. This in turn casts doubt on the validity of previous criticisms of Prototype theory based on simulation studies using MS78, and possibly other category structures as well. Rather, those criticisms or unsuccessful simulation results should be attributed to the mismatch of the theory and integrated attention system in modeling, but not the theory itself. Incorporating more realistic attention mechanisms in prototype model may facilitate fairer comparisons between Prototype and Exemplar theories in order to better understand the nature of human categorization processes.

Acknowledgements This research was supported in part by the Office of Naval Research, Grant # N0001405-1-00632.

References 1. Medin, D. L., Schaffer, M. M.: Context Theory of Classification Learning. Psychological Review 85 (1978) 207-238 2. Nosofsky, R. M.: Attention, Similarity and the Identification-Categorization Relationship. Journal of Experimental Psychology: General 115 (1986) 39-57 3. Matsuka, T.: Generalized Exploratory Model of Human Category Learning. International Journal of Computational Intelligence 1 (2004) 7-15 4. Aha, D. W., Goldstone, R. L.: . Concept Learning and Flexible Weighting. In: Proceedings of the 14h Annual Meeting of the Cognitive Science Society Lawrence Erlbaum Associates, Hillsdale, NJ. (1992) 534-539 5. Anderson, J. R., Fincham, J. M.: Categorization and Sensitivity to Correlation. Journal of Experimental Psychology: Learning, Memory & Cognition, 22 (1996) 119-128 6. Matsuka, T.: Simple, Individually Unique, and Context-Dependent Learning Method for Models of Human Category Learning. Behavior Research Methods 37 (2005) 240-255 7. Matsuka, T.: Modeling Human Learning as Context Dependent Knowledge Utility Optimization. In: Wang, L., Chen, K., Ong, Y. S. (eds.): Advances in Natural Computation, Lecture Notes on Computer Science, Vol. 3610. Springer-Verlag, Berlin (2005) 933-946 8. Kohonen, T. Self-Organizing Maps, 3rd edn. Springer, Berlin (2001) 9. Love, B. C., Medin, D. L., Gureckis, T. M.: SUSTAIN: A Network Model of Category Learning. Psychological Review 111 (2004) 309-332 10. Kruschke, J. E. ALCOVE: An Exemplar-Based Connectionist Model of Category Learning. Psychological Review 99 (1992) 22-44 11. Smith, J. D., Minda, J. P.: Thirty Categorization Results in Search of a Model. Journal of Experimental Psychology: Learning, Memory, and Cognition 26 (2000) 3-27.

On the Learning Algorithms of Descriptive Models of High-Order Human Cognition Toshihiko Matsuka1 and Arieta Chouchourelou2 1

Center for Decision Technologies, Stevens Institute of Technology, Hoboken, NJ 07030, USA 2 Rutgers University, Newark, NJ 07102, USA

Abstract. The primary focus in computational modeling research in high order human cognition is to compare how realistically embedded algorithm describes human cognitive processes. However, several current models incorporated learning algorithms that apparently have questionable descriptive validity or qualitative plausibleness. The present research attempts to bridge this gap by identifying five critical issues overlooked by previous modeling research and then introducing a modeling framework that addresses the issues and offers better qualitative plausibleness. A simulation study utilizing the present framework with two distinctive implementation approaches shows their descriptive validity.

1 Introduction Artificial Intelligence by its nature is an interdisciplinary science, and thus there are many sub-fields emphasizing different, sometimes competing issues. While a majority of AI research has emphasized solving complex problems with algorithms that qualitatively have no implication with regard to real human intelligence, there is a line of research that strictly focuses on how algorithms or models mimic human behaviors of interest in order to test theories of human cogntive processes. Several simulation studies suggested the predictive validity of some cognitive models (e.g., [1]). However, in spite of their success, some simulated cognitive processes incorporated in many cognitive models appear to have a normative justification (i.e., how humans should think), or their algorithms do not necessarily give qualitative interpretations that resemble real human cognitive processes. The present paper is an attempt to raise concerns regarding explicit and implicit theories and assumptions integrated in cognitive models of human learning that appear to have debatable descriptive validity (i.e., not describing real human cognitive processes). In particular, the present work focuses on the descriptive validity of learning algorithms embedded in traditional models of human category learning or concept formation and then introduces a new framework that offers more realistic qualitative interpretation and better descriptive validity. Note that the terms ”category learning” and ”concept formation” will be used interchangeably throughout this paper, because conceptual knowledge is considered to be categorically organized. 1.1 Model of Category Learning Forward Algorithm. There have been several models of category learning introduced, but the majority of recent successful models (e.g., [1]) can be considered to be a variant of Radial Basis Functions. A generic model may be formulated as follows: J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 41–49, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Ok =

 j

wkj R (d(xi , ψji , αi )) ;

(1)

where d is a function characterizing perceived psychological distance (scaled by dimensional selective attention coefficient α) between input stimulus x and the j-th mentally represented reference points, ψj , (e.g., memorized exemplars or category prototypes) both of which are characterized by I feature dimensions. R is an activation function for reference points transforming the perceived distance to activation (e.g., an exponential function). Each reference point is connected to all category output nodes, and the activations of output nodes (Ok ) are determined by the weighted sum, weighted by association weight coefficient w, of the activation of reference points. The output activations will be  used to obtain response probability by the following function: P (k) = exp(φOk )/ m exp(φOm ), where φ scales decisiveness of response. In short, the models assume that people utilize psychological similarity between input object (x) and some reference points (ψ) as evidence for categorizing the input instance into the most probable category. Traditional Learning Algorithm. The majority of cognitive models of human category ∂E learning incorporate an online version of gradient descent, or Δθi = −λ ∂θ , where θ i indicates ”learnable” concepts or model coefficients (e.g., wkj and αi in Eq. 1). The almost  exclusive choice of the error function is the sum of squared error or E(θ) = 1/2 k (dk − Ok )2 where dk is the desired output for category node k.

2 Descriptive Validity of the Traditional Learning Algorithm Probability of ”Optimal” Learning: Although it is not explicitly assumed, the gradient type learning algorithm can be qualitatively interpreted as that humans try to learn concepts and category by adjusting all possible relevant knowledge (i.e., model coefficients) locally optimally in order to minimize misclassification whenever they receive feedback on accuracy of their concepts or knowledge. Thus, the probability of finding locally better concept for any given learning trial and throughout the entire learning phase is always 1, or  t

P (E(θ t + Δθ) < E(θt + Δϑ), ∀Δϑ  = Δθ|xt , θ t , {|Δθ|, |Δϑ|} < ) = 1 (2)

where t indicates time, and  is a small number, implying that it holds only locally. This implicit assumption does not appear to hold in real human concept formation. In reality, it is most likely true that some knowledge is updated correctly while other is not necessarily done so. Objectives of Learning: Human cognitive competence is often characterized by adaptability, which manifests itself prominently in context dependent thinking skills. But, this type of cognitive ability has been largely overlooked in high-order cognitive modeling research. In particular, as described in Section 1.1, virtually all models of human category learning assume that humans are motivated and capable of minimizing categorization error in every training instance. In other words, the models assume that everyone

On the Learning Algorithms of Descriptive Models

43

undertakes exactly the same computational operation for updating coefficients irrespective of a learner’s personal contextual factors (e.g., motivation). Recent empirical and simulation studies, however, do not support this assumption. Inadequacy of error minimization as a sole mechanism of real human concept formation was empirically suggested [3]. Real-world human learning is more likely to be explained as a process of multi-objective optimization of utility of knowledge, where objectives are not only defined by categorization errors, but also by each learner’s intra-, inter-, and extra contextual factors. For example, one laboratory experiment showed that in ordinary situation humans have a preference for simpler, yet sufficiently accurate concepts over complex but marginally more accurate concepts, suggesting their objectives in learning the concept were defined by both accuracy and simplicity [3]. Matsuka [2] showed that only models that incorporated contextual characteristics in their learning algorithms were capable of reliably replicating some empirical phenomena. Arbitrary Decision in Learning: Another important finding of the aforementioned empirical study [3] that showed humans being biased toward simple sufficient knowledge1 is that each participant’s preference to pay attention to one diagnostic feature dimension over the other equally diagnostic yet highly collinear or redundant dimension seemed to be the result of arbitrary decision or random choice. The traditional gradient descent algorithm, again, does not allow cognitive models to have such arbitrary decision in learning process, because the algorithm assumes and actually does ’correctly’ update all coefficients, resulting in paying attention to the two diagnostic dimensions. Degrees of Locality in Learning: A most commonly used learning algorithm in models of category learning is an online version of gradient descent. Thus, the traditional learning models do not necessarily assume strictly local learning because of its stochastic approximation, its learning strategy is still highly locally oriented. More precisely, there is no mechanism that explicitly influences the degree of locality (or globality) in search areas. This suggests that the capability of finding the global minimum is highly influenced by the initial coefficient configuration (i.e., initial mental state) and by the stimuli presentation order. Much of real life and ordinary learning are probably based on heuristic and locally-oriented learning. But, there are rarer instances where humans would make an attempt for globally-oriented learning probably in the case of non-ordinary situations where the maximization of knowledge utility is crucial (e.g., preparing for exams). In order words, humans seem to have the capability of adjusting their search areas in the conceptual space. This mechanism has not been explicitly incorporated (e.g., parameterized) in previous cognitive models. The Nature of Error Hypersurface: In order to find the gradient for the error hypersurface, the objective function for the traditional gradient descent learning needs to be differentiable or its first derivative needs to be defined. Thus, these learning models assume the error hypersurface to be smooth and continuous. Some previous studies, however, suggest the possibility of a discontinuous property in hypothetical error surface in human cognition. Bower and Trabasso [4] showed evidence that individuals could 1

This bias, however, is probably limited to certain circumstances, and thus it does not have to be always present. Rather, context factors determine the presence or absence of the bias.

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have learned category in an all-or-none manner (i.e., accuracy suddenly increased from a near base-rate accuracy), which they referred to as ”insightful” learning. This type of learning may be explained by either that the error hypersurface has the discontinuous or all-or-none property (i.e., step functions) or that learning utilizes Hessian matrix with model coefficients θ being sufficiently close to a minimum. However, it appears impossible for ordinary human to consciously obtain an inverse of Hessian matrix while performing a categorization task. Thus, those empirical studies might have suggested that at least in some situations the error hypersurface for multi-objective category learning is discontinuous. Although insightful learning is mainly supported by very simple learning tasks, these empirical findings raise concerns about the descriptive validity of the standard gradient descent as a model of human learning in some cases.

3 Toward Descriptive Models of Category Learning We raised five issues on the descriptive validity of the traditional learning algorithm applied to cognitive model of category learning. The following sections introduce a new framework that can, at least partly, overcome the issues, followed by two possible implementation approaches. The first approach is a simple extension of the gradient descent, and the other is based on a derivative-free stochastic optimization method. In spite of the differences in their optimization processes, the two new learning models fundamental assumptions are the same – human learning is a multi-objective optimization process with stochastic characteristics. 3.1 Fundamental Principles Multi-objective in Learning: The present framework assumes that human learning is not only driven by error-minimization, but by optimization of subjectively and contextually defined multidimensional knowledge utility. Operationally, it is framed as a minimization problem, and thus a smaller utility value implies better utility. The generic utility function can be formulated as: Υ (θ) = λE E(θ) +

M m

λm · Qm (θ)

(3)

where the first term defines accuracy of concepts (e.g., SSE), and each Qm function in the second term can be interpreted as a function defining a particular contextual factor m, λm is a scalar weighting importance of context factor m, and M is the number of contextual factors. There are virtually infinitely many contextual functions appropriately defined for Eq. 3. Concept abstraction, domain expertise, conservation, and knowledge commonality are few examples of contextual, mainly motivational, factors possibly influencing learning processes and outcomes [2]. Stochasticity in Learning: One criticism regarding the descriptive validity of the traditional gradient descent methods is its absolute certainty in finding locally optimal knowledge for any given single learning trial (i.e., Eq. 2). The present framework overcame this issue by assuming that concept update consists of two elements, structured, G, and stochastic, ν, processes, where the latter can be considered as noise, ”random

On the Learning Algorithms of Descriptive Models

45

thoughts,” or random hypotheses. Thus, models’ coefficients (i.e., concepts) are updated as follows:   θ(t+1) = θ(t) + G θ(t) + ν; (4) The following two sections introduce two approaches implementing these two characteristics (Eqs. 3 & 4) in learning. 3.2 Noisy Gradient Descent with Multi-objective Functions The present learning model is a very simple, yet possibly effective extension of the traditional gradient descent. Here, the structured learning process, G, is a function of context-dependent multi-dimensional concept utility, and stochastic process, ν is timedecreasing random noise perturbation added to each coefficient update. Specifically,   ∂Υ ∂E  ∂Qm (θi ) (t) G θi = −λ∇(Υθ ) = −λ = −λE − λQm (5) m ∂θi ∂θi ∂θi where, λs are step sizes weighting different contextual factors (including categorization accuracy). In the present model, ν is random noise, i.e., νi ∼ Φ(0, ζ(·)), in learning, where ζ is a decreasing function. ζ needs to be a decreasing function for convergence. In addition, this function should return sufficiently large in the early learning phases to explore wider ”conceptual” space. The present model will be referred to as NGD. Remarks on Descriptive Validity: The added random noise in coefficient updates can be interpreted as follows: People try to revise their relevant knowledge (i.e., coefficient) about categories, but revision of each coefficient succeeds probabilistically due to the somewhat coarse and imperfect nature of human high-order cognitive processes. Furthermore, the added noise provides arbitrary decisions or random choices in learning. In addition, the degree of locality in learning can be controlled by the level of noise and its decaying speed. (e.g., it becomes globally oriented learning when the noise level is high and noise decrement is sufficiently slow). Because the present model utilizes utility gradient, the error hypersurface for the multi-objective utility needs to be continuous or the contextual factors need to be defined by some differentiable functions, and thus it probably would not sufficiently replicate ”insightful learning”. 3.3 Stochastic Hypothesis-Testing The present model is re-formulation of Matsuka’s [2] stochastic context-dependent learning model termed SCODEL, and will be referred to as SCO-R. The fundamental concepts underlying the present and the previous models are very similar. But, as compared with NGD, SCO-R has a more heuristic-oriented and hypothesis-testing like interpretations. In SCO-R, G process is very simple, i.e., to maintain status quo. Instead, it contains an elaborated and heuristically controlled ν process. In particular, it assumes that people form a hypotheses set, η, about category knowledge (i.e., model coefficients) in a random fashion. Operationally, this process is accomplished by randomly drawing hypothetical coefficient update values by an independently sampled term (i.e., ηi ) from a prespecified, in general, zero-mean symmetric distribution. The hypothetical update terms may be accepted to become ν terms (i.e., νi = ηi ) or discarded (i.e.,

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νi = 0) depending on their utility (i.e., Eq 3.) relative to that of the currently accepted concept of coefficients (θ(t) ). Note that in SCO-R, a smaller value of Υ indicates better utility or less futility. Therefore,  ηi ∼ Φ(0, ζ(T (t) )), if P (η) = P (η|θ (t) , T (t) ) ≥ U (0, 1); νi = (6) 0, otherwise. ⎧  (t)  −1 ⎨ (θ (t) ) 1 + exp Υ (θ +η)−Υ , if Υ (θ (t) + η) > Υ (θ (t) ); t T P (η) = (7) ⎩1, otherwise. where the superscript t indicates time, T indicates a parameter called ”temperature” (see [5]) that decreases over time, U is the Uniform distribution with min and maximum being 0 and 1 respectively, Φ(0, T t ) is a symmetric distribution function, The random distributions, Φ takes a time-decreasing parameter T that controls width of the random distributions (it also affects the probability of accepting a new hypotheses set, see Eq. 7). The temperature decreases across training blocks according to the following annealing schedule function: T t = δ(υ, t), where δ is the temperature decreasing function that takes temperature decreasing rate, υ, and time t as inputs. A choice of the annealing function may depend on the particular choice of the random number generation functions [6]. Note that the decrease in the temperature causes decreases in width of the distributions Φ. The effect of transition in distribution widths controlled by the temperature can be interpreted as follows: In early stages of learning SCODEL is quite likely to produce ”radical” hypotheses (i.e., the new set of hypotheses thus coefficients are very different from the currently valid and accepted concepts). But, as learning progresses, the widths of the random distribution decrease, so that it increasingly stabilizes its hypotheses and establishes more concrete and stable knowledge about the category. In sum, SCO-R does not assume learning involves computation intensive (back) propagations of error in the multi-faceted coefficient space, or calculation of partial derivative for each coefficient for the error hypersurface. Rather, a very simple operation (e.g., comparison of two utility values) along with the operation of stochastic processes is assumed to be the key mechanism in human learning. Remarks on Descriptive Validity: As in NGD, SCO-R assumes that the probability of having ”optimal” learning in the concept space for a given trial (and the entire learning phases) can be and often is less than 1. By its algorithmic nature, it also incorporates arbitrary decision in learning, and it can be global optimization-oriented, depending on parameter configurations. Unlike the gradient descent method, the utility functions for SCODEL do not need to be differentiable or continuous, and thus it allows more flexibility in defining its contextual factors. Furthermore, the present type of learning model has shown to be capable of replicating rapid changes in both categorization accuracies and attentional shift in a simple categorization task [7].

4 Simulations In order to investigate the capabilities of the present framework operating as different types of hypothetical yet plausible learners, a simulation study is conducted. The

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47

Table 1. Schematic representation of stimulus set used in simulations Category Dim1 Dim2 Dim3 Dim4 Dim5

A 0 0 0 1 1

A 0 0 1 2 2

A 0 0 2 3 3

A 0 0 0 4 4

A 0 0 1 5 5

A 0 0 2 6 6

B 1 1 0 7 7

B 1 1 1 8 8

B 1 1 2 9 9

B 1 1 0 10 10

B 1 1 1 11 11

B 1 1 2 12 12

Table 2. Functions used in Simulation Study Function & Objective Output Accuracy Weight Decay Attn. Elim.

Expertise ψiM ax

and

ψiM ax−1

Implementation

   Ok = j wkj · exp −1 · i αi |ψji − xi |  U E (w, α) = λE1/2 k (dk − Ok )2 2 U W (w) = λW k j wkj  −1 I 2 2 αi · αl  l U S (α) = λS i  −1 I 2 1 + α2i · l αl  

 exp −c · i αi ψiM ax − xi  Ξ  

U (α) = λΞ  exp −c · αi ψ M ax−1 − xi  i

(Eq. 8) (Eq. 9) (Eq. 10) (Eq. 11)

(Eq. 12)

i

indicates the maximally and 2nd maximally activated exemplars

present simulation study does not emphasize the present frameworks’ accuracies in learning per se. Instead, greater emphasis is placed on its capabilities in showing different expectable context dependent learning by hypothetical people engaged in hypothetical category learning tasks with the aim of evaluating its potential as an alternative or more descriptive model of high-order human learning processes. In particular, we test the two implementation approaches for the stochastic multi-objective learning framework using an extended version of a problem (Table 1) suggested by Matsuka [2]. Noted that all feature values are treated as nominal values differentiating each element within the dimension, and thus their numeric values and differences do not have any meaning. Methods: Five types of hypothetical human subjects with different kinds of hypothetical yet realistic learning objectives or combinations of objectives were involved in the present study. The five types of learners are: EMN - error minimizer whose objective in learning is to minimize categorization error, thus Υ = U E (i.e., Eq 9 in Table 2); ABS - abstractive learner who tries to acquire simple yet accurate concepts, or Υ = U E + U W + U S ; ATT - attentive learner who pays attention to many feature dimensions (e.g., being obsessive compulsive or trying to understand relationships among feature dimensions), thus Υ = U E − U S ; DEX - domain expertise whose objective includes acquiring accurate concept in addition to ability to distinguish different exemplars, thus Υ = U E +U Ξ ; and finally EDX, efficient domain expertise who would try to acquire simple accurate knowledge and the ability to distinguish between different exemplars, thus Υ = U E + U S + U Ξ . These five types of learners are empirically observed or suggested [2] [3] [8] [9].

48

T. Matsuka and A. Chouchourelou Table 3. Results of Simulation Study Types of Learner EMN ABS ATT DEX EDX

Model Accuracy NGC SCO NGC SCO NGC SCO NGC SCO NGC SCO

0.94 0.93 0.92 0.93 0.94 0.92 0.94 0.93 0.94 0.91

Dim 1 0.34 (0.20) 0.34 (0.18) 0.48 (0.48) 0.43 (0.38) 0.21 (0.03) 0.22 (0.04) 0.19 (0.08) 0.20 (0.08) 0.02 (0.13) 0.11 (0.11)

Mean Predicted Attention (StdDev) Dim2 Dim3 Dim4 Dim 5 0.36 (0.21) 0.17 (0.16) 0.07 (0.11) 0.06 (0.10) 0.32 (0.19) 0.14 (0.15) 0.10 (0.11) 0.10 (0.11) 0.43 (0.48) 0.02 (0.11) 0.04 (0.17) 0.03 (0.15) 0.42 (0.39) 0.07 (0.12) 0.04 (0.08) 0.04 (0.10) 0.21 (0.03) 0.20 (0.04) 0.19 (0.04) 0.19 (0.04) 022 (0.04) 0.20 (0.05) 0.18(0.05) 0.18 (0.06) 0.19 (0.08) 0.02 (0.03) 0.30 (0.07) 0.31 (0.07) 0.19 (0.08) 0.05 (0.06) 0.28 (0.06) 0.28 (0.06) 0.02 (0.12) 0.01 (0.05) 0.48 ((0.47) 0.46 (0.47) 0.11 (0.11) 0.04 (0.05) 0.37 (0.18) 0.37 (0.18)

Eq. 8 described forward categorization process (i.e., Eq. 1). The five types of learners were implemented with both NGD and SCO-R, and thus there were a total of 10 models simulated. All 10 models were trained with corrective feedback to learn to categorize 12 instances into either categories A or B. Note that we do not test the models abilities to learn (although all models successfully learned to categorize inputs), but instead we test their abilities to show different expectable context dependent learning outcome, particularly in learned attention allocations. For NGD, Gaussian distribution with mean equals to 0 and the standard deviation equals to the median changes in the t−1 previous training trial was used for ν process or νw ∼ N (0, M EDIAN (Δwkj )), t−1 να ∼ N (0, M EDIAN (Δαi )) For SCO-R, Cauchy distribution and exponential decay functions were used for random hypothesis generations and temperature decrement, respectively. There were 500 simulated subjects for each model. The parameters were selected arbitrary. Results: Table 3 shows the results of the simulation study. All models learned the category structure accurately at approximately the same precision. Note that predicted classification accuracies can be easily adjusted by manipulating model’s classification decisiveness parameter φ (see Section 1.1) and thus are not of the greatest importance other than the fact that all 10 models reached similar classification accuracies with the same φ value. For each type of learners, NGC-MO and SCO-R’s predictions on relative attention allocations (both means and standard deviations) were very similar, except EDX. For EDX, NGC-MO predict more variability in the amounts of attention allocated to Dimensions 4 and 5. Otherwise, NGC-MO and SCO-R showed similar patterns of learned attention distributions. These predictions are basically consistent with empirical observations [3] and cognitive theory [8] [9], indicating the two implementation approaches in the present framework successfully capture the essence of human learning.

5 Conclusion The present research raised five critical issues associated with the descriptive validity of the most widely used learning algorithm in high-order human cognitive processes,

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49

namely, the online version of gradient descent. These issues are the probability of ”optimal” learning, objectives in learning, arbitrary decision in learning, the degree of locality in learning, and the nature of error hypersurface. To overcome these limitations, a framework with two implementation approaches for models of human category learning or concept formation were introduced. The present framework assumes that human concept formation is not only driven by error minimization, but also by optimization of subjectively and contextually define concept utility. It also assumes that stochasticity plays a key role in the optimization process. Qualitative interpretations of the two learning models significantly expand the descriptive validity of currently available cognitive models of human category learning. Our simulation study quantitatively demonstrated both predictive and descriptive validities of both approaches.

Acknowledgements This work was supported by the Office of Naval Research (# N00014-05-1-00632).

References 1. Kruschke, J. E.: ALCOVE: An Exemplar-Based Connectionist Model of Category Learning. Psychological Review 99 (1992) 22-44 2. Matsuka, T.: Simple, Individually Unique, and Context-Dependent Learning Method for Models of Human Category Learning. Behavior Research Methods 37 (2005) 240-255. 3. Matsuka, T., Corter, J. E.: Process Tracing of Attention Allocation in Category Learning. (2006) Under review. 4. Bower, G. H., Trabasso, T. R.: Reversals Prior to Solution in Concept Identification. Journal of Experimental Psychology 66 (1963) 409-418. 5. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E.: Equation of State Calculations by Fast Computing Machines. Journal of Chemical Physics 21 (1953) 1087-1092. 6. Ingber, L.: Very Fast Simulated Re-Annealing. Journal of Mathematical Computer Modelling 12 (1989) 967–973. 7. Matsuka, T., Corter, J. E.: Stochastic Learning Algorithm for Modeling Human Category Learning. International Journal of Computational Intelligence 1 (2004) 40-48. 8. Tanaka, J., Taylor, M.: Object Categories and Expertise: Is the Basic Level in the Eye of the Beholder? Cognitive Psychology 23 (1991) 457-482 9. Tanaka, J., Gauthier, I.: Expertise in Object and Face Recognition. In: Medin, Goldstone & Schyns (eds.): Perceptual Learning: The psychology of Learning and Motivation Vol. 36. Academic Press, San Diego, CA: (1997) 83-125

A Neural Model on Cognitive Process Rubin Wang1,2, Jing Yu2, and Zhi-kang Zhang1 1

Institute for Brain Information Processing and Cognitive Neurodynamics, School of Information Science and Engineering, East China University of Science and Technology, Meilong Road 130, Shanghai 200237, P.R. China {rbwang, zhikangb}@163.com 2 School of Science, Donghua University, West Yan-An Road 1882, Shanghai 200051, P.R. China {rbwang, seychelle}@163.com

Abstract. In this paper we studied a new dynamic evolution model on phase encoding in population of neuronal oscillators under condition of different phase, and investigated neural information processing in cerebral cortex and dynamic evolution under action of different stimulation signal. It is obtained that evolution of the averaging number density along with time in space of three dimensions is described in different cluster of neuronal oscillators firing action potential at different phase space by means of method of numerical analysis. The results of numerical analysis show that the dynamic model proposed in this paper can be used to describe mechanism of neurodynamics on attention and memory.

1 Introduction In early published papers, we have studied dynamic models of phase resetting with neural population, and these models were generalized to cognitive neural dynamic system. These models extended the analysis in which interactive neurons in dynamic model of phase resetting is limited in one neural population [1-3]. In those research results, amplitude of neural population was extended to changeable amplitude from original amplitude of limit cycle, nonlinear dynamic model on two neural populations was obtained under case of changeable amplitude. We further studied nonlinear stochastic dynamic models in terms of phase resetting in which coupling intensity of synapses among neurons is changeable along with time. Accordingly, models of these types can reveal the dynamic mechanism between evolution models of learning, memory and neuronal plasticity [4,5]. Particularly, we studied the selfsynchronization evolution model in noise presence and the desynchronization evolution model in periodic stimulation presence for neural population possessing stochastic amplitude, and many conclusions possessing biological sense were obtained [6-8]. Due to eigenfrequency of each neuronal oscillator to be different in neural population, the phase and the amplitude of each neuronal oscillator are also different in neuronal activities [9]. Why does our attention focus a certain event we simultaneously ignore other events in unconsciousness? Why does our attention easily memorize some events which were been paid attention, and events are not paid attention to be easily J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 50 – 59, 2006. © Springer-Verlag Berlin Heidelberg 2006

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51

forgotten. In order to deeply describe the neural information processing under action of stimulation and to answer questions above mentioned, in this paper we present the neural dynamic model that possesses different phase. The results of numerical computation show that the dynamic model can satisfactorily describe mechanism of neurodynamics on attention and memory.

2 The Dynamic Model Assuming a population of neural oscillator possessing periodic activity is composed by N oscillators as in figure 1. There are two clusters in the population of neural oscillators, oscillators’ numbers and phase in the first cluster are denoted to N1 and θ1, respectively. And oscillators’ numbers and phase in the second cluster are denoted to N- N1 and θ2, respectively. The interaction of neurons between two clusters is denoted to M which will be given in next part.

Fig. 1. Two clusters of neural oscillators on different phase in one population

Where Ω1 is eigenfrequency of N1 neural oscillators, amplitude is R1(t) and external stimulation is S1(Ψj, R1). Ω2 is eigenfrequency of N–N1 neural oscillators, amplitude is R2(t)and external stimulation is S2(Ψj, R1). The dynamic equations under both noise and periodic stimulation presence are given by ⎧ 1 N ∑ M (ψ j −ψ k , rj , rk ) + Fj ,1 (t ), ( j = 1, 2L N1 ) ⎪ψ& j = Ω1 + S1 (ψ j , rj ) + N k =1 ⎪ ⎪ ⎪ 1 N ⎪ψ& j = Ω 2 + S2 (ψ j , rj ) + ∑ M (ψ j −ψ k , rj , rk ) + Fj ,2 (t ), ( j = N1 + 1,L N ) ⎨ N k =1 ⎪ ⎪r& = α r − β r 3 , j = 1, 2L N 1 j 1 j ( 1) ⎪ j ⎪ ⎪⎩r&j = α 2 rj − β 2 rj3 , ( j = N1 + 1,L N )

(1)

where Fji is the term of white noise. Stimulation term is S k (ψ j , rj ) =

L

∑

m =1

I km rjm + 2 cos ( mψ j + γ km ), k = 1, 2

(2)

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And coupling term is

(

M ψ j −ψ k , rj , rk

)＝－m∑=1r L

m m j rn

( K sin m (ψ j −ψ k ) + C cos m(ψ −ψ )) m

m

The Fokker-Plank equation on probabilistic density

j

k

(3)

f in equation (1) is given by

N1 ∂ 2 N ∂f ∂f ∂ ⎡ Q N ∂2 = − ∑ g k ( Rk ) + f − ∑ T f⎤ ∑ [T1 f ] − ∑ 2 ⎣ 2 ⎦ k =1 ψ ψ ∂t ∂Rk 2 j =1 ∂ψ j ∂ ∂ j =1 j = N1 +1 j j

(4)

The probabilistic density and coefficients on vector {ψ l } of phase and vector {Rl } of amplitude in Fokker-Planck equation (4) can be written in the form

f = f (ψ 1 , Lψ N , R1，L RN , t )

1 N T = T11 1 N k∑ =1 1 N T = T22 2 N k∑ =1

( (

) ( ) ) ( ) ⎛ ⎧Ω + S (ψ , R ) + M (ψ j − ψ k , R2 , R1 )(k = 1,L N1 ) ⎞ ⎜T = ⎪ 2 2 j 2 ⎟ ⎜ 22 ⎨⎪Ω + S (ψ j , R ) + M (ψ j −ψ , R , R )(k = N + 1,L N )⎟ 2 1 k 2 2 ⎩ 2 2 ⎝ ⎠

⎛ ⎧Ω + S ψ , R + M ψ j − ψ k , R1, R1 (k = 1,L N1 ) ⎞ ⎜T = ⎪ 1 1 j 1 ⎟ ⎜ 11 ⎨Ω + S ψ , R + M ψ − ψ , R , R (k = N + 1, L N )⎟ ⎪ j j 1 1 k 1 2 ⎩ 1 1 ⎝ ⎠

One obtains an expression on function of amplitude with time t from equation of amplitude in equations (1) as follows Ri (t ) =

Ri ,0 1−

βi 2 R α i i ,0

αit

e

×

β 1+ i αi

⎛ ⎞ ⎜ ⎟ ⎜ Ri ,0 αi t ⎟ ×e ⎟ ⎜ βi 2 ⎜ ⎟ ⎜⎜ 1 − α Ri ,0 ⎟⎟ i ⎝ ⎠

2

(5)

In Figure 2 one observes that the amplitude reached the limit cycle R=1 after neuron is activated to 5s.

Fig. 2. Amplitude of limit cycles: R0 = 0.5 ， α = β = 1

Therefore, the amplitude becomes a constant after persisting a period of time.

A Neural Model on Cognitive Process

A globe number density is defined as follows n (θ1 ,θ 2 , R1 , R2 , t ) = n1 (θ1 , R1 , R2 , t )

＋n (θ , R , R , t ) 2

2

1

2

53

(6)

According to equation (5) we know the globe number density is a function on phase and time, and the function satisfies the following equation:

∫

2π

0

n (θ ′,θ ′, R1 , R2 , t ) dθ ′ = 1

(7)

Inserting (6) into (4) yields equation on the averaging number density 2 ∂n 2π 2π ⎛ ∂f ∂n = ∫ ∫ dψ l ⎜ n%1 + n%2 ∂f ⎞⎟ = − ∑ (α Rk − β Rk3 ) ∂t ⎠ ⎝ ∂t ∂t 0 0 ∂ Rk k =1 ⎞ ⎛ ⎞ Q ⎛ ∂2 ∂2 ∂ ∂ n1 + Ω 2 n2 ⎟ ⎜ 2 n1 + 2 n2 ⎟ − ⎜ Ω1 2 ⎝ ∂θ1 ∂θ 2 ⎠ ⎝ ∂θ1 ∂θ 2 ⎠ ∂ ∂ − ( S1 (θ1 , R1 ) n1 ) − ∂θ ( S2 (θ 2 , R2 ) n2 ) ∂θ1 2 +

−

∂ ⎡ 2π n1 (θ1 ) ∫0 ( M (θ1 −ψ ′, R1 , R1 ) n1 (ψ ′) + M (θ1 −ψ ′, R1 , R2 ) n2 (ψ ′) ) dψ ′⎤ ⎥⎦ ∂θ1 ⎢⎣

−

∂ ⎡ 2π n (θ 2 ) ∫0 ( M (θ 2 − ψ ′, R2 , R1 ) n1 (ψ ′) + M (θ 2 −ψ ′, R2 , R2 ) n2 (ψ ′)) dψ ′⎤ ⎦⎥ ∂θ 2 ⎣⎢ 2

(8)

Using Fourier transformation, the stimulation and coupling terms can be written as S j (ψ , R j ) = ∑ S jk ( R ) eikx ,

( j = 1, 2 )

k

⎧ I jk k + 2 iγ jk , ( k = 1,L, L ) ⎪ Rj e ⎪ 2 ⎪I S jk ( R ) = ⎨ jk R j −k + 2e−iγ jk , ( k = −1,L , − L ) ⎪ 2 ⎪ ⎪ other ⎩0,

M (ψ − ψ ′, R j , Rn ) =

－∑ M k

k

( R , R )eik (ψ −ψ ′) j

n

⎧π R kj Rnk ( −Ck + iK k ) , ( k = 1,L L ) ⎪⎪ M k ( R j , Rn ) = ⎨−π R −j k Rn− k ( C− k + iK − k ) , ( k = −1, L − L ) ⎪ other ⎪⎩0,

(9)

(10)

(11)

(12)

The averaging number density is expanded to Fourier series as follows:

ikθ n j (θ j , R1 , R2 , t ) = ∑ nˆ j ( k , R1 R2 , t ) e j , j = (1, 2 ) k

(13)

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One obtains N1 N − N1 , nˆ2 ( 0 ) = 2π N 2π N Under the above condition the averaging number density is rewritten in the following form nˆ1 ( 0 ) =

n (θ1 ,θ 2 , R1 , R2 , t ) = ∑ n j (θ j , R1 , R2 , t ) 2

j =1

(

= ∑ nˆ1 ( k , R1 , R2 , t ) eikθ1 + nˆ2 ( k , R1 , R2 , t ) eikθ 2 k

)

(14)

Inserting (14) into (8) yields the Fokker-Planck equation expressed by the averaging number density

⎧ ∂nˆ1 ( k , t ) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ⎪ ∂nˆ2 (k , t ) ⎪ ∂t ⎪ ⎪ ⎩

2

∂nˆ1

m =1

∂Rm

= − ∑ g m ( Rm )

−

k

2

2

Qnˆ1 − ik Ω1nˆ1 − ik ∑ S1m ( R1 ) nˆ1 ( k − m ) m

−ik ∑ nˆ1 ( k − m ) ⎡⎣ M m ( R1 , R1 ) nˆ1 ( m ) + M m ( R1 , R2 ) nˆ2 ( m ) ⎤⎦ L

m =±1

= − ∑ g m ( Rm ) 2

m =1

∂nˆ2 ∂Rm

−

k

2

2

Qnˆ2 − ik Ω 2 nˆ2 − ik ∑ S 2 m ( R2 ) nˆ2 ( k − m )

(15)

m

−ik ∑ nˆ2 ( k − m ) ⎡⎣ M m ( R1 , R2 ) nˆ1 ( m ) + M m ( R2 , R2 ) nˆ 2 ( m ) ⎤⎦ L

m=±1

Taking into account equation (5), one obtains dnˆ1 ∂nˆ1 ∂nˆ1 & ∂nˆ1 & ∂nˆ ∂nˆ ∂nˆ = + R1 + R2 = 2 + g1 ( R1 ) 1 + g 2 ( R2 ) 1 dt ∂t ∂R1 ∂R ∂t ∂R1 ∂R2 2

And then equation (15) can be rewritten as ⎧ dnˆ1 (k , t ) ⎪ ⎪ dt ⎪ ⎪ ⎨ ⎪ dnˆ2 (k , t ) ⎪ dt ⎪ ⎪ ⎩

=−

k2 Qnˆ1 − ik Ω1nˆ1 − ik ∑ S1m ( R1 ) nˆ1 ( k − m ) m 2

−ik ∑ nˆ1 ( k − m ) ⎡⎣ M m ( R1 , R1 ) nˆ1 ( m ) + M m ( R1 , R2 ) nˆ2 ( m )⎤⎦ L

m =±1

k2 = − Qnˆ2 − ik Ω 2 nˆ2 − ik ∑ S 2 m ( R2 ) nˆ2 ( k − m ) m 2

(16)

−ik ∑ nˆ2 ( k − m ) ⎡⎣ M m ( R1 , R2 ) nˆ1 ( m ) + M m ( R2 , R2 ) nˆ2 ( m )⎤⎦ L

m =±1

3 Evolution Results in Case of No Stimulation In case of no stimulation, if all parameters of in two clusters are completely the same, then neural model (16) is reduced to the result obtained in P.A. Tass’s monograph [9]. Therefore, we do not analysis for this case. In order to describe clearly the interaction of between two clusters in neural oscillators, we give evolution process of the averaging number density in space of three dimension in the below. The averaging number

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55

density on the first cluster’s neurons is described in (ψ1, n) plane, and averaging number density on the second cluster’s neurons is described in (ψ2, n) plane. The neural oscillators in the first and the second cluster satisfy each normalization condition in two planes, respectively.

Fig. 3. Evolution of the averaging number density of different phase N1:N2=10:1.Initial condition： n (ψ 1 ,0) = 1 2π + 0.012 cosψ 1 + 0.012 cos(2ψ 1 ) ， n(ψ 2 ,0) = 1 2π + 0.012 cosψ 2 + 0.012 cos(2ψ 2 ) ， Q1=Q2= 0.4, Ω1=Ω2=2π

Fig. 4. Evolution of the averaging number density of different phase N1:N2=1:10 (Initial condition is the same as Fig.3)

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R. Wang, J. Yu, and Z.-k. Zhang

In Figures 3 and 4 we clearly observe that the larger is difference of quantity of neurons, the more salience is to tend the evolution of which quantify of neural oscillators is the most of that cluster under condition of the same parameters. The proportion of quantity of neurons in the two clusters is very large to impact of evolution result, and neurons that the cluster with the most of quantity can dominate evolution result of system (the globe averaging number density at diagonal line). It is necessary to emphasize that we first give the dynamic evolution on neural encoding in phase space of two dimensions by means of numerical analysis, and numerically proved that early results on phase coding of neuronal population in phase space of one dimension lose abundant useful neural information [10]. Hence, theory of phase coding in phase space of one dimension cannot be applied to research of cognitive process. Actually, the theory of phase coding in high-dimension case only can preferably describe neural encoding and evolution in neural population in cerebral cortex.

4 Attention and Memory Under Stimulation Presence In this section, we research stimulation to impact of neural encoding in phase space of three dimensions. The desynchronization process caused instantaneously stimulation and the resynchronization process after end of stimulation will be interested in two questions in the section. The stimulation is composed by three steps in the numerical analysis. 1. In the first step there is no stimulation in neuronal activities, i.e. S1 = S 2 = 0 . We have studied the synchronization process and corresponding firing pattern on spontaneous behavior of large-scale neuronal population in reference [7]. Hence here does not describe in detail. 2. In the second step there is stimulation presence in neuronal activities, i.e. S1 ≠ 0 or S 2 ≠ 0 , S1 ≠ 0 and S 2 ≠ 0 . The question of stress is that different stimulation acts on different neural cluster, so that we can further research response of population of neural oscillators to different stimulation in nature. 3. In the third step there is stimulation stop in a synchronization process, i.e. S1 = S 2 = 0 . In the process we study cluster of neuronal oscillators how transit to a stable state from a non-stable state of the desynchronization process, i.e. how to return to the stable synchronization state by resynchronization process. Actually, in the procedure of the third step, stimulation result is an initial condition of resynchronization process. For this case, we had discussed impact of result of neural encoding to different initial condition in reference [8]. Figure 5 describes how neural oscillators of two clusters that quantitative ration of neurons is 1:10 evolve to a non-stable state of desynchronization from stable state of the synchronization under impulse’s stimulation of different order. Due to dominating the system’s the neural cluster receiving stimulation of one-order harmonic, there is to appear distribution of the averaging number density of periodic single peck. After instantaneous stimulation 0.3s, we observe to appear to a new distribution of averaging number density as in figure 5 (c), and this new distribution is completely different

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57

with the initial distribution as in figure 5(a). This stimulation result reflects that the cluster of the most of neural oscillators in quantity can dominate dynamic behavior of system in cognitive procedure. Figure 5(d) describes that the distribution of neuronal activities after the resynchronization procedure of 18s under coupling action among neurons tends to the configuration in figure 5(b). The distribution of the desynchronization under action of couple in figure 5(c) restores again the stable state of the synchronization both the first phase and the second phase as in figure 5(b). This phenomenon shows that memorial state as stimulation effect is very stable after end of stimulation.

Spontaneous behavior 9s

(a)

(b) Stimulation time 0.3s

Spontaneous

behavior 18s (d)

(c)

Fig. 5. Quantitative ration of neurons N1:N2＝1： 10 (Initial condition and parameters are the same as Fig.3) Si (ψ , Ri ) = I i1 Ri 2 cos ψ + I i 2 Ri 2 cos( 2ψ ) , I11 = 0 I12 = 7 , I 21 = 7 , I 22 = 0 （ a） Initial

distribution of averaging number density of oscillators （ b） distribution after spontaneous behavior of 9s （ c） distribution after end of instantaneous stimulation 0.3s （ d） distribution after spontaneous behavior 18s after stimulation stop

There is a complete evolution process of on neural encoding in phase space in figure 5. These processes illustrated some biological phenomena, such as why we can identify a familiar sound from noisy environment, simultaneously, and noise of various frequencies from the environment does not disperse our attention. This is because stimulating effect with cognition and size of quantity of those neurons of acting as cognitive process are close relation, such as N1 cluster. In N2 cluster, coupling resonance with cognitive process can be produced from neural oscillators. And only small neurons act reaction to from noise in environment, this reaction does not reach the level of coupling resonance, such as neural activities in N1 cluster. Due to the encod-

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R. Wang, J. Yu, and Z.-k. Zhang

ing to external stimulation in N1 cluster to be not reached conscious level, this kind of neural activities only remained some faint response in cerebral cortex. The faint response will be not deposited in cortex by the dynamic mechanism of attention. But the memory as effect of attention was saved from observation of stimulation result in figure 5(d). Comparison with figure 5 (c) we observe that the size on wave in the first phase space becomes to very small in figure 5 (d), but that the size on wave in the second phase space becomes to very large, this phenomenon shows memorial effect on cognition.

5 Concluding Remarks In this paper we proposed a new dynamic model of phase resetting on population of neural oscillators possessing different phase, and performed stability and numerical analysis for spontaneous behavior of large-scale neural population. Meanwhile, we performed numerical analysis for the synchronization and the desynchronization activity of neurons in two case of symmetry (quantity of neural oscillator in two clusters are completely the same) and asymmetry (quantity of neural oscillators in two clusters are different). The numerical results firstly showed that early results on phase encoding of neuronal population in phase space of one dimension lost abundant useful neural information. Hence, theory of phase encoding in phase space of one dimension cannot be applied to research of cognitive process. And then the results of numerical computation showed that the neural dynamic model given in high-dimension phase space can deeply describe dynamic mechanism on attention and memory. Next we showed that cluster of the most of neural oscillators in quantity can dominate dynamic behavior of system in cognitive procedure in evolution procedure of distribution of the averaging number density. The conclusions above mentioned can explain these biological phenomena, i.e. why events noticed can be easily memorized, and memorial contents as stimulation effect are very stable. The conclusions can help us to comprehend mechanism of neurodynamics on attention and memory.

Acknowledgement The work reported in this paper was supported by the National Natural Science Foundation of China under Grant No. 30270339.

References 1. Wang, R.B., Zhang, Z.K.: Nonlinear stochastic models of neurons activities. Neurocomputing 51 (2003) 401-411 2. Wang, R.B., Zhang, Z.K.: A Dynamic Evolution Model for the Set of Populations of Neurons. Int. J. Nonlinear Sci. and Num. Simul. 4(3) (2003) 203-208 3. Wang, R.B., Yu, W., Jiao, X.F.: A Review of Application on Stochastic Dynamics in Brain Information Processing. Fudan Lecture in Neurobiology 307 (2004) 193-206 4. Wang, R.B., Jiao, X.F.: Stochastic Model and Neural Coding of Large-scale Neural Population with Variable Coupling Strength. Neurocomputing (2005) (in press)

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5. Jiao, X.F., Wang, R.B.: Nonlinear Evolution Model and Neuronal Coding of Neuronal Population with the Variable Coupling Strength in the Presence of External Stimuli. Appl. Phys. Lett. 87 (2005) 083901 6. Wang, R.B.: A New Nonlinear Phase Setting Models in Neurons Activities. In: Wang, L.P., Rajapakse, J.C. (eds.): Proceedings of 9th International Conference on Neural Information Processing, Vol. 5. Fudan University Press, Shanghai (2002) 2497-2501 7. Wang, R.B., Yu, W.: A Stochastic Nonlinear Evolution Model and Dynamic Neural Coding on Spontaneous Behavior of Large-scale Neuronal Population. In: Wang, L.P., Chen, K. (eds.): Advances in Natural Computation. Part 1. Lecture Notes in Computer Science, Vol. 3610. Springer-Verlag, Berlin Herdelberg (2005) 490-497 8. Wang, R.B., Yu, W.: Stochastic Nonlinear Evolutional Model of the Large-Scaled Neuronal Population and Dynamic Neural Coding Subject to Stimulation. J. Biol. Med. Eng. (2006) (in press) 9. Tass, P.A.: Phase Resetting in Medicine and Biology. Springer-Verlag, Berlin Herdelberg Germany (1999) 10. Wang, R.B.: Some Advances in Nonlinear Stochastic Evolution Models of Neuron Population. In: Zhu, W.Q. (eds.): Advance in Stochastic Structural Dynamics. CRC Press. USA (2003) 453-461

Approximation Bound of Mixture Networks in Lpω Spaces Zongben Xu, Jianjun Wang, and Deyu Meng Institute for Information and System Science, Xi’an Jiaotong University, Xi’an, Shaan’xi, 710049, P.R. China {zbxu, wangjianjun}@mail.xjtu.edu.cn

Abstract. The approximation order estimation problem for multidimensional functions by the mixture of experts neural networks is studied. It is shown that under very mild condition on activation functions, the mixture neural networks have the same approximation order with that of the normal feedforward sigmoid neural networks. The obtained result sharpens the estimation developed by Maiorov and Meir in IEEE Trans. on Neural Networks (9(1998),969-978) over the compact region in Lpω Spaces and underlies applicability of the mixture neural networks.

1

Introduction

Universal approximation capabilities of a broad range of neural networks have been established by many researchers (see, e.g., Cybenko [1], Mhaskar [5] and Chen [3]). The works that have been conducted mainly concentrate on the problem of denseness or feasibility of using feedword neural networks (FNNs) as universal approximation. But from application point of view, a more worthnoticing problem is what is the degree of approximation by FNNs for specific type of functions to be approximated. Recently, several authors have derived approximation bounds for the three layer of FNNs for many different classes of functions (see, e.g., [4]-[9]). The obtained results characterize quantitatively the approximation accuracy of the FNNs. In [10], Jordan and Jacobs have introduced a new type of neural networks called the mixture networks or the mixture of experts. Such type of neural networks, endowed with a certain probabilistic interpretation, consists of a finite number of sub-networks with each accomplishing a specific subtask, and requires the activations of the units in the hidden-layer to sum to unity. It is shown in [10] that the mixture networks have several distinct advantages over the normal FNNs, for instance, it is easier and more efficient in training and it has much stronger generalization capability. It is, however, an unclear problem whether or not such type networks can keep the same approximation accuracy with the normal FNNs. Maiorov and Meir ([6]) have studied this problem in a specific setting. They, through using the theory of weighted polynomial approximation, have developed an approxαr imation order estimation of the form O(n− d ) with 0 < α = 1 − β1 < 1 in the measure of Lpω , where the weight ω(x) = exp(−a|x|β ) with β ≥ 2 and J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 60–65, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Approximation Bound of Mixture Networks in Lpω Spaces

61

a > 0. The obtained approximation order is the same with that of FNNs in αr the setting of Lpω . This result, however, does not suggest that O(n− d ) is optimal in order for approximation of the mixture networks. In this paper, our aim is to provide a sharpened, nearly optimal approximation order estimation for the mixture networks though using a new weight in the measure of Lpω . The obtained result provides not only an affirmative answer to the approximation accuracy estimation problem, but also underlies the applicability of the mixture networks. We first briefly state the mixture network structure to be studied( For more details, we refer to [6], [10]). Consider the problem of modeling a regression functions E(y|x) in a probabilistic setting. A mixture network, or called a mixture of experts model, is composed of n expert networks, each of which solves a function approximation problem over a local region of the input space. The probability model of each expert is p(y|x; θj )(j = 1, 2, · · · , n), where x ∈ Rd is the input vector, y ∈ R is the output vector and θj ∈ Θ is parameter vector associated with each expert. Thus, the overall probabilistic model assumes the form of a mixture density n  p(y|x; Θ) = gj (x; θg )p(y|x; θj ), (1) j=1

where gj (x; θg ) ≥ 0

n 

and

gj (x; θg ) = 1.

(2)

j=1

The regression function E(y|x; Θ) is obtained by taking the expectation with the random variable y in (1), that is E(y|x; Θ) =

n 

gj (x; θj )μ(x; θj ).

(3)

j=1

where μ(x; θj ) = Ep(y|x;θj ) (y). In application, p(y|x; θj ) or equivalently, μ(x; θj ) and gj (x; θg ) assume specific forms. For the convergence or approximation order estimation purpose, however, one can assume μ(x; θj ) to be constant (for, gj (x; θj ) is a general nonlinear function in x and μ(x; θj ) can be absorbed into it whenever necessary ). With this understanding, the regression function defined by (3) can be equivalently defined as E(y|x; Θ) =

n 

cj gj (x; θj )

(4)

j=1

In [6], with sigmoid function φ(x) = (1 + e−x )−1 , gj (x; θj ) are taken in the form of φ(aTj x + bj ) gj (x; θj ) =  , j = 1, 2, ..., n n φ(aTj x + bj ) j=1

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Z. Xu, J. Wang, and D. Meng

Thus the regression functions defined by the mixture networks constitute the following family of functions: ⎧ ⎫ n  ⎪ ⎪ ⎪ ⎪ ck φ(ak · x + bk ) ⎨ ⎬ k=1 d Fn = fn (x) : fn (x) =  , a ∈ R , b , c ∈ R (5) k k k n ⎪ ⎪ ⎪ ⎪ φ(aj · x + bj ) ⎩ ⎭ j=1

In the subsequent study we do not limit the function φ to be sigmoid, but, instead, we assume that φ is a monotonic activation function such that there exists a constant b satisfying |φ(k) (b)| > c > 0 for k = 1, 2, · · · . From [2], any monotonic sigmoid function must have this property.

2

Notation and Main Results

Following [11], we will take the Chebyshev orthogonal weights in the measure of Lpω . That is, for any x = (x1 , x2 , . . . , xd ) ∈ [−1, 1]d , we take ω(x) = 1 d Πi=1 ω(xi ), ω(xi ) = (1 − x2i )− 2 . With so chosen weight function, we define a weighted norm of a function f by  1 f p,ω = ( | ω(x)f (x) |p dx) p , 1 ≤ p < ∞ (6) [−1,1]d

where m = (m1 , m2 , . . . , md ) ∈ Z d , dx = dx1 dx2 · · · dxd . We denote the class of functions for which f p,ω is finite by Lpω . This is a Banach space. The function class we wish to use the mixture neural networks to approximate is r,d Ψp,ω = {f : f (λ) p,ω ≤ M, |λ| ≤ r},

(7) |λ|

f where λ = (λ1 , λ2 , . . . , λd ), |λ| = λ1 + λ2 + · · · + λd , f (λ) = ∂x λ∂1 ···∂x λ , r is any 1 d d natural number, and M < ∞. For any two subsets F and G in Lpω , we define the distance from F to G by

dist(F, G, Lpω ) = sup inf f − gp,ω . f ∈F g∈G

and we define a class of multivariate polynomials by ⎧ ⎫ ⎨ ⎬  Pm = P : P (x) = bi1 ,i2 ,...,id xi11 · · · xidd , bi1 ,i2 ,...,id ∈ R, ∀i1 , . . . , id . ⎩ ⎭ 0≤|i|≤|m|

The main results of this work is the following theorem. Theorem. For any 1 ≤ p < ∞, there holds the following estimation: r,d −d dist(Ψp,ω , Fn , Lω p ) < Cn r

Here and hereafter C is a positive constant independent of n and x (Its value may be different in different contexts).

Approximation Bound of Mixture Networks in Lpω Spaces

63

r,d This theorem reveals two things: (i) for any multivariate functions f ∈ Ψp,ω , there is a mixture neural network of the form (4) that approximates f arbitrarily well in Lpω . That is, the mixture neural networks can be used as the universal r,d approximator of functions in Ψp,ω ; (ii) quantitatively, the approximation accur racy of a mixture network of the form (4) can attain the order of (n− d ), where d is the dimension of input space and r is the smoothness of the function to be approximated. This estimation is exactly the same with that for the normal FNNs proved in [11]. r Clearly, the approximation order estimation O(n− d ) developed here sharpens αr that provided in [6] in the sense that the parameter α in the estimation O(n− d ) in [6] has been dismissed. This shows that our estimation is more fundamental. Furthermore, we can show that in L2ω the approximation order estimation developed here is optimal. So it can be conjectured that the estimation presented in this work can not be improved further.

3

Proof of the Main Theorem

The proof of Theorem requires several lemmas, which we will present first. r,d Lemma 1. ( see [11] ). For any 1 ≤ p < ∞, f ∈ Ψp,ω , and m = (m1 , m2 , ..., md ) with mi ≤ m, there holds the following.

inf  f − p p,ω ≤ Cm−r .

p∈Pm

Lemma 2. The function ϕ(x) defined by  −d ϕ(x) = (2ρ) φ(t · x)dt,

0 < ρ < 1.

[−ρ,ρ]d

has the following properties: d d (i) |ϕ(x)| ≥ | i=1 φ(−ρ|xi |)| or |ϕ(x)| ≥ | i=1 φ(ρ|xi |)|; r,d (ii) Let s = (s1 , · · · , sd ), Then ∀f ∈ Ψp,ω ,   f ϕ p,ω =  Ds (f ϕ) p,ω ≤ C1 < ∞, |s|≤r r,d where C1 = C1 (r, ρ, d) < ∞, i.e.,f ϕ ∈ Ψp,ω .

Lemma 3. For the following class of functions Γn = {un (x)|un (x) =

 0≤k≤N

dk

pk (x) , ϕ(x)

d dk ∈ R, k ∈ Z+ },

1

where the polynomials pk (x) ∈ Pk ,and N = (N, N, · · · , N ) with N = n d , we have r p −d dist{Φr,d (8) p,ω , Γn , Lω )} ≤ Cn The Lemma 2 and Lemma 3 are obvious, here we omit the proof.

64

Z. Xu, J. Wang, and D. Meng

Lemma 4. Let s = (s1 , s2 , · · · , sd ) and s =

d

(1 + si ). Then there exists ar,k

i=1

such that for h sufficiently small 

ar,k φ(h(2r − s) · x)

p (x)

0≤r≤s

k

− −1 

ϕ(x) s φ(h(2r − s) · x)

≤ Mk CΩ(φ, h),

(9)

p,ω

0≤r≤s

where Mk is a constant dependent only on k, p, d, Ω(f, h) = sup|t|≤h |f (x + t) − f (x)| is the modulus of continuity of f . Proof. Let Ah (x, k) =



T

ar,k φ(h(2r − s) x),

Bh (x) = s−1

0≤r≤s

Since



T

φ(h(2r − s) x).

0≤r≤s



p (x) A (x, k)

p (x)

k

h k − ≤ (ϕ(x) − Bh (x))

ϕ(x)

ϕ(x)Bh (x)

Bh (x) p,ω p,ω

1

.

+ (pk (x)−Ah (x, k)) = I1 + I2

Bh (x)

p,ω

We will estimate the bound of I1 and I2 respectively. First,











−d −1 φ(t · x)dt − s φ(h(2r − s) · x)

ϕ(x) − Bh (x) = (2ρ)



d [−ρ,ρ] 0≤r≤s ∞ ∞

 

= s−1 (2h)−d {φ((t + h(2r − s)) · x) − φ(h(2r − s) · x)}dt

[0,2h]d ∞

0≤r≤s

≤ Ω(φ, t · x) ≤ 2d(1 + x∞ )Ω(φ, h).

(10)

By in [6], we can express the polynomials pk (x) as pk (x) = using jthe approach bj x , x ∈ Rd , and by M inkowski s inequality, we derive 0≤j≤k



p (x)

k

ϕ(x)Bh (x)

≤ p,ω

 0≤j≤k

d



j x 2 |bj | x (1 + e )

i=1

≤ Mk .

(11)

p,ω

From (10) and (11), we thus obtain I1 ≤ Mk CΩ(φ, h). For I2 , we can follow the proof of Theorem 2 in [11], to yield that for any  > 0, I2 ≤ . Since  is arbitrary, combining the estimation of I1 , I2 , and then gives Lemma 4. dk Ah (x,k)

Now we prove the Theorems. Consider the function fN,s,h (x) = 1 d

1 d

1 d

k≤N

Bh (x)

,

where N = (n , n , · · · , n ), k, r, s ∈ Clearly, fN,s,h(x) ∈ Fn . Using the triangle inequality, Lemma 3, M inkowski s inequality and Lemma 4, we have d Z+ . 

Approximation Bound of Mixture Networks in Lpω Spaces





f (x) − fN,s,h (x)

p,ω



  pk (x) pk (x)

≤ f (x) − dk dk ( −

+

ϕ(x) ϕ(x) k≤N k≤N p,ω  r ≤ Cn− d + CΩ(φ, h) Mk |dk |

 k≤N

dk Ah (x, k)

)

Bh (x)

65

p,ω

k≤N

Since the above equation holds  for any pk (x), we may take the minimum over r pk ∈ PN , set Ω(φ, h) ≤ n− d /C Mk |dk |, and then we get the theorems. k≤N

Acknowledgement This work was supported by National Science Foundation Nos.10371097 and 70531030.

References 1. Cybenko, G.: Approximation by Superpositions of a Sigmoidal Function. Math. Contr. Signals Syst. 2(1989) 303-314 2. Xu, Z.B., Cao, F.L.: Simultaneous Lp -Approximation Order for Neural Networks. Neural Networks 18(7) (2005) 914-923 3. Chen, T.P., Chen, H.: Approximation Capability to Functions of Several Variables, Nonlinear Functions, and Operators by Radial Function Neural Networks. IEEE Trans. Neural Networks 6(1995) 904-910 4. Barron, A.R.: Universal Approximation Bound for Superpositions of a Sigmoidal Function. IEEE Trans. Inform. Theory (39)(1993) 930-945 5. Mhaskar, H.N.: Neural Networks for Optimal Approximation for Smooth and Analytic Functions. Neural Comput. 8(1996) 164-177 6. Maiorov, V., Meir, R.S.: Approximation Bounds for Smooth Functions in C(Rd ) by Neural and Mixture Networks. IEEE Trans. Neural Networks 9 (1998) 969-978 7. Burger, M., Neubauer, A.: Error Bounds for Approximation with Neural Networks. J. Approx. Theory 112 (2001) 235-250 8. Kurkova, V., Sanguineti, M.: Comparison of Worst Case Errors in Linear and Neural Network Approximation. IEEE Trans. Inform.Theory 48(2002) 264-275 9. Wang, J.L., Sheng, B.H., Zhou, S.P.: On Approximation by Non-periodic Neural and Translation Networks in Lpω Spaces. ACTA Mathematica Sinica (in chinese) 46(2003) 65-74 10. Jordan, M.I., Jacobs, R.A.: Hierarchical Mixtures of Experts and the EM Algorithm. Neural Comput. 6 (1994) 181-214 11. Wang, J.J., Xu, Z.B., Xu, W.J.: Approximation Bounds by Neural Networks in Lpω . Lecture Notes in Computer Science, Vol.3173. Springer-Verlag, Berlin Heidelberg New York (2004) 1-6

Integral Transform and Its Application to Neural Network Approximation Feng-jun Li and Zongben Xu Faculty of Science, Institute for Information and System Science, Xi’an Jiaotong University, Xi’an,710049, P.R. China [email protected]

Abstract. Neural networks are widely used to approximate nonlinear functions. In order to study its approximation capability, a theorem of integral representation of functions is developed by using integral transform. Using the developed representation, an approximation order estimation for the bell-shaped neural networks is obtained. The obtained result reveals that the approximation accurately of the bell-shaped neural networks depends not only on the number of hidden neurons, but also on the smoothness of target functions.

1

Introduction

Many authors (see, e.g.,[1-5]) have concluded that a three-layered feed-forward neural network (FNN) can approximate an arbitrary function with desired accuracy whenever the FNN has sufficiently many hidden neurons. The approximation can be realized through many different ways like using Fourier transform and convolution. Recently, integral transformation has been applied to resolve this question. Murata ([6]) introduced an integral transformation of sigmoidal functions and proved the universal approximation capability of FNNs and give an approximation theorem (Theorem 2) when p = 2. In this work, we study the approximation bound estimation problem of neural networks. The aim of this is to reveal the quantitative relation between the approximation accuracy of a FNN and the number of its hidden neurons. We also prove an approximation theorem when 2 ≤ p < ∞. As the first step towards this goal, we focus in the present work on the case when the complete information about the target function is known (thus, some analytic tools like integral transformation can be applied). Throughout the paper we will use boldface characters to stand for vectors, and regular characters for scalars. In section 2 we introduce the integral transform and inverse transform of functions by using ridge functions. And then, in section 3, we apply the obtained approximate representation theorem to derive an approximation bound estimation of the bell-shaped three-layered FNNs. Finally, in section 4 we summarize our current research. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 66–71, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Integral Transform and Its Application

2

67

An Integral Transformation and Its Approximate Inverse

A ridge function (Martin and Allan [7]) is a multivariate function h : Rn −→ R of the form h(x1 , x2 , · · · , xn ) = g(a1 x1 + a2 x2 + · · · + an xn ) = g(a · x − b),

(1)

where g : R → R, b ∈ R, a = (a1 , a2 , · · · , an ) ∈ Rn \{0} and a · x is the inner product of a and x. That is to say, a ridge function is such a multivariate function that takes constant values on the parallel hyper-planes a · x = c, c ∈ R. The vector a ∈ Rn \{0} in Eq.(1) is generally called the direction. In the following, we discuss the possibility and method of approximating a given function f : Rn → R by a linear combination of ridge functions. We assume that the function f ∈ L1 (Rn )∩Lp (Rn ) (2 ≤ p < ∞). Two bounded functions ϕ, ψ ∈ L1 (R) ∩ L2 (R) are said to be a permitted pair whenever they satisfy ⎧ ˆ ˆ ˆ ψ(−y) = ϕ(y) ˆ ψ(y) ⎪ ⎨ ϕ(−y)  ∞ |ϕ(y) ˆ ˆ ψ(y)| dy < ∞ (2) 0 yn ⎪ ˆ ⎩  ∞ ϕ(y) ˆ ψ(y) dy = 0, yn 0 where “ˆ” denotes the Fourier transform and “ − ” denotes the conjugate. Let   ∞ ˆ ˆ ϕ(y) ˆ ψ(y) ϕ(y) ˆ ψ(y) C= dy = 2 dy. (3) n n |y | y R 0 Using the ridge function ϕ, an integral transform Wϕ of function f is introduced by Murata ([6]) and Widder ([8]) as follows:  1 (Wϕ f )(a, b)x = ϕ(a · x − b)f (x)dx. (4) (2π)n C Rn For the transformation Wϕ , we can prove the following estimation in terms of Lp —norm. Theorem 1. For any  > 0 and a real number δ, let f ∈ L1 (Rn ) ∩ Lp (Rn ) (2 ≤ p < ∞), if δ → 0, then there holds the following inequality:  2 ||f (x) − (Wϕ f )(a, b)x ψ(a · x − b)e−δ|a| dadb||p ≤ . (5) Rn+1

Proof. Let us define 1 F (x) = (2π)n C



ψ(a · x − b)ϕ(a · z − b)f (z)e−δ|a| dadbdz. 2

(6)

R2n+1

Since f (z) ∈ L1 (Rn ), ϕ is bounded, and ψ ∈ L1 (R), we can see that the right 2 n side of Eq.(6) is absolutely integrable. Let ρ(x, δ) = 1/(4πδ) 2 e−|x| /4δ , according

68

F.-j. Li and Z. Xu

to Fubini theorem, parseval’s formula and properties of convolution operation, we obtain  ∞ 2 ˆ 1 ∗ f (x)dw, F (x) = ϕ(w) ˆ ψ(w)ρ (7) n δ2 Cw 0 where “∗” denotes the convolution. Using H¨older inequality, we then get  ∞  ˆ 2 ˆ ψ(w) δ  ϕ(w)  ||F (x) − f (x)||p ≤  ||ρ(x, 2 ) ∗ f (x) − f (x)||p dw n |C| 0 w w  r  ˆ 2 ˆ ψ(w)  δ  ϕ(w) =  ||ρ(x, 2 ) ∗ f (x) − f (x)||p dw |C| 0 wn w  ∞  ˆ 2 ˆ ψ(w) δ  ϕ(w)  +  ||ρ(x, 2 ) ∗ f (x) − f (x)||p dw |C| r wn w  A1 + A2 .

(8)

Since lim ||ρ(x, δ) ∗ f (x) − f (x)||p = 0

(9)

δ→0

and ||ρ(x, δ) ∗ f (x)||p ≤ ||f (x)||p , holds in any case (Lu and Wang [9]), therefore, ||ρ(x, δ) ∗ f (x) − f (x)||p ≤ ||f (x)||p . In addition



∞

0

 ϕ(y) ˆ   ˆ ψ(y)  dy < ∞. yn

(10)

(11)

From Eqs.(10) and (11), we conclude that for any  > 0, we may choose r so that |A1 | < 2 . From Eqs. (9) and (11), we may also choose δ0 so that rδ2 < δ0 . Thus we can conclude |A2 | < 2 . This finishes the proof of Theorem 1.

3

Application to Neural Networks

By using the above approximate representation theorem, we can estimate approximation bound of some special type of feed-forward neural networks (FNNs). 3.1

Three-Layered FNNs with Bell-Shaped Activation Functions

A sigmoidal function σ with the properties ⎧ σ(x) → 1 as x → ∞ ⎪ ⎪ ⎨ σ(x) → 0 as x → −∞ 

σ (x) > 0 as x ∈ R ⎪ ⎪ ⎩  σ (x) → 0 as |x| → ∞

(12)

has been extensively used as the activation function of hidden neurons of a FNN, but it does not belong to L1 (R), so the previous approximate integral representation theorem can not be directly applied to the sigmoidal FNNs. Thus we

Integral Transform and Its Application

69

introduce a so-called bell-shaped activation function. A bell-shaped function ψ is  such a function that has the following properties: R ψ(x)dx < ∞, maxx ψ(x) = 1 and ψ(x) ≥ 0 as x ∈ R, and ψ(x) → 0, as |x| → ∞. It is clear that any bellshaped function is unmodal. In the following, we will consider the FNNs with bell-shaped functions as the activation functions of the hidden neurons. Such a general FNN can be expressed as N (x) =

m

ci ψ(ai ·x − bi ),

(13)

i=1

for any x ∈ Rn , where ai ∈ Rn , bi , ci ∈ R and m denotes the number of hidden neurons. According to Katsuyuki, Taichi and Naohiro ([10]), a bell-shaped function can be constructed from two sigmoidal functions. For example, ψ(x) = c(σ(x + d) − σ(x − d)), is a bell-shaped function, where c is a constant to normalize the maximum value and d is a positive constant. In this case, a bell-shaped hidden neuron can be constructed from two sigmoidal hidden neurons. So, in this sense, a bell-shaped FNNs can be explained as a sigmoidal FNN with twice as many as sigmoidal hidden neurons. 3.2

Function Approximation by Bell-Shaped FNNs

In this subsection, by using Theorem 1, we prove a theorem on approximation bound estimation of a bell-shaped FNNs. The obtained result will quantitatively characterize approximation capability of the bell-shaped FNNs to any integrable functions. We assume that input x is generated according to a probability density μ(x), and the approximation bound of a three-layered bell-shaped FNN, N (x), with m hidden neurons will be estimated in terms of Lpμ —norm. That is, we are going to estimate the error function defined by

  p1 ||N (x) − f (x)||p,μ = (N (x) − f (x))p μ(x)dx , (2 ≤ p < ∞), (14) Rn

where f is a target function to be approximated. Let  CW = |Re(Wϕ f )(a, b)x |dadb < ∞.

(15)

Rn+1

Our main result is the following theorem. Theorem 2. Assume that the absolute integral CW defined by (21) is bounded, and ψ is a bell-shaped function, ϕ is a function such that(ψ, ϕ) is a permitted pair. If a three layered FNN N (x) with ψ being their activation functions of the hidden neurons, then, for any input distribution μ(x), there holds the following approximation order estimation: 1 ||N (x) − f (x)||p,μ = O( √ ), (2 ≤ p < ∞), p m where“O” is independent of x, m, n, f, N .

(16)

70

F.-j. Li and Z. Xu

Proof. Define (a,b) = Sign(Re(Wϕ f )(a, b)x )CW , C

(17)

and 1 |Re(Wϕ f )(a, b)x )|. CW

(18)

|(Wϕ f )(a, b)x |dadb < ∞.

(19)

μ(a, b) = By the assumption, we obtain  Rn+1

Without loss of generality, we assume f and ψ are both real functions. From [6] we get  (a,b) ψ(a · x − b)μ(a, b)dadb. f (x) = C (20) Since μ(a, b) is positive and its integral over Rn+1 is equal to 1, μ(a, b) can be viewed as a probability density of x and b. (x) = 1 m C Let us consider the function defined by: N i=1 (ai ,b) ψ(ai ·x − bi ), m where (ai , bi ), (i = 1, 2, · · · , m) are independently chosen according to probability (x) − f (x) is a bounded density μ(a, b). From Eqs.(15)—(20), we know that N (x) − f (x)| ≤ M . function. So there exists a positive constant M , such that |N The expectation and variance of N (x) are given by ( Murata [6] ) (x)) = f (x), E(N (x)) ≤ 1 (C 2 − f (x)2 ). V (N m W

(21) (22)

Combining Eqs.(21) and (22), and make use of properties of expectation and variance, we thus have   (x) − f (x))p μ(x)dx = E[((N (x) − f (x))2 ) p2 ]μ(x)dx E( (N 

(x) − f (x) p N )2 ) 2 ]μ(x)dx M  (x) − f (x) N p ≤M E(( )2 )μ(x)dx M C1 2 ≤ (C − ||f ||22,μ ) m W C2 ≤ , (2 ≤ p < ∞), (23) m = Mp

E[((

where C1 and C2 are different positive constants. This finishes the proof of Theorem 2.

Integral Transform and Its Application

4

71

Conclusion

We have developed an approximate representation theorem of any integrable functions by using ridge functions. By using the developed representation theorem, we have established an approximation order estimation for the bell-shaped FNNs. The developed approximation order estimation reveals that the approximation accuracy of the bell-shaped neural networks is inversely proportional to the number of hidden neurons. Since it is known that the bell-shaped FNNs have specially important application in dynamical system simulation, the obtained result is of great significance in understanding and clarifying approximation capability of the bell-shaped FNNs.

Acknowledgement This work was supported by NSFC projects under contract Nos. 10371097 and 70531030.

References 1. Lewicki, G., Marino, D.: Approximation of Functions of Finite Variation by Superpositions of a Sigmoidal Function. Applied Mathematics Letters 17 (12)(2004) 1147–1152 2. Barron, A. B.: Universal Approximation Bounds for Superpositions of a Sigmoidal Function. IEEE Trans. Inform. Theory 3 (6)(1993) 930–945 3. Funahashi, K.: On the Approximate Realization of Continuous Mappings by Neural Networks. Neural Networks 2 (1)(1989 ) 183–192 4. Xin, L.: Simultaneous Approximations of Multivariate Functions and Their Derivatives by Neural Networks with One Hidden Layer. Neurocmputing 12 (8)(1996) 327–343 5. Makovoz, Y.: Uniform Approximation by Neural Networks. Journal of Approximation Theory 95 (11)(1998) 215–228 6. Murata, N.: An Integral Representation of Functions Using Three-layered Networks and Their Approximation Bounds . Neural Networks 9 (6)(1996) 947–956 7. Martin, D.B., Allan, P.: Identifying Linear Combination of Ridge Functions. Advances in Applied Mathematics 22 (1)(1999) 103–118 8. Widder,D.V.: An Intrduction to Transform Theoy. Academic Press, New York(1971) 9. Lu, S. Z., Wang, K. Y.: Real Analysis. Bei Jing Normal University Pess, Bei Jing( in chinese)(1997) 10. Katsuyuki, H., Taichi, H., Naohiro, T.: Upper Bound of the Expected Training Error of Neural Network Regression for a Gaussian Noise Sequence. Neural Networks 14 (10)(2001) 1419–1429

The Essential Approximation Order for Neural Networks with Trigonometric Hidden Layer Units Chunmei Ding1 , Feilong Cao1,2 , and Zongben Xu2 1

Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou, Zhejiang, 310018, P.R. China [email protected] 2 Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P.R. China [email protected]

Abstract. There have been various studies on approximation ability of feedforward neural networks. The existing studies are, however, only concerned with the density or upper bound estimation on how a multivariate function can be approximated by the networks, and consequently, the essential approximation ability of networks cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability of a class of feedforward neural networks with trigonometric hidden layer units is clarified in terms of the second order modulus of smoothness of approximated function.

1

Introduction

Function approximation is the most fundamental capability of feedforward neural networks (FNNs). Various studies on the density or feasibility of FNNs approximating continuous or integrable functions have been made in past years. Typical results can see [7], [9], [10], [17], [4] and [6]. Among those studies, it is shown in particular that any continuous multivariate function defined on a compact set of Rd can be approximated arbitrarily well by one hidden layer FNN of the form ⎛ ⎞ m d   N (x) = ci σi ⎝ wij xj + θi ⎠ , x ∈ Rd , d ≥ 1, (1) i=1

j=1

where for any 1 ≤ i ≤ m, θi ∈ R is the threshold, wi = (wi1 , wi2 , ..., wis )T ∈ Rd is the connection weight of neuron i in the hidden layer with the input neurons, and ci ∈ R is its connection weight with the output neuron, and σi (·) is the sigmoidal activation function. In (1), the number m has special importance: it determines the number of hidden units used and correspondingly specify the topology structure of the sigmoidal, one hidden layer FNN. In most of the existing studies, such number m is justified to be existed, finite and sufficiently large. It has, however, hardly been explicitly specified. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 72–79, 2006. c Springer-Verlag Berlin Heidelberg 2006 

The Essential Approximation Order for Neural Networks

73

In recent years, a quantitative study on the approximation order or approximation speed of the neural networks (1) has attracted much attention(see [1], [12], [15], [13], [5], [3] and [16]). In all these studies, some upper bound estimations on the approximation order of the neural networks were established. Such upper bound estimations can, on one side, imply the convergence of the neural networks to the approximated function, and, on the other side, provide quantitative estimations on how accurate the neural networks approximate the function. These estimations cannot, however, completely characterize the approximation capability of the neural networks in general, because an established upper bound estimation might be too loose to reflect their inherent approximation capability. Given an FNN, let us refer the highest approximation accuracy the FNN may achieve to as the essential approximation order of the FNN. Then, clearly, the inherent approximation capability of an FNN can be uniquely characterized by its essential approximation order. Thus to study the essential approximation order of an FNN is fundamental and important in clarifying an FNN’s inherent approximation capability. In the present paper, we will derive such an essential approximation order estimation for a special class of neural networks with trigonometric hidden layer units. Our approach will be based on establishing upper and lower bound estimations on the approximation order of the FNN simultaneously, and then derive the essential approximation order of the network when the upper and lower bounds become identical. We will take multivariate function approximation theory as tools in general and take the second order modulus of smoothness as the measures of approximation order in particular. Thus, for the FNNs, both their essential approximation order and the relationship between approximation speed and the number of hidden units are clarified. The paper is organized as follows. Some notations are introduced in section 2 below. Section 3 states the main results and their significations. In section 4, the proof of the main results is given by using some techniques of approximation theory.

2

Some Notations

Let N be the set of positive integral number, and R the set of real number. ith

Assume N0 = N ∪ {0} and 0 = (0, 0, . . . , 0), 1i = (0, 0, . . . , 0, 1 , 0 . . . , 0) ∈  1/2  d 2 N0d . Let |r| = di=1 |ri | for r = (r1 , r2 , . . . , rd ) ∈ N0d , t = for i=1 ti d d t = (t1 , t2 , . . . , td ) ∈ R , and rt = i=1 ri ti . Let Lp2π , 1 ≤ p < ∞, be the Banach space consisting of all p-th Lebesgue integrable functions with 2π periodic for each variable defined on Rd . We identify L∞ 2π = C2π with the space of continuous functions with 2π periodic for each variable defined on Rd . For f ∈ Lp2π , 1 ≤ p ≤ ∞, its norm is defined by



1/p π p |f (u)| du ,1≤p<∞ , −π f p = maxx∈[−π,π]d |f (x)| , p = ∞ and the r-th symmetric difference of f along direction h is given by

74

C. Ding, F. Cao, and Z. Xu (r)

Δh f (x) =

r     r r (−1)i f x + ( − i)h . i 2 i=0

(r)

In terms of Δh f (·), the r-th modulus of smoothness of f is then defined by    (r)  ωr (f, t)p = sup Δh f (·) . (2) p

0<h≤t

A function f is said to belong to Lipschitian function class with order up to r, r ∈ N , denoted by f ∈ Lip(α)r , if its r-th modulus of smoothness ωr (f, t)p satisfies ωr (f, t)p = O(tα ), with an α ∈ (0, r]. The modulus of smoothness of f defined by (2) could be regarded to as a direct multivariate extension of the univariate modulus of smoothness. Such modulus has some properties similar to univariate modulus of smoothness. For example, (1) (2) (3) (4)

ωr (f, t)p monotonically decreases to zero as t goes to zero; ωr (f, t + s)p ≤ ωr (f, t)p + ωr (f, s)p for any t, s ∈ (0, ∞); ωr (f, t)p ≤ 2r−s ωs (f, t)p for any s ≤ r; ωr (f, λt)p ≤ (λ + 1)r ωr (f, t)p for λ > 0. π 1 We denote by f ∗ g(x) = (2π) f (t)g(x − t)dt the convolution of f and d −π ∧ −irt g, and by f (r) =< f, e > the Fourier transformation of function f , where π 1 < f, g >= (2π) f (t)g(t)dt is the inner product of f and g. Furthermore, we d −π introduce a concept of r-th K-functional for a function f ∈ Lp2π , 1 ≤ p ≤ ∞, and δ > 0, r ∈ N , i.e.,

 Kr (f, δ r )p =

f − gp + δ r sup Dβ gp

inf

Dβ g∈Lp 2π

where |β| = β1 + β2 + · · ·+ βd ,

,

(3)

|β|=r

β = (β1 , β2 , . . . , βd ) ∈ N0d , and Dβ =

∂ |β| β β ∂x1 1 ···∂xd d

is differential operator. The theory of K-functional was first introduced by Peetre [14] and was further developed by Johnen and Scherer [11] and Ditzian and Totik [8]. It is usually used to measure the distance from a normed space to its density subspace. It was shown in [11] that the K-functional given by (2) is equivalent to the modulus of smoothness defined in (3), i.e., there are constants C1 and C2 , such that C1 ωr (f, δ)p ≤ Kr (f, δ r )p ≤ C2 ωr (f, δ)p . (4) In the following, we will give some notations used in [15]. We denote for λ ∈ N and each fi ∈ Lp2π , 1 ≤ p ≤ ∞, p = (p1 , p2 , . . . , pd ) ∈ N0d ,  Bλ =

2 λ+2

q = (q1 , q2 , . . . , qd ) ∈ N0d ,

d ,

bλ,r =

d  i=1

sin

ri + 1 π, λ+2

The Essential Approximation Order for Neural Networks

75

and αλ,p,q [fi ] = 2Bλ bλ,p bλ,q < fi , cos(p − q)t >, βλ,p,q [fi ] = 2Bλ bλ,p bλ,q < fi , sin(p − q)t >, θλ [fi ] = < fi , 1 >, With those notations, Suzuki [15] constructed the following three-layer networks with (2λ + 1)d − 1 trigonometric hidden layer units: T Nλ [fi ](x) = θλ [fi ] +

0≤p u ,qv ≤λ

Combinations of

{αλ,p,q [fi ] cos(p − q)x + βλ,p,q sin(p − q)x} p =q

where the summation is over combinations of p = (pu )du=1 and q = (qv )dv=1 ∈ N0d such that p = q, 0 ≤ pu , qv ≤ λ. That is, if it is added in the case of (p, q), it is not added in the case (q, p).

3

The Main Results

In [15], Suzuki obtained the estimation of upper bound on approximation error in terms of the first order modulus of smoothness, that is,     π2 √ 1 fi − T Nλ [fi ]p ≤ 1 + d ω1 fi , , fi ∈ Lp2π , 1 ≤ p ≤ ∞. (5) 2 λ+2 p This result is more important since it offers not only an estimation of upper bound on approximation order of the constructed networks T Nλ [fi ], but also bridges the gap of the approximation speed of the networks and the number of hidden units (2λ + 1)d − 1. The estimation cannot, however, completely characterize the capability of the constructed neural network in general, because such upper bound estimation might be too loose to reflect their inherent approximation capability. Furthermore, such upper bound estimation cannot to answer whether or not the first order modulus of smoothness estimated in (5) is the essential approximation order of the networks T Nλ [fi ]. In this paper, we first use the second order modulus of smoothness of function as a metric, and prove that the modulus can be used not only to estimate the upper bound of approximation error, but also can characterize the lower bound of the approximation speed. In fact, we will prove the following results. Theorem 3.1. For fi ∈ Lp2π , 1 ≤ p ≤ ∞, we have   2   1 d 1 2 fi − T Nλ [fi ]p ≤ 1+π ω2 fi , . 2 2 λ+2 p

(6)

The above theorem offers an upper bound estimation on approximation order of T Nλ [fi ], which describes clearly how approximation speed of the network is

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C. Ding, F. Cao, and Z. Xu

affected by the number of hidden units m, or equivalently, parameter λ (for m = (2λ + 1)d − 1 is positively proportional to λ). In particular, it shows that the approximation speed of the networks is proportionally upper controlled by the second order modulus of smoothness and how many hidden units are used. Whenever fi ∈ Lp2π , it then further shows that T Nλ [fi ] − fi p → 0 as λ (equivalently, m = (2λ + 1)d − 1) goes to infinity. That is, any continuous or integrable function with 2π periodic can be approximated arbitrarily well by T Nλ [fi ]. Now we give a lower bound estimation on the approximation speed for the networks. Theorem 3.2. If fi ∈ Lp2π , 1 ≤ p ≤ ∞, then  ω2 fi ,

1 λ+2

 p

≤ M λ−2

λ 

kT Nk [fi ] − fi p ,

(7)

k=1

here and in the following M is a positive constant independent of f and λ. Theorem 3.2 provides a lower bound estimation of approximation accuracy of T Nλ [fi ], which shows that the weighted arithmetic average of T Nλ [fi ] over pa rameters λ, λ12 λk=1 k T Nk [fi ] − fi p , is lower controlled by the second order modulus of smoothness of fi . It is noted that there holds always the following identity λ 1  lim T Nλ [fi ] − fi p = lim 2 k T Nk [fi ] − fi p . λ→∞ λ→∞ λ k=1

So, whenever λ (or, equivalently, the number of hidden units m = (2λ+1)d −1) is λ sufficiently large, T Nλ [fi ] − fi p and λ12 k=1 k T Nλ [fi ] − fi p can be viewed approximately as the same. In such case, the statements (6) and (7) then imply     1 1 C2 ω2 fi , ≤ T Nλ [fi ] − fi p ≤ C1 ω2 fi , , λ+2 p λ+2 p where the positive constants C1 and C2 are independent of fi and λ. The following theorem reflects the relation between the speed of approximation of the networks and the constructive properties of approximated function. Theorem 3.3. Suppose fi ∈ Lp2π , 1 ≤ p ≤ ∞, then   T Nλ [fi ] − fi p = O λ−α , 0 < α < 2, if and only if fi ∈ Lip(α)2 . This theorem shows particularly that when the functions to be approximated are Lipschitzian with order up to 2 (i.e., fi ∈ Lip(α)2 , 0 < α < 2), the approximation speed of T Nλ [fi ] is determined by the second order modulus of smoothness of the function fi . So, for the given activation function and number of units in the hidden layer of the networks T Nλ [fi ], the better smoothness the approximated function holds, the faster approximation speed of the networks is. Inversely,

The Essential Approximation Order for Neural Networks

77

in order to get faster approximation speed of the networks approximating a function, the approximated function must have better smoothness. This reveals that the smoothness of the approximated function essentially determines the approximation speed of the networks.

4

The Proof of Main Results

Now we are in position to prove our main results stated in the section 3. We first prove Theorem 3.1. Let r = (r1 , r2 , . . . , rd ) ∈ N0d , λ ∈ N . The d-dimensional F´ejer-Korovkin  2   kernel Kλ is defined by Kλ (t) = Bλ  0≤ri ≤λ bλ,r eirt  , where bλ,r

⎛

d 

ri + 1 = sin π, λ+2 i=1

Bλ = ⎝

⎞−1



(bλ,r )2 ⎠

,

0≤ri ≤λ

then(see [15]) ⎛ Bλ = ⎝



0≤ri ≤λ

and



d 

ri + 1 sin π λ+2 i=1

2 ⎞−1 ⎠

 =

2 λ+2

d ,

 2   0≤p u ,qv ≤λ    irt   Kλ (t) = Bλ  bλ,r e  = 1 + 2Bλ bλ,p bλ,q cos(p − q)t. 0≤ri ≤λ  p =q∈N d 0

Recalling the convolution Kλ ∗ fi (x) =

1 (2π)d



π −π

Kλ (t)fi (x − t)dt,

gives(see [15]) T Nλ [F ] = (T Nλ [f1 ], T Nλ [f2 ], . . . , T Nλ [fd ]) = (Kλ ∗ f1 , Kλ ∗ f2 , . . . , Kλ ∗ fd ). From the expression of Kλ (t) and the fact that Kλ (t) is an odd function for t, it follows that  π 1 1 Kλ ∗ fi (x) = (fi (x + t) + fi (x − t)) Kλ (t)dt. (2π)d −π 2 On the other hand, by calculation we obtain(see also [15]) Kλ∧ (0) = 1,

Kλ∧ (r) = Bλ

0≤p u ,qv ≤λ

bλ,p bλ,q

p−q=r,p,q∈N0d π and Kλ∧ (1i ) = cos λ+2 . Therefore, by the properties of the modulus of smoothness, we find

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C. Ding, F. Cao, and Z. Xu

   π  1  1  T Nλ [fi ] − fi p =  (fi (· + t) + fi (· − t) − 2fi (·)) Kλ (t)dt  d (2π) −π 2 p  π  π  2 1 1 1 1 ≤ Kλ (t)ω2 (fi , t)p dt ≤ ω2 (fi , δ)p Kλ (t) 1 + δ −1 t dt d d (2π) −π 2 (2π) −π 2 ⎛   1/2 d  1 2 1  π 2 ≤ ω(fi , δ) ⎝1 + t Kλ (t)dt 2 δ (2π)d i=1 −π k  d  1 1  π 2 + 2 tk Kλ (t)dt . δ (2π)d −π k=1

Since

t t ≤ π sin , 2

we see

0 ≤ t ≤ π;

π sin

t t2 ≤ π 2 sin2 , 2

t ≤ t, 2

−π ≤ t ≤ 0,

−π ≤ t ≤ π.

Consequently,    π  π 1 1 π 2 2 2 2 t Kλ (t)dt ≤ π sin tk Kλ (t)dt = π 1 − cos . (2π)d −π k (2π)d λ+2 −π Finally, we take δ = to find

1 λ+2

1 T Nλ [fi ] − fi p ≤ 2



d

and recall

k=1

π 1 − cos λ+2 ≤

d 2



π λ+2

2 , it is easy



 2   d 1 2 1+π ω2 fi , . 2 λ+2 p

This completes the proof of Theorem 3.1. The proof of Theorem 3.2 is similar to [2], we omit the detail. Now, we prove Theorem fi ∈ Lip(α)2 , from Theorem 2.1 we obtain  α 3.3. If −α T Nλ [fi ] − fi p ≤ M λ1 = O(λ ) and, inversely, whenever the above estimation holds, we obtain from Theorem 3.2  ω2 fi ,

1 λ+2

 p

≤ M λ−2

λ 

k 1−α ≤ M (λ + 2)−α

k=1

1 2 Since, for any t ∈ (0, 1) there always is an integer λ such that λ+2 ≤ t ≤ λ+2 , the above estimation implies fi ∈ Lip(α)2 . This shows that the necessary and sufficient condition for T Nλ[fi ]−fi p = O(λ−α ) is fi ∈ Lip(α)2 . This completes the proof of Theorem 3.3.

Acknowledgement This work was supported by the Nature Science Foundation of China (60473034) and China Postdoctoral Science Foundation (2004035225).

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References 1. Barron, A.R.: Universial Approximation Bounds for Superpositions of a Sigmodial Function. IEEE Trans. Inform. Theory 39(1993)930-945 2. Cao, F.L., Xiong, J.Y.: Steckin-Marchaud-Type Inequality in Connection with Lp Approximation for Multivariate Bernstein-Durrmeyer Operators. Chinese Contemporary Mathematics 22(2)(2001)137-142 3. Cao, F.L., Li, Y.M., Xu Z.B.: Pointwise Approximation for Neural Networks. In: Wang, J., Liao, X.F., Zhang,Y.(eds): Advance in Neural Networks-ISNN 2005. Lecture Notes in Computer Science, Vol.3496. Springer-Verlag, Berlin Heidelberg New York(2005)39-44 4. Chen, T.P., Chen, H.: Universal Approximation to Nonlinear Operators by Neural Networks with Arbitrary Activation Functions and Its Application to Dynamical System. IEEE Trans. Neural Networks 6(1995)911-917 5. Chen, X.H., White, H.: Improved Rates and Asymptotic Normality for Nonparametric Neural Network Estimators. IEEE Trans. Inform. Theory 45(1999)682-691 6. Chui, C.K., Li, X.: Approximation by Ridge Functions and Neural Networks with One Hidden Layer. J. Approx. Theory 70(1992)131-141 7. Cybenko, G.: Approximation by Superpositions of Sigmoidal Function. Math. of Control Signals, and System 2(1989)303-314 8. Ditzian, Z., Totik, V.: Moduli of Smoothness, Springer-Verlag, Berlin Heidelberg New York(1987) 9. Funahashi, K.I.: On the Approximate Realization of Continuous Mappings by Neural Networks. Neural Networks 2(1989)183-192 10. Hornik, K., Stinchombe, M., White, H.: Multilayer Feedforward Networks are Universal Approximation. Neural Networks 2(1989)359-366 11. Johnen, H., Scherer, K.: On the Equivalence of the K-Functional and the Moduli of Continuity and Some Applications. In: Schempp, W., Zeller, K. (eds.): Constructive Theory of Function of Several Variables. Lecture Notes in Mathematics, Vol. 571, Springer-Verlag, Berlin Heidelberg New York(1977)119-140 12. K˚ urkova, V., Kainen, P.C., Kreinovich, V.: Estimates of the Number of Hidden Units and Variation with Respect to Half-Space. Neural Networks 10(1997)10681078 13. Maiorov, V., Meir,R.S.: Approximation Bounds for Smooth Functions in C(Rd ) by Neural and Mixture Networks. IEEE Trans. Neural Networks 9(1998)969-978 14. Peetre, J.: On the Connection Between the Theory of Interpolation Spaces and Approximation Theory. In: Alexits, G., Stechkin, S.B.(eds): Proc. Conf. Construction of Function. Budapest(1969)351-363 15. Suzuki, Shin: Constructive Function Approximation by Three-Layer Artificial Neural Networks. Neural Networks 11(1998) 1049-1058 16. Xu, Z.B., Cao, F.L.: Simultaneous Lp -Approximation Order for Neural Networks. Neural Networks 18(2005)914-923 17. Ito, Y.: Approximation of Functions on a Compact Set by Finite Sums of Sigmoid Function without Scaling. Neural Networks 4(1991)817-826

Wavelets Based Neural Network for Function Approximation Yong Fang1 and Tommy W.S. Chow2 1

School of Communication and Information Engineering, Shanghai University, Shanghai, 200072, China [email protected] 2 Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China [email protected]

Abstract. In this paper, a new type of WNN is proposed to enhance the function approximation capability. In the proposed WNN, the nonlinear activation function is a linear combination of wavelets, that can be updated during the networks training process. As a result the approximate error is significantly decreased. The BP algorithm and the QR decomposition based training method for the proposed WNN is derived. The obtained results indicate that this new type of WNN exhibits excellent learning ability compared to the conventional ones.

1 Introduction The approximation of a general continuous function by neural network (NN) has been widely studied because of its outstanding capability of fitting nonlinear models for input/output data. A three-layer NN is usually represented by the following finite sums of the form: N

g ( x ) = ∑ wi σ ( aiT x + bi ) ,

(1)

i =1

where wi , bi ∈ R , ai ∈ R n , σ (⋅) is a given function from R to R , x ∈ R n is the input vector. It has been proved that the output, g ( x ) , is dense in the space of continuous function defined on [0,1]n if σ (⋅) is a continuous, discriminating function. Generally, σ (⋅) is adopted as a sigmoid function that is discriminatory. Because wavelet decomposition has been emerged as a new powerful tool for representing nonlinearity, a class of network combining wavelets and neural networks have recently been investigated [1-10]. It has shown that this class of wavelet networks can provide better function approximation ability than ordinary basis function networks. It is noticed that the development of the above WNNs are all theoretically based upon the wavelet frame theory given by Daubechies [4] in one-dimensional (1-D) case and generalized by Kugarajah and Zhang [7] in a multi-dimensional (M-D) case. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 80 – 85, 2006. © Springer-Verlag Berlin Heidelberg 2006

Wavelets Based Neural Network for Function Approximation

81

A rather complex network architecture is inevitably resulted when the construction of an MWNN is solely in accordance with the wavelet frame theory. For practical applications, a relative large network size is often required. In this paper, we study the possibility of using a 1-D wavelet function for the development of an MWNN. Based upon the theories of approximation of nonlinear functions using neural networks [8], our proposed WNN theorems are derived in accordance with the theories that the sigmoid function can be replaced by any continuous or discontinuous function [8], if certain conditions are satisfied (the wavelet function satisfy these conditions too) [8]. Following the development of this new MWNN, we based upon the discrete wavelet transforms to further develop another new type of MWNN, called MWNN-DWT. The activation function of the MWNNDWT is a linear combination of wavelet bases rather than the sigmoid function or the wavelet function.

2 Wavelets and Wavelet Neural Networks (WNN) Wavelets are functions whose dilations and translations form a frame of L2 ( R) . That is, for some a > 0, b > 0 , the family ψ lk ( x ) = a − l / 2ψ ( a − l x − bk ) , for l , k ∈ Z

(2)

satisfies the frame property [4]. The sufficient conditions of wavelet frames were fiven in [4]. For instance, the “Morlet” wavelet with a = 2,b = 1 can build a frame of L2 ( R) . g (x ) =

Hence, the collection of all linear combination of elements of the frame

∑ wlkψ lk ( x )

(where I ⊂ Z 2 ) is dense in L2 ( R) . This implies that for any

( l ,k )∈Ι

f ∈ L2 ( R) and ε > 0 , there exists a positive integer N and constants such that N

f ( x ) − ∑ wiψ ( a i x + bi ) < ε .

(3)

i =1

Pati and Krishnaprasad [3] connected the wavelet with NN by applying Daubechies’ results [4]. They proposed the following network form g( x) =

∑ wlkψ lk ( x ) ,

(4)

( l ,k )∈Ι

where the index set Ι is the integer translation and the integer dilation, which are determined by using the time-frequency localization properties under a given accuracy [3-4], [6]. Kugarajah and Zhang [7] firstly built the M-D wavelet frames by a single mother wavelet as following form ψ l , k ( x ) = a − nl / 2ψ ( a − l x − bk ) for l ∈ Z , k ∈ Z n

(5)

where ψ ( x ) ∈ L2 ( R n ) , x ∈ R n , a, b ∈ R and a > 1 . Also, they gave the sufficient conditions of wavelet frames by generalizing Daubechies’ theorem [4]. In the sequence, l the dilation index l is a scalar and the scalar dilation parameter a is shared by all the dimensional of a wavelet. They proposed a methodology to construct a M-D wavelet function leading to frames [7], but not all 1-D wavelet can be extended as a M-D

82

Y. Fang and T.W.S. Chow

wavelet. The special conditions can be found in [7]. Zhang and Benveniste [1], and Kugarajah and Zhang [7] respectively gave the M-D wavelet frames in the following multi-scaling forms: ψ l , k ( x ) = (det D j )1 / 2ψ ( D j x − bk ) for j, k ∈ Z n ,

(6)

ψ l ,k ( x ) = det D 1j / 2ψ ( D j x − Tk ) for j, k ∈ Z n ,

(7)

where D j = diag ( a j1 ,L , a jn ) , j = ( j1 ,L j n ) T ∈ Z n and T = diag (b1 , L, bn ) , a > 1, bi > 0, i = 1, L , n , b > 0 . They proved that if a 1-D wavelet function ψ (x ) can consti-

tute frames, the tensor product of the 1-D wavelet function can also constitute frames. Zhang and Benveniste [1] presented a network structure in the form of −

N

g ( x ) = ∑ wiψ [ Di Ri ( x − t i )] + g ,

(8)

i =1

where Ri are rotation matrices, and g is introduced to deal with nonzero mean functions on finite domains.

3 MWNN Based on Discrete Wavelet Transform (MWNN-DWT) In the MWNN described in (12), the activation function is a 1-D mother wavelet function. As it is well known that the dilations and translations of wavelet can provide a good representation of nonlinearity [4],[6], a 1-D dilations and translations for the wavelet in the MWNN is then developed to enhance the approximation capability of the network. In this section we derive a new type of MWNN, called MWNN-DWT, ∧

ψ

ψ 00

b1

........ ψ lk

+

w00 wlk

−

+

g

a1

w1

x

+

........................................ aN

bN

y

wN ψ 00

+

w00

..........

+

wlk

ψ lk

Fig. 1. The architecture of the MWNN

which is based upon the discrete wavelet transform. The output, g (x ) , of MWNNDWT depicted in Fig. 1 is represented by the following form: N

∧

−

g ( x ) = ∑ wi ψ ( aiT x + bi ) + g ,

(9)

∑ wlkψ (a l x − kb) , {a l / 2ψ (a l x − kb) l , k ∈ Z }

is a frame for the L2 ( R) and

i =1

where ψˆ ( x ) =

( l ,k )∈Ι

Ι is the index set of pairs (l,k) of integer translation and integer dilation. The parame-

Wavelets Based Neural Network for Function Approximation

83

ter g is introduced so that the approximation of functions with nonzero average is possible [1]. It is clear that the activation function in MWNN-DWT (9) is expressed in a linear combination of the dilating and translating wavelets. Through adaptively adjusting the 1-D wavelet frames, the MWNN-DWT is capable of providing an excellent approximation capability. The (13) can be rewritten as the following two equivalent forms: N

g ( x ) = ∑ wi i =1

g( x) =

∑

−

∑ wlkψ [a l (aiT x + bi ) − kb] + g N

( l ,k )∈Ι

(10)

( l ,k )∈Ι

−

wlk ∑ wiψ [a l (aiT x + bi ) − kb ] + g .

(11)

i =1

To demonstrate the approximate capability of the MWNN-DWT, it is applied to approximate the function f ( x ) = 0.5e − x sin(6 x ) over the domain [-1,1]. The parameter g was initialized by the mean of the function observations, and other parameters were simply randomized between -0.5 and 0.5. It should also be noted that a rather special procedure in initializing the parameters is used in [1], while the parameters of the proposed MWNN-DWT is simply randomized in a general fashion. In order to compare the approximation capability of the networks described in (8) and (11), the initial parameters of these networks are all similarly randomized. Note that no rotation parameter is required for the wavelet network (8) in a 1-D case [1]. The training set consist of 100 points uniformly sampled in f (x ) . The function approximation performance of an MWNN-DWT in (11) with 5 neurons and 9 wavelet coefficients is compared to an WNN in (8) with 8 wavelons. In both networks, 25 parameters are used and trained by the standard BP algorithm. Fig.2 is the total squared errors over 2000 iterations, where the solid line shows the error of the MWNN-DWT and the dashed line represents that of the WNN. Clearly, the proposed MWNN-DWT provides much better results compared to that of the WNN. It is noted that the parameters of the MWNN-DWT consists of two parts: the weights of the network and the coefficients of the wavelets. In order to determine the optimal index set I and to speed up the convergence of the MWNN-DWT, we divide the training process into two stages. Firstly, let w00 = 1, wlk = 0 , l , k ≠ 0 and apply the BP algorithm for the network training. In this case, ψˆ ( x ) = ψ ( x ) , the network is the same as the MWNN in (13) but less parameters is required. In this training stage, the activation function is fixed and the standard BP algorithm is used. After a number of iterations and the network converges to a specified level of error, all the parameters are fixed. Therefore, the (11) can be expressed as g( x) =

∑ wlk Glk ( x ) + g ,

(12)

( l ,k )∈I N

where Glk ( x ) = ∑ wiψ [a l ( aiT x + bi ) − kb] . The second training stage is to determine i =1

the coefficients of the wavelets wlk . As wlk appears to be linear in the output equa-

84

Y. Fang and T.W.S. Chow

tion (12) of the network, most optimization techniques can be used to minimize the measure function if the index set is given. In this paper, the QR decomposition is used to adjust the activation function according to the form in (16). This process is equivalent to solving a least squares minimization problem for a given training set TP = {( x i , f ( x i )}iP=1 . By choosing an order of the index set the equation (12) can be written as F = GW ,

(13)

where F = ( f ( x1 ),L , f ( x P )) T is a column vector, G = (L , Gˆ lk , L ,1) is the P×# ( I ) matrix, #(I) denotes the number of the elements of the index set I, Gˆ lk = (Glk ( x1 ),L, Glk ( x P )) T , and W is the coefficient vector which needs to be determined. To obtain the matrix G, search the Gˆ lk in accordance with the order: l, k = 0,± 1,± 2, L . If the rank (G , Gˆ lk ) > rankG and max j Glk ( x j ) > ε , then G ⇐ (G , Gˆ lk ,1) ,

where G = (G ,1) and the given ε is a very small number. Following the above procedures, QR decomposition is used to determine the coefficients and the averaged W. The weights can then evaluated. Obviously, the overall convergence rate is substantially speeded up. In the above example, it required 5000 iterations to converge to a total squared error of 0.0112. In order to speed up the rate of convergence and to enhance the approximation capability of our network, the proposed training scheme was used for the function f ( x ) = 0.5e − x sin(6 x ) . The proposed network with 5 hidden units was firstly trained by BP algorithm. After 100 iterations, we obtain the matrix G with 12 columns, which implies that there are 12 coefficients of wavelets required to be determined by QP decomposition. Our results show that a total squared error of 7.9954 × 10 −5 was obtained by only 100 training iterations together with the QR decomposition. The computational time required by this training scheme is substantially reduced compared to other MWNN training schemes. For the case of a 2-D function given in above section: f ( x, y ) = sin( x ) / x sin( y ) / y , an MWNN-DWT with 50 hidden units was trained by the proposed training scheme. In this example, only 400 training iterations (with BP algorithm) together with the QR decomposition was required to provide a total squared error of 0.1688. Fig. 3 and Fig. 4 show the original function and approximation results respectively. Total Squared Error

1 1 .5 9 .5 7 .5 5 .5 3 .5 1 .5 -0 .5 0

400

800

1200 1600 2000 N u m b e r o f Ite ra tio n s

Fig. 2. The total squared errors of the WNN (8) and the proposed network for first function

Wavelets Based Neural Network for Function Approximation

1 .0 0

1 .0 0

0 .8 0

0 .8 0

0 .6 0

85

0 .6 0

0 .4 0

10 6

0 .2 0 2 0 .0 0

-2 -6

-0 .2 0 10

6

2

-2

-6

-10 -1 0

Fig. 3. Original 2-D function

0 .4 0

10 6

0 .2 0 2 0 .0 0

-2 -6

-0 .2 0 10

6

2

-2

-6

-1 0 -1 0

Fig. 4. Resulting approximation

4 Conclusion In this paper, we use Discrete Wavelet Transform to extend the MWNN to a new type of network called, MWNN-DWT. This enables us to minimize the measure function by adjusting the wavelet bases activation function. Our results indicate that the proposed MWNN-DWT can deliver an enhanced function approximation capability. The proposed training algorithm, which is based on the standard BP algorithm and QR decomposition, has an outstanding convergence rate compared to other MWNN training algorithms.

Acknowledgement This work is supported by the National Natural Science Foundation of China (60472103， Shanghai Excellent Academic Leader Project (05XP14027)， and Shanghai Leading Academic Discipline Project (T0102).

References 1. Zhang, Q., Benveniste, A.: Wavelet Networks. IEEE Trans. on Neural Networks 3(1992) 889-898 2. Yamakawa, T., Uchino, E., Samatsu, T.: Wavelet Neural Networks Employing Overcomplete Number of Compactly Supported Non-orthogonal Wavelete and Their Applications. IEEE Int. Conf. on Neural Network 1(1994) 1391-1396 3. Pati, Y. C., Krishnaprasad, P.S.: Analysis and Synthesis of Feedforward Neural Networks Using Discrete Affine Wavelet Transformations. IEEE Trans. on Neural Networks 4(1993) 73-85 4. Daubechies, I.: The Wavelet Transform, Time-frequency Localization and Signal Analysis. IEEE Trans. on Information Theory 36(1990) 961-1005 5. Zhang, J., Walter, G.G., Miao, Y., Wayne, W.N.: Zhang, Q., Benveniste, A.: Wavelet Neural Networks for Function Learning. IEEE Trans. on Signal Processing 43(195) 1485-1496 6. Daubechies, I, Ten Lectures on Wavelets. Philadelphia, PA: SIAM Press(1992) 7. Kugarajah, T. Zhang, Q.: Multidimensional Wavelet Frames. IEEE Trans. on Neural Networks 6(19995) 1552-1556 8. Chen, T., Chen, H.: Approximarions of Continuous Functionals by Neural Networks with Application to Dynamic System. IEEE Trans. on Neural Networks 4(1993) 910-918 9. Chow, T.W.S., Fang, Y.: Two-dimensional Learning Strategy for Multiplayer Feedforward Neural Network. Neurcomputing 34(2000) 195-206

Passivity Analysis of Dynamic Neural Networks with Different Time-Scales Alejandro Cruz Sandoval and Wen Yu Departamento de Control Automtico, CINVESTAV-IPN, A.P. 14-740, Av. IPN 2508, Mxico D.F., 07360, M´exico [email protected]

Abstract. Dynamic neural networks with different time-scales include the aspects of fast and slow phenomenons. Some applications require that the equilibrium points of the designed network be stable. In this paper, the passivity-based approach is used to derive stability conditions for dynamic neural networks with different time-scales. Several stability properties, such as passivity, asymptotic stability, input-to-state stability and bounded input bounded output stability, are guaranteed in certain senses. Numerical examples are also given to demonstrate the effectiveness of the theoretical results.

1

Introduction

A wide class of physical systems in engineering contains both slow and fast dynamic phenomena occurring in separate time-scales. Resent results show that neural network techniques seem to be very effective to model a wide class of complex nonlinear systems with different time-scales when we have no complete model information, or even when we consider the plant as a ”black box” [4]. This model-free approach uses the nice features of neural networks, but the lack of a model for the controlled plant makes it hard to obtain theoretical results on the stability. Some of neural networks applications, such as patterns storage and solving optimization problem, require that the equilibrium points of the designed network be stable [5]. So, it is important to study the stability of neural networks. Dynamic neural networks with different time-scales can model the dynamics of the short-term memory (neural activity levels) and the long-term memory (dynamics of unsupervised synaptic modifications). Their capability of storing patterns as stable equilibrium points requires stability criteria which includes the mutual interference between neuron and learning dynamics. The dynamics of dynamic neural networks with different time-scales are extremely complex, exhibiting convergence point attractors and periodic attractors [1]. Networks where both short-term and long-term memory are dynamic variables cannot be placed in the form of the Cohen-Grossberg equations [3]. However, a large class of competitive systems have been identified as being ”generally” convergent to point attractors even though no Lyapunov functions have found for their flows. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 86–92, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Passivity Analysis of Dynamic Neural Networks with Different Time-Scales

87

There are not many results on the stability analysis of neural networks in spite of their successful applications. The global asymptotic stability (GAS) of dynamic neural networks has been developed during the last decade. Negative semi-definiteness of the interconnection matrix may make Hopfield-Tank neuro circuit GAS [2]; The stability of neuro circuits was established by the concept of diagonal stability [6]. By the frameworks of the Lur’e systems, the absolute stability of multilayer perceptrons (MLP) and recurrent neural networks were proposed in [13] and [7]. Input-to-state stability (ISS) analysis method is an effective tool for dynamic neural networks, and in [14] it is stated that if the weights are small enough, neural networks are ISS and GAS with zero input. Stability of identification and tracking errors with neural networks was also investigated. [4] and [9] studied the stability conditions when multilayer perceptrons are used to identify and control a nonlinear system. Lyapunov-like analysis is a popular tool to prove the stability. [10] and [14] discussed the stability of signal-layer dynamic neural networks. For the case of high-order networks and multilayer networks the stability results may be found in [7] and [10]. Passivity theory is another effective tool to analyze the stability of a nonlinear system. It may deal with nonlinear systems using only the general characteristics of the input-output dynamics, and offers elegant solutions for the proof of absolute stability. Passivity framework is a promising approach to the stability analysis of neural networks, because it can lead to general conclusions on the stability using only input-output characteristics. Passivity properties of MLP were examined in [9]. By means of analyzing the interconnection of error models, they derived the relationship between passivity and closed-loop stability. To the best of our knowledge, open loop analysis based on the passivity method for dynamic neural networks with different time-scales has not yet been established in the literature. In this paper we apply passivity techniques for two time-scales neural networks, which have two types of states variables (short-term and long-term memory), describing the slow and fast dynamics of the systems. We prove that a gradient-like learning law will make the dynamic neural network with different time-scales stability by passivity method. With additional conditions, the neural networks are global asymptotic stability (GAS) and input-to-state stability (ISS).

2

Stability Properties of Dynamic Neural Networks with Different Time-Scales

A general dynamic neural network with two time-scales can be expressed as ·

T

T

x = Ax + W1 σ 1 (V1 [x, z] ) + W3 φ1 (V3 [x, z] )u · T T z = Bz + W2 σ 2 (V2 [x, z] ) + W4 φ2 (V4 [x, z] )u

(1)

where x ∈ Rn and z ∈ Rn are slow and fast states, , Wi ∈ Rn×2n (i = 1 · · · 4) are the weights in output layers, Vi ∈ R2n×2n (i = 1 · · · 4) are the weights in hidT den layers, σ k = [σ k (x1 ) · · · σ k (xn ) , σ k (z1 ) · · · σ k (zn )] ∈ R2n (k = 1, 2), φ(·) ∈

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A xt

³

V3

V4 zt

³

V1

V1

W1

I1

W3

u

I2 V2

¦

W4

u

V2

W2

¦

1

H

B

Fig. 1. Dynamic neural network with two time-scales

R2n×2n is diagonal matrix,φk (x, z) = diag [φk (x1 ) · · · φk (xn ) , φk (z1 ) · · · φk (zn )] (k = 1, 2), u(k) = [u1 , u2 · · · um , 0, · · · 0]T ∈ R2n . A ∈ Rn×n , B ∈ Rn×n are stable matrices (Hurwitz).  is a small positive constant. The structure of the dynamic neural networks (2) is shown in Fig. 1. When  = 0, the dynamic neural networks (1) have been discussed by many authors, for example [7], [10] and [14]. One may see that Hopfield model is a special case of this kind of neural networks with A = diag {ai } , ai := −1/Ri Ci , Ri > 0 and Ci > 0. Ri and Ci are the resistance and capacitance at the ith node of the network respectively. The subT T structure W1 σ 1 (V1 [x, z] ) + W3 φ1 (V3 [x, z] )u is a multilayer perceptron structure. In order or simplify the theory analysis, we let the hidden layers Vi = I. We discuss a single layer neural network ·

x = Ax + W1 σ 1 (x, z) + W3 φ1 (x, z)u · z = Bz + W2 σ 2 (x, z) + W4 φ2 (x, z)u

(2)

Because A and B are Hurwitz matrices, there exist positive definite matrices Q1 and Q2 ∈ n×n such that the following Lyapunov functions have positive definite solutions P1 and P2 P1 A + AT P1 = −Q1 P2 B + B T P2 = −Q2

(3)

The following theorem gives a basic passivity condition for the dynamic neural network with two time-scales. Theorem 1. If the weights of the dynamic neural networks defined by (2) are updated as ·

W 1 = −P1 xσ T1 + 12 xuT ·

W 2 = − 1 P2 zσ T2 + 12 zuT ·

T

W 3 = −P1 x (φ1 u) + 12 xuT ·

T

W 4 = − 1 P2 z (φ2 u) + 12 zuT

(4)

Passivity Analysis of Dynamic Neural Networks with Different Time-Scales

89

where P1 and P2 are the solution of (3), then the dynamic neural network (2) is strictly passive from input u to output y ∈ R2n y = xT (W1 + W3 ) + z T (W2 + W4 )

(5)

Proof. Select a Lyapunov function St (or we call it storage function) as         St (x, z) = xT P1 x + z T P2 z + tr W1T W1 + tr W2T W2 + tr W3T W3 + tr W4T W4 (6)

where P ∈ n×n is positive definite matrix, tr {·} stands for the ”trace” and is defined as the sum of all the diagonal elements of a matrix. According to (2), its derivative is ·     S t (x, z) = xT P1 A + AT P1 x + 1 z T P2 B + B T P2 z 2 T +2xT P1 W1 σ 1  + 2 z T P 2xT P1 W 2 W2 σ 2 + 4 φ2 u  z P2 W 3 φ1 u +  · T

+2tr W 1 W1

· T

+ 2tr W 2 W2

· T

+ 2tr W 3 W3

· T



+ 2tr W 4 W4

Adding and subtracting xT (W1 + W3 ) u + z T (W2 + W4 ) u and using (3), and the updating law as (4), we have ·  1 S t (x, z) = −xT Q1 x − z T Q2 z + xT (W1 + W3 ) + z T (W2 + W4 ) u 

(7)

From  T Definition 1 Twe see that if the input is defined as u and the output as x (W1 + W3 ) + z (W2 + W4 ) , then the dynamic neural network given by (2) is strictly passive with 1 Vt = xT Q1 x + z T Q2 z ≥ 0  Remark 1. When we have hidden layers Vi , since σ(·) and φ(·) are bounded, the passivity property has no relationship with Vi . The weights of the hidden layer may be fixed. Furthermore, we can also conclude that the stability properties of the dynamic neural networks (2) are not influenced by the hidden layers. Corollary 1. If the dynamic neural networks (2) is unforced (u = 0), the updating law (4) will make the equilibriums x = z = 0 stability. If the control input of the dynamic neural networks (2) is selected as  u = −μy = −μ xT (W1 + W3 ) + z T (W2 + W4 ) , μ>0 (8) the updating law (4) will make the equilibrium xt = 0 asymptotic stability. Theorem 2. If the upper bound of the weights Wi (i = 1 · · · 4) satisfies   λmin (Q1 ) ≥ λmax W 1 + W 3   λmin (Q2 ) ≥ λmax W 2 + W 4 

(9)

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where W i = sup Wi 2Λ−1 , matrix norm is defined as A2B := AT BA , Λi is i positive definite matrix, λmax (·) and λmin (·) are the maximum and minimum eigenvalues of the matrix. The updating law (4) will make the dynamic neural networks (2) input-to-state stability (ISS). Remark 2. Consider Property 2; ISS means that the behavior of the dynamic neural networks should remain bounded when their inputs are bounded. If bounded disturbances are also regarded as inputs, the neural networks are also bounded with respect to the disturbances. So the dynamic neural networks (2) are bounded-input bounded-output (BIBO) stability. If P1 and P2 are selected large enough, then condition (9) is not difficult to be satisfied. From (3) we have T1 P1 A + T AT P1 = −T1 Q T2 P2 B + T B T P2 = −T2 Q

(10)

where Ti = TiT > 0. If the positive definite matrix Ti is large enough, (9) is satisfied, i.e. ( is a small positive constant)   λmax W 1 + W 3 ≤ λmin (T1 Q1 )   2 Q2 ) λmax W 2 + W 4 ≤ λmin (T 

3

Simulation

To illustrate the theoretical results, we give the following simulations. The dynamic neural network is ·

x = Ax + W1 σ 1 (x, z) + W3 φ1 (x, z)u · z = Bz + W2 σ 2 (x, z) + W4 φ2 (x, z)u

(11)



 −5 0 −5 0 where x ∈  , z ∈  , A = , B = ,  = 0.05. The active 0 −5 0 −5 functions σ k and φk are selected as tanh(). First, let check the passivity of the 2

2

5 x1

4 3

x2 2 1

z1 z2

0 -1 0

1

2

3

Time (s) 4

5

Fig. 2. Dynamic neural network with two time-scales

Passivity Analysis of Dynamic Neural Networks with Different Time-Scales

91

5 x1

4 3 2

x2 z1

1 0

z2 Time (s)

-1 0

1

2

3

4

5

Fig. 3. Zero-input responses

 10 0 neural networks (11). If we select Q1 = Q2 = , the solution of Lyapunov 0 10

 10 equation (3) is P1 = P2 = . The updating law for the weights W are used as 01 (4). Fig.2 and Fig.3 show the bounded inputs responses (u = [3 sin (0.5t) , 0, 0, 0]), zero-input responses and additional output feedback (8) u = −μy (μ = 1).

4

Conclusion

By means of passivity technique, we have given some new results on the stability of dynamic neural networks with different time-scales. A simple gradient learning algorithm can make the neural networks passive, stability, asymptotic stability, input-to-state stability and bounded input bounded output stability in certain senses. Although it is not easy to construct the special passive circuits with the passivity technique used in this paper, it is possible to extend these results to neuro identification and neuro control.

References 1. Amari, S.: Field Theory of Self-Organizing Neural Nets. IEEE Trans. Syst., Man, Cybern. 13(1) (1983) 741–748 2. Forti, M., Manetti, S., Marini, M.: Necessary and Sufficient Condition for Absolute Stability of Neural Networks. IEEE Trans. on Circuit and Systems-I 41(4) (1994) 491-494 3. Grossberg, S.: Competition, Decision and Consensus. J. Math. Anal. Applicat. 66(5) (1978) 470–493 4. Jagannathan, S., Lewis, F.L.: Identification of Nonlinear Dynamical Systems Using Multilayered Neural Networks. Automatica 32(12) (1996) 1707-1712 5. Jin, L., Gupta, M.: Stable Dynamic Backpropagation Learning in Recurrent Neural Networks. IEEE Trans. Neural Networks 10(9) (1999), 1321–1334 6. Kaszkurewics, K., Bhaya, A.: On a Class of Globally Stable Neural Circuits. IEEE Trans. on Circuit and Systems-I 41(1) (1994) 171-174

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7. Kosmatopoulos, E.B., Polycarpou, M.M.,: Christodoulou, M.A., Ioannou, P.A. High-Order Neural Network Structures for Identification of Dynamical Systems. IEEE Trans. on Neural Networks 6(2) (1995) 442-431 8. Meyer-B¨ ase, A., Ohl, F., Scheich, H.: Singular Perturbation Analysis of Competitive Neural Networks with Different Time-Scales. Neural Comput. 8(3) (1996) 545–563 9. Lewis, F.L., Liu, K., Yesildirek, A.: Multilayer Neural-Net Robot Controller with Guaranteed Tracking Performance. IEEE Trans. on Neural Network 7(2) (1996) 388-398 10. Rovithakis, G.A., Christodoulouand, M.A.: Adaptive Control of Unknown Plants Using Dynamical Neural Networks. IEEE Trans. on Syst., Man and Cybern. 24(2) (1994) 400-412 11. Sontag, E.D., Wang, Y.: On Characterization of the Input-to-State Stability Property. System & Control Letters 24(3) (1995) 351-359 12. Shao, Z.H.: Robust Stability of Two-Time-Scale Systems with Nonlinear Uncertainties. IEEE Transactions on Automatic Control 49(1) (2004) 135-140 13. Suykens, J., Moor, B., Vandewalle, J.: Robust Local Stability of Multilayer Recurrent Neural Networks. IEEE Trans. Neural Networks 11(1) (2000) 222–229 14. Yu, W., Poznyak, A.S.: Indirect Adaptive Control via Parallel Dynamic Neural Networks. IEE Proceedings - Control Theory and Applications 37(1) (1999) 25-30

Exponential Dissipativity of Non-autonomous Neural Networks with Distributed Delays and Reaction-Diffusion Terms Zhiguo Yang1,2 , Daoyi Xu1 , and Yumei Huang1 1

Institute of Mathematics, Sichuan University, Chengdu 610064, China [email protected] 2 College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China

Abstract. In this paper, a class of non-autonomous neural networks with distributed delays and reaction-diffusion terms is considered. Employing the properties of diffusion operator and the techniques of inequality, we investigate positive invariant set, global exponential stability, and then obtain the exponential dissipativity of the neural networks under consideration. Our results can extend and improve earlier ones. An example is given to demonstrate the effectiveness of these results.

1

Introduction

Dynamics of autonomous neural networks based on Hopfield architecture have been intensively studied in the recent years. Many important results have been obtained ([1]-[3]). However, non-autonomous neural network model can even accurately depict the evolutionary processes of the networks ([4]-[5]). Therefore, it is important to study the dynamics of non-autonomous neural networks. On the other hand, delay and diffusion effects[6] cannot be avoided in neural networks when electrons are moving in asymmetric electromagnetic fields. Thus, the study to neural networks with delays and reaction-diffusion terms becomes extremely important to manufacture high quality neural networks. As is well known, the stability of neural networks with or without delay has received much attention in the literature([1]-[6]). As pointed out in [7], the dissipativity is also an important concept in dynamical neural networks. And it has found applications in the areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control. In the literature [7], some new sufficient conditions for the global dissipativity of autonomous neural networks with time delay are obtained. However, to the best of our knowledge, there is no investigation to the dissipativity of non-autonomous neural networks with delays and reaction-diffusion terms. In this article, we consider a class of non-autonomous neural networks with distributed delays and reaction-diffusion terms. By using the properties of diffusion operator and the techniques of inequality, we firstly investigate positive invariant set, the global exponential stability, and then obtain the exponential J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 93–99, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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dissipativity of the networks. Our results can extend and improve earlier ones. An example is given to demonstrate the effectiveness of these results.

2

Model Description and Preliminaries

In this article, we consider the following non-autonomous reaction-diffusion neural networks with distributed delays and reaction-diffusion terms ⎧ m n   ∂ui ∂ui ∂ ⎪ ⎪ bij (t)gj (uj (t − τij (t), x)) ⎪ ∂t = ∂xk (Dik ∂xk ) − ai (t)ui + ⎪ ⎪ j=1 k=1 ⎪ ⎨ n t  + cij (t) −∞ kij (t − s)fj (uj (s, x))ds + Ji (t), (1) ⎪ j=1 ⎪ ⎪ ∂ui ∂ui ∂ui ⎪ ⎪ := ( ∂x1 , · · · , ∂xm ) = 0, t ≥ t0 , x ∈ ∂Ω, ⎪ ⎩ ∂n ui (t0 + s, x) = φi (s, x), −∞ < s ≤ 0, x ∈ Ω, i ∈ 1, 2, · · · , n. where 0 ≤ τij (t) ≤ τ , τ is a constant, Ω is a bounded domain in real mdimensional linear vector space Rm with smooth boundary ∂Ω and measure μ = mesΩ > 0; Smooth function Dik = Dik (t, x) ≥ 0 corresponds to the transmission diffusion operator; kij are continuous delay kernels functions. We assume that functions ai (t), bij (t), cij (t) and Ji (t) are continuous and the solution u(t, x) = u(t; t0 , φ) of system (1) is uniquely determined by the initial value φi (s, x). Δ Let C = C((−∞, 0] × Ω, Rn ). Then for φ(s, x) ∈ C, we define [φ]+ s =(φ1 2s , Δ

Δ

φ2 2s , · · · , φn 2s )T , where ||φi (s, x)||2s = max ||φi (s, x)||2 and ||φi (s, x)||2 = −∞<s≤0  1 ( Ω φ2i (s, x)dx) 2 . For any real matrices A = (aij )n×m and B = (bij )n×m , we write A ≥ B if aij ≥ bij , ∀i ∈ N , j ∈ {1, 2, · · · , m}. Definition 1. A set S ⊂ C is called to be a positive invariant set of Eq.(1) if for any initial value φ ∈ S, ut (t0 , φ) ∈ S, ∀t ≥ t0 , where ut (t0 , φ) = u(t + s; t0 , φ)) for s ∈ (−∞, 0]. Definition 2. Eq.(1) is said to be globally exponentially stable, if there are constants λ > 0 and M ≥ 1 such that for any two solutions u(t; t0 , φ) and u(t; t0 , ψ) with the initial functions φ, ψ ∈ C, respectively, one has u(t; t0 , φ) − u(t; t0 , ψ) ≤ M φ − ψs e−λ(t−t0 ) , ∀t ≥ t0 , Δ

where, ||u|| =(

n 

i=1

1

Δ

||ui (t, x)||22 ) 2 , ||φ||s =(

clidean norm of Rn .

n 

i=1

1

||φi (s, x)||22s ) 2 = |[φ]+ s |, |.| is Eu-

Definition 3. The neural network defined by (1) is called a exponentially dissipative system, if there are three positive constants M1 , M2 and λ such that ||u(t; t0 , φ)|| ≤ M1 φs e−λ(t−t0 ) + M2 ,

∀ φ ∈ C, t ≥ t0 .

For convenience, we introduce the following assumptions.

Exponential Dissipativity of Non-autonomous Neural Networks

95

(A1 ) There exist σj > 0, lj > 0 such that |gj (u) − gj (v)| ≤ σj |u − v|, |fj (u) − fj (v)| ≤ lj |u − v|, ∀j ∈ N , u, v ∈ R. (A2 ) There exists a positive constants β such that the continuous delay kernels functions kij : [0, ∞) → [0, ∞) satisfy  ∞ kˆij = eβξ kij (ξ)dξ < ∞, i, j ∈ N . 0

(A3 ) There exist continuous functions hi (t) > 0 and constants a ˆi > 0, ˆbij ≥ 0, ˆ Ji ≥ 0, i, j ∈ N , such that ai (t) ≥ a ˆi hi (t), |bij (t)| ≤ ˆbij hi (t), |cij (t)| ≤ cˆij hi (t), |Ji (t)| ≤ Jˆi hi (t). ˆ − CL ˆ is an M -matrix [8], where Cˆ = (ˆ (A4 ) Aˆ − Bσ c∗ij )n×n , cˆ∗ij = cˆij kˆij , ˆ = (ˆbij )n×n , σ =diag{σ1 , · · · , σn }, L =diag{l1, · · · , ln }, Aˆ =diag{ˆ B a1 , · · · , a ˆn }.

3

Main Results

In this section, we investigate positive invariant, the global exponential stability, and obtain the exponential dissipativity of non-autonomous system (1). Theorem 1. Assume that conditions (A1 )-(A4 ) are satisfied. Let μ = mesΩ and Jˆ = (Jˆ1 , · · · , Jˆn )T , gˆ = (|g1 (0)|, · · · , |gn (0)|)T , fˆ = (|f1 (0)|, · · · , |fn (0)|)T , Δ ˆ ˆ g + Cˆ fˆ + Jˆ , S = {φ ∈ C|[φ]+ ≤ N = ˆ − CL) ˆ −1 Iμ}. Then S is a I = Bˆ (A − Bσ s positive invariant set of system (1). ˆ − CL ˆ is an Proof. Without loss of generality, we let Jˆ > 0. Since Aˆ − Bσ −1 T Δ ˆ ˆ ˆ ˆ ˆ − M -matrix, we have (A − Bσ − CL) ≥ 0, and N = (N1 , · · · , Nn ) = (A − Bσ −1 + ˆ CL) Iμ > 0. We now prove for φ ∈ C, when [φ]s ≤ N , Δ

[u(t, x)]+ =(u1 (t, x)2 , u2 (t, x)2 , · · · , un (t, x)2 )T ≤ N, ∀t ≥ t0 ,

(2)

where u(t, x) = u(t; t0 , φ) is the solution of (1) with the initial functions φ ∈ C. First, we shall prove that for p > 1, [φ]+ s < pN implies [u(t, x)]+ < pN,

t ≥ t0 .

(3)

If not, there must be l and t1 > t0 such that ul (t1 , x)2 = pNl , ul (t, x)2 < pNl , −∞ < t < t1 ,

(4)

ui (t, x)2 ≤ pNi , ∀i ∈ N , −∞ < t ≤ t1 .

(5)

Multiply both sides of the equation in (1) with ui (t, x), and integrate

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1 d 2 dt



m 

u2i dx = Ω

ui

Ω

k=1

∂ ∂ui (Dik )dx − ai (t) ∂xk ∂xk

 u2i dx + Ω

×[gj (uj (t − τij (t), x)) − gj (0)]dx +  × +

n

−∞ n

bij (t)

ui Ω

j=1



cij (t)

ui Ω

j=1 t



n

n kij (t − s)[fj (uj (s, x)) − fj (0)]dsdx + [ bij (t)gj (0)



cij (t)fj (0)

j=1

j=1



t

−∞

kij (t − s)ds + Ji (t)]

ui dx, i ∈ N . (6) Ω

From the boundary condition, we get m  k=1

ui

Ω

∂ ∂ui (Dik )dx = ∂xk ∂xk



∂ui m )k=1 ·ds − ∂xk m

(ui Dik ∂Ω

=−

k=1

m 

Dik ( Ω

k=1

 Dik ( Ω

∂ui 2 ) dx ∂xk

∂ui 2 ) dx, ∀i ∈ N , ∂xk

(7)

∂ui m ∂ui ∂ui T where ∇ is the gradient operator, (Dik ∂x ) = (Di1 ∂x , · · · , Dim ∂x ) . 1 m k k=1 By the conditions(A1 )-(A2 ), equation (6)-(7) and Holder inequality, we have n d ˆbij σj ||uj (t − τij (t), x)||2 ||ui ||2 ≤ −ˆ ai hi (t)||ui ||2 + hi (t)[ dt j=1  t n + cˆij kij (t − s)lj uj (s, x)2 ds + Ii μ], i ∈ N , t ≥ t0 . (8) j=1

where Ii =

−∞

n n  ˆbij |gj (0)| +  cˆij kˆij |fj (0)| + Jˆi . From [φ]+ < pN , (4) and (5), s

j=1

j=1

||ul (t1 , x)||2 ≤ e−



t1 t0

 a ˆl hl (s)ds

||φl ||2s +

t1

e−



t1 s

a ˆl hl (ξ)dξ

n ˆblj σj ||uj (s hl (s)[

t0

−τlj (s), x)||2 +

n

 cˆlj

j=1

< e−

+



t1 t0

a ˆl hl (s)ds

[pNl −

j=1

s

−∞

klj (s − ξ)lj uj (ξ, x)2 dξ + Il μ]ds

n n 1 ˆ ( blj σj pNj + cˆlj kˆlj lj pNj + Il μ)] a ˆl j=1 j=1

n n 1 ˆ ( blj σj pNj + cˆlj kˆlj lj pNj + Il μ). a ˆl j=1 j=1

(9)

ˆ − CL ˆ is an M -matrix and N = (Aˆ − Bσ ˆ − CL) ˆ −1 Iμ, one can get Since Aˆ − Bσ

Exponential Dissipativity of Non-autonomous Neural Networks

a ˆi pNi ≥

n

ˆbij σj pNj +

j=1

Noting that e−



t1 t0

a ˆl hl (s)ds

n

cˆij kˆij lj pNj + Ii μ, ∀i ∈ N , p > 1.

97

(10)

j=1

≤ 1, from (9) and (10), we obtain ||ul (t1 , x)||2 < pNl ,

which contradicts the equality in (4). Let p → 1 in (3), then the proof is complete. Theorem 2. Suppose that the conditions (A1 )-(A4 ) hold. And there exist a positive constant λ such that continuous functions hi (t) satisfy  t a ˆi hi (ξ)dξ ≥ λ(t − s), t ≥ s ≥ t0 , i ∈ N . (11) s

Then system (1) is globally exponentially stable and the exponential convergent rate δ ≤ min{λ, β} is determined by (15). Proof. For any φ, ψ ∈ C, we denote y(t, x) = u(t, x) − v(t, x),

(12)

where u(t, x) = u(t; t0 , φ) and v(t, x) = v(t; t0 , ψ) are the solutions of (1) with the initial functions φ, ψ ∈ C, respectively. Then from (1), y(t, x) must satisfy ⎧ m n   ∂yi ∂ i ⎪ = (D ) − a (t)y + bij (t)¯ gj (yj (t − τij (t), x)) ⎪ ∂y ik i i ⎪ ∂t ∂xk ∂xk ⎪ j=1 ⎪ k=1 ⎪ ⎨ n t  + cij (t) −∞ kij (t − s)f¯j (yj (s, x))ds, (13) ⎪ j=1 ⎪ ⎪ ∂yi ∂yi ∂yi ⎪ ⎪ := ( ∂x , · · · , ∂x ) = 0, t ≥ t0 , x ∈ ∂Ω, ⎪ 1 m ⎩ ∂n yi (t0 + s, x) = φi (s, x) − ψi (s, x), −∞ < s ≤ 0, x ∈ Ω, i ∈ N . where, g¯j (yj (t − τij (t), x)) = gj (uj (t − τij (t), x)) − gj (vj (t − τij (t), x)) and f¯j (yj (s, x)) = gj (uj (s, x)) − gj (vj (s, x)). Similar to the proof of the inequality (8), we obtain n d ˆbij σj ||yj (t − τij (t), x)||2 ||yi ||2 ≤ −ˆ ai hi (t)||yi ||2 + hi (t)[ dt j=1  t n + cˆij kij (t − s)lj yj (s, x)2 ds], ∀i ∈ N , t ≥ t0 . j=1

(14)

−∞

ˆ − CL ˆ is an M -matrix [8], there exist r = (r1 , · · · , rn )T > 0 and Since Aˆ − Bσ a δ > 0 such that ˆ − CL)r ˆ (Aˆ − Bσ > 0, ˆ − (Bσe ˆ τ δ + CL)r(1 ˆ Ar + Next, we shall prove that for any ε > 0,

δ ) > 0. λ−δ

(15)

98

Z. Yang, D. Xu, and Y. Huang

||yi (t, x)||2 < ri R(||φ − ψ||s + ε)e−δ(t−t0 ) = zi (t), t ∈ (−∞, +∞), i ∈ N . (16) where R is a positive constant satisfying Rri ≥ 1, i ∈ N . Obviously, when t ∈ (−∞, t0 ], we have ||yi (t, x)||2 < ri R(||φ − ψ||s + ε)e−δ(t−t0 ) = zi (t), i ∈ N . So, if (16) is not, there must be l and t1 > t0 such that yl (t1 , x)2 = zl (t1 ), yl (t, x)2 < zl (t), −∞ < t < t1 ,

(17)

yi (t, x)2 ≤ zi (t), ∀i ∈ N , −∞ < t ≤ t1 .

(18)

From (A2 ), (11), (14), (15), (17) and (18) we obtain ||yl (t1 , x)||2 ≤e

−



t1 t0

a ˆ l hl (s)ds

 ||yl (t0 , x)||2 +

t1

e−



t1 s

a ˆ l hl (ξ)dξ

t0

×||yj (s − τlj (s), x)||2 +

n

 cˆlj lj

j=1

n ˆblj σj hl (s)[ j=1

s

−∞

klj (s − ξ)yj (ξ, x)2 dξ]ds,

t  1 1 t1 d(e− s aˆl hl (ξ)dξ ) <e rl R(||φ − ψ||s + ε) + a ˆl t0 ds n n ˆblj σj rj eδτ + ×[ cˆlj kˆlj lj rj ]R(||φ − ψ||s + ε)e−δ(s−t0 ) ds −



t1 t0

a ˆ l hl (s)ds

j=1

j=1

≤ R(||φ − ψ||s + ε){e− ×

n

 t1 t0

a ˆ l hl (s)ds

[rl −

n 1 ˆ 1 (blj σj rj eδτ + cˆlj kˆlj lj rj )] + a ˆl j=1 a ˆl

(ˆblj σj rj eδτ + cˆlj kˆlj lj rj )[e−δ(t1 −t0 ) +

j=1

< R(||φ − ψ||s + ε){e +

−λ(t1 −t0 )

δ (e−δ(t1 −t0 ) − e−λ(t1 −t0 ) )]} λ−δ

n 1 ˆ [rl − (blj σj rj eδτ + cˆlj kˆlj lj rj )(1 a ˆl j=1

n δ 1 ˆ δ )] + (blj σj rj eδτ + cˆlj kˆlj lj rj )(1 + )e−δ(t1 −t0 ) } λ−δ a ˆl j=1 λ−δ

≤ R(||φ − ψ||s + ε)e−δ(t1 −t0 ) rl = zl (t1 ). which contradicts the equality in (17). Then (16) is true. Let ε → 0, we have [y(t, x)]+ ≤ rR||φ − ψ||s e−δ(t−t0 ) , t ∈ (−∞, +∞). u(t; t0 , φ) − u(t; t0 , ψ) ≤ |r|R||φ − ψ||s e−δ(t−t0 ) , t ∈ (−∞, +∞).

(19) (20)

Therefore, system (1) is globally exponentially stable. The proof is complete. Using Theorem 1 and Theorem 2, we easily obtain the follow theorem. Theorem 3. Suppose all conditions in Theorem 2 are satisfied. Then the neural network (1) is exponentially dissipative.

Exponential Dissipativity of Non-autonomous Neural Networks

4

99

Example

Example. Consider non-autonomous reaction-diffusion neural networks ⎧ ∂u (t,x) 1 ⎪ = Δu1 (t, x) − (4 + e−t )u1 (t, x) − sin(t2 )g1 (u1 (t − 12 , x)) ⎪ ∂t t ⎪ ⎪ ⎪ +g2 (u2 (t − | cos t|, x)) + −∞ e−(t−s) u1 (s, x)ds ⎪ ⎪ t ⎪ ⎪ ⎪ +2 −∞ e−(t−s) u2 (s, x)ds + 4π1 2 arctan(t), ⎪ ⎪ ⎨ ∂u2 (t,x) = Δu2 (t, x) − (5 + e−t )u2 (t, x) + 12 cos(t2 )g1 (u1 (t − 34 , x)) ∂t (21)  1 t −(t−s) ⎪ −2g (u (t − 1, x)) + e u (s, x)ds ⎪ 2 2 1 2 −∞ ⎪ t ⎪ ⎪ ⎪ + −∞ e−(t−s) u2 (s, x)ds + 4π1 2 arctan(t), ⎪ ⎪ ⎪ ⎪ ui (t0 + s, x) = φi (s, x), −∞ < s ≤ 0, x ∈ Ω, ⎪ ⎪ ⎩ ∂ui ∂ui ∂ui ∂ui t ≥ t0 , x ∈ ∂Ω, i = 1, 2. ∂n := ( ∂x1 , ∂x2 , ∂x3 ) = 0, where Ω = {(x1 , x2 , x3 ) ∈ R3 |x21 + x22 + x23 = 1}, and the sigmoid function g1 (s) = g2 (s) = tanh(s). It is easy to check that the conditions (A1 )-(A4 ) are satisfied and we may take τ = 1, h1 (t) = h2 (t) ≡ 1, kˆ11 = kˆ12 = kˆ21 = kˆ22 = 1,









 40 11 10 12 10 ˆ ˆ ˆ A= , B= 1 , σ= , C= 1 , L= , 05 01 01 2 2 2 1

1  ˆ g + Cˆ fˆ + Jˆ = Jˆ = 8π I = Bˆ , μ = mesΩ = 4π 1 3 , 8π



5 2 −3 −1 ˆ ˆ ˆ ˆ ˆ A − Bσ − CL = is an M -matrix , N = (A − Bσ) Iμ = 61 , −1 2 2 S1 = {φ ∈ C | φ1 2s ≤ 56 , φ2 2s ≤ 12 }. It follows from Theorem 1-3 that system (21) is exponentially dissipative.

References 1. Zhang, Y.: Global Exponential Convergence of Recurrent Neural Networks with Variable Delays. Theoretical Computer Science 312(1) (2004) 281-293 2. Wang, L.: Stability of Cohen-Grossberg Neural Networks with Distributed Delays. Applied Mathematics and Computation 160(1) (2005) 93-110 3. Zhang, Q., Wei, X., Xu, J.: Delay-dependent Exponential Stability of Cellular Neural Networks with Time-varying Delays. Chaos Solitons & Fractals 23(4) (2005) 13631369 4. Xu, D., Zhao, H.: Invariant and Attracing Sets of Hopfield Neural Networks with Delay. International Journal of Systems Science 32(7) (2001) 863-866 5. Jiang, H., Li, Z., Teng, Z.: Boundedness and Stability for Nonautonomous Cellular Neural Networks with Delay. Physics Letters A 306(1) (2003) 313-325 6. Cao, J.: An Estimation of the Domain of Attraction and Convergence Rate for Hopfield Continuous Feedback Neural Networks. Physics Letters A 325(4) (2004) 370-374 7. Arik, S.: On the Global Dissipativity of Dynamical Neural Networks with Time Delays. Physics Letters A 326(4) (2004) 126 -132 8. Liao, X.: Theory and Applications of Stability for Dynamical Systems. Defence Industry Publishing House, Beijing (2000)

Convergence Analysis of Continuous-Time Neural Networks Min-Jae Kang1, Ho-Chan Kim1, Farrukh A. Khan2, Wang-Cheol Song2, and Jacek M. Zurada3 1

Faculty of Electrical & Electronic Engineering, Cheju National University, Jeju 690-756, South Korea {minjk, hckim}@cheju.ac.kr 2 Department of Computer Engineering, Cheju National University, Jeju 690-756, South Korea {farrukh, philo}@cheju.ac.kr 3 Department of Electrical Engineering, University of Louisville, Louisville, KY, 40292 USA [email protected]

Abstract. The energy function of continuous-time neural network has been analyzed for testing the existence of stationary points and the global convergence of network. The energy function always has only one stationary point which is a saddle point in the unconstrained space when the total conductance of neuron's input is zero ( Gi = 0 ). However, the stationary points exist only inside the hypercube R n ∈ [0,1] when the total conductance of neuron's input is not zero ( Gi ≠ 0 ). The Hessian matrix of the energy function is used for testing the global convergence of the network.

1 Introduction Single-layer feedback neural networks, also called gradient-type networks, have received widespread attention in the technical literature [1-4]. Most of the published results, however, deal with the discrete-time, discrete-output networks. The discretetime networks represent limit case of continuous-time networks which involve continuous activation function and changing the output values continuously rather than discretely. Some of the postulates regarding both the convergence and performance of discrete-time networks remain valid for their continuous-time counterparts. There are also numerous differing aspects in both types of networks. In this paper, the energy function of continuous-time neural networks has been analyzed for proving the existence of stationary points and the networks’ convergence.

2 Description of Continuous-Time Actual Neural Networks Hopfield neural network consists of n neurons each mapping the input voltage ui of the i-th neuron into the output voltage vi through the activation function ƒ (ui). Conductance Wij connects the output of the j-th neuron to the input of the i-th neuron. The J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 100 – 108, 2006. © Springer-Verlag Berlin Heidelberg 2006

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101

postulated network is symmetric, i.e. wij =wji. Also, since it has been assumed that wii = 0, the output of neurons are not connected to their own inputs. The input conductance and capacitance of i-th neuron is of non zero value and is equal to gi and Ci respectively. Applying KCL (Kirchoff’s Current Law) at the input node having potential ui gives the equilibrium equation

Ci

dui = dt

n

∑W v

ij j

+ ii − Gi ui

i = 1,..., n.

(1)

j =1

Denoting the total conductance connected to the input node i as Gi, where n

Gi = gi +

∑W

ij

(2)

j =1

The most common choice is the so-called sigmoid function with the finite value of gain λ as follows:

vi = f (ui ) =

1 1 + exp(−λ ui )

(3)

Hopfield (1984) [3] introduced the postulated energy function for his neural network as follows:

1 E (v ) = − 2

n

n

n

vj

n

∑ ∑W v v − ∑ i v + ∑ G ∫ f ij i j

i =1

j =1

i i

i =1

i

i =1

−1 i

( z )dz

(4)

0.5

The negative derivative of the energy function for symmetric W can be computed as:

−

n

∑W v

dE = dvi

ij j

+ ii − Gi ui

(5)

j =1

By comparing equation (5) with the right hand side of the equation (1) it can be shown that

−

du dE = Ci i dvi dt

(6)

Since W is symmetric and f (u ) is a monotonic increasing function, it can be easily shown that the energy function E (v) is a Lyapunov like function [1].

dE = dt

n

∑ i =1

dE dvi =− dvi dt

n

∑ i =1

n

Ci

∑

dui dvi du = − Ci ( i ) 2 f ' (u ) dt dt dt i =1

≤

0

(7)

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3 Energy Function Analysis of Continuous-Time Neural Networks The Hopfield type neural network converges by following a path that decreases the Lyapunov like energy function. By analyzing this energy function, the existence of stationary points has been proved and the global convergence of the system has also been tested. 3.1 Existence of Stationary Points

The continuous-time neural network has the neuron's transfer sigmoid gain λ of finite value whereas sigmoid gain is infinite for the discrete-time neural network. The energy function has been analyzed for proving existence of stationary points. It is shown that the total conductance of neuron's input node is a major factor for existence of stationary points. Proposition 1. If the neuron's transfer sigmoid gain λ is finite, then the following statements are true. n

(1) If the total conductance of the i-th neuron's input is zero (G = g + W = 0) , then ∑ ij i i j =1

st

there exists one stationary point ( v ) of the energy function in the space R n ∈ (−∞, ∞) (2) If the total conductance of the i-th neuron's input is not zero ( Gi ≠ 0 ), then st

there could exist more than one stationary points ( v ), and all stationary points are inside the space R n ∈ (0,1) . Proof (1) The stationary points of the postulated energy function E(v) in equation (4) can be expressed as follows:

∇E (v ) = Wv + i − Gu = 0

(8)

vi ) 1 − vi

(9)

where,

ui = f −1 (vi ) =

1

λ

ln(

Using equation (9), the third term of equation (8) can be written as follows:

G

λ

ln(

v ) = Wv + i 1− v

(10)

If the i-th neuron's total conductance is zero ( Gi = 0 ), then equation (10) can be expressed as follows:

Wv = −i

(11)

Convergence Analysis of Continuous-Time Neural Networks

And it is well-known that if W is symmetric and wii

103

= 0 , the inverse of W exists

[10]. Therefore, only the single stationary point exists in the space R n ∈ ( −∞, ∞ ) and is given as follows:

v st = −W −1i

(12)

Proof (2) Let's define the left hand side of equation (10) as follows

S (vi ) =

Gi

λ

ln(

vi ), i = 1, 2,..., n 1 − vi

(13)

If Gi ≠ 0 and λ > 0 , then S (vi ) is differentiable in vi ∈ (0,1) and if Gi > 0 , S (vi ) is strictly increasing, therefore,

∀vix1 , vix 2 ∈ (0,1), vix1 > vix 2 ⇒ S (vix1 ) ≥ S (vix 2 ), and, lim + S (vi ) = −∞, v j →0

lim S (vi ) = ∞ .

v j →1−

If Gi < 0 , S (vi ) is strictly decreasing, therefore,

∀vix1 , vix 2 ∈ (0,1), vix1 > vix 2 and

lim S (vi ) = ∞,

V j → 0+

⇒ S (vix1 ) ≤ S (vix 2 )

lim S (vi ) = −∞

V j →1−

In other words, S (vi ) exists only in the space R n ∈ (0,1) and its value changes to positive infinite or negative infinite near the boundary of the hypercube R n ∈ (0,1) . However, the right hand side of equation (10) is a finite value in the space Rn ∈ (0,1) .Therefore, the solution of equation (10) always exists in the space R n ∈ (0,1) for all i. This means that the stationary points of the energy function with n Gi ≠ 0 always exist inside the hypercube R ∈ (0,1) .

Case Study The 2-bit A/D converter is also selected for a case study. The energy function for 2bit A/D converter including weights and bias currents can be expressed as follows [3], where x is analog input.

1 E (v) = − [v1 2

⎛ 0 −2 ⎞ ⎡ v1 ⎤ v2 ] ⎜ ⎟ ⎢ ⎥ − [v1 ⎝ −2 0 ⎠ ⎣ v2 ⎦

⎡ v1 ⎤ ⎢ G1 fi −1 ( z ) dz ⎥ 1⎤ ⎡ ⎥ x − ⎥ ⎢ 0.5 ⎥ v2 ] ⎢ 2 +⎢ v ⎢ ⎥ ⎢ 2 ⎥ ⎣⎢ x − 2 ⎦⎥ ⎢G fi −1 ( z ) dz ⎥ 2 ⎢ ⎥ ⎣ 0.5 ⎦

∫

∫

(14)

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If G1 = G2 = 0 , the stationary points of E can be obtained by using equation (12) as follows: 1⎤ ⎡ −0.5 ⎞ ⎢ x − ⎥ ⎡ v1 ⎤ ⎛ 0 2 ⎟⎢ ⎢v ⎥ = − ⎜ ⎥ 0 ⎠ ⎣ 2⎦ ⎝ −0.5 ⎣⎢ x − 2 ⎦⎥

(15)

Also, in case of Gi ≠ 0 , the stationary points of E can be obtained by using equation (10) as follows: v1 ⎤ ⎡ G1 1⎤ ⎡ ⎢ λ ln 1 − v ⎥ −2 ⎞ ⎡ v1 ⎤ ⎢ x − ⎥ 1 ⎥ ⎛ 0 ⎢ =⎜ + 2 ⎟ ⎢ ⎥ ⎥ ⎢ G2 v2 ⎥ ⎝ −2 0 ⎠ ⎣ v2 ⎦ ⎢ ⎢⎣ x − 2 ⎦⎥ ⎢ ln ⎥ v 1 λ − 2⎦ ⎣

(16)

Fig.1a shows the stationary points of E for the analog input x=0, 1, 3, 4 in the case of G1 = G2 = 0 . It has shown that the stationary points can exist outside the space R 2 ∈ (0,1) . However, in the case of Gi ≠ 0 , all stationary points exist inside the 2-dimensional space R2 ∈ (0,1) as shown in Fig. 1b. The stationary points for the analog input x=0, 1, 2, 3 in the case of Gi ≠ 0 have been obtained graphically by using equation (16) as shown in Fig. 1b.

(a)

(b)

Fig. 1. The locations of the stationary points of E for analog input x=0, 1, 2, 3 and the case of (a) G1 = G2 = 0 , (b) G1 = G2 = 0.5

λ = 2 , in

3.2 Global Convergence

The continuous actual neural network has the sigmoid gain λ with finite value whereas the discrete neural network has it with the infinite value. The energy function with the sigmoid gain of finite value has been analyzed for proving system's global

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105

convergence. The Hessian matrix of the energy function has been tested for judging the stationary points. The Hessian matrix has to be positive definite, negative definite and indefinite if the stationary point is minima, maxima and saddle point respectively. Proposition 2. If the sigmoid gain λ is finite, then the following statements are true.

(1) if the total conductance is positive Gi > 0 , then the stationary points are minima near the boundary of the hypercube R n ∈ (0,1) . (2) if the total conductance is zero Gi = 0 , then the stationary point is a saddle point. (3) if the total conductance is negative Gi < 0 , then the stationary points are maxima near the boundary of the hypercube R n ∈ (0,1) . Proof (1) The Hessian matrix of the energy function has to be positive definite if the stationary point is minimum near the boundary of the hypercube R n ∈ (0,1) . Using the chain rule, the Hessian matrix of the energy function can be derived as below:

∂u ∂v

H = ∇ 2 E (v) = −W + G

(17)

Using the partial derivative of the activation function of the neuron, equation (17) can be derived as follows H = ∇2 E (v) = −W + G

1

(18)

λ (v − v 2 )

The Hessian matrix (18) can be expressed in matrix form as: G1 ⎡ ⎢ 2 ⎢ λ (v1 − v1 ) ⎢ ⎢ − w21 2 ∇ E (v ) = ⎢ ⎢ M ⎢ ⎢ ⎢ − wn1 ⎣

⎤ ⎥ ⎥ ⎥ G1 K − w2 n ⎥ 2 λ (v2 − v2 ) ⎥ ⎥ O M ⎥ G1 ⎥ K − wn 2 2 ⎥ λ (vn − vn ) ⎦ − w12

By using Gerschgorin’s Circle Theorem [9], n

Gi

λ (vi − vi ) 2

−

∑

wij

≤ λi

≤

j =1, j ≠ i

− w1n

K

(19)

∃λi (eigenvalue) s.t. n

Gi

λ (vi − vi ) 2

+

∑

wij

i = 1,..., n

j =1, j ≠ i

(20)

Let us now define the diagonal element of the Hessian matrix as follows:

D(vi ) =

Gi

λ (vi − vi2 )

(21)

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If the total conductance is positive ( Gi > 0 ), then D (vi ) is positive infinite near the boundary of the hypercube R n ∈ (0,1) , that is, (∀vi , lim D (vi ) = ∞, + v j →0

lim D (vi ) = ∞) .

vi →1−

Therefore, all eigenvalues of the Hessian matrix become positive near the boundary of the hypercube R n ∈ (0,1) . Proof (2) Since, the second term of right hand side in equation (18) disappears if Gi

= 0 , the

Hessian matrix of the energy function can be written as follows:

H = ∇ 2 E (v) = W

(22)

where, W = W T , Wii = 0 i = 1, 2,..., n

Since the sum of all eigenvalues of a matrix equals its trace [9], the trace of the Hessian matrix can be expressed as follows: n

Tr ( H ) = −

∑

n

Wii =

i =1

∑λ

i

(23)

i =1

where, λi are the eigenvalues of the Hessian matrix. Since, all diagonal elements of the Hessian matrix are zero, ( Wii = 0, i = 1, 2,..., n ,), equation (23) can be rearranged as follows n

Tr ( H ) =

∑λ

i

=0

(24)

i =1

This means that all eigenvalues are zero, or that some eigenvalues are negative and the others are positive for the summation to be zero. If all eigenvalues of the Hessian matrix are zero, then the Hessian matrix is zero rank. Therefore, all eigenvalues of the Hessian matrix can not be zero because the Hessian matrix of zero rank is meaningless. Therefore, since the summation of all eigenvalues yields zero, it is obvious that some eigenvalues are negative and the others are positive which proves that the Hessian matrix is indefinite. This proves completely that the stationary point of the energy function is a distinct saddle point if the total conductance is zero ( Gi = 0 ). Proof (3) If the total conductance is negative ( Gi < 0 ), then D(vi ) in equation (21) is negative

infinite

near

(∀Vi , lim+ D (vi ) = −∞, vi → 0

the

boundary

of

the

hypercube

R n ∈ (0,1)

lim D (vi ) = −∞) . Therefore, all eigenvalues of the Hessian

vi →1−

matrix become negative near the boundary of the hypercube R n ∈ (0,1) . This means that the Hessian matrix becomes negative definite near the boundary of the hypercube.

Convergence Analysis of Continuous-Time Neural Networks

107

Case Study Fig. 2a provides the energy map of the 2-bit A/D converter for the case x=1.6, λ =2, G1 = G2 = 0.5 . There are three stationary points which are at (0.0288, 0.9897), (0.6005, 0.499) and (0.9827, 0.0447). It can be seen that the stationary points near the corners are minima and exist indeed inside the 2-dimensional space R 2 ∈ (0,1) . The obtained eigenvalues at the minimum (0.0288, 0.9897) are 8.6854 and 24.777. Also the eigenvalues at the other minimum (0.9827, 0.0447) are 15.1362 and 5.4236. Fig. 2b shows the energy map of the 2-bit A/D converter for the case x=1.6, λ =2, G1 = G2 = 0 . There is only one stationary point at (0.6, 0.55) and the eignenvalues at this point are -2 and 2. Therefore, the Hessian matrix of this case is indefinite as expected. It can be seen in Fig. 2b that this stationary point is a saddle point. The energy function of the 2-bit A/D converter for the case x=1.6, λ =2, G1 = G2 = −0.5 has three stationary points as shown Fig. 2c. As expected, two maxima occur at (0.0130, 0.0091) and (0.9599, 0.9468) inside near the 2-dimensional space R 2 ∈ (0,1) . And the eigenvalues are -19.0412 and -28.1855 at (0.0130, 0.0091) and -7.8707 and -3.5875 at (0.9599, 0.9468).

(a)

(b)

(c)

Fig. 2. The energy map for the analog input x=1.6, and λ = 2 , in the cases of (a) G1 = G2 = 0.5, (b) G1 = G2 = 0, (c) G1 = G2 = -0.5

4 Conclusion The continuous-time neural network has the sigmoid gain of finite value, and neuron's output changes continuously. The discrete-time network is limit case of continuoustime networks, because the sigmoid gain of the discrete-time network is infinite. In this paper, by using the energy function of the continuous-time neural network, the existence of stationary points has been proved and the global convergence of network has also been analyzed. The stationary points can exist in any space when the total conductance of neuron's input is zero ( Gi = 0 ), however, the stationary points can exist only inside of hypercube Rn ∈ (0,1) when the total conductance of neuron's input is not zero ( Gi ≠ 0 ). The Hessian matrix of the energy function is used for testing the global convergence of the

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network. The Hessian matrix of the energy function is indefinite when the total conductance of neuron's input is zero ( Gi = 0 ). Therefore, the stationary point is a saddle point and the continuous-time neural network converges to the constrained minimum. When the total conductance of neuron's input is positive ( Gi > 0 ), the Hessian matrix is positive definite near the boundary of the hypercube Rn ∈ (0,1) for the stationary points to be minima. However, there exist no minima near the boundary of the hypercube R n ∈ (0,1) when Gi < 0 , because the Hessian matrix is negative definite.

References 1. Hopfield, J.J., Tank, D.W.: Neural Computation of Decisions in Optimization Problems. Biological Cybernetics 52 (1985) 141-152 2. Hopfield, J.J., Tank, D.W.: Computing with Neural Circuits: A Model. Science 233 (1986) 625-633 3. Friedberg, S., Insel, A.: Introduction to Linear Algebra with Application. 1st edn. PrenticeHall, Englewood Cliffs N.J. (1986) 4. Michel, A.N., Farrel, J.A., Porod, W.: Qualitative Analysis of Neural Networks. IEEE Trans. Circuits Systems 36 (1989) 229-243 5. Zurada, J. M.: Introduction to Artificial Neural Systems. 1st edn. West, St. Paul MN. (1992) 6. Peng, M., Gupta, N., Armitage, A.F.: An investigation into the Improvement of Local Minima of the Hopfield Network. Neural Networks 9(7) (1996) 1241-1253 7. Golub, G.H., Van Loan, C.F.: Matrix Computations. 3rd edn. John Hopkins Univ. Press, (1996) 8. Chen, T., Amari, S.I.: Stability Asymmetric Hopfield Networks. IEEE Trans. Neural Networks 12(1) (2001) 159-163

Global Convergence of Continuous-Time Recurrent Neural Networks with Delays Weirui Zhao1,2 and Huanshui Zhang1 1

Shenzhen Graduate School, Harbin Institute of Technology, D424,HIT Campus of Shenzhen University Town, Xili, Shenzhen, 518055, P.R. China 2 Department of Mathematics, Wuhan University of Technology, 122 Luoshi road, Wuhan, Hubei, 430070, P.R. China [email protected]

Abstract. We present new global convergence results of neural networks with delays and show that these results partially generalize recently published convergence results by using the theory of monotone dynamical systems. We also show that under certain conditions, reversing the directions of the coupling between neurons preserves the global convergence of neural networks.

1

Introduction

Recently, there has been much activity to study the global stability of neural networks with delays and many stability criteria are proposed [1]-[9]. In this paper, we present a new criterion for the global convergence of neural networks with delays. We show that this criterion is more explicit and easily verified than those presented in the past. Our criteria is proved using a different strategy than the recent literature. In present papers, the most general case is proved by incorporating all the generalities into Lyapunov functional, resulting in a complicated Lyapunov functional. Instead, we deduce the global convergence of neural networks with delays by the theory of monotone dynamical systems.

2

Global Convergence of Neural Networks with Delays

Consider a neural networks with delays described by the state equations: n n   dxi (t) = −di xi (t) + aij gj (xj (t)) + bij gj (xj (t − τj )) + ui . dt j=1 j=1

(1)

where n denotes the number of neurons in the network; xi (t) is the state of the ith neuron at time t; gj (xj (t)) denotes the activation function of the jth neuron at time t; the constant weight coefficient aij denotes the connection of the jth neuron to the ith neuron at time t; the constant weight coefficient bij denotes the connection of the jth neuron to the jth neuron at time t − τj ; τj J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 109–114, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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is the transmission delay along the axon of the jthe unit and is a nonnegative constant; ui is the external bias on the ith neuron; di is the positive rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and the external inputs ui . The activation function gj (·) is continuously differentiable and monotonically nondecreasing with gj (0) = 0. In our analysis, we will employ the following class of activation functions |gj (x)| < kj , gj (x) > 0, for allj. where 0 ≤ kj < +∞. The sigmoid activation functions are used in the Hopfield networks [10] are typical representatives of this classical bounded functions. An initial condition of systems (1) is constituted by the history of the activation of each neuron i during a time interval corresponding to delay τi . Initial conditions for (1) are of the form φ = (φ1 , φ2 , · · · , φn ), where each φi is a continuous function defined on the interval [−τi , 0] of length equal to the delay τi . n  Thus the phase space for (1) is the product space Cτ = C([−τi , 0], R), where i=1

C(I, R) denotes the space of continuous functions  from the interval I on the real R. Cτ is a Banach space with the norm |φ| = |φi |. For φ = (φ1 , φ2 , · · · , φn ) in Cτ , there exists a unique solution of systems (1), denoted x(t, φ) = (x1 (t, φ), x2 (t, φ), · · · , xn (t, φ)) such that xi (t, φ) = φi (t) for τi ≤ t ≤ 0, and x(t, φ) satisfy systems (1) for t ≥ 0. We denote xt the (semi)flow associated with systems (1), that is, xt ∈ Cτ as xt = (x1t , x2t , · · · , xnt ) where xit = xi (t + θ) for θ ∈ [−τi , 0]. To simplify the notations, the dependence on the initial condition φ will not be indicated unless necessary. Definition 1. A square matrix A = (aij ) is a quasipositive matrix if it is nonnegative except perhaps on its main diagonal, i.e., aij ≥ 0 for i  = j. A square matrix A = (aij ) is a nonnegative matrix if aij ≥ 0 for all i, j. Definition 2. A n × n square matrix A is a irreducible matrix if for every nonempty, proper subset I of the set N = {1, 2, · · · , n}, there is an i ∈ I and j ∈ J ≡ N \ I such that aij  = 0. Thank to the monotonicity of system it is also possible to draw a picture of the global behavior of the trajectories in the phase space. We first verify that system (1) satisfy a boundedness condition. Lemma 1. (i) For φ = (φ1 , φ2 , · · · , φn ), we define F (φ) = (F1 (φ), F2 (φ), · · · , Fn (φ))T , where Fi = −di φi (0)+

n  j=1

aij gj (φj (0))+

n 

bij gj (φj (−τj ))+ui for i = 1, 2, · · · , n,

j=1

then, F maps bounded subsets of Cτ to bounded subsets of Rn . (ii)There is a bounded subset D of Cτ such that for all φ in Cτ , there is T > 0 such that xt (Φ) ∈ D for all t > T .

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Proof. (i) Let fi (x1 , x2 , · · · , xn , y1 , y2 , · · · , yn ) = −di xi +

n 

aij gj (xj ) +

j=1

n 

bij gj (yj ).

j=1

The first result of this lemma stems from the fact that each fi maps bounded subsets of R2n to bounded subsets of R. (ii) For φ = (φ1 , φ2 , · · · , φn ) in Cτ , let x(t) = (x1 (t), x2 (t), · · · , xn (t)) be the solution of systems (1) satisfying xt0 = φ. First, we prove that there exists ti > t0 n  with xi (ti ) < γi ≡ d1i [ (|aij | + |bij |)kj + |ui | + 1]. Otherwise xi (t) ≥ γi for all t ≥ t0 . Therefore,

j=1

n 

x˙ i (t) = −di xi (t) +

aij gj (xj (t)) +

j=1

≤ −di γi +

n 

|aij ||gj (xj (t))| +

j=1 n 

< −di γi + [

n  j=1 n 

bij gj (xj (t − τij )) + ui |bij ||gj (xj (t − τij ))| + |ui |

j=1

(|aij | + |bij |)kj + |ui |] = −1.

j=1

t Then xi (t) = xi (t0 ) + t0 x˙ i (s)ds < xi (t0 ) − (t − t0 ) for t > t0 . Thus, xi (t) < γi for t > t0 + xi (t0 ) − γi , which are contradict to xi (t) < γi for all t > t0 . Thus there exists ti > 0 with x(ti ) < γi . It follows that xi (t) < γi for all t > ti . Otherwise there exists si > ti with xi (si ) > γi , then there exists ri ∈ (ti , si ) with xi (ri ) = γi and x˙ i (ri ) ≥ 0. On the n n   other hand, x˙ i (ri ) = −di γi + aij gj (xj (ri )) + bij gj (xj (ri − τij )) + ui < 0. j=1

j=1

This is a contradiction. The case that there exists Ti > ti such that xi (t) > −γi for all t > Ti is analogous. Hence  |xi (t)| < γi for all t > T = max{T1 , T2 , · · · , Tn }. Let D ≡ {φ ∈ Cτ : n  |φ| ≤ γi2 }, then xt (φ) ∈ D for all t > T + max τi . i=1

1≤i≤n

Using above lemma, we can give the global convergence of the solutions of (1). Theorem 1. Let A = (aij )n×n is irreducible and quasipositive, B = (bij )n×n is nonnegative and for every j for which τj > 0, there exist i such that bij  = 0, then the set of convergent points in Cτ contains an open and dense subset. If systems (1) has a unique equilibrium point, then all solutions of (1) converge to it. Proof. For all (x1 , x2 , · · · , xn , y1 , y2 , · · · , yn ), under theorem’s assumptions, we have ∂fi (x1 , x2 , · · · , xn , y1 , y2 , · · · , yn ) = aij gj (xj ) ≥ 0 ∀i  = j, ∂xj

112

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∂fi (x1 , x2 , · · · , xn , y1 , y2 , · · · , yn ) = bij gj (yj ) ≥ 0 ∀i, j. ∂yj

Thus systems (1) are cooperative systems [13]. Obviously,  ∂fi ∂fi −di + aii gi (xi ) + bii gi (yi ), for i = j; + = aij gj (xj ) + bij gj (yj ), for i =  j. ∂xj ∂yj Because A is irreducible and quasipositive, for every nonempty, proper subset I of the set N = {1, 2, · · · , n}, there is an i ∈ I and j ∈ J ≡ N \ I such that ∂f aij > 0. Thus aij gj (xj ) + bij gj (yj ) > 0, i.e., the Jacobian matrix ∂f ∂x + ∂y is irreducible for every φ ∈ Cτ . For each j and each (x1 , x2 , · · · , xn , y1 , y2 , · · · , yn ), n ,y1 ,y2 ,···,yn ) there exists i such that ∂fi (x1 ,x2 ,···,x = bij gj (yj ) ≥ 0 because bij > 0 ∂yj and gj (yj ) > 0. Therefore systems (1) is irreducible. The fact that the systems (1) is cooperative and irreducible together with the previous lemma imply that, the set of convergent points in Cτ contains an open and dense subset[Theorem 4.1 (p.90)[13]]. If systems (1) has a unique equilibrium point, then all solutions converge to it [Theorem 3.1 (p.18)[13]], this complete the proof of this theorem. Note AT is irreducible and quasipositive if A is irreducible and quasipositive, and B T is nonnegative if B T is nonnegative, this implies the following proposition: Proposition 1. Let A = (aij )n×n is irreducible and quasipositive, B = (bij )n×n is nonnegative and for every j for which τj > 0, there exist i such that bij  = 0, then the set of convergent points in Cτ contains an open and dense subset. If the equilibrium of systems is unique, then all solutions of systems converge to it for each of the following four state equations: dxi (t) dt

= −di xi (t) +

dxi (t) dt

= −di xi (t) +

dxi (t) dt

= −di xi (t) +

dxi (t) dt

= −di xi (t) +

n  j=1 n  j=1 n  j=1 n  j=1

aij gj (xj (t)) + aji gj (xj (t)) + aij gj (xj (t)) + aji gj (xj (t)) +

n  j=1 n  j=1 n  j=1 n 

bij gj (xj (t − τj )) + ui ; bij gj (xj (t − τj )) + ui ; bji gj (xj (t − τj )) + ui ; bji gj (xj (t − τj )) + ui .

j=1

The above result gives condition under which such reversals do not change the global convergence of the neural networks. As a special case, we consider the following one dimensional system[1]: x(t) ˙ = −dx(t) + ag(x(t)) + bg(x(t − τ )) + I,

(2)

where g(x) = tanh(x), d > 0. As all conditions of theorem are trivially satisfied for scalar equation except on cooperativity of systems (2), and systems (2) obviously is cooperative when b > 0, we have

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Proposition 2. Suppose b > 0 and the equilibrium of systems (2) is unique, then all solutions of systems (2) converge to it. We also have the following corollary: Corollary 1. Suppose b > 0 and one of the following conditions satisfy (i) a + b ≤ d;    a+b−d a+b (ii) a + b > d and (−d ln( a+b − ) − (a + b) a+b−d + I)I > 0; d  d  a+b−d a+b (iii) a + b > d and (−d ln( a+b ) + (a + b) a+b−d + I)I > 0; d + d Then all solutions of systems (2) converge to its unique equilibrium. Proof. Let H(u) = −du + (a + b)g(u) + I, if a + b ≤ d, then H  (u) = −d + (a + b)g  (u) < −d point is unique; else let + a + b < 0, thus the  equilibrium 

a+b−d a+b−d u1 = ln( a+b ) and u2 = ln( a+b ), then H  (u) < 0 for d − d d + d u ∈ (−∞, u1 ) ∪ (u2 , +∞) and H  (u) > 0 for u ∈ (u1 , u2 ), and (−du1 + (a + b)g(u1 ) + I)I > 0 or (−du2 + (a + b)g(u2 ) + I)I > 0. Thus, the equilibrium point also is unique. Therefore the result of this corollary stems from the above proposition.

3

Comparison with Previous Results

Several results on the global asymptotical stability of neural networks with delays have appeared in recent years[[1]-[9]], each one improving upon and generalizing on previous results. The results in [6] and [7] are the most general among them, but there exist two main difference their and ours. Firstly, their proof require the fact that F (x1 , x2 , · · · , xn , x1 , x2 , · · · , xn ) is a homeomorphism of Rn onto itself in [6] and [7], in our paper, we only require that the equilibrium of systems is unique. Secondly, it is not easy to select P, Q in [6] and P, K in [7] directly satisfy corresponding inequality, but our conditions are easily verified except on the uniqueness of equilibrium. Here we give an example for which Chen’s methods fails and ours is successful. Considering the following one dimension systems: x(t) ˙ = −x(t) + g(x(t)) + g(x(t − τ )) + 5,

(3)

where g(x) = tanh(x).Thus A=1,B=1,D=1,G=1. Firstly, for all P > 0, Q > 0, we have 2P DG−1 − (P A + AT P ) − P BQ−1 (P B)T − Q = −P 2 Q−1 − Q < 0. So Chen’s method fails to show the convergence of this equation. Secondly, for any factorization of B = B1 B2 , any positive number K and P , R = 2P D − P A − AT P − P B1 KB1T − B2T K −1 B2 = −P 2 KB12 − B22 K −1 < 0, So Huang’s method also fails to show the convergence of this equation. In our paper, as A + B = 2, B > 0, systems (3) √ is cooperative √ and irreducible; Let H(u) = −u + 2g(u) + 5, then H(u1 ) = − ln( 2 − 1) − 2 + 5 > 0. Thus the system (3) has a unique equilibrium and all solutions of systems (3) converge to it.

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Conclusion

This letter studies the global convergence of continuous-time recurrent neural networks with time delays using monotone dynamical theory. Several results are presented to characterize the global convergence. Those criterion is more explicit and easily verified than those presented in the past. We also show that under certain conditions, reversing the directions of the coupling between neurons preserves the global convergence of neural networks. Some examples are given to demonstrate the difference of the new results.

Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 60574016. The authors would like to thank the anonymous reviewers and the editor for their constructive comments.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9]

[10]

[11] [12] [13]

Gopalsamy, K., Lung,I.: Convergence under Dynamical Thresholds with Delays. IEEE Trans. Neural Networks 8(2) (1997) 341-348 Pakdaman, K., Malta, C.P.: A Note on Convergence under Dynamical Thresholds with Delays. IEEE Trans. Neural Networks 9(1) (1998) 231-234 Chen, T.: Convergence of Delayed Dynamcal Systems. Neural Processing Letters 10(2) (1999) 267-271 Chen, T.: Global Convergence of Delayed Dynamcal Systems. IEEE Transactions on Neural Networks 12(6) (2001) 1532-1536 Chen, T.: Global Exponential Stability of Delayed Dynamcal Systems. Neural Networks 14(8) (2001) 977-980 Lu, W., Rong, L., Chen, T.: Global Convergence of Delayed Dynamcal Systems. International Journal of Neural Systems 13(3)(2003) 1-12 Huang, Y., Wu, C.: A Unifying Proof of Global Asymptotical Stability of Neural Networks with Delay. IEEE International Symposium on Circuits and Systems-II 52(4) (2005) 181-184 Arik, S.: An Improved Global Stability Result for Delayed Cellular Neural Networks. IEEE Trans. on Circ.and Syst.-I 49(8) (2002) 1211-1214 Arik, S.: Global Asymptotical Stability of a Larger Class of Delayed Neural Networks. In: Proc. of IEEE International Symposium on Circuits and Systems, Vol. 5. Thailand (2003) 721-724 Hopfield, J.: Neurons with Graded Response have Collective Computational Properties like Those of Two-state Neurons, In: Proc. National Academy of Sciences, Vol. 81. USA (1984) 3088-3092 Hale, J.(ed): Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence RI (1988) Hale, J., Verduyn Lunel S. (eds): Introduction to Functional Differential Equations, Springer, New York (1993) Smith, H. (ed): Monotone Dynamical Systems An Introduction to the theory of Competitive and Cooperative systems, Vol.41, Mathematical Surveys and Monographys (American Mathematical Society, Providence (1995)

Global Exponential Stability in Lagrange Sense of Continuous-Time Recurrent Neural Networks Xiaoxin Liao1 and Zhigang Zeng2 1

2

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China xiaoxin [email protected] School of Automation, Wuhan University of Technology, Wuhan, Hubei, 430070, China [email protected]

Abstract. In this paper, global exponential stability in Lagrange sense is further studied for continuous recurrent neural network with three different activation functions. According to the parameters of the system itself, detailed estimation of global exponential attractive set, and positive invariant set is presented without any hypothesis on existence. It is also verified that outside the global exponential attracting set; i.e., within the global attraction domain, there is no equilibrium point, periodic solution, almost periodic solution, and chaos attractor of the neural network. These theoretical analysis narrowed the search field of optimization computation and associative memories, provided convenience for application.

1

Introduction

Lyapunov stability is one of the important properties for dynamic systems. Form a systems-theoretic point of view, the global stability of recurrent neural networks is a very interesting issue for research because of the special nonlinear structure of recurrent neural networks. From a practical point of view, the global stability of recurrent neural networks is also very important because it is a prerequisite in many neural network applications such as optimization, control, and signal processing. In recent years, the Lyapunov stability of continuous-time recurrent neural networks has received much attention in many paper [1]-[9], [12]-[14] and monographs [10],[11]. However, the stability in Lagrange sense for neural network is almost not studied. In [15], we firstly studied the global dissipativity and global exponential dissipativity of neural network. And we also provided the estimation of global attractive set and global exponential attractive set. On the basis of [15], global exponential stability in Lagrange sense is further studied for continuous recurrent neural network with three different activation functions. According to the parameters of the system itself, detailed estimation of global exponential attractive set, and positive invariant set is presented. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 115–121, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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We considered that the study of the global exponential stability in Lagrange sense at least has the following meanings: First, one of the core applications of neural network is optimization computation by finding out equilibrium point of neural networks. Once the global exponential attractive set is found, that the rough boundary of the optimization solution is set, thus provided prior knowledge. Second, the study of period solution and almost periodic solution is always the important topic of dynamic system as well as the important research topic of neural network. The necessary condition of existence on period solution and almost periodic solution is that the system is ultimately bounded. The period solution and almost periodic solution should be within the global attractive set, so that it provided more narrow boundary of the existence of them. Third, it is generally considered that continuous dynamic system will has the situation of chaos when the system is dissipative (or ultimately bounded) in large scale and has positive Lyapunov index in small scale. Recently people found the existence of chaos in neural network. Therefore, it is very necessary to study the stability in Lagrange sense for neural networks. Fourth, the global stability in Lyapunov sense on unique equilibrium point can be counted as an special case of stability in Lagrange sense by regarding equilibrium point as attractive set. Hence, form both theory and application, it is significative to study the global stability in Lagrange sense for neural networks.

2

Preliminaries

Consider a general recurrent neural network model with multiple time delays ci

n  dxi (t) = −di xi (t) + (˜ aij gj (xj (t)) + bij gj (xj (t − τij ))) + Ii , dt j=1

(1)

where i = 1, · · · , n, ci > 0 and di > 0 are positive parameters, xi is a state variable of the i−th neuron, Ii is an input (bias), a ˜ij , ˜bij are connection weights from neuron i to neuron j, and gi (·) is an activation function with gi (0) = 0, continuous and monotonically nondecreasing. When (˜bij = 0), we have ci

n  dxi (t) = −di xi (t) + a ˜ij gj (xj (t)) + Ii , i = 1, 2, · · · , n. dt j=1

(2)

Next, we define three classes of activation functions. (1) The set of sigmoidal activation functions is defined as def

S =

  g(x)|g(0) = 0, |gi (x)| ≤ ki , i = 1, 2, · · · , n, D+ gi (x) ≥ 0 .

(3)

(2) The set of Lurie type activation functions is defined as def

L = {g(x) | 0 < xi gi (xi ) ≤ ki x2i , xi  = 0, ∀xi ∈ , i = 1, 2, · · · , n}.

(4)

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117

It is obvious that the activation functions satisfied Lipschitz condition or partial Lipschitz condition are typical representatives of L-type. But there exist some L-type functions which do not satisfy Lipschitz condition or partial Lipschitz condition. (3) The set of monotone nondecreasing activation functions is denoted as def

G = {g(x) | g(x) ∈ C[R, R], D+ gi (xi ) ≥ 0, i = 1, 2, · · · , n}.

(5)

Evidently, S ⊂ G. But, S is not subset of L, L is also not subset of S. G is not subset of L, L is also not subset of G. Let Ω ⊆ n be compact set in n , Ωε ⊆ n be ε-neighborhood of Ω. Denote n \Ω be complement set of Ω in n . ρ(x, Ω) := inf y∈Ω ||x − y|| is distance between x and Ω. Let CH (t0 ) be the set of continuous function ψ : [t0 − τ, t0 ] → n with ||ψ||t0 = sups∈[t0 −τ,t0 ] |ψ(s)| ≤ H, where τ = max1≤i,j≤n {τij }. Denote x(t; t0 , ψ) as solution of (1), it means that x(t; t0 , ψ) is continuous and satisfies (1) and x(s; t0 , ψ(s)) = ψ(s), for s ∈ [t0 − τ, t0 ]. Definition 1. If there exists a compact set Ω ⊆ n such that ∀x0 ∈ n \Ω, limt→+∞ ρ(x(t), Ω) = 0, then Ω is said to be a globally attractive set. A set Ω is called positive invariant, if ∀x0 ∈ Ω implies x(t) ⊆ Ω for t ≥ t0 . An equivalent definition on globally attractive set is as follows: Definition 2. If there exist a compact set Ω ⊆ n and T > 0, such that ∀x0 ∈ n \Ω, when t ≥ t0 + T, x(t) ⊆ Ωε , then Ω is said to be a globally attractive set. Definition 3. If there exist a compact set Ω ⊆ n and α > 0, such that ∀x0 ∈ n \Ω and x(t) ∈ n \Ω, ρ(x(t), Ω) ≤ exp{−α(t − t0 )}ρ(x(0), Ω), then Ω is said to be a globally exponentially attractive set of (1). An equivalent definition on globally exponentially attractive set is as follows: Definition 4. If there exist a radically unbounded and positive definite Lyapunov function and  > 0, α > 0, such that when V (x0 ) > , V (x(t)) > , (V (x(t)) − ) ≤ exp{−α(t − t0 )}(V (x0 ) − ), then {x|V (x) ≤ } is said to be a globally exponentially attractive set of (1). Definition 5. System (1) with globally exponentially attractive set is said to be globally exponentially stable in Lagrange sense, or be globally exponentially dissipative, or be ultimately bounded.

3

g(x) ∈ S (1)

Let |gi (xi )| ≤ ki , 2Mi  Ω1 =

x|

n  i=1

=

n

˜

j=1 ((|aij |+|bij |)kj +|Ii |),

(2)

2Mi

=

n

j=1 (|aij |+|Ii |),



n ci x2i /2

≤

(1) 2 i=1 (Mi ) /εi

2 min1≤i≤n (di − ci )/ci

, where 0 < εi < di

,

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 (1) Ω2 = x| |xi | ≤ 2Mi /di , i = 1, 2, · · · , n ,  n  n   (1) Ω3 = x| ci |xi | ≤ 2Mi / min {dj /cj } , i=1

 Ω4 =

x|

n 

ci x2i /2

≤

i=1

n

1≤j≤n

i=1

(2) 2 i=1 (Mi ) /εi

2 min1≤i≤n (di − ci )/ci

 , where 0 < εi < di

,

 (2) Ω5 = x| |xi | ≤ 2Mi /di , i = 1, 2, · · · , n ,  n  n   (2) Ω6 = x| ci |xi | ≤ 2Mi / min {2dj /cj } , i=1

i=1

⎧ ˜ii , ⎨a aij = 0, ⎩ a ˜ij ,

where

1≤j≤n

i = j, a ˜ii > 0, i = j, a ˜ii ≤ 0, i = j.

Theorem 1. If g(x) ∈ S, then Ωi (i = 1, 2, 3) are globally exponentially attractive

sets and positive invariant sets of (1). Hence, Ω123 = Ω1 Ω2 Ω3 is a globally exponentially attractive set and positive invariant set of (1). Theorem 2. If g(x) ∈ S, then Ωi (i = 4, 5, 6) are globally exponentially attractive

sets and positive invariant sets of (2). Hence, Ω456 = Ω4 Ω5 Ω6 is a globally exponentially attractive set and positive invariant set of (2). Remark 1. Globally attractive sets, which obtained by using Theorems 1 and 2, may not be the best; i.e., people can obtain better estimation by other methods. But, it is not importance to improve on the size of attractive set. It is importance to judge whether neural network (1) or (2) is globally exponentially stable in Lagrange sense or not.

4

g(x) ∈ L

n Theorem 3. If g(x) ∈ L, and −dj + i=1 |˜ aij |kj < 0, then Ω7 is a globally exponentially attractive set and positive invariant set of (2), where  n  n  i=1 |Ii | Ω7 = x| ci |xi | ≤ , ξ1 min1≤i≤n di /ci i=1 ⎧ ⎨ ξ1 = sup

⎩

ξ, −

dj (1 − ξ) + a ˜jj + kj

n  i=1,i =j

|˜ aij | ≤ 0, j = 1, 2, · · · , n, 0 < ξ < 1

⎫ ⎬ ⎭

.

Theorem 4. If g(x) gn (·), then k = k1 = k2 = · · · = kn . n∈ L, g1 (·) = g2 (·) = · · · = + If −dj + a ˜jj + i=1,i |˜ a |k < 0, and D g(x) ≥ 0, then Ω8 is a globally ij =j exponentially attractive set and positive invariant set of (2), where

Global Exponential Stability in Lagrange Sense

 Ω8 =

5

n 

max1≤i≤n |Ii | max1≤i≤n ci n x| ci |xi | ≤ min {d − aij |k} 1≤i≤n i j=1 |˜ i=1

119

 .

g(x) ∈ G

Theorem 5. If g(x) ∈ G, and A˜ + A˜T is negative definite, then Ω9 is a globally exponentially attractive set and positive invariant set of (2), where  n  n 2   xi i=1 Ii Ω9 = x| ci gi (xi )dxi ≤ , 4ε min1≤i≤n di /ci 0 i=1  ε = sup , A˜ + A˜T + En is negative semidefinite, 0 < < 1 . Theorem 6. If g(x) ∈ G, and A˜ + A˜T is negative semidefinite, then Ω10 is a globally exponentially attractive set and positive invariant set of (2), where  n    xi 2 Ω10 = x| ci gi (xi )dxi ≤ , min1≤i≤n di /ci 0 i=1 =

max

|xi |=2|Ii |

n  i=1

 ci

xi

gi (xi )dxi . 0

 Theorem 7. If g(x) ∈ G, and a ˜jj + ni=1,i aij | ≤ 0, then Ω11 is a globally =j |˜ exponentially attractive set and positive invariant set of (2), where  n  n  i=1 |Ii | Ω11 = x| ci |xi | ≤ . min1≤i≤n di /ci i=1 Remark 2. According to Theorems 1-7, in outside the global exponential attracting set; i.e., within the global attraction domain, there is no equilibrium point, periodic solution, almost periodic solution, and chaos attractor of the neural network.

6

Example

Consider the neural networks studied by Q. Li and X. Yang in [16]: x(t) ˙ = −x(t) + W F (x(t)), where F (x(t)) = (f (x1 (t)), f (x2 (t)), f (x3 (t)))T , f (x) = (|x + 1| − |x − 1|)/2, ⎛ ⎞ ⎛ ⎞ w11 , w12 , 0 1.2, −1.6, 0 A = ⎝ w21 , w22 , w23 ⎠ = ⎝ 1.2, 1 + p, 0.9 ⎠ . 0, w32 , w33 0, 2.2, 1.5

(6)

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(2)

When p = −0.12, 2M1 = 2(|w11 | + |w12 |) = 5.6, 2M2 = 2(|w21 | + |w22 | + (2) |w23 |) = 5.96, 2M3 = 2(|w32 | + |w33 |) = 7.4. Hence Ω12 = {(x1 , x2 , x3 ), |x1 | ≤ 5.6, |x2 | ≤ 5.96, |x3 | ≤ 7.4, } is a globally exponentially attractive set and positive invariant set of (6). Similarly, when p = −0.03, p = 0, p = 0.12, Ω13 = {(x1 , x2 , x3 ), |x1 | ≤ 5.6, |x2 | ≤ 6.14, |x3 | ≤ 7.4, }, Ω14 = {(x1 , x2 , x3 ), |x1 | ≤ 5.6, |x2 | ≤ 6.4, |x3 | ≤ 7.4, } Ω15 = {(x1 , x2 , x3 ), |x1 | ≤ 5.6, |x2 | ≤ 6.44, |x3 | ≤ 7.4, } are globally exponentially attractive sets and positive invariant sets of (6), respectively.

Acknowledgement This work was supported by the Natural Science Foundations of China under Grant 60274007 and 60474011.

References 1. Hopfield, J.J.: Neurons with Graded Response Have Collective Computational Properties Like Those of Two-state Neurons. Proc. Natl. Academy Sci. 81 (1984) 3088-3092 2. Chua, L.O., Yang, L.: Cellular Neural Networks: Theory. IEEE Trans. Circuits Syst. 35 (1988) 1257-1272 3. Liao, X.X.: Stability of Hopfield-type Neural Networks (I). Science in China, Series A 38 (1995) 407-418 4. Liao, X.X.: Mathematical Theory of Cellular Neural Networks (I). Science in China, Series A 24 (1994) 902-910 5. Liao, X.X.: Mathematical Theory of Cellular Neural Networks (II). Science in China, Series A 24 (1994) 1037-1046 6. Liao, X.X., Wang, J.: Algebraic Criteria for Global Exponential Stability of Cellular Neural Networks with Multiple Time Delays. IEEE Trans. Circuits and Systems I. 50 (2003) 268-275 7. Liao, X.X., Wang, J., Zeng, Z.G.: Global Asymptotic Stability and Global Exponential Stability of Delayed Cellular Neural Networks. IEEE Trans. on Circuits and Systems II, Express Briefs 52 (2005) 403-409 8. Shen, Y., Jiang, M.H., Liao, X.X.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Time-varying Delays and Continuously Distributed Delays. In: Wang, J., Liao, X.F., Zhang, Y. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 156-161 9. Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Syst. I 51 (2004) 2313-2324 10. Yi, Z., Tan, K.K.: Convergence Analysis of Recurrent Neural Networks. Kluwer Academic Publishers (2004) 11. Michel, A.N., Liu, D.: Qualitative Analysis and Synthesis of Recurrent Neural Networks. Marcel Dekker, New York (2002) 12. Xu, D.Y., Yang, Z.C.: Impulsive Delay Differential Inequality and Stability of Neural Networks. J. Math. Anal. and Appl. 305 (2005) 107-120

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13. Liao, X.F., Yu, J.B.: Robust Stability for Interval Hopfield Neural Networks with Time Delay. IEEE Transactions on Neural Networks 9 (1998) 1042-1045 14. Forti, M., Manetti, S., Marini, M.: Necessary and Sufficient Condition for Absolute Stability of Neural Networks. IEEE Trans. Circuits and Syst. I 41 (1994) 491-494 15. Liao, X.X., Wang, J.: Global Dissipativity of Continuous-time Recurrent Neural Networks with Time Delay. Physical Review E 68 (2003) 016118 16. Li, Q.D., Yang, X.S.: Complex Dynamics in A Simple Hopfield-type Neural Network. In: Wang, J., Liao, X.F., Zhang, Y. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 357-362

Global Exponential Stability of Recurrent Neural Networks with Time-Varying Delay Yi Shen1 , Meiqin Liu2 , and Xiaodong Xu3 1

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China [email protected] 2 College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China [email protected] 3 College of Public Administration, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China [email protected]

Abstract. A new theoretical result on the global exponential stability of recurrent neural networks with time-varying delay is presented. It should be noted that the activation functions of recurrent neural network do not require to be bounded. The presented criterion, which has the attractive feature of possessing the structure of linear matrix inequality, is a generalization and improvement over some previous criteria.

1

Introduction

In recent years, recurrent neural networks are widely studied in [1-2], because of their immense potentials of application perspective. The Hopfield neural networks and cellular neural networks are typical representative recurrent neural networks among others, and have been successfully applied to signal processing, especially in image processing, and to solving nonlinear algebraic and transcendental equations. Such applications rely heavily on the stability properties of equilibria. Therefore, the stability analysis of recurrent neural networks is important from both theoretical and applied points of view[1-8]. In this paper, a new theoretical result on the global exponential stability of recurrent neural networks with time-varying delay is presented. It should be noted that the activation functions of recurrent neural network do not require to be bounded. The presented criterion, which has the attractive feature of possessing the structure of linear matrix inequality, is a generalization and improvement over some previous criteria in Refs.[3-8]. The model of recurrent neural network to be considered herein is described by the state equation x(t) ˙ = −Dx(t) + Af (x(t)) + Bf (x(t − δ(t))) + u,

(1)

where x(t) = [x1 (t), · · · , xn (t)]T denotes the state vector, f (x(t)) = [f1 (x1 (t)), · · · , fn (xn (t))]T is the output vector, D = diag(d1 , · · · , dn ) > 0 is the self feedback J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 122–128, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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123

matrix, A = (aij )n×n is the feedback matrix, B = (bij )n×n is the delayed feedback matrix, u = [u1 , · · · , un ]T is the constant vector, and δ : R+ → [0, τ ] is the delay, τ > 0 and δ  (t) ≤ 0. The activation function fi (·) satisfies the following condition (H) There exist ki > 0 such that for any θ, ρ ∈ R 0 ≤ (fi (θ) − fi (ρ))(θ − ρ)−1 ≤ ki .

(2)

It is easy to see that these activation functions satisfied (H) are not necessarily bounded. However, some usual sigmoidal functions and piecewise linear functions which are employed in [3-8] are bounded on R.

2

Main Results

Theorem 2.1. If there exist positive definite symmetric matrices P = [pij ] ∈ Rn×n and G = [gij ] ∈ Rn×n and a positive definite diagonal matrix Q = diag(q1 , · · · , q2 ) such that ⎛ ⎞ P D + DT P −P A −P B 2QDK −1 − QA − AT Q − G −QB ⎠ > 0, (3) M = ⎝ −AT P T T −B P −B Q G where K = diag(k1 , . . . , kn ) (see (2)), then the network (1) has the unique equilibrium point x∗ and it is globally exponentially stable. Proof. By (3), it follows  2QDK −1 − QA − AT Q − G −B T Q

−QB G

 > 0.

(4)

From the Schur Complement (see [1]) and (4), we obtain 2QDK −1 − QA − AT Q − G − QBG−1 B T Q > 0.

(5)

By (5) and G + QBG−1 B T Q − B T Q − QB ≥ 0, we get 2QDK −1 − Q(A + B) − (A + B)T Q > 0.

(6)

Set J(x) = −Dx + (A + B)f (x) + u, then J is injective on Rn . Otherwise, there is x  = y, x, y ∈ Rn such that J(x) = J(y), that is to say D(x − y) = (A + B)(f (x) − f (y)).

(7)

It is obvious that f (x)  = f (y), otherwise, x = y by (7). Multiplying the both sides of (7) by 2(f (x) − f (y))T Q, we obtain 2(f (x) − f (y))T QD(x − y) = 2(f (x) − f (y))T Q(A + B)(f (x) − f (y)).

(8)

By (2), we have 2(f (x) − f (y))T QD(x − y) ≥ 2(f (x) − f (y))T QDK −1 (f (x) − f (y)).

(9)

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Therefore, by (8) and (9), we get (f (x) − f (y))T (2QDK −1 − Q(A + B) − (A + B)T Q)(f (x) − f (y)) ≤ 0. (10) Since f (x) − f (y)  = 0, by (6), we have (f (x) − f (y))T (2QDK −1 − Q(A + B) − (A + B)T Q)(f (x) − f (y)) > 0. (11) It is obvious that (10) contradicts (11). Therefore, J is injective. Now one can prove lim|x|−→+∞ |J(x)| = +∞. Because u and f (0) are constants, it only needs to prove lim|x|−→+∞ |J(x)| = +∞, where J(x) = −Dx + (A + B)(f (x) − f (0)). Firstly, we will show that limx−→+∞ |J(x)| = +∞. If |f (x)| is bounded on [0, +∞), then limx−→+∞ |J(x)| = +∞. Assume that |f (x)| is unbounded on [0, +∞), so there exist series xn ∈ [0, +∞) such that limn−→+∞ |f (xn )| = +∞. By (2), f (x) is continuous, monotonic and increasing, and assume that {xn } is monotonic and increasing, then we have limn−→+∞ xn = +∞ , limn−→+∞ f (xn )= +∞. Therefore, there is m for any sufficiently large x ∈ [0, +∞) such that xm ≤ x, f (xm ) ≤ f (x). Since limm−→+∞ xm = +∞, and limm−→+∞ f (xm ) = +∞, we can get limx−→+∞ f (x) = +∞, that is to say, limx−→+∞ |f (x)| = +∞. Therefore, limx−→+∞ |f (x) − f (0)| = +∞. And by (2), we have 2(f (x) − f (0))T QJ(x) ≤ −μ|f (x) − f (0)|2 ,

(12)

where μ = λmin (2QDK −1 − Q(A + B) − (A + B)T Q), it follows from (6) that μ > 0. So, by (12), we get μ|f (x) − f (0)|2 ≤ 2Q|f (x) − f (0)||J(x)|.

(13)

limx−→+∞ |f (x) − f (0)| = +∞ together with (13) implies limx−→+∞ |J(x)| = +∞. Similarly, we can prove limx−→−∞ |J (x)| = +∞. Hence, we obtain lim|x|−→+∞ |J(x)| = +∞. By Ref.[2], it follows that J(x) is a homeomorphism from Rn to itself. Therefore, the equation J(x) = 0 has a unique solution; i.e., the system (1) has a unique equilibrium x∗ . One will prove that the unique equilibrium x∗ = (x∗1 , · · · , x∗n )T of the system (1) is globally exponentially stable. Since x∗ is the equilibrium of the system (1), −Dx∗ + (A + B)f (x∗ ) + u = 0, by (1), we have y(t) ˙ = −Dy(t) + Ag(y(t)) + Bg(y(t − δ(t))),

(14)

where y(·) = [y1 (·), · · · , yn (·)]T = x(·) − x∗ is the new state vector, g(y(·)) = [g1 (y1 (·)), · · · , gn (yn (·))]T represents the output vector of the transformed system, and gi (yi (·)) = fi (yi (·) + x∗i ) − fi (x∗i ), i = 1, 2, · · · , n. Obviously, g(0) = 0, y ∗ = 0 is a unique equilibrium of the system (14). Therefore the stability of the equilibrium x∗ of the system (1) is equivalent to the one of the trivial solution of the system (14). Choose the following Lyapunov functional yi (t) t n  V (y(t)) = y T (t)P y(t) + 2 qi gi (s)ds + g(y(s))T Gg(y(s))ds, i=1

0

t−δ(t)

Global Exponential Stability of Recurrent Neural Networks

125

The time derivative of V (y(t)) along the trajectories of (14) turns out to be V˙ (y(t)) ≤ −λ|y(t)|2 , (15) where λ = λmin (M ) > 0, (15) has been obtained by (2) and δ  (t) ≤ 0. Set h(ε) = −λ + λmax (P )ε + QKε + GK2eετ τ ε. Owing to h (ε) > 0, h(0) = −λ < 0, h(+∞) = +∞, there exists a unique ε such that h(ε) = 0; i.e., there is ε > 0 satisfies − λ + λmax (P )ε + QKε + GK2eετ τ ε = 0.

(16)

From the definition of V and (2), it follows |V (y(t))| ≤ (λmax (P ) + QK)|y(t)| + GK 2

t

|y(s)|2 ds. (17)

2 t−δ(t)

For ε > 0 satisfying (16), by (15) and (17), we get (eεt V (y(t))) = εeεt V (y(t)) + eεt V˙ (y(t)) ≤ eεt (−λ + λmax (P )ε + QKε)|y(t)|2 t εt 2 +εe GK |y(s)|2 ds.

(18)

t−δ(t)

Integrating the both sides of (18) from 0 to an arbitrary t ≥ 0, we can obtain eεt V (y(t)) − V (y(0)) t ≤ eεs (−λ + λmax (P )ε + QKε)|y(s)|2ds 0 t s + εeεs GK2ds |y(r)|2 dr. And



t



s

εeεs GK2ds 0

(19)

s−δ(s)

0

|y(r)|2 dr ≤ s−δ(s)

t

−τ

εGK2τ eε(r+τ ) |y(r)|2 dr. (20)

Substituting (20) into (19) and applying (16), we have eεt V (y(t)) ≤ C, t ≥ 0. where C = V (y(0)) + (21), we get

0 −τ

(21)

εGK2τ eετ |y(r)|2 dr. By the definition of V and

|y(t)| ≤

C e−εt , t ≥ 0. λmin (P )

(22)

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This implies the trivial solution of (14) is globally exponentially stable and its exponential convergence rate ε is decided by (16). Remark 1. In Theorem 2.1, we do not suppose that the activation function is bounded and delay is constant. However, in [3], several asymptotic stability results are derived by the bounded piecewise linear function and constant delay while our result ensures the globally exponential stability of the system (1) in similar conditions. In addition, the network (1) is even more general system than the one in [3]. Corollary 2.2. Assume that K = D = I (identity matrix) in (1), then the neural network (1) has the unique equilibrium point and it is globally exponentially stable if there exist matrices Q, T and an nonnegative constant β such that one of the following conditions is satisfied (A1) QA+AT Q+T < 0, 2Q+T −I −QBB T Q ≥ 0, where Q = diag(q1 , . . . , qn ) is a positive definite diagonal matrix; and T = (tij )n×n is a symmetric positive definite matrix. √ (A2) A + AT + βI < 0, B ≤ √2β, β > 1. T (A3) A + A + βI < 0, B ≤√ 1 + β, 0 ≤ β ≤ 1. (A4) A + AT + βI < 0, B ≤ 1 + γ + β, β > 0, where γ is the minimum eigenvalue of −(A + AT + βI) (A5) A + AT < 0, B ≤ 1. √ (A6) A + AT + βI < 0, B ≤ √1 + β, β ≥ 0. (A7) A + AT + βI < 0, B ≤ 2β, β > 0. Proof. It is obvious that (A4) is deduced by one of the conditions (A3),(A5) and (A6), furthermore, (A2) is obtained by (A7). Therefore, it only needs to prove that the conclusion holds if one of the conditions (A1), (A2) and (A4) is satisfied. Let K = D = I, P = αI, G = αB T B + I, α > 0, by Lemma A (see Appendix), then M > 0 in (3) is equivalence to     S −QB αAT 1 −1  αA αB − α −B T Q αB T B + I αB T 2   1 −1  αAT αA −αB > 0, =U+ α (23) −αB T 2   S − αAT A −QB where U = , S = 2Q−QA−AT Q−αB T B−I. Obviously, −B T Q I if U > 0, then (23) holds. By Lemma A, U > 0 is equivalence to 2Q − QA − AT Q − αB T B − I − αAT A − QBB T Q > 0.

(24)

Case 1: It is obvious that if α is sufficient small, then, by (A1), there exists a matrix Q such that (24) holds. Case 2: Take Q = β1 I, β > 1. If α is sufficient small, then, there is a matrix Q satisfying (24) by (A2).

Global Exponential Stability of Recurrent Neural Networks

Case 3: Set Q =

1 β+1 I, β

127

> 0, then

2Q − QA − AT Q − αB T B − I − αAT A − QBB T Q I A + AT + βI BB T β(A + AT + βI) = − − − − αB T B − αAT A 2 2 1+β (1 + β) (1 + β) (1 + β)2 Take α sufficiently small, by the above formula and the condition (A4), it is easy to follow that there exists a matrix Q such that (24) holds. The proof is completed. Remark 2. Assume that D = I, fi (xi ) = 12 (|xi + 1| − |xi − 1|), δ(t) = τ (constant) in (1), then Corollary 2.2 reduces to the main results in [3-8]. However, Refs.[3-8] only prove the asymptotic stability of the network (1). Furthermore, the following example 3.1 satisfies the conditions of the Theorem 2.1 while it doesn’t satisfy the ones of Corollary 2.2. Therefore, the Theorem 2.1 is improvement and generalization of Refs.[3-8].

3

Conclusion

In this paper, we present theoretical results on the global exponential stability of a general class of recurrent neural networks with time-varying delay. These conditions obtained here are easy to be checked in practice, and are of prime importance and great interest in the design and the applications of network. The criterions are more general than the respective criteria reported in [3-8].

Acknowledgments The work was supported by Natural Science Foundation of Hubei (2004ABA055) and National Natural Science Foundation of China (60574025, 60074008).

References 1. Cao, J., Wang, J.: Global Asymptotic and Robust Stability of Recurrent Neural Networks with Time Delays. IEEE Trans. Circuits Syst. I 52 (2005) 417-426 2. Zeng, Z., Wang, J., Liao, X.: Global Exponential Stability of a General Class of Recurrent Neural Networks with Time-Varying Delays. IEEE Trans. Circuits Syst. 50 (2003) 1353-1359 3. Singh, V.: A Generalized LMI-Based Approach to the Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Networks 15 (2004) 223225 4. Arik, S.: An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Networks 13 (2002) 1239-1242 5. Liao, T., Wang, F.: Global Stability for Cellular Neural Networks with Time Delay. IEEE Trans. Neural Networks 11 (2000) 1481-1484

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6. Arik, S., Tavsanoglu, V.: On the Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47 (2000) 571-574 7. Cao, J.: Global Stability Conditions for Delayed CNNs. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48 (2001) 1330-1333 8. Arik, S.: An Improved Global Stability Result for Delayed Cellular Neural Networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49 (2002) 1211-1214

New Criteria of Global Exponential Stability for a Class of Generalized Neural Networks with Time-Varying Delays Gang Wang, Hua-Guang Zhang, and Chong-Hui Song School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110004, People’s Republic of China wanggang [email protected], [email protected]

Abstract. In this paper, we essentially drop the requirement of Lipschitz condition on the activation functions. By employing Lyapunov functional and several new inequalities, some new criteria concerning global exponential stability for a class of generalized neural networks with time-varying delays are obtained, which only depend on physical parameters of neural networks. Since these new criteria do not require the activation functions to be differentiable, bounded or monotone nondecreasing and the connection weight matrices to be symmetric, they are mild and more general than previously known criteria.

1 Introduction In recent years, the stability problem of time-delayed neural networks has been deeply investigated for the sake of theoretical interest as well as applications (see, e.g., [1]-[4], [7]). Due to the finite switching speed of amplifies, time delays are likely to be present in models of electronic networks. Thus a delay parameter must be introduced into the system model. All of the results above assume that the activation functions satisfy the Lipschitz condition. However, in many evolutionary processes as well as optimal control models and flying object motion, there are many bounded monotone-nondecreasing signal functions which do not satisfy the Lipschitz condition. For instance, in the simplest case of the pulse-coded signal function which has received much attention in many fields of applied sciences and engineering, an exponentially weighted time average of sampled pulses is often used which does not satisfy the Lipschitz condition. Therefore, it is very important and, in fact, necessary to study the issue of the stability of such a dynamical neural systems whose activation functions do not satisfy the Lipschitz condition. Feng and Plamondon [5] first proposed the hypothesis (H1) (see Sec. 2) which was different from the Lipschitz condition. The proposed hypothesis (H1) is significant, and allows us to include a variety of activation functions whether satisfying Lipschitz condition or not. In [5], some global asymptotic stability criteria are obtained which only depend on physical parameters of neural networks. Compared with the criteria with lots of unknown parameters, due to no degrees of freedom to be tuned, these criteria can be checked easily and quickly. Therefore the criteria only depending on physical parameters of neural networks are more worth being improved. In the spirit of [5], J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 129–134, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Zhang et al. [6] proposed some new global asymptotic stability criteria which are less restrict than those in [5]. But due to lots of unknown parameters, it is difficult to tune the degrees of freedom and tastes plenty of time. On the other hand, [5] and [6] only obtained some global asymptotic stability criteria with constant delays. To the best of our knowledge, global exponential stability with time-varying delays under the hypothesis (H1) are seldom considered. The purpose of this paper is to improve the results in [5]. We present some new criteria concerning the global exponential stability for a class of generalized neural networks with time-varying delays, which only depend on physical parameters of neural networks under the hypothesis (H1). The main advantages of the obtained exponential stability conditions include: (i) We essentially drop the requirement of Lipschitz condition on the activation functions, and do not require the activation functions to be differentiable, bounded or monotone nondecreasing and the connection weight matrices to be symmetric. (ii) The stability conditions only depend on physical parameters of neural networks, which can be checked easily and quickly. (iii) The conditions obtained here are less restrictive, and are a generalization of some previous works.

2 Preliminary We consider a class of generalized neural networks with time-varying delays described by the following nonlinear differential equations n n   dxi (t) = −di (xi (t)) + aij fj (xj (t)) + bij gj (xj (t − τij (t))) + Ii , dt j=1 j=1

xi (s) = ϕi (s), s ∈ [−τ, 0],

i = 1, 2, · · · , n,

(1)

where xi (t) is the state, fj (·) and gj (·) denote the activation functions, aij and bij denotes the constant connection weight and the constant delayed connection weight, respectively, τij (t) is the time-varying transmission delay, which is nonnegative, bounded, and differentiable, τ = max τij (t), di (·) > 0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs with di (0) = 0. Suppose system (1) has at least an equilibrium. Let zi (t) = xi (t) − x∗i , where x∗ = ∗ (x1 , x∗2 , · · · , x∗n ) is an equilibrium of system (1), then system (1) reduces to n n   dzi (t) = −Di (zi (t)) + aij αj (zj (t)) + bij βj (zj (t − τij (t))), dt j=1 j=1

(2)

where Di (zi (t)) = di (zi (t) + x∗i ) − di (x∗i ),αj (zj (t)) = fj (zj (t) + x∗j ) − fj (x∗j ), βj (zj (t − τij (t))) = gj (zj (t − τij (t)) + x∗j ) − gj (x∗j ). Remark 1. Obviously, (0, 0, · · · , 0)T is an equilibrium point of (2). Therefore, the global exponential stability analysis of the equilibrium point x∗ of (1) can now be transformed to the global exponential stability analysis of the trivial solution z = 0 of (2).

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131

Throughout this paper, we assume that the activation functions αj , βj (j = 1, 2, · · · , n) satisfy the following property: (H1) the function αj , βj (j = 1, 2, · · · , n) satisfy , zj αj (zj ) > 0(zj  = 0); zj βj (zj ) > 0(zj  = 0) and that there exist real numbers mj and nj such that αj (zj ) , zj zj  =0

βj (zj ) . zj zj  =0

mj = sup

(3)

nj = sup

Remark 2. Note that the hypothesis (H1) is weaker than the Lipschitz condition which is mostly used in literature. Further, if fj , gj for each j = 1, 2, · · · , n are Lipschitz functions, then fj and gj for each j = 1, 2, · · · , n can be replaced by the respective Lipschitz constants. (H2) There exist positive constants Di and Di , i = 1, 2, · · · , n such that 0 < Di ≤

di (xi ) − di (yi ) ≤ Di , xi − yi

We denote ψ − ϕ∗ 2 =

sup

n 

−τ ≤s≤0 i=1

f or all xi  = yi .

|ψi (s) − ϕ∗i |2 ,

(4)

where ϕ∗ = (ϕ∗1 , ϕ∗2 , · · · , ϕ∗n ) is the initial value of (1). Definition 1. The equilibrium x∗ = (x∗1 , x∗2 , · · · , x∗n ) of (1) is said to be globally exn  ponentially stable, if there exist ε > 0 and M (ε) ≥ 1, such that (xi (t) − x∗i )2 ≤ i=1

M (ε)ψ − ϕ∗ 2 e−εt for all t > 0, and ε is called the degree of exponential stability. Definition 2. Let f (t) : R → R be a continuous function. The upper right dini derivative D+ f (t) is defined as 1 (f (t + h) − f (t)). h Lemma 1. Suppose that a, b and c are constant, a > 0, b ≥ 0, and 0 ≤ c ≤ ∞ then D+ f (t) = lim

h→0+

1 b2 − ac2 + bc ≤ − ac2 + . 2 2a

(5)

− (ac − b)2 ≤ 0.

(6)

1 b2 − ac2 + bc ≤ − ac2 + . 2 2a

(7)

Proof. According to

we obtain

The proof is completed. Lemma 2. (Cauchy inequality) Suppose that ai and bi (i = 1, 2, ...n) are positive real numbers, then n n n    ( ai bi )2 ≤ ( a2i )( b2i ) (8) i=1

i=1

i=1

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3 Main Results Theorem 1. Assume that fj , gj (j = 1, 2, · · · , n) satisfy the hypothesis (H1) above, τij (t) ≤ 0, and suppose further that n  n  1 2 2 1 (ajk mk + b2jk n2k ) < D i , D 2 j j=1

(9)

k=1

for all t > 0 and i = 1, 2, · · · , n, then the equilibrium point x∗ of the system (1) is globally exponentially stable. Proof. (9) implies that we can always choose a small constant ε > 0 such that n  n  a2jk m2k + b2jk n2k eετ 1 ε − (D i − ) + < 0, 2 2 Dj − ε2 j=1

(10)

k=1

where τ = max τij (t), i, j = 1, 2, · · · , n. We choose a Lyapunov functional for system (2) as V (z, t) =

n 

(11)

(Vi1 (z, t) + Vi2 (z, t)),

i=1

where Vi1 (z, t) = zi2 (t)eεt ,

(12)

and 2 Vi2 (z, t) =

n  j=1

b2ij n2j



t t−τij (t)

n  j=1

Di −

zj2 (s)eε(s+τij (t)) ds (13)

.

ε 2

Calculating the upper right derivative Vi1 (z, t) along the solutions of system (2). From (H1) and (H2), one can derive that D+ Vi1 (z, t)(2)   n n   1 ε 2 εt ≤ 2e − 2( Di − )zi (t) + |zi (t)| |aij |mj |zj (t)| + |zi (t)| |bij |nj |zj (t − τij (t))| . 2 4 j=1 j=1 According to Lemma 1, i.e., −ac2 + bc ≤ − 21 ac2 +

b2 2a ,

we obtain

D+ Vi1 (z, t)(2)  ≤ 2eεt

1 ε − ( Di − )zi2 (t) + 2 4

(

n 

|aij |mj |zj (t)|)2

j=1

Di −

ε 2

( +

n 

j=1

|bij |nj |zj (t − τij (t))|)2  Di −

ε 2

. (14)

New Criteria of Global Exponential Stability

133

By Lemma 2, we obtain D+ Vi1 (z, t)(2)  ≤ 2e

εt

n 

1 ε − ( Di − )zi2 (t) + 2 4

j=1

a2ij m2j

n 

j=1 Di − 2ε

n 

zj2 (t) +

j=1

b2ij n2j

n  j=1

zj2 (t − τij (t)) 

Di −

.

ε 2

(15) Calculating the upper right derivative Vi2 (z, t) along the solutions of system (2), one can derive that  n  n n   2  2 b2ij n2j eεt zj (t)eετ − zj2 (t − τij (t)) D+ Vi2 (z, t)(2) ≤

j=1

j=1

j=1

Di −

.

ε 2

(16)

It follows from (11),(15) and (16) that  n  n  n   a2jk m2k + b2jk n2k eετ 2 1 ε + εt D V (z, t)(2) ≤ 2e − (D i − ) + zi (t) < 0. 2 2 Dj − 2ε i=1 j=1 k=1

i.e., V (z, t) ≤ V (z, 0),

t ≥ 0.

(17)

Note that V (z, t) ≥

n 

zi2 (t)eεt ,

(18)

 b2ij n2j ||ψ − ϕ∗ ||2 . ε(Di − 2ε )

(19)

i=1

and  V (z, 0) ≤ 1 + 2e

ετ

n  n  i=1 j=1

From (17), (18) and (19), we have n 

zi2 (t)

∗ 2 −εt

≤ M (ε)||ψ − ϕ || e

 ,

i=1

M (ε) = 1 + 2e

ετ

n  n  i=1 j=1

 b2ij n2j ≥ 1. ε(Di − 2ε )

Because zi = xi − x∗i , we obtain n 

(xi (t) − x∗i )2 ≤ M (ε)||ψ − ϕ∗ ||2 e−εt .

(20)

i=1

So, for all t > 0, the equilibrium point x∗ of system (1) is globally exponentially stable. The proof is completed.

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If we derive di (xi ) = di xi , di > 0, system (1) can be described as follows n n   dxi (t) = −di xi (t) + aij fj (xj (t)) + bij gj (xj (t − τij (t))) + Ii . (21) dt j=1 j=1

Then we have the following result. Corollary 1. Assume that fj , gj (j = 1, 2, · · · , n) satisfy the hypothesis (H1) above, τij (t) ≤ 0, and suppose further that n  n  1 2 2 1 (ajk mk + b2jk n2k ) < di d 2 j j=1 k=1

for all t > 0 and i = 1, 2, · · · , n, then the equilibrium point x∗ of the system (21) is globally exponentially stable.

4 Conclusion In this paper, some new criteria concerning global exponential stability of a class generalized neural networks with time-varying delays are obtained. All these new criteria only depend on physical parameters of neural networks, so can be checked easily. Furthermore, the obtained results improve upon some existing results, and are less conservative than some previous results.

Acknowledgment This work was supported by the National Natural Science Foundation of China (60325311, 60534010, 60572070, 60504006) and the Program for Changjiang Scholars and Innovative Research Team in University.

References 1. Arik, S., Tavanoglu, V.: Equilibrium Analysis of Delayed CNNs. IEEE Trans. Circuits syst. I. 45(2) (1998) 168-171 2. Liang, J.L., Cao, J.D.: Global Asymptotic Stability of Bi-Directional Associative Memory Networks with Distributed Delays. Applied Mathematics and Computation. 152(2) (2004) 415-424 3. Zeng, Z.G., Wang, J., Liao, X.X.: Global Asymptotic Stability and Global Exponential Stability of Neural Networks with Unbounded Time-varying Delays. IEEE Trans. on Circuits and System II. 52(3) (2005) 168-173 4. Ensari, T., Arik, S.: Global Stability of a Class of Neural Networks with Time-varying Delay. IEEE Trans. on Circuits And System Ii. 52(3) (2005) 126-130 5. Feng, C.H., Plamondon, R.: On the Stability of Delay Neural Networks System. Neural Networks. 14 (2001) 1181-1118 6. Zhang, Q., Ma, R.N., Wang, C., Xu, J.: On the Global Stability of Delayed Neural Networks. IEEE Trans. On Automatic Control. 48(5) (2003) 794-797 7. Zhang, H.G., Wang, G.: Global Exponential Stability of a Class of Generalized Neural Networks with Variable Coefficients and Distributed Delays. In: Huang, D.S., Zhang, X.P, Huang, G.B. (eds.): Advances in Intelligent Computing. Lecture Notes in Computer Science, Vol. 3644. Springer-Verlag, Berlin Heidelberg He Fei (2005) 807-817

Dynamics of General Neural Networks with Distributed Delays Changyin Sun1,2 and Linfeng Li1 1

College of Electrical Engineering, Hohai University, Nanjing 210098, P.R. China 2 Research Institute of Automation, Southeast University, Nanjing 210096, P.R. China [email protected]

Abstract. The paper introduces a general class of neural networks with periodic inputs. By constructing a Lyapunov functional and the Halanay-type inequality separately, we obtain easily verifiable sufficient conditions ensuring that every solutions of the delayed neural networks converge exponentially to the unique periodic solutions. The results obtained can be regarded as a generalization to the discrete case of previous results.

1 Introduction The dynamical properties of Hopfield neural networks play primarily important roles in their applications [1-8]. Therefore, one of the most investigated active issues has been to obtain conditions ensuring neural networks possesses a unique equilibrium point which globally converges [9-21]. Since an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period [12-14], the analysis of periodic solutions of neural networks may be considered to be more general sense than that of equilibrium points. In practice, time delays are unavoidably encountered in implementation of neural networks. They are often the source of oscillation and instability. Hence, exploring convergence of delayed neural networks is an important topic, see, e.g., [18-20], and references therein. The property of global convergence prevents a delayed neural network from the risk of getting stuck at some local minimum of the energy function, which has been demonstrated in a number of practical applications reported in the literature [20]. Among the most previous results of the dynamical analysis of neural systems, a frequent assumption is that the time delays are discrete. However, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, there will be a distribution of propagation delays. This means that the signal propagation is no longer instantaneous and is better represented by a model with continuously distributed delays. In this paper, our focus is on periodicity of continuous-time neural networks with continuously distributed delays. Further, we point out that the discrete delay case can be included in our models by choosing suitable kernel functions. In this paper, our focus is on periodicity of general delayed neural networks. By using the Lyapunov functional and the Halanay-type inequality separately, we will J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 135 – 140, 2006. © Springer-Verlag Berlin Heidelberg 2006

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derive some easily checkable conditions to guarantee exponential periodicity of neural networks with delays. The rest of the paper is organized as follows. Hopfield neural network model with distributed delays and preliminaries will be given in Section 2. Main results about Periodicity of neural networks will be presented and discussed in Section 3. In Section 4, two examples are given to illustrate the correctness of our results witch are independent of each other. Finally, Section 5 is the conclusion.

2 Neural Network Model and Preliminaries Consider the model of continuous-time neural networks described by the following integrodifferential equations of the form

(

n

)

∞

x&i (t) = −di xi (t) + ∑aij f j ∫0 Kij (s)xj (t − s)ds + Ii (t) , t ≥ 0 j=1

(1)

where x i (t ) is the state vector of the i th unit at time t , a ij are constants. For simplicity, let D be an n × n constant diagonal matrix with diagonal elements d i >0, A = (a ij )

i = 1,2, K, n,

are

n×n

constant

interconnection

matrices,

f (x) = ( f1(x1), f2 (x2 ),K, fn (xn ))T : R → R is a nonlinear vector-valued activation function, n

n

I(t) =(I1(t),I2(t),K, In(t))T ∈R n is an input periodic vector function with period ω , i.e., there exists a constant ω > 0 such that I i (t + ω ) = I i (t ) (i = 1,2, L , n) for all t ≥ 0 , and x = ( x1 , x 2 , K , x n ) T ∈ R

n

. Suppose further f ∈ GLC in R

n

, i.e., for each

j ∈{1,2,L, n} , g j : R → R is globally Lipschitz continuous with Lipschitz con-

stant L j , i.e. f j (u) − f j (v) ≤ Lj u − v for all u, v ∈ R. K ij (⋅) , i, j = 1,2, L , n are the delay kernels which are assumed to satisfy the following conditions simultaneously: (i) K ij : [0, ∞) → [0, ∞); (ii) K ij are bounded and continuous on [0, ∞] ; (iii)

∞

∫K 0

ij

( s )ds = 1 ;

(iv) There exists a positive number μ such that

∫

∞

0

K ij ( s )e μs ds < ∞ .

The literature [22] has given some examples to meet the above conditions. The initial conditions associated with the system (1) are given by x i ( s ) = γ i ( s ) , s ≤ 0 , i = 1,2, L , n, where γ i (⋅) is bounded and continuous on (−∞, 0] . As a special case of neural system (1), the delayed neural networks with constant input vector

I = ( I 1 , I 2 , K , I n ) T ∈ R n have been studied widely in [22]. This

system is described by the following functional differential equations

Dynamics of General Neural Networks with Distributed Delays n

x&i (t) = −di xi (t) + ∑aij f j j=1

( ∫ K (s)x (t − s)ds) + I , t ≥ 0 ∞

0

ij

j

i

137

(2)

Let x * = (x1* , x 2* , L, x n* ) be the equilibrium point of system (2), i.e., n

d i xi* = ∑ aij f j ( xi* ) + I i .

(3)

j =1

If delay kernel functions kij ( s ) are of the form kij ( s ) = δ ( s − τ ij ) , i, j = 1, 2,L , n where τ ij > 0 is discrete delays. Then systems (2) reduces to Hopfield neural networks with discrete delays [15]. Therefore the discrete delay case can be included in our models by choosing suitable kernel functions. For the proof of the main results in this paper, the following Halanay-type inequality is needed. Halanay-type inequality in comparison to Lyapunov functional is rarely used in the literature on stability of dissipative dynamical systems in spite of their possible generalizations and applications (see [22] and its references therein for detail). This equality is narrated from [22] as follows. Lemma 1. Let v(t ) > 0 for t ∈ R , τ ∈ [0, ∞) and t 0 ∈ R . Assume

d v(t ) ≤ −av(t ) + b⎛⎜ sup v( s ) ⎞⎟ for t > t 0 . dt ⎠ ⎝ s∈[t −τ , t ] If a > b > 0 , then there exist constants γ > 0 and k > 0 such that v(t ) ≤ ke − γ ( t − t0 ) , for t > t 0 .

3 Main Results In this section, let C = C ([−τ ,0], R n ) be the Banach space of all continuous function from [−τ ,0] to R n with the topology of uniform convergence. Given any φ , ϕ ∈ C , let x(t, φ ) = ( x1 (t, φ ), x 2 (t , φ ), L, x n (t, φ )) T and x(t , ϕ ) = ( x1 (t , ϕ ), x 2 (t , ϕ ), L , x n (t , ϕ )) T be the solutions of (1) starting from φ and ϕ respectively. Using Lyapunov functional method, similar to the proof of Theorem 1 of literature [9], we can easily obtain the following theorem. Theorem 1. Suppose f ∈ GLC. If the system parameters d i , a ij (i, j = 1,2, L , n) satisfy the following conditions n

d i − L i ∑ a ji > 0 , j =1

for i = 1, L , n ,

(4)

then for every periodic input I (t ) , the delayed neural system (1) is robustly exponentially periodic.

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By Lemma 1, we use a method comprising of Halanay-type inequality to establish another set of sufficient conditions for the network (1) to converge exponentially towards its periodic solution as follows. Theorem 2. Suppose f ∈GLC. If the system parameters d i , a ij (i, j = 1,2, L , n) satisfy the following conditions n

d i − ∑ a ij L j > 0 , j =1

for i = 1, L , n ,

(13)

then for every periodic input I (t ) , the delayed neural system (1) is exponentially periodic. Remark 1. When I = ( I 1 , I 2 , L , I n ) is a constant vector, then for any constant T ≥ 0

we have I = I (t + T ) = I (t ) for t ≥ 0 . Thus by the results obtained, when the sufficient conditions in Theorem 1 or Theorem 2 are satisfied, a unique periodic solution becomes a periodic solution with any positive constant as its period. So, the periodic solution reduced to a constant solution, that is, an equilibrium point. Furthermore, all other solutions globally exponentially converge to this equilibrium point as t → +∞ . The unique equilibrium point of the delayed neural system (2) globally exponentially convergence. Obviously, the results obtained are consistent with the exponential stability results that were recently reported in [11,15,16,20].

4 Examples In this section, we give two examples to illustrate the correctness of our results. We also show that the conditions for these two theorems are independent of each other by providing the following Example 1 and 2. Example 1. Consider the dynamical systems (1) where d 1 = 3, d 2 = 4 , a11 = 1 , a 22 = 2 , a12 = 4 , a 21 = 2 , L1 = 1 / 2 , L 2 = 3 / 4 , for i = 1,2 . By Theorem 2, we can check 2

d 1 = 3 < ∑ a ij L j = 3.5 . j =1

Hence the sufficient conditions in Theorem 2 are not satisfied. However, it is easy to verify that all the hypotheses of Theorem 1 are satisfied. Therefore, the criteria in Theorem 1 are effective. Example 2. Consider the dynamical systems (1) where d 1 = 4, d 2 = 4 , a11 = 1 / 2 , a 22 = 1 / 5 , a12 = 1 / 3 , a 21 = 1 / 3 , L1 = 8 , L 2 = 12 ,

for i = 1,2 . By Theorem 1, we can check 2

d 1 = 4 < L1 ∑ a ji = 20 / 3 . j =1

Dynamics of General Neural Networks with Distributed Delays

139

Hence the sufficient conditions in Theorem 1 are not satisfied. However, it is easy to verify that all the hypotheses of Theorem 2 are satisfied. Therefore, the criteria in Theorem 2 are applicable.

5 Conclusions In this paper, a general class of neural networks with distributed delays and periodic inputs are investigated. Exponential periodicity of neural systems has been investigated using Lyapunov functional and Halanay-type inequality respectively. The easily checked conditions ensuring the exponential periodicity of neural systems are obtained.

References 1. Bouzerman, A., Pattison, T.: Neural Network for Quadratic Optimization with Bound Constraints. IEEE Trans. Neural Networks 4 (1993) 293-303 2. Forti, M., Nistri, P.: Global Convergence of Neural Networks with Discontinuous Neuron Activations. IEEE Trans. Circuits and Systems I 50 (11) (2003) 1421-1435 3. Forti, M., Tesi, A.: New Conditions for Global Stability of Neural Networks with Application to Linear and Quadratic Programming Problems. IEEE Trans. Circuits and Systems I 42 (1995) 354-366 4. Venetianer, P., Roska, T.: Image Compression by Delayed CNNs. IEEE Trans. Circuits Syst. I 45 (1998) 205-215 5. Sun, C., Feng, C.: Neural Networks for Nonconvex Nonlinear Programming Problems: A Switching Control Approach. In: Wang, J., Liao, X., Yi, Z. (eds.): Advance in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg (2005) 694 699 6. Morita, M.: Associative Memory with Non-montone Dynamics. Neural networks 6 (1993) 115-126 7. Tank, D., Hopfield, J.: Simple “Neural” Optimization Networks: An A/D Converter, Signal Decision Circuit, And A Linear Programming Circuit. IEEE Trans. Circuits & Systems 33 (1996) 533-541 8. Kennedy, M., Chua, L.: Neural Networks for Linear and Nonlinear Programming. IEEE Trans. Circuits & Systems 35 (1988) 554-562 9. Liao, X., Wong, K., Yang, S.: Convergence Dynamics of Hybrid Bidirectional Associative Memory Neural Networks with Distributed Delays. Physics Letters A 316 (2003) 55-64 10. Cao, J., Wang, J.: Global Asymptotic Stability of Recurrent Neural Networks with Lipschitz-continuous Activation Functions and Time-Varying Delays. IEEE Trans. Circuits Syst. I 50 (2003) 34-44 11. Liao, X., Wang, J.: Algebraic Criteria for Global Exponential Stability of Cellular Neural Networks with Multiple Time Delays. IEEE Trans. Circuits Systems I 50 (2) (2003) 268275 12. Sun, C., Feng, C.: Exponential Periodicity of Continuous-time And Discrete-time Neural Networks with Delays. Neural Processing Letters 19(2) (2004) 131-146 13. Sun, C., Feng, C.: On Robust Exponential Periodicity of Interval Neural Networks with Delays. Neural Processing Letters 20 (1) (2004) 53-61

－

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14. Sun, C., Feng, C.: Exponential Periodicity And Stability of Delayed Neural Networks. Mathematics and Computers in Simulation 66 (2004) 469-478 15. Sun, C., Zhang, K., Fei, S., Feng, C.: On Exponential Stability of Delayed Neural Networks with A General Class of Activation Functions. Physics Letters A 298 (2002) 122132 16. Sun, C., Feng, C.: Global Robust Exponential Stability of Interval Neural Networks with Delays. Neural Processing Letters 17 (1) (2003) 107-115 17. Zhang, J., Jin, X.: Global Stability Analysis in Delayed Hopfield Neural Network Models. Neural Networks 13 (2000) 745-753 18. Wang, L.: Stability of Cohen–Grossberg Neural Networks with Distributed Delays. Applied Mathematics and Computation 160 (2005) 93–110 19. Zeng, Z., Wang, J., Liao, X.: Global Exponential Stability of A General Class of Recurrent Neural Networks with Time-Varying Delays. IEEE Trans. Circuits System I 50 (2003) 1353-1359 20. Yi, Z., Tan, K.: Convergence Analysis of Recurrent Neural Networks. Kluwer Academic Publishers, Dordrecht (2003) 21. Mohamad, S.: Global Exponential Stability in Continuous-time And Discrete-time Delay Bi-directional Neural Networks. Physical D 159 (2001) 233-251 22. Mohamad, S., Gopalsamy, K.: Exponential Stability of Continuous-time And Discretetime Cellular Neural Networks with Delays. Applied Mathematics and Computation 135 (2003) 17-38

On Equilibrium and Stability of a Class of Neural Networks with Mixed Delays Shuyong Li1 , Yumei Huang2,3 , and Daoyi Xu2 1

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China [email protected] 2 Institute of Mathematics, Sichuan University, Chengdu 610064, China {huangyumei181, daoyixu}@163.com 3 School of Mathematics and Computer Engineering, Xihua university, Pixian 610039, China

Abstract. In this paper the authors analyze problems of existence and global asymptotic stability of the equilibrium for the neural networks with mixed delays. Some new sufficient conditions ensuring the existence, uniqueness, and global asymptotic stability of the equilibrium are established by means of Leray-Schauder principle, arithmetic-mean-geometricmean inequality and vector delay differential inequality technique. These conditions are less restrictive than previously known criteria.

1

Introduction

In the design of a delayed neural networks(DNN’s), stability is a main concern. In many applications such as linear program or pattern recognition, for example, it is desired to design a DNN with unique equilibrium that is globally asymptotically stable(GAS). Therefore, the GAS of the equilibrium of DNN has become a subject of intense research activities [1],[2]. Among many existing research works, some works consider the discrete delays [1], and others continuous delays [2]. To the best of our knowledge, neural networks with mixed delays (both continuous and discrete) are seldom considered. However, mixed delays in practice are more common [3]. Therefore, the studies of neural networks with mixed delays are more important than those with continuous or discrete delays. Consider a class of neural networks with mixed delays described by the following nonlinear integral differential equations [3],[7]: ui (t) = −μi ui (t) +  +bij gj (

n 

[aij fj (uj (t − τj (t)))

j=1 t

−∞

kj (t − s)uj (s)ds)] + Ii , t ≥ t0 ,

ui (t) = φi (t), t ∈ (−∞, t0 ], t0 ≥ 0, i = 1, · · · , n, J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 141–146, 2006. c Springer-Verlag Berlin Heidelberg 2006 

(1)

142

S. Li, Y. Huang, and D. Xu

where xi (t) is the state, fj and gj denote the activation functions, aij and bij represent the constant connection weight. Ii is the external bias, μi > 0 represents the rate, kj is the refractoriness and the delay τj (t) ≥ 0 is continuous with t − τj (t) ≥ r(t) and r(t) → ∞ as t → ∞. The initial function φi (i = 1, · · · , n) is bounded and continuous on (−∞, 0]. In this paper, fj ’s, gj ’s and kj ’s satisfy the following conditions. (A1 ) There exist Kj > 0 and Lj > 0 such that for any u, v ∈ R, |fj (u) − fj (v)| ≤ Kj |u − v|, |gj (u) − gj (v)| ≤ Lj |u − v|, i = 1, · · · , n. (A2 ) All kj ’s satisfy |kj | ∈ L1 [0, ∞), i.e., there is a number cj such that  ∞ |kj (s)|ds = cj > 0, i = 1, · · · , n. 0

2

Existence of Equilibrium

Lemma 1. Let U is a nonempty bounded open set in the Banach space X. ¯ ⊂ X → Rn is compact and let 0 ∈ U and ∂U is Assume that the operator h : U the boundary of U . If there is an i such that hi (u)sgnui > 0, ∀u ∈ ∂U,

(2)

¯ then the equation h(u) = 0 has a solution u on U. Proof. We define a mapping T (u) = u − h(u). Then u − λT (u) = λh(u) + (1 − λ)u

(3)

is a homotopic mapping (or homotopy [4]), and for all u ∈ ∂U , |ui − λTi (u)| = (1 − λ)|ui | + λhi (u)sgnui . Condition (2) implies that for all u ∈ ∂U , we have ui = 0 and  (1 − λ)|ui | > 0 if λ ∈ [0, 1), |ui − λTi (u)| ≥ hi (u)sgnui > 0 if λ = 1.

(4)

(5)

That is, λT (u) = u for all (u, λ) ∈ ∂U × [0, 1]. By Leray-Schauder principle [4, ¯ This implies that pp.556], T has a fixed point u∗ on U. ¯, T (u∗ ) = u∗ − h(u∗ ) = u∗ , i.e., h(u∗ ) = 0, u∗ ∈ U and the proof is completed. Lemma 2.  (Arithmetic-mean–geometric-mean inequality [5, p.13]). For xi ≥ 0, αi > 0 and ni=1 αi = 1, n n   i xα ≤ αi xi , (6) i i=1

i=1

On Equilibrium and Stability of a Class of Neural Networks

143

the sign of equality holding if and only if xi = xj for all i, j = 1, · · · , n. By M, we denote the “M -matrix” class [6]. For any real matrices A = (aij ) and B = (bij ), we write A ≥ B if aij ≥ bij , ∀i, j = 1, · · · , n. Theorem 1. Suppose that conditions (A1 ) and (A2 ) are satisfied. If the following condition (A3 ) holds. (A3 ) There exist positive constants rkij , k = 1, 2, 3, 4; i, j = 1, 2, · · · , n, and α ≥ 1 such that Δ − M ∈ M, where Δ =diag{δ1, δ2 , · · · , δn }, M = (mij ) and ⎧ for α = 1, ⎨ μi , n

1 1  δi = αμ − (α − 1)  −1 1−α −1 1−α |aij |Kj r1ij r2ij + |bij |Lj cj r3ij r4ij , for α > 1, (7) ⎩ i j=1

⎧ ⎨ |aij |Kj + |bij |Lj cj , mij =

α = 1, (8)

⎩ |a |K rα−1 r + |b |L c rα−1 r , α > 1. ij j 1ij 2ij ij j j 3ij 4ij

Then there exists at least an equilibrium u∗ of system (1). Proof. We know that u∗ is an equilibrium of Eq.(1) if and only if u∗ = (u∗1 , · · · , u∗n )T is a solution of equations

 t  n  μi ui = aij fj (uj ) + bij gj kj (t − s)uj ds + Ii −∞

j=1 Δ

= Fi (u), i = 1, · · · , n.

(9)

For i = 1, · · · , n, we let hi (u) = (α|ui |α−1 + 1)(μi ui − Fi (u)),

(10)

and h(u) = [h1 (u), · · · , hn (u)] . Then, equations (9) and h(u) = 0 have exactly the same solutions. It follows from (A1 ) and from (9), we have T

|Fi (u)| ≤

n  

n   |aij |Kj |uj | + |bij |Lj cj |uj | + ξi (or = ηij |uj | + ξi ), (11)

j=1

j=1

where ξi =

n 

(|aij ||fj (0)| + |bij ||gj (0)|) + |Ii |, ηij = (|aij |Kj + |bij |Lj cj ). (12)

j=1

For α > 1, by using (7), (8), (11) and Lemma 2, we obtain α|ui |

α−1

|Fi (u)| ≤

n



α

−1 1−α α |aij |Kj r1ij r2ij (r1ij |uj |α + (α − 1)r2ij |ui |α )

j=1

 α −1 1−α α +|bij |Lj cj r3ij r4ij (r3ij |uj |α + (α − 1)r4ij |ui |α ) + αξi |ui |α−1 = (αμi − δi )|ui |α +

n  j=1

mij |uj |α + αξi |ui |α−1 .

(13)

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When α = 1, from (7), (8) and (12), we have δi = μi and mij = ηij , this implies that (13) holds for α = 1. Combining (10), (11) and (13), we have hi (u)sgnui ≥ δi |ui |α + μi |ui | −

n 

[mij |uj |α + ηij |uj |] − αξi |ui |α−1 − ξi

j=1

⎧ n  α ⎪ ⎪ [mij |uj |α + ηij |uj |] − αξi |ui |α−1 − ξi , for α > 1, ⎨ δi |ui | − j=1 ≥ (14) n  ⎪ ⎪ 2δ |u | − 2 m |u | − 2ξ , for α = 1. ⎩ i i ij j i j=1

¯ i (u) = hi (u)sgnui , h(u) ¯ ¯ 1 (u), h ¯ 2 (u) · · · , h ¯ n (u)]T , H = (ηij ), Ξ = Let h = [h T diag{ξ1 , ξ2 , · · · , ξn } and ξ = [ξ1 , ξ2 , · · · , ξn ] . Thus, we can rewrite the above inequality in the following matrix-vector form:  (Δ − M )[uα ]+ − H[u]+ − αΞ[uα−1 ]+ − [ξ]+ , for α > 1, ¯ h(u) ≥ (15) 2(Δ − M )[u]+ − 2[ξ]+ , for α = 1, where [uα ]+ = [|u1 |α , · · · , |un |α ]T for α > 0. From Condition (A3 ) and properties of M -matrix, there exists a positive n×n diagonal matrix D =diag{d1 , d2 . · · · , dn } Δ such that matrix D(Δ − M ) + (Δ − M )T D = A is positive definite, that is, the minimal eigenvalue λ(A) of A is positive. By means of inequality (15) as well as [uα ]+ ≥ 0 and D ≥ 0, we can easily obtain 2

n 

¯ i (u) = ([uα ]+ )T Dh(u) ¯ ¯ T (u)D[uα ]+ di |ui |α h +h

i=1

⎧ α 2 α E ( H E u E ⎪ ⎨ λ(A) u E − 2 u E D

α−1 Δ +α Ξ

u

E + ξ E ), α > 1, E ≥ Γ (u) = ⎪ ⎩ 2λ(A) u 2E − 4 u E D E ξ E , α = 1,

(16)

where · E is the Euclidean norm. It follows from (16) that Γ (u) → ∞ as u E → ∞. Then there is a R0 such that for all u E ≥ R0 , 2

n 

¯ i (u) ≥ Γ (u) > 0. di |ui |α h

(17)

i=1

Consider the following set: U (R0 ) = {u : u E < R0 }. It is easy to verify that U (R0 ) is a nonempty bounded open set, 0 ∈ U (R0 ) and h defined by (10) ¯ 0 ). From (17), there is at least an i such that is compact on U(R ¯ i (u) = hi (u)sgnui > 0 for all u ∈ ∂U (R0 ). h (18) By Lemma 1, we can conclude that equation h(u) = 0 or (9) has at least a ¯ 0 ). Hence, neural network model (1) has at least an equilibrium. solution in U(R Remark 1. When α > 1, the existing results on the existence of equilibrium always require the boundedness of the activation functions [1],[2]. The boundedness condition has been dropped in our results.

On Equilibrium and Stability of a Class of Neural Networks

3

145

Uniqueness of Equilibrium and Its Stability

Let K∞ is the class of continuous functions φ : R+ → R+ with φ(0) = 0, φ(s) > 0 if s > 0, φ is strictly increasing, and φ(s) → +∞ as s → +∞. C = C[(−∞, 0], Rn ] is the Banach space of continuous bounded functions φ : (−∞, 0] → Rn with φ = sup−∞≤s≤0 |φ(s)| and | · | is any norm on Rn . Equations (1) are categorized as retarded functional differential equations, x˙ i (t) = fi (t, xt ), i = 1, · · · , n,

(19)

where xt = [x1t , · · · , xnt ]T ∈ C is defined as xit (s) = xi (t + s) for s ∈ (−∞, 0] and i = 1, · · · , n, fi : R × C → R is continuous and fi (t, 0) ≡ 0. From Theorem 1 in [7], we can easily get the following lemma. Lemma 3. Suppose there exist functions vi : R+ × R → R+ with the following properties: vi (t) ≡ vi (t, xi ) is continuous and locally Lipschitzian in xi , w1i (|xi |) ≤ vi (t, xi ) ≤ w2i (|xi |), and the Dini derivative of vi (t) along Equations (19) satisfy  t n n   + D vi (t, xi ) ≤ −δi vi + pij vjt r + qij ψi (t − s) vjs r ds, (20) j=1

j=1

−∞

where δi , pij , qij ∈ R+ are constants, w1i , w2i ∈ K∞ , ψi ≥ 0 and ψi ∈ L1 [0, ∞),

vjt r = supr≤u≤t |vjt (u)|, r = r(t) ≤ t and r(t) → ∞ as t → ∞. If Δ − M ∈ M, (21) ∞ where M = (mij ) and mij = pij + qij 0 ψi (s)ds. Then the equilibrium x∗i = 0 of system (19) is globally asymptotically stable. Theorem 2. Assume that assumptions (A1 ), (A2 ) and (A3 ) hold. Then system (1) has a unique equilibrium u∗ , which is globally asymptotically stable. Proof. Since assumptions (A1 ), (A2 ) and (A3 ) hold, system (1) has an equilibrium u∗ . Let y(t) = u(t) − u∗ , system (1) can be written as  t n  yi (t) = −μi yi (t) + [aij f¯j (yj (t − τj (t))) + bij g¯j ( kj (t − s)yj (s)ds)], (22) −∞

j=1

where i = 1, · · · , n, fj (yj ) = fj (yj + u∗j ) − fj (u∗j ), and  g¯j (

t −∞

 kj (t − s)yj (s)ds) = gj (

t

−∞

kj (t − s)(yj (s) + u∗j )ds) − gj (cj u∗j ).

Then f¯j (0) = g¯j (0) = 0 and system (22) has an equilibrium at u = 0. To apply Lemma 3, we let vi (t) = |yi (t)|α for α ≥ 1 and   α−1 α−1 |aij |Kj r1ij r2ij , for α > 1, |bij |Lj r3ij r4ij , for α > 1, pij = , qij = (23) |aij |Kj , for α = 1, |bij |Kj , for α = 1.

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Calculating the Dini derivative of vi (t) along the solution of Equation (22), from condition (A1 ), we have n  D+ vi (t) ≤ α[−μi |yi (t)|α + |yi (t)|α−1 ( |aij |Kj |yj (t − τj (t))|

 +|bij |Lj

j=1 t

−∞

|kj (t − s)||yj (s)|ds)].

(24)

For α > 1, using (24) and Lemma 2, we have D+ vi (t) ≤ −αμi vi (t) +

n

 α −1 1−α α |aij |Kj r1ij r2ij (r1ij

vjt r + (α − 1)r2ij vi (t)) j=1

−1 +|bij |Lj r3ij r4ij

= −δi vi (t) +

n

 j=1



t

−∞

α

1−α α |kj (t − s)|(r3ij

vjs r + (α − 1)r4ij vi (t))ds

pij vjt r + qij



t

−∞

 |kj (t − s)| vjs r ds .



(25)

This implies that vi (t) satisfies condition (20) with ψi (s) = ki (s). It is obvious that (25) holds for α = 1. On the other hand, condition (21) can be obtained by using assumptions (A2 ) and (A3 ), together with (23). Now, by Lemma 3, the trivial solution of Equation (22) is globally asymptotically stable, and therefore the equilibrium u∗ is globally asymptotically stable. Since all of the solutions of (1) tend toward u∗ as t → ∞, system (1) has a unique equilibrium.

Acknowledgments The work is supported by the National Natural Science Foundation of China under Grant 10371083.

References 1. Cao, J.D.: A Set of Stability Criteria for Delayed Cellular Neural Networks. IEEE Trans. Circuits Syst. I 48 (2001) 494-498 2. Guo, S., Huang, L.: Exponential Stability and Periodic Solutions of Neural Networks with Continuously Distributed Delays. Phys. Rev. E 67 (2003) 011902 3. Kolmanovskii, V., Myshkis, A.: Intoduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht (1999) 4. Zeidler, E.: Nonlinear Functional Analysis and its Application. I: Fixed-Point Theorems. Springer-Verlag, New York (1986) 5. Beckenbach, E.F., Bellman, R.: Inequalities. Springer-Verlag, New York (1961) 6. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979) 7. Xu, D.Y.: Integro-Differential Equations and Delay Integral Inequalities. Tohoku Math. J. 44 (1992) 365-378

Stability Analysis of Neutral Neural Networks with Time Delay Hanlin He1 and Xiaoxin Liao2 1

College of Sciences, Naval University of Engineering, Hubei, Wuhan, 430033, China [email protected] 2 Department of Control Science and Engineering, Huazhong University of Science and Technology, Hubei, Wuhan, 430074, China

Abstract. A method is proposed for asymptotic stability analysis of neutral neural networks with delayed state. The delay is assumed unknown, but constant. Sufficient conditions on delay-independent asymptotic stability are given in terms of the existence of symmetric and positive definite solutions of a constant Riccati algebraic matrix equation coupled with a discrete Lyapunov equation.

1

Introduction

The stability of time-delay neural networks is a of practical and theoretical interests since the existence of a delay in a neural network may induce instability, and various results were reported [1, 3, 5-15]. However, due to the complicated dynamic properties of the neural cells, in many cases the existing neural network models cannot characterize the properties of a neural reaction process precisely. A natural and important problem is to how to further describe and model the dynamic behavior for such complex neural reactions. Such problems naturally introduces the neutral neural networks. For example, in the biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process [2]. A different example is proposed in [8], where the neutral phenomena exist in large-scale integrated circuits. Some criteria for mean square exponential stability and asymptotic stable of stochastic neutral neural networks are provided in [13-14]. This paper considers the stability of a particular class of neutral neural networks with time-delay. We are interested in giving conditions for delayindependent stability conditions. By solving some appropriate algebraic Riccati equation, based on the Lyapunov-Razumikhin type theory and an appropriate Lyapunov-Razumikhin functional, this paper achieves a sufficient criterion for globally asymptotic stability of the neutral neural networks with time-delays. The organization of this paper is as follows: Section 2 gives a stability result for a functional differential equation of neutral type. Section 3 gives the description of the neutral neural networks in this paper and derives the stability result. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 147–152, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Section 4 and 5 give the singular value test and LMI formulation for the stability result derived in Section 3. Section 6 is an example to explain the stability result. Some conclusions of Section 7 ends this paper.

2

Preliminaries: Stability Theory

Throughout this paper, the following notations are used. λmin (P ), λmax (P ) denote the minimal and maximal eigenvalues of matrix P . Cτ = C([−τ, 0], Rn ) denotes the Banach space of continuous vector functions mapping the interval [−τ, 0] into Rn with the topology of uniform convergence. . refers to either the Euclidean vector norm or the induced matrix 2-norm. Consider the following functional differential equation of neutral type: d [D(xt )] = f (xt ) dt

(1)

with an appropriate initial condition: x(t0 + θ) = φ(θ), ∀θ ∈ [−τ, 0], (t0 , φ) ∈ R+ × Cτ

(2)

where D : D(xt ) = x(t) − Cx(t − τ ). We say that the operator D is stable if the zero solution of the corresponding homogeneous difference equation is uniformly asymptotically stable. Then we have the following lemma: Lemma 1 [4]. Suppose D is stable, f : R × Cτ → Rn takes bounded sets of Cτ in bounded sets of Rn and suppose u(s), v(s) and w(s) are continous, nonnegative and nondecreasing functions with u(s), v(s) > 0,for s  = 0 and u(0) = 0, v(0) = 0. If there is a differential continuous functional V (xt ) : R × Cτ → Rn such that 1)u(D(xt ) ≤ V (xt ) ≤ v(xt ), 2)V˙ (xt ) ≤ −w(xt ) where xt = x(t + θ), then the solution x = 0 of the neutral equation (1)-(2) is uniformly asymptotically stable.

3

Main Result

We consider the neutral neural networks with linear and nonlinear time delay described by the differential-difference equation of the form  x(t) ˙ − C x(t ˙ − τ ) = Ax(t) + Ad x(t − τ ) + Bf (x(t − τ )) (3) x(t) = φ(t), t ∈ [−τ, 0] Where x(t) ∈ Rn is the state of the neuron; A, B are real constant matrices with appropriate dimensions; τ is constant time-delay; φ(t) is a continuous vectorvalued initial function defined on [−τ, 0]; f ∈ C(Rn , Rn ) is bounded function, the boundedness is l, that is f (x(t)) ≤ l x(t) and f (0)) = 0; Then, we have the following theorem. Theorem 1. The neutral neural networks system (3) is delay-independent asymptotically stable if

Stability Analysis of Neutral Neural Networks with Time Delay

149

(i) C is Schur-Cohn stable matrix; (ii) There exist two symmetric and positive definite matrices R and Q such that the following Riccati equation has a symmetric and positive definite matrix solution P > 0. AT P + P A + P BB T P + S + Q + [P (AC + Ad ) + SC]R−1 [C T S + (ATd + C T AT )P ] = 0

(4)

where S > 0 is the symmetric and positive definite matrix solution of the Lyapunov discrete equation C T SC − S + l2 I + R = 0

(5)

Proof. Take the following Lyapunov functional candidate:  t T V (xt ) = (x(t) − Cx(t − τ )) P (x(t) − Cx(t − τ )) + x(θ)T Sx(θ)dθ

(6)

t−τ

Introducing the operator D : Cτ → Rn : D(xt ) = x(t) − Cx(t − τ )

(7)

Clearly V (xt ) satisfies the following inequalities u(D(xt )) ≤ V (xt ) ≤ v(xt C ) where xt C = max−τ ≤θ≤0 xt (θ), u(s) = λmin (P )s2 and v(s) = (λmax (P ) + τ λmax (S))s2 . The time derivative of V (xt ) along the system (1) is given by: V˙ (xt ) = (x(t) − Cx(t − τ ))T [P A + AT P + S](x(t) − Cx(t − τ )) +xT (t − τ )ATd P (x(t) − Cx(t − τ )) + (x(t) − Cx(t − τ ))T P Ad x(t − τ ) +2(x(t) − Cx(t − τ ))T P Bf (x(t − τ )) +x(t − τ )T C T SCx(t − τ ) − x(t − τ )T Sx(t − τ ) +(x(t) − Cx(t − τ ))T P ACx(t − τ ) +x(t − τ )T C T AT P (x(t) − Cx(t − τ )) +(x(t) − Cx(t − τ ))T SCx(t − τ ) +x(t − τ )T C T S(x(t) − Cx(t − τ ))

(8)

By the condition of f and Schwarz inequality 2(x(t) − Cx(t − τ ))T P Bf (x(t − τ )) ≤ 2l(x(t) − Cx(t − τ ))T P Bx(t − τ ) ≤ (x(t) − Cx(t − τ ))T P BB T P (x(t) − Cx(t − τ )) + l2 x(t − τ )T x(t − τ ) Since S is the positive solution of the discrete Lyapunov equation (5) and using the operator form (7), we have V˙ (xt ) ≤ D(xt )T [P A + AT P + S + P BB T P ]D(xt ) +D(xt )T (P (AC + Ad ) + SC)x(t − τ )) +x(t − τ )T ((C T AT + ATd )P + C T S)D(xt ) −x(t − τ )T Rx(t − τ )

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Since P is the symmetric and positive definite solution of the Ricati equation (4), we have V˙ (xt ) ≤ −D(xt )T QD(xt ) −[((C T AT + ATd )P + C T S)D(xt ) − Rx(t − τ )]T R−1 [((C T AT + ATd )P + C T S)D(xt ) − Rx(t − τ )] ≤ −D(xt )T QD(xt )

(9)

By Lemma 1, the solution x = 0 of the system (3) is globally uniformly asymptotically stable. Remark 1. Since P BB T P + S + Q + [P (AC + Ad ) + SC]R−1 [C T + (ATd + C T AT )P ] > 0, from standard results on Lyapunov equations, A is stable iff P ≥ 0. Hence, the necessary condition for the equation (4) has solution is that A is stable. Remark 2. The Schur-Cohn stability of the matrix C ensures the stability of the operator D : Cτ → Rn : D(xt ) = x(t) − Cx(t − τ )

4

Singular Value Test

Equation (4) can be rewritten as (A + (AC + Ad )R−1 C T S)T P + P (A + (AC + Ad )R−1 C T S) +P [BB T + (AC + Ad )(AC + Ad )T ]P + S + Q + SCR−1 C T S = 0 (10) Hence the associated Hamiltonian matrix is given by   A + (AC + Ad )R−1 C T S BB T + (AC + Ad )(AC + Ad )T −(S + Q + SCR−1 C T S) −(A + (AC + Ad )R−1 C T S)T

(11)

In order to have a symmetric and positive definite solution to the Ricati equation (4), the associated Hamiltonian matrix (11) has no eigenvalues on the imaginary axis.

5

LMI Formulation

Pay attention that R is the solution of (5), via Schur complement, the theorem 1 can be easily converted into an LMI feasibility problem. Corollary 1. The neutral neural networks (3) is delay-independent asymptotically stable if there exist two symmetric and positive definite matrices P > 0 and S > 0 such that the the following LMIs hold:  T  A P + P A + P BB T P + S P (AC + Ad ) + SC <0 (12) C T S + (ATd + C T AT )P C T SC − S + l2 I C T SC − S + l2 I < 0

(13)

Stability Analysis of Neutral Neural Networks with Time Delay

6

151

Illustrative Example

Example: Suppose the system (3) with the following data       −2 0 0 −1 0.1 0.2 A= ; Ad = ;B = 0 −2 0 0 0.3 0.1  C=

   sin x2 (t − τ ) 0.1 0 ; f (x(t − τ ) =  0 0.2 1 − cos x1 (x(t − h(t))

Hence l = 1, the inequalities (12)-(13) are satisfied for P = I, R = 2I. From Theorem 1 the zero solution of neural networks (3) is globally uniformly asymptotically stable for the given data.

7

Conclusions

A novel type of neutral neural networks is studied in this paper. Following the recent development on the stability result for systems of neutral type and the algorithm and software for LMI, some new criteria on neutral neural networks are provided. Further researches include more elegant stability analysis approaches with less conservativeness and some control strategies for the neutral differential equations.

Acknowledgments This work is supported by the Natural Science Foundation of China (60474011), and Academic Foundation of Naval University of Engineering, China.

References 1. Cao, J.D. and Li,Q.: On the exponential stability and periodic solution of delayed cellular neural networks. J. Math. Anal. and Appl. 252 (1) (2000) 50-64 2. Curt, W.: Reactive Molecules: The Neutral Reactive Intermediates in Organic Chemistry. Wilay Press, New York (1984) 3. Fu, C.J., He, H.L., Liao, X.X.: Globally Stable Periodic State of Delayed CohnGrossberg Neural Networks. In: Wang J. et al (eds): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 276-281 4. Hale, J.K., Verduyn Lunel, S.M.: Introduation to Functional Differential Equations Applied Math. Sciences, Vol. 99, Springer-verlag, New York (1991) 5. He, H.L., Wang, Z.S., Liao, X.X.: Stability Analysis of Uncertain Neural Networks with Linear and Nonlinear Time Delays. In Wang J. et al (eds): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 199-202 6. Liao, X.X., Xiao, D.M.: Globally Exponential Stability of Hopfield Neural Networks with Time-varying Delays. ACTA Electronica Sinica, 28 (1) (2000) 87-90

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7. Liao, X.X., Wang, J.: Algebraic Criteria for Global Exponential Stability of Cellular Neural Networks with Multiple Time Delays. IEEE Trans. Circuits and Systems I. 50 (2) (2003) 268-275 8. Shen, Y., Zhao, G.Y., Jiang, M.H., Mao, X.R.: Stochastic Lotka-Volterra Competitive Systems with Variable Delay. In: Huang, D.S., Zhang, X.P., Huang, G.B. (eds.): Advances in Intelligent Computing. Lecture Notes in Computer Science, Vol. 3645. Springer-Verlag, Berlin Heidelberg New York (2005) 238-247 9. Shen, Y., Zhao, G.Y., Jiang, M.H., Hu, S.G.: Stochastic High-order Hopfield Neural Networks. In: Wang, L.P., Chen, K., Ong, Y.S. (eds.): Advances in Natural Computation. Lecture Notes in Computer Science, Vol. 3610. Springer-Verlag, Berlin Heidelberg New York (2005) 740-749 10. Zeng, Z.G., Wang, J., Liao, X.X.: Global Exponential Stability of A General Class of Recurrent Neural Networks with Time-varying Delays. IEEE Trans. Circuits and Syst. I 50 (10) (2003) 1353-1358 11. Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Syst. I 51 (11) (2004) 2313-2324 12. Shen, Y., Jiang, M.H., Liao, X.X.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Time-varying Delays and Continuously Distributed Delays. In: Wang, J., Liao, X.F., Zhang, Y. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 156-161 13. Zhang, Y.M., Guo, L. Wu, L.Y. et al: On Stochastic Neutral Neural Networks. In: Wang J. et al (eds): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 69-74 14. Zhang, Y.M., Shen, Y., Liao, X.X.: Exponential for Stochastic Interval Delayed Hopfield Neural Networks. Control Theory and Applications 20 (12) (2003) 12991300 15. Wang, Z.S., He, H.L., Liao, X.X.: Stability Analysis of Uncertain Neural Networks with Delay. In: Yin F. et al (eds): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3173. Springer-Verlag, Berlin Heidelberg New York (2004) 44-48

Global Asymptotical Stability in Neutral-Type Delayed Neural Networks with Reaction-Diffusion Terms Jianlong Qiu1,2 and Jinde Cao1 1 2

Department of Mathematics, Southeast University, Nanjing 210096, China Department of Mathematics, Linyi Normal University, Linyi 276005, China {qjl9916, jdcao}@seu.edu.cn

Abstract. In this paper, the global uniform asymptotical stability is studied for delayed neutral-type neural networks by constructing appropriate Lyapunov functional and using the linear matrix inequality (LMI) approach. The main condition given in this paper is dependent on the size of the measure of the space, which is usually less conservative than space-independent ones. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed criteria.

1

Introduction

It is well known that the dynamics of delayed recurrent neural networks such as cellular neural networks (CNNs) and Hopfield neural networks (HNNs) have been deeply investigated, in recent years, many researchers have studied the global asymptotical stability and global exponential stability analysis for delayed neural networks. A great number of results on this topic have been reported in the literature; see, [1, 2, 3] and the references therein. However, strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields. In [5, 6, 7], the authors considered the stability of neural networks with reaction-diffusion terms. In [8], the authors investigated the exponential stability of a class of delayed neural networks described by nonlinear delay differential equations of neutral type without reaction-diffusion terms. Motivated by the above discussions, the aim of the present paper is to consider a class of neutral-type neural networks with time delays and reaction-diffusion terms described by a nonlinear delay differential equation of neutral type.   m n  ∂ui (t, x)  ∂ ∂ui (t, x) = Dik − ai ui (t, x) + bij fj (uj (t, x)) ∂t ∂xk ∂xk j=1 k=1 n 

+

j=1

cij gj (uj (t − τ, x)) +

n  j=1

hij

∂uj (t − τ, x) + Ii , x ∈ Ω, (1) ∂t

where x = (x1 , x2 , · · · , xm )T ∈ Ω ⊂ Rm , Ω is a compact set with smooth boundary ∂Ω and mesΩ > 0 in space Rm ; u = (u1 , u2 , · · · , un )T ∈ Rn , ui (t, x) J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 153–158, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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denotes the state of the ith neuron at time t and in space x; fj ,  gj are activation functions of the jth neuron; the scalar ai > 0 is the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t and in space x; bij , cij , hij , i, j = 1, 2, · · · , n are known constants denoting the strength of the ith neurons on the jth neurons; smooth functions Dik = Dik (t, x, u) ≥ 0 are transmission diffusion operator of the ith neurons; τ is non-negative constant, which correspond to finite speed of axonal signal transmission delay; Ii denote the ith component of an external input source. The initial and boundary condition of the system (1) are given by  T ∂ui ∂ui ∂ui ∂ui = , ,··· , = 0, ∀t ≥ 0, x ∈ ∂Ω, i = 1, 2, · · · , n, (2) ∂n ∂x1 ∂xk 2 ∂xm ∂ ∂φi (s, x) ui (s, x) = φi (s, x), ui (s, x) = , −τ ≤ s ≤ 0, i = 1, 2, · · · , n. (3) ∂t ∂t in which φi (s, x) are bounded and first order continuous differentiable on [−τ, 0]. Notation. Throughout this paper, vector X ∈ Rn (n-dimensional Euclidean √ T space), its norm is defined as X = X X. C 1,2 (R+ ; Rm ) is a Banach space. n Ω ⊆ Rm is simply connected compact subset . Y ∈ C 1,2 : R+ ×  Ω → R is2a real 1 Lebesgue measurable function on Ω, for the norm Y (t)2 = [ Ω Y (t, x) dx] 2 . E denotes identity matrix. The notation A > B means that the matrix A − B is symmetric positive definite. For an arbitrarily real matrix H  (hij )n×n , + H +  (h+ ij )n×n , where hij = max(hij , 0), i, j = 1, 2, · · · , n.

2

Preliminaries

Throughout the paper we need the following assumption and lemmas. (H) The neurons activation functions fj , constants lj > 0, kj > 0 such that 0≤

fj (ξ1 ) − fj (ξ2 ) ≤ lj ; ξ1 − ξ2

gj are continuous, and there exist

0≤

gj (ξ1 ) − gj (ξ2 ) ≤ kj ξ1 − ξ2

for any ξ1 , ξ2 ∈ R, ξ1 = ξ2 , j = 1, 2, · · · , n. Definition. A vector U ∗ = (u∗1 , u∗2 , · · · , u∗n )T ∈ Rn is said to be an equilibrium point of system (1) if it satisfies −ai u∗i +

n  j=1

bij fj (u∗j ) +

n 

cij gj (u∗j ) + Ii = 0, i = 1, 2, · · · , n.

j=1

Lemma 1.[8] Suppose W, U are any matrices,  is a positive number and matrix H = H T > 0, then the following inequality holds W T U + U T W ≤ W T HW + −1 U T H −1 U.

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Lemma 2. For any constant matrix Σ ∈ Rn×n , Σ = Σ T > 0, Ω ⊂ Rn , mesΩ > 0, if ω : Ω → Rn is vector function such that the integration is well defined, then 

T ω(s)ds

 Σ

Ω



 ≤ |Ω|

ω(s)ds Ω

ω T (s)Σω(s)ds.

(4)

Ω

Remark: Obviously, we assume that the matrix Σ is identity matrix in the proof. If not, we letv(s) = P w(s), where P is a invertible orthogonal matrix Er 0 such that P T ΣP = . And by H¨ older inequality, it is easily prove it. 0 0 Lemma 3.[9] The equilibrium of the system is globally uniformly asymptotically stable if there exists a C 1 function V : R+ × Rn → R such that (i) V is a positive definite, decrescent and radially unbounded, and (ii) −V˙ is positive definite. Now, we let yi (t, x) = ui (t, x)−u∗i , i = 1, 2, · · · , n. It is easy to see that system (1) can be transformed into ∂yi (t, x)  ∂ = ∂t ∂xk m

k=1 n 

+

  n  ∂yi (t, x) Dik − ai yi (t, x) + bij fj (yj (t, x)) ∂xk j=1

cij gj (yj (t − τ, x)) +

j=1

n  j=1

hij

∂yj (t − τ, x) , x ∈ Ω, ∂t

(5)

where fj (yj (t, x)) = fj (yj (t, x) + u∗j ) − fj (u∗j ), j = 1, 2, · · · , n, gj (yj (t − τ, x)) = gj (yj (t − τ, x) + u∗j ) −   gj (u∗j ), j = 1, 2, · · · , n. Then it is easy to see that fj (0) = 0, gj (0) = 0, and fj (·), gj (·) satisfy assumption (H).

3

Main Results

Theorem. Under the assumption (H), the equilibrium point U ∗ of system (1) is globally uniformly asymptotically stable if there exist positive definite matrix Q1 and diagonal matrices P > 0 and Q2 > 0 such that ⎛ ⎞ Γ1 Γ2T Γ3T Ψ = ⎝ Γ2 Σ1 Γ4T ⎠ < 0, (6) Γ3 Γ4 Σ2 where T

Γ1 = −2P A + KQ1 K + 2P BL + AQ2 A − 2AQ2 B + L + |Ω|LB + Q2 B + L, T

T

Γ2 = P C − AQ2 C + + |Ω|LB + Q2 C + , Γ3 = P H − AQ2 H + + |Ω|LB + Q2 H + , T

T

T

Γ4 = |Ω|C + Q2 H + , Σ1 = −Q1 + |Ω|C + Q2 C + , Σ2 = −Q2 + |Ω|H + Q2 H + .

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Proof: Consider a Lyapunov functional as 



t

T

V (t) =

GT (Y (α, x))Q1 G(Y (α, x))dα

[Y (t, x)P Y (t, x) + Ω



t−τ t

+

( t−τ

∂ ∂ Y (α, x))T Q2 Y (α, x)dα]dx, ∂α ∂α

(7)

where Y (t, x) = (y1 (t, x), · · · , yn (t, x))T , F (Y (t, x)) = (f1 (y1 (t, x)), · · · , fn (yn (t, x)))T , G(Y (t, x)) = (g1 (y1 (t, x)), · · · , gn (yn (t, x)))T . Computing the time-derivative of V (t), along the solution of system (5), we have      n m  ∂ ∂yi (t, x) V˙ (t) = 2pi yi (t, x) Dik dx − 2 Y T (t, x)P AY (t, x)dx ∂x ∂x k k Ω k=1 Ω i=1  ∂ +2 Y T (t, x)[P BF (Y (t, x)) + P CG(Y (t, x)) + P H Y (t − τ, x)]dx ∂t  Ω + GT (Y (t, x))Q1 G(Y (t, x)) − GT (Y (t − τ, x))Q1 G(Y (t − τ, x)) Ω

∂ ∂ ∂ ∂ +( Y (t, x))T Q2 Y (t, x) − ( Y (t − τ, x))T Q2 Y (t − τ, x)]dx. ∂t ∂t ∂t ∂t

(8)

From boundary condition (2) and Green formula, we have   m

yi (t, x)

Ω k=1

∂ ∂xk

   2   m ∂yi (t, x) ∂yi (t, x) Dik dx = − Dik dx. (9) ∂xk ∂xk Ω k=1

Using (9) and assumption (H), we have  T V˙ (t) ≤ Y (t, x)[(−2P A + KQ1 K + 2P BL)Y (t, x) + 2P CG(Y (t − τ, x)) Ω

∂ Y (t − τ, x)] − GT (Y (t − τ, x))Q1 G(Y (t − τ, x)) ∂t

∂ ∂ ∂ ∂ −( Y (t − τ, x))T Q2 Y (t − τ, x)( Y (t, x))T Q2 Y (t, x) dx. ∂t ∂t ∂t ∂t

+2P H

(10)

We multiply system (5) by yi (t, x), and then integral on x in Ω 1 d 2 dt

   ∂yi (t, x) = Dik dx − ai yi2 (t, x)dx ∂xk Ω Ω k=1 Ω     n n + bij yi (t, x)fj (yj (t, x))dx + cij yi (t, x)gj (yj (t − τ, x))dx



yi2 (t, x)dx

  m

Ω j=1

+

  n Ω j=1

∂ yi (t, x) ∂xk

Ω j=1

hij yi (t, x)

∂yj (t − τ, x) dx, i = 1, 2, · · · , n. ∂t

(11)

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Taking into account (H), (9) and by Schwarz inequality, it is easy to see that the equality (11) can be transformed into n n   d + yi (t, x)2 ≤ −ai yi (t, x)2 + bij lj yj (t, x)2 + c+ ij kj yj (t − τ, x)2 dt j=1 j=1

+

n 

h+ ij y˙ j (t − τ, x)2 , i = 1, 2, · · · , n.

(12)

j=1

By Lemma 2 and some classical inequalities, we have T   n  ∂ ∂ d Y (t, x) Q2 Y (t, x)dx ≤ q2i ( ||yi (t, x)||2 )2 ∂t ∂t dt Ω i=1   T ≤ Y T (t, x)[(AQ2 A − 2AQ2 B + L + |Ω|LB + Q2 B + L)Y (t, x) Ω T

+2(−AQ2 C + + |Ω|LB + Q2 C + )G(Y (t − τ, x)) + 2(−AQ2 H + ∂ T T +|Ω|LB + Q2 H + )] Y (t − τ, x) + GT (Y (t − τ, x))[|Ω|C + ∂t ∂ T + Q2 C G(Y (t − τ, x)) + 2|Ω|C + Q2 H + Y (t − τ, x)] ∂t   T ∂ ∂ +T + +|Ω| Y (t − τ, x) H Q2 H Y (t − τ, x) dx. ∂t ∂t Substituting (13) into (10) and the condition (6), we have  V˙ (t) ≤ ΛT (t, x)Ψ Λ(t, x)dx,

(13)

(14)

Ω

 T ∂ where Λ = Y T (t, x), GT (Y (t − τ, x)), ( ∂t Y (t − τ, x))T )T . From Lemma 3, we follow that the equilibrium of system (1) is globally uniformly asymptotically stable. 

4

Simulation Example

1 Example. Let k = n = 2, Dik = 1, i, k = 1, 2, |Ω| = 16 , τ = 1, and fj = gj , j = 1, 2, with the Lipschtiz constant l1 = k1 = 0.5, l2 = k2 = 0.7. Consider the following delayed neural network in (1) with parameters as     2.8346 0 0.3346 −1.8921 A= ,B = , 0 9.7352 1.7243 0.5769     −0.8765 3.5647 0.3346 0.1745 C= ,H = . −0.7324 −2.0125 −0.0765 0.4217

By Theorem 1, we can conclude that this delayed neural network is globally asymptotically stable. using the Matlab LMI Control Toolbox to solve the LMI in (6), we obtain P, Q1 , Q2 as follows

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 P =

5

0.2312 0 0 0.0949



 , Q1 =

0.9522 0.0425 0.0425 1.0600



 , Q2 =

0.0700 0 0 0.0151

 .

Conclusion

In this paper, we have investigated the global uniform asymptotical stability for neutral-type neural networks with time delays and reaction-diffusion terms. Using Lyapunov functional method and the LMI approach, we gave a sufficient criterion ensuring the global asymptotical stability of the equilibrium point. The obtained result improves and extend several earlier publications and is useful in applications of manufacturing high quality neural networks.

Acknowledgement This work is supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grant BK2003053.

References 1. Cao, J. and Zhou, D.: Stability Analysis of Delayed Cellular Neural Networks. Neural Networks 11 (1998) 1601-1605 2. Cao, J. and Wang, J.: Global Exponential Stability and Periodicity of Recurrent Neural Networks with Time Delays. IEEE Trans. Circuits Syst. I 52(5) (2005) 925931 3. Cao, J. and Ho, D.W.C.: A General Framework for Global Asymptotic Stability Analysis of Delayed Neural Networks Based on LMI Approach. Chaos, Solitons and Fractals 24(5) (2005) 1317-1329 4. Gu, K.: An Integral Inequality in the Stability Problems of Time-delay Systems. Proceedings of 39th IEEE CDC, Sydney, Australia (2000) 2805-2810 5. Liao, X.X., Fu, Y., Gao, J. and Zhao, X.: Stability of Hopfield Neural Networks with Reaction-diffusion Terms. Acta Electron. Sinica 28 (2002) 78-80 6. Liang, J. and Cao, J.: Global Exponential Stability of Reaction-diffusion Recurrent Neural Networks with Time-varying Delays. Phys. Lett. A 314 (2003) 434-442 7. Wang, L. and Xu, D.: Global Exponential Stability of Reaction-diffusion Hopfield Neural Networks with Time-varying Delays. Science in China E 33 (2003) 488-495 8. Xu, S., Lam, J., Ho, D.W.C. and Zou, Y.: Delay-dependent Exponential Stability for a Class of Neural Networks with Time Delays. J. Comput. Appl. Math. 183 (2005) 16-28 9. Vidyasagar, M.: Nonliear Systems Analysis. Englewood Cliffs, New Jersey (1993)

Almost Sure Exponential Stability on Interval Stochastic Neural Networks with Time-Varying Delays Wudai Liao1 , Zhongsheng Wang1 , and Xiaoxin Liao2 1

School of Electrical and Information Engineering, Zhongyuan University of Technology, 450007, Zhengzhou, Henan, P.R. China {wdliao, zswang}@zzti.edu.cn 2 Department of Control Science and Engineering, Huazhong University of Science and Technology, 430074, Wuhan, Hubei, P.R. China [email protected]

Abstract. Because of VLSI realization of artificial neural networks and measuring the elements of the circuits, noises coming from the circuits and the errors of the parameters of the network systems are therefore unavoidable. Making use of the stochastic version of Razumikhin theorem of stochastic functional differential equation, Lyapunov direct methods and matrix analysis,almost sure exponential stability on interval neural networks perturbed by white noises with time varying delays is examined, and some sufficient algebraic criteria which only depend on the systems’ parameters are given. For well designed deterministic neural networks, the results obtained in the paper also imply that how much tolerance against perturbation they have.

1

Introduction

Exponential stability of deterministic neural networks have been widely studied and many useful results are obtained, such as [1, 2, 3, 4, 5]. In fact, neural networks work in noised environment considering unavoidable noises coming from VLSI realization of them and noised process of synapsis transmission of brain neural networks. So, for reality, we need introduce noises into the neural networks, which we call them stochastic neural networks. The stability of stochastic neural networks is also discussed by some scholars and some criteria [6, 7, 8, 9] ensuring them to be exponentially stable are gained. In practice, the weights between neurons of neural networks and noised intension must be estimated by using confidence intervals, which give us an interval stochastic neural networks. Based on the finite presentation of interval matrices, we get some criteria [10] which ensure the interval stochastic neural networks to be almost sure exponentially stable. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 159–164, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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In the present paper, we consider the stability of the interval stochastic neural network with time varying delays of the form dx(t) = [−(B + ΔB)x(t) + (A + ΔA)f (x(t − τ (t)))]dt + [(σ + Δσ)x(t − τ (t))]dw(t) .

(1)

Where B = diag.(b1 , b2 , · · · , bn ) is positive diagonal matrix, A, σ ∈ IR are weight matrix and perturbed matrix respectively, the error matrices ΔB ∈  B],  ΔA ∈ [−A,  A],  Δσ ∈ [−  A, σ [−B, σ, σ  ], and B,  ∈ IRn×n are up-bound man trices of errors, x = (x1 , x2 , · · · , xn )T ∈ IR is the state vector of the neural networks, x(t − τ (t)) = (x1 (t − τ1 (t)), x2 (t − τ2 (t)), · · · , xn (t − τn (t)))T , τi (t) ≥ 0 is the delay of the neuron i and 0 ≤ τi (t) ≤ τ, i = 1, 2, · · · , n. f (·) is the vector of the output functions of the neurons, f (x) = (f1 (x1 ), f2 (x2 ), · · · , fn (xn ))T , fi (·) satisfies that there exists a positive number αi > 0 such that n×n

|fi (u)| ≤ 1 ∧ αi |u|, −∞ < u < +∞, i = 1, 2, · · · , n .

(2)

w(·) is a one-dimension standard Brownian motion. We establish some sufficient algebraic criteria which ensure the neural network (1) to be almost sure exponentially stable and they also show that in what extent of noises the corresponding deterministic neural network can tolerate.

2

Notation

Let | · | denote the Euclidean norm of a vector x. The matrix norm  ·  and  · F are defined in the following way respectively:   A = λmax (AT A), AF = trace(AT A) . Where A ∈ IRn×n . We know that the two matrix norms are consistent to Euclidean norm of a vector x ∈ IRn , and Frobenius norm is consistent to itself, that is |Ax| ≤ A · |x|, |Ax| ≤ AF · |x|, ABF ≤ AF · BF .

(3)

C([−τ, 0]; IRn ) the space of all continuous IRn -valued functions φ defined on [−τ, 0], C2,1 (IRn × [t0 − τ, ∞); IR+ ) the family of all non-negative real-valued functions V (x, t) defined on IRn × [t0 − τ, ∞) which are continuously twice differentiable in x ∈ IRn and once differentiable in t ∈ [−τ, 0], LpFt ([−τ, 0]; IRn ) the family of all Ft -measurable C([−τ, 0]; IRn )-valued random variables φ such that Eφp < ∞. For any t ≥ t0 , define xt ∈ C([−τ, 0]; IRn ) : xt (θ) = x(t + θ), θ ∈ [−τ, 0], considering the following stochastic functional differential equation dx(t) = f (xt , t)dt + g(xt , t)dw(t) , t ≥ t0 , xt0 = ξ ∈ LpFt ([−τ, 0]; IRn ) . 0

(4)

For Lyapunov function V (x, t) ∈ C2,1 (IRn × [t0 − τ, ∞); IR+ ), define the differential operator from C2,1 (IRn × [t0 − τ, +∞)) to IR: 1 LV (φ, t) = Vt (φ(0), t)+Vx (φ(0), t)f (φ, t)+ trace[g T (φ(t), t)Vxx (φ(0), t)g(φ, t)] . 2

Almost Sure Exponential Stability on Interval Stochastic Neural Networks

3

161

Lemmas

In this section, we give two lemmas used in the paper latter. Lemma 1. Assume that there exists a Lyapunov function V (x, t) ∈ C2,1 (IRn × [t0 − τ, ∞); IR+ ) and positive numbers λ, p, c1 , c2 , q > 1 such that c1 |x|p ≤ V (x, t) ≤ c2 |x|p holds for any (x, t) ∈ IRn × [t0 − τ, ∞), and ELV (φ, t) ≤ −λEV (φ(0), t) holds for any t ≥ t0 and φ ∈ LpFt ([−τ, 0]; IRn ) satisfying EV (φ(θ), t + θ) < qEV (φ(0), t), −τ ≤ θ ≤ 0, then we have (1) For all ξ ∈ LpFt ([−τ, 0]; IRn ), the corresponding solution of the system (4) 0 has the following estimation p

E|x(t, ξ)| ≤

c2 p E|ξ| e−γ(t−t0 ) , t ≥ t0 . c1

Where γ = min{λ, logq/τ }. (2) For any solution x(t) of System (4), we have the following inequality 1 γ lim sup log|x(t, ξ)| ≤ − , a.s. t p t→∞ so long as p ≥ 1 and there exists positive constant k > 0 such that E(|f (xt , t)|p + |g(xt , t)| ≤ k

sup E|x(t + θ)|p , t ≥ t0

−τ ≤θ≤0

(5)

holds. That is the trivial solution of System (4) is almost sure exponential stability, and the Lyapunov exponent is γ/p. The proof of Lemma 1 is in Reference [11] and we can easily deduce that the condition (5) holds for System (1).

4

Main Results

We will set up some sufficient criteria ensuring the neural network (1) almost sure exponential stability in this section. For a symmetric positive definite matrix Q, select the Lyapunov function V (x) = xT Qx and for φ ∈ C([−τ, 0]; Rn ), the differential operator L with respect to the solution of neural network (1) has the form: LV (φ) = 2φT (0)Q[−(B + ΔB)φ(0) + (A + ΔA)f (φ(−τ (t)))] +φT (−τ (t))(σ + Δσ)T Q(σ + Δσ)φ(−τ (t)) .

(6)

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Theorem 1. Assume that there exist a symmetric positive definite Q and a positive number , such that  Q α2  2 + (σ +  min{bi } > max{bi } + + [ (A + A) σ )2 ] 2 2λmin (Q) 

(7)

holds, then for any ξ ∈ L2F0 ([−τ, 0]; IRn ), the corresponding solution of neural network (1) has the following properties lim sup t→∞

1 log q log(E|x(t; ξ)|2 ) ≤ − . t τ

(8)

1 log q log(|x(t; ξ)|) ≤ − . t 2τ

(9)

lim sup t→∞

Where α = max{αi } and q > 1 is the unique root of equation Q α2 log q  2 +(σ+ −2 min{bi }+2 max{bi }++ [ (A+A) σ)2 ]q = − . λmin(Q)  τ Proof (of Theorem 1). Because matrix Q is positive definite, so Q1/2 exists and is reversible, and λmin (Q1/2 (B + ΔB)Q−1/2 ) = λmin (B + ΔB), considering that two matrices (Q1/2 (B + ΔB)Q−1/2 ) and (B + ΔB) are similar. In the following, three terms in the differential operator (6) are estimated. − 2φT (0)Q(B + ΔB)φ(0) = −2φT (0)Q1/2 [Q1/2 (B + ΔB)Q−1/2 ]Q1/2 φ(0) ≤ −2λmin(Q1/2 (B + ΔB)Q−1/2 )φT (0)Qφ(0) = −2λmin(B + ΔB)V (φ(0)) . By using λmin (B +ΔB) = min{bi +Δbi } ≥ min{bi }+min{Δbi }, −bi ≤ Δbi ≤ bi and min{Δbi } ≥ min{−bi } = − max{bi }, we have the estimation of the first term in (6) as follows − 2φT (0)Q(B + ΔB)φ(0) ≤ −2(min{bi } − max{bi })V (φ(0)) .

(10)

Similarly , the estimation of the second term in (6) is as follows: 2φT (0)Q(A + ΔA)f (φ(−τ (t))) ≤ 2|φT (0)Q1/2 | · |Q1/2 (A + ΔA)f (φ(−τ (t)))| ≤ |φT (0)Q1/2 |2 + (1/)|Q1/2 (A + ΔA)f (φ(−τ (t)))|2

(11)

≤ V (φ(0)) + (1/)Q(A + ΔA) |f (φ(−τ (t)))|  2 )V (φ(−τ (t))) . ≤ V (φ(0)) + (α2 Q/λmin(Q))(A + A 2

2

And the third term in (6) can be estimated as follows φT (−τ (t))(σ + Δσ)T Q(σ + Δσ)φ(−τ (t)) Q(σ +  σ )2 ≤ Q(σ + Δσ)2 |φ(−τ (t))|2 ≤ V (φ(−τ (t))) . (12) λmin (Q)

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163

Substituting (10), (11) and (12) into (6), we have LV (φ) ≤ (−2 min{bi } + 2 max bi + )V (φ(0)) Q α2  2 + (σ +  + [ (A + A) σ )2 ]V (φ(−τ (t))) . λmin (Q) 

(13)

Therefore, for any t ≥ 0 and φ ∈ L2Ft ([−τ, 0]; IRn ) satisfying EV (φ(−τ (t))) ≤ qEV (φ(0)), we have ELV (φ) ≤ {−2 min{bi } + 2 max{bi } +  qQ α2  2 + (σ +  + [ (A + A) σ )2 ]}EV (φ(0)) . λmin (Q)  According to the condition (7) of the theorem, the equation qQ α2 log q  2 + (σ +  −2 min{bi } + 2 max{bi } +  + [ (A + A) σ )2 ] = − λmin (Q)  τ has an unique root q > 1, and therefore, log q EV (φ(0)) . τ By using Lemma 1, the desired estimations (8) and (9) hold. The proof is complete. ELV (φ) ≤ −

Corollary 1. Assume that there exits a symmetric positive-definite matrix Q, such that  1 Q  + Q (σ +  min{bi } > max{bi } + (1 + α2 )(A + A) σ )2 2 λmin (Q) 2λmin (Q) holds, then the estimations (8) and (9) hold, where α = max αi and q > 1 is the unique root of equation  Q   √q −2 min{bi } + 2 max{bi } + (1 + α2 )(A + A) λmin (Q) +

Q log q (σ +  σ )2 q = − . 2λmin (Q) τ

Proof (of Corollary 1). Similar to the proof of Theorem 1, by choosing    = qQ/λmin(Q)(A + A) in (13), then, we have   LV (φ) ≤ [−2 min{bi } + 2 max{bi } + qQ/λmin (Q)(A + A)]V (φ(0))  Q  + + [α2 Q/qλmin (Q)(A + A) (σ +  σ )2 ]V (φ(−τ (t))) . λmin (Q)

The remain of the proof is same as in Theorem 1. The proof is complete.

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By choosing the symmetric matrix Q equals unit matrix I in Corollary 1, we have the following corollary. Corollary 2. For the neural network (1), if 1  + 1 (σ +  min{bi } > max{bi } + (1 + α2 )(A + A) σ 2 ) (14) 2 2 holds, then the properties (8) and (9) hold, where α = max{αi } and q > 1 is the unique root of equation log q  √q + (σ +  −2 min{bi } + 2 max{bi } + (1 + α2 )(A + A) σ )2 q = − . τ Remark 1. Considering that the properties (3) of Frobenius norm, after carefully examining the proof of Theorem 1, Euclidean matrix norm can be replaced by Frobenius norm in all theorem and corollaries of this section. Remark 2. All conclusions obtained in the paper ensure exponential stability for deterministic neural network when perturbed item is zero in neural network (1).

Acknowledgement The work is supported by National Natural Science Foundation of China (60274007, 60474001) and Foundation of Young Bone Teacher of Henan Province of China.

References 1. Cao, J., Zhou, D.: Stability Analysis of Delayed Celluar Neural Networks. Neural Networks 11 (1998) 1601-1605 2. Liao, X.: Robust Stability for Interval Hopfield Neural Networks with Time Delay. IEEE Tran. Neural Networks 9(5) (1998) 1042-1045 3. Wang, J., Wu, G.: A Multilayer Recurrent Neural Network for Solving ContinuousTime Algebraic Riccati Equations. Neural Networks 11 (1998) 939-950 4. Lu, H.: On Stability of Nonlinear Continuous-Time Neural Networks with Delays. Neural Networks 13 (2000) 1135-1143 5. Liang, X., Wang, J.: A Proof of Kaszkurewicz and Byaya’s Conjecture on Absolute Stability of Neural Networks in Two-Neuron Case. IEEE Trans. Circuits and Systems-1: Fundamental Theory and Its Applications 47(4) (2000) 609-611 6. Liao, X., Mao, X.: Stability of Stochastic Neural Networks. Neual, Parallel and Scientific Computations 14(4) (1996) 205-224 7. Blythe, S., Mao, X.: Stability of Stochastic Delay Neural Networks. Journal of the Franklin Institute 338(2001) 481-495 8. Shen, Y., Liao, X.: Robust Stability of Nonlinear Stochastic Delayed Systems. Acta Automatica Sinic 25(4) (1999) 537-542 9. Liao, X., Mao, X.: Exponential Stability of Stochastic Delay Interval Systems. Systems and Control Letters 40(2000) 171-181 10. Liao, W., Liao, X.: Robust Stability of Time-Delayed Interval CNN in Noisy Environment. Acta Automatica Sinic 30(2)(2004) 300-305 11. Mao, X.: Stochastic Differential Equations and Their Applications. 1st edn. Horwood Pub. Chichester (1997)

Stochastic Robust Stability of Markovian Jump Nonlinear Uncertain Neural Networks with Wiener Process Xuyang Lou and Baotong Cui Research Center of Control Science and Engineering, Southern Yangtze University, 1800 Lihu Rd., Wuxi, Jiangsu 214122, P.R. China [email protected], [email protected]

Abstract. This paper deals with the stochastic robust stability problem for Markovian jump nonlinear uncertain neural networks (MJNUNNs) with Wiener process. Some criteria for stochastic robust stability of Markovian jump nonlinear uncertain neural networks are derived, even if the system contains Wiener process. All the derived results are presented in terms of linear matrix inequality.

1

Introduction

In recent years, considerable efforts have been devoted to the analysis of neural networks due to their potential applications in many areas [1]. The stability analysis on various aspects of neural networks has received considerable attentions in the past few years (see, for instance, [1-13] and the references cited therein). In ref [8], the authors introduced linear matrix inequality (LMI) approach for the exponential stability of BAM neural networks with constant or time-varying delays. In ref [7], we studied global asymptotic stability of delay bi-directional associative memory (BAM) neural networks with distributed delays and reaction-diffusion terms. The absolute exponential stability (AEST) of a class of BAM neural networks is analyzed in ref [3]. It was pointed out [14, 15] that a neural network could be stabilized or destabilized by certain stochastic inputs. However, the stability analysis for stochastic neural networks is difficult. Recently, although the stability analysis of neural networks has received much attention, the stability of stochastic neural networks has not been widely studied. Some results related to this issue have been reported in the literatures (see for instance [15-20]). In [16, 17], the stochastic stability for Hopfield neural networks were studied. The problem of stochastic robust stability (SRST) for uncertain delayed neural networks with Markovian jumping parameters is investigated via linear matrix inequality (LMI) technique in [18]. Liao and Mao [19] mainly discussed the mean square exponential stability of the stochastic delay neural network. To the best of our knowledge, only few works have been done on the SRST analysis for MJNUNNs. However, the SRST analysis for MJNUNNs with Wiener J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 165–171, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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process disturbance has never been tackled. The aim of this paper is to derive sufficient conditions for SRST analysis for MJNUNNs with Wiener process disturbance. All the results obtained in this paper are in terms of LMI. Notations and facts. In the sequel, we denote AT and A−1 the transpose and the inverse of any square matrix A. We use A > 0 (A < 0) to denote a positive- (negative-) definite matrix A; and I is used to denote the n × n identity matrix. Let R denote the set of real numbers, Rn denotes the n-dimensional Euclidean space and Rn×m the set of all n × m real matrices. diag[·] denotes a block diagonal matrix. Γ ([−d, 0]; Rn ) denotes the family of continuous functions φ(·) from [−d, 0] to Rn with the norm φ = sup |φ(θ)|. The symbol “” −d≤θ≤0

within a matrix represents the symmetric term of the matrix. The mathematical expectation operator with respect to the given probability measure P is denoted by E{·}. Fact 1. (Schur complement). Given constant matrices Ω1 , Ω2 , Ω3 , where 0 < Ω1 = Ω1T and 0 < Ω2 = Ω2T , then Ω1 + Ω3T Ω2−1 Ω3 > 0, if and only if     Ω1 Ω3T −Ω2 Ω3 < 0, or < 0. Ω3 −Ω2 Ω3T Ω1 Fact 2. For any real matrices Σ1 , Σ2 , Σ3 with appropriate dimensions and Σ3T Σ3 ≤ I, it follows that Σ1T Σ2 + Σ2T Σ1 ≤ εΣ1T Σ3 Σ1 + ε−1 Σ2T Σ3−1 Σ1 , ∀ε > 0.

2

Problem Statement

Consider a nonlinear uncertain neural network with time delay, which is described by a set of functional differential equations as follows x˙ m (t) = −am xm (t) − bdm xm (t − d) +

L 

bmn fn (xn (t))

n=1

+

L 

bdmn fn (xn (t − d)) + cm gm (xm (t), xm (t − d)),

(1)

n=1

for m, n = 1, 2, · · · , L. System (1) can be represented in vector state space form as follows x(t) ˙ = −Ax(t) − Ad x(t − d) + Bf (x(t)) + Bd f (x(t − d)) + Cg(x(t), x(t − d)), (2) where x(t) = (x1 (t), . . . , xL (t))T is the state vector of the neural network, d is the transmission delay, x(t − d) = (x1 (t − d), . . . , xL (t − d))T is the delayed state vector of the neural networks, A = diag(a1 , · · · , aL ), Ad = diag(ad1 , · · · , adL ) are positive diagonal matrices, B = (bmn )L×L , Bd = (bdmn )L×L , denote the connection weights, C = (cmn )L×L , denote the perturbation weights, g(x(t), x(t − d)) is

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 the uncertain perturbation with the form of g(x(t), x(t − d)) = g1 (x1 (t), x1 (t − T  d)), . . . , gn (xn (t), xL (t − d)) , the activation function is f (x(t)) = f1 (x1 (t)), T . . . , fL (xL (t)) . Though out this paper, we assume that the activation function fm (xm (t)) (m = 1, 2, . . . , L) and the nonlinear uncertain perturbation function g(x(t), x(t− d)) satisfy the following conditions: (H1 ) If there exist positive constants km , m = 1, . . . , L, such that 0<

fm (ξ1 ) − fm (ξ2 ) ≤ km ξ1 − ξ2

for all ξ1 , ξ2 ∈ R, ξ1  = ξ2 , m = 1, . . . , L. As we know, dynamical systems whose structures vary in response to random changes, which may result from abrupt phenomena such as parameter shifting and so on, are frequently occurring in practical situations. So it is necessary to consider the follow system (3). In this paper, let us consider the class of MJNUNNs with Wiener process defined on the probability space (Ω, Υ, P), with the following dynamics: ⎧  ⎪ − d) + B(ηt )f (x(t)) + Bd (ηt )f (x(t − d)), ⎪ ⎨ dx(t) = − A(ηt )x(t) − Ad (ηt )x(t  (3) +C(ηt )g(x(t), x(t − d)) dt + W (ηt )x(t)dω(t) ⎪ ⎪ ⎩ x(t) = φ0 , t ∈ [−d, 0], where φ0 ∈ Γ ([−d, 0]; Rn ), ω(t) is a standard Wiener process on the given probability space, W is the Wiener process matrix that is supposed to be known, for more details related to Wiener process, see ([21, 22, 23]). Given a probability space (Ω, Υ, P), where Ω is the sample space, Υ is the algebra of events and P is the probability measure defined on Υ. Let the random form process {ηt , t ∈ [0, +∞)} be a homogeneous, finite-state Markovian process with right continuous trajectories with generator  = (πij ) and transition probability from mode i at time t to mode j at time t + δ, i, j ∈ S, (S = {1, · · · , N }) :  πij δ + o(δ), if i  = j, pij = P r(ηt+δ = j|ηt = i) = (4) 1 + πij δ + o(δ), if i = j, with transition probability rates πij ≥ 0 for i, j ∈ S, i  = j and πii = −

N

πij ,

j=1,j =i

where δ > 0 and lim o(δ)/δ = 0. Note that the set S comprises the various δ→0

operational modes of the system under study. Assumption 1. The mode ηt is available at time t. Assumption 2. There exist positive constant matrices M and M1 , such that g(x(t), x(t − d)) ≤ M x(t) + M1 x(t − d).

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To proceed, we need the following definitions. Definition 1. The Markovian jump nonlinear neural networks (MJNNNs) (perturbation-free) is said to be stochastic stable if for all initial state φ0 and mode η0 such that lim E{x(t, φ0 , η0 )2 } = 0. (6) t→∞

Definition 2. The MJNUNNs with Wiener process is said to be SRST if it is stochastic stable for any admissible perturbation and Wiener process disturbance. Lemma 1. The infinitesimal generator of the Markov process (x(t), ηt ) is given by: LV (t, x(t), ) =

 ∂V (t, x(t), i)  − Ai x(t) + Bi f (x(t)) + Ci f (x(t − d)) ∂x N   1  + tr xT (t)WiT Pi Wi x(t) + πij V (t, x(t), j). 2 j=1

For stochastic systems, Itˆ o’s formula has an important role in the analysis of stochastic systems. The details can be found in [24] and omitted here. In the sequel, for simplicity, while ηt = i, the matrices A(ηt ), Ad (ηt ), B(ηt ), Bd (ηt ), C(ηt ), are represented by Ai , Adi , Bi , Bdi , Ci . Hence, we extract from system (3) for ηt = i ∈ S the following MJNUNN ⎧  ⎪ dx(t) = − Ai x(t) − Adi x(t − d) + Bi f (x(t)) ⎪ ⎨  (7) +B f (x(t − d)) + C g(x(t), x(t − d)) dt + Wi x(t)dω(t), di i ⎪ ⎪ ⎩ x(t) = φ0 , t ∈ [−d, 0].

3

Main Results

In this section, SRST criteria for MJNUNNs with Wiener process are given. Theorem 1. Suppose the activation function f (·) satisfy (H1 ). If there exist matrices Xi > 0, Qi > 0, Ri > 0, and Si > 0, such that the following LMIs holds for all i ∈ S:





 Xi Xi ATdi X i Xi K T Xi Xi MdT ≥ 0, ≥ 0, ≥0 (8)  Ri  Si  εi I and

⎡

⎤ (1, 1) Xi K T Xi M T ⎣  −Qi 0 ⎦ < 0, i = 1, · · · , N,   −ε1 I

where T (1, 1) = −(Ai Xi + Xi ATi ) + Ri + Bi Qi BiT + Bdi Si Bdi N +Ci (ε1 + ε2 )CiT + 3Xi + WiT Xi Wi + πij Xj . j=1

Then, the MJNUNN (7) is SRST.

(9)

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Proof. Consider the following generalized Lyapunov functional for mode i ∈ S : V (t, x(t), i) = xT (t)Pi x(t).

(10)

where Pi = PiT is symmetric and positive-definite matrix. Then, from Lemma 1 and Assumption 2, the infinitesimal generator of the Markov process (t, x(t), i) becomes: LV (t, x(t), i) ≤ −xT (t)(Pi Ai + ATi Pi )x(t) + xT (t)Pi Ri Pi x(t) +xT (t − d)ATdi Ri−1 Adi x(t − d) + xT (t)Pi Bi Qi BiT Pi x(t)

T T +xT (t)K T Q−1 i Kx(t) + x (t)Pi Bdi Si Bdi Pi x(t) +xT (t − d)K T Si−1 Kx(t − d) + xT (t)Pi Ci (ε1 + ε2 )CiT Pi x(t) −1 T T T T +ε−1 1 x (t)M M x(t) + ε2 x (t − d)Md Md x(t − d)

+xT (t)WiT Pi Wi x(t) +

N 

πij xT (t)Pj x(t).

(11)

j=1

Pre- and post-multiply (11) with Xi = Pi−1 , and by the Schur complement, if and only if inequality (8) and (9) hold, LV (t, x(t), i) < 0 exist, that is the MJNUNN (7) is SRST. This completes the proof. Theorem 2. Suppose the activation function f satisfy (H1 ). If there exist matrices Xi > 0, Qi > 0, and Si > 0, such that the following LMI holds for all i ∈ S: ⎡ ⎤ G11 G12 G13 G = ⎣  G22 G23 ⎦ > 0 (12)   G33 and

⎡

(1, 1) G12 + G13 + GT23 ⎢  (2, 2) ⎢ ⎢   ⎢ ⎣    

⎤ KT 0 KT 0 0 0 ⎥ ⎥ −Si 0 0 ⎥ ⎥ < 0, i = 1, · · · , N,  dG33 0 ⎦   −Qi

(13)

where T (1, 1) = −(Pi Ai + ATi Pi ) + Pi Bi Qi BiT Pi + Pi Bdi Si Bdi Pi + Ci (ε1 + ε2 )CiT N −1 T T +WiT Pi Wi + πij Pj + ε−1 1 M M + ε2 Md Md + dG11 + 2G13 , j=1

T (2, 2) = ε−1 2 Md Md + dG22 + 2G23 Then, the MJNUNN (7) is SRST.

Proof. Consider the following generalized Lyapunov functional for mode i ∈ S : V (t, x(t), i) = V1 + V2 + V3 ,

(14)

tθ 0 t V1 = xT (t)Pi x(t), V2 = 0 θ−d rT Grdsdθ, V3 = −d t+θ x˙ T (s)G33 x(s)dsdθ, ˙  T ˙ where r = x(θ) x(θ − d) x(s) .

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Now, in this case, for the first term V1 , the infinitesimal generator of the Markov process (t, x(t), i) becomes: LV (t, x(t), i) = LV1 + LV2 + LV3 ≤ η T (t)Ωη(t),

(15)

T T T where η(t) ˙ ⎤ , ⎡ = [x (t) x (t − d)T x(t)] (1, 1) G12 + G13 + G23 0 (2, 2) 0 ⎦ < 0, i = 1, · · · , N, Ω=⎣    dG33 where T (1, 1) = −(Pi Ai + ATi Pi ) + Pi Bi Qi BiT Pi + K T Q−1 i K + Pi Bdi Si Bdi Pi N T +Ci (ε1 + ε2 )CiT + WiT Pi Wi + πij Pj + ε−1 1 M M j=1

T +ε−1 2 Md Md + dG11 + 2G13 , T −1 T (2, 2) = K Si K + ε−1 2 Md Md + dG22 + 2G23 . By the Schur complement, Ω < 0 if and only if inequality (12) and (13) hold, LV (t, x(t), i) < 0 exist, that is the MJNUNN (7) is SRST. This completes the proof.

Remark 1. In the same way, some similar Markovian jump nonlinear uncertain neural networks can also be established, such as MJNUNNs with time-varying delays, MJNUNNs with multiple delays, Markovian jump uncertain BAM neural networks, etc.

4

Conclusions

The problems of SRST of MJNUNNs with Wiener process are investigated. Some criteria for SRST of MJNUNNs with Wiener process are derived. The sufficient conditions are expressed in terms of LMIs, which make them computationally efficient and flexible.

Acknowledgments The authors would like to thank the anonymous reviewers and the Associate Editor for their comments on improving the overall quality of the paper.

References 1. Rong, L.B.: LMI Approach for Global Periodicity of Neural Networks with TimeVarying Delays. IEEE Trans. Circuits Systems I: Regular Papers 52(7) (2005) 1451–1458 2. Cao, J., Liang, J.: Boundedness and Stability for Cohen-Grossberg Neural Network with Time-Varying Delays. J. Math. Anal. Appl. 296(2) (2004) 665–685 3. Lu, H.T.: Absolute Exponential Stability Analysis of Delayed Neural Networks. Physics Letters A 336(2-3) (2005) 133–140

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4. Lu, H.T., He, Z.Y.: Global Exponential Stability of Delayed Competitive Neural Networks with Different Time Scales. Neural Networks 18(3) (2005) 243–250 5. Cao, J.D., Huang, D.S., Qu, Y.Z.: Global Robust Stability of Delayed Recurrent Neural Networks. Chaos, Solitons and Fractals 23(1) (2005) 221–229 6. Cao, J.D., Chen, T.P.: Globally Exponentially Robust Stability and Periodicity of Delayed Neural Networks. Chaos, Solitons and Fractals 22(4) (2004) 957–963 7. Lou, X.Y., Cui, B.T.: Global Asymptotic Stability of BAM Neural Networks with Distributed Delays and Reaction-Diffusion Terms. Chaos, Solitons and Fractals 27(5) (2006) 1347–1354 8. Huang, X., Cao, J.D., Huang, D.S.: LMI-Based Approach for Delay-dependent Exponential Stability Analysis of BAM Neural Networks. Chaos, Solitons and Fractals 24(3) (2005) 885–898 9. Cao, J.D.: On Exponential Stability and Periodic Solutions of CNNs with Delays. Phys. Lett. A 267(5-6) (2000) 312–318 10. Cao, J.D., Zhou, D.M.: Stability Analysis of Delayed Cellular Neural Networks. Neural Networks 11(9) (1998) 1601–1605 11. Cao, J.D., Jiang, Q.H.: An Analysis of Periodic Solutions of Bi-Directional Associative Memory Networks with Time-Varying Delays. Physics Letters A 330(3-4) (2004) 203–213 12. Liang, J.L., Cao, J.D., Ho, D.W.C.: Discrete-Time Bidirectional Associative Memory Neural Networks with Variable Delays, Physics Letters A 335(2-3) (2005) 226–234 13. Li, C.D., Liao, X.F., Zhang, R.: Delay-Dependent Exponential Stability Analysis of Bi-Directional Associative Memory Neural Networks with Time Delay: an LMI Approach. Chaos, Solitons and Fractals 24(4) (2005) 1119–1134 14. Mao, X.R.: Stochastic Differential Equations and Applications. Chichester, U.K. Horwood (1997) 15. Liao, X.X., Mao, X.R.: Exponential Stability and Instability of Stochastic Neural Networks. Stochast. Anal. Appl. 14(2) (1996) 165–185 16. Wan, L., Sun, J.H.: Mean Square Exponential Stability of Stochastic Delayed Hopfield Neural Networks. Physics Letters A 343(4) (2005) 306–318 17. Huang, H., Ho, D.W.C., Lam, J.: Stochastic Stability Analysis of Fuzzy Hopfield Neural Networks with Time-Varying Delays. IEEE Trans. Circuits Systems II: Express Briefs 52(5) (2005) 251–255 18. Xie, L.: Stochastic Robust Stability Analysis for Markovian Jumping Neural Networks with Time Delays. The 2005 IEEE International Conference on Networking, Sensing and Control (2005) 923–928 19. Liao, X.X., Mao, X.R.: Stability of Stochastic Neural Networks. Neural, Para. Sci. Comput. 4(2) (1996) 205–224 20. Blythe, S., Mao, X.R., Shah, A.: Razumikhin-Type Theorems on Stability of Stochastic Neural Networks with Delays. Stochast. Anal. Appl. 19(1) (2001) 85– 101 21. Amold, L.: Stochastic Differential Equations: Theory and Applications. Loh Wilry and Sons, New York (1974) 22. Boukas, E.K., Liu, K.: Deterministic and Stochastic Systems with Time-Delay. Birkhauser, Boston (2002) 23. Raouf, J., Boukas, E.K.: Robust Stabilization of Markovian Jump Linear Singular Systems with Wiener Process. Proc. of the 2004 American Control Conference. Boston, Massachusetts (2004) 3170–3175

Stochastic Robust Stability Analysis for Markovian Jump Discrete-Time Delayed Neural Networks with Multiplicative Nonlinear Perturbations Li Xie1, Tianming Liu3, Guodong Lu2, Jilin Liu1, and Stephen T.C. Wong3 1

Department of Information and Electronic Engineering, Zhejiang University 2 State Key Lab of CAD CG, Zhejiang University, 310027 Hangzhou, P.R. China [email protected] 3 Center for Bioinformatics, HCNR, Harvard Medical School, 02115 Boston, MA, USA [email protected]

＆

Abstract. The problem of stochastic robust stability for Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbation is investigated via Lyapunov stability theory in this paper. Based on the linear matrix inequality (LMI) methodology, a novel analysis approach is developed. The sufficient conditions of stochastically robust stable are given in terms of coupled linear matrix inequalities. The stable criteria represented in LMI setting are less conservative and more computationally efficient than the methods reported in the literature.

1 Introduction In recent years, considerable research results on various aspects of neural networks are presented. Most of the literature on this subject was focused on the stability analysis of neural networks. Also, the existing studies on dynamical properties of neural networks are predominantly concentrated on the neural network with delay-free [1-3]. Since there exist transmission-delays among the neurons, the stability and the convergence properties of neural networks would be greatly dependent on their intrinsic time delays. In ref [4-6], by using the method of Liapunov functional, the stability criteria of bidirectional associative memory networks and recurrent neural networks with time delays are derived. When the neural networks are subject to the influence of the Markovian jump parameters, standard approaches for the behavior analysis of the neural networks are not applicable. The stochastic Lyapunov-Karsoviskii functional approach was adopted in studying of Markovian jump systems. In ref [7, 8], the stability, stablizability and controllability of jump systems were studied. The results are presented in the form of matrix measures or a set of coupled algebraic Riccati equations, which are too conservative or are difficult to obtain the solutions. It is well known that the linear matrix inequality (LMI) approach is a less conservative alternative to facilitate the solution process. The LMI-based techniques have been successfully employed in a variety of stability analysis for neural networks [9]. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 172 – 178, 2006. © Springer-Verlag Berlin Heidelberg 2006

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In this paper, we report the stochastic robust stable criteria for Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbation that are developed based on the LMI approach. The organization of the remaining part of this paper is as follows. In Section 2, the system model of Markovian jump discretetime delayed neural networks is provided. In Section 3, the stochastic robust stability of the neural network is investigated. Finally, we present the conclusions in Section 4.

2 Systems Descriptions Consider the discrete-time delayed neural networks with Markovian jumping parameters and multiplicative nonlinear perturbations, which are described by a set of discrete-time functional difference equations as follows: L

xn (k + 1) = −an (rk ) xn (k ) − a1n (rk ) xn (k − d ) + ∑ bnm (rk )σ m [ xm (k )] m =1

L

L

+ ∑ b1nm (rk )σ m [ xm (k − d )] + ∑ cnm (rk ) f m [ xm (k ), xm (k − d )]w(k ) , m =1

n, m = 1,K , L .

m =1

(1)

where xn (k ) denotes the state of the nth neurons at time k ; σ m [ xm (k )] denotes the

activation functions of the mth neuron on the nth neuron; f m [ xm (k ), xm (k − d )] denotes the multiplicative nonlinear perturbation of the mth neuron on the nth neuron; n, m = 1,K , L , and L denotes the number of neurons in a neural network; d is a positive integer denoting the constant time delay of the state of the neurons; the variable {w(k )} is a standard random scalar sequence satisfying: Ε{w(k )} = 0 , Ε{w2 (k )} = 1 .

(2)

where w(0), w(1), K , are independent; an (rk ) , a1n (rk ) denote the charging time constants or passive decay rates of the nth neuron, bnm (rk ) , b1nm (rk ) denote the connection weights of the jth neuron on the ith neuron, cnm (rk ) are parameters of multiplicative nonlinear noises. These coefficients are the functions of the random process { rk , k ∈ Z }, which is a discrete-time discrete-state Markov chain taking values in a finite set S = {1, 2,K , s} , with transition probabilities from mode i at time k to mode j at time k + 1 , i.e. rk = i , rk +1 = j ∈ S . For any i, j ∈ S , we have: pij = Pr{rk +1 = j | rk = i} , pij ≥ 0 . s

∑p j =1

ij

= 1.

(3) (4)

We can rewrite the Markovian jump discrete-time delayed neural network with multiplicative nonlinear perturbation (1) in the following vector form: xk +1 = − A(rk ) xk − A1 (rk ) xk − d + B(rk )σ ( xk ) + B1 (rk )σ ( xk − d ) + C (rk ) f ( xk , xk − d )wk .

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where A(rk ) = diag{a1 (rk ),K , aL (rk )} ; A1 (rk ) = diag{a11 (rk ),K , a1L (rk )} ;

B(rk ) = {bnm (rk )}L× L ; B1 (rk ) = {b1nm (rk )}L× L ; C (rk ) = {cnm (rk )}L× L ;

xk = x(k ) = [ x1 (k ),K , xL (k )] ; xk − d = x(k − d ) = [ x1 (k − d ),K , xL (k − d ) ] ; T

T

σ ( xk ) = {σ 1[ x1 (k )],K , σ L [ xL (k )]}T ; σ ( xk − d ) = {σ 1[ x1 (k − d )],K , σ L [ xL (k − d )]}T ; f ( xk , xk − d ) = { f1 ( x1 (k ), x1 (k − d )),K , f L ( xL (k ), xL (k − d ))}T . To obtain our results, we state centain assumptions and definitions as follows: Assumption 1. The activation functions σ n (.) satisfy globally Lipschitz conditions,

i.e. there exist positive constants wn such that

0<

σ n (u ) − σ n (v) u −v

≤ wn , n = 1,K , L .

(6)

Assumption 2. The perturbation function f ( x(k ), x(k − d )) satisfies norm-bounded

condition, i.e., there exist positive matrices M and M 1 , such that

f ( x(k ), x(k − d )) ≤ M x(k ) + M 1 x (k − d ) .

(7)

Definition 1. The Markovian jump discrete-time delayed neural network (5) (perturbation-free) is said to be stochastically stable if for every initial state ( x0 , r0 ) , there exist a finite number M% ( x , r ) > 0 , such that 0

0

K

2 lim E{∑ x(k ) x0 , r0 } ≤ M% ( x0 , r0 ) .

K →∞

k =0

(8)

Definition 2. The Markovian jump discrete-time delayed neural network (5) is said to be stochastic robust stable if it is stochastically stable for any admissible perturbation.

3 Main Results In this section, the sufficient conditions for stochastic robust stability of Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbations are presented. Theorem 1. The Markovian jump discrete-time delayed neural network (5) is stochastic robust stable if there exist matrices Pi > 0 ( i = 1,K , s ), S > 0 , R > 0 , and

constants ε n > 0 (n = 1,K , 6) , ρ j > 0 ( j = 1,K , s ) satisfying the coupled LMIs ⎡ Ξi11 Ξ i = ⎢⎢ Ξi 21 ⎢⎣ Ξi 31

Ξi12 Ξ i 22 0

Ξi13 ⎤ 0 ⎥⎥ < 0 , Ξ i 33 ⎥⎦

⎡ ρjI ⎢ ⎢⎣ Pj Ci

CiT Pj ⎤ ⎥ > 0 , i, j = 1,K , s . Pj ⎥⎦

(9)

Stochastic Robust Stability Analysis

175

where Ξ i11 = − Pi + (ε1 + ε 2 + ε 3 ) AiT Ai + (ε 4 + ε 5 ) A1Ti A1i + ε 6W T BiT BiW +2 ρ% ij ( M T M + M 1T M 1 ) , Ξ i12 = ΞTi 21 = ⎡⎣ AiT P%ij , A1Ti P%ij , W T BiT P%ij , W T B1Ti P%ij ⎤⎦ , Ξ i13 = ΞTi 31 = ⎡⎣ A1Ti P%ij , W T BiT P%ij , W T B1Ti P%ij , W T BiT P%ij , W T B1Ti P%ij , W T B1Ti P%ij ⎤⎦ , Ξ i 22 = diag{− P%ij , − P%ij , − P%ij , − P%ij } , Ξ i 33 = diag{−ε1 I , − ε 2 I , − ε 3 I , − ε 4 I , − ε 5 I , − ε 6 I } , s

s

j =1

j =1

W = diag{w1 ,K , wL } , P%ij = ∑ pij Pj , ρ%ij = ∑ pij ρ j .

(10)

Proof. Introduce the Lyapunov function candidate for the neural network (5)

Vk ( xk , rk ) = xkT P(rk ) xk +

k −1

∑

i=k −d

xiT Sxi +

k −1

∑σ

i=k −d

T

( xi )Rσ ( xi ) .

(11)

Let Pi denotes P(rk ) when rk = i , and Pi is a const matrix for each i ∈ S . Then ΔVk ( xk , rk ) = E{Vk +1 ( xk +1 , rk +1 ) xk , rk = i ) − Vk ( xk , rk = i ) = {− xkT AiT − xkT− d A1Ti + σ T ( xk ) BiT + σ T ( xk − d ) B1Ti }P%ij {− Ai xk − A1i xk − d + Biσ ( xk ) + B1iσ ( xk − d )} + f T ( xk , xk − d )CiT P%ij Ci f ( xk , xk − d ) + xkT Sxk − xkT− d Sxk − d + σ T ( xk ) Rσ ( xk ) −σ T ( xk − d ) Rσ ( xk − d ) − xkT Pi xk ≤ xkT AiT P%ij Ai xk + ε1 xkT AiT Ai xk + ε1−1 xkT− d A1Ti P%ij P%ij A1i xk − d +ε 2 xkT AiT Ai xk + ε 2−1σ T ( xk ) BiT P%ij P%ij Biσ ( xk ) + ε 3 xkT AiT Ai xk +ε 3−1σ T ( xk − d ) B1Ti P%ij P%ij B1iσ ( xk − d ) + xkT− d A1Ti P%ij A1i xk − d +ε 4 xkT− d A1Ti A1i xk − d + ε 4−1σ T ( xk ) BiT P%ij P%ij Biσ ( xk ) +ε 5 xkT− d A1Ti A1i xk − d + ε 5−1σ T ( xk − d ) B1Ti P%ij P%ij B1iσ ( xk − d ) +σ T ( xk ) BiT P%ij Biσ ( xk ) + ε 6σ T ( xk ) BiT Biσ ( xk ) +ε 6−1σ T ( xk − d ) B1Ti P%ij P%ij B1iσ ( xk − d ) + σ T ( xk − d ) B1Ti P%ij B1iσ ( xk − d ) +2 ρ% ij ( xkT M T Mxk + xkT− d M 1T M 1 xk − d ) − xkT Pi xk +σ T ( xk ) Rσ ( xk ) − σ T ( xk − d ) Rσ ( xk − d ) + xkT Sxk − xkT− d Sxk − d .

(12)

Let S = A1Ti P%ij A1i + ε1−1 A1Ti P%ij P%ij A1i + (ε 4 + ε 5 ) A1Ti A1i + 2 ρ% ij M 1T M 1 . R = B1Ti P%ij B1i + (ε 3−1 + ε 5−1 + ε 6−1 ) B1Ti P%ij P%ij B1i .

(13)

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We have ΔVk ( xk , rk ) ≤ xkT Θi xk .

(14)

where Θi = − Pi + AiT P%ij Ai + A1Ti P%ij A1i + (ε1 + ε 2 + ε 3 ) AiT Ai + (ε 4 + ε 5 ) A1Ti A1i +ε1−1 A1Ti P%ij P%ij A1i + W T BiT P%ij BiW + W T B1Ti P%ij B1iW + ε 6W T BiT BiW +2 ρ% ij ( M T M + M 1T M 1 ) + (ε 3−1 + ε 5−1 + ε 6−1 )W T B1Ti P%ij P%ij B1iW + (ε 2−1 + ε 4−1 )W T BiT P%ij P%ij BiW .

(15)

By using Schur complement lemma, if the coupled LMIs (9) exist, Θi < 0 hold. Then it follows that ΔVk ( xk , rk ) ≤ xkT Θi xk < 0 .

(16)

Here, for x(k ) ≠ 0 , ΔVk ( xk , rk ) ≤− Vk ( xk , rk )

− xkT Θi xk xkT P(rk ) xk +

k −1

∑

i =k −d

xiT Sxi +

k −1

∑

i=k −d

≤ α −1 . xiT W T RWxi

(17)

where ⎧ λmin (−Θi ) ⎫ ⎬ <1. ⎩ λmax ( EP(rk )) ⎭

(18)

α = 1 − min ⎨ rk ∈S

and ⎡ S + W T RW L ⎤ 0 ⎡ P(rk ) 0 ⎤ ⎢ ⎥ ; S =⎢ EP(rk ) = ⎢ M O M ⎥ ⎥ . S⎦ ⎣ 0 T ⎢ 0 L S + W RW ⎥⎦ ⎣ d ×d From (17), we have 0<

E{Vk +1 ( xk +1 , rk +1 ) xk , rk = i ) Vk ( xk , rk )

≤α .

(19)

that is 0 < α < 1 , therefore

E{Vk +1 ( xk +1 , rk +1 ) xk , rk = i ) ≤ αVk ( xk , rk ) .

(20)

which implies that K

E{∑ Vk ( xk , rk ) x0 , r0 ) ≤ (1 + α + L + α K )V0 ( x0 , r0 ) ≤ k =0

1 − α K +1 V0 ( x0 , r0 ) . 1−α

(21)

thus, we have K

lim E{∑ xkT P (rk ) xk x0 , r0 ) ≤

K →∞

k =0

1 V0 ( x0 , r0 ) . 1−α

(22)

Stochastic Robust Stability Analysis

177

Define V0 ( x0 , r0 ) . M% ( P(r0 ))  (1 − α ) max { ρ [ P(ri )]} i∈S

(23)

then N

lim E{∑ Vk ( xk , rk ) x0 , r0 ) ≤ M% ( P(r0 )) .

N →∞

k =0

(24)

which implies that the Markovian jump discrete-time delayed neural network (5) is stochastic robust stable for all admissible nonlinear perturbation. Remark. In Theorem 1, stochastic robust stability criterion for Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbation is derived in terms of LMIs. The result can be simplified to delay-free case. Corollary 1. The Markovian jump discrete-time neural network (5) with delay-free is stochastic robust stable if there exist matrices Pi > 0 ( i = 1,K , s ), S > 0 , R > 0 , and

constants ε > 0 , ρ j > 0 ( j = 1,K , s ) satisfying the coupled LMIs

⎡ Ξ i11 ⎢ Ξ i = ⎢ P%ij Ai ⎢ P%ij BiW ⎣

AiT P%ij − P% ij

0

W T BiT P%ij ⎤ ⎥ 0 ⎥ <0, −ε I ⎥⎦

⎡ ρjI ⎢ ⎣⎢ Pj Ci

CiT Pj ⎤ ⎥ > 0 , i, j = 1,K , s . Pj ⎦⎥

(25)

where Ξ i11 = − Pi + ε AiT Ai + 2 ρ%ij M T M , s

s

j =1

j =1

W = diag{w1 ,K , wL } , P%ij = ∑ pij Pj , ρ%ij = ∑ pij ρ j .

(26)

4 Conclusions In this paper, we consider the problem of stochastic robust stability analysis for Markovian jump discrete-time delayed neural networks with multiplicative nonlinear perturbation. The sufficient condition of the robust stochastically stable are given in terms of linear matrix inequalities. Our approach is believed to be computationally more flexible and efficient then other existing approaches. Acknowledgement. This work was supported by Chinese Natural Science Foundation (60534070, 60473129), and by HCNR, Harvard Medical School, USA.

References 1. Michel, A.N., Farrell, J.A., Porod, W.: Qualitative Analysis of Neural Networks, IEEE Trans. CAS 36(2) (1989) 229-243 2. Kelly, D.G.: Stability in Contractive Nonlinear Neural Networks, IEEE Trans. Bio. Eng., 37(3) (1990) 231-242

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3. Liang, X.B., Wang, J.: An Additive Diagonal Stability Condition for Absolute Exponential Stability of a General Class of Neural Networks, IEEE Trans. CAS—I 48 (2001) 1308-1317 4 Cao, J., Wang, J.: Global Asymptotic and Robust Stability of Recurrent Neural Networks with Time Delays, IEEE Trans. CAS—I 52(2) (2005) 417-426 5 Liao, X., Wong, K.W.: Robust Stability of Interval Bidirectional Associative Memory Neural Network With Time Delays, IEEE Trans. SMC—B 34(2) (2004) 1142-1154 6. Feng, C.H., Plamondon, R.: Stability Analysis of Bidirectional Associative Memory NetWorks with Time Delays, IEEE Trans. NN 14(6) (2003) 1560-1565 7. Ji, Y., Chizeck, H.J.: Controllability, Stabilizability, and Continuous-Time Markovian Jump Linear Quadratic Control, IEEE Trans. AC 35(7) (1990) 777–788 8. Feng, X., Loparo, K.A., Ji, Y. et. al.: Stochastic Stability Properties of Jump Linear System, IEEE Trans. AC 37(1) (1992) 38–53 9. Liao, X., Chen, G.R., Sanchez, E.N.: LMI-Based Approach for Asymptotically Stability Analysis of Delayed Neural Networks. IEEE Trans. CAS-I 49(7) (2002) 1033-1039

Global Robust Stability of General Recurrent Neural Networks with Time-Varying Delays Jun Xu1,2 , Daoying Pi1 , and Yong-Yan Cao1 1

National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, P.R. China 2 College of Information & Management, Jiangxi University of Finance & Economics, Nanchang 330013, P.R. China {xujung, yongyancao}@gmail.com, [email protected]

Abstract. This paper is devoted to global robust stability analysis of the general recurrent neural networks with time-varying parametric uncertainty and time-varying delays. To remove the dependence on the size of time-delays, Lyapunov-Razumikhin stability theorem and LMI approach are applied to derive the global robust stability conditions for the neural networks. Then delay-dependent global robust stability criteria are developed based on integrating Lyapunov-Krasovskii functional method and LMI approach. These stability criteria are in term of the solvability of linear matrix inequalities.

1

Introduction

Robust stability have been extensively studied by researches recently ( see for example [5], [8], [9], [10] and the references therein) because the stability of a neural networks may often be destroyed by its unavoidable uncertainty due to the existence of modeling errors, external disturbance and parameter fluctuation during the implementation on very-large-scale-integration (VLSI) chip [10]. Global robust stability of interval neural networks with time-varying delays was studied in [10], [11]. Robust stability of cellular neural networks with constant parametric uncertainties and single time delays was studied in [8]. Robust stability of cellular neural networks with time-varying parametric uncertainties and time-varying delays was studied in [9]. LMI is an efficient approach which is globally used in neural networks’ stability study, see for example in [5], [6], [8], [9] and the references therein. Motivated by these, in this paper, we study robust stability of a general recurrent neural networks with norm-bounded time-varying parametric uncertainties and time-varying delays by integrating Lyapunov-Krasovskii functional method (Lyapunov-Razumikhin function method) and LMI approach. We can divide the robust stability problem of time-delay system into two categories according to their dependence on the size of the delays, namely delay-independent stability and delay-dependent stability. While delay-independent stability criteria guarantee the asymptotic stability of the system irrespective of the size of the J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 179–184, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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time-delay, delay-dependent stability criteria provide an upper bound of timedelay size which assures the asymptotic stability of the system. In this work we derive both delay-independent and delay dependent stability conditions.

2

Model Description and Preliminaries

The recurrent neural networks with time-varying delays and time-varying parametric uncertainties model is described by the following differential equations: dx(t) = −(C + C(x, t))x(t) + (A + A(x, t))f (x(t)) dt + (B + B(x, t))l(x(t − τ (t))) + I, x(t) =φ(t), t ∈ [−¯ τ , 0]

(1)

where x(t) ∈ Rn is the neuron state vector; x(t − τ (t)) = [x1 (t − τ1 (t)), x2 (t − τ2 (t)), . . . , xn (t − τn (t))]T , τj (t) : [0, ∞) → [0, τ¯](j = 1, 2, . . . , n) are continuous and differentiable functions; A = {aij }n×n and B = {bij }n×n are feedback matrix and delayed feedback matrix, respectively; C = diag(c1 , c2 , . . . , cn ) > 0 is self-feedback matrix; C(x, t), A(x, t) and B(x, t) are matrix functions representing the uncertainties in the matrix C, A and B, respectively; I denotes the external bias; τ (t) denotes the time-varying delays which is bounded by τ (t) ≤ τ¯ satisfying τ˙ (t) ≤ d < 1. Throughout this paper, we assume that each neuron activation function in system (1), satisfies following Assumption (H) 0≤

fj (xj ) − fj (yj ) lj (xj ) − lj (yj ) ≤ G¯j and 0 ≤ ≤ H¯j , xj − yj xj − yj

for all xj , yj ∈ R, where xj = yj (j = 1, 2, . . . , n). In this paper, we assume that the uncertainties can be described by C(x, t) = D0 F0 (x, t)E0 ; A(x, t) = D1 F1 (x, t)E1 ; B(x, t) = D2 F2 (x, t)E2 , where Di , Ei (i = 0, 1, 2) are known real constant matrices , and Fi (t), i = 0, 1, 2, are unknown real time-varying matrix with Lebesgue measurable elements bounded by FiT (x, t)Fi (x, t) ≤ I, ∀t (i = 0, 1, 2). For simplicity, in what follows, we will denote ˜ = C + C(x, t); A(t) ˜ = A + A(x, t); B(t) ˜ = B + B(x, t). C(t) We assume that the system (1) has an equilibrium point xe = (xe1 , xe2 , . . . , xen )T . In order to simplify our proofs, we shall shift the equilibrium point to the origin using the transformation y(t) = x(t) − xe . By denoting g(y(t)) = f (y(t) + xe ) − f (xe ) and h(y(t − τ (t))) = l(y(t − τ (t)) + xe ) − l(xe ), we can transform system (1) into the form dy(t) ˜ ˜ ˜ = −C(t)y(t) + A(t)g(y(t)) + B(t)h(y(t − τ (t))). dt

(2)

Global Robust Stability of General Recurrent Neural Networks

181

With the assumption (H), we have the following inequalities ¯ Gy(t), ¯ g T (y(t))Qg(y(t)) ≤ y T (t)GQ T ¯ Hy(t ¯ h (y(t − τ (t)))h(y(t − τ (t))) ≤ y (t − τ (t))H − τ (t)), T

(3) (4)

where Q > 0 is any diagonal matrix.

3

Main Results

3.1

Delay-Independent Global Robust Stability

Firstly, we study the global robust stability of the general neural networks with Lyapunov-Razumikhin function approach (Theorem 1.2 in [3]). Theorem 1. Consider system (2),if there exist diagonal marix Q > 0, and scalars γ0 , γ1 , γ2 > 0, ν > 1 such that ¯G ¯ − νQ − γ −1 QD0 DT Q − γ1 QD1 DT Q Ξ = CQ + QC − γ0 E0T E0 − G 0 1 0 −γ2 QD2 D2T Q − QAAT Q − QBB T Q − QAE1T (γ1 I1 − E1 E1T )−1 E1 AT Q −QBE2T (γ2 I2 − E2 E2T )−1 E2 B T Q > 0 namely,

⎡

Π1 QAE1T QBE2T QD0 QD1 QD2 ⎢ E1 AT Q Π2 0 0 0 0 ⎢ ⎢ E2 B T Q 0 Π 0 0 0 3 ⎢ ⎢ D0T Q 0 0 γ I 0 0 0 3 Ξ=⎢ ⎢ D1T Q 0 0 0 γ1−1 I4 0 ⎢ ⎢ DT Q 0 0 0 0 γ2−1 I5 2 ⎢ ⎣ AT Q 0 0 0 0 0 BT Q 0 0 0 0 0

⎤ QA QB 0 0 ⎥ ⎥ ⎥ 0 ⎥ 0 0 ⎥ ⎥>0 0 0 ⎥ ⎥ 0 0 ⎥ ⎥ I6 0 ⎦ 0 I7

¯H ¯ > 0, Q−H

(5)

(6)

¯ G, ¯ Π2 = γ1 I1 −E1 E1T > 0, Π3 = γ2 I2 − where Π1 = CQ+QC −γ0E0T E0 −νQ− G T E2 E2 > 0. And I1 , I2 , I3 , I4 , I5 , I6 , I7 are identity matrices. Then the trivial solution of system (1) is globally robustly stable which is delay-independent. Proof: Given diagonal matrix Q > 0, consider a Lyapunov-Razumikhin function candidate as V (y(t)) = y T (t)Qy(t). First, it is easy to have π1 y(t) 2 ≤ V (y(t)) ≤ π2 y(t) 2

(7)

where π1 = λmin (Q) and π2 = λmax (Q). The derivative of V (y(t)) along the solution of (2) can be computed as follows V˙ (y(t)) = 2y T (t)Qy(t) ˙ = −y T (t)(CQ + QC)y(t) − 2y T (t)E0T F0T (x, t)D0T Qy(t) ˜ ˜ +2y T (t)QA(t)g(y(t)) + 2y T (t)QB(t)h(y(t − τ (t)).

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Using Lemma 2.2 in [1], we have − 2y T (t)E0T F0T (t)D0T Qy(t) ≤ γ0 y T (t)E0T E0 y(t) + γ0−1 y T (t)QD0 D0T Qy(t). (8) By applying Lemma 1 in [4], Lemma 2.2 in [1] and (3), we have ˜ 2y T (t)QA(t)g(y(t)) ≤y T (t)[QAAT Q + QAE1T (γ1 I1 − E1 E1T )−1 E1 AT Q (9) ¯ G]y(t). ¯ + γ1 QD1 DT Q + G 1

Similarly, we can obtain ˜ 2y T (t)QB(t)h(y(t − τ (t)) ≤ y T (t)[QBB T Q + QBE2T (γ2 I2 − E2 E2T )−1 (10) ¯ Hy(t ¯ · E2 B T Q + γ2 QD2 D2T Q]y(t) + y T (t − τ (t))H − τ (t)). With the inequations (8)–(10) and condition (6) in Theorem 1, we have ¯G ¯ − QAE1T (γ1 I1 − E1 E1T )−1 V˙ (y(t)) ≤ −y T (t)[CQ + QC − γ0 E0T E0 − G · E1 AT Q − QBE2T (γ2 I2 − E2 E2T )−1 E2 B T Q − γ0−1 QD0 D0T Q − γ1 Q ·

D1 D1T Q

−

γ2 QD2 D2T Q

(11)

− QAA Q − QBB Q]y(t) + y (t − τ (t))Qy(t − τ (t)). T

T

T

By using Lyapunov-Razumikhin stability theorem (Theorem 1.2 in [3]), we assume that there exists a real ν > 1 such that y T (t − τ (t))Qy(t − τ (t)) = V (y(t − θ)) < νV (y(t)) = νy T (t)Qy(t).

(12)

With (9) and (10), we can obtain that V˙ (yt ) ≤ −y T (t)Ξy(t). So, by conditions (5), (6) in Theorem 1, inequalities (7), (10), (11), and Lyapunov-Razumikhin stability theorem (Theorem 1.2 in [3]), we have that the zero solution of system (2) is globally uniformly asymptotically stable. By Definition 2.1 in [1], the zero solution of system (2) is globally robustly stable and hence the trivial solution of system (1) is globally robustly stable. The proof is completed. 3.2

Delay-Dependent Global Robust Stability

To reduce conservativeness in the analysis when the information on the size of time-delays is available, we will establish some delay-dependent robust stability conditions for the general neural networks with Lyapunov-Krasovskii functional method. Theorem 2. Consider system (2), if there exist P > 0, and scalars ε0 , ε1 , ε2 > 0 such that T T ¯H ¯ −G ¯G ¯ − ε0 E0T E0 − ε−1 Γ = CP + P C − H 0 P D0 D0 P − ε1 P D1 D1 P

−δ −1 ε2 P D2 D2T P − P AAT P − δ −1 P BB T P − P AE1T (ε1 I8 − E1 E1T )−1 E1 AT P −δ −1 P BE2T (ε2 I9 − E2 E2T )−1 E2 B T P > 0

Global Robust Stability of General Recurrent Neural Networks

183

namely, ⎡

Λ1 P AE1T P BE2T P D0 P D1 P D2 ⎢ E1 AT P Λ2 0 0 0 0 ⎢ ⎢ E2 B T P 0 Λ 0 0 0 3 ⎢ ⎢ D0T P 0 0 ε I 0 0 0 10 Γ =⎢ ⎢ D1T P 0 0 0 ε−1 0 1 I11 ⎢ −1 ⎢ DT P 0 0 0 0 δε2 I12 2 ⎢ ⎣ AT P 0 0 0 0 0 BT P 0 0 0 0 0

⎤ PA PB 0 0 ⎥ ⎥ ⎥ 0 ⎥ 0 0 ⎥ ⎥ > 0, 0 0 ⎥ ⎥ 0 0 ⎥ ⎥ I13 0 ⎦ 0 δI14

(13)

¯H ¯ −G ¯ G, ¯ Λ2 = ε1 I8 − E1 E T > 0, Λ3 = where Λ1 = CP + P C − ε0 E0T E0 − H 1 T δ(ε2 I9 − E2 E2 ) > 0, δ = 1 − d > 0. And I8 , I9 , I10 , I11 , I12 , I13 , I14 are identity matrices. Then the trivial solution of system (1) is globally robustly stable for all time delay τ (t) ≤ τ¯ and τ˙ (t) ≤ d < 1, which is delay-dependent. The Lyapunov-Krasovskii functional is chosen as 

t

T

hT (y(ξ))h(y(ξ))dξ,

V (yt ) = V1 + V2 = y (t)P y(t) +

(14)

t−τ (t)

where P > 0 to be determined. Proof: Consider the Lyapunov-Krasovskii functional (14). First, we have π3 yt (0) 2 ≤ V (yt ) ≤ π4 yt 2c ,

(15)

¯ 2 . Taking the where π3 = λmin (P ). Note (4), we have π4 = λmax (P0 ) + τ¯ H derivative of V2 leads to V˙2 = hT (y(t))h(y(t)) − (1 − τ˙ (t))hT (y(t − τ (t)))h(y(t − τ (t))). By (4), we have ¯ Hy(t) ¯ V˙2 ≤ y T (t)H − δhT (y(t − τ (t))h(y(t − τ (t))).

(16)

As the proof of Theorem 1, the derivative of V1 along the solution of (2) can be computed as follows V˙1 = − y T (t)(CP + P C)y(t) − 2y T (t)E0T F0T (x, t)D0T P y(t) ˜ ˜ + 2y T (t)P A(t)g(y(t)) + 2y T (t)P B(t)h(y(t − τ (t)).

(17)

Similar to the proof of Theorem 1, we have V˙ (yt ) ≤ −y T (t)Γ y(t). Similarly, by Lyapunov-Krasovskii stability theorem (Theorem 1.1 in [3]), we have that the trivial solution of system (1) is globally robustly stable. The proof is completed.

184

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Conclusion

In this paper, both the delay-independent and delay-dependent global robust stability are discussed by Lyapunov-Razumikhin function approach and LyapunovKrasovskii functional method for the general recurrent neural networks with time-varying parametric uncertainties and time-varying delays. By integrating Lyapunov-Razumikhin function approach and LMI approach, delay-independent stability conditions which do not involve the time derivative of the time-delays are studied. Then delay-dependent stability criteria which involve the time derivative of the time-delays are obtained by Lyapunov-Krasovskii functional method and LMI approach.

Acknowledgments This work is supported by the 973 program of China under grant No. 2002CB312200; the National Science Foundation of China under grant No. 60574019 and No. 60474045.

References 1. Cao, Y. Y., Sun, Y. X. and Cheng, C.: Delay-Dependent Robust Stabilization of Uncertain Systems with Multiple State Delays. IEEE Trans. Automatic Control 43 (1998) 1608-1612 2. Hale, J. K. and Lunel, S. M. V.: Introduction to Functional Differential Equations. Applied Math. Scinces, New York: Springer-Verlag 99 (1993) 3. Niculescu, S. I.: Delay Effects on Stability: An Robust Control Approach. Lecuture Notes in Control and Information Sciences, Springer-Verlag London Limited (2001) 4. Liao, X. F., Chen, G. and Sanchez, E. N.: Delay-dependent Exponential Stability Analysis of Delayed Neural Networks: An LMI Approach. Neural Networks 15 (2002) 855-866 5. Cao, J. and Wang, J.: Global Asymptotic and Robust Stability of Recurrent Neural Networks with Time Delays. IEEE Trans. on Circuits and systems-I 52 (2005) 417425 6. Cao, J. and Ho, Daniel W. C.: A General Framework for Global Asymptotic Stability Analysis of Delayed Neural Networks Based on LMI Approach. Chaos, Solitons and Fractals 24 (2005) 1317-1329 7. Huang, H., Ho, D.W.C. and Cao, J.: Analysis of Global Exponenital Stability and Periodic Solutions of Neural Networks with Time-varying Delays. Neual Networks 18 (2005) 161-170 8. Singh, V.: Robust Stability of Celluar Neural Networks with Delay: linear matrix inequality approach. IEE Proc.-Control Theory Appl. 151 (2004) 125-129 9. Zhang, H., Li, C. and Liao, X. F.: A Note on the Robust Stability of Neural Networks with Time Delay. Chaos Solitons and Fractals 25 (2005) 357-360 10. Li, C., Liao, X. F., Zhang, R. and Prasad, A.: Global Robust Exponential Stability Analysis for Interval Neural Networks with Time-varying Delays. Chao, Solitons and Fractals 25 (2005) 751-757 11. Chen, A., Cao, J. and Huang, L.: Global Robust Stability of Interval Cellular Neural Networks with Time-varying Delays. Chaos, Solitons and Fractals 23 (2005) 787-799

Robust Periodicity in Recurrent Neural Network with Time Delays and Impulses Yongqing Yang1,2 1

2

School of Science, Southern Yangtze University, Wuxi 214122, China [email protected] Department of Mathematics, Southeast University, Nanjing 210096, China

w Abstract. In this paper, the robust periodicity for recurrent neural netmorks with time delays and impulses is investigated. Based on Lyapunov n ethod and fixed point theorem, a sufficient condition of global expoential robust stability of periodic solution is obtained.

1

Introduction

The subject of neural networks has become one of the important technical tools for solving a variety of problems such as associative memories, optimization problems, pattern recognition, signal processing. Such applications rely heavily on the stability of networks. Refs [1-2,14] studied the existence and stability of the equilibrium point of neural networks. It is well known that studies on neural networks not only involve discussion of stability, but also involve other dynamics behaviors such as periodic solution, bifurcation and chaos. In many theory and application researches, the property of periodic solutions is of great interest [7,8,13,15]. In hardware implementation of neural networks, time delays and parameter fluctuation are unavoidable. Various sufficient conditions have been presented for global robust stability,[4-6,16]. On the other hand, many physical systems undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. The stability of some neural networks with time delays and impulses were studied [9-12]. However, the robust stability of periodic solution of neural networks with delays and impulses is seldom considered. Motivated by the above discussions, the aim of this paper is to discuss the global exponential robust stability of periodic solution of recurrent neural networks with time delays and impulses. The organization of the paper is as follows: In Section 2, we introduce some definitions needed in later sections. In Section 3, a sufficient condition for the global exponential robust stability of periodic solution is derived by using Lyapunov method and fixed point theorem. Conclusions are given in Section 4. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 185–191, 2006. c Springer-Verlag Berlin Heidelberg 2006 

186

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Y. Yang

Preliminaries

In the paper, we consider the following neural networks with time delays and impulses. ⎧ n n   ⎪ aij fj (xj (t)) + bij fj (xj (t − τij )) + Ii (t), t ≥ 0 t  = tk , ⎪ ⎪ x˙ i (t) = −ci xi (t) + ⎪ j=1 j=1 ⎪ ⎪ ⎨ Δx (t ) = x (t+ ) − x (t− ) = I (x (t )), i = 1, 2, ..., n, k ∈ N = {1, 2, ...}, i

k

i

k

i

ik

k

i

k

CI := {C = diag(ci ) : 0 < C ≤ C ≤ C, i.e., 0 < ci ≤ ci ≤ ci , i = 1, 2, ..., n} ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ AI := {A = diag(aij ) : A ≤ A ≤ A, i.e., aij ≤ aij ≤ aij , i, j = 1, 2, ..., n} ⎩ BI := {B = diag(bij ) : B ≤ B ≤ B, i.e., bij ≤ bij ≤ bij , i, j = 1, 2, ..., n}

(1)

where n corresponds to the number of units in a neural network; xi (t) is the state of the ith unit at time t; fj (xj (t)) is the output of the jth unit at time t; aij , bij , ci , (ci > 0) are constant; Ii (t) is a periodic external input with periodic ω; transmission delays τij are nonnegative constants with 0 ≤ τij ≤ τ (i, j = 1, 2, ..., n) . Impulsive moments {tk , k ∈ N } satisfy 0 = t0 < t1 < t2 < ..., lim tk = ∞, k→∞

xi (tk ) corresponds to the abrupt changes of the state at fixed impulsive mo+ ment tk ; xi (tk ) = xi (t− k ) and xi (tk ) exist. Accompanying system (1) are the initial conditions of the form xi (t) = φi (t), t ∈ [−τ, 0], i = 1, 2, ..., n, where φi (t) is continuous for t ∈ [−τ, 0]. Throughout this paper, we assume that the activation functions fi , i = 1, 2, ..., n and impulsive operator satisfy the following conditions (H1 ) fi (x) (i = 1, 2, ..., n) are bounded on R (H2 ) |fi (u) − fi (v)| ≤ ui |u − v|, ∀u, v ∈ R, u = v. (H3 ) Iik (xi (tk )) = −γik xi (tk ), 0 < γik < 2, i = 1, 2, ..., n, k ∈ N. For the sake of convenience, we introduce the following notations: + a+ ij = max(|aij |, |aij |), bij = max(|bij |, |bij |), i, j = 1, 2, ..., n.   n 

 φ = sup |φi (t)|2 . −τ ≤t≤0

i=1

For any φ ∈ C([−τ, 0] → R ), H1 and H2 guarantee the existence and uniqueness of the solution of Eq.(1)[3]. n

Definition 1. A piecewise continuous function x(t) : [0, +∞) → Rn is called an ω− periodic solution of Eq.(1), if + (I) x(t) is continuous at t = tk , x(tk ) = x(t− k ) and x(tk ) exist for each tk ; (II) x(t) satisfies Eq.(1) for t ∈ [0, +∞) ; (III) x(t) = x(t + ω), ∀t ∈ R. Definition 2. The periodic solution x(t, ϕ) of Eq.(1) is said to be globally exponentially stable, if there exist some constants ε > 0 and M ≥ 1 such

Robust Periodicity in Recurrent Neural Network

187

that for every solution x(t, φ) of Eq.(1) with any initial value φ ∈ C([−τ, 0]; Rn ),  x(t, φ) − x(t, ϕ) ≤ M  φ − ϕ  e−εt , ∀t ≥ 0, Definition 3. The periodic solution of Eq.(1) is said to be globally exponentially robust stable if Eq.(1) has exactly one global exponential robust stable periodic solution for all C ∈ CI , A ∈ AI , B ∈ BI .

3

Global Robust Periodicity

In the section, we will derive a sufficient condition for the global exponential robust stability of periodic solution of Eq.(1). Theorem 1. Under the assumption (H1 ), (H2 ) and (H3 ), if there exist positive constants wi > 0 and αij , βij ∈ R, i, j = 1, 2, ..., n such that n

2(1−αij )  + 2(1−αij ) aij

uj

+ b+ ij

2(1−βij )

j=1

+

n

wj j=1

wi

2αji  + 2αji aji

ui

+ b+ ji

2βji

< 2ci

(2)

then for every periodic input I(t) of Eq.(1), there exists global exponential robust stable ω−periodic solution. Proof. By inequality (2) , choose a small ε > 0 such that 2(ε − ci ) +

n

2(1−αij )  + 2(1−αij ) aij

uj

+ b+ ij

2(1−βij )

j=1 n

wj 2αji  + 2αji 2βji 2ετji ui aji + b+ e < 0, (3) ji wi j=1

 Let P C([−τ, 0], Rn ) = x(t)|x : [−τ, 0] → Rn , where x(t) is piecewise continun ous function, then P C([−τ, 0],

R ) is a Banach space.n  n Let P C([−τ, +∞), R ) = x(t)|x : [−τ, +∞) → R , where x(t) is piecewise + continuous periodic solution at t = tk , and x(tk ) = x(t− k ), x(tk ) exist at t = tk . n n If x ∈ P C([−τ, +∞), R ), we define xt ∈ P C([−τ, 0], R ) by xt (s) = x(t + s) for −τ ≤ s ≤ 0. Moreover, we define xt+ ∈ P C([−τ, 0], Rn ) by xt+ (s) = x(t + s) for −τ ≤ s < 0 and xt+ (0) = x(t+ ). For ∀φ, ψ ∈ P C([−τ, 0], Rn ), we denote the solution (1) through (0, φ) and (0, ψ) as x(t, φ) and x(t, ψ) respectively. Thus, it follows from Eq.(1) that +

n

d (xi (t, φ) − xi (t, ϕ)) = −ci (xi (t, φ) − xi (t, ϕ)) + aij [fj (xj (t, φ)) − fj (xj (t, ϕ))] dt j=1

+

n

bij [fj (xj (t − τij , φ)) − fj (xj (t − τij , ϕ))], t ≥ 0, t  = tk ,

j=1 − + − Δ[xi (tk , φ) − xi (tk , ϕ)] = xi (t+ k , φ) − xi (tk , φ) − xi (tk , ϕ) + xi (tk , ϕ)

= Iik (xi (tk , φ) − xi (tk , φ)), i = 1, 2, ..., n, k ∈ N = 1, 2, ... (4)

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Consider the following Lyapunov function: V (t) =

n

n

 2α wi |xi (t, φ) − xi (t, ψ)|2 e2εt + uj ij |bij |2βij

i=1



j=1

t

|x(t, φ) − x(t, ψ)|2 e2ε(s+τij ) ds



t−τij

Calculating the upper right Dini-derivative D+ V of V along the solution of Eq.(1) at the continuous points t = tk , we have D+ V |(1) =

n

 wi 2εe2εt|xi (t, φ) − xi (t, ψ)|2 + 2e2εt |xi (t, φ) − xi (t, ψ)|D+ |xi (t, φ) − xi (t, ψ)|

i=1

+ −

n

j=1 n

2αij + 2βij bij |xj (t, φ)

uj

2αij + 2βij bij |xj (t

uj

− xj (t, ψ)|2 e2ε(t+τij )

− τij , φ) − xj (t − τij , ψ)|2 e2εt



j=1

≤

n

 wi e2εt 2(ε − ci )|xi (t, φ) − xi (t, ψ)|2

i=1

+2

n

u j a+ ij |xi (t, φ) − xi (t, ψ)||(xj (t, φ) − xj (t, ψ))|

j=1

+2

+

n

j=1 n

u j b+ ij |xi (t, φ) − xi (t, ψ)||(xj (t − τij , φ) − xj (t − τij , ψ))| 2αij + 2βij bij |xj (t, φ)

uj

− xj (t, ψ)|2 e2ετij

j=1

−

n

2αij + 2βij bij |xj (t

uj

− τij , φ) − xj (t − τij , ψ)|2



j=1

≤ +

n

n

2(1−βij )  2(1−αij )  + 2(1−αij ) wi 2(ε − ci ) + uj aij + b+ ij

i=1 n

j=1

wj 2αji  + 2αji 2βji 2ετji  ui aji + b+ e |xi (t, φ) − xi (t, ψ)|2 e2εt ≤ 0, t = tk . (5) ji w i j=1

Also, we can calculate right limits of Lyapunov function V (t) at impulsive moments {tk , k ∈ N } as follows: V (t+ k)=

n

 + 2 2εtk wi |xi (t+ k , φ) − xi (tk , ψ)| e

i=1

+

n

j=1

2α uj ij |bij |2βij



tk

tk −τij

|xj (s, φ) − xj (s, ψ)|2 e2ε(s+τij ) ds



Robust Periodicity in Recurrent Neural Network

=

n

 wi |1 − γik |2 |xi (tk , φ) − xi (tk , ψ)|2 e2εtk

i=1

+

189

n

2βij 2α uj ij b+ ij



tk

tk −τij

j=1

|xj (s, φ) − xj (s, ψ)|2 e2ε(s+τij ) ds

≤ V (tk ), k ∈ N,

(6)

which implies that V (t) ≤ V (0) for t ≥ 0 where V (0) =

n

n

 2βij 2α wi |xi (0, φ) − xi (0, ψ)|2 + uj ij b+ ij

i=1



−τij

j=1

≤ [ max {wi } + τ e2ετ 1≤i≤n

n

0

2αij + 2βij bij }]

wi max {uj 1≤j≤n

i=1

|xj (s, φ) − xj (s, ψ)|2 e2ε(s+τij ) ds  φ − ϕ 2

(7)

Furthermore, we have  xt (φ) − xt (ϕ) ≤ M e−εt  φ − ϕ , t ≥ 0

where M =

  n ij + 2βij  max {wi }+τ e2ετ  wi max {u2α bij } j 1≤j≤n 1≤i≤n i=1 min {wi }

≥ 1 is a constant.

1≤i≤n

Choose a positive integer m such that M m e−mεω ≤

1 4

Define a Poincare mapping T : P C([−τ, 0], Rn ) → P C([−τ, 0], Rn ) by xω (s) = x(ω + s) Then we can obtain  T m φ − T m ψ ≤

1 φ−ψ . 4

This means that T m is a contraction mapping, hence there exists a unique fixed point φ∗ ∈ P C([−τ, 0], Rn ) such that T m φ∗ = φ∗ . Note that T m (T φ∗ ) = T (T m φ∗ ) = T φ∗ . By the uniqueness of fixed point of mapping T m , we have T φ∗ = φ∗ , which means xω (φ∗ ) = φ∗ . Let x(t, φ∗ ) be the solution of Eq.(1) through (0, φ∗ ); obviously, x(t + ω, φ∗ ) be the solution of Eq.(1), and xt+ω (φ∗ ) = xt (xω (φ∗ )) = xt (φ∗ ), Therefore

x(t + ω, φ∗ ) = x(t, φ∗ )

f or

f or

t ≥ 0.

t ≥ 0.

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Y. Yang

This shows that x(t, φ∗ ) is exactly one unique ω−periodic solution of Eq.(1), and it is easy to see that all other solutions of Eq.(1) converge exponentially to it as t → +∞. This completes the proof. Corollary 1. Under the assumption (H1 ), H2 and (H3 ), if the following inequality holds n n



 + + u j a+ + b + u i a+ (8) ij ij ji + bji < 2ci j=1

j=1

then for every periodic input I(t) of Eq.(1), there exists global exponential robust stable ω−periodic solution.

4

Conclusions

In this paper, we discuss the dynamical behaviors of recurrent neural networks with time delays and impulses. By using Lyapunov method and fixed point theorem, we prove the global exponential robust stability and existence of robust periodic solution of Eq.(1). The criteria obtained are easily verifiable and possess many adjustable parameters, which provides flexibility for the design and analysis of recurrent neural networks with time delays and impulses.

Acknowledgement This work was supported by the Natural Science Foundation of Jiangsu Province, China under Grants BK 2004021.

References 1. Roska, T., Wu, C. W., Balsi, M. and Chua, L. O.: Stability and Dynamics of Delaytype General and Cellular Neural Networks. IEEE Trans. Circuits Syst. -I 39 (6) (1992) 487-490 2. Cao, J.: Global Stability Analysis in Delayed Cellular Neural Networks, Phys. Rev. E 59 (5) (1999) 5940-5944 3. Lakshmikanthan, V., Bainov, D. D., Simeonov P. S., Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 4. Wang, L. and Lin, Y.: Global Robust Stability for Shunting Inhibitory CNNs with Delays. J. Neural Syst. 14 (4) (2004) 229-235 5. Xu, S., Lam, J., Ho, D. W. C. and Zou, Y.: Global Robust Exponential Stability Analysis for Interval Recurrent Neural Networks. Phys. Lett. A 325 (2) (2004) 124-133 6. Cao, J. and Chen, T.: Globally Exponentially Robust Stability and Periodicity of Delayed Neural Networks. Chaos, Solitons and Fractals 22 (4) (2004) 957-963 7. Zhao, H.: Global Exponential Stability and Periodicity of Cellular Neural Networks with Variable Delays. Phys. Let. A 336 (3) (2005) 331-341 8. Guo, S. and Huang, L.: Periodic Oscillation for a Class of Neural Networks with Variable Coefficients. Nonlinear Analy. 6 (6) (2005) 545-561

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9. Xu, D. and Yang, Z.: Impulsive Delay Differential Inequality and Stability of Neural Networks. J. Math. Anal. Appl. 305 (1) (2005) 107-120 10. Li, Y. and Lu, L.: Global Exponential Stability and Existence of Periodic Solution of Hopfield-type Neural Networks with Impulses. Phys. Lett. A 333 (11) (2004) 62-71 11. Akca, H., Alassar, R., Covachev, V., Covachev, Z. and Zahrani, E. A.: Continuoustime Additive Hopfield-type Neural Networks with Impulses. J. of Math. Anal. Appl. 290 (2) (2004) 436-451 12. Gopalsamy, K.: Stability of Artificial Neural Networks with Impulses. Appl. Math. Comput. 154 (3) (2004) 783-813 13. Cao, J. and Wang, J.: Global Exponential Stability and Periodicity of Recurrent Neural Networks with Time Delays. IEEE Trans. Circuits Syst.-I 52 (5) (2005) 920-931 14. Cao, J. and Wang, J.: Global Asymptotic and Robust Stability of Recurrent Neural Networks with Time Delays. IEEE Trans. Circuits Syst.-I 52 (2) (2005)417-426 15. Cao, J., Chen, A. and Huang, X.: Almost Periodic Attractor of Delayed Neural Networks with Variable Coefficients. Phys. Lett. A 340 (6) (2005) 104-120 16. Cao, J., Huang, D.-S. and Qu, Y.: Global Robust Stability of Delayed Recurrent Neural Networks. Chaos, Solitons and Fractals 23 (1) (2005) 221-229

Global Asymptotical Stability of Cohen-Grossberg Neural Networks with Time-Varying and Distributed Delays Tianping Chen1 and Wenlian Lu1,2 1 Key Laboratory of Nonlinear Science of Chinese Ministry of Education, Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China [email protected] 2 Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany [email protected], {wenlian, Wenlian.Lu}@mis.mpg.de

Abstract. In this paper, we discuss delayed Cohen-Grossberg neural networks with time-varying and distributed delays and investigate their global asymptotical stability of the equilibrium point. The model proposed in this paper is universal. A set of sufficient conditions ensuring global convergence and globally exponential convergence for the CohenGrossberg neural networks with time-varying and distributed delays are given. Most of the existing models and global stability results for CohenGrossberg neural networks, Hopfield neural networks and cellular neural networks can be obtained from the theorems given in this paper.

1

Introduction

Cohen-Grossberg neural networks, initially proposed and studied in [7], have attracted increasing interest. The Cohen-Grossberg neural networks can be described by the following differential equations n  dui (t) = −ai (ui )[bi (ui ) − aij gj (uj (t)) + Ii ] i = 1, · · · , n dt j=1

(1)

In this paper, we investigate the following generalized Cohen-Grossberg neural networks with time-varying delays and distributed delays  n  dui (t) = −ai (ui ) bi (ui ) − aij gj (uj (t)) dt j=1  n  ∞  − fj (uj (t − τij (t) − s))dKij (s) + Ii j=1

(2)

0

where  ∞ dKij (s) are Lebesgue-Stieljies measures for each i, j = 1, · · · , n, and satisfy 0 |dKij (s)| < ∞. ai (x) > 0 are continuous, bi (x) are Lipshitz continuous with bi (x) ≥ γi > 0, i = 1, · · · , n, J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 192–197, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Global Asymptotical Stability of Cohen-Grossberg Neural Networks

193

The initial conditions for system (2) are ui (s) = φi (s)

f or

s ∈ [−∞, 0]

(3)

where φi ∈ C([−τ, 0]), i = 1, · · · , n, are bounded continuous functions. τij = supt>0 τij (t) < ∞. This model was first proposed in [1] for delayed neural networks and investigated in [2, 3, 4]. For convenience of description, for any vector v = [v1 , · · · , vn ]T , we denote ||v|| = max |vi |

(4)

i=1,···,n

and for any continuous function f (t) D+ f (t) =

lim

h>0,h→0

sup

|f (t + h)| − |f (t)| h

(5)

denotes its right upper-derivative.

2

Main Results

In this section, we investigate the globally asymptotical stability and globally exponential stability of the generalized Cohen-Grossberg neural networks with time-varying delays and distributed delays (2). We will prove two theorems. Theorem 1 addresses the globally asymptotical stability. Theorem 2 is about the globally exponential stability. Theorem 1. Suppose that gi ∈ Lip(Gi ), fi ∈ Lip(Fi ). If there are positive constants ξ1 , · · · , ξn such that   n n   min ξi γj − ξj Gj |aij | − ξj Fj i

j=1

j=1

∞

 |dKij (s)| = η > 0

(6)

0

Then the dynamical system (2) has a unique equilibrium v ∗ , which is globally asymptotically stable. Remark 1. By a simple linear transform ui (t) = ξi vi (t), without loss of generality, in the proof, we assume that ξi = 1. We first give two lemmas. Lemma 1. (see [2]) Under the assumptions made in Theorem 1, the dynamical system (2) has at least an equilibrium v ∗ = [v1∗ , · · · , v1∗ ]T . Lemma 2. Under the assumptions made in Theorem 1, every solution u(t) of the system (2) is bounded. Proof. Suppose u(t) is a solution of (2), v ∗ is the equilibrium given in Lemma 1. Let w(t) = u(t) − v ∗ , M (t) = sup−∞<s≤t ||w(s)||. It is clear that ||w(t)|| ≤ M (t).

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If for some t0 , ||w(t0 )|| = M (t0 ), and it0 = it0 (t0 ), which depends on t0 , is an index such that |wit0 (t0 ))| = ||w(t0 )|| Then, we have D+ |wi0 (t)| ≤ (−γi + η)|wi0 (t)| +

  ai0 j Gj +

dKi0 j (s)Fj |wj (t)|

0

j

  n   ≤ (−γi + η) + |ai0 j |Gj +

∞

∞

 |dKi0 j (s)|Fj ||w(t0 )|| < 0

(7)

0

j=1

which means that M (t) is non-increasing at all t > 0 and thus M (t) = M (0). ¯ = max{M (0) + |v ∗ |}. Lemma 2 is proved. Thus |ui (t)| ≤ M i ¯ , we can Because ai (ui ) > 0. Therefore, in a bounded closed region |ui | ≤ M find a constant αi > 0 such that 0 < αi ≤ ai (x). Proof of Theorem 1. Suppose u(t) is a solution of (2), v ∗ is the equilibrium given in Lemma 1, w(t) = u(t) − v ∗ . We will prove that for arbitrarily small  > 0, there exists a sufficient large T¯ such that ||w(t)|| <  for all t > T¯. First, for any  > 0, pick a sufficiently large T , such that  ∞ n  η M ξj Fj |dKij (s)| < (8) 2 T j=1 where M is the constant given in Lemma 2. Then, due to (6), we can pick a small positive α = α(T ), which is dependent of T , such that for all i = 1, · · · , n, (−γi + αα−1 i ) +

n 

Gj |aij | +

j=1

+

n 



j=1 ∞

ξj Fj

j=1

n 

T



T

eα(τij +s) |dKij (s)|

Fj 0

|dKij (s)| > −

η 2

(9)

Let y(t) = eαt w(t) and M1 (t) = supt−T <s≤t ||y(s)||. We claim that if ||w(t0 )|| > , then M1 (t) is non-increasing at t0 . In fact, at any t0 , there are two possible cases: Case 1. ||y(t0 )|| < M1 (t0 ). In this case, by the continuity of w(t), M1 (t) is nonincreasing at t0 . Case 2. ||y(t0 )|| = M1 (t0 ), and it0 = it0 (t0 ), which depends on t0 , is an index such that |yit0 (t)| = ||y(t0 )|| Then, by some algebra, we have  n  D |yit0 (t0 )| ≤ αit0 (uit0 (t0 )) (−γit0 + αα−1 Gj |ait0 j | it ) + +

0

j=1

Global Asymptotical Stability of Cohen-Grossberg Neural Networks

+

+

n  j=1 n 



T

Fj 0

 eα(τit0 j +s) |dKit0 j (s)| M1 (t0 ) 

Fj e

αt0

∞ T

j=1

≤ αit0 (uit0 (t0 ))e +

n  j=1

 Fj 0

T

195

 |wj (t0 − τit0 j (t0 ) − s)||dKit0 j (s)|

αt0

  n  −1 |wit0 (t0 )| (−γit0 + ααit ) + Gj |ait0 j | 0

  n  eα(τit0 j +s) |dKit0 j (s)| + M Fj j=1

j=1 ∞ T

 |dKit0 j (s)|

(10)

By the choice of T and α, we have ∞ n   M j=1 Fj T |dKij (s)| max <    i (−γi + αα−1 ) + n Gj |aij | + n Fj T eα(τij +s) |dKij (s)| i j=1 j=1 0 Therefore, if |wit0 (t0 )| > , we have D+ yit0 (t0 ) < 0 Thus, M1 (t) is nonincreasing at t0 . In summary, we conclude that if ||w(t)|| >  for all t > 0, then M1 (t) is non-increasing for all t > 0, i.e. M1 (t) ≤ M1 (0) for all t > 0, which implies ||y(t)|| ≤ M1 (0) and ||w(t)|| ≤ M1 (0)e−αt (11) It contradicts ||w(t)|| > . Therefore, we can find a T¯, such that ||w(T¯)|| < . Therefore, for all t > T¯, we have ||w(t)|| ≤ . Because  is arbitrary, we conclude that lim u(t) = v ∗

(12)

t→∞

Theorem 1 is proved completely. Following theorem gives the result of globally exponential stability. Theorem 2. Suppose that gi ∈ Lip(Gi ), fi ∈ Lip(Fi ). If there are positive constants α, ξ1 , · · · , ξn such that    ∞ n n   α(τij +s) min ξi (−γi +αα−1 )− ξ G |a |− ξ F e |dK (s)| ≥ 0 (13) j j ij j j ij i i

j=1

j=1

0

Then the dynamical system (2) has a unique equilibrium v ∗ such that ||u(t) − v ∗ || = O(e−αt )

(14)

Proof of Theorem 2. With the same notations and arguments used in the proof of Theorem 1, we can prove that under condition (13), M1 (t) is bounded. Therefore, ||w(t)|| ≤ M1 (0)e−αt (15)

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Remark 2. It should be mentioned that the existence of the constant αi > 0 such that 0 < αi ≤ ai (x) is a direct consequence of the assumptions made in Theorem 1. It is not a prerequisite requirement. All the assumptions 0 < αi ≤ ai (x) ≤ α ¯ i and τ˙ij (t) ≤ 1 needed in [5] are removed.

3

Comparisons

It is easy to see that if dKij (0) = bij δ(0), where δ(0) = 1 and δ(s) = 0, for s  = 0, then (2) reduces to the Cohen-Grossberg neural networks with timevarying delays n n   dui (t) = −ai (ui )[bi (ui ) − aij gj (uj (t)) − bij fj (uj (t − τij (t))) + Ii ] (16) dt j=1 j=1

Instead, if dKij (s) = bij kij (s)ds, τij (t) = 0, then (2) reduces to the CohenGrossberg neural networks with distributed delays n n  ∞   dui (t) = −ai (ui )[bi (ui ) − aij gj (uj (t)) − fj (uj (t − s))kij (s)ds + Ii ] dt j=1 j=1 0

(17) Then, the stability analysis for the Cohen-Grossberg neural networks with timevarying delays (see [5, 6, 8, 9, 10, 12, 15, 16] and others) or distributed delays (see [13] and others) given in literature can be derived from the Theorems in this paper. Recently, several researchers also investigated stability criteria with Lp (1 ≤ p ≤ ∞) norm. However, in [17], it has been proved that the stability criteria with L1 norm is the best. More precisely, most of the stability criteria with Lp (1 ≤ p ≤ ∞) norm implies (6). It means that the stability analysis given there is a direct consequence of Theorem 1 and Theorem 2. In particular, the results given in [14] can be derived by that in [12] and all the results can be obtained by Theorem 1 and Theorem 2 given in this paper.

4

Conclusions

In this paper, we investigate the stability of a universal model of generalized Cohen-Grossberg neural networks. Asymptotical stability and exponential stability are given. Most of the models can be dealt with by this universal methods easily.

Acknowledgements This work is supported by National Science Foundation of China under Grant 60374018, 60574044.

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References 1. Chen, T., Lu, W.: Stability Analysis of Dynamical Neural Networks. IEEE Conf. Neural Networks and Signal Processing, Nanjing, China (2003) 14-17 2. Chen, T.: Universal Approach to Study Delayed Dynamical Syatems, ICNC2005, Lecture Notes on Computer Sciences, Vol. 3610 (2005) 245-253 3. Lu, W., Chen, T.: On Periodic Dynamical Systems. Chinese Annals of Mathematics 25B(4) (2004) 455-462 4. Chen, T., Lu, W., and Chen, G.: Dynamical Behaviors of a Large Class of General Delayed Neural Networks. Neural Computation 17 (2005) 949-968 5. Chen, T., Rong, L.: Robust Global Exponential Stability of Cohen-Grossberg Neural Networks With Time Delays. IEEE Trans. Neural Networks 15(1) (2004) 203206 6. Chen, Y.: Global Stability of Neural Networks with Distributed Delays. Neural Networks 15 (2002) 867-871 7. Cohen, M. A., Grossberg, S.: Absolute Stability and Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks. IEEE Trans. Syst. Man Cybern. B. (13) (1983) 815-821 8. Deng, J., Xu, D., and Yang, Z.: Global Exponental Stability Analysis of CohenGrossberg Neural Networks with Multiple Time-Varying Delays. Proceedings of 11th IEEE International Conference on Electronics, Circuits and Systems (2004) 230-233 9. Hwang, C., Cheng, C., Liao, T.: Globally Exponential Stability of Generalized Cohen-Grossberg Neural Netqorks with Delays. Physics Letters A 319 (2003) 157166 10. Liu, J.: Global Exponential Stability of Cohen-Grossberg Neural Netqorks with Time-Varying Delays. Chaos, Solitons and Fractals 26 (2005) 935-945 11. Lu, H.: Global Exponental Stability Analysis of Cohen-Grossberg Neural Networks. IEEE Trans. on Circuits and Systems II 52(9) (2005) 476-479 12. Liao, X., Li, C., Wong, K.: Criteria for Exponential Stability of Cohen-Grossberg Neural Networks. Neural Networks 17 (2004) 1401-1414 13. Sun, J., Wan, L.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Continuous Distributed Delays. Physics Letters A 342 (2005) 331-340 14. Tu, F., Liao, X.: Harmless Delays for Global Asymptotic Stability of CohenGrossberg Neural Networks. Chaos, Solitons and Fractals 26 (2005) 927-933 15. Wang, L., Zou, X.: Exponential Stability of Cohen-Grossberg Neural Networks. Neural Networks 15 (2002) 415-422 16. Ye, H., Michel, A. and Wang, K.: Qualitative Analysis of Cohen-Grossberg Neural Networks with Multiple Delays. Physical Review E 51 (1995) 2611-2618 17. Zheng, Y., Chen, T.: Global Exponential Stability of Delayed Periodic Dynamical Systems. Physics Letters A 322(5-6) (2004) 344-355

LMI Approach to Robust Stability Analysis of CohenGrossberg Neural Networks with Multiple Delays Ce Ji, Hua-Guang Zhang, and Chong-Hui Song Institute of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, 110004, P.R. China [email protected], [email protected]

Abstract. Robust stability analysis of a class of Cohen-Grossberg neural networks with multiple delays is given. An approach combining the Lyapunov functional with the linear matrix inequality (LMI) is taken to obtain the sufficient condition for the globally robust stability of equilibrium point. The result is established without assuming the differentiability and monotonicity of activation functions.

1 Introduction In recent years, considerable efforts have been devoted to the stability analysis of Cohen-Grossberg neural networks. By Lyapunov functional, [1] established some criteria for the globally asymptotic stability of this model. [2] investigated CohenGrossberg neural networks with multiple delays and obtained several exponential stability criteria. However, there are few existing results on the robust stability for Cohen-Grossberg neural networks. In [3], we analyzed the globally robust stability of Cohen-Grossberg neural networks with multiple delays and perturbations of interconnecting weights, and obtained the sufficient conditions for the globally robust stability of this model. However, the results given by [3] comprised the uncertainties ΔTk ∈ ℜn× n , k = 0,L , K , thus the practicability of the theory and corollary was debased. Aiming at this case, based on the assumption that the uncertainties are norm-bounded and satisfy a certain condition, we will give the robust stability criterion expressed by linear matrix inequality (LMI) in this paper. In contrast to the results in [3], the LMI approach has the advantage that it can be solved numerically and very effectively using the interior-point method. The result is very practical in the process of design and implementation of Cohen-Grossberg neural networks. The paper is organized as follows. In Section 2, the network model and preliminaries will be described as the basis of later sections. In Section 3, we will establish the sufficient condition for the globally robust stability of this model.

2 Model Description and Preliminaries The Cohen-Grossberg neural network model with multiple delays can be described by equation J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 198 – 203, 2006. © Springer-Verlag Berlin Heidelberg 2006

LMI Approach to Robust Stability Analysis of Cohen-Grossberg Neural Networks

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K ⎡ ⎤ x&(t ) = − A( x(t )) ⎢ B( x(t )) − T0 S ( x(t )) − ∑Tk S ( x(t −τ k )) ⎥ , k =1 ⎣ ⎦

(1)

where x = ( x1 ,L , xn )T ∈ ℜn , A( x) = diag[a1 ( x1 ),L , an ( xn )] , B( x) = [b1 ( x1 ),L , bn ( xn )]T , and S ( x) = [ s1 ( x1 ),L , sn ( xn )]T . Amplification function ai (⋅) is continuous. Behaved

function bi (⋅) is continuous and satisfies bi (0) = 0 . si (⋅) is the activation function

representing the ith neuron and si (0) = 0 . T0 = [t0 ij ] ∈ ℜn×n denotes that part of the

interconnecting structure which is not associated with delay, Tk = [tkij ] ∈ ℜ n×n denotes that part of the interconnecting structure which is associated with delay τ k , where τ k denotes kth delay, k = 1,L , K and 0 < τ 1 < L < τ K < +∞ . Considering the influences of parameter perturbations for model (1), then we can describe (1) as K ⎡ ⎤ x&(t ) = − A( x(t )) ⎢ B( x(t )) − (T0 + ΔT0 )S ( x(t )) − ∑ (Tk + ΔTk )S ( x(t − τ k ))⎥ , ⎣ ⎦ k =1

(2)

where ΔTk = ⎡⎣ Δtkij ⎤⎦ ∈ ℜ n× n , k = 0,L , K are time-invariant matrices representing the norm-bounded uncertainties. Throughout this paper, we have the following assumptions. Assumption 1. For the uncertainties ΔTk , k = 0,L , K , we assume

[ ΔT0 L ΔTK ] = HF [U 0 LU K ] ,

(3)

where F is an unknown matrix representing parametric uncertainty which satisfies

FT F ≤ E ,

(4)

where E is an identical matrix, and H , U 0 ,L , U K can be regarded as the known structural matrices of uncertainty with appropriate dimensions. Assumption 2. For i = 1,L , n , we assume

(H1) ai ( xi ) is positive and bounded, i.e., 0 < α im ≤ ai ( xi ) ≤ α iM . (H2) bi ( xi ) is continuous and differentiable, and satisfies

bi ( xi ) xi ≥ β im > 0.

(5)

(H3) si ( xi ) is bounded and satisfies 0 ≤ si ( xi ) xi ≤ σ iM .

(6)

Definition. The equilibrium point of system (1) is said to be globally robustly stable with respect to the perturbation ΔT0 and ΔTk , k = 1,L , K , if the equilibrium point of system (2) is globally asymptotically stable.

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C. Ji, H.-G. Zhang, and C.-H. Song

Obviously, the origin is an equilibrium point of (1) and (2). In order to study the globally robust stability of the equilibrium point x = 0 for (1), it suffices to investigate the globally asymptotic stability of equilibrium point x = 0 of system (2). Lemma. [4] If U , V and W are real matrices of appropriate dimension with M

satisfying M = M T , then M + UVW + W T V T U T < 0 ,

for all V T V ≤ E , if and only if there exists a positive constant

(7)

ε

such that

M + ε −1UU T + ε W T W < 0 .

(8)

In the following section, we will give the sufficient conditions for the globally asymptotic stability of equilibrium point x = 0 of system (2).

3 Robust Stability Theorem. For any bounded delay τ k , k = 1,L, K , the equilibrium point x = 0 of system (2) is globally asymptotically stable, if there exist a positive definite matrix P , a positive constant ε , and positive diagonal matrixes Λk = diag [ λk1 ,L, λkn ] ,

where λki > 0, i = 1,L, n, k = 0,L, K , such that the following linear matrix inequality (LMI) holds

⎡ γ11 PAMTK EM ⎢ M T M M T M ⎢E TK A P −ΛK + ε E UKUK E ⎢ M M ⎢ M M ⎢ M T ⎢ EMT T AM P ε E U0 UK EM ⎢ T0 M 0 ⎣⎢ H A P

PAMT0 EM PAM H ⎤ L ⎥ ε EMUKTU0 EM 0 ⎥ L L M M ⎥ ⎥ < 0, O M M ⎥ L −Λ0 + ε EMU0TU0 EM 0 ⎥ ⎥ L L −ε I ⎦⎥ 0 L L O L L

(9)

K

where γ 11 = − B m Am P − PAm B m + ∑ Λk , AM = diag[α1M ,L , α nM ] , Am = diag[α1m ,L , k =0

α ], B = diag[ β ,L , β ], and E M = [σ 1M ,L , σ nM ] . m n

m

m 1

m n

Proof. We can rewrite (2) as K ⎡ ⎤ x& (t ) = − A( x(t )) ⎢ B( x(t )) − (T0 + ΔT0 ) E( x(t )) x(t ) − ∑(Tk + ΔTk ) E( x(t −τ k )) x(t − τ k )⎥ , k =1 ⎣ ⎦

(10)

where

E( x) = diag [ε1 ( x1 ),L, ε n ( xn )] , εi ( xi ) = si ( xi )/ xi , i = 1,L, n . Then from (6), we have ε i ( xi ) ∈ [0, σ iM ] .

(11)

LMI Approach to Robust Stability Analysis of Cohen-Grossberg Neural Networks

201

Here, we introduce the following Lyapunov functional K

V ( xt ) = xT (t ) Px(t ) + ∑ ∫ k =1

0

−τ k

xtT (θ )Λ k xt (θ )dθ .

(12)

where xt (⋅) denotes the restriction of x(⋅) to the interval [t − τ K , t ] translated to

[ −τ K , 0] . For

s ∈ [ −τ K , 0] , we define xt ( s ) = x(t + s ) , where t > 0 .

The derivative of V ( xt ) with respect to t along any trajectory of system (10) is given by K

K

V& ( xt ) = x& T (t ) Px(t ) + xT (t ) Px& (t ) + ∑ xtT (0)Λ k xt (0) − ∑ xtT (−τ k )Λ k xt (−τ k ) k =1

k =1

K

= −xT (t ) BT AT Px(t ) − xT (t ) PABx(t ) + ∑ xT (t )Λk x(t ) + xT (t )Λ0 x(t ) k =1

−1 0

+ x (t ) PA(T0 + ΔT0 ) E ( x(t ))Λ E ( x(t ))(T0 + ΔT0 )T AT Px(t ) T

T

1 − ⎡ 1 ⎤ − ⎢Λ02 x(t ) − Λ0 2 ET ( x(t ))(T0 + ΔT0 )T AT Px(t )⎥ ⎣ ⎦

T

1 − ⎡ 1 ⎤ × ⎢Λ02 x(t ) − Λ0 2 ET ( x(t ))(T0 + ΔT0 )T AT Px(t ) ⎥ ⎣ ⎦ K

+∑ xT (t )PA(Tk + ΔTk )E( x(t −τ k ))Λk−1ET (x(t −τ k ))(Tk + ΔTk )T AT Px(t ) k =1

T

1 − ⎡ 1 ⎤ −∑⎢Λk2 x(t −τk ) −Λk 2 ET (x(t −τk ))(Tk +ΔTk )T AT Px(t)⎥ k =1 ⎣ ⎦ K

1 − ⎡ 1 ⎤ ×⎢Λk2 x(t −τk ) −Λk 2 ET (x(t −τk ))(Tk +ΔTk )T AT Px(t)⎥ ⎣ ⎦ K

≤ − xT (t )( B T AT P + PAB ) x(t ) + ∑ xT (t )Λ k x(t ) k =0

+ x (t ) PA(T0 + ΔT0 ) E ( x(t ))Λ 0−1 E T ( x(t ))(T0 + ΔT0 )T AT Px(t ) T

K

+∑ xT (t ) PA(Tk + ΔTk ) E ( x(t − τ k ))Λ−k 1 ET ( x(t − τ k ))(Tk + ΔTk )T AT Px(t ) ,

(13)

k =1

where B = diag[b1 ( x1 ) x1 ,L , bn ( xn ) xn ] . For convenience, we denote A( x(t )) and B( x(t )) as A and B, respectively. Since AM , B , Am , B m are all positive diagonal matrixes, and P is a positive definite matrix, then by the Assumption 2 and the relevant matrix theory, we can prove K K ⎧ ⎫ V& (xt ) ≤ xT (t ) ⎨− ( Bm Am P + PAm Bm ) + ∑Λk + ∑ PAM (Tk + ΔTk )EM Λ−k1EM (Tk + ΔTk )T AM P⎬ x(t) k =0 k =0 ⎩ ⎭

= xT (t ) S M x(t ) .

(14)

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C. Ji, H.-G. Zhang, and C.-H. Song

The detailed proof is given in [3]. From (14), we know V& ( xt ) < 0 if S M < 0 . By the functional differential equations theory [5], for any bounded delay τ k > 0 , the equilibrium point x = 0 of system (2) is globally asymptotically stable. According to the Schur Complement [6], S M < 0 can be expressed by the following LMI K ⎡ m m m m ⎢− ( B A P + PA B ) + ∑ Λk k =0 ⎢ E M (TK + ΔTK )T AM P ⎢ ⎢ M ⎢ ⎢ M ⎢ M ⎢ M ⎢ ( E T T0 )T AM P + Δ 0 ⎣

⎤ PAM (TK + ΔTK )EM L L L PAM (T0 + ΔT0 )EM ⎥ ⎥ 0 0 0 0 −ΛK ⎥ ⎥ 0 O 0 0 0 ⎥ < 0. ⎥ 0 0 O 0 0 ⎥ 0 0 0 O 0 ⎥ ⎥ 0 0 0 0 −Λ0 ⎦

(15)

Because of [ ΔT0 L ΔTK ] = HF [U 0 LU K ] , (15) can be expressed as K ⎡ m m m m ⎢ −B A P − PA B + ∑ Λk k =0 ⎢ E M TK T P ⎢ ⎢ M ⎢ ⎢ M ⎢ E M T0T P ⎢⎣

⎡ PAM H ⎤ ⎢ ⎥ ⎢ 0 ⎥ + ⎢ M ⎥ F ⎡⎣0 U K E M ⎢ ⎥ ⎢ M ⎥ ⎢ 0 ⎥ ⎣ ⎦

⎤ PTK E M L L PT0 E M ⎥ ⎥ − ΛK 0 L 0 ⎥ ⎥ 0 O O M ⎥ 0 ⎥ M O O ⎥ − Λ0 ⎥⎦ 0 L 0 ⎡ 0 ⎤ ⎢ EMU T ⎥ K⎥ ⎢ L L U0 E M ⎤⎦ + ⎢ M ⎥ F T ⎡⎣ H T AM P 0 L L 0⎤⎦ < 0. ⎢ ⎥ ⎢ M ⎥ ⎢⎣ E MU0T ⎥⎦

(16)

Using the Lemma, we know (16) holds for all F F ≤ E if and only if there exists a constant ε > 0 such that T

K ⎡ m m m m ⎢ − B A P − PA B + ∑ Λk k =0 ⎢ E M TK T AM P ⎢ ⎢ M ⎢ ⎢ M ⎢ M T M E T 0 A P ⎣⎢

⎡PAM HHT AM P ⎢ 0 1⎢ + ⎢ M ε⎢ M ⎢ ⎢ 0 ⎣

PAM TK E M − ΛK 0 M 0

0 0⎤ ⎡0 ⎥ ⎢ M T M 0⎥ ⎢0 E UKUK E M M ⎥ + ε ⎢M ⎥ ⎢ M M ⎥ ⎢M M T M ⎥ ⎢ 0 L L 0⎦ ⎣0 E U0 UK E 0 0 M M

L L O O

L L O O

⎤ L L PAM T0 E M ⎥ ⎥ 0 L 0 ⎥ ⎥ O O M ⎥ ⎥ 0 O O ⎥ L 0 − Λ0 ⎦⎥

L L O L L

0 L ⎤ M T M⎥ L E UKU0 E ⎥ ⎥ < 0. L M ⎥ O M ⎥ L EMU0TU0 EM ⎥⎦

(17)

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203

Rearranging (17), we get ⎡ ς11 PTK EM ⎢ M T M T M ⎢E TK P −ΛK + ε E UKUK E ⎢ M M ⎢ M ⎢ M M T ⎢ EM T T P ε E U0 UK EM 0 ⎣ K

L L L L O L L O L L

where ς 11 = − B m Am P − PAm B m + ∑ Λk + k =0

1

ε

⎤ ⎥ ε E U U0 E ⎥ ⎥ <0, M ⎥ M ⎥ −Λ0 + ε EMU0TU0 EM ⎥⎦ PT0 EM

M

T K

M

(18)

PHH T P.

By use of the Schur Complement, (18) is equivalent to condition (9). The proof is completed.

Acknowledgements This work was supported by the National Natural Science Foundation of China (60325311, 60534010, 60572070, 60504006) and the Program for Changjiang Scholars and Innovative Research Team in University.

References 1. Chen, T.P., Rong, L.B.: Delay-Independent Stability Analysis of Cohen-Grossberg Neural Networks. Physics Letters A 317 (2003) 436–449 2. Hwang, C.C., Cheng, C.J., Liao, T.L.: Globally Exponential Stability of Generalized Cohen-Grossberg Neural Networks with Delays. Physics Letters A 319 (2003) 157–166 3. Ji, C., Zhang, H.G., Guan, H.X.: Analysis for Global Robust Stability of Cohen-Grossberg Neural Networks with Multiple Delays. Lecture Notes in Computer Science, Vol. 3173. Springer-Verlag, Berlin Heidelberg (2004) 96–101 4. Singh, V.: Robust Stability of Cellular Neural Networks with Delay: Linear Matrix Inequality Approach. IEE Proc.-Control Theory Appl. 151(1) (2004) 125–129 5. Michel, A.N., Wang, K., Hu, B.: Qualitative Theory of Dynamical Systems–The Role of Stability Preserving Mappings. Second Edition. NY: Marcel Dekker, New York (2001) 6. Boyd, S., El Ghaoui, L., Feron, E.: Linear Matrix Inequalities in System and Control Theory. Studies in Applied Mathematics, Philadelphia: SIAM (1994)

Existence and Global Stability Analysis of Almost Periodic Solutions for CohenGrossberg Neural Networks Tianping Chen, Lili Wang, and Changlei Ren Key Laboratory of Nonlinear Science of Educational Chinese Ministry of Education, Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China {tchen, 042018047, 042018045}@fudan.edu.cn

Abstract. In this paper, we investigate a large class of periodic CohenGrossberg neural networks and prove that under some mild conditions, there is a unique almost-periodic solution, which is globally stable. As special examples, we derive many results given in literature.

1

Introduction

It is well known that Cohen-Grossberg neural networks, proposed by Cohen and Grossberg in [1, 2], have been extensively studied both in theory and applications. The Cohen-Grossberg neural networks can be described by the following ordinary differential equations n  dui (t) = −ai (ui (t))[bi (ui (t)) − aij gj (uj (t)) + Ii ], i = 1, . . . , n dt j=1

(1)

Recently, there are lots of papers discussing Cohen-Grossberg neural networks in literature (see [3-17] and others). In this paper, we investigate the almost periodic solutions of the following generalized Cohen-Grossberg neural networks n  dui (t) = −ai (ui (t))[bi (ui (t)) − aij (t)gj (uj (t)) dt j=1 n  ∞  − fj (uj (t − τij (t) − s))dKij (t, s) + Ii (t)], i = 1, . . . , n (2) j=1

o

where τij ≥ 0, dsKij (t, s),for any fixed t ≥ 0, are Lebesgue-Stieljies measures.The initial conditions are ui (s) = φi (s)

for s ∈ (−∞, 0]

where φi ∈ C(−∞, 0], i = 1, . . . , n. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 204–210, 2006. c Springer-Verlag Berlin Heidelberg 2006 

(3)

Existence and Global Stability Analysis of Almost Periodic Solutions

2

205

Definitions

In this section, we present some definitions and denotations required throughout the paper. Definition 1. Let x(t) : R → Rn is a continuous function, we call it an almost periodic function, if for any  > 0, there exist l = l() > 0 and ω = ω() in any interval with the length of l(), such that for all t ∈ R, x(t + ω) − x(t) <  Definition 2. {ξ, ∞}-norm: u(t) = u(t){ξ,∞} = max | ξi−1 ui (t)|, where i=1,...,n

ξi > 0, i = 1, . . . , n.

3

Main Results

Throughout this article, we make the following assumptions. Assumption 1: gi (·) ∈ Lip(Gi ), fi (·) ∈ Lip(Fi ), i = 1, . . . , n where Gi , Fi > 0, where Lip(G) denotes Lipshitz functions with Lipschitz constant G > 0. i (v) Assumption 2: 0 < αi ≤ ai (x) ≤ α¯i , bi (t)is continuous and bi (u)−b ≥ u−v γi > 0, i = 1, . . . , n, τij (t) ≥ 0, aij (t), Ii (t) are almost periodic functions, i, j = 1, . . . , n.

Assumption 3: ∀s ∈ R, Kij (t, s) : t → Kij (t, s) is continuous. Moreover, for any fixed t ∈ R, dKij (t, s) : s → dKij (t, s) are Lebesgue-Stieljies measures. Suppose that |ds Kij (t, s)| ≤ |dKij (s)|, where dKij (s) is a Lebesgue-Stieljies ∞ measure independent of t and there exists  > 0, o es |dKij (s)| < ∞, i, j = 1, . . . , n. Assumption 4: For any  > 0, there exists l = l() > 0 and ω = ω() in any interval with the length of l, such that  ∞ |dKij (t + ω, s) − dKij (t, s)|ds < , ∀ t ∈ R, i, j = 1, 2, . . . , n (4) 0

Remark 1. By Bochner Theorem, Assumption 4 is equivalent that aij (t), τij (t), Ii (t) are continuous almost periodic functions. Therefore, they are all bounded. Denote that |a∗ij | = sup |aij (t)| < ∞, t∈R

|Ii∗ |

= sup |Ii (t)| < ∞,

τij∗ = sup τij (t) < ∞, t∈R

i, j = 1, . . . , n.

t∈R

Lemma 1. Suppose that Assumption 1-4 are satisfied. If there are positive constants ξ1 , . . . , ξn , such that for every t > 0, i = 1, . . . , n,  ∞ n n   − αi γi ξi + |aij (t)|¯ αi Gj ξj + α ¯ i Fj ξj |dKij (t, s)|ds < −η < 0 (5) j=1

j=1

0

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T. Chen, L. Wang, and C. Ren

Then any solution u(t) of the dynamical system (2) is bounded. Proof: Define M (t) = max u(s). For any fixed t0 , there are two possibilities: s≤t

(i) u(t0 ) < M (t0 ). In this case, there exists δ > 0, u(t) < M (t0 ) for t ∈ (t0 , t0 + δ). Thus M (t) = M (t0 ) for t ∈ (t0 , t0 + δ). (ii)u(t0 ) = M (t0 ). In this case, let it0 be an index such that ξi−1 |uit0 (t0 )| = t0 u(t0 ). Notice that |gj (s)| ≤ Gj |s| + |gj (0)|,

|fj (s)| ≤ Fj |s| + |fj (0)|,

j = 1, . . . , n,

and denote   n n   ∗ ¯ I = max α ¯ i |bi (0)| + α ¯ i [|aij ||gj (0)| + |fj (0)| i

j=1

By some algebra, we have   d|uit0 | (t) dt +

n 

 − αit0 γit0 ξit0 +

t=t0



α ¯it0 Fj ξj

j=1

∞

0





|dKij (s)|ds +

|Ii∗ | ]

,

0

j=1

≤

∞

s ∈ R.

n 

|ait0 j (t0 )|¯ αit0 Gj ξj

j=1

|dKit0 j (t0 , s)|ds M (t0 ) + I¯ ≤ −ηM (t0 ) + I¯

¯ If M (t0 ) ≥ 2|I|/η. There exists δ1 > 0, such that M (t) = M (t0 ) for t ∈ (t0 , t0 + δ1 ). ¯ On the other hand, if M (t0 ) < 2|I|/η. Then there exist δ2 > 0, such that ¯ M (t) < 2|I|/η, t ∈ (t0 , t0 + δ2 ). ¯ Let δ = min{δ1 , δ2 }, then, M (t) ≤ max{M (t0 ), 2|I|/η} holds for every t ∈ (t0 , t0 + δ). ¯ In summary, u(t) ≤ M (t) ≤ max{M (0), 2|I|/η}. Lemma is proved. Lemma 2. Suppose that Assumption 1-4 are satisfied. If there are positive constants ξ1 , . . . , ξn , β > 0,such that for every t > 0, i = 1, . . . , n, (−αi γi + β)ξi +

n 

|aij (t)|¯ αj Gj ξj

j=1

+

n 

∗



∞

α ¯ j Fj ξj eβτij

eβs |dKij (t, s)|ds < −η < 0

0

j=1

Then for any  > 0, there exists T > 0, l = l() > 0, and ω = ω() in any interval with the length of l, such that for any solution u(t) of the dynamical system (2), u(t + ω) − u(t){ξ,∞} ≤ , Proof: Denote i (ω, t) = +

n   j=1

∀t > T

 aij (t + ω) − aij (t) gj (uj (t + ω))

Existence and Global Stability Analysis of Almost Periodic Solutions

+

n   j=1

∞

207

 fj (uj (t − τij (t + ω) + ω − s))

0

 − fj (uj (t − τij (t) + ω − s)) dKij (t + ω, s) +

n   j=1



∞

  fj (uj (t − τij (t) + ω − s))d Kij (t + ω, s) − Kij (t, s)

0

 + Ii (t + ω) − Ii (t)

(6)

By Lemma 1, u(t) is bounded. Thus, the right side of (5) is also bounded, which implies that u(t) is uniformly continuous. Therefore, by the assumption A, for any  > 0, there exists l = l() > 0 such that every interval [α, α + l], α ∈ R, contains an ω for which |i (ω, t)| ≤

1 η 2α ¯

f or all t ∈ R , i = 1, 2, · · · , n

(7)

where α ¯ = max α ¯i . i

Define



ui (t+ω)

xi (t) = ui (t)

1 dρ. ai (ρ)

Then, we have   n  dxi (t) = − bi (ui (t + ω)) − bi (ui (t))] + aij (t)[gj (uj (t + ω)) − gj (uj (t)) dt j=1    n ∞  + fj (uj (t − τij (t) + ω − s)) − fj (uj (t − τij (t) − s)) dKij (t, s) j=1

0

+ i (ω, t) Let it be an index such that ξi−1 |xit (t)| = x(t). Differentiating eβs |xit (s)|, we t have   d e−βt eβs |xit (s)| dt s=t n  ≤ β|xit (t)| − γit |uit (t + ω) − uit (t)| + |ait j (t)|Gj |uj (t + ω) − uj (t)| +

n   j=1

j=1 ∞

Fj |uj (t − τit j (t) + ω − s) − uj (t − τit j (t) − s)||dKit j (t, s)|

0

+ |it (ω, t)| ≤ −(γit αit − β)|xit (t)| + +

n   j=1

0

n 

|ait j (t)|Gj α ¯ j |xj (t)| + η/2α ¯

j=1 ∞

∗

Fj α ¯j |xj (t − τit j (t) − s)|e−β(τit j (t)+s) eβ(τij +s) |dKit j (t, s)|

208

T. Chen, L. Wang, and C. Ren

Denote Ψ (t) = max{eβs x(s)}. If there is such a point t0 > 0 that Ψ (t0 ) = s≤t

eβt0 x(t0 ){ξ,∞} . Then we have   d eβt |xit0 (t)| < −ηΨ (t0 ) + ηeβt0 /2α ¯ dt t=t0

(8)

In addition, if Ψ (t0 ) ≥ eβt0 /2α ¯, then Ψ (t) is non-increasing in a small neighborhood (t0 , t0 + δ) of t0 . On the other hand, if Ψ (t0 ) < eβt0 , then in a small neighborhood (t0 , t0 + δ) of t0 , eβt x(t) < eβt0 /2α ¯ , and Ψ (t) ≤ max{Ψ (t0 ), eβt0 /2α ¯ }. By the same reasonings used in the proof of Lemma 1, for all t > t0 , we have eβt x(t) ≤ max{Ψ (t0 ), eβt /2α ¯}. Therefore, there exists t1 > 0, for all t > t1 , x(t) ≤ . In summary, there must exist T > 0 such that x(t) ≤  holds for all t > T . Lemma 2 is proved. Theorem 1. Suppose that Assumptions 1-4 are satisfied. If there are positive constants ξ1 , . . . , ξn , β > 0, such that for every t > 0, i = 1, . . . , n, (−αi γi + β)ξi +

n 

|aij (t)|¯ αj Gj ξj +

j=1

n 

∗



α ¯j Fj ξj eβτij

j=1

∞

eβs |dKij (t, s)|ds < 0

0

Then the system (2) has a unique almost periodic solution v(t). Furthermore, every solution u(t) of the system (2) satisfying u(t) − v(t) = O(e−βt ). Proof: Define   n  i,k (t) = ai (ui (t + tk )) aij (t + tk ) − aij (t) gj (uj (t + tk )) +

n   j=1

∞

j=1



fj (uj (t − τij (t + tk ) + tk − s))

0

 −fj (uj (t − τij (t) + tk − s)) dKij (t + tk , s) +

n   j=1



∞

  fj (uj (t − τij (t) + tk − s))d Kij (t + tk , s) − Kij (t, s)

0

 − Ii (t + tk ) − Ii (t) where {tk } is a sequence of real numbers. Because u(t) is bounded. we can select a sequence {tk } such that |i,k (t)| ≤ 1/k, ∀i, t. By Arzala-Ascoli Lemma and diagonal selection principle, we can select a subsequence tkj of tk , such that u(t + tkj ) (for convenience, we still denote by

Existence and Global Stability Analysis of Almost Periodic Solutions

209

u(t + tk )) uniformly converges to a continuous function v(t) = [v1 (t), v2 (t), · · · , vn (t)]T on any compact set of R. Using Lebesgue dominated convergence theorem, we have   t+h n  vi (t + h) − vi (t) = −ai (vi (σ)) bi (vi (σ)) − aij (σ)gj (vj (σ)) t

−

j=1

n  

∞

 fj (vj (σ − τij (σ) − s))dKij (σ, s) + Ii (σ) dσ,

0

j=1

for any t > 0 and h ∈ R. Hence, v(t) is a solution of the system (2). Meanwhile, with Lemma 2, for any  > 0, there exists T > 0, l = l() > 0, and ω = ω() in any interval with the length of l, such that |ui (t + tk + ω) − ui (t + tk )| ≤ ,

∀t > 0

Let k → ∞, it is easy to get |vi (t + ω) − vi (t)| ≤ ,

∀t > 0.

Hence, v(t) is almost periodic. Finally, we will prove that for every solution u(t) = [u1 (t), . . . , un (t)]T , u(t) − v(t) = O(e−βt ). Similar to the proof of the previous lemmas, define  ui (t) 1 zi (t) = dρ, vi (t) ai (ρ) Then,   n  dzi (t) = − bi (ui (t)) − bi (vi (t))] + aij (t)[gj (uj (t)) − gj (vj (t)) dt j=1  n  ∞  + fj (uj (t − τij (t) − s)) − fj (vj (t − τij (t) − s)) dKij (t, s) j=1

0

  Let Φ(t) = max eβs z(s) . by the same arguments used in the proof of the s≤t

Lemma, we can prove Φ(t) = Φ(0) for all t ≥ 0. Then, x(t){ξ,∞} ≤ Φ(0)e−βt which implies u(t) − v(t){ξ,∞} ≤ α ¯ Φ(0)e−βt , Theorem 1 is proved.

α ¯ = max α ¯i i

210

4

T. Chen, L. Wang, and C. Ren

Conclusions

In this paper, a thorough analysis of existence, uniqueness, and global stability of the almost periodic solution for a class of Cohen-Grossberg neural networks has been presented. Under some general conditions, we get a set of sufficient conditions guaranteeing the existence and the global stability of the almost periodic solutions for these systems.

Acknowledgements This work is supported by National Science Foundation of China under Grant 60374018, 60574044.

References 1. Cohen, M. A., and Grossberg, S.: Absolute Stability and Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks. IEEE Trans. Syst. Man Cybern. B. 13 (1983) 815-821 2. Grossberg, S.: Nonlinear Neural Networks: Principles, Mechanisms, and Architectures. Neural Networks 1 (1988) 17-61 3. Lu, W., Chen, T.: Dynamical Behaviors of Cohen-Grossberg Neural Networks with Discontinuous Activation Functions. Neural Networks 18 (2005) 231-242 4. Cao, J.: New Results Concerning Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks. Physics Letters A 307 (2003) 136-147 5. Zhou, J., Liu, Z., Chen, G.: Dynamics of Delayed Periodic Neural Networks. Neural Networks 17(1) (2004) 87-101 6. Zheng, Y., Chen, T.: Global Exponential Stability of Delayed Periodic Dynamical Systems. Physics Letters A 322(5-6) (2004) 344-355 7. Lu, W., Chen, T.: On Periodic Dynamical Systems. Chinese Annals of Mathematics 25B(4) (2004) 455-462 8. Chen, T. Lu, W. Chen, G.: Dynamical Behaviors of a Large Class of General Delayed Neural Networks. Neural Computation 17 (2005) 949-968 9. Cao, J., Wang, J.: Global Exponential Stability and Periodicity of Recurrent Neural Networks With Time Delays. IEEE Transactions on Circuits and Systems-I: Regular Paper 52(5) (2005) 920-931 10. Li, Y.: Existence and Stability of Periodic Solutions for Cohen-Grossberg Neural Networks with Multiple Delays. Chaos, Solitons and Fractals 20(3) (2004) 459-466 11. Chen, T., Lu, W.: Stability Analysis of Dynamical Neural Networks. IEEE Inl Conf. Neural Networks and Signal Processing, Nanjing, China, December. (2003) 14-17 12. Lu, W., Chen, T.: Global Exponential Stability of Almost Periodic Solution for a Large Class of Delayed Dynamical Systems. Science In China Series A-Mathematics 48(8) 1015-1026 2005 13. Liu, B., Huang, L.: Existence and Exponential Stability of Almost Periodic Solutions for Cellular Neural Networks with Time-Varying Delays. Physics Letters A 341(1-4) (2005) 135-144

A New Sufficient Condition on the Complete Stability of a Class Cellular Neural Networks Li-qun Zhou1,2 and Guang-da Hu1 1

2

Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R. China Department of Mathematics, QiQihar University, QiQihar, 161006, P.R. China {zhouliqun, ghu}@hit.edu.cn

Abstract. In this paper, a sufficient condition is presented to ensure the complete stability of a cellular neural networks (CNNs) that output functions are a set of piecewise sigmoid nonlinear functions. The convergence theorem of the Gauss-Seidel method and Gauss-Seidel method, which is an iterative technique for solving a linear algebraic equation, plays an important role in our discussion.

1

Introduction

In this paper we consider the standard model of the CNNs: x˙ = −x + Ay(x) + b or x˙ i = −xi +

n 

aij yj (xj ) + bi , ∀ i,

(1)

j=1

where x = [x1 , x2 , . . . , xn ]T is the state vector, A = [aij ] is feedback matrix of the system. b = [b1 , b2 , . . . , bn ]T is a constant input vector. y(x) = [y1 (x1 ), y2 (x2 ), . . . , yn (xn )]T is the output vector, where ⎧ ⎨ f (r0 ), xi > r0 , yi (xi ) = f (xi ) = f (xi ), xi ∈ [−r0 , r0 ], (2) ⎩ −f (r0 ), xi < r0 . we will assume that self-feedback coefficients aii > 1/β, ∀ i, where β denotes the maximum slope of f (xi ); It is important to note that under the condition aii > 1/β, ∀ i, a stable equilibrium point can only be in total saturation region (|xi | ≥ r0 , ∀ i); f (xi ) are sigmoid and monotonically increasing in [−r0 , r0 ]; (xi ) f (xi ) ∈ C  (R1 ) and f (xi ) is bounded, that is 0 < dfdx ≤ β, 0 < β ≤ 1 and i |f (xi )| ≤ f (r0 ) = α; 0 < r0 < r, where r represents the positive abscissa of maximum value point of (1), when f (xi ) are a set of sigmoid nonlinear functions. Studies on completely stability of CNNs have been vigorously done and many criteria have been obtained so far [1-5]. In this paper, we give a new sufficient condition for a CNNs whose output functions are piecewise sigmoid nonlinear functions to be completely stable. The main result of the present paper is based J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 211–216, 2006. c Springer-Verlag Berlin Heidelberg 2006 

212

L.-q. Zhou and G.-d. Hu

on the stability condition for standard CNNs given in [4]. A convergence theorem of the Gauss-Seidel method [5], which is an iterative technique for solving linear algebraic equations plays a important role. Now we give some definitions and theorems for CNNs. Definition 1. Let P be a matrix with positive diagonal elements. The comparison matrix S of P is defined as sii = pii and sij = −|pij |, if i  = j. Definition 2. An n×n matrix P with nonpositive off-diagonal elements is called M -matrix if all its principal minors are positive. Theorem 1. ([4]) A standard CNN is completely stable if the comparison matrix of A − I is an M -matrix. Where I denotes the identity matrix. Theorem 2. ([3][5]) There exists a stable equilibrium point in the total saturation region if the comparison matrix of A − I is a M -matrix.

2

Complete Stability of CNNs

Theorem 3. Let W = (wij ) be the n × n matrix as follows: a β ii min − 1, if i = j wij = , −|aij|, if i  =j

(3)

where βmin = min{f  (xi )|xi ∈ [−r0 , r0 ]}. If W is an M -matrix, then the CNN is completely stable. The theorem 3 is the main result of this paper. First we give the following lemma. Lemma 1. The CNNs (1) is completely stable if 1) Every (n − 1)-cell CNN x y( defined by d x + A x) + b is completely stable independent of the value of dt = − b, where x  is any , y( x) ∈ Rn−1 , b is an n − 1 dimensional constant vector and A principal submatrix of A of order n − 1. (m) (m) (m) 2) The sequence {d(m) }∞ = [d1 , d2 , . . . , m=0 of n dimensional vector d (m) T dn ] generated by Algorithm 1 given below converges to the vector. Algorithm 1 (0) 1) Set m = 1 and initialize the n-dimensional vector u(0) , l(0) , d(0) , βi,min , (0)

(0)

(0)

(0)

βi,max , Bi,min , Bi,max and βi (0)

lj

(0)

= −α, uj

(0)

Bi,min =

max { (0)

(0)

(0)

(0)

= α, Ij

f (xj )∈Ij

Bi,max =

as follows:

min

(0)

f (xj )∈Ij (0)

aii βi,min =

i−1 

aij f (xj ) +

j=1

{

i−1 

aij f (xj ) +

j=1

(0)

(0)

n 

= 2α,

aij f (xj ) + bi },

n 

aij f (xj ) + bi },

j=i+1

∗(0)

∗(0) −xi,max

(0)

j=i+1

Bi,min −xi,max

aii βi

(0)

= [lj , uj ], dj

(0)

(0)

, aii βi,max = (0)

∗(0)

Bi,max −xi,min

(0)

∗(0) −xi,min

= min{aii βi,max , aii βi,min }.

,

A New Sufficient Condition on the Complete Stability

213

2) For ∀ i, compute (m) pi0

(m)

qi0

=

=

(m)

−1

[

(m)

aii βi,min − 1

min

(m−1)

yj ∈Ij0

−1 (m)

(m)

i−1 

(m−1)

yj ∈Ij0

(m)

yj ∈Ij0

n 

(m)

yj ∈Ij0

(m)

(m)

aij yj + bi ],

(4)

aij yj + bi ],

(5)

j=i+1

aij yj + max

j=1

(m)

n 

aij yj + min

j=1

[ max

aii βi,max − 1 (m)

i−1 

j=i+1

(m)

(m)

(m)

(m)

where ui0 = f (pi0 ), li0 = f (qi0 ), Ii0 = [li0 , ui0 ], di0 = ui0 − li0 . (m) (m) (m) (m−1) 3) Set aii βi = min{aii βi,min , aii βi,max , aii βi }, ∀ i compute (m) pi

(m)

qi

(m)

=

−1 (m)

aii βi =

−1

−1 (m)

aii βi

−1

(m)

[

i−1 

min

(m−1)

yj ∈Ij

(m)

i−1 

(m−1)

yj ∈Ij

(m)

aij yj + min yj ∈Ij

j=1

[ max

n 

n 

aij yj + max

(m)

yj ∈Ij

j=1 (m)

(m)

(m)

aij yj + bi ],

(6)

aij yj + bi ],

(7)

j=i+1

j=i+1

(m)

(m)

(m)

(m)

where ui = f (pi ), li = f (qi ), Ii = [li , ui ], di = ui − li . (m) (m) (m) 4) If d(m) = [d1 , d2 , . . . , dn ]T is sufficiently close to the zero vector, go to Step 4 Stop. Otherwise, add 1 to m and go to Step 2. Secondly, we show that if W = (wij ) is an M -matrix, then {d(m) }∞ m=0 in Algorithm 1 converges to the zero vector. In order to do that, we consider the Gauss-Seidel method [6] for solving a linear algebraic equation W z˜ = 0 with the (0) initial condition z˜i = 2α, for ∀ i. The algorithm is given as follows: Algorithm 2 (0) 1) Set m = 1 and z˜j = 2α, ∀j. 2) Set βmin = min{f  (xi )|xi ∈ [−r0 , r0 ]}. For all i compute i−1  (m)

z˜i (m)

=

j=1

(m)

|aij |˜ zj

n 

+

j=i+1

(m−1)

|aij |˜ zj

.

aii βmin − 1

(m)

(8)

(m)

3) If z˜(m) = [˜ z1 , z˜2 , . . . , z˜n ]T is sufficiently close to the zero vector, go to Step 4 Stop. Otherwise, add 1 to m and go to Step 2. Lemma 2. If the matrix W is an M -matrix, then the sequence {˜ z (m) }∞ m=0 in Algorithm 2 converges to the zero vector. Proof: See [4] Theorem 4, we get immediately lemma 2. Algorithm 3 (0) 1) Set m = 1 and zj = 2α, ∀j. i−1 

2) For all i, compute (m)

(m) zi

(m)

=

j=1

(m)

|aij |zj

(m)

+

n 

(m−1)

|aij |zj

j=i+1 (m) aii βi −1

.

3) If z (m) = [z1 , z2 , . . . , zn ]T is sufficiently close to the zero vector, go to Step 4 Stop. Otherwise, add 1 to m and go to Step 2.

214

L.-q. Zhou and G.-d. Hu

Lemma 3. If the matrix W is an M -matrix, then the sequence {z (m) }∞ m=0 in Algorithm 3 converges to the zero vector. (0)

Proof: Since zi (m−1) zj

≤

(0)

(m)

= z˜i , ∀i. Suppose that zj

(m−1) z˜j ,

(m)

i−1 

=

, j = 1, 2, . . . , i − 1 and

j = i + 1, . . . , n holds. Since for all m and for ∀i, aii βi

aii βmin > 1, thus we have that (m) zi

(m)

≤ z˜j

j=1

(m)

|aij |zj

+

n 

(m) aii βi

− 1 > aii βmin − 1 > 0. Then

≤

j=1

i−1 

(m−1)

|aij |zj

≤

j=i+1 (m) aii βi −1 i−1 

(m)

|aij |˜ zj

+

>

n 

j=1

(m)

|aij |˜ zj

+

n 

(m−1)

|aij |˜ zj

j=i+1 (m) aii βi −1

(m−1)

j=i+1

|aij |˜ zj

(m)

= z˜i

aii βmin −1

.

Thus for ∀i and for all m ≥ 0, zi ≤ z˜i holds, then the sequence {z (m) }∞ m=0 converges to the zero vector faster than the sequence {˜ z (m) }∞ m=0 . (m)

(m)

Lemma 4. If the matrix W is an M -matrix, then the sequence {d(m) }∞ m=0 in Algorithm 1 converge to the zero vector. (0)

Proof: Since di (m−1)

dj

(m)

di

(m−1)

≤ zj

(0)

ui i−1 

≤

j=1

(m)

≤ zj

, j = 1, 2, · · · , i − 1 and

, j = i + 1, · · · , n. holds. Then (m)

=

(m)

= zi , ∀i. suppose that dj

(m)

|aij |dj

+

(m)

− li n 

(m)

= f (pi

i−1 

(m−1)

|aij |dj

j=i+1 (m) aii βi −1

(m)

) − f (qi

≤

j=1

(m)

|aij |zj

(m)

) ≤ pi +

n 

(m)

− qi

(m−1)

|aij |zj

j=i+1 (m) aii βi −1

(m)

= zi

.

Thus for ∀i and for all m ≥ 0 , di ≤ zi holds, then the sequence {d(m) }∞ m=0 converges to the zero vector faster than the sequence {z (m) }∞ m=0 . (m)

(m)

Proof of Theorem 3: Since Algorithm 1 is completed if W = (wij ) is an M matrix, the sufficient condition for the n-cell CNN to be completely stable is that every (n − 1)-cell CNN whose feedback matrix is an (n − 1) × (n − 1) principal submatrix of A is completely stable. Note that for any (n − 1) × (n − 1) principal ˜ is an M -matrix. Thus a sufficient condition for submatrix A˜ of A, the matrix W complete stability of an n-cell CNN is that every (n−2)-cell CNN whose feedback matrix is an (n − 2) × (n − 2) principal submatrix of A is completely stable. Continuing the above discussion, we can finally state as follows: a sufficient condition for complete stability of an n-cell CNN is that every 1-cell CNN whose feedback matrix is a 1 × 1 principal submatrix of A is completely stable. Since every one-cell CNN is completely stable, the n-cell CNN is completely stable. In fact, if f (xi ) = xi , ∀i, in [−r0 , r0 ] for the function (2), then βmin = β = 1. In Theorem 3 the matrix W is nicely the comparison matrix of A − I. Thus we show that the comparison matrix of A − I in [3,4] is M -matrix which is only a special case in our sufficient condition.

A New Sufficient Condition on the Complete Stability

3 3.1

215

Stability of a Class CNNs and Numerical Example Output Function Is Piecewise tanh Function

Now we choose r0 . let f (xi ) = tanh(xi ) = Bi , where h(xi ) = −xi + aii f (xi ), Bi =

exi −e−xi exi +e−xi n 

in (1), we have x˙ i = h(xi ) +

aij f (xj ) + bi . Since

j=1,j =i 4 , (exi +e−xi )2

then 0 <

df (xi ) dxi

df (xi ) dxi

=

≤ 1. We consider two following cases.

1. If aii > 1. i) Let Bi = 0, then x˙ i = h(xi ). Let r > 0, 4aii −1 + (er +e −r )2 = 0. We get that

dh(xi ) dxi |xi =r

= 0, that is

√ √ 1 r = ln( aii + aii − 1) and f  (xi )|xi =±r = . aii

(9)

We choose 0 < r0 < r, βmin = f  (xi )|xi =±r0 . Then x˙i has the characteristics shown in Fig.1(a). ii) Bi = constant  = 0 is valid for cases shown in Fig.1(b). Bi>r1 x’ (t) i

x’i

x’i Bi=r1

Bi>0 Bi
r1 −r2

r

2

Bi=0 Bi<0

xi

x (t)

xi

i

r0 r

Bi<−r1 B =−r i

(a)

1

Bi>−r1 (b)

(c)

Fig. 1. (a) Bi = 0, there are three equilibrium points : x∗i = 0 is unstable, x∗i = −r2 , x∗i = r2 are stable. (b) Bi  = 0, all of the stable equilibrium points share the common property |x∗i | > r0 . (c) Bi = 0, there is only one stable equilibrium point x∗i = 0; Bi  = 0, there is only one stable equilibrium point x∗i  = 0.

Theorem 4. We assume that aii > 1, then lim |xi (t)| = |x∗i | > r0 ,

t→∞

(10)

√ √ where r0 < ln( aii + aii − 1). 2. If aii ≤ 1, then h(xi ) has the characteristic shown in Fig.1(c). We can make a conclusion that when aii ≤ β1 , if W = (wij ) is an M -matrix, then 1 − a β ii min , if i = j the CNNs is completely stable. Where wij = , β = −|aij|, if i  = j min min{f  (xi )|xi ∈ [−r0 , r0 ]}. The discussion shows that completely stability and global asymptotic stability for CNNs are equivalence when CNNs has only an equilibrium point.

216

L.-q. Zhou and G.-d. Hu 10

8

6

4

2

0

−2

−4

−6

−8

−10 −10

−8

−6

−4

−2

0

2

4

6

8

10

Fig. 2. The CNN has five equilibrium points. where there are four stable equilibrium points(4.0755, 2.9717), (−4.0755, 2.9717), (2.7170, −3.8208),(−2.7170, 3.8208) in the total saturation region, and (0, 0) is unstable equilibrium.

3.2

Numerical Example and Simulation Result

Let us consider the two-cell CNNs state equation  x˙ = −x + 4f (x ) + 0.8f (x ), 1 1 1 2 (11) x˙ 2 = −x2 − 0.5f (x1 ) + 4f (x2 ). √ By (9), we have r = ln(2 + 3) and choose r0 = ln 3.5. Since βmin = 0.2791, 2 |W | = (16βmin − 1)2 − 0.4 = 0.0136 > 0, hence W is an M -matrix. So the CNNs is completely stable. We see the simulation result Fig.2.

Acknowledgement We would like to thank the referees for their careful comments and helpful suggestions which improve this paper significantly.

References 1. Wu, C.W., Chua, L.O.: A More Rigorous Proof of Complete Stability of Cellular Neural Networks. IEEE Trans. Circuit Syst. I. 44(4) (1997) 370-371 2. Chua, L. O., Roska, T.: Stability of a Class of Nonreciprocal Cellular Neural Networks: Theory. IEEE Trans. Circuit Syst. 37(12) (1990) 1520-1527 3. Arik, S., Tavsanoglu, V.: Equilibrium Analysis of Nonsymmetric CNNs. Int. J. Circuit Theory Appl. 24(2) (1996) 269-274 4. Takahashi, N., Chua, L.O.: On the Complete Stability of Nonsymmetric Cellular Neural Networks. IEEE Trans. Circuit Syst. I. 45(7) (1998) 754-758 5. Takahashi, N., Chua, L.O.: A New Sufficient Condition for Nonsymmetric CNNs to Have a Stable Equilibrium Point. IEEE Trans. Circuit Syst. I. 44(11) (1997) 1092-1095 6. Young, D.M.: Iterative Solution of Linear Solutions. New York, Academic (1971)

Stability Analysis of Reaction-Diffusion Recurrent Cellular Neural Networks with Variable Time Delays Weifan Zheng, Jiye Zhang, and Weihua Zhang Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu 610031, China {wfzheng, jyzhang}@home.swjtu.edu.cn

Abstract. In this paper, the global exponential stability of a class of recurrent cellular neural networks with reaction-diffusion and variable time delays was studied. When neural networks contain unbounded activation functions, it may happen that equilibrium point does not exist at all. In this paper, without assuming the boundedness, monotonicity and differentiability of the active functions, the algebraic criteria ensuring existence, uniqueness and global exponential stability of the equilibrium point of neural networks are obtained.

1 Introduction In recent years, the dynamic behaviors of neural networks have been deeply investigated due to its applicability in some signal processing, imaging processing, and pattern recognition problems, especially in some parallel computation and difficult optimization problems [1-5]. Such applications rely on the qualitative properties of system. In hardware implementation, time delays are unavoidable due to finite switching speeds of the amplifiers and communication time etc. Time delays, especially variable delays and unbounded delay, may lead to an oscillation and furthermore, to instability of networks [6]. In a complex dynamical system, absolute invariable delays is few; and in some cases such as the processing of moving images it requires the introduction of delays in the signal transmitted through the networks [7]. Therefore, the stability of neural networks with variable delays is practically important, and has been extensively studied in the literatures [8-11]. However, diffusion effect cannot be avoided in the neural networks model when electrons are asymmetric electromagnetic field. The asymptotical stability of neural networks with reaction-diffusion was discussed in [12, 17]. But many of systems in these literatures require increasingly monotonic neuron activation functions and nonnegative diffusion functions. In this paper, we relax some limits on activation functions and diffusion functions of neural network, which are similar to those discussed in [12-17]. By constructing the proper Liapunov function, we analyze the conditions ensuring the existence, uniqueness and global exponential stability of the equilibrium of the models. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 217 – 223, 2006. © Springer-Verlag Berlin Heidelberg 2006

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2 Notations and Preliminaries In this paper, we analyze the stability of reaction-diffusion recurrent cellular neural networks with variable delays described by the following differential equations n m ∂ui ∂u ∂ [ Dik (t , x, ui ) i ] − ci ui (t ) + ∑ aij f j (u j (t )) =∑ ∂t k =1 ∂xk ∂xk j =1 n

− ∑ bij g j (u j (t − τ ij (t ))) + J i , i = [1, n] .

(1)

∂u ∂u ∂ui = col( i ,..., i ) = 0, i = [1, n] , t ∈ I ， x ∈ ∂Ω ， ~ ∂x1 ∂xm ∂n

(2)

j =1

where ui is the state of neuron i, (i = 1,2,L, n) and n is the number of neurons; Di (t , x, ui ) is smooth reaction-diffusion function, A = (aij ) n×n , B = (bij ) n×n are connection

matrices,

J = ( J 1 , J 2 , L , J n )T

is

the

constant

input,

respectively.

Τ

f (u ) = ( f1 (u1 ), f 2 (u2 ),..., f n (un )) , g (u ) = ( g1 (u1 ), g 2 (u2 ),L, g n (un )) are the activation functions of the neurons, C = diag(c1 , c2 ,L, cn ) > 0 . The delays τ ij (t ) T

( i, j = 1,2, L , n ) are bounded functions, i.e. 0 ≤ τ ij (t ) ≤ τ .The initial conditions of equation (1) are of the form ui ( s ) = φi ( s) , −τ ≤ s ≤ 0 , where φi is bounded and continuous in [−τ ,0] . Equation (2) is the boundary condition of equation (1), in which x ∈ Ω ⊂ R m , Ω is a compact set with smooth boundary and mesΩ > 0 , ∂Ω is the boundary of Ω , t ∈ I = [0,+∞) . For convenience, we introduce some notations. The notation u = (u1 , u2 ,..., un )T ∈ R n represents a column vector (the symbol ( )Τ denotes transpose). For matrix

A = (aij ) n×n , | A | denotes absolute-value matrix given by | A |= (| aij |) n×n , i, j = 1, 2… n; [ A]S is defined as ( AT + A) / 2 . For x ∈ R n , | x |= (| x1 |,..., | xn |) Τ , || x || denotes a vector norm defined by || x ||= max{| xi |} . Let 1≤i ≤ n

∂ ∂u ( D1 j (t , x, u1 ) 1 ),..., ∂xk k =1 ∂xk model (1) becomes the following system: m

D(t , x, u ) = col(∑

m

∂

∑ ∂x k =1

( Dnj (t , x, un ) k

∂un )) , ∂xk

u& (t ) = D(t , x, u (t )) − Cu (t ) + Af (u (t )) + Bg (u (t − τ ij (t )) + J .

(3)

System (3) and (1) have the same properties of stability. If there is a constant u0 = u0* =const (const denotes invariable constant) which is the solution of the following equation:

Cu (t ) = Af (u (t )) + Bg (u (t − τ ij (t )) + J , then

∂u0* = 0 . From equation (3), we can get Dik (t , x, ui (t )) =0, and ∂x

(4)

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219

D (t , x, u (t )) − Cu (t ) + Af (u (t )) + Bg (u (t − τ ij (t )) + J = 0 . It implies that equation (4) and (3) has the same equilibrium, and system (4) has the same equilibrium as that of system (1).Supposing activation functions satisfy Assumption 1. There exist the real numbers p j > 0 , q j > 0 , such that

p j = sup |

f j ( y) − f j ( z)

| , q j = sup |

g j ( y) − g j ( z )

y−z y≠ z Let P = diag( p1 ,..., pn ) , Q = diag(q1 ,..., qn ) . y≠ z

y−z

| . ( j = 1,2, L, n) .

3 Existence and Uniqueness of the Equilibrium Point In this section, we will give a necessary and sufficient condition ensuring the existence and uniqueness of the equilibrium point with respect to the class of unbounded activation functions satisfying Assumption 1. Definition 1[18]. A real matrix A = ( aij ) n×n is said to be an M-matrix if aij ≤ 0 i, j = 1,2,..., n , i ≠ j , and all successive principal minors of A are positive. We firstly define a map associated with (1) as follows: H (u ) = −Cu + Af (u ) + Bg (u ) + J .

(5)

It is known that the solutions of H (u ) = 0 are equilibriums in (1). If map H (u ) is a homeomorphism on R n , then there exists a unique point u* such that H (u*) = 0 , i.e., system (1) has a unique equilibrium u * . Lemma

[8]

. If H (u ) ∈ C 0 satisfies (i) H (u ) is injective on R n ; (ii) H (u ) → ∞ as

u → ∞ ; then H (u ) is a homeomorphism of R n . Theorem 1. If Assumption 1 is satisfied, and α = C − (| A | P + | B | Q ) is an M-matrix, then, for every input J , system (1) has a unique equilibrium u * . Proof. In order to prove that for every input J , system (1) has a unique equilibrium point u * , it is only to prove that H (u ) is a homeomorphism on R n . In following, we shall prove it in two steps.

Step 1. We shall prove that condition (i) in Lemma is satisfied. Suppose that there exist x, y ∈ R n with x ≠ y such that H ( x ) = H ( y ) . From (5), we get C ( x − y ) = A( f ( x ) − f ( y )) + B ( g ( x ) − g ( y )) .

(6)

From Assumption 1, there exist matrices K = diag( K1 ,..., K n ) ( − P ≤ K ≤ P ), and

L = diag( L1 ,..., Ln ) (−Q ≤ L ≤ Q ) , such that g ( x) − g ( y ) = L( x − y ) . So (6) can be written as:

f ( x) − f ( y ) = K ( x − y )

[−C + ( AK + BL)]( x − y ) = 0 .

and (7)

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In the following, we prove that det[−C + ( AK + BL)] ≠ 0 . Considering the system

z& = [−C + ( AK + BL)]z .

(8)

Due to α being an M-matrix, using property of M-matrix ξ i > 0 (i = 1,2,..., n) such that

[18]

, there exist

n

− ciξ i + ∑ ξ j (| a ji K j | + | b ji L j |) < 0 (i = 1,2,..., n) .

(9)

j =1

n

Constructing a Liapunov function V ( z ) = ∑ ξ i | zi | . Calculating the upper right dei =1

rivative D +V of V along the solutions of (8), we get n

n

i =1

j =1

D +V ( z ) ≤ ∑ [−ciξ i + ∑ ξ j (| a ji K j | + | b ji L j |)] | zi | <0

(|| z ||≠ 0) .

By stability Theorem [18], the zero solution of (8) is globally asymptotically stable. Thus matrix −C + ( AK + BL) is a stable matrix and det[−C + ( AK + BL)] ≠ 0 . From (7), x = y , which is a contradiction. So map H is injective. Step 2. We now prove that condition (ii) in Lemma is satisfied. Let H (u ) = −Cu + A( f (u ) − f (0)) + B( g (u ) − g (0)) .To show that H is homeomorphism, it suffices to show that H (u ) is homeomorphism. From Assumption 1, we get | f i (u ) − f i (0) |≤ pi | u | (i=1, 2,…,n), | gi (u ) − gi (0) |≤ qi | u | (i=1, 2,…,n). Since α is an M-matrix, from property of M-matrix [18], there exists a positive definite diagonal matrix D such that [ D (−C + | A | P + | B | Q )]S ≤ −ε En < 0 , for sufficiently small ε > 0 , where En is an identity matrix. By calculating, we have [ Du ]Τ H (u ) ≤ −ε u . By using Schwartz inequality, we get ε u 2

2

≤ D u H (u ) ,

namely, ε || u || / || D || ≤|| H (u ) || . So， || H (u ) ||→ +∞ , i.e. || H (u ) ||→ +∞ as || u ||→ +∞ . From Lemma, we know that H (u ) is homeomorphism on R n . So systems (1) have a unique equilibrium point u * . The proof is completed.

4 Global Exponential Stability of Equilibrium Point In this section, we shall apply the ideal of vector Liapunov method [18, 19] to analyze global exponential stability of model (1). Definition 2. The equilibrium point u * of (1) is said to be globally exponentially stable, if there exist constant λ > 0 and β > 0 such that || u (t ) − u* ||

≤ β || φ − u* || e − λ t for all t ≥ 0 , where || φ − u* ||= max sup | φi ( s ) − ui* | . 1≤ i ≤ n s∈[ −τ , 0 ]

Theorem 2. If Assumption 1 is satisfied and α = C − (| A | P + | B | Q ) is an M-matrix, then for each input J, systems (1) have a unique equilibrium point, which is globally exponentially stable.

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Proof. Since α is an M-matrix, from Theorem 1, system (1) has a unique equilib-

rium point u * . Let z (t ) = u (t ) − u * , Eqs. (1) can be written as ∂z ∂zi (t ) m ∂ [ Dik (t , x, ( z (t ) + u * )) i ] =∑ ∂xk ∂t k =1 ∂xk n

n

j =1

j =1

− ci zi (t ) + ∑ aij f j ( z j (t )) + ∑ bij g j ( z j (t − τ ij (t ))) ,

(10)

where f j ( z j (t )) = f j ( z j (t ) + u j *) − f j (u j *) and g j ( z j (t − τ ij (t ))) = g j ( z j (t − τ ij (t )) + u j *) − g j (u j *) , ( j = 1,L, n) .

The initial condition of equations (10) is ψ ( s ) = φ ( s ) − u * , −τ ≤ s ≤ 0 , and equations (10) have a unique equilibrium at z = 0 . From Assumption 1, we get | f j ( z j (t )) |≤ p j | z j (t ) | , | g j ( z j (t − τ ij (t ))) | ≤ q j | z j (t − τ ij (t )) | ( j = 1,L, n) . Due to α being an M-matrix [18], there exist ξ i > 0 (i = 1,2,..., n) and λ > 0 satisfy n

− ξ i (ci − λ ) + ∑ ξ j (| aij | p j + e λτ | bij | q j ) < 0 ,

(11)

j =1

where, τ is a fixed number. Constructing Vi (t ) = eλt zi (t ) , and Liapunov functional Vi (t ) = ∫Ω | Vi (t ) | dx (i = 1,2,..., n) . Calculating the upper right derivative D + Vi (t ) of Vi (t ) along the solutions of equation (10), and considering Assumption 1, boundary condition (2) and mesΩ > 0 ， we get m ∂u ∂ D + Vi (t ) = ∫Ω e λt sgn zi ∑ [ Dik (t , x, ( zi (t ) + u * )) i ]dx + ∫ λeλt | zi (t ) | dx Ω ∂xk k =1 ∂xk n

n

j =1

j =1

+ ∫ e λt {sgn zi (t )[ −ci zi (t ) + ∑ aij f j ( z j (t )) + ∑ bij g j ( z j (t − τ ij (t )))]}dx Ω

∂u ≤ e λt sgn zi (t )∑ [ Dik (t , x, ( zi (t ) + u * )) i ] |∂Ω ∂xk k =1 m

n

n

j =1

j =1

+ ∫Ω e λt [(−ci + λ ) | zi (t ) | + ∑ (| aij || f j ( z j (t )) |) + ∑ | bij || g j ( z j (t − τ ij (t ))) |]dx ≤

n

∫ { − (c − λ ) | V (t ) | +∑ [| a Ω

+e ≤

λτ ij ( t )

i

i

| bij | q j e

λ ( t −τ ij ( t ))

j =1

ij

| z j (t − τ ij (t )) |]}dx n

∫ { − (c − λ ) | V (t ) | +∑ [| a Ω

i

i

| p j | V j (t ) |

j =1

ij

| p j | V j (t ) | + eλτ | bij | q j sup | V j ( s) |]}dx . (12) t −τ ≤ s ≤t

Defining the curve γ = { y (l ) : yi = ξ i l , l > 0, i = 1,2,..., n} , and the set Ω( y ) = {u : 0 ≤ u ≤ y, y ∈ γ } . Let ξ min = min{ξi } , ξ max = max{ξ i } and taking 1≤i ≤ n

1≤i ≤ n

l0 = (1 + δ ) || ψ || / ξ min , where δ > 0 is a constant number. Then

{| V |: | V |= eλs | ψ ( s) |, − τ ≤ s ≤ 0} ⊂ Ω( z0 (l0 )) ,

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namely, | Vi ( s) |= eλs | ψ i ( s ) |< ξi l0 , −τ ≤ s ≤ 0 ， i=1,2,…,n. We claim that | Vi (t ) |< ξi l0 , for t ∈ [0,+∞) ， i=1, 2, … , n . If it is not true, then there exist some index i and time

t1 ( t1 > 0 ), such that | Vi (t1 ) |= ξil0 , D + | Vi (t1 ) |≥ 0 , and | V j (t ) |≤ ξ j l0 , for −τ < t ≤ t1 , j =1, 2, … , n. So we get D + Vi (t1 ) = ∫Ω D + | Vi (t1 ) | dx ≥ 0 (i=1, 2, … , n). (13) However, from (11) and (12), we get n

D + Vi (t1 ) = ∫Ω D + | Vi (t1 ) | dx ≤ [−ξ i (ci − λ ) + ∑ ξ j (| aij | p j + e λτ | bij | q j )]l0 < 0 , j =1

which contradicts with inequalities (13). So | Vi (t ) |< ξi l0 , for t ∈ [0,+∞) , therefore, | zi (t ) |< ξ il0 e − λt ≤ β || ψ || e − λt (i=1, 2, … , n), where β = (1 + δ )ξ max / ξ min . From Definition 2, the zero solution of systems (10) is globally exponentially stable, i.e., the equilibrium point of systems (1) is globally exponentially stable. The proof is completed. Remark: The similar system is discussed in reference [12-17], but all of them require nonnegative diffusion function or increasing activation functions in systems, however, those conditions are not required in Theorem 2 in this paper.

5 Conclusion In the paper, a thorough analysis of existence, uniqueness, and global exponential stability of the equilibrium point for a class of reaction-diffusion recurrent cellular neural networks with variable delays have been presented. By applying idea of vector Liapunov function method and M-matrix theory, sufficient conditions independent of the delays for global exponential stability are obtained. Moreover, the conditions are easily tested.

Acknowledgments This work is supported by Youth Science Foundation of Sichuan (No. 05ZQ026-015), National Program for New Century Excellent Talents in University (No. NCET-040889), and Natural Science Foundation of China (No.10272091).

References 1. Chua, L. O., Yang, L.: Cellular Neural Networks: Theory. IEEE Trans. Circ. Syst. 35 (1988) 1257-1272 2. Zhang, J., Jin, X.: Global Stability Analysis in Delayed Hopfield Neural Networks Models. Neural Networks 13 (2000) 745-753 3. Zhang, J., Yang, Y.: Global Stability Analysis of Bidirectional Associative Memory Neural Networks with Time Delay. Int. J. Circ. Theor. Appl. 29 (2001) 185-196

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4. Zhang, Y., Peng, P. A., Leung, K. S.: Convergence Analysis of Cellular Neural Networks with Unbounded Delay. IEEE Trans. Circ. Syst.-I 48 (2001) 680-687 5. Zhang, J.: Globally Exponential Stability of Neural Networks with Variable Delays. IEEE Trans. Circ. Syst.-I 50 (2003) 288-291 6. Civalleri, P. P., Gill, L. M., Pandolfi, L.: On Stability of Cellular Neural Networks with Delay. IEEE Trans. Circ. Syst.-I. 40 (1993) 157-164 7. Roska, T., Chua, L. O.: Cellular Neural Networks with Delay Type Template Elements and Nonuniform Grids. Int. J. Circ. Theor. Appl. 20 (1992) 469-481 8. Forti, M., Tesi, A.: New Conditions for Global Stability of Neural Networks with Application to Linear and Quadratic Programming Problems. IEEE Trans. Circ. Syst.-I 42 (1995) 354-366 9. Zheng, W., Zhang, J.: Global Exponential Stability of a Class of Neural Networks with Variable Delays. Computers and Mathematics with Applications 49 (2005) 895-902 10. Zhang, J., Suda, Y., Iwasa, T.: Absolutely Exponential Stability of a Class of Neural Networks with Unbounded Delay. Neural Networks 17 (2004) 391-397 11. Zhang, J.: Absolute Stability Analysis in Cellular Neural Networks with Variable Delays and Unbounded Delay. Computers and Mathematics with Applications 47 (2004) 183-194 12. Liao, X., Fu, Y. et al.: Stability of Hopfield Neural Networks with Reaction-diffusion Terms. Acta Electronica Sinica 28 (2000) 78-80(in Chinese) 13. Liao, X., Li, J.: Stability in Gilpin-Ayala Competition Models with Diffusion. Nonlinear Anal. 28 (1997) 1751-1758 14. Wang, L., Xu, D.: Global Stability of Reaction-diffusion Hopfield Neural Networks with Variable Time Delay. Science in China (serial E) 33 (2003) 488-495 15. Liang, J., Cao, J.: Global Exponential Stability of Reaction-Diffusion Recurrent Neural Networks with Time-varying Delays. Physics letters A 314 (2003) 434-442 16. Song, Q., Cao, J.: Global Exponential Stability and Existence of Periodic Solutions in BAM Networks with Delays and Reaction-diffusion Terms. Chaos Solitons and Fractals 23 (2005) 421-430 17. Song, Q., Zhao, Z., Li, Y.: Global Exponential Stability of BAM Neural Networks with Distributed Delays and Reaction-diffusion Terms. Physics Letters A 335 (2005) 213-225 18. Siljak, D. D.: Large-scale Dynamic Systems — Stability and Structure. Elsevier NorthHolland, New York (1978) 19. Zhang, J., Yang, Y., Zeng, J.: String Stability of Infinite Interconnected System. Applied Mathematics and Mechanics 21 (2000) 791-796

Exponential Stability of Delayed Stochastic Cellular Neural Networks Wudai Liao1 , Yulin Xu1 , and Xiaoxin Liao2 1

2

School of Electrical and Information Engineering, Zhongyuan University of Technology, 450007, Zhengzhou, Henan, P.R. China {wdliao, xuyulin}@zzti.edu.cn Department of Control Science and Engineering, Huazhong University of Science and Technology, 430074, Wuhan, Hubei, P.R. China [email protected]

Abstract. In view of the character of saturation linearity of output functions of neurons of the cellular neural networks, the method decomposing the state space to sub-regions is adopted to study almost sure exponential stability on delayed cellular neural networks which are in the noised environment. When perturbed terms in the model of the neural network satisfy Lipschitz condition, some algebraic criteria are obtained. The results obtained in this paper show that if an equilibrium of the neural network is the interior point of a sub-region, and an appropriate matrix related to this equilibrium has some stable degree to stabilize the perturbation, then the equilibrium of the delayed cellular neural network can still remain the property of exponential stability. All results in the paper is only to compute eigenvalues of matrices.

1

Introduction

The stability problem on cellular neural networks is widely studied from origin to now, including the original work [1], delayed cases [2] and other studies. For reality, we should consider the case that the cellular neural networks are in noised environment, this is because the network realization is through VLSI approach and the information transmission among real brain neuron cells ia a noised process. We call these neural networks stochastic neural networks. The stability analysis of these neural networks originates in [3], and after that, lots of results [4, 5, 6] are obtained by some scholars. Because of the characteristic of saturation linearity about the output functions of neurons in cellular neural networks, we are going to use this to study the stability problem on stochastic cellular neural networks, it has the form dx(t) = [−Bx(t) + Af (x(t − τ )) + I]dt + σ (x(t), x(t − τ ))dw(t) .

(1)

Where x = (x1 , x2 , · · · , xn )T ∈ IRn is the state vector of the neural networks, x (t − τ ) = (x1 (t − τ1 ), x2 (t − τ2 ), · · · , xn (t − τn ))T , τi ≥ 0 is the time delay of the J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 224–229, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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225

neuron i and 0 ≤ τi ≤ τ, i = 1, 2, · · · , n.f (·) is the vector of the output functions of the neurons, f (x) = (f1 (x1 ), f2 (x2 ), · · · , fn (xn ))T , fi (·) has the form fi (u) =

1 (|u + 1| − |u − 1|) − ∞ < u < ∞, i = 1, 2, · · · , n . 2

(2)

σ (·, ·) ∈ IRn×m is the perturbed matrix satisfied Lipschtz condition, that is there exists a positive constant L such that  σ (x, y) − σ  (¯ x, y¯)2 ≤ L(x − x ¯2 + y − y¯2 ) .

(3)

· in this paper denotes the Frobenius norm of a matrix. w(·) is an m-dimension Brownian motion, B = diag.(b1 , b2 , · · · , bn ) is a positive diagonal matrix, A = (aij )n×n is the weight matrix between neurons, I is the bias of the neuron.

2

Main Results

We will set up some sufficient criteria ensuring the neural network (1) almost sure exponential stability in this section. According to IR = (−∞, ∞) being decomposed into three intervals (−∞, −1), [−1, 1] and (1, ∞), the n-space IRn can be divided into 3n sub-regions. Suppose that x∗ = (x∗1 , x∗2 , · · · , x∗n )T is an equilibrium of the system (1), which is the interior point in one of the sub-regions, N (x∗ ) is the greatest neighborhood of the point x∗ which is in the same sub-region that the point x∗ is in. Take the transformation z = x − x∗ , and a function  1, |u| < 1 φ(u) = . 0, |u| > 1 From the assumptions above and Property (2) of the output functions, we have f (z(t − τ ) + x∗ ) − f (x∗ ) = Φ(x∗ )z(t − τ ) holds for points in N (x∗ ), where the matrix Φ(x∗ ) = diag.(φ(x∗1 ), φ(x∗2 ), · · · , φ (x∗n )) is a diagonal matrix which the elements are either 0 or 1. Thus, in order to discuss the stability of the equilibrium x∗ of System (1), we need only study the same property of the trivial equilibrium z = 0 of the following system dz(t) = [−Bz(t) + AΦ(x∗ )z(t − τ )]dt + σ(z(t), z(t − τ ))dw(t) .

(4)

Where σ(z(t), z(t − τ )) = σ (z(t) + x∗ , z(t − τ ) + x∗ ) − σ (x∗ , x∗ ). n For ∀z, y ∈ IR and a symmetric positive definite matrix Q, we have the following estimation on σ(z(t), z(t − τ )) by using the condition (3). trace(σ T (z, y)Qσ(z, y)) ≤ trace(Q)L(z2 + y2 ) .

(5)

We first give a definition and a lemma [4] which plays an important role in this paper.

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Definition 1. For a Lyapunov function V ∈ C 2 (IRn ; IR+ ), that is, the function is continuously twice differentiable with respect to its variables, IR+ = [0, ∞), for any z, y ∈ IRn , defines the operator L generated by the system (4) as following 1 LV (z, y) = Vx [−Bz + AΦ(x∗ )y] + trace((σ(z, y))T Vxx σ(z, y)) . 2 Lemma 1. For System (4), if there exit functions V ∈ C 2 (IRn ; IR+ ), μ ∈ C(IRn ; IR+ ), μi ∈ C(IR+ ; IR+ ), i = 1, 2, · · · , n, and constants λ1 > λ2 ≥ 0, such that (1)LV (z, y) ≤ −λ1 μ(z) + λ2

n 

μi (yi ), (2)V (z) ≤ μ(z), (3)

i=1

n 

μi (zi ) ≤ μ(z)

i=1

hold for any z, y ∈ IRn , then, the trivial equilibrium z = 0 is almost sure exponential stability. In the following, we will get the main results. Theorem 1. For a diagonal positive definite matrix Q = diag.(q1 , q2 , · · · , qn ), select a positive diagonal matrix R = diag.(r1 , r2 , · · · , rn ), if the matrix ⎛ ⎞ −2QB + αU + R QAΦ(x∗ ) ⎠ H=⎝ (QAΦ(x∗ ))T −R + αU is negative definite, then the trivial equilibrium z = 0 of System (4) is almost sure exponential stability, where α = trace(Q)L, U is the unit matrix. Proof (of Theorem 1). For ∀z, y ∈ IRn , choose the Lyapunov function V (z) = z T Qz, and then, the operator L generated by System (4) has the form LV (z, y) = 2z T Q(−Bz + AΦ(x∗ )y) + trace((σ(z, y))T Qσ(z, y)) . By using condition (5) and denoting −λ = λmax (H), λ > 0, we have the following estimation LV (z, y) ≤ −2z T (QB)z + 2z T [QAΦ(x∗ )]y + α(z2 + y2 )  n n  z T T = (z , y )H − ri zi2 + ri yi2 y i=1

≤−

n 

(λ + ri )zi2 +

i=1

n 

i=1

(ri − λ)yi2 .

i=1

From the construction of the matrix H, we can easily deduce that ri − λ ≥ α > 0, i = 1, 2, · · · , n. Denote that λ1 = min { 1≤i≤n

1 ri − λ (λ + ri )}, λ2 = max { } . 1≤i≤n λ + ri qi

Exponential Stability of Delayed Stochastic Cellular Neural Networks

227

Obviously, λ1 > 0, 0 < λ2 < 1 (use λ > 0) and (λ + ri )/λ1 ≥ qi , i = 1, 2, · · · , n. Let n 1  1 μ(z) = (λ + ri )zi2 , μi (yi ) = (λ + ri )yi2 , λ1 i=1 λ1 then, we have (1)LV (z, y) ≤ −λ1 μ(z) + λ1 λ2

n  i=1

μi (yi ), (2)V (z) ≤ μ(z), (3)

n 

μi (yi ) = μ(y) .

i=1

According to Lemma 1, the trivial equilibrium z = 0 of System (4) is almost sure exponential stability. The proof is complete. Corollary 1. If there exist positive diagonal matrices Q = diag.(q1 , q2 , · · · , qn ) and R = diag.(r1 , r2 , · · · , rn ) such that the matrix ⎛ ⎞ −2QB + R QAΦ(x∗ ) ⎠ H1 = ⎝ ∗ T (QAΦ(x )) −R has the stable degree α, that is λmax (H1 ) < −α, then the trivial equilibrium z = 0 System (4) is almost sure exponential stability. Proof (of Corollary 1). From Theorem 1, in order to prove Corollary 1, we need only to verify the matrix H in Theorem 1 to be negative definite. For ∀z, y ∈ IRn ,



z z (z T , y T )H = (z T , y T )H1 + αz2 + αy2 y y ≤ (λmax (H1 ) + α)z2 + (λmax (H1 ) + α)y2 . Because λmax (H1 ) < −α, the matrix H in Theorem 1 is negative definite. The proof is complete. In following, we give an approach to choose the optional matrix R and the matrix Q, so, it is easy to use in system synthesis. Corollary 2. For an appropriate positive number m > 0, If the matrix ⎛ ⎞ 2m − m+1 U B −1 AΦ(x∗ ) ⎠ H2 = ⎝ 2 (B −1 AΦ(x∗ ))T − m+1 U n has the stable degree α = L i=1 b−1 i , then the trivial equilibrium z = 0 of System (4) is almost sure exponential stability. Proof. Let −2QB + R = −mR and Q = B −1 in matrix H1 , this implies that R = [2/(m + 1)]U , hence, the matrix H1 becomes the matrix H2 . The proof is complete.

228

W. Liao, Y. Xu, and X. Liao

Corollary 3. For equilibrium x∗ = (x∗1 , x∗2 , · · · , x∗n )T of System (1), |x∗i | < 1, i = 1, 2, · · · , n. If there exists a number m > 0, such that the matrix ⎛ ⎞ 2m − m+1 U B −1 A ⎠ H3 = ⎝ 2 (B −1 A)T − m+1 U has the stable degree α = L exponential stability.

n

−1 i=1 bi ,

then the equilibrium x∗ is almost sure

Proof. In this case, Φ(x∗ ) = U , by using Corollary 2, Corollary 3 holds. The proof is complete. Corollary 4. For equilibrium x∗ = (x∗1 , x∗2 , · · · , x∗n )T of System (1), |x∗i | > 1, i = 1, 2, · · · , n. If there exists a number m > 0, such that ⎧ 2 n −1 −1 ⎨ m+1 ( i=1 bi ) , m > 1 0≤L< ⎩ 2m n −1 −1 , m≤1 i=1 bi ) m+1 ( then, the equilibrium x∗ can still remain exponential stability while x∗ of the deterministic neural network corresponding to System (1) is exponential stability. Proof. In this case, Φ(x∗ ) = 0, H2 has the following form 2m

− m+1 U 0 2 0 − m+1 U its biggest eigenvalue is   2m 2 λmax (H2 ) = max − ,− . m+1 m+1 So, the condition of this corollary implies λmax (H2 ) < −α = −L using Corollary 2, this corollary holds. The proof is complete.

n

−1 i=1 bi ,

by

Corollary 5. For equilibrium x∗ = (x∗1 , x∗2 , · · · , x∗n )T of System (1), |x∗j0 | < 1, |x∗i | > 1, i = 1, · · · , j0 − 1, j0 + 1, · · · , n. If there exists positive number m, such that

2 n  aij0 4m cj0 := < b (m + 1)2 i i=1 and 0 ≤ L < λ(m) hold, then, the equilibrium x∗ is almost sure exponential stability, where    

−1 n 2 2m m−1 2 −1 λ(m) = min , , 1 − cj0 + ( ) ) · bi . m+1 m+1 m+1 i=1

Exponential Stability of Delayed Stochastic Cellular Neural Networks

229

Proof. Without loss generality, let |x∗1 | < 1, |x∗i | > 1, i = 2, 3, · · · , n. Hence, Φ(x∗ ) = diag.(1, 0, · · · , 0), and then the character polynomial of matrix H2 is as following f (λ) = (λ +

2 2m n−1 2 2m )n−1 (λ + ) [(λ + )(λ + ) − c1 ] . m+1 m+1 m+1 m+1

Its all eigenvalues satisfy the following equations λ+

2 2m 4m = 0, λ + = 0, λ2 + 2λ + − c1 = 0 . m+1 m+1 (m + 1)2

The condition c1 < 4m/(m+ 1)2 implies that the bigger root of the last equation above is negative, it has the form  m−1 2 λ = −1 + c1 + ( ) . m+1 Then, from the condition of this corollary, the condition of Corollary 2 holds. The proof is complete.

3

Conclusions

From discussion above, we conclude the results as following: 1. All results obtained in our paper hold for the deterministic case corresponding to neural network (1); 2. If the perturbed intensity is pre-estimated, then we can choose the parameter matrices B and A to design a deterministic neural network which has enough robustness to stabilize the perturbed intensity.

Acknowledgements This work is supported by National Natural Science Foundation of China (60274007, 60474001, 70572050).

References 1. Chua, L., Yang, L.: Cellular Neural Networks: Theory. IEEE Trans. Circuits and Systems 35 (1988) 1257-1272 2. Cao, J., Zhou, D.: Stability Analysis of Delayed Celluar Neural Networks. Neural Networks 11 (1998) 1601-1605 3. Liao, X., Mao, X.: Stability of Stochastic Neural Networks. Neual, Parallel and Scientific Computations 14(4) (1996) 205-224 4. Blythe, S., Mao, X.: Stability of Stochastic Delay Neural Networks. Journal of The Franklin Institute 338 (2001) 481-495 5. Shen, Y., Liao, X.: Robust Stability of Nonlinear Stochastic Delayed Systems. Acta Automatica Sinic 25(4) (1999) 537-542 6. Mao, X.: Stochastic Differential Equations and Their Applications. 1st edn. Horwood Pub., Chichester (1997)

Global Exponential Stability of Cellular Neural Networks with Time-Varying Delays and Impulses Chaojin Fu and Boshan Chen Department of Mathematics, Hubei Normal University, Huangshi, Hubei, 435002, China [email protected]

Abstract. In this paper, global exponential stability of cellular neural networks with time-varying delays and impulses is studied. By estimating the delay differential inequality with impulses, some sufficient conditions for global exponential stability of impulsive neural networks are obtained. The obtained results are new and they complement previously known results. Finally, an example is given to illustrate the theory.

1

Introduction

In recent years, cellular neural networks (CNNs) have attracted great attention due to their significant potential in applications. In many engineering applications and hardware implementations of neural networks, time delays even varying delays in neuron signals are often inevitable, because of internal or external uncertainties. In addition, it is well known that the stability of CNNs and cellular neural networks with time-varying delays (DCNNs) are critical for their applications, especially in signal processing, image processing, and solving nonlinear algebraic and transcendental equations, and optimization problems. In the past decade, the stability of CNNs and DCNNs has been widely investigated; e.g., [1]-[10]. Moreover, in order to improve the convergence rate of the neural networks toward equilibrium points, and reduce the calculating time, such network as having globally stable equilibrium is prior to be considered. As CNNs and DCNNs are large-scale nonlinear dynamical systems, their stability analysis is a nontrivial task. However, besides delay effect, impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics and telecommunications, etc. Many interesting results on impulsive effect have been gained; e.g., [11]-[13]. In this paper, global exponential stability of cellular neural networks with time-varying delays and impulses is studied. Scopes of impulses are magnified. By estimating the delay differential inequality with impulses, some sufficient conditions for global exponential stability of impulsive neural networks are obtained. The obtained results are new and they complement previously known results. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 230–235, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Global Exponential Stability of Cellular Neural Networks

2

231

Preliminaries

Considers a CNN with time-varying delays and impulses (IDCNN) 

 ¯ ¯ = −ci ui (t) + n = tk ij f (uj (t)) + bij f (uj (t − τj (t)))) + Ii , t > t0 , t  j=1 (a  (1) n ¯(uj (t− )) + Ii |, |Δui (tk )| ≤ | − ci ui (t− ) + (a + b ) f t = t , ij ij k k k j=1 dui (t) dt

where i = 1, 2, · · · , n, k = 1, 2, · · · , aij and bij are constant connection weights, ci is a positive constant, Ii is an external input or bias, f (·) is the activation function defined by f¯(uj (t)) = (|uj (t) + 1| − |uj (t) − 1|)/2,

(2)

τj (t) ≤ τ is the time delay upper-bounded by a constant τ, i, j ∈ {1, 2, · · · , n}, − Δui (tk ) = ui (t+ k ) − ui (tk ) is the impulse at moments tk and t1 < t2 < t3 < · · · is a strictly increasing sequences such that limk→+∞ tk = +∞. ui (t+ k ) denotes the right limit of ui (t) at t = tk . ui (t− ) denotes the left limit of u (t) at t = tk . i k Throughout of this paper, we denote C = diag{c1 , c2 , · · · , cn }, A = (aij )n×n , B = (bij )n×n , C([t0 −τ, t0 ], n ) is the set of continuous function φ : [t0 −τ, t0 ] → n . Since f¯ is bounded, it can be easily proved that IDCNN (1) has at least one equilibrium point u∗ = (u∗1 , u∗2 , · · · , u∗n )T such that for i = 1, 2, · · · , n, Ii =

ci u∗i

−

n 

(aij + bij )f¯(u∗j ).

(3)

j=1

Let x(t) = (x1 (t), x2 (t), · · · , xn (t))T = (u1 (t)−u∗1 , u2 (t)−u∗2 , · · · , un (t)−u∗n )T , f (ri ) = f¯(ri + u∗i ) − f¯(u∗i ), then IDCNN (1) can be rewritten as 

 = −ci xi (t) + nj=1 (aij + bij )f (xj (t)), t > t0 , t = tk n − |Δxi (tk )| ≤ | − ci xi (t− ) + (a + b )f (x (t ))|, t = tk . ij ij j k j=1 k dxi (t) dt

(4)

The initial condition of IDCNN (1) (or IDCNN (4)) is assumed to be φ(ϑ) = (φ1 (ϑ), φ2 (ϑ), · · · , φn (ϑ))T ,

(5)

where φ(ϑ) ∈ C([t0 −τ, t0 ], n ). Denote ||φ||t0 = maxi∈{1,···,n},s∈[t0 −τ,t0 ] {|φi (s)|}. Denote u(t; t0 , φ) as the solution of (1) with initial condition (5). It means that u(t; t0 , φ) satisfies (1) and u(s; t0 , φ) = φ(s), for s ∈ [t0 − τ, t0 ]. Also simply denote u(t) as the solution of (1). Lemma 1. [10] Let P be a given n × n matrix. If λ is an eigenvalue of P˜ , then −λ is also an eigenvalue of P˜ , and λ2 is an eigenvalue of P P T and P T P, where   0 P ˜ P := . PT 0

232

3

C. Fu and B. Chen

Main Results

In the following discussions, let λ A+AT , λBB T and λB be respectively the maxi2 mum eigenvalues of matrices A + AT  := , BB T and B 2



0 B BT 0

 .

According to Lemma 1, λ2B = λBB T .  Let M = (max1≤i≤n {1 + ci + nj=1 (|aij + bij |)})2 . If − cmin + λ A+AT + λB < 0,

(6)

2

where cmin := min1≤i≤n ci , then there exists θ > 0, such that θ − 2cmin + 2λ A+AT + λB + λB exp{θτ } = 0.

(7)

2

Theorem 1. If (6) holds, and there exists β > 0 such that for k = 1, 2, · · · , tk − t0 ≥ βk; nM ≤ exp{θβ/2}, then IDCNN (1) is globally exponentially stable. Proof. Denote u∗ = (u∗1 , u∗2 , · · · , u∗n )T as an equilibrium point of IDCNN (1). Let x(t) = (x1 (t), x2 (t), · · · , xn (t))T = (u1 (t) − u∗1 , u2 (t) − u∗2 , · · · , un (t) − u∗n )T . From (2), for ∀t ≥ T, IDCNN (1) can be rewritten as (4). Let V (t) = V (x(t)) = xT (t)x(t), then for t = tk , dV (x(t)) |(4) ≤ −2xT (t)Cx(t) + xT (t)(A + AT )x(t) dt  T  xT (t), xT (t − τ (t)) +(xT (t), xT (t − τ (t)))B ≤ (−2cmin + 2λ A+AT + λB )V (x(t)) + λB V (x(t − τ (t))).

(8)

2

Let G(t) = V (t) − ||φ||t0 exp{−θ(t − t0 )}, Then ∀t ∈ [t0 , t1 ), G(t) ≤ 0. Otherwise, on the one hand, ∀s ∈ [t0 − τ, t0 ], G(s) ≤ 0, then there exist t∗1 , t∗2 and ε > 0 such that t1 > t∗2 > t∗1 ≥ t0 , G(t∗1 ) = 0, G(t∗2 ) = ε, dG(t) dG(t) |t=t∗1 ≥ 0, |t=t∗2 > 0; dt dt

(9) (10)

for s ∈ (t∗1 , t∗2 ], G(s) > 0; for t ∈ [t0 − τ, t∗2 ], G(t) ≤ ε. On the other hand, by (7), dG(t) dV (t) |t=t∗2 = |t=t∗2 + θ||φ||t0 exp{−θ(t∗2 − t0 )} dt dt ≤ (θ − 2cmin + 2λ A+AT + λB + λB exp{θτ })||φ||t0 exp{−θ(t∗2 − t0 )} 2

+(−2cmin + 2λ A+AT + λB )ε.

(11)

2

By (6), (8) and (11),

dG(t) ∗ dt |t=t2

≤ 0. This contradicts (10), thus

∀t ∈ [t0 , t1 ), V (t) ≤ ||φ||t0 exp{−θ(t − t0 )}.

(12)

Global Exponential Stability of Cellular Neural Networks

Hence, ∀i ∈ {1, 2, · · · , n}, ∀t ∈ [t0 , t1 ), xi (t) ≤ (4) and (12), − − |xi (t+ 1 )| ≤ |xi (t1 )| + | − ci xi (t1 ) +



n 

233

||φ||t0 exp{−θ(t − t0 )}. Form

(aij + bij )f (xj (t− 1 ))|

j=1

≤ M ||φ||t0 exp{−θ(t1 − t0 )}. Hence, V (t+ 1 ) ≤ nM ||φ||t0 exp{−θ(t1 − t0 )}.

(13)

Similar to the proofs of (12) and (13), for k ≥ 2, ∀t ∈ [t1 , t2 ), V (t) ≤ nM ||φ||t0 exp{−θ(t − t0 )}.

(14)

k V (t+ k ) ≤ (nM ) ||φ||t0 exp{−θ(tk − t0 )}.

(15)

∀t ∈ [tk , tk+1 ), V (t) ≤ (nM )k ||φ||t0 exp{−θ(t − t0 )}.

(16)

Hence, for k ≥ 3,

Since for k = 1, 2, · · · , tk − t0 ≥ βk; nM ≤ exp{θβ/2}, from (16), for ∀t ∈ [tk , tk+1 ), V (t) ≤

(nM )k θ θ ||φ||t0 exp{− (t − t0 )} ≤ ||φ||t0 exp{− (t − t0 )}. exp{βθ(tk − t0 )/2} 2 2

Hence, IDCNN (1) is globally exponentially stable. Corollary 1. If (6) holds, and for all k = 1, 2, · · · , |Δui (tk )| = 0, then IDCNN (1) is globally exponentially stable. Proof. Since |Δui (tk )| = 0, tk − t0 = +∞. Hence, for ∀n, M, there exists β > 0 such that for k = 1, 2, · · · , tk −t0 > βk; nM ≤ exp{θβ/2}. According to Theorem 1, IDCNN (1) is globally exponentially stable. In the following discussion, we always assume that for ∀i = 1, 2, · · · , n, ci = 1. Remark 1. If |Δui (tk )| = 0, (1) is a cellular neural network without impulse. Arik ([14]) has shown that if −(A + AT ) is positive definite and ||B|| ≤ 1 then the DCNN (1) is globally exponentially stable. √ Liao ([15]) has shown that if −(A + AT + βE) is positive definite and ||B|| ≤ 1 + β then the DCNN (1) is globally exponentially stable.√Arik ([16]) has shown that if −(A + AT + βE) is positive definite and ||B|| ≤ 2β then the DCNN (1) is globally exponentially stable. If −(A+AT +βE) is positive definite, then λA+AT +βE < 0; i.e., λ(A+AT )/2 < −β/2, where E is unit matrix. Hence, it is necessary that −(A + AT ) is positive definite in [14], [15], [16]

234

C. Fu and B. Chen

According to Lemma 1, λBB T ≤ ||B||2 implies that λB ≤ ||B||. So, if conditions of [ Theorem 1 in [14]] hold, then λB + λ A+AT ≤ ||B|| + λ A+AT < 1 + 0 = 1. 2

2

Hence, when condition (6) holds, the result of Corollary 1 improve upon the existing one in [14]. If conditions of [Theorem 1 in [15]] hold, then

λB + λ A+AT ≤ ||B|| + λ A+AT < 1 + β − β/2 ≤ 1. 2 2 √ (Since (1 + β/2)2 = 1 + β + β 2 /4 > 1 + β, 1 + β − β/2 < 1.) Hence, when condition (6) holds, the result of Corollary 1 improve upon the existing one in [15]. If conditions of [Theorem 1 in [16]] hold, then

λB + λ A+AT ≤ ||B|| + λ A+AT < 2β − β/2 ≤ 1. 2

2

2 2 (Since √ (1 − β/2) = 1 − β + β /4 ≥ 0, 1 + β + β /4 ≥ 2β, then (1 + β/2) ≥ 2β; i.e., 2β − β/2 ≤ 1.) Hence, when condition (6) holds, the result of Corollary 1 improve upon the existing one in [16]. 2

4

2

Simulation Results

In this section, we give one example to illustrate the new results. Example 1. Consider IDCNN (1), where     0.2 0.4 0.3 −0.4 A= , B= , −0.4 0.2 0.4 0.3 T

I = (0.1, 0) , ci = 1, the time delays τi (t) = 1 (i = 1, 2), t0 = 0, tk = 13.6k. Then λ A+AT = 0.2, λB = 0.5, M = 6.25, θ = 0.3736 such that 2

θ − 2cmin + 2λ A+AT + λB + λB exp{θτ } ≤ 0. 2

Since exp{βθ/2} = exp{13.6×0.3736/2} > 12.5 = nM, according to Theorem 1, this cellular neural network is globally exponentially stable. However, since −(A + AT ) is not positive definite, even if for all k = 1, 2, · · · , Δxi (tk ) = 0, i = 1, 2, the conditions in [14], [15], [16] cannot be used to ascertain the stability of this cellular neural network.

5

Concluding Remarks

In this paper, global exponential stability of cellular neural networks with timevarying delays and impulses is studied. Scopes of impulses are magnified. By estimating the delay differential inequality with impulses, some sufficient conditions for global exponential stability of impulsive neural networks are obtained. Those sufficient conditions independent on the equilibrium point. Hence, when it is difficult to obtain the equilibrium point, the given results in this paper are with more reason. Finally, an example is given to illustrate the theory.

Global Exponential Stability of Cellular Neural Networks

235

Acknowledgement This work was supported by the Natural Science Foundation of China under Grant 60474011 and the Young Foundation of Hubei Provincial Education Department of China under Grant 2003B001.

References 1. Forti, M., Tesi, A.: New Conditions for Global Stability of Neural Networks with Application to Linear and Quadratic Programming Problems. IEEE Trans. Circ. Syst. I 42 (1995) 354-366 2. Liao, X.X., Wang, J.: Algebraic Criteria for Global Exponential Stability of Cellular Neural Networks with Multiple Time Delays. IEEE Trans. Circuits and Systems I. 50 (2003) 268-275 3. Yi, Z., Heng, A., Leung, K.S.: Convergence Analysis of Cellular Neural Networks with Unbounded Delay. IEEE Trans. Circuits Syst. I 48 (2001) 680-687 4. Chen, T.P., Rong, L.B.: Robust Global Exponential Stability of Cohen- Grossberg Neural Networks with Time-Delays. IEEE Transactions on Neural Networks 15 (2004) 203-206 5. Liao, X.F., Li, C.G., Wong, K.W.: Criteria for Exponential Stability of CohenGrossberg Neural Networks. Neural Networks 17 (2004) 1401-1414 6. Zeng, Z.G., Wang, J., Liao, X.X.: Global Asymptotic Stability and Global Exponential Stability of Neural Networks with Unbounded Time-varying Delays. IEEE Trans. on Circuits and Systems II, Express Briefs 52 (2005) 168-173 7. Cao, J.: Results Concerning Exponential Stability and Periodic Solutions of Delayed Cellular Neural Networks. Physics Letters A 307 (136-147) 2003 8. Zhou, J., Liu, Z., Chen, G.R.: Dynamics of Delayed Periodic Neural Networks. Neural Networks 17 (2004) 87-101 9. Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Syst. I 51 (2004) 2313-2324 10. Zeng, Z.G., Wang, J., Liao, X.X.: Global Exponential Stability of A General Class of Recurrent Neural Networks with Time-varying Delays. IEEE Trans. Circuits and Syst. I 50 (2003) 1353-1358 11. Xu, D., Yang, Z.: Impulsive Delay Differential Inequality and Stability of Neural Networks. J. Math. Anal. Appl. 305 (2005) 107-120 12. Guan, Z., Chen, G.: On Delayed Impulsive Hopfield Neural Networks. Neural Networks 12 (1999) 273-280 13. Guan, Z., Lam, J., Chen, G.: On Impulsive Autossociative Neural Networks. Neural Networks 13 (2000) 63-69 14. Arik, S.: On the Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Circuits Syst. I 47 (2000) 571-574 15. Liao, T.L., Wang, F.C.: Global Stability for Cellular Neural Networks with Time Delay. IEEE Trans. Neural Networks 11 (2000) 1481-1484 16. Arik, S.: An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Networks 13 (2002) 1239-1242

Global Exponential Stability of Fuzzy Cellular Neural Networks with Variable Delays Jiye Zhang, Dianbo Ren, and Weihua Zhang Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu 610031, China [email protected]

Abstract. In this paper, the global exponential stability of fuzzy cellular neural networks with time-varying delays is studied. Without assuming the boundedness and differentiability of the activation functions, based on the properties of M-matrix, by constructing vector Liapunov functions and applying differential inequalities, the sufficient conditions ensuring existence, uniqueness, and global exponential stability of the equilibrium point of fuzzy cellular neural networks with variable delays are obtained.

1 Introduction Recently, artificial neural networks have been intensively studied and widely applied to various information processing problems. Cellular neural networks (CNNs), introduced by Chua and Yang in 1988, have been successfully applied to signal processing systems, especially in static image treatment [1], and to solve nonlinear algebraic equations [2-3]. Hopfield-type neural networks (HNNs) and their various generalizations have been applied to deal with the tasks of classification, associative memory, parallel computation and optimization problems [4-5]. Yang extended the CNNs from classical to fuzzy sets, and proposed the fuzzy cellular neural networks (FCNNs), and applied it to the image processing [6,7]. Such applications rely on the existence of equilibrium points and qualitative properties of the neural networks. The stability of HNNs and CNNs were widely studied, and some results were given [8-17]. In hardware implementation, time delays occur due to finite switching speeds of the amplifiers and communication time. Some conditions ensuring the global exponential stability of FCNNs with variable time delays were given in [18], but in which differentiability of the delays terms were assumed. In the paper, we study the fuzzy neural networks, which contains variable delays. By constructing proper nonlinear integro-differential inequalities involving both variable delays, applying the idea of vector Liapunov method, we obtain the sufficient conditions of global exponential stability.

2 Notation and Preliminaries For convenience, we introduce some notations. x Τ and AΤ denote the transpose of a vector x and a matrix A , where x ∈ R n and A ∈ R n×n . [ A]s is defined as J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 236 – 242, 2006. © Springer-Verlag Berlin Heidelberg 2006

Global Exponential Stability of Fuzzy Cellular Neural Networks with Variable Delays

[ A]s = [ AΤ + A] 2

| x|

.

| x |= (| x1 |, | x 2 |, L | xn |)

Τ

denotes

the

absolute-value

vector

given

237

by

and | A | denotes the absolute-value matrix given by

| A |= (| aij |) n×n . || x || denotes the vector norm defined by || x ||= ( x12 + L + xn2 )1/ 2 and

|| A || denotes the matrix norm defined by || A ||= (max{λ : λ is an eigenvalue of

∧ ∨

AΤ A })1 / 2 . and denote the fuzzy AND and fuzzy OR operation, respectively. The dynamical behavior of FCNNs with variable time delays can be described by the following nonlinear differential equations: n

n

j =1

j =1

n

x&i = −d i xi (t ) + ∑ aij f j ( x j (t )) + ∑ bij u j + J i + ∧ α ij f j ( x j (t − τ ij (t ))) n

n

j =1

n

+ ∨ β ij ( f j ( x j (t − τ ij (t ))) + ∧ Rij u j + ∨ S ij u j j =1 j =1 j =1

( i = 1,2,L n ).

(1)

where xi is the state of neuron i, i = 1,2,L, n , and n is the number of neurons; ui and

J i denote input and bias of the ith neuron, respectively; f i is the activation function of the ith neuron; d i is the damping constants, and d i >0; aij , bij are elements of feedback template and feedforward template; α ij , β ij , Rij and S ij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feedforward MIN template and fuzzy feedforward MAX template, respectively; τ ij (t ) denote the variable time delays. Assume that delays τ ij (t ) are bounded, continuous with

τ ij (t ) ∈ [0,τ ] for all t ≥ 0 , where τ is a constant, i, j = 1,2,L, n . The initial conditions associated with equation (1) are of the form xi ( s ) = φi ( s ) , s ∈ [−τ ,0] , φi ∈ C ([−τ ,0], R) , i = 1,2, L , n ,

where it is assumed that φi ∈ C ([−τ ,0], R) , i = 1,2, L , n .

Let D = diag(d1 , d 2 ,L , d n ) , A = (aij ) n×n , B = (bij ) n×n , α = (α ij ) n×n , β = ( β ij ) n×n ,

u = (u1 , u 2 ,..., u n) Τ , J = ( J 1 , J 2 , L, J n) Τ , f ( x) = ( f 1 ( x1), f 2 ( x2),L , f n ( x n))Τ . Assumption 1. For each j ∈ {1,2,..., n} , f j : R → R is globally Lipschitz with

Lipschitz constants L j > 0 , L = diag( L1 , L 2 , L, L n ) > 0 .

i.e. | f j ( x j ) − f j ( y j ) |≤ L j | x j − y j | for all x j , y j . We let

Definition 1. The equilibrium point x * of (1) is said to be globally exponentially stable, if there exist constant λ > 0 and M > 0 such that | xi (t ) − xi * |≤ M || φ − x* || e − λt for all t ≥ 0 , where || φ − x* ||= max{ sup | φi ( s ) − xi* |} . 1≤ i ≤ n

s∈[ −τ , 0 ]

Lemma 1 [19]. Let A =( aij ) be a matrix with non-positive off-diagonal elements.

Then the following statements are equivalent: (i) A is an M-matrix, (ii) The real parts of all eigenvalues of A are positive, Τ (iii) There exists a vector ξ > 0 , such that ξ A > 0 ,

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(iv) A is nonsingular and all elements of A−1 are nonnegative, (v) There exists a positive definite n × n diagonal matrix Q such that matrix

AQ + QAΤ is positive definite. Lemma 2 [6]. Suppose x and x ' are two states of system (1), then n

n

j =1

j =1

n

n

j =1

j =1

| ∧ α ij f j ( x j ) − ∧ α ij f j ( x j ' ) | ≤

n

∑ |α

ij

|| f j ( x j ) − f j ( x j ' ) |

ij

|| f j ( x j ) − f j ( x j ' ) |

j =1

| ∨ β ij f j ( x j ) − ∨ β ij f j ( x j ' ) | ≤

n

∑ |β j =1

Lemma 3 [5]. If H ( x) ∈ C 0 is injective on R n , and || H ( x) ||→ ∞ as || x ||→ ∞ , then

H ( x) is a homeomorphism of R n .

3 Existence and Uniqueness of the Equilibrium Point In the section, we study the existence and uniqueness of the equilibrium point of (1). We firstly study the nonlinear map associated with (1) as follows: n

n

n

j =1

j =1

H i ( xi ) = − d i xi + ∑ aij f j ( x j ) + ∧ α ij f j ( x j ) + ∨ β ij f j ( x j ) + I i , i = 1,2,L n , j =1

n

(2)

n

where I i = ∧ Rij u j + ∨ S ij u j + J i , i = 1,2,L, n . j =1 j =1 Let H ( x) = ( H 1 ( x1 ), H 2 ( x2 ),..., H n ( xn ))Τ . It is known that the solutions of H ( x) = 0 are equilibriums in (1). If map H ( x) is a homeomorphism on Rn , then there exists a unique point x * such that H ( x*) = 0 , i.e., systems (1) have a unique equilibrium x * . Based on the Lemma 3, we get the conditions of the existence of the equilibrium for system (1) as follows. Theorem 1. If Assumption 1 is satisfied, and D − (| A | + | α | + | β |) L is an M- matrix, then for each u , system (1) has a unique equilibrium point. Proof. In order to prove that systems (1) have a unique equilibrium point x * , it is only need to prove that H ( x) is a homeomorphism on Rn . In the following, we shall

prove that map H ( x) is a homeomorphism in two steps. Step 1. We prove that H ( x) is an injective on Rn . For purposes of contradiction, suppose that there exist x, y ∈ R n with x ≠ y , such that H ( x) = H ( y ) . From (2), Lemma 2, and Assumption 1, we get

[ D − (| A | + | α | + | β |) L] | x − y |≤ 0 .

(3)

Because of D − (| A | + | α | + | β |) L being an M-matrix, from Lemma 1, we know that all elements of ( D − (| A | + | α | + | β |) L) −1 are nonnegative. Therefore | x − y |≤ 0 , i.e.,

Global Exponential Stability of Fuzzy Cellular Neural Networks with Variable Delays

239

x = y . From the supposition x ≠ y , thus this is a contradiction. So map H ( x) is injective. Step 2. We prove that || H ( x) ||→ ∞ as || x ||→ ∞ .

Let H ( x) = H ( x) − H (0) . From (2), we get n

n

n

j =1

j =1

H i ( xi ) = −d i xi + ∑ aij ( f j ( x j ) − f j (0)) + ∧ α ij f j ( x j ) − ∧ α ij f j (0) j =1

n

n

j =1

j =1

+ ∨ β ij f j ( x j ) − ∨ β ij f j (0) ( i = 1,2,L n ).

(4)

To prove that H ( x) is a homeomorphism, it only suffices to show that H ( x) is a homeomorphism. Since D − (| A | + | α | + | β |) L is an M-matrix, from Lemma 1, there exists a positive definite diagonal matrix T = diag{T1 , T2 ,L , Tn } , such that [T (− D + (| A | + | α | + | β |) L)]s ≤ −ε E n < 0 , (5) where ε is a sufficiently small positive number and E n is the identity matrix. From (4) and Lemma 2, we get n

∑ x T {− d

[Tx]Τ H ( x) =

i =1

i

i

n

i

xi + ∑ a ij [( f j ( x j ) − f j (0)] j =1

n

n

n

+ ∧ α ij f j ( x j ) − ∧ α ij f j (0) + ∨ β ij f j ( x j ) − ∨ β ij f j (0)} j =1 j =1 j =1 j =1 n

n

≤ ∑ Ti {− d i xi2 + | xi | ∑ | a ij || ( f j ( x j ) − f j (0)) | n

j =1

i =1

n

n

+ | xi | ∑ |α ij || f j ( x j ) − f j (0) | + | xi | ∑ | β ij || f j ( x j ) − f j (0) |} j =1

j =1

≤| x | [T (− D + (| A | + | α | + | β |) L)] | x | ≤ −ε || x ||2 . Τ

s

(6)

Using Schwarz inequality, and from (6), we get ε || x || ≤|| T || || x || || H ( x) || , so 2

|| H ( x) ||≥ ε || x || / || T || . Therefore, || H ( x) ||→ +∞ , i.e., || H ( x) ||→ +∞ as || x ||→ +∞ . From steps 1 and 2, according to Lemma 3, we know that for every input u , map H ( x) is a homeomorphism on Rn , so system (1) has a unique equilibrium point. The proof is completed.

4 Global Exponential Stability of the Equilibrium Point Theorem 2. If Assumption 1 is satisfied and D − (| A | + | α | + | β |) L is an M-matrix, then for each u , system (1) has a unique equilibrium point, which is globally exponentially stable. Proof. Since D − (| A | + | α | + | β |) L is an M-matrix, from Theorem 1, systems (1)

have a unique equilibrium x * . Let y (t ) = x(t ) − x * , systems (1) can be written as

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J. Zhang, D. Ren, and W. Zhang n

n

y& i (t ) = − d i yi (t ) + ∑ aij f j ( y j (t ) + x j *) − ∑ aij f j ( x j *)) j =1

j =1

n

n

+ ∧ α ij f j ( y j (t − τ ij (t )) + x j *) − ∧ α ij f j ( x j *) j =1 j =1 n

n

j =1

j =1

+ ∨ β ij f j ( y j (t − τ ij (t )) + x j *) − ∨ β ij f j ( x j *) ( i = 1,2,L, n ).

(7)

The initial conditions of equation (7) are Ψ ( s ) = φ ( s ) − x* , s ∈ [−τ ,0] . Systems (7) have a unique equilibrium at y = 0 . Let Vi (t ) = e λt | yi (t ) | , (8)

where λ is a constant to be given. Calculating the upper right derivative of Vi (t ) along the solutions of (7), we have D + (Vi (t )) = e λt sgn( yi (t ))[ y& i (t ) + λyi (t )] n ≤ e λt {− d i | y i (t ) | + ∑ | aij || f j ( y j (t ) + x j *) − f j ( x*j ) | j =1

n

n + | ∧ α ij f j ( y j (t − τ ij (t )) + x*j ) − ∧ α ij f j ( x*j ) | j =1 j =1 n n + | ∨ β ij f j ( y j (t − τ ij (t )) + x*j ) − ∨ β ij f j ( x*j ) | +λ | yi (t ) |} , ( i = 1,2,L, n ). j =1

j =1

From Assumption 1 and Lemma 2, we get n

D + (Vi (t )) ≤ (− d i + λ )Vi (t ) + ∑ L j [| aij |V j (t ) + e λτ (| α ij | + | β ij |) sup V j ( s )] , j =1

t −τ ≤ s ≤t

(9)

where τ is a fixed number. Due to D − (| A | + | α | + | β |) L is an M-matrix, from the Lemma 1, we get that there exist positive constant numbers ξ i , i = 1,2,L n, satisfying n

− ξ i d i + ∑ ξ j (| aij | + | α ij | + | β ij |) L j < 0 ( i = 1,2,L n ).

(10)

j =1

Constructing functions as follows n

Fi (η ) = − ξ i (d i − η ) + ∑ ξ j[| aij | + eητ (| α ij | + | β ij |)] L j , ( i = 1,2,L n ). j =1

(11)

Fi (η ) is a continuous function about η , and from (10) we know that n

Fi (0) = − ξ i d i + ∑ ξ j (| aij | + | α ij | + | β ij |) L j < 0 ( i = 1,2,L n ). j =1

So there exists a constant λ > 0 such that n

− ξ i (d i − λ ) + ∑ ξ j[| aij | + eλτ (| α ij | + | β ij |)] L j < 0 ( i = 1,2,L n ). j =1

(12)

In order to prove our results, we define the curve γ = {z (l ) : zi = ξ i l , l > 0, i = 1,2,L , n} and the set Ω( z ) = {u : 0 ≤ u ≤ z , z ∈ γ } . Let ξ M = max ξ i , ξ m = min ξ i , taking i =1,....,N

i =1,....,N

Global Exponential Stability of Fuzzy Cellular Neural Networks with Variable Delays

241

l0 = (1 + δ ) e λτ || Ψ || / ξ m ， δ > 0 be a constant. Defining set O = {V : V = e λτ || Ψ1 ( s) ||,

|| Ψ2 ( s) |, L, || Ψn ( s ) ||) Τ ,−τ ≤ s ≤ 0} . So, O ⊂ Ω( z0 (l0 )) , namely Vi (s ) ≤ eλτ || Ψi ( s ) ||< ξ il0 , −τ ≤ s ≤ 0

( i = 1,2,L n ).

(13)

In the following, we shall prove Vi (t ) < ξ il0 ， t > 0

( i = 1,2,L n ).

(14)

If (14) is not true, then from (13), there exist t1 > 0 and some index i such that Vi (t1 ) = ξ il0 , D + (Vi (t1 )) ≥ 0 , V j (t ) ≤ ξ j l0 , j = 1,2,L n , t ∈ [−τ , t1 ] .

(15)

However, from (9), and (12), we get n

D + (Vi (t1 )) ≤ {− ξ i (d i − λ ) + ∑ ξ j[| aij | + eλτ (| α ij | + | β ij |)] L j}l0 < 0 . j =1

This is a contradiction. So Vi (t ) < ξ il0 , for t > 0 ( i = 1,2,L n ). Furthermore, from (8), and (14), we obtain | yi (t ) | ≤ ξ il0 e − λt ≤ (1 + σ ) e λτ ξ M / ξ m || Ψ || e − λt ≤ M || Ψ || e − λt , t ≥ 0 ( i = 1,2,L n ). where M = (1 + σ ) e λτ ξ M / ξ m . So | xi (t ) − xi * |≤ M || xi (t ) − xi * || e − λt , and the equilibrium point of (1) is globally exponentially stable. The proof is completed. In [18], the global exponential stability of systems (1) was studied, and the stability conditions were given under the assumption 0 ≤ dτ ij (t ) / dt ≤ 1 . In fact, in the proof of results in [18], the following Lyapunov functionals were used, n

n

i =1

j =1

V ( y (t )) = ∑ ri (| yi (t ) | e lt + ∑ ∫t −τ t

ij

(t )

(| α ij | + | β ij |) k j | y j ( s ) | e ls ds .

It is obvious that the assumptions of the boundedness and for dτ ij (t ) / dt are necessary. However, in this paper, the constraint for the time delays is unnecessary.

4 Conclusions In this paper, without assuming the boundedness and differentiability of the Activation functions, we analyze the existence, uniqueness, and global exponential stability of the equilibrium point of fuzzy cellular neural networks with variable delays. Applying the idea of vector Liapunov function method, we obtain sufficient conditions for global exponential stability independent of the delays. Yang in [6] derived some results on stability of FCNNs, however, in which the model of FCNNs do not involved time delays and the activation function is bounded. Ref. [18] studied the exponential stability of the equilibrium point of FCNNs with variable delays under Assumption 1, but in which the boundedness and differentiability for the delays is assumed. Those assumptions for the delays are unnecessary in this paper. So the results in this paper extent the results in [6, 18].

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Acknowledgments This work is supported by the National Program for New Century Excellent Talents in University (No. NCET-04-0889), and Natural Science Foundation of China (No.50375127, 10272091)

References 1. Chua, L. O., Yang, L.: Cellular Neural Networks: Theory. IEEE Trans. Circ. Syst. 35 (1988) 1257-1272 2. Arik, S., Tavsanoglu, V.: On the Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Circ. Syst.- I 47 (2000) 571-574 3. Arik, S.: An Improved Global Stability Result for Delayed Cellular Neural Networks. IEEE Trans. Circ. Syst. 49 (2002) 1211-1214 4. Gopalsamy, K., He, X.: Stability in Asymmetric Hopfield Nets with Transmission Delays. Physica D 76 (1994) 344-358 5. Forti, M., Tesi, A.: New Conditions for Global Stability of Neural Networks with Application to Linear and Quadratic Programming Problems. IEEE Trans. Circ. Syst.-I 42 (1995) 354-366 6. Yang, T., Yang, L. B.: Exponential Stability of Fuzzy Cellular Neural Networks with Constant and Time-Varying Delays. IEEE Trans. Circ. Syst.-I 43 (1996) 880-883 7. Yang, T., Yang, L. B.: Fuzzy Cellular Neural Networks: A New Paradigm for Image Processing. Int. J. Circ. Theor. Appl. 25 (1997) 469-481 8. Roska, T., Wu, C. W., Balsi, M., Chua, L. O.: Stability of Cellular Neural Networks with Dominant Nonlinear and Delay-Type Templates. IEEE Trans. Circ. Syst. 40 (1993) 270272. 9. Zhang, J.: Global Stability Analysis in Delayed Cellular Neural Networks. Computers and Mathematics with Applications 45 (2003) 1707-1720 10. Zhang, J.: Absolutely Exponential Stability in Delayed Cellular Neural Networks. Int. J. Circ. Theor. Appl. 30 (2002) 395-409 11. Zhang, J.: Globally Exponential Stability of Neural Networks with Variable Delays. IEEE Trans. Circ. Syst.-I 50(2003) 288-291 12. Yucel, E., Arik, S.: New Exponential Stability Results for Delayed Neural Networks with Time Varying Delays. Physica D 191 (2004) 314–322 13. Zhang, J.: Absolute Stability Analysis in Cellular Neural Networks with Variable Delays and Unbounded Delay. Computers and Mathematics with Applications 47 (2004) 183-194 14. Van Den Driessche, P., Zou, X.: Global Attractivity in Delayed Hopfield Neural Networks Models. SIAM J. Appl. Math. 58 (1998) 1878-1890 15. Chen, T.: Global Exponential Stability of Delayed Hopfield Neural Networks. Neural Networks 14 (2001) 977-980 16. Xu, D., Zhao, H., Zhu, H.: Global Dynamics of Hopfield Neural Networks Involving Variable Delays. Computers and Mathematics with Applications 42 (2001) 39-45 17. Zhang, J., Suda, Y., Iwasa, T.: Absolutely Exponential Stability of a Class of Neural Networks with Unbounded Delay. Neural Networks 17 (2004) 391-397 18. Liu, Y., Tang, W.: Exponential Stability of Fuzzy Cellular Neural Networks with Constant and Time-Varying Delays. Physics Letters A 323 (2004) 224-233 19. Siljak, D. D.: Large-Scale Dynamic Systems-Stability and Structure. Elsevier, New York (1978)

Stability of Fuzzy Cellular Neural Networks with Impulses Tingwen Huang1 and Marco Roque-Sol2 1

2

Texas A&M University at Qatar, Doha, P. O. Box 5825, Qatar [email protected] Mathematics Department, Texas A&M University, College Station, TX77843, USA [email protected]

Abstract. In this paper, we study impulsive fuzzy cellular neural networks. Criteria are obtained for the existence and exponential stability of a unique equilibrium of fuzzy cellular neural networks impulsive state displacements at fixed instants of time.

1

Introduction

Fuzzy cellular neural networks (FCNN) is a generalization of cellular neural networks (CNN) by using fuzzy operations in the synaptic law calculation allowing us to combine the low level information processing capability of CNN’s with the high level information processing capability, such as image understanding, of fuzzy systems. Yang el al. in [13]-[15] introduced FCNN and investigated existence and stability of equilibrium point of FCNN. After the introduction of FCNN, some researchers studied stability of FCNN with constant time delays and time-varying delays (see [9]), others considered stability of FCNN with distributed delays (see [4], [6]) and exponential stability of FCNN with diffusion effect (see [5]). However, in real world, many evolutionary processes are characterized by abrupt changes at certain time. These changes are called to be impulsive phenomena, which are encountered in many fields such as physics, chemistry, population dynamics, optimal control, etc. In this paper, we will study FCNN model incorporating impulses: n n n    dxi (t) = −di xi (t) + bij μj + Ii + αij fj (xj (t)) + Tij μj dt j=1 j=1 j=1

+

n  j=1

βij fj (xj (t)) +

n 

Hij μj ,

t = tk ,

j=1

n x(t+ 0 ) = x0 ∈ R , − Δxi (tk ) = xi (t+ k ) − xi (tk ) = −γik xi (tk ),

i = 1, · · · , n,

k = 1, 2, · · · , (1)

where αij , βij , Tij and Hij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feedJ. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 243–248, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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forward  MAX template respectively; bij are elements of feed-forward template; and denote the fuzzy AND and fuzzy OR operation respectively; xi , μi and Ii denote state, input and bias of the ith neurons respectively; fi is the activation − function; Δxi (tk ) = xi (t+ k ) − xi (tk ), k = 1, 2, · · · , are the impulses at moments tk and t1 < t2 < · · · is a strictly increasing sequence such that limk→∞ tk = ∞. As usual in the theory of impulsive differential equations, at the points of discontinuity tk of the solution t → xi (t) we assume that xi (tk ) = xi (t− k ). According to (1), there exist the limits x˙ i (t− ˙ i (t+ ˙ i (tk ) = x˙ i (t− k ) and x k ), we assume x k ). In this paper, we assume that H: fi is a bounded function defined on R and satisfies |fi (x) − fi (y)| ≤ li |x − y|,

i = 1, · · · , n,

(2)

for any x, y ∈ R.

2

Main Results

In order to obtain the main results on the existence and stability of the equilibrium point of FCNN with impulses, we would like to cite the following lemma first. Lemma 1. ([14]). For any aij ∈ R, xj , yj ∈ R, i, j = 1, · · · , n, we have the following estimations, |

n 

aij xj −

j=1

and |

n 

n 

aij yj | ≤

j=1

aij xj −

j=1

n 



(|aij | · |xj − yj |)

(3)

(|aij | · |xj − yj |)

(4)

1≤j≤n

aij yj | ≤

j=1

 1≤j≤n

Now, we are ready to state and prove the main results. First, we establish sufficient conditions to guarantee the existence and uniqueness of the equilibrium point for the system (1). Theorem 1. Suppose that assumption H is satisfied. Suppose further that (i) di > 0; (ii) the following inequalities hold: di − li

n  j=1

|αji | − li

n 

|βji | > 0,

i = 1, · · · , n;

j=1

(iii) γik = 0, i = 1, · · · , n, k = 1, 2, · · · ; Then impulsive FCNN (1) exists equilibrium point x∗ = (x∗1 , · · · , x∗n ).

(5)

Stability of Fuzzy Cellular Neural Networks with Impulses

245

We can similarly prove this theorem by imitating the proof of Theorem 2.1 in [2] through constructing a contraction map, then by the contraction mapping principle, there exists a unique fixed point of the map. Existence of a unique solution of system (1) follows. Theorem 2. Assume H is satisfied and the two conditions: (i) and (ii) in Theorem 1 are satisfied too. Further assume that the impulsive operators Ii (xi (t)) satisfy Ii (xi (t)) = −γik (xi (tk ) − x∗i ),

0 < γik < 2,

i = 1, · · · , n,

k ∈ Z +.

(6)

where x∗ is the unique equilibrium point of system (1). The proof of the theorem is similar to the proof of Theorem 1. An additional difference is the consideration of the impulse effect. From the imposed condition on impulsive operator, it is clear that constructing mapping is still contracted on those discontinuous points. Thus, the results follow. We omit the rigorous proofs for Theorem 1 and Theorem 2. Theorem 3. Assume that all conditions of Theorem 2 hold. Then there exists a positive constant λ such that all solutions of system (1) satisfy the following inequality: n n   |xi (t) − x∗i | ≤ e−λt |xi (0) − x∗i | t > 0. (7) i=1 ∗

where x =

(x∗1 , · · · , x∗n )T

i=1

denotes the equilibrium.

Proof. By x∗ is the equilibrium of (1), we have the following: n  d (xi (t) − x∗ ) = −di (xi (t) − x∗ ) + αij (fj (xj (t)) − x∗ ) dt j=1

+

n 

βij (fj (xj (t)) − x∗ )

j=1

(8) for i = 1, 2, · · · , n, t > 0, t  = tk , k ∈ Z + , and hence by assumption H and Lemma 1, n  d+ |xi (t) − x∗ | ≤ −di |xi (t) − x∗ | + αij lj |xj (t) − x∗ | dt j=1

+

n 

βij lj |xj (t) − x∗ |

j=1

≤ −di |xi (t) − x∗ | +

n  j=1

|αij |lj |xj (t) − x∗ |

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+

n 

|βij |lj |xj (t) − x∗ |

j=1

(9) d+ dt

for i = 1, 2, · · · , n, t > 0, t  = tk , k ∈ Z + , where Also, from the conditions, we have

so

is the upper right derivative.

∗ ∗ ∗ xi (t+ k ) − xi = xi (tk ) + Ii (x(tk )) − xi = (1 − γik )(xi (tk ) − xi ),

(10)

∗ ∗ ∗ |xi (t+ k ) − xi | = |(1 − γik )(xi (tk ) − xi )| ≤ |xi (tk ) − xi |,

(11)

for i = 1, · · · , n, k ∈ Z . Let us define the Lyapunov function V (·) by +

V (t) = V (x1 , x2 , · · · , xm )(t) =

n 

|xi (t) − x∗i |,

(12)

i=1

for t > 0 and by (9), we can obtain  d+ d+ V (t) = |xi (t) − x∗i | dt dt i=1 n

≤

n 

{−di |xi (t) − x∗ | +

i=1

+

=

n 

|αij |lj |xj (t) − x∗ |

j=1

n 

|βij |lj |xj (t) − x∗ |}

j=1 n 

n 

i=1

j=1

{−di + li

|αji | + li

n 

|βji |}|xi (t) − x∗ |

j=1

(13) By the condition: di − li

n 

|αji | − li

j=1

n 

|βji | > 0,

i = 1, · · · , n;

(14)

i = 1, · · · , n.

(15)

j=1

there exists a positive number λ, such that di − li

n  j=1

|αij | − li

n 

|βij | ≥ λ,

j=1

Thus, by (13) and (15), we have the following: d+ V (t) ≤ −λV (t), dt

t > 0, t = tk .

(16)

Stability of Fuzzy Cellular Neural Networks with Impulses

247

Also, we have the following: V (t+ k) =

n 

∗ |xi (t+ k ) − xi | ≤

i=1

n 

|xi (tk ) − x∗i | = V (tk ),

k ∈ Z +.

(17)

i=1

Now, using the stability theorem in [11] and (16), (17), we obtain d+ V (t) ≤ −λV (t), dt Therefore, we have

V (t) ≤ e−λt V (0),

t > 0.

(18)

t > 0.

(19)

i.e., n 

|xi (t) − x∗i | ≤ e−λt

i=1

n 

|xi (0) − x∗i |,

t > 0.

(20)

i=1

Thus, we have completed the proof of this theorem.

3

Conclusion

In this paper, we discuss the existence and exponential stability of the equilibrium of FCNN with impulses. Several sufficient conditions set up here are easily verified.

Acknowledgments The first author is grateful for the support of Texas A&M University at Qatar.

References 1. Cao, J., Wang, J., Liao, X.: Novel Stability Criteria of Delayed Cellular Neural Networks. International Journal of Neural Systems 13(2003)365-375 2. Gopalsamy, K.: Stability of Artificial Neural Networks with Impulses. Applied Mathematics and Computation 154(2004)783-813 3. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Cambridge University Press, Cambridge (1999) 4. Huang, T., Zhang, L.: Exponential Stability of Fuzzy Cellular Neural Networks. In: Wang, J., Liao X., Yi Z. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg, New York (2005)168-173 5. Huang, T.: Exponential Stability of Delayed Fuzzy Cellular Neural Networks with Diffusion, to appear in Chaos, Solitons and Fractrals. 6. Huang, T.: Exponential Stability of Fuzzy Cellular Neural Networks with Unbounded Distributed Delay, to appear in Physics Letters A.

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7. Li, Y.: Global Exponential Stability of BAM Neural Networks with Delays and Impulses. Chaos, Solitons and Fractals 24(2005)279-285 8. Li, C., Liao, X., Zhang, R.: Impulsive Synchronization of Nonlinear Coupled Chaotic Systems. Physics Letters A 328(2004)47-50 9. Liu, Y., Tang, W.: Exponential Stability of Fuzzy Cellular Neural Networks with Constant and Time-varying Delays. Physics Letters A 323(2004)224-233 10. Liao, X.F., Wong, K, Li, C: Global Exponential Stability for a Class of Generalized Neural Networks with Distributed Delays. Nonlinear Analysis: Real World Applications 5(2004)527-547 11. Samoilenko A., Perestyuk N.: Impulsive Differential Equations, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, Vol. 14. World Scientific, Singapore (1995) 12. Yang, T., Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Theory. In Proc. of IEEE International Workshop on Cellular Neural networks and Applications (1996)181-186 13. Yang, T. , Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Applications. In Proc. of IEEE International Workshop on Cellular Neural Networks and Applications (1996)225-230 14. Yang, T., Yang, L.B.: The Global Stability of Fuzzy Cellular Neural Network. Circuits and Systems I: Fundamental Theory and Applications 43(1996)880 - 883 15. Yang, X., Liao, X., Evans, D., Tnag, Y.: Existence and Stability of Periodic Solution in Impulsive Hopfield. Neural Networks with Finite Distributed Delays 343(2005)108-116

Absolute Stability of Hopfield Neural Network Xiaoxin Liao1 , Fei Xu2 , and Pei Yu2 1 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China xiaoxin [email protected] 2 Department of Applied Mathematics, The University of Western Ontario, London, Ontario, N6A 5B7, Canada [email protected], [email protected]

Abstract. This paper presents some new results for the absolute stability of Hopfield neural networks with activation functions chosen from sigmoidal functions which have unbounded derivatives. Detailed discussions are also given to the relation and difference of absolute stabilities between neural networks and Lurie systems with multiple nonlinear controls. Although the basic idea of the absolute stability of neural networks comes from that of Lurie control systems, it provides a very useful practical model for the study of Lurie control systems.

1

Introduction

As we know, the fundamental theoretical basis for Hopfield neural networks [1] being widely used is because it uses electric circuit simulations to solve nonlinear algebraic or transcendental equations, and the process can be automated. This great advantage has attracted many researchers to study neural networks. However, in general, when a neural network has multiple equilibrium points, it is very difficult to solve global optimization problems. To solve a global optimization problem based on neural networks, it is better for the system to have unique equilibrium point which is yet globally attractive. Based on this idea, Forti et al. [2], [3] first proposed the concept of absolute stability of neural networks. That is, for a given type of activation functions (usually S-type functions (sigmoidal functions) or L-type functions satisfying Lipschitz condition), for an arbitrarily chosen activation function g(x) and an arbitrary input current I, the equilibrium point of the network is unique, and local stable in the sense of Lyapunov and globally attractive. The absolute stability for Hopfield neural networks is defined in [2], in which the sufficient and necessary condition for S-type functions was first obtained, and was later generalized to a general class of neural networks [4], [5]. For L-type functions, on the other hand, the sufficient condition for exponentially absolute stability was given in [5], and later sufficient and necessary condition was obtained [6]. In this paper, we will present some new results for the exponentially absolute stability of Hopfield neural networks with activation functions chosen from Stype functions which have unbounded derivatives. The results are different from that reported in the literature so far. We have also improved the results presented J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 249–254, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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in the above mentioned publications, but, due to the page limit, will not discuss in this paper. We will discuss the relation and difference of the absolute stabilities between neural networks and Lurie systems with multiple nonlinear controls. It will be shown that on one hand, the absolute stability problem of neural networks provides a practically useful new model for further study of Lurie systems with multiple nonlinear feedback controls, while On the other hand, the theory and methodology of absolute stability of Lurie control systems are helpful in the study of absolute stability of neural networks.

2

Relation and Difference Between Hopfield Neural Networks and Lurie Systems with Multiple Nonlinear Controls

Consider the following Hopfield neural networks: ui  + Tij gj (uj ) + Ii , Ri j=1 n

ci u˙ i = −

(1)

where the dot denotes differentiation with respect to time t, u ∈ Rn , T = (Tij )n×n is a constant matrix, g(u) = (g1 (u1 ), . . . , gn (un ))T : Rn → Rn is a nonlinear diagonal mapping, I = (I1 , . . . , In )T ∈ Rn is a constant vector. gj (uj ) : R → R is C 1 function and gj (uj ) > 0 satisfying gi (R) ≡ (ai , bi ), ai , bi ∈ R, ai < bi . For convenience in comparison of Hopfield neural networks and Lurie systems with multiple nonlinear controls, we first introduce the transformations: di = Tij 1 Ci Ri and bij = ci into Eq. (1) to obtain u˙ i = −di ui +

n 

bij gj (uj ) +

j=1

Ii . Ci

(2)

Since the original system (1) assumes the function gj (uj ) being a sigmoidal type, we have gi ∈ F∞ = {gi |0 < ui gi (ui ) < ∞, gi is continuous and gi (0) = 0} . Therefore, the nonlinear functions in neural networks and Lurie control systems are basically same. More precisely, the nonlinear functions in neural networks require that the functions are monotone and have the maximum slope at the origin. So strictly speaking, the nonlinear functions in neural networks belong to a subset of the nonlinear functions in Lurie control systems. Let ui = u∗i be an equilibrium point of (2), and f (xi ) = gi (ui ) − gi (u∗i ) = gi (xi + u∗i ) − gi (u∗i ). Then (2) can be rewritten as x˙ i = −di xi +

n  j=1

bij fj (xj ).

(3)

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251

Since fj (0) = gi (u∗i ) − gi (u∗i ) = 0, and fj = gj > 0, we have fj ∈ F∞ = {fj |0 < xj fj < ∞, fj (0) = 0 fj is continuous} . Remarks: 1) Hopfield neural network is actually a particular case of the more general Lurie systems with multiple nonlinear controls, given by [7][8] x˙ i =

n  j=1

aij xj +

n  j=1

bij fj (σj ),

σj =

n 

ci xi ,

(4)

i=1

in which letting aij = 0, i  = j, aii = −di < 0, and σj = xj , cj = 1, ci = 0, i  = j yields the Hopfield neural network (3). 2) For the Lurie control system (4), it does not require aii < 0 for all i, nor does the nonlinear function fj (σj ) have to be monotone. It only requires σi fk (σi ) ≥ 0 (i.e., the function is located in the first and third quadrants of the σ-f (σ) plane), and is thus more general than the neural network (3). 3) The study of the Lurie systems with multiple nonlinear controls is far behind that of the Lurie systems with single control. As pointed out in a SIAM conducted research report (1988) The Future of Control Theory – Mathematical Prospect that although the study on the stability of nonlinear control systems has received considerable attentions and many results have been obtained, most of achievements are limited to the systems with single control, such as Popov criterion, Lyapunov method, etc. The problem for the Lurie systems with multiple controls has not been solved with satisfactory. This indicates the difficulty in generalizing the study from the case of single control to multiple controls. 4) Note that the Hopfeild neural network (3) contains an input Ii , while the input in the Lurie system (4) is zero. However, this non-zero input can be removed by a simple translation and thus (3) becomes (4). Since the activation functions gi (ui ) and fi (xi ) are the same type of functions, the absolute stabilities of the equilibrium point u = u∗ of (3) and x = 0 of (4) are the same. The study of the global stability of the neural network (3) has opened a new area for optimal computation. Therefore, it is natural to combine the studies on the stability of neural networks and the absolute stability of Lurie control systems.

3

Some Results on Hopfield Neural Networks with Infinite Gains to Be Absolutely Exponentially Stable

In this section, we will consider sufficient conditions for the absolute stability of Hopfield neural networks with a class of sigmoidal nonlinear simulation functions (i.e., the simulation functions have unbounded gains). Let SUB := {g(x)|g(x) ∈ C[R, R], D+ gi (ui ) ≥ 0, i = 1, · · · , n}, i.e., sup D+ gi (ui ) < ∞. In the following, we consider (2) with gi (ui ) ∈ SUB .

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Definition 1. ∀gi ∈ SUB and ∀Ii ∈ R, if the equilibrium point of (2) is globally exponentially stable, then it is said that system (2) is absolutely exponentially stable with respect to SUB . Theorem 1. If there exist two groups of constants ξi > 0, ηi > 0, i = 1, · · · , n such that the matrix A is negative definite, then we have the following conclusions: 1) system (2) is absolutely exponentially stable; and 2) the Lyapunov exponent is − λμ , where   A11 A12 A= , AT12 A22 2n×2n

  A11 = diag − Rξ11 , · · · , − Rξnn ,

η1 ηn A12 = diag(− 2R , · · · , − 2R )n×n 1 n

A22 = (

+(  B=

ηi Tij +ηj Tji )n×n , (i = 1, · · · , n), 2

ξi Tij +ξj Tji )n×n , 2

B11 B12 T B12 B22

T B12 = B12 =



 ,

B11 = diag

2n×2n diag( C12η1 , · · · , Cn2ηn )n×n ,

C1 ξ1 Cn ξn 2 ,···, 2

 , n×n

B22 = On×n .

−λ denotes the maximum eigenvalue of matrix A and μ is the maximum eigenvalue of matrix B, the superscript T denotes transpose. Proof. Since g ∈ S, the existence of equilibrium point of system (2) is a wellknown result. Let u = u∗ be an equilibrium point of (2). 1) Let x = (x1 , · · · , xn )T = (u1 − u∗1 , · · · , un − u∗n )T . Then fi (xi ) = gi (xi + u∗i ) − gi (u∗i ), and system (2) can be rewritten as Ci x˙ i =

n 

Tij fj (xj ) −

j=1

xi Ri

(i = 1, · · · , n).

(5)

The global stability of the equilibrium u = u∗ of system (2) is equivalent to the equilibrium x = 0 of system (5). Construct the radially unbounded Lyapunov function: V (x) =

n  Ci ξi i=1

2

x2i

+

n  i=1

 ηi Ci

xi

fi (xi )dxi .

(6)

0

2) Set V ∗ = et V,  > 0 is a sufficiently small constant. Then we have dV ∗   ≤0 dt (5)

for 0 <  1.

(7)

Absolute Stability of Hopfield Neural Network

253

Hence, n  Ci ξi i=1

2

x2i ≤ V (x(t)) ≤ e−t V0∗ ,

(8)

which results in n  i=1

x2i ≤

V0∗ min1≤i≤n

Ci ξi 2

( > 0).

(9)

Inequality (9) implies that the equilibrium x = 0 of system (5), or the equilibrium u = u∗ of system (2) is globally exponentially stable, i.e., system (2) is absolutely exponentially stable. Also, the positive constant  can be considered as a Lyapunov exponent. 3) Let −λ, μ be the maximum eigenvalues of matrices A, B, respectively. n

n   dV ∗  t 2 2 xi + fi (xi ) ≤ 0. (10)  ≤ e (−λ + μ) dt (5) i=1 i=1 Thus, the inequality (8) holds and  = λμ is an estimation of the Lyapunov exponent. The is complete.  proof  ◦ ◦ A A B◦ B◦ 11 12 Let A◦ = , and B ◦ = [ ◦ 11 ◦ 12 ]2n×2n , where ◦ ◦ A 21 A 22 B 21 B 22   ◦ ◦ 1 1 Tij + Tji = −diag , · · · , , = A11 A22 R1 Rn 2 n×n,   ◦ 1 1 Tij + Tji ,···, + A12 = −diag 2R1 2Rn n×n 2 n×n,  ◦ C1 Cn ,···, B 11 = diag 2 2 n×n,  T ◦ ◦ ◦ C1 Cn ,···, B 12 = B 12 = diag B 22 = On×n. 2 2 n×n, Corollary 1. If matrix A◦ is negative definite, then the equilibrium u = u∗ of system (2) is unique and globally exponentially stable, and the Lyapunov exponential can be chosen as  = λ◦ /μ, where −λ◦ and μ are the maximum eigenvalues of matrices A◦ and B ◦ , respectively. Proof. Taking ξi = ηi = 1 (i = 1, · · · , n) in Theorem 1 leads to the corollary. Theorem 2. If there exist two groups of constants ξi > 0, ηi > 0, (i = 1, · · · , n) such that the following matrix G is negative definite, then the conclusion of theorem 1 holds, where     G11 G12 G= , G11 = −diag Rξ11 , · · · , Rξnn , T G12 G22 2n×2n     ξ T +ξ T η T +η T G12 = GT12 = (1 − δij ) i ijR1 j ji , G22 = i ij 2 j ji . 2n×2n

n×n

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Also the estimation (9) is satisfied, where  = min min1≤i≤n is the maximum eigenvalue of matrix G, and μ∗ =

ηi Ri

−ξi Tii

ηi Ci Ci ξi max1≤i≤n 2 .

 ∗ , μλ∗ , −λ∗

Proof. From the negative definite property of matrix G, we know that Tii < 0, (i = 1, · · · , n). Using Lyapunov function (6), The remaining part of the proof is similar to Theorem 1.

4

Conclusion

In this paper, we have presented some new results for the absolutely exponential stability of Hopfield neural networks which have activation function chosen from sigmoidal functions with unbounded derivatives. We have also discussed the relation and difference of the absolute stabilities between neural networks and Lurie systems with multiple nonlinear controls. It has been shown that the basic idea of the absolute stability of neural networks is the same as that of Lurie control systems, and thus the theory and methodology of Lurie control systems can promote the study of absolute stability of neural networks.

Acknowledgement This work was supported by the Natural Science Foundation of China (NSFC No. 60274007 and No. 60474001), the Premier’s Research Excellence Award (PREA), and the Natural Science and Engineering Research Council of Canada (NSERC No. R2686A02).

References 1. Tank, D.W., Hopfield, J.J.: Simple Neural Optimization Networks: An A/O Converter Signal Decision Circuit and a Linear Programming Network. IEEE Trans. Circuits Syst. 23(5) (1986) 533-541 2. Forti, M., Manetti, S., Marini, M.: Necessary and Sufficient Condition for Absolute Stability of Neural Networks. IEEE Trans. Circuits Syst. 41(7) (1994) 491-494 3. Forti, M., Liberatore, S.A., Manetti, J., Marini, M.: On Absolute Stability of Neural Networks. Proc. 1994 IEEE Int. Symp. Circuits Syst. 241-244 4. Kaszkurewicz, E., Bhaya, A.: Comments on ‘Necessary and Sufficient Condition for Absolute Stability of Neural Networks’. IEEE Trans. Circuits Syst. I 42 (1995) 497-499 5. Liang, X., Wang, J.: An Adaptive Diagonal Stability Condition for Absolute Exponential Stability of a General Class of Neural Networks. IEEE Trans. Circuits Syst. 48(11) (2001) 1038-1317 6. Hu, S., Wang, J.: Absolute Exponential Stability of a Class of Continuous Time Recurrent Neural Networks. IEEE Trans. Neural Networks 14(1) (2003) 35-45 7. Liao, X.X.: Absolute Stability of Nonlinear Control Systems. Kluwer Academic China Science Press, Beijing (1993) 8. Liao, X.X., Yu, P.: Sufficient and Necessary Conditions for the Absolute Stability of Time-Delayed Lurie Control Systems. J. Math. Anal. Appl. (2006) (in press)

Robust Stability Analysis of Uncertain Hopfield Neural Networks with Markov Switching Bingji Xu and Qun Wang School of Information Engineering, China University of Geosciences, Beijing, 100083, China [email protected]

Abstract. The robust stability of uncertain Hopfield neural networks with Markov switching is analyzed, the parametric uncertainty is assumed to be norm bounded. Sufficient conditions for the exponential stability are established by constructing suitable Lyapunov functionals. The stability criteria represented in terms of linear matrix inequalities (LMIs), and are computationally efficient.

1

Introduction

In the past two decades, neural networks has been received considerable attention, and there have been extensive research results presented about the stability analysis of neural network and its applications (See,e.g.,[1-6]). The stability research related to Hopfield neural networks have been extensively studied and developed in recent years (e.g.,[7,8]). As is well known, parametric uncertainty occur due to the modelling inaccuracies and/or changes in the environment of the model, which can affect the stability of a network by creating oscillatory and unstable characteristics. It is important to investigate the robust stability of neural networks with parametric uncertainties. With the development of intelligent control,switched systems have been widely studied. In this paper, we will study a class of switched Hopfield neural networks, the individual subsystems of the switched Hopfield neural networks are a set of Hopfield neural networks, and a Hopfield neural networks switched to the next one according to a continuous Markov process with a finite state space. In addition, the parametric uncertainty is considered and assumed to be norm bounded. Lyapunov functionals are employed to investigate the sufficient conditions for the robust exponential stability. The remainder of this paper is organized as follows. In section 2, the mathematical model of the neural networks is described and some necessary assumptions are given. Section 3 is dedicated to the analysis of robust stability for the uncertain Hopfield neural networks with Markov switching. Some exponential stability criteria for these neural networks are derived in terms of LMIs. Conclusions are given in Section 4. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 255–260, 2006. c Springer-Verlag Berlin Heidelberg 2006 

256

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B. Xu and Q. Wang

Systems Descriptions

We consider the uncertain Hopfield neural networks with Markov switching described by du(t) = −[D(r(t)) + ΔD(t, r(t))]u(t) + [A(r(t)) + ΔA(t, r(t))]g(u(t)) + I(r(t)), dt (1)  T where u(t) = u1 (t), u2 (t), · · · , un (t) is the neuron state vector, and g(u) =  T g1 (u1 ), g2 (u2 ), · · · , gn (un ) is the neuron activation function vector. {r(t), t ≥ 0} is a time homogeneous Markov process with right continuous trajectories taking values from a finite set S = {1, 2, · · · , N }, with transition probabilities:  γij Δt + o(Δt) i =j P {r(t + Δt) = j|r(t) = i} = 1 + γii Δt + o(Δt) i = j, o(Δt) Δt→0 Δt

where Δt > 0, lim

= 0, and γij ≥ 0 for i  = j, while γii = −

N 

γij for

j=1,i =j

i ∈ S.   For each r(t) = i ∈ S, D(r(t)) = diag d1 (r(t)), d2 (r(t)), · · · , dn (r(t)) is a positive diagonal matrix, A(r(t)) is interconnection weight matrix, I(r(t)) is a constant input vector, ΔD(t, r(t)) and ΔA(t, r(t)) are unknown real valued functions representing time-varying parametric uncertainties. To simplify the notation, M (t, i) and M (i) will be denoted by Mi (t) and Mi , respectively. Throughout this paper, we assume that the parametric uncertainties ΔDi (t) and ΔAi (t) are norm-bounded and can be described as ΔDi (t) = Bi Fi (t)Ci , ΔAi (t) = Ei Gi (t)Hi , i ∈ S,

(2)

where Bi , Ci , Ei , Hi , i ∈ S are known constant matrices of appropriate dimensions, and Fi (t), Gi (t), i ∈ S are unknown matrices representing the parametric uncertainties,which satisfies FiT (t)Fi (t) ≤ I, GTi (t)Gi (t) ≤ I, ∀ t ∈ , i ∈ S,

(3)

in which I is the identify matrix of appropriate dimension. Since the matrix Di in system (1) is diagonal, it is reasonable to assume that ΔDi is also diagonal, but not necessary positive definite. The neuron activation functions gi (u), i = 1, 2, · · · , n, are continuous and satisfy the following conditions: 0≤

gi (ui ) − gi (vi ) ≤ Ki , ∀ ui  = vi , ui , vi ∈ , i = 1, 2, · · · , n, ui − vi

(4)

Let K = diag(K1 , K2 , · · · , Kn ), u∗ = (u∗1 , u∗2 , · · · , u∗n , )T be an equilibrium point of system (1), and set x(t) = u(t) − u∗ , fi (xi (t)) = gi (xi (t) + u∗i ) − gi (u∗i ), i = 1, 2, · · · , n. Then, fi (z), i = 1, 2, · · · , n, satisfies |fi (z)| ≤ Ki |z|, zfi(z) ≥ 0, ∀z ∈ , i = 1, 2, · · · , n.

(5)

Robust Stability Analysis of Uncertain Hopfield Neural Networks

257

System (1) may be rewritten as follows. dx(t) = −[D(r(t)) + ΔD(t, r(t))]x(t) + [A(r(t)) + ΔA(t, r(t))]f (x(t)), dt or dx(t) = −[Di + ΔDi (t)]x(t) + [Ai + ΔAi (t)]f (x(t)), i = 1, 2, · · · , N. dt

3

(6)

(7)

Exponential Stability

If u∗ is an equilibrium point of system (1), then x = 0 is an equilibrium point of system (6) and system (7). Definition 1. The trivial solution for system (6) is said to be exponentially stable in mean square if for all uncertainties ΔDi (t), ΔAi (t) satisfies (2) and (3), lim sup

t→∞

  1 log Ex(t)2 < 0, t ≥ 0. t

Definition 2. The trivial solution for system (6) is said to be almost surely exponentially stable if for all uncertainties ΔDi (t), ΔAi (t) satisfies (2) and (3), lim sup

t→∞

  1 log x(t) ≤ −r, r > 0, t ≥ 0, a.s. t

Theorem 3.1. If there exist matrices Pi > 0, diagonal matrices Qi = diag(qi1 , qi2 , · · · , qin ) > 0, and positive scalars εi , μi , i = 1, 2, · · · , N , such that the following LMIs hold for i = 1, 2, · · · , N, ⎡ ⎤ Φi (1, 1) Pi Ai − Qi Di −Pi Bi Pi Ei ⎢ ATi Pi − Di Qi Qi Ai + ATi Qi + μi HiT Hi −Qi Bi Qi Ei ⎥ ⎥ < 0, Φi = ⎢ ⎣ −BiT Pi −BiT Qi −εi I 0 ⎦ EiT Pi EiT Qi 0 −μi I in which Φi (1, 1) =

N  j=1

  γij KQj + Pj − Pi Di − Di Pi + εi CiT Ci . Then, the trivial

solution for system (6) is both exponentially stable in mean square and almost surely exponentially stable. Proof. Take the Lyapunov functional T

V (x, i) = x Pi x + 2

n 



xj

qij

j=1

fj (s)ds. 0

It is easy to verify that     min λmin (Pi ) x2 ≤ V (x, i) ≤ max λmax (Pi ) + λmax (KQi ) x2 . 1≤i≤N

1≤i≤N

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The infinitesimal operator L for the Lyapunov functional V (x, i) at point {x(t), t, r(t) = i} is given by [9]  N   ∂V ∂V  LV (x, i) = + x(t) ˙  + γij V (t, j) ∂t ∂t r(t)=i j=1 = −xT (Pi Di + Di Pi )x + 2xT Pi Ai f (x) − 2xT Pi Bi Fi (t)Ci x +2xT Pi Ei Gi (t)Hi f (x) − 2xT Qi Di f (x) + 2f T (x)Qi Ai f (x) −2f T Qi Bi Fi (t)Ci x + 2f T (x)Qi Ei Gi (t)Hi f (x)  xl N N n    + γij xT Pj x + 2 γij qjl fl (s)ds. j=1

j=1

(8)

0

l=1

It follows from (3) and (5) that 2

n 

 qjl

xl

fl (s)ds ≤ xT KQj x,

(9)

0

l=1

[Fi (t)Ci x]T [Fi (t)Ci x] ≤ xT CiT Ci x, [Gi (t)Hi f (x)] [Gi (t)Hi f (x)] ≤ f T

T

(10)

(x)HiT Hi f (x).

(11)

Substituting (9),(10) and (11) into (8), we obtain LV (x, i) ≤ −xT (Pi Di + Di Pi )x + 2xT Pi Ai f (x) − 2xT Pi Bi Fi (t)Ci x +2xT Pi Ei Gi (t)Hi f (x) − 2xT Qi Di f (x) + 2f T (x)Qi Ai f (x) −2f T Qi Bi Fi (t)Ci x + 2f T (x)Qi Ei Gi (t)Hi f (x) +

+

N 



N 

γij xT Pj x

j=1

 γij xT KQj x + εi xT CiT Ci x − [Fi (t)Ci x]T [Fi (t)Ci x]

j=1

  +μi f T (x)HiT Hi f (x) − [Gi (t)Hi f (x)]T [Gi (t)Hi f (x)] = y T Φi y,

(12)

 T where y = xT , f T (x), xT CiT FiT (t), f T (x)HiT GTi (t) . By Φi < 0, i = 1, 2, · · · , N, we know that there exist a scalar δ > 0 such that LV (x, i) ≤ −δx2 . Thus, by [9] we known that the trivial solution for system (6) is both exponentially stable in mean square and almost surely exponentially stable.This completes the proof. Theorem 3.2. If there exist matrices Pi > 0, diagonal matrices Ri = diag(ri1 , ri2 , · · · , rin ) > 0, and positive scalars εi , μi , i = 1, 2, · · · , N , such that the following LMIs hold for i = 1, 2, · · · , N,

Robust Stability Analysis of Uncertain Hopfield Neural Networks

⎡

N 

⎢ j=1 ⎢ Ψi = ⎢ ⎢ ⎣

γij Pi − Pi Di − Di Pi + εi CiT Ci ATi Pi + KRi −BiT Pi EiT Pi

259

⎤ −Pi Bi Pi Ei ⎥ ⎥ μi HiT Hi − 2KRi 0 0 ⎥ ⎥ < 0. 0 −εi I 0 ⎦ 0 0 −μi I Pi Ai + Ri K

Then, the trivial solution for system (6) is both exponentially stable in mean square and almost surely exponentially stable. Proof. Take the Lyapunov functional V (x, i) = xT Pi x. By (8), one has LV (x, i) = −xT (Pi Di + Di Pi )x + 2xT Pi Ai f (x) − 2xT Pi Bi Fi (t)Ci x +2xT Pi Ei Gi (t)Hi f (x) +

N 

γij xT Pj x.

(13)

j=1

It follows from (5) that   fj (xj ) Kj xj − fj (xj ) ≥ 0, j = 1, 2, · · · , n. This yields LV (x, i) ≤ −xT (Pi Di + Di Pi )x + 2xT Pi Ai f (x) − 2xT Pi Bi Fi (t)Ci x n    +2xT Pi Ei Gi (t)Hi f (x) + 2 rij fj (xj ) Kj xj − fj (xj )

+

N 

j=1

 T

T

γij x Pj x + εi x

CiT Ci x

 − [Fi (t)Ci x] [Fi (t)Ci x] T

j=1

  +μi f T (x)HiT Hi f (x) − [Gi (t)Hi f (x)]T [Gi (t)Hi f (x)] = y T Ψi y,

(14)

 T where y = xT , f T (x), xT CiT FiT (t), f T (x)HiT GTi (t) . The remainder of the proof is similar to that of Theorem 3.1. The proof is complete. Remark. If the parametric uncertainties ΔDi (t) and ΔAi (t) are norm-bounded and can be described as ΔDi (t) ≤ αi , ΔAi (t) ≤ βi , i ∈ S, we can obtain the criteria for the √ exponential stability of systems (6) by let √ Bi = Ci = αi I and Ei = Hi = βi I.

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Conclusions

In this paper, we studied the uncertain Hopfield neural networks with Markov switching. Some sufficient conditions are derived to ensure the exponential stability in mean square and almost sure exponential stability by constructing suitable Lyapunov functionals. The stability criteria are given in terms of LMIs which can be easily solved by some existing software.

Acknowledgments This paper is supported by National Science Foundation of China under Grant 60474011 and 60574025.

References 1. Liao, X.X., Wang, J., Zeng, Z.G.: Global Asymptotic Stability and Global Exponential Stability of Delayed Cellular Neural Networks. IEEE Trans. on Circuits and Systems II 52 (2005) 403-409 2. Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. on Circuits and Systems I 51 (2004) 2313-2324 3. Zeng, Z.G., Wang, J., Liao, X.X.: Global Asymptotic Stability and Global Exponential Stability of Neural Networks With Unbounded Time-Varying Delays. IEEE Trans. on Circuits and Systems II 52 (2005) 168-173 4. Shen, Y., Zhao, G.Y., Jiang, M.H., Mao, X.R.: Stochastic Lotka-Volterra Competitive Systems with Variable Delay. In: Huang, D.S., Zhang, X.P., Huang, G.B. (eds.): Advances in Intelligent Computing. Lecture Notes in Computer Science, Vol. 3645. Springer-Verlag, Berlin Heidelberg New York (2005) 238-247 5. Shen, Y., Zhao, G.Y., Jiang, M.H., Hu, S.G.: Stochastic High-order Hopfield Neural Networks. In: Wang, L.P., Chen, K., Ong, Y.S. (eds.): Advances in Natural Computation. Lecture Notes in Computer Science, Vol. 3610. Springer-Verlag, Berlin Heidelberg New York (2005) 740-749 6. Shen, Y., Jiang, M.H., Liao, X.X.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Time-varying Delays and Continuously Distributed Delays. In: Wang, J., Liao, X.F., Zhang, Y. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 156-161 7. Xu, B.J., Liu, X.Z., Liao, X.X.: Global Asymptotic Stability of High-Order Hopfield Type Neural Networks with Time Delays. Computers and Mathematics with Applications 45 (2003) 1729-1737 8. Liu, X.Z., Teo, K.L., Xu, B.J.: Exponential Stability of Impulsive High-Order Hopfield-Type Neural Networks With Time-Varying Delays. IEEE Trans. on Neural Networks 16 (2005) 1329 - 1339 9. Mao, X.R.: Stability of Stochastic Differential Equations With Markovian Switching. Stochastic Processes and Their Applications 79 (1999) 45-67

Asymptotic Stability of Second-Order Discrete-Time Hopfield Neural Networks with Variable Delays Wei Zhu1,2 and Daoyi Xu2 1 Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, 400065, Chongqing, China [email protected], [email protected] 2 Institute of Mathematics, Sichuan University, 610064, Chengdu, China

Abstract. This paper studies the problem of asymptotic stability of second-order discrete-time Hopfield neural networks with variable delays. By utilizing inequality techniques, we obtain sufficient conditions for the existence and asymptotic stability of an equilibrium point and estimate the region of existence and the attraction domain of the equilibrium point. A numerical example is given to illustrate our theoretical results.

1

Introduction

Hopfield neural networks with time delays have been extensively investigated over the years, and various sufficient conditions for the stability of the equilibrium point of this class of neural networks have been presented in [1]-[3] and the references cited therein. However, there are very few results on the stability of the equilibrium point for high order Hopfield type neural networks with time delays ([4]-[8]),. Especially, for the high order discrete-time Hopfield neural networks, to the best of our knowledge, no results related have appeared in literature. From the fact that high order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks and S. Mohamad et al. s papers (see, [9], [10]) on both continuous and discrete neural networks, we find the necessary of studying the stability of high order discrete neural networks. So, in this paper, by utilizing inequality techniques, we present sufficient conditions for the existence and asymptotic stability of an equilibrium point and estimate the region of existence and the attraction domain of the equilibrium point for second-order discretetime Hopfield neural networks. A numerical example is given to illustrate our theoretical results.

2

Preliminary

Consider the following second-order discrete-time Hopfield neural networks x(k + 1) = A0 x(k) + Ag(x(k − τ1 (k))) + G(x(k − τ2 (k)))Bg(x(k)) + J, J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 261–266, 2006. c Springer-Verlag Berlin Heidelberg 2006 

(1)

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with k = 0, 1, · · · , where x(k) = col{xi (k)} ∈ Rn , A0 = diag{ai } ∈ Rn×n , A = (aij )n×n , g(x(k)) = col{gj (xj (k))} ∈ Rn , J = col{Ji } ∈ Rn , xi (k) is the output of the ith neuron; aij are the first-order synoptic weights of the neural networks, which are not necessary symmetric; and Ji is the external input of the ith neuron; τ1 (k), τ2 (k) are the delays of neuron, which satisfy 0 ≤ τ1 (k), τ2 (k) ≤ τ , τ is a positive integer; and gj : R → R is the activation function and satisfies gj (0) = 0; G(x(k)) and B T are n × n2 matrices of the forms G(x(k)) = [G1 (x(k)), · · · , Gn (x(k))], B T = [B1 , · · · , Bn ], with Gi (x(k)) being the matrix whose ith row is g T (x(k)) = (g1 (x1 (k)), · · · , gn (xn (k))), and the other elements are all zeros, i.e. ⎡ ⎤ 0 0 ··· 0 ⎢ ··· ⎥ ··· ··· ··· ⎢ ⎥ ⎢ ⎥ 0 0 ··· 0 ⎢ ⎥ ⎢ Gi (x(k)) = ⎢g1 (x1 (k)) g2 (x2 (k)) · · · gn (xn (k))⎥ ⎥, ⎢ ⎥ 0 0 · · · 0 ⎢ ⎥ ⎣ ··· ⎦ ··· ··· ··· 0 0 ··· 0 while ⎡

ai11 ⎢ ai21 ⎢ Bi = ⎣ ··· ain1

ai12 ai22 ··· ain2

··· ··· ··· ···

⎤ ai1n ai2n ⎥ ⎥. ··· ⎦ ainn

where aijl (i, j, l = 1, · · · , n) are the second-order synoptic weights of the neural networks. In this paper, we will use the following notations. A solution of (1) with initial function φ(k) = col{ψi (k)} (k = −τ, · · · , 0) is denoted by x(k, 0, φ) or simply by x(k) if no confusion should occur. N (a) = {a, a + 1, · · · }, N (a, b) = {a, a + 1, · · · , b}. C = {φ | φ : N (−τ, 0) → Rn }. For vector x ∈ Rn and matrix n 1 1 A ∈ Rn×n , we define ||x|| = { i=1 x2i } 2 , ||A|| = {λmax (AAT )} 2 , where λmax (·) is the largest eigenvalues of the relevant matrix, clearly, the vector norm and matrix norm defined as above are adopted, and under the above choice of norm, we have ||G(x(k))|| = ||g(x(k))||. Throughout this paper, we always suppose that (H1) ||g(u) − g(v)|| ≤ L||u − v||, for all u, v ∈ Rn , L > 0. (H2) 1 − ||A0 || − ||A||L > 0, (1 − ||A0 || − ||A||L)2 − 4||B||L2 ||J|| > 0. 2 m (H3) 0 < h < 1−||A0 ||−||A||L−2||B||L , ||B||L2 1

where m =

1−||A0 ||−||A||L−{(1−||A0||−||A||L)2−4||B||L2 ||J||} 2 2||B||L2

> 0.

Asymptotic Stability of Second-Order Discrete-Time

3

263

Existence and Uniqueness of Equilibrium Point

In this section, we first show that system (1) has a positive invariant set and then give an existence and uniqueness result about the equilibrium of system (1) in the set. Definition 1. A set S ⊂ C is called to be a positive invariant set of system (1) if for any initial value φ ∈ S, we have xk (0, φ) ∈ S for all k ∈ N (1), where xk (0, φ) = x(k + s, 0, φ), s ∈ N (−τ, 0). Theorem 1. If (H1) and (H2) hold, then the set Sm = {φ(k) | ||φ(k)|| ≤ m} is a positive invariant set of (1). Proof. It suffices to verify that ||x(k)|| ≤ m(k ∈ N (1)) holds for any ||φ(k)|| ≤ m(k ∈ N (−τ, 0)). If this is not true, then there must be some kl such that ||x(kl + 1)|| > m

and ||x(k)|| ≤ m,

k ∈ N (−τ, kl ).

(2)

Noticing that, from (1) we can derive x(k + 1) = Ak+1 x(0) + 0

k

Ak−s Ag(x(s − τ1 (s))) 0

s=0

+

k

Ak−s G(x(s − τ2 (s)))Bg(x(s)) + 0

s=0

k

Ak−s J. 0

s=0

Hence, we have m < ||x(kl + 1)|| ≤ ||A0 ||

kl +1

||x(0)|| +

kl

||A0 ||kl −s ||A|| · ||g(x(s − τ1 (s)))||

s=0

+

kl

||A0 ||kl −s ||G(x(s − τ2 (s)))|| · ||B|| · ||g(x(s))|| +

s=0

kl

||A0 ||kl −s · ||J||

s=0

≤ ||A0 ||kl +1 ||x(0)|| +

kl

||A0 ||kl −s ||A|| · L||x(s − τ1 (s))||

s=0

+

kl

||A0 ||kl −s L||x(s − τ2 (s))|| · ||B|| · L||x(s)|| +

s=0

kl

||A0 ||kl −s · ||J||

s=0

1 − ||A0 ||kl +1 1 ≤ {||A0 ||kl +1 + (||A||L + ||B||L2 m + ||J||)}m, 1 − ||A0 || m 1

since m =

1−||A0 ||−||A||L−{(1−||A0||−||A||L)2 −4||B||L2 ||J||} 2 2||B||L2

is a root of the equation

||B||L2 m2 + (||A||L + ||A0 || − 1)m + ||J|| = 0, then we have ||B||L2 m + ||A||L +

1 m ||J||

= 1 − ||A0 ||, so we can derive

m < ||x(kl + 1)|| ≤ m,

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which is a contradiction. So ||x(k)|| ≤ m for k ∈ N (−τ ), which implies that Sm is a positive invariant set of (1). Theorem 2. If (H1) and (H2) hold, then system (1) has a unique equilibrium point x∗ ∈ Rn in the positive invariant set Sm . Remark 1. The proof is trivial, so be omitted here.

4

Asymptotic Stability of Equilibrium Point

In this section, we will study the asymptotic stability and attraction domain of the equilibrium point x∗ . Let y(k) = x(k) − x∗ , then (1) becomes y(k + 1) = A0 y(k) + Af (y(k − τ1 (k))) + G(x∗ )Bf (y(k)) + F (y(k − τ2 (k)))Bg(y(k) + x∗ ), y(k) = ϕ(k), k ∈ N (−τ, 0),

k ∈ N (0),

(3)

where ϕ(k) = φ(k) − x∗ , f (y) = g(y + x∗ ) − g(x∗ ), F (y) = G(y + x∗ ) − G(x∗ ). Clearly, x∗ is asymptotically stable for (1) if and only if the equilibrium point 0 of (3) is asymptotically stable. Theorem 3. If (H1), (H2) and (H3) hold, then the set Dh = {ϕ(k) | ||ϕ(k)|| ≤ h} is a positive invariant set of (3). Remark 2. The proof is similar with that of Theorem 1, so be omitted. Theorem 4. If (H1), (H2) and (H3) hold, then the equilibrium point 0 of (3) is asymptotically stable, and the attraction domain is Dh . Proof. From Theorem 3, there must be a nonnegative constant σ such that lim sup ||y(k)|| = σ ≤ h.

(4)

k→+∞

According to the definition of superior limit, for arbitrarily small constant δ > 0, there is a K > 0, such that for any k ≥ K, ||y(k − τ )|| ≤ σ + δ. In view of the procedure of the proof of Theorem 1, for a similar argument, when k ≥ K, we obtain k ||y(k + 1)|| ≤ ||A0 ||k+1−K ||y(K)|| + ||A0 ||k−s · ||A|| · ||f (y(s − τ1 (s)))|| s=K

+

k

||A0 ||k−s · ||G(x∗ )|| · ||B|| · ||f (y(s))||

s=K

+

k

||A0 ||k−s ||F (y(s − τ2 (s)))|| · ||B|| · ||g(y(s) + x∗ )||

s=K

≤ {||A0 ||k+1−K +

1 − ||A0 ||k+1−K 1 − ||A0 ||

(||A||L + 2||B||L2 m + ||B||L2 h)}(σ + δ)

Asymptotic Stability of Second-Order Discrete-Time

265

Combining (4) with the definition of superior limit again, there are kl ≥ K, l ∈ N (1) such that limkl →+∞ ||y(kl + 1)|| = σ. Letting kl → +∞ and δ → 0, then we derive 1 σ≤ (||A||L + 2||B||L2 m + ||B||L2 h)σ (5) 1 − ||A0 || If σ = 0, then we have

1 1−||A0 || (||A||L

h≥

+ 2||B||L2 m + ||B||L2 h) ≥ 1, i.e.,

1 − ||A0 || − ||A||L − 2||B||L2 m , ||B||L2

which contradicts (H3). Hence, σ must be zero and the proof is completed.

5

An Illustrate Example

Example. Consider the following second-order discrete-time Hopfield neural networks x(k + 1) = A0 x(k) + Ag(x(k − τ1 (k))) +G(x(k − τ2 (k)))Bg(x(k)) + J, k ∈ N (1),

(6)

where







 1/9 0 3/10 1/5 1/8 2/15 2/9 3/10 1/3 T A0 = ,A = ,B = ,J = , 0 1/10 1/15 1/4 1/10 3/20 1/20 1/4 1/4 g1 (x1 ) = 12 sin(x1 ), g2 (x2 ) = 12 x2 , τ1 (k) = k − 1(mod 5), τ2 (k) = k − 1(mod 4). By simple calculation, we derive that L = 12 , ||A0 || = 0.1111, ||A|| = 0.4186, ||B|| = 0.5095, ||J|| = 0.4167, so 1 − ||A0 || − ||A||L = 0.6796 > 0, (1 − ||A0 || − ||A||L)2 − 4||B||L2 ||J|| = 0.2495 > 0 and m = 0.7067, h < 3.9218. It follows from Theorem 4 that system (6) has an asymptotic stability equilibrium point x∗ in domain {x(k) | ||x(k)|| ≤ 0.7067}, and by numerical calculation, we derive that x∗ = (0.5352.0.3990)T and the relevant attraction domain is Dh = {x(k) | ||x(k) − x∗ || ≤ h < 3.9218}.

6

Conclusion

In this article, the second-order discrete-time Hopfield neural networks model involving variable delays is investigated under the assumption that activation function gj satisfies gj (0) = 0 (In fact, if gj (0) = 0, we also can discuss it in a similar method.). For the model, we have derived criteria of existence and uniqueness of equilibrium point in some domain, and given a sufficient condition independent of delays that ensure the equilibrium point is asymptotically stable, and estimated the attraction domain of the relevant equilibrium point.

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Acknowledgement The work is supported by National Natural Science Foundation of China under Grant 10371083 and the Young Teachers’ Foundation of Chongqing University of Posts and Telecommunications A2004-12.

References 1. Xu, D.Y., Zhao, H.Y., Zhu, H.: Global Dynamics of Hopfield Neural Networks Involving Variable Delays. Computers and Mathematics with Applications 42(12)(2001)39-45 2. Liao, X.F.,Wong, K.W.: Global Exponential Stability for A Class of Retarded Functional Differential Equations with Applications in Neural Networks. J. Math. Anal. Appl. 293(1)(2004)125-148 3. Guo, S.J., Huang, L.H., Wang, L.: Exponential Stability of Discrete Hopfield Neural Networks. Computers and Mathematics with Applications 47(8-9)(2004)1249-1256 4. Dembo, A., Farotimi, O., Kailath, T.: High-order Absolutely Stable Neural Networks. IEEE Trans. Circ. Syst. II 38(1)(1991)57-65 5. Xu, B.J., Liu, X.Z., Liao, X.X.: Global Asymptotic Stability of High-Order Hopfield Type Neural Networks with Time Delays. Computers and Mathematics with Applications 45(10-11)(2003)1729-1737 6. Zhang, Y.: Robust Stabilization of Bilinear Uncertain Delay Systems. Journal of UEST of China 22(4)(1993)414-419(In Chinese) 7. Cao, J., Liang, J., Lam, J.: Exponential Stability of High-order Bidirectional Associative Memory Neural Networks with Time Delays. Physica D 199(3-4)(2004)425436 8. Cao, J.: Global Exponential Stability of Hopfield Neural Networks. Int. J. Syst. Sci. 32(2)(2001)233-236 9. Mohamad, S.: Gopalsamy, K.: Exponential Stability of Continuous-time and Discrete-time Cellular Neural Networks with Delays. Applied Mathematics and Computation 135(1)(2003)17-38 10. Mohamad, S., Gopalsamy, K.:Dynamics of A Class of Discrete-time Neural Networks and Their Continuous-time Counterparts. Mathematics and Computers in Simulation 53(1-2)(2000)1-39

Convergence Analysis of Discrete Delayed Hopfield Neural Networks Sheng-Rui Zhang1,2 and Run-Nian Ma2,3 1

School of Highway, Chang’an University, Xi’an, 710064 China [email protected] 2 University Key Lab of Information Sciences and Engineering, Dalian University, 116622, China 3 Telecommunication Engineering Institute, Air Force Engineering University, Xi’an, 710077, China [email protected]

Abstract. The convergence of recurrent neural networks is known to be bases of successful applications of the networks. In this paper, the convergence of discrete delayed Hopfield neural networks is mainly investigated. One new sufficient and necessary condition for the delayed network having no one stable state is given. Also, several new sufficient conditions for the delayed networks converging towards a limit cycle with 2-period and with 4-period are respectively obtained. All results established here partly extend the previous results on the convergence of both discrete Hopfield neural networks and discrete delayed Hopfield neural networks in parallel updating mode.

1 Introduction Discrete Hopfield neural networks (DHNN) are one of the famous neural networks with a wide range of applications, such as content addressable memory, pattern recognition, and combinatorial optimization [1-6]. Discrete delayed Hopfield neural network (DDHNN) is an extension of DHNN. The convergence of DDHNN is investigated and some results are given in references [7-10]. The main purpose of this paper is to obtain some new results on the stability of DDHNN.

2 Basic Model Discrete delayed Hopfield neural networks having n neurons can be determined by two n×n real matrices W 0 = ( wij0 ) n× n , W 1 = ( wij1 ) n× n , and an n-dimensional column

vector θ = (θ 1 , L , θ n ) T , denoted by N=(W0⊕W1,θ). There are two possible values for

…

the state of each neuron i: 1 or –1. Denote the state of neuron i at time t∈{0,1,2, } as xi(t), the vector X(t)=(x1(t), ,xn(t))T is the state of the whole neurons at time t. The dynamic behavior of DDHNN can be described by the following state equations

…

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 267 – 272, 2006. © Springer-Verlag Berlin Heidelberg 2006

268

S.-R. Zhang and R.-N. Ma

∑ w x (t ) + ∑ w x (t − 1) + θ ) , i ∈ I = {1,2, L , n} , n

xi (t + 1) = sgn(

n

0 ij

1 ij

j

j =1

where t∈{0,1,2,

j

(1)

i

j =1

…}, and the sign function is defined as follows ⎧ 1, if u ≥ 0 sgn(u ) = ⎨ . ⎩− 1, if u < 0

(2)

We rewrite equation (1) in the compact form X(t+1)=sgn(W0X(t)+W1X(t-1)+θ).

(3)

If the state X* satisfies the following condition X*=sgn(W0X*+W1X*+θ),

(4)

the state X* is called a stable state (or an equilibrium point). Let N=(W0⊕W1,θ) start from any initial states X(0),X(1). For t≥2, if there exists time t1∈{0,1,2, } such that the updating sequence X(0),X(1),X(2),X(3), satisfies that X(t+T)=X(t) for all t≥t1, where T is the minimum value which satisfies the above condition, then we call that the initial states X(0),X(1) converges towards a limit cycle with T-period. This means that a limit cycle with 1-period is a stable state. The delayed network is called to be strict, if the condition (5) is satisfied for all states X, Y∈{-1,1}n, where X=(x1, ,xn)T, Y=(y1, ,yn)T

…

…

…

n

…

n

∑w x +∑w 0 ij

1 ij

j

j =1

y j + θi ≠ 0 , i∈I.

(5)

j =1

In this paper, the stability of DDHNN is mainly studied. The main contribution of this paper is expressed by the following results.

3 Main Results

…

Theorem 1. If the network N=(W0⊕W1,θ) is strict, then the state X=(x1, ,xn)T is a stable state if and only if there exists a positive diagonal matrix D=diag(d1, ,dn) (di>0, i=1,2, ,n) such that

…

(W0+W1-D)X+θ=0. Proof. If X=(x1, fied. Let

…,x ) n

T

…

(6)

is a stable state of the network N=(W0⊕W1,θ), then (4) is satisn

n

j =1

j =1

d i = xi (∑ wij0 x j + ∑ wij1 x j + θi ) , i∈I.

…

(7)

Since the state X=(x1, ,xn)T being a stable state of the strict network N=(W0⊕W1,θ), we know that di>0 for each i∈I by the definition of sign function. And

Convergence Analysis of Discrete Delayed Hopfield Neural Networks n

n

j =1

j =1

269

xi d i = ∑ wij0 x j + ∑ wij1 x j + θi , i∈I.

(8)

We note that the matrix form of (8) is (6). So, the necessary condition is proved. On the other hand, if there exists a positive diagonal matrix D=diag(d1, ,dn) (di>0, i=1,2, ,n) such that the state X=(x1, ,xn)T satisfies (6), then (8) is true. So, (7) is true too, i.e.,

…

…

…

n

n

j =1

j =1

xi (∑ wij0 x j + ∑ wij1 x j + θ i ) > 0 , i∈I.

(9)

This then means that the state X satisfies (4), so X is a stable state of the delayed network. Hence, the sufficient condition is proved. The proof is completed. Example 1. Consider N=(W0⊕W1,θ), where the expressions of matrices W0, W1 and θ are respectively in the following

⎛ 1 − 1⎞ ⎛0 0⎞ ⎛0⎞ ⎟⎟ , W 1 = ⎜⎜ ⎟⎟ , θ = ⎜⎜ ⎟⎟ . W 0 = ⎜⎜ ⎝ 1 − 1⎠ ⎝0 0⎠ ⎝0⎠

(10)

We can test that the network has only one stable state X=(1,1)T, and other states can converge towards the stable state X=(1,1)T. Also, there exists a nonnegative diagonal matrix D=diag(0, 0) such that states X=(1,1)T and -X=(-1,-1)T satisfy (6) respectively. Obviously, -X is not a stable state of the network, but it satisfies (6). From example 1, we know that, even if there exists a nonnegative diagonal matrix D=diag(d1, ,dn) (di≥0, i=1,2,…,n) such that (6) is satisfied, the state X=(x1, ,xn)T can not guaranteed to being a stable state. This means that, if the delayed network is not strict, we have no corresponding results to theorem 1. But based on the proof of theorem 1, we can prove that, if the state X=(x1, ,xn)T is a stable state, then there exists a nonnegative diagonal matrix D=diag(d1, ,dn) (di≥0, i=1,2, ,n) such that (6) is satisfied. From all above, we know that, if the delayed network is not strict, then (6) is only necessary condition for the state X being a stable state, but not sufficient condition. Based on theorem 1, we can easily give the following theorem.

…

…

… …

…

Theorem 2. If the network N=(W0⊕W1,θ) is strict, then the delayed network has no one stable state if and only if for any state X=(x1, ,xn)T there does not exist a positive diagonal matrix D=diag(d1, ,dn) (di>0, i=1,2, ,n) such that (6) is true.

…

…

…

Corollary 1. Let the network N=(W ⊕W ,θ) be strict. If matrix W0+W1 and θ satisfies the following condition for each X∈{-1,1}n 0

X T (W 0 + W 1 ) X ≤ −

1

∑θ j∈I

i ,

X∈{-1,1}n,

(11)

then the delayed network has no one stable state. Especially, if one of the following conditions (12), (13) and (14) is satisfied, then the delayed network has no one stable state

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S.-R. Zhang and R.-N. Ma

∑w

0 ij

+ wij1 − θ i , i ∈ I ,

(12)

∑w

0 ji

+ w1ji − θ i , i ∈ I ,

(13)

wii0 + wii1 ≤ −

j∈I ( j ≠ i )

wii0 + wii1 ≤ −

j∈I ( j ≠ i )

wii0 + wii1 ≤ −

1 2

∑w

0 ij

+ wij1 + w 0ji + w1ji − θ i , i ∈ I.

(14)

j∈I ( j ≠ i )

Proof. Without loss generality, we assume that the network N=(W0⊕W1,θ) has a stable state X=(x1, ,xn)T, then, by theorem 1, there exists a positive diagonal matrix D=diag(d1, ,dn) (di>0, i=1,2, ,n) such that (6) is satisfied. This then means

…

…

…

n

X T (W 0 + W 1 ) X + X Tθ = ∑ d j .

(15)

j =1

On the one hand, we know that XT(W0+W1)X+XTθ≤0 because of the condition (11) being true. On the other hand, d1+ dn>0. This conflicts (15). Consequently, the network has no one stable state. Especially, if one of the above conditions (12), (13) and (14) is satisfied, we can easily prove that the condition (11) is true too. So the delayed network has no one stable state. The proof is completed.

…+

Remark: In corollary 1, let θ=0. If the condition (12) is satisfied, then matrix W0+W1 is called to be nonpositive definite (not necessarily symmetric) on the set {-1,1}. If conditions (13) and (14) are respectively satisfied, matrix W0+W1 is respectively called to be negative row-diagonally dominant and negative column-diagonally dominant. Hence, if N=(W0⊕W1,0) is strict and matrix W0+W1 is negative row-diagonally dominant or negative column-diagonally dominant or satisfies the condition (14), then the delayed network has no one stable state.. Corollary 2. Let the network N=(W0⊕W1,θ) be strict and θ=0. If matrix W0+W1 is antisymmetric, then the delayed network has no one stable state. Especially, if matrices W0 and W1 are antisymmetric, then the delayed network has neither a stable state nor a limit cycle with 2-period. Proof. If the network has a stable state X, then there exists a positive diagonal matrix D=diag(d1, ,dn) (di>0, i=1,2, ,n) such that equation (6) is satisfied. So (15) is true, where θ=0. On the one hand, we know that XT(W0+W1)X=0 because of the matrix W0+W1 being antisymmetric. On the other hand, d1+ dn>0. This conflicts (15). So, the delayed network has no one stable state. Especially, if matrices W0 and W1 are antisymmetric, so matrix W0+W1 is antisymmetric too. The proof is completed. In the following, without loss generality, we assume that the network has a limit cycle (X1,X2) with 2-period. Then, we have

…

…

…+

X1=sgn(W0X2+W1X1), X2=sgn(W0X1+W1X2).

(16)

Convergence Analysis of Discrete Delayed Hopfield Neural Networks

271

Let r=(X1)T(W0X2+W1X1), s=(X2)T(W0X1+W1X2). Since the delayed network is strict, we have r>0, s>0 by the definition of sign function. So, r+s>0. Also, from the expression of r and s, we have r+s=(X1)T(W0X2+W1X1)+(X2)T(W0X1+W1X2) =(X1)TW0X2+(X1)TW1X1+(X2)TW0X1+(X2)TW1X2 =(X1)TW0X2+(X2)TW0X1=0.

(17)

Obviously, this conflicts r+s>0. Consequently, the delayed network has no one limit cycle with 2-period. The proof is completed. In the following, we give some conditions for the delayed network converging respectively towards a limit cycle with 2-period and 4-period. Theorem 3. Let the network N=(W0⊕W1,θ) be strict, we have two results as follows. 1). If the condition (18) is satisfied

max{wii0 + w1ii , wii0 − w1ii } ≤ −

∑w

0 ij

j∈I ( j ≠ i )

∑w

−

− θi , i ∈ I ,

1 ij

j∈I ( j ≠i )

(18)

then the delayed network converges towards a limit cycle with 2-period. 2). If the condition (19) is satisfied max{wii0 + wii1 , − wii0 + wii1 } ≤ −

∑w

0 ij

−

j∈I ( j ≠ i )

∑w

1 ij

− θi , i ∈ I ,

j∈I ( j ≠i )

(19)

then the delayed network converges towards a limit cycle with 4-period. Proof. 1). For any initial states X(0),X(1), if xi(0)=xi(1), since the delayed network being strict, based on (1) and (18), we have n

n

j =1

j =1

xi (2) = sgn( ∑ wij0 x j (1) + ∑ wij1 x j (0) + θ i ) = sgn(( wii0 + wii1 ) xi (0)) = − xi (0) .

(20)

As the same reason, we can calculate xi(3) in the following n

n

j =1

j =1

xi (3) = sgn( ∑ wij0 x j (2) + ∑ wij1 x j (1) + θ i ) = sgn(−( wii0 − wii1 ) xi (0)) = xi (0) .

(21)

And so on, the state updating sequence of neuron i updating process can be interpreted as follows xi(0), xi(0), -xi(0), xi(0), -xi(0), xi(0), -xi(0),

….

If xi(0)≠xi(1), as the same reason, we can calculate the state updating sequence of neuron i updating process, which can be interpreted as follows xi(0), -xi(0), xi(0), -xi(0), xi(0), -xi(0), xi(0),

….

As proved above, we know that N=(W ⊕W ,θ) converges towards a limit cycle with 2-period for any initial states X(0),X(1). 0

1

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2). For any initial states X(0),X(1), if xi(0)=xi(1), based on the same reason, we can calculate the state updating sequence of neuron i updating process, which can be interpreted as follows xi(0), xi(0), -xi(0), -xi(0), xi(0), xi(0), -xi(0), -xi(0),

….

If xi(0)≠xi(1), as the same reason, we can calculate the state updating sequence of neuron i updating process, which can be interpreted as follows xi(0), -xi(0), -xi(0), xi(0), xi(0), -xi(0), -xi(0), xi(0),

….

As proved above, we know that N=(W ⊕W ,θ) converges towards a limit cycle with 4-period for any initial states X(0),X(1). The proof is completed. 0

1

Acknowledgments. The work is partly supported by open Foundation of University Key Lab of Information Sciences and Engineering, Dalian University. The authors would like to thank the referees for their valuable comments and suggestions to improve this paper.

References 1. Hopfield, J.J.: Neural Networks and Physical Systems Emergent Collective Computational Abilities. Proc. Nat. Acad. Sci. USA 79(1982) 2554-2558 2. Bruck, J.: On the Convergence Properties of the Hopfield Model. Proceedings IEEE 78 (1990) 1579-1585 3. Xu, Z., Kwong, C.P.: Global Convergence and Asymptotic Stability of Asymmetrical Hopfield Neural Networks. J. Mathematical Analysis and Applications 191(1995) 405426 4. Lee, D.: New Stability Conditions for Hopfield Neural Networks in Partial Simultaneous Update Mode. IEEE, Trans. Neural Networks 10(1999) 975-978 5. Goles, E.: Antisymmetrical Neural Networks. Discrete Applied Mathematics 13(1986) 97100 6. Ma, R., Zhang, Q., Xu, J.: Convergence of Discrete-time Cellular Neural Networks. Chinese J. Electronics 11(2002) 352-356 7. Ma, R., Xi, Y., Gao, H.: Stability of Discrete Hopfield Neural Networks with Delay in Serial Mode. In: Yin, F., Wang, J. and Guo, C.(eds.): Advance in Neural Networks-ISNN 2004. Lecture Notes in Computer Science 3173 Springer-Verlag Berlin Heidelberg (2004) 126-131 8. Qiu, S., Tang, E.C.C. Yeung, D.S.: Stability of Discrete Hopfield Neural Networks with Time-delay. Proceedings of IEEE International Conference on System, Man Cybernetics(2000) 2546-2552 9. Ma, R., Xi, Y., Lei, S.: Dynamic Behavior of Discrete Hopfield Neural Networks with Time-Delay. Chinese J. Electronics 14(2005) 187-191 10. Ma, R., Lei, S., Zhang, S.: Dynamic Behavior Analysis of Discrete Neural Networks with Delay. In: Wang J., Liao X. and Zhang Y.(eds.): Advance in Neural Networks-ISNN 2005. Lecture Notes in Computer Science 3496 Springer-Verlag Berlin Heidelberg (2005) 259264

An LMI-Based Approach to the Global Stability of Bidirectional Associative Memory Neural Networks with Variable Delay Minghui Jiang1 , Yi Shen2 , and Xiaoxin Liao2 1

Institute of Nonlinear Complex Systems, Three Gorges University, Yichang, Hubei, 443000, China [email protected] 2 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China [email protected], [email protected]

Abstract. Based on the linear matrix inequality (LMI), new sufficient conditions on the global exponential stability and asymptotic stability of bidirectional associative memory neural networks with variable delay are presented, and exponential converging velocity index is estimated. Furthermore, the results in this paper are less conservative than the ones reported so far in the literature. One example is given to illustrate the feasibility of our main results.

1

Introduction

Bidirectional associative memory (BAM) neural networks, firstly proposed by Kosko [1] for their various applications such as designing associative memories, solving optimization problems and automatic control engineering, are a class of two-layer hetero-associative networks with delays. Such applications rely heavily on the stability properties of equilibria of neural networks [2-7]. As is well known, in both biological and man-made neural networks, the delays arise because of the processing of information. Thus, the study of neural network with delays becomes extremely important to manufacture high quality neural networks. Unfortunately, most of the existing literatures on BAM neural networks with delays were restricted to models with constant delays[8-11]. Therefore, the stability analysis of BAM neural networks with variable delays is important from both theoretical and applied points of view. In this paper, based on the linear matrix inequality, sufficient conditions on the global exponential stability and asymptotical stability of bidirectional associative memory neural networks with variable delay are presented. It should be noted that the results in this paper are less conservative than the ones reported so far in the literature. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 273–278, 2006. c Springer-Verlag Berlin Heidelberg 2006 

274

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Preliminaries

The BAM neural network model is described by the following differential equation  q x˙ i (t) = −ai xi (t) + j=1 wij fj (yj (t − τ (t)) + Ii , i = 1, 2, . . . , p, p (1) y˙ j (t) = −bj yj (t) + i=1 vji gi (xi (t − δ(t)) + Jj , j = 1, 2, . . . , q, for t ≥ 0, where x(t) = [x1 (t), · · · , xp (t)]T ∈ Rp , y(t) = [y1 (t), · · · , yq (t)]T ∈ Rq , ai ≥ 0 and bi ≥ 0 are constant, Ii and Jj are external inputs, wij and vji are connection weights of the network, the variable delays τ (t) and δ(t) continuous and differential functions which satisfy 0 ≤ τ (t), δ(t) ≤ τ and τ  (t), δ  (t) ≤ 0. The activation functions fj (·) and gi (·) are assumed to satisfy the following conditions: (H1) There exist positive constants Mj , j = 1, · · · , q and Li , i = 1, · · · , p such that for any θ, ρ ∈ R 0 ≤ (fj (θ) − fj (ρ))(θ − ρ)−1 ≤ Mj , θ  = ρ, 0 ≤ (gi (θ) − gi (ρ))(θ − ρ)−1 ≤ Li , θ  = ρ. (H2) There exist positive constants Nj , j = 1, · · · , q and Ri , i = 1, · · · , p such that |fj (u)| ≤ Nj and |gi (u)| ≤ Ri for all u ∈ R. Throughout this paper, unless otherwise specified, we let A ≥ 0 denote nonnegative definite matrix, A > 0 denote positive definite symmetric matrix and etc. AT , A−1 , λmin (A), λmax (A), denotes, respectively, the transpose of, the inverse of, the minimum eigenvalue of, the maximum eigenvalue of a square matrix A. Let | · | and A denote the Euclidean norm in Rn and the 2-norm in Rn×n , respectively.

3

Main Results

In this section, in order to investigate the stability of equilibria of the BAM neural network (1), we present the following existence of the equilibrium point of the network (1). Theorem 1. The BAM neural network (1) has at least one equilibrium point if the network (1) satisfies Assumption (H1) and (H2). Proof. Obviously, under the Assumption (H1) and (H2), the right-hand side of the BAM neural network (1) satisfies the global Lipschitz condition. It is well known (see [14]) that the network (1) with initial condition has at least one continuous solution on t ≥ −τ . Consider a mapping F : Rp+q → Rp+q defined by   F1 (x, y) F (x, y) = , (2) F2 (x, y)

A LMI-Based Approach to the Global Exponential Stability of BAM

where (F1 (x, y))i = a−1 i {

q 

275

wij fj (yj (t − τ (t)) + Ii }, i = 1, · · · , p,

j=1

(F2 (x, y))j = b−1 j {

p 

vji gi (xi (t − δ(t)) + Jj }, j = 1, · · · , q.

j=1

The rest of the proof is similar to the one in Ref.[8] and omitted. In order to show the uniqueness of the equilibrium point, we use the transformation ui (t) = xi (t) − x∗i , i = 1, · · · , p, zj (t) = yj (t) − yj∗ , j = 1, · · · , q to shift the equilibrium point of the BAM neural network (1) to the origin, and get the following form of the BAM neural network (1):   u˙ i (t) = −ai ui (t) + qj=1 wij sj (zj (t − τ (t)), i = 1, 2, . . . , p, p (3) z˙j (t) = −bj zj (t) + j=1 vji ei (ui (t − δ(t)), j = 1, 2, . . . , q, where

sj (zj (t)) = fj (zj (t) + yj∗ ) − fj (yj∗ ), j = 1, 2, . . . , q, ei (ui (t)) = gi (ui (t) + x∗i ) − gi (x∗i ), i = 1, 2, . . . , p.

It is obvious that the functions sj and ej satisfy the same assumptions (H1) and (H2). The BAM neural network (3) can be rewritten as the following vector form:  u(t) ˙ = −Au(t) + W S(z(t − τ (t)), (4) z(t) ˙ = −Bz(t) + V E(u(t − δ(t)). where u(t) = (u1 (t), · · · , up (t))T , z(t) = (z1 (t), · · · , zq (t))T , A = diag(a1 , · · · , ap ), B = diag(b1 , · · · , bq ), W = (wij )p×q , V = (vji )q×p , S(z) = (s1 (z1 ), · · · , sq (zq ))T , E(u) = (e1 (u1 ), · · · , ep (up ))T . We next show that the origin of the BAM neural network (4) is unique and global asymptotically stable in the following Theorem 2. Theorem 2. Assume that Assumption (H1) and (H2) hold. The origin of the BAM neural network (4) is the unique equilibrium point and global asymptotically stable if there exist positive constants α, β, θ, κ such that the following LMIs hold − αL−1 AL−1 − 2θAL−1 + βV T B −1 V + κI + θW W T < 0

(5)

− βM −1 BM −1 − 2κBM −1 + αW T A−1 W + θI + κV V T < 0

(6)

and

where M = diag(M1 , · · · , Mq ), L = diag(L1 , · · · , Lp ), and I is an identity matrix. Proof. Let (u∗ , z ∗ ) be the equilibrium of the BAM neural network (4), we have  −Au∗ + W S(z ∗ ) = 0, (7) −Bz ∗ + V E(u∗ ) = 0.

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By Assumption (H1), we know that u∗ = z ∗ = 0 if S(z ∗ ) = E(u∗ ) = 0, and vice versa. Therefore, by (7), if S(z ∗ ) = 0, then E(u∗ ) = 0, and vice versa. We can prove that S(z ∗ ) = E(u∗ ) = 0 by contradiction method and omitted. Next we will prove that the origin of the BAM neural network (4) is global asymptotically stable. Choose the following Lyapunov functional  ui (t)  t p p   V (u(t), z(t)) = αuT (t)u(t) + 2 θ ei (s)ds + κ e2i (ui (s))ds i=1 q 

+βz T (t)z(t) + 2 

i=1 t

+α 

t−τ (t) t

+β

0

 κ

zi (t)

i=1 q 

si (s)ds + 0

i=1

t−δ(t)



t

s2i (zi (s))ds

θ t−τ (t)

S(z(s))T W T A−1 W S(z(s))ds E(u(s))T V T B −1 V E(u(s))ds.

(8)

t−δ(t)

The time derivative of V (u(t), z(t)) along the trajectories of (4) turns out to be V˙ (u(t), z(t)) = −2αuT (t)Au(t) + 2αuT (t)W S(z(t − τ (t))) − 2θE T (u(t))Au(t) +2θE T (u(t))W S(z(t − τ (t))) + κE T (u(t))E(u(t)) −(1 − δ  (t))κE T (u(t − δ(t)))E(u(t − δ(t))) − 2βz T (t)Bz(t) +2βz T (t)V E(U (t − δ(t))) − 2κS T (z(t))Bz(t) +2κS T (z(t))V E(u(t − δ(t))) + θS T (z(t))S(z(t)) −(1 − τ  (t))θS T (z(t − τ (t)))S(z(t − τ (t))) +αS(z(t))T W T A−1 W S(z(t)) −(1 − τ  (t))αS(z(t − τ (t)))T W T A−1 W S(z(t − τ (t))) +βE(u(t))T V T B −1 V E(u(t)) −(1 − δ  (t))βE(u(t − δ(t)))T V T B −1 V E(u(t − δ(t)))

(9)

Using the conditions δ  (t), τ  (t) ≤ 0 and Assumption (H1), we can derive by (5), (6) and (9) V˙ (u(t), z(t)) ≤ E(u∗ )T (−αL−1 AL−1 − 2θAL−1 + βV T B −1 V + κI +θW W T )E(u∗ ) + S(z ∗ )T (−βM −1 BM −1 − 2κBM −1 +αW T A−1 W + θI + κV V T )S(z ∗ ) < 0.

(10)

Therefore, by the standard Lyapunov-type theorem [13], we can conclude the origin of the BAM neural network (4) is globally asymptotically stable. This complete the proof. Remark 1. The Theorem 2 in this paper reduces to the Theorem 2 in [8] while α = β = θ = κ = 1 and δ(t) = τ (t) = τ . Therefore, the The Theorem 2 in this paper generalized and improved the results in [8].

A LMI-Based Approach to the Global Exponential Stability of BAM

277

In similar method, we can prove the the origin of the BAM neural network (4) is globally exponentially stable in the following result. Theorem 3. Assume that Assumption (H1) and (H2) hold. The origin of the BAM neural network (4) is the unique equilibrium point and global exponentially stable if there exist positive constants α, β, θ, κ such that the following LMIs hold − 2θAL−1 + βV T B −1 V + κI + θW W T < 0

(11)

− 2κBM −1 + αW T A−1 W + θI + κV V T < 0

(12)

and

where M = diag(M1 , · · · , Mq ), L = diag(L1 , · · · , Lp ), and I is an identity matrix. Remark 2. The Theorem 3 provides the sufficient condition on the globally exponential stability of the origin of the BAM neural network (4). However, Refs.[7-11] only prove the asymptotic stability of the BAM neural network (4).

4

Illustrative Example

Example 1. Assume that the network parameters of system the BAM neural network (1) are chosen as:   1.41 1.41 τ (t) = δ(t) = 2 + 1/t, A = B = I, W = V = 1.41 −1.41 and

1 1 tanh(zi ), i = 1, 2, ei (ui ) = tanh(ui ), i = 1, 2 2 2 1 1 with M = L = diag( 2 , 2 ) and all inputs Ii = Jj = 0. In this case, there exists at least one feasible solution α = β = 963.7426, θ = κ = 8.0695 of the LMIs (5) and (6) in Theorem 2. Therefore, the equilibrium of the above system is global asymptotically stable. However, the result of Theorem 2 in [8] yields   0.95240 0 −1 −1 −1 T −1 T −L AL − 2AL + V B V + I + W W = > 0. 0 0.95240 si (zi ) =

Hence, the criteria in [8] can not applied in the Example 1.

5

Conclusion

In this paper, we present theoretical results on the global exponential stability and asymptotic stability of the BAM neural network with variable delay. These conditions obtained here are easy to be checked in practice, and are of prime importance and great interest in the design and the applications of network. The criterions are more general than the respective criteria reported in [8]. Finally, illustrative examples are also given to compare the results with existing ones.

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Acknowledgments The work was supported by National Natural Science Foundation of China (60274007, 60574025) and Natural Science Foundation of Hubei (2004ABA055).

References 1. Kosko, B.: Bidirectional Associative Memories Systems. IEEE Trans. Man and Cybernetics 18(1988) 49-60 2. Cao, J., Wang, J.: Global Asymptotic Stability of Recurrent Neural Networks with Lipschitz-continuous Activation Functions and Time-Varying Delays. IEEE Trans. Circuits Syst. I 50 (2003) 34-44 3. Zeng, Z., Wang, J., Liao X.: Global Exponential Stability of a General Class of Recurrent Neural Networks with Time-Varying Delays. IEEE Trans. Circuits Syst. 50 (2003) 1353-1359 4. Singh, V.: A Generalized LMI-Based Approach to the Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Networks 15 (2004) 223-225 5. Arik, S.: An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Networks 13 (2002) 1239-1242 6. Liao, T., Wang, F.: Global Stability for Cellular Neural Networks with Time Delay. IEEE Trans. Neural Networks 11 (2000) 1481-1484 7. Zhao, H.: Global Stability of Bidirectional Associative Memory Neural Networks with Distributed Delays. Phys. Lett. A 30 (2002) 519-546 8. Arik, S., Tavsanoglu, V.: Global Asymptotic Stability Analysis of Bidirectional Associative Memory Neural Networks with Constant Time Delays. Neurocomputing 68 (2005) 161-176 9. Liao, X., Yu, J.: Qualitative Analysis of Bi-directional Associative Memory with Time Delay. Int. J. Circuit Theory Appl. 26 (1998) 219-229 10. Liao, X., Yu, J., Chen, G.: Novel Stability Criteria for Bidirectional Associative Memory Neural Networks with Time Delays. Int. J. Circuit Theory Appl. 30 (2002) 519-546 11. Mohamad, S.: Global Exponential Stability in Continuous-time and Discrete-time Delayed Bidirectional Neural Networks. Physica D 159 (2001) 233-251 12. Yi, Z., Tan, K.K.: Convergence Analysis of Recurrent Neural Networks. Kluwer Academic Publishers, Dordrecht (2003) 13. Chen, M., Chen, Z., Chen, G.: Approximate Solution of Operation Equations. World Scientific, Singapore (1997) 14. Driver, R. D.: Ordinary and Delay Differential Equations. Springer-Verlag NewYork Inc. (1977)

Existence of Periodic Solution of BAM Neural Network with Delay and Impulse Hui Wang1,2, Xiaofeng Liao1, Chuandong Li1, and Degang Yang1,3 1 College of Computer Science, Chongqing University, Chongqing, 400030, China 2 Department of Mathematics, Leshan Normal College, 614003, China 3 College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China

Abstract. By using the continuation theorem for Mawhin’s coincidence degree and some analytical techniques, several sufficient conditions are obtained ensuring existence of periodic solution of BAM neural networks with variant coefficients, delays and impulse.

1 Introduction Since BAM neural network was first introduced by Kosko [1], it has been paid much attention in past decade due to its applicability in many fields. There are many studying results about the periodic solution and their exponential convergence properties [2-7]. To the best of our knowledge, the periodic oscillatory solution is seldom considered for the BAM networks with delays and impulse. In this paper, we investigate the existence of positive periodic solution of impulsive and delay BAM systems:

⎧ . n ⎛ ⎛ ⎞⎞ ⎪ x ( t ) = −a ( t ) x ( t ) + ∑ α ( t ) f ⎜ y ⎜ t − τ ⎟ ⎟ + I ( t ) i i i ji j j ji ⎠⎠ i ⎝ ⎝ ⎪ j =1 ⎪ Δx t = x t+ − x t− = I x t ⎪⎪ i k i k i k ik i k ⎨ . p ⎛ ⎪ y ( t ) = −b ( t ) y ( t ) + ∑ β ( t ) g x ⎛⎜ t − σ ⎞⎟ ⎞ + J ( t ) j j i ⎜⎝ j ⎝ ij ⎠ ⎟⎠ j ⎪ j j = 1 ij ⎪ ⎪ Δ y t = y t + − y t − = J ⎛⎜ y t ⎞⎟ j k j k j k jk ⎝ j k ⎠ ⎪⎩

( ) ( ) ( )

( ( ))

( ) ( )

( )

( )

(1)

where x ( t ) and y ( t ) are the activations of the i th neuron and the j th neuron, i j respectively. α I

i

ji

(t )

and β

ij

(t )

are the connection weights at the time t,

( t ) , J j ( t ) denote the external inputs at time t. τ ji

and σ correspond to the finite ij

( )

( ) are the impulses

speed of the axonal signal transmission at time t, Δ x t , Δ y t j k i k J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 279 – 284, 2006. © Springer-Verlag Berlin Heidelberg 2006

280

H. Wang et al.

and t < t < L is a strictly increasing sequence such that 1 2 k lim t = +∞ . k →∞ k Throughout this paper, we assume that: (H1) a ( t ) > 0 ,b ( t ) > 0,a ( t ) ,α ( t ) , I ( t ) ,b ( t ) , β ( t ) , J ( t ) are positive i j i ji i j ij j

at moments t

continuous periodic functions of period ω . (

min a ( t ) > 0, min b ( t ) > 0, ) 0≤t ≤ω i 0≤t ≤ω j

(H2) f , g are bounded; i j (H3) There exists a positive integer m such that t

J

j (k + m)

=J

jk

k +m

= t + ω,I =I , k ik i ( k + m)

,i = 1, 2 ,L , p; j = 1, 2 L ,n , k = 1, 2 ,L

2 Preliminaries We state Mawhin’s continuous theorem [8] as follows. Theorem 1.1. Let X and Y be two Banach spaces and L : domL I X → Y be a

Fredholm mapping of index 0. Suppose that Ω is open bounded in X and N : Ω → Y is L-compact on Ω . Furthermore, suppose that (i)

(ii)

For each λ ∈ ( 0 ,1) ,x ∈ ∂Ω , Lx ≠ λ Nx

For each x ∈ ∂Ω I KerL, QNx ≠ 0 , and deg ( JQN ,Ω I KerL,0 ) ≠ 0

where J : Im Q → KerL is isomorphism. Then Lx = Nx has at least one solution in

Ω I domL .

1ω ∫ f ( t ) dt . For any nonω0

In the following, we shall use the notation f =

⎧⎪ p + n (q) ,t t ⎤ = z : 0,ω ] → R z ( t ) exist 1 2,L m ⎥⎦ ⎨⎪ [ ⎩ ( q ) ( t + 0 ) , z ( q ) ( t − 0 ) exist at t ,t t ; and z( j ) t = for t ≠ t ,t t ;z 1 2,L m 1 2 ,L m k

negative integer q, let C

( q ) ⎡0,ω ;t ⎢⎣

( )

z

( j)

(tk − 0,) k = 1,L m, j = 0,1,L q ⎫⎬⎭ with the norm

( j ) ( t ) ⎫⎪ z

q

, where ⎬ ⎪⎭ j = 1

C

( q ) ⎡0,ω ;t ⎢⎣

. is any norm of R

,t t ⎤ is a Banach space. 1 2,L m ⎥⎦

z

q

p+n

⎧⎪ = max ⎨ sup ⎪⎩ t ∈ [ 0,ω ] . It is easy to see that

Existence of Periodic Solution of BAM Neural Network with Delay and Impulse

⎧ (1) Let domL = ⎨ Z ∈ C ⎩

⎫ ⎡ 0,ω ;t ,t t ⎤⎬ ⎢⎣ ⎥ 1 2,L m ⎦ ⎭

(

N : Z → Y , Nz =

⎡ n ⎢ − a1 t x1 t + ∑ α j1 t f j ⎛ y j t −τ j1 ⎞ + I1 t ⎜ ⎟ ⎢ ⎝ ⎠ j =1 ⎢ ⎢ M ⎢ n ⎛ ⎞ ⎢−a t x t + p n ∑ α jp t f j ⎜ y j t −τ jp ⎟+ I p ⎢ ⎝ ⎠ ⎢ j =1 ⎢ p ⎢ −b t y t + 1 1 ∑ βi1 t gi ⎛⎜⎝ xi t −σ i1 ⎞⎟⎠ + J1 t ⎢ i =1 ⎢ ⎢ M ⎢ p ⎢ ⎛ ⎞ ⎢ −bn t yn t + ∑ βin t gi ⎜⎝ xi t −σ in ⎟⎠ + J n t ⎢⎣ i =1

()

() ()

()

(

)

(

( (

))

Δ x1(tm )

M

M

Δ x p (tm )

Δ y1(t1) L Δ y1(tm ) M

Δ yn (t1)

()

{

M

Δ yn (tm )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

1ω ∫ z ( t ) dt ω0 m ⎛1 ⎞ = ⎜ ∫ ω ψ ( s )ds + ∑ C + d , 0 ,L 0, 0 ⎟ 0 ⎜ω ⎟ k k =1 ⎝ ⎠

where vector ψ is the first column of the right of equation (3) Obviously KerL = z : z ∈ C ∈ R

(2)

Δ x1(t1) Δ x p (t1)

(t )

()

)

Pz = QNz = Q M = ψ ,C ,C L C ,d 1 2 m

()

)

(

()

() ()

)

(

()

() ()

( ) )

( ) ( )

z → z ′, Δ z t , Δ z t ,L Δ z t , 0 m 1 2

L : domL → Y

() ()

281

p+n

}

,t ∈ [ 0,ω ] , Im L = Z × R

(3)

(4)

(5)

( p + n ) × m × {0}

And dim KerL = p + n = co dim Im L , then P and Q are continuous projectors satisfying Im P = KerL , Im L = KerQ = Im ( I - Q ) . It’s easy to see that Im L is closed in Y, thus we have the following results. Lemma 2.1. Let L and N be defined by (2); then L is a Fredholm operator of index 0. Lemma 2.2. Let L and N be defined by (2) and (3), respectively, suppose Ω is an open bounded subset of domL , then N is L-compact on Ω . Proof. For any z ∈ Ω , we denote the inverse of the map L

domL ∩ KerP

→ Y by

K , through an easy computation, we can find P m 1ωt K M = ∫ t ϕ ( s )ds + ∑ C − ∫ ∫ ϕ ( s )dsdt − ∑ C 0 P k ω k t >t k =1 00 k

Thus, QN( Ω ) is bounded and N is a completely continuous mapping, by using

( )

( I − Q ) N Ω is relatively compact, thus N P is L-compact on Ω . The proof of Lemma2.2 is complete. Arzela-Ascoli theorem, we know that K

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3 Existence of Periodic Solution In this section, based on the Mawhin’s continuation theorem, we shall study the existence of at least one periodic solution of (1). Theorem 3.1 Assume that (H1), (H2) and (H3) hold, then system (1) has at least one ω -periodic solution. Proof. Based on the Lemma1.1 and Lemma 1.2, now, what we need to do is just to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lx = λ Nx, λ ∈ ( 0,1) ,

we have ⎧ ⎡ ⎤ n ⎪ dxi t ⎛ ⎞ ⎪ =λ ⎢⎢ −ai t xi t + ∑ α ji t f j ⎜⎜ y j ⎛⎜ t −τ ji ⎞⎟ ⎟⎟+ Ii t ⎥⎥ ,t ≠tk ,t∈⎡⎣⎢0,ω ⎤⎦⎥ ⎝ ⎠⎠ ⎪ dt ⎝ ⎢ ⎥ j =1 ⎪ ⎣ ⎦ ⎪ ⎛ +⎞ ⎛ −⎞ ⎛ x t ⎞, = − = = Δ x t x t x t λ I i 1 , L p,x ⎪⎪ i k i⎜ k ⎟ i⎜ k ⎟ ik ⎜⎝ i k ⎟⎠ i 0 = xi ω ⎝ ⎠ ⎝ ⎠ ⎨ ⎡ ⎤ p ⎪ dy j t ⎛ ⎞ ⎪ =λ ⎢⎢−b j t y j t + ∑ βij t gi ⎜⎜ x j ⎛⎜ t −σ ij ⎞⎟ ⎟⎟ + J j t ⎥⎥ ,t ≠tk ,t∈⎡⎣⎢0,ω ⎤⎦⎥ ⎪ dt ⎝ ⎠⎠ ⎝ i =1 ⎢⎣ ⎥⎦ ⎪ ⎪ ⎛ +⎞ ⎛ −⎞ ⎛ ⎞ j =1,Ln,y j 0 = y j ω ⎪ Δ y j tk = y j ⎜ t ⎟ − y j ⎜ t ⎟ =λ J jk ⎜ y j tk ⎟ , ⎝ ⎠ ⎪⎩ ⎝ k ⎠ ⎝ k ⎠

()

() ()

()

( )

()

() ( )

( )

()

() ()

()

( )

()

()

( )

(

(6)

( )

)

T Suppose that Z ( t ) = xT , yT ∈ Z is a solution of system (6) for some λ ∈ ( 0,1) . Integrating (6), over the interval [ 0,ω ] , we obtain

m ⎛ ⎞ ω ω n ⎛ ⎞ ∫0 ai ( s )xi ( s ) ds = ∫0 ∑ α ji ( s ) f j ⎜⎜ y j ⎛⎜⎝ s −τ ji ⎞⎟⎠ ⎟⎟+ Ii ( s )ds + ∑ Iik ⎝⎜ xi (tk ) ⎠⎟ ⎝ ⎠ k =1 j =1 m ⎛ ⎞ ω ω p ⎛ ⎞ ∫0 b j ( s ) y j ( s ) ds = ∫0 ∑ βij ( s ) gi ⎜⎜ xi ⎛⎜⎝ s −σ ij ⎞⎟⎠ ⎟⎟+ J j ( s )ds + ∑ J jk ⎜⎝ y j (tk ) ⎟⎠ ⎝ ⎠ k =1 i =1

(7)

Let ξ + ,η + ∈ [ 0,ω ] , such that i j

( )

x ξ + = sup x ( t ) , y ⎛⎜η + ⎞⎟ = sup y (t ) i i j⎝ j ⎠ t ∈ [ 0,ω ] i t ∈ [ 0,ω ] j m Then, by (7), we have x ξ + a ≥ −ω nα M − ∑ I x i i i i ik i k =1 m y ⎜⎛η + ⎟⎞ b ≥ −ω n β N - ∑ J ⎜⎛ y j⎝ j ⎠ j j jk ⎝ j k =1

( )

Hence,

( )

⎞ ( ( )) ⎟⎟⎠ / ai := Ti+ ,

m ⎛ x ξ + ≥ ⎜ −ω nα M + ∑ I x t ⎜ i i i ik i k k =1 ⎝

( (tk )), similarly, (tk ) ⎟⎠⎞.

(8)

Existence of Periodic Solution of BAM Neural Network with Delay and Impulse

m ⎛ y ⎛⎜η + ⎞⎟ ≥ ⎜ −ω n β N + ∑ J j⎝ j ⎠ ⎜ j jk k =1 ⎝ Let ξ − ,η − ∈ [ 0 ,ω ] , such that i j

( )

⎛ y t ⎞ ⎞ / b := T + . ⎜ j k ⎟ ⎟⎟ j j ⎝ ⎠⎠

( )

( )

x ξ − = inf x ( t ) , y η − = inf y (t ) , i i j i t ∈ [ 0 ,ω ] i t ∈ [ 0,ω ] j

( )

(9)

( ( ))

m Then, by (7), we have x ξ − a ≤ ω nα M + ∑ I x t ,i = 1,L p. i i i i ik i k k =1 m Similar to (10), we have y ⎛⎜η − ⎞⎟ b ≤ ω n β N + ∑ J ⎛⎜ y t ⎞⎟. j⎝ j ⎠ j j jk ⎝ j k ⎠ k =1

( )

( )

⎞ ( ( )) ⎟⎟⎠ / ai := Ki− ,

m ⎛ x t Hence, x ξ − ≥ ⎜ ω nα M + ∑ I ⎜ i i i ik i k k =1 ⎝ From (9) and (12), we obtain

283

{

− ⎛ ⎞ y j ⎜η −j ⎟ ≥ K j . ⎝ ⎠

sup z ( t ) < max T + , K − i i i t ∈ [ 0 ,ω ]

}

(10)

(11)

(12)

define = H . i

p+n Denote A = ∑ H + E , clearly, A is independent of λ , and we select i i =1 T ⎧⎪ T ⎛ = ⎜ x , x L x , y , y L y ⎞⎟ ∈ Z : z( t ) < H , Ω = ⎨ z( t ) = φ T ,ψ T p 1 2 n⎠ ⎝ 1 2 ⎪⎩ z t + 0 ∈ Ω ,k = 1, 2 L m . k It is clear that Ω verifies the requirement (i) in Theorem1.1. When z ∈ KerL ∩ ∂Ω , z is a constant vector in R p + n with z = H . Then QNz =

(

)

(

)

}

⎡ ⎧ ⎫ n m ⎢ 1 ∫ ω ⎨⎪−a ( t ) x ( t ) + ∑ α ( t ) f ⎜⎛ y t − τ ⎟⎞ + I ( t ) ⎬⎪ dt + 1 ∑ I x t j1 j⎝ j 1 1i ⎠ 1 1k 1 k ⎢ ω 0 ⎪ 1 ω ⎪⎭ j =1 k =1 ⎩ ⎢ M ⎢ ⎫⎪ n ⎢ 1 ω ⎧⎪ 1 m ⎛ ⎛ ⎞⎞ ∑ I x t ⎢ ∫0 ⎨−a p ( t ) x p ( t ) + ∑ α jp ( t ) f j ⎜ y j ⎜ t − τ pi ⎟ ⎟ + I p ( t )⎬ dt + ω k = 1 pk 1 k ⎠⎠ ⎝ ⎝ ⎢ω j =1 ⎩⎪ ⎭⎪ ⎢ p ⎫⎪ 1 ω ⎧⎪ 1 m ⎢ ⎢ ω ∫0 ⎨−b1 ( t ) y1 ( t ) + ∑ βi1 ( t ) gi xi t − σ i1 + J1 ( t )⎬ dt + ω ∑ J1k y1 tk i =1 k =1 ⎩⎪ ⎭⎪ ⎢ M ⎢ p ⎧ ⎢ 1 1 m ⎪⎫ ω⎪ ⎢ ∑ J y t ∫0 ⎨−bn ( t ) yn ( t ) + ∑ βin ( t ) gi xi t − σ in + J n ( t ) ⎬ dt + nk 1 k ω ω i =1 k =1 ⎩⎪ ⎭⎪ ⎣⎢

(

)

( )

( )

≠0

( (

))

( )

( (

))

( )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0,L , 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦⎥

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Let J : Im Q → KerL, ( r, 0,L , 0, 0 ) → r . Then, in view of the assumption in

Theorem (6), it is easy to prove that deg { JQNz,Ω ∩ KerL,0} ≠ 0 . By now, we have

proved that ω verifies all the requirements in Theorem1.1. Hence, system (1) has at least one ω − periodic solution in Ω . The proof is complete. □

4 Conclusions In this paper, we used the continuation theorem of coincidence degree theory to study the existence of periodic solution for BAM model of neural network with impulses and time-delays. A set of easily verifiable sufficient conditions for the existence of periodic solution was obtained. The Global exponential asymptotic stability of the periodic solution of this system will be discussed in future papers.

Acknowledgements This work was jointly supported by the National Natural Science Foundation of China (Grant no. 60574024, 60573047), the Scientific Research Fund Projects for Youngsters of Sichuan Provincial Education Administration Office (Grant no. 2005B039), and the Natural Science Foundation of Chongqing (Grant no. 8509).

References 1. Kosko, B.: Adaptive Bi-directional Associative Memories. Appl. Opt. 26 (23) (1987) 4947– 4960. 2. Li, Y., Lu, L.: Global Exponential Stability and Existence of Periodic Solution of Hopfieldtype Neural Networks with Impulses. Phys Lett A 333 (2004):62-71. 3. Liao, X., Wong, K.W., and Yang, S.Z.: Convergence Dynamics of Hybrid Bidirectional Associative Memory Neural Networks with Distributed Delays. Phys. Lett. A 316 (2003) 55-64. 4. Li, C., Liao, X.: New Algebraic Conditions for Global Exponential Stability of Delayed Recurrent Neural Networks. Neurocomputing 64(2005) 319-333. 5. Li, C., Liao, X.: Delay-dependent Exponential Stability Analysis of Bi-directional Associative Memory Neural Networks: an LMI Approach. Chaos, Solitons & Fractals 24(4) ( 2005) 1119-1134. 6. Li, C., Liao, X., Zhang, R.: A Unified Approach for Impulsive Lag Synchronization of Chaotic Systems with Time Delay. Chaos, Solitons and Fractals 23 (2005) 1177-1184 7. Song, Q., Cao, J.: Global Exponential Stability and Existence of Periodic Solutions in BAM Networks with Delays and Reaction–diffusion Terms. Chaos, Solitons and Fractals 23 (2005) 421–430 8. Gains, R.E., Mawhin, J.L.: Coincidence Degree and Nonlinear Differential Equation. Lecture Notes in Math., vol. 586. Springer- Verlag, Berlin (1977)

On Control of Hopf Bifurcation in BAM Neural Network with Delayed Self-feedback Min Xiao1,2 and Jinde Cao1 1

2

Department of Mathematics, Southeast University, Nanjing 210096, China Department of Mathematics, Nanjing Xiaozhuang College, Nanjing 210017, China [email protected]

Abstract. In this paper, the control of Hopf bifurcations in BAM neural network with delayed self-feedback is presented. The asymptotic stability theorem and the relevant corollary for linearized nonlinear dynamical systems are stated. For BAM neural network with delayed self-feedback, a control model based on washout filter is proposed and analyzed. By applying the stability lemma, we investigate the stability of the control system and state the relevant theorem for choosing the parameters of the stabilized control system. Some numerical results are also given to illustrate the correctness of our results.

1

Introduction

A neural network, especially a time delayed neural network, is usually a largescale dynamic system that possesses rich dynamical behaviors. Therefore there has been increasing interest in the study of the dynamics of neural networks (see [5-7, 13, 15, 16]). Bidirectional associate memorial (BAM) neural network with or without axonal signal transmission delays is a type of network with the neurons arrayed in two layers (see [1, 2]). Networks with such a bidirectional structure have practical applications in storing paired patterns or memories and possess the ability of searching the desired patterns via both directions: forward and backward (see [1, 3, 4]). The delayed BAM neural network can be described by the following system ⎧ m  ⎪ ⎪ cji fi (yj (t − τji )) + Ii , ⎨ x˙ i (t) = −xi (t) + j=1 (1) n  ⎪ ⎪ y ˙ (t) = −y (t) + d g (x (t − σ )) + J , ⎩ j j ij j i ij j i=1

where cji , dij (i = 1, 2, · · · , n, j = 1, 2, · · · , m) are the connection weighs through the neurons in two layers: the X-layer and the Y-layer. On the X-layer, the neurons whose states are denoted by xi (t) receive the input Ii and the inputs outputted by those neurons in the Y-layer via activation function fi , while on the Y-layer, the neurons whose associated states are denoted by yj (t) receive the input Jj and the inputs outputted by those neurons in the X-layer via activation function gj . J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 285–290, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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On the other hand, recent work (see [9]) has shown that inhibitory selfconnections can play a role in a network under some conditions on delays. This motivates us to incorporate inhibitory self-connections terms into the model system (1), and study the following system ⎧ m  ⎪ ⎪ cji fi (yj (t − τji )) + Ii , ⎨ x˙ i (t) = −xi (t) + aii fi (xi (t − rii )) + j=1 (2) n  ⎪ ⎪ dij gj (xi (t − σij )) + Jj . ⎩ y˙ j (t) = −yj (t) + bjj gj (yj (t − mjj )) + i=1

Although the system (2) can be mathematically regarded as a Hopfied-type neural network, it really produces many nice properties due to the special structure of connection weights and has many practical applications. For the sake of implicity, we assume that there are only one neuron in the X-layer and Y-layer respectively, the time delay from the X-layer to Y-layer is τ , which is the same as the time delay from the Y-layer back to X-layer, and self-feedback delay is also τ . This simplified BAM neural network with delayed self-feedback model can be described by the following system ⎧ ⎨ x˙ 1 (t) = −x1 (t) + a1 f (x1 (t − τ )) − a1 b1 f (y1 (t − τ )), (3) ⎩ y˙ 1 (t) = −y1 (t) + a2 f (x1 (t − τ )) − a2 b2 f (y1 (t − τ )), where ai (i = 1, 2), bi (i = 1, 2) are real constant parameters. Bifurcation phenomena can be found in many time-delayed dynamical systems, which may have richer dynamical behaviors [6, 11, 12, 8, 10]. System (3) may create Hopf bifurcation when active function f (·) and the system parameters ai , bi satisfy some specific conditions. Bifurcation control generally refers to the design of a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some specific dynamical behaviors. Recently, Chen et al. studied the problem of “anti-control” of bifurcations [14]. The term “anti-control” means that a certain bifurcation is created with desired location and properties by control approach. In [14], the authors presented a washout filter based anti-control Hopf bifurcations control law for a general control system.

2

Main Results

In this section, firstly, we investigate the stability of linear system with delay x˙ i (t) =

n 

(aij xj (t) + bij xj (t − τ )),

i = 1, 2, . . . , n, τ > 0.

j=1

We first introduce the following two lemmas in for system (4).

(4)

On Control of Hopf Bifurcation in BAM Neural Network

287

Lemma 1. The equilibrium point of system (4) is asymptotically stable if the following conditions hold: (1) There exists τ0 and the real parts of all eigenvalues of characteristic equation Δ(λ; τ0 ) = 0 of Eq. (4) are rigorously negative when τ = τ0 ;   (2) For an arbitrary real number ξ and τ ∈ [τ0 , τ ](τ0 < τ ) or τ ∈ [τ, τ0 ](τ0 ≥  τ ), Δ(iξ; τ )  = 0. In particular, system (4) is degenerated as a system without time-delay when τ0 = 0. As a result, the eigenvalues of the characteristic equation are easy to obtain without the term e−τ λ . We describe this special case in the following lemma. Lemma 2. The equilibrium point of system (4) is asymptotically stable if the following conditions hold: (1) The real parts of all eigenvalues of characteristic equation Δ(λ; τ ) = 0 of Eq. (4) are rigorously negative when τ = 0;   (2) For an arbitrary real number ξ and τ ∈ [0, τ ], Δ(iξ; τ )  = 0. In the remain part of this paper, Lemmas 1 and 2 will be used frequently. We consider a general form of dynamical system with parameter μ: y˙ = f (t, y, μ),

(f (t, 0, μ) = 0).

(5)

Adding a control action u to the equation on one component yi , and taking u as the following form [14]: ˆ u = f(x, K), w˙ = yi − αw = x,

(6)

where K is the control parameter, fˆ is a nonlinear function. In order to keep the structure of all equilibriums of Eq. (6), the following conditions must be satisfied • α > 0 for the stability of the washout filter. • fˆ(0, K) = 0 for preservation of equilibrium points. By using the above approach, the control system for Eq. (3) is designed as follows: ⎧ x˙ 1 (t) = −x1 (t) + a1 f (x1 (t − τ )) − a1 b1 f (y1 (t − τ )) + fˆ(ζ), ⎪ ⎪ ⎨ y˙ 1 (t) = −y1 (t) + a2 f (x1 (t − τ )) − a2 b2 f (y1 (t − τ )), (7) u˙ = x1 (t) − αu = ζ, ⎪ ⎪ ⎩ ˆ f (ζ) = −K1 ζ. where f (0) = 0, α > 0, K1 is a constant scaling parameter. The linearized equation of Eq. (7) is in the following form: ⎧ ⎨ x˙ 1 (t) = −x1 (t) + a1 f  (0)x1 (t − τ ) − a1 b1 f  (0)y1 (t − τ ) − K1 (x1 (t) − αu), y˙ 1 (t) = −y1 (t) + a2 f  (0)x1 (t − τ ) − a2 b2 f  (0)y1 (t − τ ), (8) ⎩ u˙ = x1 (t) − αu.

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The characteristic equation for Eq. (8) is: ⎛ ⎞ −1 − K1 + a1 f  (0)e−λτ − λ −a1 b1 f  (0)e−λτ K1 α det ⎝ a2 f  (0)e−λτ −1 − a2 b2 f  (0)e−λτ − λ 0 ⎠ = 0. (9) 1 0 −α − λ Therefore, Δ(λ; τ ) = −λ3 − [2 + α + K1 + (a2 b2 − a1 )f  (0)e−λτ ]λ2 −[(1 + α + a2 b2 f  (0)e−λτ )(1 + K1 − a1 f  (0)e−λτ ) +α(1 + a2 b2 f  (0)e−λτ ) − K1 α + a1 b1 a2 f 2 (0)e−2λτ ]λ −α(1 + a2 b2 f  (0)e−λτ )(1 − a1 f  (0)e−λτ ) − αa1 b1 a2 f 2 (0)e−2λτ . When τ = 0, we have Δ(λ; 0) = −λ3 − [2 + α + K1 + (a2 b2 − a1 )f  (0)]λ2 −[(1 + α + a2 b2 f  (0))(1 + K1 − a1 f  (0)) + α(1 + a2 b2 f  (0)) − K1 α +a1 b1 a2 f 2 (0)]λ − α(1 + a2 b2 f  (0))(1 − a1 f  (0)) − αa1 b1 a2 f 2 (0). Denote p = 2 + α + K1 + (a2 b2 − a1 )f  (0), q = (1 + α + a2 b2 f  (0))(1 + K1 − a1 f  (0)) + α(1 + a2 b2 f  (0)) −K1 α + a1 b1 a2 f 2 (0), r = α(1 + a2 b2 f  (0))(1 − a1 f  (0)) + αa1 b1 a2 f 2 (0).

(10)

From Δ(λ; 0) = 0, we obtain λ3 + pλ2 + qλ + r = 0.

(11)

According to the Routh-Hurwitz criterion, it is implied that if (H) p > 0, r > 0, pq − r > 0, then the real parts of all eigenvalues of Eq. (11) is rigorously negative. Based on the above discussions and Lemma 2, the following result for the control system (7) is obtained immediately. Theorem 1. Let p, q, r are defined by (10). Suppose that (H) holds, and for   an arbitrary real number ξ and τ ∈ [0, τ ], characteristic equation Δ(iξ; τ )  = 0, then the control system (7) is asymptotically stable.

3

Illustrative Example

For the control system (7), if we take a1 = 1, b1 = 2.3, a2 = 1.8, b2 = 0.6, f (·) = tanh(·), τ = 1, K1 = 3, α = 2, then f (0) = 0, f  (0) = 1 and all the conditions of Theorem 1 are satisfied. The waveforms x1 (t), y1 (t) are shown in Fig. 1, respectively. The phase portraits of system (7) without and with control are shown in Fig. 2, respectively. Fix a1 , a2 , b2 and let b1 vary, then we obtain the bifurcation figures (see Fig. 3 ), which cross x1 (t) = x1 (t − τ ) and y1 (t) = y1 (t − τ ) with parameter b1 of system (7) with and without control. By comparing the two bifurcation plots, we observe that the bifurcation occurrence is postponed as the system is being controlled.

On Control of Hopf Bifurcation in BAM Neural Network Without control

1.5

289

With control

1.5

1 1

x1

x1

0.5 0

0.5

−0.5 0 −1 −1.5

0

20

40

60

80

−0.5

100

0

20

40

time t

1.5

80

100

60

80

100

1 Without control

With control

0.8

1

0.6

y1

0.5

y1

60 time t

0

0.4 0.2

−0.5

0

−1 −1.5

−0.2 0

20

40

60

80

−0.4

100

0

20

40

time t

time t

Fig. 1. Waveform of System (7) without and with control 1.5

1 Without control

With control 0.8

1

0.6 0.5

y1

y1

0.4 0

0.2 −0.5 0

−1

−0.2

−1.5 −2

−1

0 x1

1

−0.4 −0.5

2

0

0.5 x1

1

1.5

Fig. 2. Phase portrait of System (7) without and with control 2 Without control

x1−y1

1

0

−1

−2

0

0.5

1

1.5

2 b1

2.5

3

3.5

4

1

1.5

2 b1

2.5

3

3.5

4

1.5 With control 1

x1−y1

0.5 0 −0.5 −1 −1.5

0

0.5

Fig. 3. Bifurcation diagram of system (7) without and with control

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M. Xiao and J. Cao

Acknowledgement This work is supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.

References 1. Gopalsamy, K., He, X.: Delay-Independent Stability in Bi-Directional Associative Memory Networks. IEEE Trans. Neural Netwoks 5 (1994) 998-1002 2. Kosko, B.: Bi-Directional Associative Memories. IEEE Trans. Syst. Man Cybern. 18 (1988) 49-60 3. Kosko, B.: Unsupervised Learning in Noise, IEEE Trans. Neural Networks 1 (1990) 44-57 4. Mohamad, S.: Global Exponential Stability in Continuous-Time and Discret-Time Delayed Bidirectional Neural Networks. Physica D 159 (2001) 233-51 5. Cao, J., Liang, J., Lam, J.: Exponential Stability of High-Order Bidirectional Associative Memory Neural Networks with Time Delays. Physica D 199 (2004) 425-36 6. Olien, L., Belair, J.: Bifurcation, Stability and Monotonocity Properties of A Delayed Neural Network Model. Physica D 102 (1997) 349-63 7. Cao, J., Wang, L.: Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays. IEEE Trans. Neural Networks 13 (2002) 457-63 8. Liao, X., Wong, K.W., Leung, C.S., Wu, Z.F.: Hopf Bifurcation and Chaos in A Single Delayed Neuron Equation with Non-monotonic Activation Function. Chaos, Solitons & Fractals 12 (2001) 1535-47 9. Driessche, P.V., Wu, J., Zou, X.: Stabilization Role of Inhibitory Self-Connection in A Delayed Neural Network. Phyisca D 150 (2001) 84-90 10. Zhuang, Z-R., Huang, J., Gao, J-Y.: Analysis of The Bifurcation Diagram of A Hybrid Bistable System with Feedback Control of Chaos. Phys. Rev. E. 60 (1999) 5422-25 11. Song, Y., Han, M., Wei, J.: Stability and Hopf Bifurcation Analysis on A Simplified BAM Neural Network with Delays. Physica D 200 (2005) 185-204 12. Lee, S.-H., Park, J.-K., Lee, B.-H.: A Study on The Nonlinear Controller to Prevent Unstable Hopf Bifurcation. Power Engineering Society Summer Meeting, 2 (2001) 978-82 13. Gopalsamy, K., Leung, I.: Delay Induced Periodicity in A Neural Network of Excitation and Inhibition. Physica D 89 (1996) 395-426 14. Chen, D., Wang, H.O., Chen, G.: Anti-control of Hopf Bifurcations through Washout Filters. Proceedings of 37th IEEE Conference on Decision and Control, 3 (1998) 3040-45 15. Zhou, J., Xiang, L., Liu, Z.: Global Dynamics of Delayed Bidirectional Associative Memory (BAM) Neural Networks. Appl Math & Mech-English Edition 25 (2005) 327-35 16. Zhou, J., Chen, T., Xiang, L.: Robust Synchronization of Delayed Neural Networks Based on Adaptive Control and Parameters Identification. Chaos, Solitons & Fractals 27 (2006) 905-13

Convergence and Periodicity of Solutions for a Class of Discrete-Time Recurrent Neural Network with Two Neurons Hong Qu and Zhang Yi Computational Intelligence Laboratory, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, People’s Republic of China {hongqu, zhangyi}@uestc.edu.cn

Abstract. Multistable neural networks have attracted much interesting in recent years, since the monostable networks are computationally restricted. This paper studies a class of discrete-time two-neurons networks with unsaturating piecewise linear activation functions. Some interesting results for the convergence and the periodicity of solutions of the system are obtained.

1

Introduction

The global stability of recurrent neural networks is a basic requirement in some practical applications of neural networks. Many researchers have been aimed to study the stability of recurrent neural networks [1, 2, 3]. While the major results have focused on the behavior studies of monostable networks. To display the computational capability of neural networks more in-depth, multistability analysis of LT [4] networks have been presented by some authors recently. In papers [5] and [6], three basic properties of multistable networks: boundedness, global attractivity and complete convergence are studied in detail for Continuous-Time and Discrete-Time form of LT networks, respectively. The qualitative analysis of continuous-time LT networks is presented in paper [7]. It is well known that, two-neurons networks sometimes display the same dynamical behavior as larger networks do, and can thus be used as prototypes to improve our understanding of the computational performance of large networks. It has been an increasing interesting in the study of the dynamics of the twoneuron networks (see for example [8]-[10]). This paper focuses on the analysis of the convergence and periodicity of solutions for the discrete-time LT networks with two-neurons. The rest of this paper is organized as follows. A brief description of the LT network model of two-neurons is presented in section 2. Convergence analysis of LT network with two-neurons is given in section 3. Periodicity of solutions for LT network with two-neurons is analyzed in sections 4. Finally, conclusions are drawn in section 5. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 291–296, 2006. c Springer-Verlag Berlin Heidelberg 2006 

292

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The Network Model with Two-Neurons

The discrete-time LT networks with two-neurons can be described as follows:  x1 (k + 1) = a11 σ(x1 (k)) + a12 σ(x2 (k)) + h1 , (1) x2 (k + 1) = a21 σ(x1 (k)) + a22 σ(x2 (k)) + h2 , or its equivalent vector form x(k + 1) = Aσ(x(k)) + h where x1 , x2 denote the activity of neuron 1 and 2, respectively, A = (aij )2×2 is a real 2 × 2 matrix, each of its elements aij ∈ R (i = 1, 2; j = 1, 2) denotes the synaptic weights and represents the strength of the synaptic connection from neuron j to neuron i, hi ∈ R (i = 1, 2) denotes external input of neuron i, h is the vector form of h1 and h2 . The activation function σ is defined as follows:  u, if u > 0 σ(u) = (2) 0, if u ≤ 0 For any k > 0, each LT neuron is either firing or silent. The whole ensemble of the neurons can be divided into a partition P + (k) of neurons with positive states xi (k) > 0 for i ∈ P + (k), and a partition P − (k) of neurons with negative states xi (k) ≤ 0 for x ∈ P − (k). Define a set     e1 0  S= e = 0 or 1, i = 1, 2 0 e2  i then, given any initial condition x(0) ∈ R2 , the trajectory of (1) starting from x(0) can be represented as ⎡ ⎤ ⎡ ⎤ k k−1 l



⎣ x(k + 1) = ⎣ AE(k − j)⎦ x(0) + AE(k − j)⎦ h + h (3) j=1

l=0

j=0

for all k ≥ 0, where each E(i) ∈ S. It is clearly that, for any (x1 (0), x2 (0)) ∈ R, system (1) has a unique solution {(x1 (k), x2 (k))} starting from (x1 (0), x2 (0)), and (x1 (0), x2 (0)) is called the initial value of {(x1 (k), x2 (k))}. The aim of this paper is to determine the limiting behavior or periodicity of {(x1 (k), x2 (k))} as n → ∞ for (x1 (0), x2 (0)) ∈ D1 ∪ D2 ∪ D3 ∪ D4 = R2 , where D1 = X ++ , D2 = X −+ , D3 = X −− , D4 = X +− and X ±± = {(x1 (0), x2 (0)) | x1 (0), x2 (0) ∈ R± }, with R+ = {u | u ∈ R and u > 0} and R− = {u | u ∈ R and u ≤ 0}. Throughout this paper, {(x1 (k), x2 (k))} denotes the solutions of system (1) with the initial value (x1 (0), x2 (0)).

Convergence and Periodicity of Solutions for a Class of Discrete-Time

3

293

Convergence of Solutions

In this section, we consider the convergence of solutions of (1) with σ satisfying (2). We only list several representative results. Theorem 1. (x1 (k), x2 (k)) → (h1 , h2 ) as k → ∞, if h1 ≤ 0, h2 ≤ 0, and (x1 (0), x2 (0)) ∈ D3 Proof. If (x1 (0), x2 (0)) ∈ D3 , then x1 (0) < 0 and x2 (0) < 0, it follows from (1) that  x1 (1) = a11 σ(x1 (0)) + a12 σ(x2 (0)) + h1 = h1 ≤ 0 (4) x2 (1) = a21 σ(x1 (0)) + a22 σ(x2 (0)) + h2 = h2 ≤ 0 let (x1 (k − 1) ≤ 0, x2 (k − 1)) ≤ 0 and h1 ≤ 0, h2 ≤ 0, then, we can obtain by (1) that  x1 (k) = a11 σ(x1 (k − 1)) + a12 σ(x2 (k − 1)) + h1 = h1 ≤ 0 (5) x2 (k) = a21 σ(x1 (k − 1)) + a22 σ(x2 (k − 1)) + h2 = h2 ≤ 0 By induction, we can get that: ∀k ∈ {1, 2, 3, ..., n, ...}, if (x1 (0), x2 (0)) ∈ D3 and h1 ≤ 0, h2 ≤ 0, then the solution starting from (x1 (0), x2 (0)) converges to (h1 , h2 ). Theorem 2. Suppose λ1 , λ2 are eigenvalues of A. Define       10 00 10 E1 = , E2 = , E4 = . 01 01 00 If λ1 < 1, λ2 < 1, then (x1 (k), x2 (k)) → A(I −A)−1 Ei h+h as k → ∞, when one of the following three conditions is satisfied: 1). (x1 (0), x⎧ 2 (0)) ∈ D1 , and  ⎨ a12 = a21 = 0 a11 h1 > 0, a12 h1 > 0 a22 > 0 T1 = , 2). (x1 (0), x2 (0)) ∈ D2 , and T2 = , a21 h2 > 0, a22 h2 > 0 ⎩ h < 0, h > 0 1 2 ⎧ ⎨ a12 = a21 = 0 a11 > 0 3). (x1 (0), x2 (0)) ∈ D4 , and T4 = . ⎩ h1 > 0, h2 < 0 Proof. There are three case of i are involved in this proof. Case 1) : When (x1 (0), x2 (0)) ∈ D1 and T1 holds, it follows from (1) that  x1 (1) = a11 x1 (0) + a12 x2 (0) + h1 (6) x2 (1) = a21 x1 (0) + a22 x2 (0) + h2 let’s consider the follows equations of x1 and x2 : a11 x1 +a12 x2 +h1 = 0. Suppose, L1 is the straight line on the plane which express the above equation. L1 join x1 -axis on point A(− ah111 , 0), join x2 -axis on point B(0, − ah121 ). It is clear that the whole plane can be divide into two region: P1 and P2 , by L1 , shown as Fig.1.

294

H. Qu and Z. Yi x2 L1

P1 A

P2

0

x1

B

Fig. 1. L1 divide R2 into two region

∀P (p1 , p2 ) ∈ R2 , if T1 is holds, then  a11 p1 + a12 p2 + h1 > 0, if P ∈ P1 a11 p1 + a12 p2 + h1 < 0, if P ∈ P2

(7)

Note that (x1 (0), x2 (0)) ∈ D1 and D1 ⊂ P1 , so we obtain: x1 (1) = a11 x1 (0) + a12 x2 (0) + h1 > 0. It is clear to see that: x2 (1) = a21 x1 (0) + a22 x2 (0) + h2 > 0,by the seem method. This implies (x1 (1), x2 (1)) ∈ D1 . Now we assume that: (x1 (k − 1), x2 (k − 1)) ∈ D1 , then equation (8) will be educed easily if T1 holds,  x1 (k) = a11 x1 (k − 1) + a12 x2 (k − 1) + h1 > 0 (8) x2 (k) = a21 x1 (k − 1) + a22 x2 (k − 1) + h2 > 0 this implies that (x1 (k), x2 (k)) ∈ D1 . Thus, we can get that: ∀k ∈ {1, 2, 3, ..., n, ...}, if (x1 (0), x2 (0)) ∈ D1 and T1 is guaranteed, then (x1 (k), x2 (k)) ∈ D1 . Case 2) : When (x1 (0), x2 (0)) ∈ D2 , it follows from (1) that  x1 (1) = a12 x2 (0) + h1 , (9) x2 (1) = a22 x2 (0) + h2 if T2 holds, then x1 (1) = h1 < 0 and x2 (1) = a22 x2 (0) + h2 > 0, which implies (x1 (1), x2 (1)) ∈ D2 . Suppose (x1 (k−1), x2 (k−1)) ∈ D2 and T2 is holds, then we obtain: x1 (k−1) < 0 and x2 (k − 1) > 0. It follows from (1) that  x1 (k) = a12 x2 (k − 1) + h1 = h1 < 0 (10) x2 (k) = a22 x2 (k − 1) + h2 > 0 which implies (x1 (k), x2 (k)) ∈ D2 . By induction, we can show that: ∀k ∈ {1, 2, 3, ..., n, ...}, if (x1 (0), x2 (0)) ∈ D2 and T2 is guaranteed, then (x1 (k), x2 (k)) ∈ D2 . Case 3) : When (x1 (0), x2 (0)) ∈ D4 . By a similar process employed in case 2), we can obtain that: ∀k ∈ {1, 2, 3, ..., n, ...}, if (x1 (0), x2 (0)) ∈ D4 and T4 holds, then (x1 (k), x2 (k)) ∈ D4 .

Convergence and Periodicity of Solutions for a Class of Discrete-Time

295

By induction, we can say that: ∀k ∈ {1, 2, 3, ..., n, ...}, if (x1 (0), x2 (0)) ∈ Di and Ti is guaranteed, then (x1 (k), x2 (k)) ∈ Di , for all i = 1, 2, 4. Notes, A ∗ Ei = Ei ∗ A if Ti holds, and Eik = Ei for all k > 1, here i = 1, 2, 4. From the above conclusion and (3), the trajectory of (1) starting from x(0) ∈ Di can be represented as ⎡ ⎤ ⎡ ⎤ k k−1 l



⎣ x(k + 1) = ⎣ AEi ⎦ x(0) + AEi ⎦ h + h j=1

l=0

= Ak Eik x(0) +

k−1

j=0

Al Eil h + h = Ak Ei x(0) +

l=0

k−1

Al Ei h + h (11)

l=0

for all k > 0, where i = 1, 2, 4. If λ1 < 1, λ2 < 1, then we obtain lim Ak = 0

k→∞

lim

k→∞

k−1

(12) 

Al Ei h = A

l=0

k−2

lim

k→∞

 Al

Ei h = A(I − A)−1 Ei h

(13)

l=0

Apply (12) and (13) to (11), we can obtain lim x(k + 1) = Ak Ei x(0) +

k−1

k→∞

 =



Al Ei h + h

l=0

lim Ak Ei x(0) + A

k→∞

 lim

k→∞

= A(I − A)−1 Ei h + h

k−2

 Al

Ei h + h

l=0

(14)

for all i = 1, 2, 4. This completes the proof.

4

Periodicity of Solutions

In this section we give some theorems for the periodicity of solutions of (1) with σ satisfying (2). Theorem 3. If h1 > 0, h2 > 0, and Ah ≤ −h, then the solution {(x1 (k), x2 (k))} of (1) starting from (x1 (0), x2 (0)) is oscillate between (h1 , h2 ) and (a11 h1 + a12 h2 + h1 , a21 h1 + a22 h2 + h2 ), when (x1 (0), x2 (0)) ∈ D3 . Proof. If (x1 (0), x2 (0)) ∈ D3 , then x1 (0) < 0, x2 (0) < 0, it follows from (1) that  x1 (1) = a11 σ(x1 (0)) + a12 σ(x2 (0)) + h1 = h1 > 0 (15) x2 (1) = a21 σ(x1 (0)) + a22 σ(x2 (0)) + h2 = h2 > 0 from (15) and (1), it follows that  x1 (2) = a11 h1 + a12 h2 + h1 x2 (2) = a21 h1 + a22 h2 + h2

(16)

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if Ah ≤ −h holds, then we can get from (16) that: x1 (2) ≤ 0 and x2 (2) ≤ 0. When k = 3, it follows from (1) that  x1 (3) = a11 σ(x1 (2)) + a12 σ(x2 (2)) + h1 = h1 (17) x2 (3) = a21 σ(x1 (2)) + a22 σ(x2 (2)) + h2 = h2 It is obviously that the solution of system (1) is oscillate between (h1 , h2 ) and (a11 h1 + a12 h2 + h1 , a21 h1 + a22 h2 + h2 ) for all k > 0, if h1 > 0, h2 > 0, and Ah ≤ −h are guaranteed and (x1 (0), x2 (0)) ∈ D3 . It is shown as {(x1 (k), x2 (k))} = {(x0 , x1 ), (h1 , h2 ), (a11 h1 + a12 h2 + h1 , a21 h1 + a22 h2 + h2 ), (h1 , h2 ), ...}. This completes the proof of Theorem 3.

5

Conclusions

Two-neurons networks often be used as the prototypes to improve our understanding of the computational performance of larger networks. This paper presents a method to analyze performance of the discrete-time LT networks with two-neurons. Some interesting sufficient conditions on the convergence and periodicity of solutions of the system are obtained.

References 1. Yi, Z., Wang, P.A. and Fu, A.W.C.: Estimate of Exponential Convergence Rate and Exponential Stability for Neural Networks. IEEE Trans. Neural Networks 10(6) (1999) 1487-1493 2. Yi, Z., Wang, P.A. and Vadakkepat, P.: Absolute Periodicity and Absolute Stability of Delayed Neural Networks. IEEE Trans. Circuits Syst I 49(2) (2002) 256-261 3. Yi, Z., and Tan, K.K.: Dynamic Stability Conditions for Lotka-Volterra Recurrent Neural Networks With Delays. Phys. Rev. E 66(1) (2002) 011910 4. Hahnloser, R.L.T.: On the Piecewise Analysis of Networks of Linear Threshold Neurons. Neural Networks 11 (1998) 691-697 5. Yi, Z., Tan, K.K. and Lee, T.H.: Multistability Analysis for Recurrent Neural Networks with Unsaturating Piecewise Linear Transfer Functions. Neural Comput. 15(3) (2003) 639-662 6. Yi, Z., and Tan, K.K.: Multistability of Discrete-Time Recurrent Neural Networks with Unsaturating Piecewise Linear Transfer Functions. IEEE Trans. Neural Networks 15(2) (2004) 329-336 7. Tan, K.C., Tang, H.J. and Zhang, W.N.: Qualitative Analysis ror Recurrent Neural Networks with Linear Threshold Transfer Functions. IEEE Trans. Circuits Syst. I 52(5) (2005) 1003-1012 8. Wei, J.J. and Ruan, S.G.: Stability and Bifurcation in a Neural Network Model with Two Delays. Physica D 130(3-4) (1999) 255-272 9. Huang, L.H., and Wu, J.: The Role of Threshold in Preventing Delay-Induced Oscillations of Frustrated Neural Networks with McCulloch-Pitts Nonlineearity. Game Theory and Algevra 11(6) (2001) 71-100 10. Gopalsamy, K. and Leung, I.: Delay Induced Periodicity in a Neural Network of Excitation an Inhibition. Physica D 89(3-4) (1996) 395-426

Existence and Global Attractability of Almost Periodic Solution for Competitive Neural Networks with Time-Varying Delays and Different Time Scales Wentong Liao1 and Linshan Wang2 1

Department of Mathematics, Ocean University of China, Qingdao 266071, People’s Republic of China [email protected] 2 Department of Mathematics, Ocean University of China, Qingdao 266071, People’s Republic of China

Abstract. The dynamics of cortical cognitive maps developed by selforganization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system, besides,this model bases on unsupervised synaptic learning algorithm. Considered the effect of time delays, we prove the existence, uniqueness and global attraction of the almost periodic solution by using fixed theorem and Variation-ofconstants formula.

1

Introduction

In the review of biology, the networks posses synapses whose synaptic weights vary in time. Recently, several papers have discussed the neural systems with time-varying weights.such as [2-5], but all these papers employ a supervised learning dynamics. In [1], A Meyer-Baese et al investigated the dynamics of cortical cognitive maps,and studied the networks model which included both the neural activity level, the short-term memory (STM), and the dynamics of unsupervised synaptic modifications, the long-term memory (LTM). Then A.MeyerBaese put forward the following neural model in [1], and this networks can be considered extensions of Grossberg’s shunting network in [6] or Amari’s model for primitive neuronal competition in [7]. Consider the following competitive neural networks (CNN): ST M : x˙i (t) = −˜ ai xi (t) +

N 

˜ ij f˜(xj (t)) + B ˜i si (t) D

(1)

j=1

LT M : s˙ i (t) = −si (t) + f˜(xi (t)) i = 1, 2, · · · , N. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 297–302, 2006. c Springer-Verlag Berlin Heidelberg 2006 

(2)

298

W. Liao and L. Wang

where xi is the current activity level, ai > 0 is the time constant of the neuron, ˜i is the contribution of the external stimulus term, f˜(xi ) is the neuron’s output. B ˜ ij represents a synaptic connection parameter si (t) is the external stimulus, D between the i − th neuron and the j − th neuron. We know that time-delays always exist in CNN networks, and they may have strong effect to the systems sometimes, so it’s necessary and very meaningful to take into account timedelays’ effect. Based on systems (1)and(2), we consider the following model: x˙i (t) = −˜ ai xi (t) +

N 

˜ ij f˜j (xj (t − τ (t))) + B ˜i si (t − τ (t)) D

(3)

j=1

s˙ i (t) = −si (t) + f˜i (xi (t − τ (t))) xi (t) = φi (t), si (t) = φi+N (t), 0 ≤ τ (t) ≤ τ , i = 1, 2, · · · , N.

(4)

Let xi+N (t) = si (t) , i = 1, 2, · · · , N ⎧ ˜ ij 1 ≤ i, j ≤ N D ⎪ ⎪ ⎨˜ Bi 1 ≤ i ≤ N, j = N + i Cij = ⎪ 1 N + 1 ≤ i ≤ 2N, and i = j + N ⎪ ⎩ 0 else 

a˜i 1 ≤ i ≤ N 1 N + 1 ≤ i ≤ 2N,  f˜i (x) 1 ≤ i ≤ N fi (x) = x N + 1 ≤ i ≤ 2N, bi =

we can rewrite the system [3] and [4] as the following model: x˙ i (t) = −bi xi (t) +

2N 

Cij fj (xj (t − τ (t)))

(5)

j=1

xi (t) = φi (t), t ∈ [−τ, 0], i = 1, 2, · · · , 2N. Where 0 ≤ τ (t) ≤ τ and τ (t) is an almost periodic function[8], fi (x) is a continuous function which satisfies fi (0) = 0, i = 1, 2, · · · , 2N ,Φ = (φ1 , φ2 , · · · , φ2N ) ∈ C([−τ, 0], R2N ) . We let φi (t) =

sup |φi (t + s)|, and we always assume that there exists a

−τ ≤s≤0

continuous solution of system (5) noted x(t) = col{xi (t)} , also we assume the following inequalities hold in this paper: |fi (x) − fi (y)| ≤ Mi |x − y|, f or all x, y ∈ R k = max{ i

2N  1

b j=1 i

d−1 i dj |Cij |Mj } < 1, di > 0, i = 1, 2, · · · , 2N

(6) (7)

Existence and Global Attractability of Almost Periodic Solution

2

299

Main Results and Proof

Theorem 1. Assume (6) − (7) hold, then there exists an unique almost periodic solution of systems (5) . Proof. For f or all x = col{xi } ∈ R2N , let x(t) = max{d−1 i |xi (t)|}, and A = i

{¯ x(t) = col{¯ xi (t)}|¯ x(t) : R → R2N , where x ¯(t) is an almost periodic f unction}, let inducing module ¯ x = sup ¯ x(t), where x ¯(t) ∈ A, and (A,  · ) is a Banach t∈R

space. For any x ¯(t) ∈ A, we consider the following system: x˙ i (t) = −bi xi (t) +

2N 

Cij fj (¯ xj (t − τ (t))), i = 1, 2, · · · , 2N

(8)

j=1

We know system (5) exists an unique almost periodic solution Xx¯ from [8], where  Xx¯ = col{

t

e

−bi (t−s)

−∞

2N  [ Cij fj (¯ xj (s − τ (s)))]ds}

(9)

j=1

We can define the map F : A → A , F (¯ x)(t) = Xx¯ (t), where x ¯ ∈ A. Let B = {¯ x|¯ x ∈ A, ¯ x ≤ k}, obviously, B is a closed convex subset of A. For any x ¯(t) ∈ B, we get  F (¯ x)(t) = col{

t

−∞

2N  e−bi (t−s) [ Cij fj (¯ xj (s − τ (s)))]ds}

≤ sup max{d−1 i { i

t∈R

≤ sup max{d−1 i { i

t∈R

j=1



t

−∞



t

−∞

2N  e−bi (t−s) [ |Cij |Mj |¯ xj (s − τ (s))|]ds} j=1 2N  e−bi (t−s) [ |Cij |Mj dj (d−1 xj (s − τ (s))|]ds)} j |¯ j=1



≤

2N t  −1 −bi (t−s) sup max{di { e [ |Cij |Mj dj ¯ x]ds)} t∈R i −∞ j=1

= max i

2N  d−1 i [ |Cij |Mj dj ¯ x] ≤ k 2 < k bi j=1

(10)

(11)

Then we conclude: F (¯ x) ∈ B . Now we prove the uniqueness of the almost periodic solution, let x ˜ ∈ B, for any x ¯ ∈ B, we obtain that F (¯ x) − F (˜ x) = sup F (¯ x)(t) − F (˜ x)(t) ≤ sup max{d−1 i { t∈R

i



t∈R t

−∞

e−bi (t−s)

2N  j=1

|Cij ||fj (¯ xj (s − τ (s))) − fj (˜ xj (s − τ (s)))|ds}

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≤

 t 2N  −1 sup max{di { e−bi (t−s) |Cij |Mj |¯ xj (s t∈R i −∞ j=1

≤

 t 2N  −1 sup max{di { e−bi (t−s) |Cij |Mj dj ¯ x− t∈R i −∞ j=1

≤ max{ i

− τ (s)) − x ˜j (s − τ (s))|ds} x ˜ds}

2N 1  −1 d |Cij |Mj dj ¯ x−x ˜} ≤ k¯ x−x ˜ bi j=1 i

(12)

Since k < 1, thus F is a compact mapping on B. By using Banach fixed theorem we know x∗ is the unique almost periodic solution of system (5). Supposed x∗ is the unique almost periodic solution of system (6, for any other solution x(t) = (x1 , x2 , · · · , x2N ), we have x˙ i (t)− x˙ ∗i (t) = −bi (xi (t)− x∗i (t))+

2N j=1

Cij [fj (xj (t− τ (t)))− fj (x∗j (t− τ (t)))]

Let yi (t) = xi (t) − x∗i (t),  gj (yj (t − τ (t))) = fj yj (t − τ (t)) + x∗j (t − τ (t)) − fj (x∗j (t − τ (t))), ϕi (t) = φi (t) − x∗i (t), t ∈ [−τ, 0]; i, j = 1, 2, · · · , 2N, from (6), we get |gi (y)| ≤ |Mi ||y| for any y ∈ R, then we can rewrite the above system as following y˙ i (t) = −bi yi (t) +

2N 

Cij gj (yj (t − τ (t)))

(13)

j=1

xi (t) = ϕi (t), t ∈ [−τ, 0], i = 1, 2, · · · , 2N. Obviously, the global attraction of x∗ is equivalent to the global attracting of system (12)’s solution y = 0, thus, we only consider the global attraction of y = 0 to system (12). Similar to literature [9], we can get the following conclusion. Theorem 2. Assumed (6)−(7) holds, then for any ϕ(t) = (ϕ1 (t), ϕ2 (t), · · · , ϕ2N (t)), system (8) exists a constant V > 0 such that y(t) < V for any t ≥ 0. Proof. For any ϕ ∈ C([−τ, 0], R2N ), by using (7) we know that there exists a constant V > 0 which satisfies ϕ < V and (1 − k)V > 0 . If y(t) < V does not hold, then y(t) ≥ V , there must be a constant t0 such that y(t0 ) = V, and y(t) < V for 0 ≤ t < t0 .By using (8) and variation-of-constants formula , we get y(t0 ) ≤

max{e−bi t0 d−1 i |ϕi (0)| i

 +

t0

e−bi (t0 −s) d−1 i .

0

|Cij |Mj dj (d−1 j |yj (s − τ (s))|)ds}  t0 2N  −bi t0 ≤ max{e V + e−bi (t0 −s) d−1 i |Cij |Mj yj (s − τ (s)))ds} i

0

j=1

Existence and Global Attractability of Almost Periodic Solution

≤ max{e

−bi t0

i



t0

e−bi (t0 −s)

V + 0

≤ max{e

−bi t0

i

2N 

301

d−1 i |Cij |Mj V ds}

j=1

V + (1 − e

−bi t0

)kV } < V

(14)

this result is contradict to y(t0 ) = V , thus conclusion holds. Theorem 3. Assume (6) − (7) holds, then the solution y(t) = 0 is global attraction to system (12). Proof. Now we prove β = 0, by using Variation-of-constants formula together with |gi (y)| ≤ |Mi ||y|, we have |y(t)| ≤ e

−bi t



t

|ϕi (0)| +

e−bi (t−s)

0

2N 

|Cij |Mj |yj (s − τ (s))|)ds}

(15)

j=1

From theorem 2, we conclude ϕ < V and y(t) < V for any t ≥ 0 .Since 2N ∞ −1 bi > 0 , then exists a constant T > 0 such that T e−bi s di dj |Cij |Mj V ds ≤ ε j=1

for any ε > 0, i = 1, 2, · · · , 2N , let lim sup y(t) = β ≥ 0 , then there exists a t→∞

constants T0 ≥ 0 such that y(t − τ (t)) ≤ β for any t ≥ T0 . Now we assume t ≥ T0 + T , from (14) we have |y(t)| |ϕi (0)| ≤ e−bi t + di di < e−bi t V +





t

e−bi (t−s)

0

t−T

e−bi (t−s)

0



t

+

e−bi (t−s)

2N  j=1



t

e−bi s

T t

e−bi (t−s)

t−T

≤e

−bi t

2N 

t

+ t−T

2N 

d−1 i |Cij |Mj dj V ds

j=1

d−1 i |Cij |Mj dj βds

j=1



∞

V +

e

−bi s

T



−1 d−1 i |Cij |Mj dj (dj |yj (s − τ (s))|)ds

−1 d−1 i |Cij |Mj dj (dj |yj (s − τ (s))|)ds

< e−bi t V + e−bi (t−T )

+

2N  j=1

t−T



2N  1 |Cij |Mj dj (d−1 j |yj (s − τ (s))|)ds d i j=1

e−bi (t−s)

2N 

d−1 i |Cij |Mj dj V ds

j=1 2N 

d−1 i |Cij |Mj dj βds

j=1

≤ e−bi t V + ε + (1 − e−bi T )

2N  j=1

−1 b−1 i di |Cij |Mj dj β

(16)

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From (14) we conclude −bi t y(t) = max(d−1 V + ε + (1 − e−bi T ) i |y(t)|) ≤ max[e i

·

i

2N 

−1 b−1 i di |Cij |Mj dj β]

j=1

< max[e−bi t V + ε + (1 − e−bi T )kβ] i

< max[e−bi t V + ε + kβ] i

(17)

Let t → ∞, ε → 0, thus β ≤ kβ, if β = 0, then k ≥ 1, this result is contradict to H2 , thus β ≡ 0, also, lim yi (t) = 0, i = 1, 2, · · · , 2N , from this result, it is easy t→∞

to prove that the almost periodic solution x∗ of system (5) is global attracting.

Acknowledgement This paper was supported by the National Natural Science Foundation of China under Grant 10171072.

References 1. Meyer-Baese, A., Pilyugin, S.S., Chen, Y.: Global Exponential Stability of Competitive Neural Networks with Different Time Scale. IEEE Trans. Neural Networks 14(5) (2003) 716-719 2. Jin, L., Gupta, M.: Stabale Dynamic Backpropagation Learning in Recurrent Neural Networks. IEEE Trans. Neural Networks 10(11) (1999) 1321-1334 3. Galicki, M., Leistitz, L., Witte, H.: Learning Continuous Trajectories in Recurrent Neural Networks with Time-dependent Weights. IEEE Trans. Neural Networks 10(5) (1999) 741-756 4. Suykens, J., Moor, B., Vandewalle, J.: Robust local Stability of Multilayer Recurrent Neural Networks. IEEE Trans. Neural Networks 11(1) (2000) 222-229 5. Wang, L.S., Xu, D.Y.: Stability Analysis of Hopfield Neural Networks with Time Delay. Applied Mathematics and Mechanics 23 (2002) 250-252 6. Grossberg, S.: Competition,Decision and Consensus. J. Math. Anal. Applicat 66 (1978) 470-493 7. Amari, S.: Competitive and Cooperater Aspects in Dynamics of Neural Excition and Self-organization. Competition Cooperation Neural Networks 20 (1982) 1-28 8. He, C.Y.: Almost Periodic Differential Equations. Higher Education Publishing House, Beijing (1992) 9. Zhao, H.Y., Wang, G.L.: Existence and Global Atrractivity of Almost Periodic Solutions for Hopfield Neural Networks with Variable Delay. Acta Mathematica Scientia A 24(6) (2004) 723-726 10. Lu, W., Chen, T.: Global Exponential Stability of Almost Periodic Trajectory of a Large Class of Delayed Dynamical Systems. Science in China A. Mathematics,(2005) 1015-1026. 11. Cao, J.D, Chen, A.P, Huang, X.: Almost Periodic Attractor of Delayed Neural Networks with Variable Coefficients. Physics letters A 340 (2005) 104 -120

Global Synchronization of Impulsive Coupled Delayed Neural Networks Jin Zhou1 , Tianping Chen2 , Lan Xiang3 , and Meichun Liu3 1

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P.R. China 2 Laboratory of Nonlinear Science, Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China {Jinzhou, Tchen}@fudan.edu.cn 3 Department of Applied Mathematics and Physics, Hebei University of Technology, Tianjin, 300130, P.R. China {Xianglanhtu, Mcliu2005}@126.com.cn

Abstract. This paper formulates and studies a model of impulsive coupled delayed neural networks. Based on stability theory of impulsive dynamical systems, a simple but less-conservative criterion is derived for global synchronization of such coupled neural networks. Furthermore, the theoretical result is applied to a typical chaotic delayed Hopfied neural networks, and is also illustrated by numerical simulations.

1

Introduction

Recently, there has been increasing interest in the study of synchronization dynamics of coupled neural networks. Because they can exhibit many interesting phenomena such as spatiotemporal chaos, auto waves, spiral waves and others. In addition, it has been found that synchronization of coupled neural networks has potential applications in many fields including secure communication, parallel image processing, biological systems, information science, etc (see [1, 2, 3, 4, 5, 6, 7]). In many evolutionary systems there are two common phenomena: delay effects and impulsive effects [4, 5, 8, 9]. For example, a signal or influence travelling through a network often is associated with time delays, and this is very common in biological and physical systems. On the other hand, many evolutionary processes, particularly some biological systems such as biological neural networks and bursting rhythm models in pathology, as well as frequency-modulated signal processing systems, and flying object motions, are characterized by abrupt changes of states at certain time instants. This is the familiar impulsive phenomena. Therefore, the investigation of synchronization dynamics for impulsive coupled delayed neural networks is an important step for practical design and application of neural networks. This paper formulates and studies a model of impulsive coupled delayed neural networks. Based on stability theory of impulsive dynamical systems, a simple but less conservative criterion is derived for global synchronization of the impulsive J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 303–308, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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coupled neural networks. It is shown that global synchronization of impulsive coupled delayed neural networks are dependent on both the time delays and the impulsive effects in the coupled neural networks.

2

Model and Preliminaries

Consider an isolate delayed neural networks, which is described by the following set of differential equations with time delays [3]: x(t) ˙ = −Cx(t) + Af (x(t)) + Aτ g(x(t − τ )) + I(t),

(1)

where x(t) = (x1 (t), · · · , xn (t)) , C = diag(c1 , . . . , cn ) with cr > 0, A = (a0rs )n×n , Aτ = (aτrs )n×n , I(t) = (I1 (t), · · · , In (t)) , f (x(t)) = [f1 (x1 (t)), · · · , fn (xn (t))] and g(x(t)) = [g1 (x1 (t), · · · , gn (xn (t))] , here we assume that the activation functions fr (x) and gr (x) are globally Lipschitz continuous, i.e., (A1 ) There exist constants kr > 0, lr > 0, r = 1, 2, · · · , n, for any two different x1 , x2 ∈ R, such that 0≤

fr (x1 ) − fr (x2 ) ≤ kr , x1 − x2

|gr (x1 ) − gr (x2 )| ≤ lr |x1 − x2 |, r = 1, 2, · · · , n.

Based on the structure of coupled delayed neural networks in [3], we formulate an array of N linearly coupled delayed neural networks with impulsive effects described by the following measure differential equation Dxi (t) = −Cxi (t) + Af (xi (t)) + Aτ g(xi (t − τ )) + I(t) +

N 

bij Γ xj (t)Dwj (t),

i = 1, 2 · · · , N.

(2)

j=1

in which xi (t) = (xi1 (t), · · · , xin (t)) is the state of the ith delayed neural networks, the operator D denotes the distribution derivative, the bounded variation functions wi : [t0 , +∞) → R are right-continuous on any compact subinterval of [t0 , +∞), Dwi depicts the impulsive effects of connecting configuration in the coupled networks. Without loss of generality, we assume that Dwj = 1 +

+∞ 

uk δ(t − tk ),

j = 1, 2 · · · , N.

(3)

k=1

where the fixed impulsive moments tk satisfy tk−1 < tk and limk→∞ tk = +∞, δ(t) is the Dirac function, uk represents strength of the impulsive effects of connection between the jth neural network and the ith neural network at time tk . For simplicity, we further assume that the inner connecting matrix Γ = diag(γ1 , · · · , γn ), and the coupling matrix B = (bij )N ×N is the Laplacian matrix, i.e. a symmetric irreducible matrix with zero-sum and real spectrum. This implies that zero is an eigenvalue of B with multiplicity 1 and all the other eigenvalues

Global Synchronization of Impulsive Coupled Delayed Neural Networks

305

of B are strictly negative. The initial conditions of Eq. (2) are given by xi (t) = φi (t) ∈ P C([t0 − τ, t0 ], Rn ), where P C([t0 − τ, t0 ], Rn ) denotes the set of all functions of bounded variation and right-continuous on any compact subinterval of [t0 −τ, t0 ]. We always assume that Eq. (2) has a unique solution with respect to initial conditions. Clearly, if uk = 0, then the model (2) becomes the well-known continuous coupled delayed neural networks [3]. Now, some definitions with respect to synchronization of impulsive coupled delayed neural networks and the famous Halanay differential inequality on impulsive delay differential inequality are listed as follows: Definition 1. The hyperplane Λ = {(xT1 , · · · , xTN )T ∈ Rn×N ; xi = xj , i, j = 1, 2 · · · , N.} is said to the synchronization manifold of the impulsive coupled delayed neural network (2). Definition 2. Synchronization manifold Λ is said to be globally exponential stable, equivalently, the impulsive coupled delayed neural network (2) is globally exponentially synchronized, if there exist constants ε > 0 and M > 0, for all φi (t) ∈ P C([t0 − τ, t0 ], Rn ), such that for i, j = 1, 2 · · · , N, xi (t) − xj (t) ≤ M e−ε(t−t0 ) .

(4)

Lemma 1. [8, 9] Suppose p > q ≥ 0 and u(t) satisfies the scalar impulsive differential inequality  + D u(t) ≤ −pu(t) + q( sup u(s)), t = tk , t ≥ t0 , t−τ ≤s≤t (5) u(tk ) ≤ αk u(t− ), u(t) = φ(t), t ∈ [t0 − τ, t0 ]. k where u(t) is continuous at t = tk , t ≥ t0 , u(tk ) = u(t+ k ) = lims→0+ u(tk + s) − u(tk + s) exists, φ ∈ P C([t0 − τ, t0 ], R). Then and u(t− ) = lim s→0 k  u(t) ≤ ( θk )e−μ(t−t0 ) ( sup φ(s)), (6) t0
t0 −τ ≤s≤t0

where θk = max{1, |αk |} and μ > 0 is a solution of the inequality μ−p+qeμτ ≤ 0.

3

Synchronization Criteria

Theorem 1. Consider the impulsive coupled delayed neural networks (2). Let the eigenvalues of its coupling matrix B be ordered as 0 = λ1 > λ2 ≥ λ3 ≥ · · · , λN . Assume that, in addition to (A1 ), the following conditions are satisfied for all i = 1, 2, · · · , n and k ∈ Z + = {1, 2, · · · , ∞} (A2 ) There exist n positive numbers δ1 , · · · , δn , and two numbers pi = δi + ci −

(a0ii )+ ki

 n  n n 1 1 τ 1 τ 0 0 − |aij |kj + |aji |ki − |aij |lj , qi = |a |li , 2 2 j=1 2 j=1 ji j=1

j =i

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such that p = min1≤i≤n {2pi } > q ⎧ = max1≤i≤n {2qi } and γi λ(γi ) + δi ≤ 0, where ⎨ λ2 , if γi > 0, (a0ii )+ = max{a0ii , 0} with λ(γi ) = 0, if γi = 0, ⎩ λN , if γi < 0. (A3 ) Let μ > 0 satisfy μ − p + qeμτ ≤ 0, and



ln θ  1 k θk = max 1, , θ = sup 1 − uk γj λ(uk γj ) k∈Z + tk − tk−1 such that uk γj λ(uk γj ) < 1 and θ < μ. Then the impulsive coupled delayed neural network (2) is globally exponentially synchronized. N Brief Proof. Let s(t) = N1 k=1 xk (t), vi (t) = xi (t) − s(t) (i = 1, 2, · · · , N ). Then, Eq. (2) can be rewritten as Dvi (t) = −Cvi (t) + A[f (xi (t)) − f (s(t))] + Aτ [g(xi (t − τ )) −g(s(t − τ ))] +

N 

bij Γ vj (t)Dwj (t) + J,

i = 1 · · · , N. (7)

j=1

where J = Af (s(t)) + Aτ g(s(t − τ )) + N1 Let us construct a Lyapunov function

N

k=1 [Af (xk (t))

+ Aτ g(xk (t − τ ))].

1  v (t)vi (t). 2 i=1 i N

V (t) =

From Condition (A1 ), and note that D V (t) ≤ +

N  n  

N

(8)

i=1 vi (t)

= 0, we can get for t = tk ,

(a0rr )+ kr

1 0 + (|a |ks + |a0sr |kr ) 2 s=1 rs n

− δ r − cr +

i=1 r=1

s =r

  n N  1 1 τ 2 2 + |aτrs |ls vir (t) + |asr |lr vir (t − τ ) + vi (t) 2 s=1 2 s=1 i=1 n 

×

N 

 bij Γ vj (t) + diag(δ1 , . . . , δn )vi (t)

j=1

≤ −pV (t) + qV (t − τ ) +

n 

v¯j (t)(γj B + δj IN )¯ vj (t)

(9)

j=1

N def  where v1j (t), · · · , v¯N j (t)) ∈ L = z = (z1 , · · · , zN ) ∈ RN | i=1 zi  v¯j (t) = (¯ n = 0 , from which it can be concluded that if γj λ(γj ) + δj ≤ 0, then j=1 v¯j (t) (γj B + δj IN )¯ vj (t) ≤ 0. This leads to D+ V (t) ≤ −pV (t) + q( sup

t−τ ≤s≤t

V (s))

(10)

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307

On the other hand, from (7) and (8), and by using the properties of Dirac measure, we have V (tk ) = V (t− k)+

n 

v¯j (t)uk γj B¯ vj (t) ≤ V (t− k ) + uk γj λ(uk γj )V (tk ), (11)

j=1

which implies that if uk γj λ(uk γj ) < 1, and k ∈ Z + , V (tk ) ≤

1 V (t− k ). 1 − uk γj λ(uk γj )

(12)

It follows from Lemma 1 that if θ < μ for all t > t0 , V (t) ≤ e−(μ−θ)(t−t0 ) (

sup

t0 −τ ≤s≤t0

V (s)).

(13)

This completes the proof of Theorem 1. Remark 1. It can be seen from (A2 ) and (A3 ) that global synchronization of the impulsive coupled delayed neural networks (2) not only depends on the coupling matrix B, the inner connecting matrix Γ, and the time delay τ, but also is determined by the strength values θk of impulsive effects and the impulsive interval tk − tk−1 . Example 1. Consider a model of impulsive coupled delayed neural networks (2),  where xi (t) = (xi1 (t), xi2 (t)) , f (xi (t)) i1 (t)), tanh(x i2 (t))) ,  = g(x  i (t)) = (tanh(x   10 2.0 −0.1 −1.5 −0.1 I(t) = (0, 0) (i = 1, 2, 3), C = ,A = and Aτ = . 01 −5.0 3.0 −0.2 −2.5 It should be noted that the isolate neural networks x(t) ˙ = −Cx(t) + Af (x(t)) + Aτ g(x(t − 1)) is actually a chaotic delayed Hopfied neural networks [4, 5] (see ⎡ ⎤ −8 2 6 Fig. 1). Let B = ⎣ 2 −4 2 ⎦ and Γ = diag(γ1 , γ2 ), then B has eigenvalues 6 2 −8 0, −6 and −14. For simplicity, we consider the equidistant impulsive interval τk − τk−1 = 0.01 and uk = −0.0005 (k ∈ Z + ). By taking kr = lr = 1 and δr = 12 (r = 1, 2), it is easy to verify that if γ1 = γ2 = 2, then all the conditions of Theorem 1 are satisfied. Hence, the impulsive coupled delayed neural networks (2) will achieve global synchronization. The simulations result corresponding to this situations is shown in Fig. 2.

4

Conclusions

In this paper, a general model of a system consisting of linearly and diffusively impulsive coupled delayed neural networks has been formulated and its synchronization dynamics have been studied. A simple criterion for global synchronization of such networks has been derived analytically. It is shown that the theoretical results can be applied to some typical chaotic neural networks such as delayed Hopfied neural networks and delayed cellular neural networks (CNN).

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3

0.6

x (t)−x (t), y (t)−y (t) ( i=2,3)

4

2

0

x3(t)−x1(t) 0.2

i

Y

1

1

x2(t)−x1(t) 0.4

0

1

−1

y2(t)−y1(t)

i

−0.2

−2

−0.4

−3 −4 −1

−0.5

0

Fig.1 X

0.5

1

−0.6 0

y3(t)−y1(t)

1

2

Fig. 2

3

4

5

time t

Fig. 1. A fully developed double-scroll-like Fig. 2. Global synchronization of the imchaotic attractors of the isolate delayed pulsive coupled delayed neural networks (2) in time interval [0, 5]. Hopfied neural networks.

Acknowledgements This work was supported by the National Science Foundation of China (Grant no. 60474071), the China Postdoctoral Science Foundation (Grant no. 20040350121) and the Science Foundation of Education Commission of Hebei Province (Grant no. 2003013).

References 1. Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives, and Applications, World Scientific Pub. Co, Singapore (1998) 2. Wu, C. W., Chua, L. O.: Synchronization in an Array Linearly Coupled Dynamical System. IEEE Trans. CAS-I. 42 (1995) 430-447 3. Chen, G., Zhou, J., Liu, Z.: Global Synchronization of Coupled Delayed Neural Networks and Applications to Chaotic CNN Model. Int. J. Bifur. Chaos. 14 (2004) 2229-2240 4. Zhou, J., Chen, T., Xiang, L.: Robust Synchronization of Delayed Neural Networks Based on Adaptive Control and Parameters Identification. Chaos, Solitons, Fractals. 27 (2006) 905-913 5. Zhou, J., Chen, T., Xiang, L.: Chaotic Lag Synchronization of Coupled Delayed Neural Networks and Its Applications in Secure Communication. Circuits, Systems and Signal Processing. 24 (2005) 599-613 6. Zhou, J., Chen, T., Xiang, L.: Robust Synchronization of Coupled Delayed Recurrent Neural Networks. In: Yin, F., Wang, J., Guo, C. (eds.): Advances in Neural Networks - ISNN 2004. Lecture Notes in Computer Science, Vol. 3173. Springer-Verlag, Berlin Heidelberg New York (2004) 144-149 7. Zhou, J., Chen, T., Xiang, L.: Adaptive Synchronization of Delayed Neural Networks Based on Parameters Identification. In: Wang, J., Liao, X., and Yi, Z. (eds.): Advances in Neural Networks - ISNN 2005. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 308-313 8. Yang, Z., Xu, D.: Stability Analysis of Delay Neural Networks with Impulsive Effects. IEEE Trans. CAS-II. 52 (2005) 517-521. 9. Yang, T.: Impulsive Control Theory. Springer-Verlag, Berlin Heidelberg New York (2001)

Synchronization of a Class of Coupled Discrete Recurrent Neural Networks with Time Delay Ping Li Computational Intelligence Laboratory, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China [email protected] http://cilab.uestc.edu.cn

Abstract. This paper studies the synchronization of a class of coupled discrete recurrent neural networks with time delay. The local and global conditions are obtained by using Lyapunov functional and linear matrix inequality methods.

1

Introduction

Recently, there has been an increasing activity studying the arrays of coupled systems in many research fields[1, 2, 4]. In fact, arrays of coupled systems not only can exhibit many interesting properties, such as synchronization[5, 6, 11], auto waves[7], etc., but can be used in many applications, such as image processing[7], secure communication[8] and pattern storage, etc. Therefore, the study of the synchronization of coupled neural networks is very significant for both theoretical research and practical use. In [9], the authors have investigated synchronization of arrays of continuous delayed neural networks with linear coupling. The model of coupled neural networks we will study in this paper is different from that used in [9], it does not contain linear terms and is discrete. The convergence of this kind of continuous neural networks has been well discussed by[10]. This kind of model studied in the paper can be written as xi (k + 1) =

n 

[aij gj (xj (k)) + bij gj (xj (k − h))] + Ii (k), (i = 1, 2, ..., n)

(1)

j=1

for t ≥ 0, where each xi (k) is the state of the ith neuron, aij and bij are constant intra-connection weights, the delay h ≥ 0, gj (xj (k)) denotes the output of the jth neuron at time t, gj (xj (k − h)) denotes the output of the jth neuron at time t − h, and Ii (k) is the input to the ith neuron. It also can be rewritten as following compact form: x(k + 1) = Ag(x(k)) + Bg(x(k − h)) + I(k) 

(2)

This work was supported by National Science Foundation of China under Grant 60471055 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20040614017.

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 309–315, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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where x(k) = (x1 (k), x2 (k), ..., xn (k))T ∈ Rn , A = (aij )n×n , B = (bij )n×n , g(x(k)) = [g1 (x1 (k)), ..., gn (xn (k))]T , g(x(k − h)) = [g1 (x1 (k − h)), ..., gn (xn (k − h))]T and I(k) = (I1 (k), I2 (k), ..., In (k))T . An array of such discrete recurrent neural networks connecting each other can be described as: xi (k + 1) = Ag(xi (k)) + Bg(xi (k − h)) + I(k) +

n 

cij Γ xj (k), i = 1, ..., N (3)

j=1

where xi (k) = (xi1 (k), ..., xin (k))T ∈ Rn , xip denotes the pth neuron of the ith neural network, C = (cij )N ×N denotes the topology of the coupled networks , it can be small-world or scale-free coupling, and Γ = diag(γ1 , ..., γn ). In this paper, we study the local and global stability of the synchronization manifold for arrays of discrete recurrent neural networks with diffuse coupling and give several sufficient condition to control the networks to synchronization by LMI approach. The rest of this paper is organized as follows: In Section 2, some necessary definitions and lemmas are given; In Section 3, we will give the local and global synchronization results which can be easily used in practice.

2

Preliminaries

In what follows, we present the definitions and lemmas are needed in this paper. Definition 1. The set S = x = (x1 (k)T , x2 (k)T , ..., xN (k)T )T is called synchronization manifold, thereinto xi (k) = xj (k), i, j = 1, ..., N . To measure the distance between the various cells, we use the definitions introduced in [6]. Definition 2. M1 (k) are matrices M with entries in Fk such that each row of M contains zeros and exactly one αIk and one −αIk for some nonzero α, where Fk is the subfield of n × n matrices as Fk = {αIk : α ∈ R} and Ik denote the n × n real identity matrix. Definition 3. M2 (k) are matrices M in M1 (k) such that for any pair of indices i and j there exist indices i1 , i2 , ..., il with i1 = i and il = j such that for all 1 ≤ q < l,Mp,iq  = 0 and Mp,iq+1  = 0 for some p. Definition 4. An irreducible matrix A = (aij )n×n is said to satisfy condition N A1 if aij ≥ 0, aii = − j=1 aij . Next, we give some lemmas: Lemma 1. if A is a matrix satisfying A1, then the following holds: 1. A has eigenvalue 0 with multiplicity 1 and corresponding eigenvalue [1, 1, ..., 1]T ; 2. all nonzero eigenvalues of A are negative. Lemma 2. [6] Let x = (x1 , x2 , ..., xN )T where xi ∈ Rn ,i = 1, 2, ..., N , then x ∈ S if and only if  M x = 0 (4) holds, for some M ∈ M2 (n).

Synchronization of a Class of Coupled Discrete Recurrent Neural Networks

3

311

Synchronization Analysis

In this section, we will mainly use Lyapunov method and linear matrix inequality approach to obtain asymptotic synchronization of the above dynamic systems. By the definition 1, we denote by s(k) the synchronization state such that: x1 (k) = x2 (k) = · · · = xN (k) = s(k), k → ∞

(5)

It is clear that the synchronization state satisfies the identical equation s(k + 1) = Ag(s(k)) + Bg(s(k − h)) + I(k)

(6)

Firstly, we investigate local stability of the synchronization manifold S for the coupled system(3). Theorem 1. Suppose that 0 = λ1 > λ2 ≥ λ3 ≥ ... ≥ λN

(7)

is the N eigenvalues of symmetric matrix C satisfying condition A1, if the following holds: w(k + 1) = (A + λi Γ )w(k) + B  w(k − h) (8) for N − 1 nonzero eigenvalues, then the system(3) is asymptotically synchronized,where A = AJ(k), B  = BJ(k − h) and J(k) := g  (s(k)) is the Jacobian of g(x(k)) at s(k). Proof : Let xi (k) = s(k) + δi (k), then we have the following variation equation for system(3): δi (k + 1) = A δi (k) + B  δi (k − h) +

N 

cij Γ δj (k)

(9)

j=1

Let δ(k) = (δ1 (k), ..., δN (k)) ∈ Rn,N , then Eq.(9) can be rewritten as follows: δ(k + 1) = A δ(k) + B  δ(k − h) + Γ δ(k)C T

(10)

Obviously, symmetric C T has the real Schur decomposition:C T = U ΛU T ,where U T = U −1 and Λ = diag(λ1 , ..., λN ). Now, Let e(k) = δU we have e(k) = A e(k) + B  e(k − h) + Γ e(k)Λ

(11)

In component form, we have ei (k) = (A + λi Γ )ei (k) + B  ei (k − h), i = 1, ..., N

(12)

Therefore, if system(12) is asymptotically stable, then δ(k) approaches origin. That is, system(3) is asymptotically synchronized. Theorem 1 is proved. Next, we give sufficient conditions for local and global stability of the system’s synchronization respectively.

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Theorem 2. If there exist two positive matrices P, Q > 0, a constant ε > 0 and an integar k0 > 0 such that  T  (A + λi Γ )P (A + λi Γ ) + (Q − P ) B T P (A + λi Γ ) ≤ −ε (13) B T P (A + λi Γ ) B T P B  − Q for all i = 2, ..., N and k ≥ k0 . then system(3) is asymptotically synchronized for any h > 0. Proof : Constructing a Lyapunov-Krasovskii functional as following: V (w(k)) = wT (k)P w(k) +

k−1 

wT ( )Qw( )

(14)

=k−h

Along system(8), then we have ΔV = V (w(k + 1)) − V (w(k)) = wT (k)(AT + λi Γ )P (A + λi Γ )w(k) + wT (k − h)B T P B  w(k − h) −wT (k)P w(k) + wT (k)Qw(k) − wT (k − h)Qw(k − h) +2wT (k − h)B T P (A + λi Γ )w(k) ⎡ ⎤  T (AT + λi Γ )P (A + λi Γ )   w(k) w(k) ⎣ +(Q − P ) B T P (A + λi Γ ) ⎦ = w(k − h) w(k − h) B T P (A + λi Γ ) B T P B  − Q ≤ −ε(w(k)2 + w(k − h)2 ) < 0

(15)

which implies system(3) is asymptotically synchronized. The proof is completed. Next, we investigate another important issue–global synchronization.We will show that the trajectory will converge to the synchronization manifold from any initial value under some mild conditions. For simplicity, assume that Γ = diag(γ1 , ..., γn ) is a diagonal matrix with γi = 1. Then, based on Lyapunov stability theory, the following result can be derived. Theorem 3. √ Suppose that C satisfies condition A1. If there exist a positive constant c ≥ 3, a semi-positive definitive matrix Q = diag(q1 , q2 , ..., qn ) , an irreducible matrix T satisfying condition A1 such that  T  2A A + Q AT B <0 (16) BT A 2B T B − Q and

1 T ≥0 (17) c2 holds, then the synchronization manifold S of system(3) is globally asymptotically stable. CT T C −

˜ ∈ M2 (1) Proof : By definition 3 and Lemma 6 in[6], there exists a p×N matrix M T ˜ ˜ ˜ such that T = −M M , and M = M ⊗ In ∈ M2 (N ). Mi = (Mi1 , Mi2 , ..., MiN )

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is the ith row of M, where Mii1 = αi In , Mii2 = −αi In and Mij = 0, if j  = i1 , j  = i2 . Denote x(k) = (x1 (k), ..., xN (k))T ∈ RnN ,where xi (k) = (xi1 (k), ..., xin (k))T ∈ Rn . Let y(k) = Mx(k)

A = IN ⊗ A

B = IN ⊗ B

C = C ⊗ In

Q = IN ⊗ Q

g(x(k)) = (g(x1 (k)), g(x2 (k)), ...., g(xN (k)))T then system(3) can be rewritten as following: y(k + 1) = MAg(x(k)) + MBg(x(k − h)) + MCx(k)

(18)

Choose a Lyapunov functional for (16) V (y(k)) = y T (k)y(k) +

k−1 

g T (x(δ))MT QMg(x(δ))

(19)

δ=k−h

then we have ΔV = V (y(k + 1)) − V (y(k)) = gT (x(k))AT MT MAg(x(k)) + gT (x(k − h))BT MT MBg(x(k − h)) +xT (k)CT MT MCx(k) + 2gT (x(k))AT MT MBg(x(k − h)) +2gT (x(k − h))BT MT MCx(k) + 2gT (x(k))AT MT MCx(k) − y T (k)y(k) +gT (x(k))MT QMg(x(k)) − gT (x(k − h))MT QT Mg(x(k − h))

(20)

By quadratic inequality, we have 2gT (x(k − h))BT MT MCx(k) ≤ gT (x(k − h))BT MT MBg(x(k − h)) +xT (k)CT MT MCx(k)

(21)

and 2gT (x(k − h))AT MT MCx(k) ≤ gT (x(k − h))AT MT MAg(x(k − h)) +xT (k)CT MT MCx(k)

(22)

By the structure of M, following equations are easy to verify[9]: MA = A1 M

MB = B1 M

where A1 = Ip ⊗ A, B1 = Ip ⊗ B. Therefore, we have ΔV ≤ 2gT (x(k))MT AT1 A1 Mg(x(k)) + 2gT (x(k − h))MT BT1 B1 Mg(x(k − h)) +3xT (k)CT MT MCx(k) + 2gT (x(k))MT AT1 B1 Mg(x(k − h)) − y T (k)y(k) +gT (x(k))MT QMg(x(k)) − gT (x(k − h))MT QT Mg(x(k − h))

(23)

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By (16) and Lemma 5 in [11], 3xT (k)CT MT MCx(k) − y T (k)y(k) = 3xT (k)CT MT MCx(k) − xT (k)MT Mx(k) 3 ≤ ( 2 − 1)xT (k)MT Mx(k) ≤ 0 c

(24)

For simplicity, we denote by H the matrix (15), we have ΔV ≤

p   j=1

+(

αj (g(xj1 (k)) − g(xj2 (k))) αj (g(xj1 (k − h) − g(xj2 (k − h))))

T

 H

αj (g(xj1 (k)) − g(xj2 (k))) αj (g(xj1 (k − h) − g(xj2 (k − h))))

3 − 1)xT (k)MT Mx(k) c2



(25)

By conditions given in theorem 3, H < 0,which implies ΔV < 0. Thus, theorem 3 is proved. By the theory of linear matrix inequality, we only need that 2B T B − Q < 0,

2AT A + Q − AT B(2B T B − Q)−1 BAT < 0

(26)

holds, then we can obtain LMI condition. So our result is convenient to use.

4

Conclusion

This paper studies the synchronization of coupled discrete recurrent neural networks with time delay, giving some sufficient conditions for guarantee the synchronization of coupled discrete neural networks. These results could be used in the design of coupled networks.

References 1. Wu, C., Chua, L.O.: A Unified Framework for Synchronization and Control of Dynamical Systems. Int. J. Bifurcation Chaos 4(4) (1994) 979-998 2. Zheleznyak, A., Chua, L.O.: Coexistence of Low- and High-Dimensional SpatioTemporal Chaos in a Chain of Dissipatively Coupled Chua’s Circuits. Int. J. Bifurcation Chaos 4(3) (1994) 639-674 3. Forti, M.: On Global Asymptotic Stability of a Class of Nonlinear Systems Arising in Neural Network Theory. Journal of Differential Equations 113 (1994) 246-264 4. Perez-Munuzuri, A., Perez-Villar, V.: Autowaves for Image Processing on a Two-Dimensional CNN Array of Excitable Nonlinear Circuits:Flat and Wrinkled Labyrinths. IEEE. Circuits Syst.-I 40 (1993) 174-181 5. Louis, M., Thomas, C.: Synchronization in Chaotic Systems. Phys. Rev. Lett. 64 (1990) 821-824 6. Wu, C., Chua, L.O.: Synchronization in An Array of Linearly Coupled Dynamical Systems. IEEE Trans. on Circuit and systems-I 42 (1995) 430-447 7. Krinsky, V., Biktashev, V., Efimov, L.: Autowaves Principle for Parallel Image Processing. Phys. D 49 (1991) 247-253

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8. Zhang, Y., He, Z.: A Secure Communication Scheme Based on Cellular Neural Networks. Proc. IEEE Int. Conf. Intelligent Processing systems 1 (1997) 521-524 9. Lu, W., Chen, T.: Synchronization of Coupled Connected Neural Networks with Delays. IEEE trans. on Circuit and systems 51 (2004) 2491-2503 10. Yi, Z., Lv, J., Zhang, L.: Output Convergence Analysis For a Class of Delayed Recurrent Neural Networks with Time Varing Inputs. IEEE Trans. on Systems, Man, and Cybernetics 36(1) (2006) 87-95 11. Lu, W., Chen, T.: Synchronization Analysis of Linearly Coupled Networks of Discrete Time Systems. Physica D 198 (2004) 148-168

Chaos and Bifurcation in a New Class of Simple Hopfield Neural Network Yan Huang 1,2 and Xiao-Song Yang 1 1

Department of Mathematics, Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, P.R. China [email protected]

2

Abstract. A class of simple Hopfield neural networks with a parameter is investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and periodic orbits for different parameters. The Lyapunov exponents are calculated, the bifurcation plot and several important phase portraits are presented as well.

1 Introduction The Hopfield neural network (HNN) [1] abstracted from brain dynamics is a significant model in artificial neural network, which is able to store certain memories or patterns in a manner rather similar to the brain - the full pattern can be recovered if the network is presented with only partial information. The smart thing about the Hopfield network is that there exists a rather simple way of setting up the connections between nodes, in such a way that any desired set of patterns can be made “stable firing patterns”. Substantial evidence has been found in biological studies for the presence of chaos in the dynamics of natural neuronal systems [2-5]. Many researchers suggested that chaos play a central role in memory storage and retrieval. From this point of view, many artificial neural networks have been proposed in order to realize more dynamical attractors such as chaos and hyperchaos in artificial neural dynamics [2, 6-9].

2 Simple 3-Neuron Hopfield Neural Networks In this paper we are concerned with a new class of simple 3-neuron Hopfield-type neural networks (3-HNNs) as follows:

x& = −cx + W ⋅ f ( x) , x = ( x1 , x2 , x3 ) ∈ R 3 , W = ( wij )3×3 , f ( x) = tanh( x) T

(1)

The connection matrix W describing the strength of connections between neurons is of the following form: J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 316 – 321, 2006. © Springer-Verlag Berlin Heidelberg 2006

Chaos and Bifurcation in a New Class of Simple Hopfield Neural Network

⎛ w11 ⎜ W = ⎜ w21 ⎜w ⎝ 31

w12 w22 w32

w13 ⎞ ⎟ 0 ⎟ w33 ⎟⎠

317

(2)

The topological connection of the weight matrix W is illustrated in Fig.1.

Fig. 1. The topological connection

Concretely, we consider the following connection matrix

u 0.995 ⎞ ⎛ 1.5 ⎜ ⎟ 0 ⎟ W = −2.1 1.68 ⎜ ⎜ 3.977 -21 1.97 ⎟ ⎝ ⎠

(3)

with a parameter u . We study periodic orbits and chaotic trajectories for different parameters u in the connection matrix (3) by means of calculating Lyapunov exponent which is very useful in studying complex dynamics of dynamical systems. To give more rigorous analysis of chaotic behavior in the 3-HNNs, we can resort to Poincaré section and Poincaré map technique and show existence of chaos by virtue of topological horseshoes theory[10-12] as [13,14], the more rigorous analysis of chaotic behavior will be given in our forthcoming work.

3 Periodic Orbit and Chaotic Attractor of the Simple Hopfield Neural Networks In this section we investigate the dynamics of (1) with the connection matrix (3), and consider the case that the parameter u varies from 1 to 7. As to be shown as the following, the dynamic behavior is very rich for u ∈ [1, 7] . We compute the Lyapunov exponents for u ∈ [1, 7] , which are illustrated in Fig.2. The Lyapunov exponents are calculated with time step size 0.01 and time length 850, following the approach by [15]. Fig.2 shows that the system (1) can exhibit rich dynamic behaviors, such as periodic orbit and chaotic attractor.

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Phase portraits of certain parameters are shown in Fig.3, the parameters, corresponding initial points, Lyapunov exponents, and dynamic property are listed in Table.1. For the meaning of the Lyapunov exponents and their relation to dynamic property, we refer the reader to [17]. It is very clear that there remains much to be done in dynamics of these simple 3-HNNs from dynamical systems point of view, especially from bifurcation viewpoint. Fig.4 is the bifurcation plot of the 3-HNNs as we adjust u ; Fig.4 (b) and Fig.4(c) are local magnifications of Fig.4 (a). Computer simulations show that the periodic orbit and chaotic behavior appear alternatively when the parameter u varies from 1 to 7, and double period bifurcation is observed when u varies from 1.5 to 1.81, which can be seen from the Fig.3 (a)-(d) and Fig.4 (b). It can be concluded from Fig.4(c) that double period bifurcation is exhibited when u varies from 2.8 to 2.9.

Lyapunov exponents

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2

1

2

3

4

5

6

7

u Fig. 2. Lyapunov exponents for u ∈ [1, 7] Table 1. The parameters, corresponding initial points, Lyapunov exponents and dynamic property of Fig.3

Fig.3

u

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

1.5 1.77 1.777 1.8 1.9 2.2 2.6 2.8 3 4.5

Lyapunov exponents 0.00, -0.433, -0.984 0.00, -0.013, -0.936 0.00, -0.004, -0.926 0.00, -0.011, -0.879 0.091, 0.00, -0.838 0.123, 0.00, -0.708 0.124, 0.00, -0.630 0.00, -0.133, -0.498 0.060, 0.00, -0.572 0.00, -0.133, -0.147

Dynamic property periodic periodic periodic periodic chaotic chaotic chaotic periodic chaotic periodic

Initial point ± (-0.130,2.424,-23.694) ± (0.906,-1.791, 13.513) ± (-0.266,-1.41, 16.89) ± (0.279, 0.1888, 4.859) ± (0.256, -0.342, -6.57) (-0.721, -0.208, 11.538) ± (-1.00,-0.314, 12.886) ± (0.104, 0.048, -4.755) ± (0.297, 0.539, -4.989) ± (-0.47,-0.449, -0.564)

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30 20

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(i)

(j)

Fig. 3. The phase portrait of system (1) with different parameter u

0.5

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3

u

(b)

3.05

3.1

3.15

3.2

3.25

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(c)

Fig. 4. The bifurcation plot of the 3-HNNs as we adjust

u

(x= 0)

For u = 1.5, 1.77, 1.777, 1.8, 2.8, 4.5, it can be concluded that one Lyapunov exponent is approximate zero; two Lyapunov exponents are negative, indicating that (1) has one or more periodic orbits. For u = 1.9, 2.2, 2.6, 3, it can be concluded from Table.1 and Fig.3 that one Lyapunov exponent is positive, one is approximately zero, one is approximate negative, indicating that (1) is chaotic. For u = 1.5, choose two different initial points ± (-0.130, 2.424,-23.694). By studying the limit sets of the two trajectories with the two initial points, we can find two periodic orbits as illustrated in Fig.3 (a). For u = 1.77, choose two different initial points ± (0.906,-1.791, 13.513), we can also find two periodic orbits as illustrated in Fig.3 (b). For u = 1.777, 1.8, 2.8, 4.5, two periodic orbits can be found with two different initial points as listed in Tab.1, respectively. For u = 1.9, two one-scroll chaotic attractors as illustrated in Fig.3 (e) are found by studying two trajectories with two different initial points ± (0.256, -0.342, -6.57). For u = 2.2, a double-scroll chaotic attractor as illustrated in Fig.3 (f) is found by studying the trajectory with initial point (-0.721, -0.208, 11.538). For u = 2.6, two one-scroll chaotic attractors as illustrated in Fig.3 (g) are found with two different initial points ± (-1.00,-0.314, 12.886). For u = 3, two chaotic attractors as illustrated in Fig.3 (i) are found with two different initial points ± (0.297, 0.539, -4.989).

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It can be seen from Fig.3 that the two one-scroll chaotic attractors turn into one double-scroll chaotic attractor when u varies from 1.9 to 2.2, then this double-scroll chaotic attractor turns into two one-scroll chaotic attractor when u varies from 2.2 to 2.6.

4 Conclusion In this paper, we study on a class of simple Hopfield neural networks with a parameter. Computer simulations show that 3-HNNs can display chaotic attractors and periodic orbits for different parameters. In order to study the rich dynamics behavior of the system, we study the Lyapunov spectrum, the bifurcation plot and several important phase portraits of the system for certain range of parameter.

References 1. Hopfield, J. J.: Neurons with Graded Response Have Collective Computational Properties like those of Two-State Neurons. Proc. Natl. Acad. Sci. USA 81(10) (1984) 3088–3092 2. Bersini, H., Sener, P.: The Connections between the Frustrated Chaos and the Intermittency Chaos in Small Hopfield Networks. Neural Networks 15(10) (2002) 1197–1204 3. Guevara, M.R., Glass, L: Chaos in Neurobiology. IEEE Trans. Syst. Man Cyb. 13 (9/10) (1983)790–798 4. Babloyantz, A. Lourenco, Brain, C.: Chaos and Computation. Int. J. Neural Syst. 7(4) (1996) 461–471 5. Dror, G., Tsodyks, M.: Chaos in Neural Networks with Dynamic Synapses. Neurocomputing 32–33 (6) (2000) 365–370 6. Wheeler, D.W., Schieve, W.C.: Stability and Chaos in an Inertial Two-Neuron Systems. Physica D 105(7) (1997) 267–284 7. Bersini, H.: The Frustrated and Compositional Nature of Chaos in Small Hopfield Network. Neural Networks 11 (6) (1998) 1017–1025 8. Li, Q.D., Yang, X.S. Yang, F.Y.: Hyperchaos in Hopfield-Type Neural Networks. Neurocomputing 67(8) (2005) 275–280 9. Yang, X.S., Yuan, Q.: Chaos and Transient Chaos in Simple Hopfield Neural Networks. Neurocomputing 69(12) (2005)232–241 10. Yang, X.S., Tang, Y.: Horseshoes in Piecewise Continuous Maps. Chaos, Solitons & Fractals 19(3) (2004) 841–845 11. Yang, X.S.: Metric Horseshoes. Chaos, Solitons & Fractals 20 (6) (2004)1149–1156 12. Yang, X.S., Li, H-M, Huang, Y.: A Planar Topological Horseshoe Theory with Applications to Computer Verifications of Chaos. J. Phys. A: Math. Gen. 38(5) (2005) 4175–4185 13. Huang, Y., Yang, X.S.: Horseshoes in Modified Chen’s Attractors. Chaos, Solitons and Fractals 26(10) (2005) 79–85 14. Huang,Y., Yang, X.S.: Chaoticity of Some Chemical Attractors: A Computer Assisted Proof. Journal of Mathematical Chemistry 38 (7) (2005) 107–117 15. Ramasubramanian, K., Sriram, M.S.: A Comparative Study of Computation of Lyapunov Spectra with Different Algorithms. Physica D 139(5) (2000) 72–86 16. Eckmann, J. P., Ruelle, D.: Ergodic Theory of Chaos and Strange Attractors. Rev. Mod. Phys. 57 (7) (1985) 617–656

Synchronization of Chaotic System with the Perturbation Via Orthogonal Function Neural Network Hongwei Wang and Hong Gu Department of Automation, Dalian University of Technology, Dalian, 116024, China [email protected]

Abstract. In the paper, the orthogonal function neural network is utilized to realize the synchronization of chaotic system. The orthogonal function neural network is Legendred orthogonal neural network. First of all, the orthogonal function neural network is trained to learn the uncertain information. The parameters of Legendred orthogonal neural network are adjusted to accomplish the synchronization of two chaotic systems with the perturbation by Lyapunov steady theorem. At last, the result of numerical example is shown to illustrate the validity of the proposed method.

1 Introduction It is a hot research topic in the field of control theories and applications that chaotic system is applied into secret communication, which shows the important value and the beautiful future. From chaotic system applying into secret communication, it is a key technology how to construct the synchronization of chaotic systems. The system can be moved into the expecting orbit according to the design requests, which is synchronization problem of chaotic system. The synchronization of chaotic system is the important field of chaotic science researching. The synchronization of chaotic system has been a familiar topic of intensive research in the past decade [1-5]. The movement of two same systems with the same initializing conditions can show the synchronization in the less time. However, the synchronization is weak because the small disturbing destroys it. In addition, the synchronization of two different chaotic systems is complicated problem. In the paper, the orthogonal function neural network is utilized to realize the synchronization of chaotic system. The orthogonal function neural network is Legendred orthogonal neural network. The orthogonal function neural network is trained to learn the uncertain information of the system. The parameters of orthogonal neural network are adjusted to accomplish the synchronization of two chaotic systems with the disturbing of parameters by Lyapunov steady theorem. At last, with the demonstration of numerical example, the simulation results can show the validity of proposed method.

2 The Description of the Problem The chaotic system is shown as the following form.

x& = F ( x) + u J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 322 – 327, 2006. © Springer-Verlag Berlin Heidelberg 2006

(1)

Synchronization of Chaotic System with the Perturbation Via Orthogonal Function

323

x ∈ R n ; F (x) is a nonlinear function, satisfying F(x) = ( f1 (x), f 2 ( x),....,f n (x))T , f i ( x)(i = 1,2,..., n) ; u is the input vector,

where the input vector satisfies

u ∈ Rn . In the actual system, the chaotic system is perturbed by the outside perturbation, and then the equation (1) can be written as the following equation.

x& = F ( x) + ΔF ( x) + u

(2)

where ΔF (x ) is the outside perturbation. Definition: the reference chaotic system is as the equation (3).

x& r = g ( x r )

(3)

x r is a state vector, satisfying x r ∈ R n , g (⋅) is the vector with many slippery nonlinear functions, having the same structure as F (⋅) or the different with F (⋅) . Let e = x − x r , we have

where

lim e(t ) = 0

(4)

t →∞

then the system (1) keeps the synchronization with the system (2). Matrix A is a matrix with negative real part of the eigenvalues. The error dynamical trajectory equation is shown as the equation (5).

e& = x& − x& r = A( x − x r ) + Ax r − Ax + F ( x) + ΔF ( x) + u − g ( x r )

(5)

The equation (5) is also defined as the following form

e& = Ae + Ax r − Ax + F ( x) + ΔF ( x) + u − g ( x r )

(6)

Because of the eigenvalues of the matrix A having negative real part, there is a positive definite and symmetric matrix P to satisfy the following Lyapunov equation.

AT P + PA = −Q where matrix Q is a positive definite and symmetric matrix. Let ing equation.

(7)

G (x) as the follow-

G ( x) = Ax r − Ax + F ( x) + ΔF ( x) − g ( x r )

(8)

The equation (5) represents as the equation (9)

e& = Ae + G ( x) + u

(9)

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The estimated value

Gˆ ( x) substitutes the function G (x) when the function

G (x) is unknown. The controller is defined as the following equation (10). u = −Gˆ ( x) − σ where

σ

(10)

is a vector with the smaller norm.

3 The Controller of Orthogonal Function Neural Network The orthogonal function neural network has simple structure, fast convergence by the comparison of the common BP neural network. The orthogonal function neural network can approach to any nonlinear function on the tight set. In the paper, the orthogonal function of neural network is Legendre orthogonal polynomial. The Legendre polynomial is defined as the following form.

P0 ( x) = 1 ⎧ ⎪ i 1 di ⎨ = P ( x ) x2 −1 i i ⎪⎩ i 2 i! dx

[(

)]

i = 1,2,....N

(11)

The global output of the orthogonal function neural network is defined as the equation (12). N

y = ∑ Φ iWi

(12)

i =1 n

where

Φi = P1i (x1)× P2i (x2 ) ×⋅⋅⋅×Pni (xn ) = ∏Pji (xj ) , Pji ( x j ) is Legendre polynoj=1

1 2 i = 1,2,..., N , j = 1,2,...,n .

mial, Pj1(xj ) = x , Pj2(xj ) = (3xj −1), Pj(i+1) (x j ) = ((2i +1)x j Pji (x j ) − iPj(i−1) (x j ))/(i +1) , 2

Lemma 1. For defining the any function g ( X ) on the section [ a, b] and any small positive number ε , there is an orthogonal function sequence

{Φ1 ( X ), Φ 2 ( X ),..., Φ N ( X )} and the real number sequence Wi (i = 1,2,..., N )

to satisfy the following equation N

g ( X ) − ∑ Wi Φ i ( X ) ≤ ε

(13)

i =1

On the basis of the lemma 1, the properties are acquired as the following form Property 1. Give a positive constant

ε0

and a continuous function G (x ) ： x ∈ R , n

exist an optimization weighted matrix W = W , W = [W1 ,W2 ,...,WN ] to satisfy the following equation. *

*

*

*

*

Synchronization of Chaotic System with the Perturbation Via Orthogonal Function

G ( x) − W * Φ ( x) ≤ ε 0 T

325

(14)

Φ satisfies Φ( x) = [Φ 1 ( x), Φ 2 ( x),..., Φ N ( x)] . Based on the equation (14), G (x ) is shown as the equation (15). T

where

G ( x ) = W * Φ ( x) + η T

(15)

where η is a vector with the smaller norm. The equation (15) is substituted into the equation (9).

e& = Ae + W * Φ ( x) + η + u (t ) T

(16)

P , satisfying T T & W = Φ ( x)e P , A P + PA = −Q , where Q is a positive definite and symmetric

Theorem 1. Exist a positive definite and symmetric matrix matrix, the controller is designed as the following form.

u = u1 + u 2

(17)

u1 is u1 = −W T Φ( x) and u 2 is u 2 = −η 0 sgn( Pe) The state of the equation (9) approaches zero, namely x → x r , the system (1) is

where

synchronization with the system (2).

~ W = W * − W , which W * an optimization weighted matrix, W is a ~ weighted matrix, W is an estimated error matrix. Define the norm of matrix R: 2 R = tr ( RR T ) = tr ( R T R ) . The lyapunov function is shown as the following form. Proof. Let

V =

1 T 1 ~ e Pe + W 2 2

2

(18)

Differentiating equation (18) with respect to the time is shown the following steps.

1 ~& ~ V& = (e&Pe + e T Pe&) + tr (W T W ) = 2 1 T T ~ ~& ~ e A P + PA e + Φ T ( x)WPe + e T Pη − e T Pη 0 sgn(e T P) + tr (W T W ) = 2 1 ~ ~& ~ − e T Qe + Φ T ( x)WPe + e T Pη − e T Pη 0 sgn(e T P) + tr (W T W ) 2 ~ ~ T T T T T Because of Φ ( x )WPe = tr ( PeΦ ( x )W ) , e Pη − e Pη0 sgn(e P) ≤ 0 1 1 ~& ~ ~ V& ≤ − e T Qe + tr (W T W + PeΦ T ( x)W ) = − e T Qe ≤ 0 2 2 ~ On the basis of the lyapunov theorem, e 、 W have the limit, then lim e = 0 .

(

)

t →∞

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4 Simulation The Lorenz chaotic system is shown as the equation (19).

⎧ v&1 = a(v 2 − v1 ) ⎪ ⎨v&2 = (b − v3 )v1 − v 2 ⎪ v& = −cv + v v 3 1 2 ⎩ 3

(19)

In the actual system, the Lorenz chaotic system is perturbed by the environment, and then the Lorenz chaotic system is represented as the following equation.

⎧ v&1 = (a + δa )(v 2 − v1 ) + d1 + u1 ⎪ ⎨v&2 = (b + δb − v3 )v1 − v 2 + d 2 + u 2 ⎪ v& = −(c + δc)v + v v + d + u 3 1 2 3 3 ⎩ 3

(20)

The system parameters are a = 10 , b = 30 , c = 8 / 3 ; The perturbation items of the parameters are δa = 0.1 , δb = 0.2 , δc = 0.2 ; The perturbation items of the system states are

d1 = 0.3 sin t , d 2 = 0.1cos t , d 3 = 02 sin(3t ) .

Fig. 1. The responding diagram of

e1

Fig. 2. The responding diagram of

Fig. 3. The responding diagram of

e3

e2

Synchronization of Chaotic System with the Perturbation Via Orthogonal Function

327

The parameters of Legendred orthogonal neural network are adjusted to accomplish the synchronization. The other parameters are A =diag(−1,−1,−1) , Q = diag(2,2,2) ,η0 = 0.02, P = diag(1,1,1) . The error responding diagrams are shown as the figure 1 to figure 3. The Based on these responding diagrams, the synchronization of the two chaotic systems is carried out by the proposed method. The validity of the proposed method is demonstrated with the result of numerical example simulation.

5 Conclusion In the paper, the orthogonal function neural network based on Legendred orthogonal polynomial is utilized to realize the synchronization of chaotic systems. The parameters of orthogonal neural network are adjusted to accomplish the synchronization of two chaotic systems with the perturbation of parameters by Lyapunov steady theorem. With the demonstration of simulating result, the proposed method can guarantee the synchronization of two chaotic systems with the perturbation of parameters.

Acknowledgement This work is supported by National Natural Science Foundation of China (60575039). The authors would like to thank the support and encourage of Prof. Xiaodong Liu.

References 1. Liu, F., Ren, Y., Shan, X.M., Qiu, Z.L.: A Linear Feedback Synchronization Theorem for a Class of Chaotic Systems. Chaos, Solution and Fractals 13(4) (2002) 723-730 2. Sarasola, C., Torrealdea, F.J.: Cost of Synchronizing Different Chaos Systems. Mathematics and Computers in Simulation 58(4) (2002) 309-327 3. Shahverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Lag Synchronization in Time-Delayed Systems. Physics Letter A 292(6) (2002) 320-324 4. Tan, W., Wang, Y.N., Liu, Z.R., Zhou, S.W.: Neural Network Control for Nonlinear Chaotic Motion. Acta Physica Sinica 51(11) (2002) 2463-2466 5. Li, Z., Han, C.S.: Adaptive Control for a Class of Chaotic Systems with Uncertain Parameters. Acta Physica Sinica 50(5) (2001) 847-850

Numerical Analysis of a Chaotic Delay Recurrent Neural Network with Four Neurons Haigeng Luo1 , Xiaodong Xu2, , and Xiaoxin Liao1 1

Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China [email protected], [email protected] 2 College of Public Administration, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China xiaodong [email protected]

Abstract. Complex dynamical behavior in a four-neuron recurrent neural network with discrete delays is investigated in this paper. The dissipativity of the system and stability of equilibrium point are studied by mens of Lyapunov theory. Stable fixed point, periodic and quasi-periodic orbits, and chaotic motion are observed in system via numerical calculation. With the change of the slope and threshold of activation function, as well as time delay and synaptic weight, the system passes from stable to periodic and then to chaotic regimes. Interestingly, the system returns to periodic or stable regimes by further changing these parameter values. Furthermore, some numerical evidences, such as phase portraits, bifurcation diagrams, power spectrum density, are given to confirm chaos.

1

Introduction

An artificial neural network (ANN) is a mathematical model that represents the biological process of human brain. These mathematical models can then be used to generate computer graphics which illustrate the usefulness of these techniques for solving problems in such areas as system control, combination optimization, data compression, pattern recognition and system identification. By far, most works focused on the stability of neural networks, see [1, 2, 3, 4, 5] and references therein. In recent years, bifurcation and chaotic phenomena have been considered in neural networks with or without time delays. For example, Das et al. [6, 7] investigated the chaotical behavior in a three dimensional neural network. Liao et al. [9] analyzed the Hopf bifurcation and chaos in a single delayed neural equation with non-monotonic activation function. Zhou et al. [10] studied chaos and synchronization in two-neuron systems with discrete delays. However, It seems to be necessary to choose a bit more complicated activation function, such as combination of sigmoid function and periodic function, to exhibit chaos in delay neural networks with one or two neurons. Moreover, most 

Corresponding author.

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 328–333, 2006. c Springer-Verlag Berlin Heidelberg 2006 

Numerical Analysis of a Chaotic Delay Recurrent Neural Network

329

works place their emphasis on synaptic weight and decay rate, and omit the influence of time delay, self-weight ( that is, wij = 0 for i = j ) and the parameters of activation function. In this paper, a small recurrent neural network consisting of four coupled neurons with discrete delays and typical sigmoid activation function is considered and their activity is determined by four differential equations as given in section 2. Dissipativity of model and stability of equilibrium point is studied in section 3. In section 4, details of numerical simulation have been given. An attempt has been made to give the model a general shape by taking all possible synaptic connections , include the self-weight, of the neurons into consideration. In numerical simulation, we have considered the influence of time delay and synaptic weight as well as the slope and the threshold of the activation function. The relevance of the finding of this work is given as a discussion in section 5.

2

Model Formulation

Consider a continuous time four-neuron recurrent neural network model with multiple time delays 4  dxi (t) = −xi (t) + wij (fi (xj (t)) − fi (xj (t − τj ))), dt j=1

(1)

for i = 1, 2, 3, 4. Where xi (t) is the state variable of the ith neuron. wij represents synaptic weight from neuron j to neuron i. When the output of neuron j excites neuron i, synaptic weight wij > 0. When the output of neuron j inhibits neuron i, synaptic weight wij < 0. When the output of neuron j has no influence on neuron i, synaptic weight wij = 0. fi (·) is a sigmoid-type activation function as fi (s) = tanh(βi (s − θi )),

(2)

where βi > 0 and θi denote the slope and the threshold of the activation function of the neuron i respectively. In the following, we choose τ = [0.8, 0.5, 0.5, τ4], β = [β1 , 1.2, 1, 1] and θ = [θ1 , 0.1, 0.1, 0.1], the connection weight matrix is taken as ⎛ ⎞ 1.5 w12 −0.9 0.5 ⎜ 1.8 1.2 0.6 1.3 ⎟ ⎟ W =⎜ (3) ⎝ 1.1 0.9 2.2 1.5 ⎠ , 0.2 −0.4 0.4 1.85 in other word, τ4 , β1 , θ1 and w12 are chosen as bifurcation parameters, other parameters values are fixed.

3

Analysis of Dissipativity and Stability

In this section, we firstly show system (1) is dissipative.

330

H. Luo, X. Xu, and X. Liao

Theorem 1. If we choose activation function as (2), then neural network model (1) is a dissipative system, the set S = S1 ∩S2 is a positive invariant and globally attractive set, where 4  4 4  4  

2 

2 

S1  x

|xi | − 2 |wij |  |wij | , i=1

j=1

i=1

(4)

j=1

4

 

2  

S2  x |xi |  4 |wij | , i = 1, 2, 3, 4 .

(5)

j=1

Proof. First, we employ 4 a radically unbounded and positive definite Lyapunov function as V (x) = i=1 x2i /2. Computing dV dt along the positive half trajectory of (1), we have

4 4  4

  dV

dxi 2 = x  2|w |x | − x i ij j i < 0, dt (1) i=1 dt i=1 j=1

(6)

when x ∈ Rn \S1 ; i.e., x ∈ / S1 . Equation (6) implies that ∀x0 ∈ S1 holds x(t, t0 , x0 ) ⊆ S1 , t  t0 . For x0 ∈ / S1 , there exist T > 0 such that x(t, t0 , x0 ) ⊆ S1 , ∀ t  T + t0 , i.e., the neural network model (1) is a dissipative system and S1 is a positive invariant and attractive set. Second, we define a radically unbounded and positive definite Lyapunov function Vi = |xi |, i = 1, 2, 3, 4. Calculating the right-upper Diniderivative D+ Vi , one obtains 4 

D+ Vi (1)  −|xi | + 2 |wij | < 0, i = 1, 2, 3, 4,

(7)

i=1

when x ∈ Rn \S2 . So, S2 is also a positive invariant and globally attractive set. Combining the above proof, we know that S = S1 ∩ S2 is a positive invariant and globally attractive set. Theorem 1 is completed. Obviously, the origin is the unique equilibrium point of neural network model (1). The Linear stability analysis is effective for investigating its stability and bifurcation. Unfortunately, the characteristic equation of (1) is a four-order transcendent equation, both analytical and numerical analysis are very difficult. Therefore, a qualitative analysis by Lyapunov theory is done. Theorem 2. The equilibrium point of neural network model (1) is globally asymptotically stable, if for all connection weights wij and slope βi , following inequality   1 2 max wij + max βi2 < , (8) 1i,j4 1i4 4 holds.

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331

Proof. Constructing a radically unbounded Lyapunov functional as follows V (x(t), x(t − τ )) =

4 4  t  1 2 xi + 2 max {βi2 }x2i (s)ds, 1i4 2 i=1 t−τ i i=1

(9)

computing the time derivative along the positive half trajectory of equation (1), using mean-value theorem and fi (s)  βi , one obtains

4 4     dV

2 = −x + x wij tanh(βi (xj − θi )) − tanh(βi (xj (t − τj ) − θi )) i i

dt (1) i=1 j=1 +2

4  i=1

  max βi2 x2i − x2i (t − τi )

1i4

4      2  − 1 − 4 max wij + max βi2 x2i  0, 1i,j4

when

1i4

max



1i,j4

i=1

  1 2 wij + max βi2 < , 1i4 4

dV

dt (1)

and = 0, if and only if xi = 0, i = 1, 2, 3, 4. This implies the neural network model is globally asymptotically stable. The proof of theorem 2 is completed.

4

Numerical Simulation

We will show numerical experiments according to the model and parameter values given in section 2. Based on numerical simulation software Matlab 7.0.4, delay differential equation solver DDE23 is used to integrate the delay neural network model (1). Phase portraits and power spectrum of the system: If the bifurcation parameters values are chosen as β1 = 12, θ1 = 0, w12 = −1.5 and τ4 = 0.8, the delay neural network model (1) possesses an unstructured chaos attractor. The phase Portraits are given in Fig. 1. The power spectrum density of x1 component is drawn in Fig. 2. It is a typical continuous spectrum and discloses system (1) is chaotic. Bifurcation diagram: To find out critical parameter domains where the activity pattern of the system takes place and to correctly establish critical dependence of the system’s stability on parameters like the synaptic weight w12 ,the time delay τ4 , the slope β1 and the threshold θ1 , we have drawn bifurcation diagrams (Figs. 3-6), by computing the local maximum Poincare section of state variable x1 . They clearly show how the system passes through period doubling bifurcation to chaos. Also the sudden change in the chaotic dynamics of the system which can be termed as crisis are clear from these figures. From these figures and a great deal of numerical calculation, we can sum up that the system is not sensitive to change of the slope, only large value of β1 can guide the system from chaotic regime to periodic orbit, and can not reach fixed point. Whereas the

H. Luo, X. Xu, and X. Liao 4

1

10

0.5

10

2

Power Spectrum Density

4

System State x (t)

332

0

−0.5

−1 4

0

10

−2

10

−4

2

10

4 2

0

0

−2 System State x3(t)

−6

10

−2 −4

−4

0

5

Fig. 1. Three dimensional view of attractor

20

2

1.5

1.5

40

45

50

x1

0.5

0

0

−0.5

−0.5 −1

−1

0

5

10

β

15

20

−1.5 −3

25

−2

−1

1

Fig. 3. The system reaches chaotic regime when the slope β1 decreases. (θ1 = 0, τ4 = 0.8, w12 = −1.5)

1

2

3

2.5

4

2

3

1.5 1 x1

2 1

0.5

0

0

−1

−0.5 −1

−2 −3 −5

0 θ1

Fig. 4. The system starts from fixed point to period orbit, then to chaotic regime, finally returns to fixed point with the change of the threshold θ1 . (β1 = 12, τ4 = 0.8, w12 = −1.5)

5

1

35

1

0.5

x

25 30 frequency (Hz)

2.5

2

1

x1

15

Fig. 2. Power spectrum of x1

2.5

−1.5

10

System State x2(t)

−1.5 0 w

5

12

Fig. 5. Bifurcation diagram of the connection weght w12 versus system state x1 . (β1 = 12, θ1 = 0, τ4 = 0.8,)

0.65

0.7

0.75

0.8 τ4/s

0.85

0.9

0.95

1

Fig. 6. Bifurcation diagram of the time delay τ4 versus system state x1 . (β1 = 12, θ1 = 0, w12 = −1.5)

time delay can guide the system from periodic motion to chaos, but the system can not reach fixed point by change of time delay. However,both of the change of connection weight and of threshold can guide the system from fixed point to periodic motion, then to chaos, and finally returning to stationary regime.

Numerical Analysis of a Chaotic Delay Recurrent Neural Network

5

333

Conclusion

In this paper, a four-neuron recurrent neural network with discrete delays is concerned. Chaotic motion is observed while changing the slope and the threshold of a sigmoid activation function, as well as the time delay and the synaptic weight of the system. Numerical simulations show these parameters impose different effect on the bifurcation situation of the system. This work can provide some reference evidence for chaos control in chaotical neural networks.

Acknowledgement The work was supported by National Natural Science Foundation of China (60274007, 60474011) and Science and Research Project of Hubei Provincial Department of Education (D200560001).

References 1. Shen, Y., Zhao, G.Y., Jiang, M.H., Mao, X.R.: Stochastic Lotka-Volterra Competitive Systems with Variable Delay. In: Huang, D.S., Zhang, X.P., Huang, G.B. (eds.): Advances in Intelligent Computing. Lecture Notes in Computer Science, Vol. 3645. Springer-Verlag, Berlin Heidelberg New York (2005) 238-247 2. Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Systems I 51 (2004) 2313-2324 3. Zeng, Z.G., Huang, D.S., Wang Z.F.: Global Stability of A General Class of Discrete-time Recurrent Neural Networks. Neural Processing Letters 22 (2005) 33-47 4. Shen, Y., Zhao, G.Y., Jiang, M.H., Hu, S.G.: Stochastic High-order Hopfield Neural Networks. In: Wang, L.P., Chen, K., Ong, Y.S. (eds.): Advances in Natural Computation. Lecture Notes in Computer Science, Vol. 3610. Springer-Verlag, Berlin Heidelberg New York (2005) 740-749 5. Shen, Y., Jiang, M.H., Liao, X.X.: Global Exponential Stability of Cohen-Grossberg Neural Networks with Time-varying Delays and Continuously Distributed Delays. In: Wang, J., Liao, X.F., Zhang, Y. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg New York (2005) 156-161 6. Das, A., das, P., Roy, A.B.: Chaos in Three Dimensional Neural Network. Applied Mathematical Modelling 24 (2000) 511-522 7. Das, A., das, P., Roy, A.B.: Chaos in A Three-dimensional Model of Neural Network. Int. J. of Bifur. and Chaos 12 (2002) 2271-2281 8. Li, C.G., Yu, J.B., Liao, X.F.: Chaos in A Three-neuron Hysteresis Hopfield-type Neural Network. Physics Letters A 285 (2001) 368-372 9. Liao, X.F, Wong, K.W., Leung, C.S., Wu, Z.F.: Hopf Bifurcation and Chaos in A Single Delayed Neuron Equation with Non-monotonic Activation Function. Chaos, Solitons and Fractals 12 (2001) 1535-1547 10. Zhou, S.B., Liao, X.F., Yu, J.B., Wong, K.W.: Chaos and Its Synchronization in Two-neuron Systems with Discrete Delays. Chaos, Solitons and Fractals 21 (2004) 133-142

Autapse Modulated Bursting Guang-Hong Wang1 and Ping Jiang1,2 1

Department of Information and Control Engineering, Tongji University, Shanghai 200092, China [email protected] 2 School of Informatics, University of Bradford, Bradford, BD7 1DP, UK [email protected]

Abstract. In this paper we present a model of autapses which are synapses connecting axons and dendrites of the same neuron and feeding back axonal action potentials to the own dendritic tree. The timely physiological self-inhibitory function they may serve, indicated by the spiketiming dependent plasticity rule, provides a potential negative feedback mechanism to control the dynamical properties of neurons. The model autapse is applied to a neuron with conductance-based minimal model to construct a fast-slow buster. Three types of bursting that the burster exhibits are analyzed geometrically through phase portraits to illustrate the modulation competence of the autapse model on bursting.

1

Introduction

Information processing depends not only on the anatomical substrates of synaptic circuits, the electrophysiological properties of neurons, but also on their dynamical properties [1]. Electrophysiologically similar neurons may respond to the same synaptic input in very different manners because of each cell’s intrinsic bifurcation dynamics. In an extensive series of experiments on the giant axon of the squid, Hodgkin and Huxley succeeded in measuring the currents responsible for action potentials and describing their dynamics in terms of differential equations [2]. H-H model is not only the starting point for detailed neuron models which account for numerous ion channels, different types of synapse, and the specific spatial geometry of an individual neuron, but also an important reference model for the derivation of simplified neuron models [3]. Further development of the geometrical analysis of neuronal models was done in [1, 4], stressing the integrator and resonator modes of operation and made the connections to other neuro-computational properties. [5, 6] developed a geometrical framework for an averaging method of singularly perturbed systems. Many mathematical models of bursters have fast-slow form (1) described in the next section. The Hodgkin-Huxley equations, FitzHugh-Nagumo equations, Morris-Lecar equations, as well as many other fast-slow models of this form, exhibit an extremely rich variety of nonlinear dynamical behaviors [7]. The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 334–343, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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generated and transitions between them can be made [5]. A top down approach was used to classify all the bursters, considering all possible pairs of co-dimension one bifurcations of resting and spiking states [4]. Resting state of neurons corresponds to a stable equilibrium, and tonic spiking state corresponds to a limit cycle attractor. Another classification framework was provided based on the observation that the bifurcations of the fast system that lead to bursting can be collapsed to a single local bifurcation, generally of higher co-dimension [7]. In [8] a neural basket cell via the Hodgkin-Huxley model was simulated with an autapse feeding back onto the soma of the neuron forced to burst by the constant spike train of 200Hz as the excitatory synaptic input. However, the synaptic level analysis of the autapse did not sufficiently reveal its dynamical role on the whole system, which in fact was not analyzed dynamically, let alone the different types of bursting. Here, we present an autapse model IK(Aut) which is driven by action potentials, and reveal its role from the perspective of dynamical system. The model autapse is applied to a neuron with conductancebased minimal model to construct a fast-slow buster. Three types of bursting that the burster exhibits are analyzed geometrically through phase portraits to illustrate the dynamics and modulation competence of the autapse model on bursting.

2 2.1

Models Fast-Slow Bursting Model

The fields of mathematical and computational neuroscience focus on modeling electrical activity in nerve cells. Fundamental processes, such as voltage changes and ion transport across cell membranes, occur on disparate time scales [7]. A neuron is a fast-slow burster if its behavior can be described by a fast-slow system of the form x˙ = f (x, u) (fast spiking) u˙ = μg(x, u) (slow modulation) ,

(1)

where, vector x describes fast variables responsible for spiking, including the membrane potential, activating and inactivating gating variables for fast currents, etc; vector u describes relatively slow variables that modulate fast spiking, e.g., gating variable of a slow K+ current, intracellular concentration of Ca2+ ions, etc; small positive parameter μ << 1 represents the ratio of time scales between spiking and modulation. At μ = 0, the fast system becomes independent of the slow one, hence they can be considered separately. The u variable can then be treated as a bifurcation parameter in the fast subsystem. This constitutes the method of dissection of neural bursting pioneered by Rinzel [9]. One should note that the fast subsystem goes through bifurcations, does not mean that the entire system (1) undergoes bifurcations during bursting. As long as parameters of (1) are fixed, the system as a whole does not undergo any bifurcations, no matter how small μ is.

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2.2

G.-H. Wang and P. Jiang

Conductance-Based Voltage-Gated INa,p +IK -Model

We select conductance-based voltage-gated INa,p +IK -model (2) as the fast spiking subsystem, which is one of the most fundamental models in computational neuroscience consisting of a fast Na+ current and a slower K+ current [1]. IL

INa,p

IK

         C V˙ = I − gL (V − EL ) − gNa m(V − ENa ) − gK n(V − EK ) m ˙ = (m∞ (V ) − m)/τm n˙ = (n∞ (V ) − n)/τn ,

(2)

where V is the membrane potential; I is the total current flowing across a patch of a cell membrane; IL is the Ohmic leak current carried mostly by Cl− ions; INa,p is the persistent inward Na+ current; IK is the outward K+ current; C is the membrane capacitance; EL , ENa and EK are the relevant Nernst equilibrium potentials; gL , gNa and gK are maximal conductance of channels for corresponding ions; m and n are the probability of channel activation gate to be open for Na+ and K+ , with time constants τm and τn , respectively; m∞ (V ) and n∞ (V ) are the steady state activation curves approximated by Boltzmann function B∞ (V ) = 1/(1 + exp((V1/2,B − V )/kB )), B = m, n ,

(3)

where the parameter V1/2 satisfies B∞ (V1/2 ) = 1/2, and k is the slope factor. Smaller values of |k| result in steeper B∞ (V ). A reasonable assumption based on experimental observations is that m is much faster than V, so that m approaches the asymptotic value m∞ (V ) instantaneously. In this case the three-dimensional system (2) can be reduced to a planar system (4) by substituting m = m∞ (V ) [1]. C V˙ = I − gL (V − EL ) − gNa m∞ (V )(V − ENa ) − gK n(V − EK ) (4) n˙ = (n∞ (V ) − n)/τn 2.3

Autapse IK(Aut) -Model

The site where the axon of a presynaptic neuron makes contact with the dendrite (or soma) of a postsynaptic cell is the synapse. When an action potential arrives at a synapse, it triggers a complex chain of biochemical processing steps that lead to the release of neurotransmitter from the presynaptic terminal into the synaptic cleft. As soon as transmitter molecules have reached the postsynaptic side, they will be detected by specialized receptors in the postsynaptic cell membrane and open specific channels so that ions from the extracellular fluid flow into the cell. The ion influx, in turn, leads to a change of the membrane potential at the postsynaptic site so that, in the end, the chemical signal is translated into an electrical response [3]. As an inhibitory synapse, autapse is assumed to activate K+ ion channels, hence its equilibrium potential EAut = EK [8], and its current can be described as [3]  IAut (t, V ) = gAut exp(−(t − t(f ) )/τa )Θ(t − t(f ) )(V − EK ) , (5) f

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337

where gAut and τa are the maximal conductance and time constant of the autapse, respectively; t(f ) denotes the arrival time of a presynaptic action potential; Θ(·) is a Heaviside step function. Based on (5), further assuming that autapses are driven by the potential difference at some higher level during a spiking period, we depict it as IAut (V) = gAut a(V  − EK ), local local local a = min 1, a + d(Vmax − Vmin ) (V = Vmax and V ≥ Vthr ) with a˙ = −a/τa (otherwise) ,

(6)

where a is similar to n, representing the probability of activation gate to be local local open for K+ driven by inhibitory autapse; Vmax and Vmin denote the local maximum and minimum of membrane potential during the latest spiking period, respectively; Vthr represents the threshold over which the potential difference can operate on the autapse; d > 0 is a proportional coefficient indicating the effect of potential difference. If τa >> τn then (6) is competent to act as the slow modulation part in (1). Combining (4) and (6), we get the whole fast-slow burster INa,p +IK +IAut -model as shown in (7). C V˙ = I − gL (V − EL ) −gNa m∞ (V )(V − ENa ) − gK n(V − EK ) − gAut a(V − EK ) local n˙ = (n∞ (V ) − n)/τn if V = Vmax and V ≥ Vthr , local local a˙ = −a/τa then a ← min 1, a + d(Vmax − Vmin ) .

(7)

C V˙ = I − gL (V − EL ) −gNa m∞ (V )(V − ENa ) − gK n(V − EK ) − gM nM (V − EK ) n˙ = (n∞ (V ) − n)/τn n˙ M = (n∞,M (V ) − nM )/τnM

(8)

The intrinsic difference between INa,p +IK +IAut -model and INa,p +IK +IK(M) model (8), which was thoroughly analyzed in [1], is that IK(M) is driven by local local potential V, but IAut is driven by potential difference Vmax − Vmin during a spiking period. From another perspective, as noted in [10], the slow variables either provide the modulation effectively without feedback from the fast system, in which case the slow variables oscillate periodically on their own, irrespective of how the coupling to the fast subsystem influences them. Or, they provide it with feedback from the fast system, in which case the switching between the two states is determined also by the fast variables. In this sense, (7) is closer to the former, while (8) belongs to the latter.

3

Numerical Results

Bifurcations that the model exhibits are illustrated in Fig. 1. One is referred to [1, 4] for details about the whole complete set of bifurcations. As shown in Fig. 1, there are two bifurcations for equilibria to transit from resting states to spiking states, and another two bifurcations for limit cycles to transit from

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Homo

Fold

Fold Cycle

subHopf

Fig. 1. Bifurcations appearing in INa,p +IK +IAut -model. Saddle-Node: A node is approached by a saddle; they coalesce and annihilate each other. Subcritical AndronovHopf : A small unstable limit cycle shrinks to a stable equilibrium and makes it lose stability. Saddle Homoclinic Orbit: A limit cycle grows into a saddle. Fold Limit Cycle: Stable limit cycle is approached by an unstable one; they coalesce and then disappear. With Fold, subHopf, Homo, and Fold Cycle for short respectively.

spiking states to resting states. The INa,p +IK +IAut -model exhibits three types of combinations of these two pairs of bifurcations: Fold /Homo, subHopf /Fold Cycle, and Fold /subHopf. Notice that in the last case the fast subsystem has two resting states: upper and lower ones, which was reported occurring in realistic neurons in [11]. The numerical results of the whole system are presented in Fig. 2, 4 and 6, with detailed geometrical analysis of the fast subsystem through phase portraits as shown in Fig. 3, 5 and 7, respectively. The values of the parameters in the three types of bursting are collected in Table 1. n = (I − gL (V − EL ) − gNa m∞ (V )(V − ENa ))/(gK (V − EK )) − gAut a/gK (9) n = n∞ (V ), n-nullcline (n˙ = 0); V -nullcline (V˙ = 0) for the above. Equation (9) depicts the two nullclines. The n-nullcline has the shape of Boltzmann function (S -shaped), while the V -nullcline is cubic (N -shaped), as shown in Fig. 3, 5 and 7. When parameters except a are fixed, the n-nullcline is fixed, and the V -nullcline can shift vertically by adjusting a to change the intersections with n-nullcline, hence change the stability of equilibria. In other words, a acts as the bifurcation parameter of the fast subsystem. But the whole system does not undergo any bifurcation. In the following, we describe the bursters by gluing the phase portraits of the frozen time points as illustrated in Fig. 3, 5 and 7. Bearing in mind the current state for each phase portrait is very important in the following description, where the term system denotes the fast subsystem. Table 1. Values of parameters of bursters in their appearance order, with units mV(V, k and E ), mS/cm2 (g), ms(τ ), mV−1 (d), μF/cm2 (C) and μA/cm2 (I) V1/2,m -20 -30 -20

km 15 7 15

gNa 20 4 20

ENa 60 60 60

V1/2,n -25 -45 -20

kn 5 5 5

gK 9 4 9

E K τn -90 0.152 -90 1 -90 0.152

τa gAut 20 2 50 3 20 10

Vthr d C I -20 0.0006 1 5 -20 0.0006 1 55 -20 0.001 1 5

gL 8 1 8

EL -80 -78 -80

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(b)

(a) 0.8

n

V (mV)

−4

−70

0 (d)

(c) 0.2

a

V (mV)

−4

−37

−70 0.2 0.8

0.1 0

0

200

0.4 0

a

0

n

time (ms)

Fig. 2. Fold/Homo bursting (a)

(b)

0.8

(c)

(d)

n−nullcline

n

V−nullcline

−0.1 −70

(e)

V (f)

0

Fold (g)

stable unstable initial state frozen time point Homo

a

0.2

a 0 30

b

c

d

e

fg

a 113

time (ms)

Fig. 3. Phase portraits of Fold/Homo bursting

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G.-H. Wang and P. Jiang (a)

(b) 1

n

V (mV)

6

delayed transition −70

0 (c)

(d)

0.12

a

V (mV)

6

−32

−70 0.12 1

0.06 0

0

0.5 0

a

400

0

n

time (ms)

Fig. 4. subHopf/Fold Cycle bursting 1

0 −70

(a)

(b)

(c)

(d)

(g)

(h)

Fold Cycle (e)

0

(f)

subHopf

a

0.12 ab c d e f

g

h

a

0 100

320 time (ms)

Fig. 5. Phase portraits of subHopf/Fold Cycle bursting

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(b)

(a) 0.7

n

V (mV)

0

−80

0 (d)

(c) 0.2

a

V (mV)

0

−40

−80 0.2 0.7

0.1 0

0

300

a

0.35 0

0

n

time (ms)

Fig. 6. Fold/subHopf bursting 0.7

(a)

(b)

(c)

Fold

0 −80

(e)

(d)

0

subHopf

a

0.2 a 0 33

b

c

de d

a 141

time (ms)

Fig. 7. Phase portraits of Fold/subHopf bursting

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Fold/Homo Burster (Fig. 3 )

(a) There are three equilibria, one stable node, one saddle and one unstable focus. The stable node represents the resting state, and the stable limit cycle surrounding the unstable focus represents the spiking state. The saddle acts as the threshold. The current state is around the resting state. a is decreasing slowly, hence V -nullcline is shifting upward. (b) The stable node is approached by the saddle with their distance decreasing. (c) They coalesce to a saddlenode. The system is undergoing the saddle-node bifurcation. (d ) The saddle-node disappears and the trajectory evolves to the limit cycle attractor to start spiking. (e) a is alternating between the fast jumping up and the slow decaying. Since the total tendency of a is increasing, V -nullcline is shifting downward, hence the limit cycle is growing into the saddle. (f ) A saddle homoclinic orbit occurs, so the system is undergoing the saddle homoclinic orbit bifurcation. (g) The stable limit cycle disappears, and the system evolves toward the stable node. Then from (a) a new bursting cycle begins. 3.2

subHopf/Fold Cycle Burster (Fig. 5 )

(a) There is a unique unstable equilibrium, which is surrounded by a stable limit cycle. The current state is spiking, and the total tendency of a is increasing. (b) The equilibrium becomes stable, which is surrounded by a new unstable limit cycle surrounded by the original stable limit cycle. It does not change the current state, spiking, since the original bigger limit cycle is still stable. However the new unstable limit cycle is growing bigger to approach onto the original stable one. (c) The two limit cycles have coalesced and annihilated each other, i.e., the system has undergone the fold limit cycle bifurcation. The trajectory is evolving toward the unique stable focus. (d )-(f ) a is decreasing but the focus is still stable. Hence the current state is on resting. (g) A new unstable limit cycle generates between the stable focus and the stable limit cycle, which does not affect the current state at present with the similar reason as in the case of (b). However, a is decreasing, so that in this case, the newly generated unstable limit cycle is shrinking to the stable focus. (h) The shrinking has made the focus lose its stability, and the system had undergone the subcritical Andronov-Hopf bifurcation. However, since the current state is still nearly on the unstable focus, it takes the system a long time to escape from the unstable focus. As shown in Fig. 4, the long time is labeled with delayed transition. Then from (a) a new bursting cycle begins. 3.3

Fold/subHopf Burster (Fig. 7 )

(a) The system has three equilibria, one stable node, one saddle and one stable focus. The current state is around the lower resting state, and V -nullcline is shifting upward with a decreasing slowly. (b) The stable node is approached by the saddle. They coalesce and annihilate each other through the saddle-node bifurcation. (c) The lower resting state disappears, and the trajectory evolves toward the upper resting state. Since the upper resting state is a focus, it will

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take a long time for the system to screw from the state corresponding to the disappeared lower resting state to the upper resting state. (d ) Before the system can stand by the upper resting state, an unstable limit cycle appears, surrounding and shrinking to the upper stable focus. Notice that the lower stable node and saddle have appeared. (e) The system has undergone the subcritical AndronovHopf bifurcation. The upper equilibrium has become unstable. Since the current state is still on the way to the original upper resting state, i.e., there is no delayed transition in this case, and the trajectory evolves toward the lower resting state quickly. Then from (a) a new bursting cycle begins.

4

Conclusions

In this paper we present a model of autapses which is driven by action potentials. The model autapse is applied to a neuron with conductance-based minimal model to construct a fast-slow buster. Three types of bursting that the burster exhibits are analyzed geometrically through phase portraits to illustrate the modulation competence of the autapse model on bursting of fast subsystem.

References 1. Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. Preprint edn. The MIT Press, Cambridge, MA (2005) 2. Hodgkin, A.L., Huxley, A.F.: A Quantitative Description of Membrane Current and Application to Conduction and Excitation in Nerve. J. Physiol. 117 (1952) 500–544 3. Gerstner, W., Kistler, W.M.: Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, New York (2002) 4. Izhikevich, E.M.: Neural Excitability, Spiking, and Bursting. Int. J. Bifurcation and Chaos 10(6) (2000) 1171–1266 5. Shilnikov, A., Calabrese, R.L., Cymbalyuk, G.: Mechanism of Bistability: Tonic Spiking and Bursting in a Neuron Model. Phys. Rev. E 71(5) (2005) 056214 6. Shilnikov, A., Cymbalyuk, G.: Transition between Tonic Spiking and Bursting in a Neuron Model via the Blue-Sky Catastrophe. Phys. Rev. L. 94(4) (2005) 048101 7. Golubitsky, M., Josic, K., Kaper, T.J.: An Unfolding Theory Approach to Bursting in Fast-Slow Systems. In: Broer, H.W., Krauskopf, B., Vegter, G. (eds.): Global Analysis of Dynamical Systems. Institute of Physics Publishing, Bristol and Philadelphia (2001) 277–308 8. Herrmann, C.S., Klaus, A.: Autapse Turns Neuron into Oscillator. Int. J. Bifurcation and Chaos 14(2) (2004) 623–633 9. Rinzel, J.: Bursting Oscillations in an Excitable Membrane Model. In: Sleeman, B.D., Jarvis, R.J. (eds.): Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, Vol. 1151. Springer-Verlag, Berlin (1985) 304–316 10. Rinzel, J., Lee, Y.S.: On Different Mechanisms for Membrane Potential Bursting. In: Othmer, H.G. (ed.): Nonlinear Oscillations in Biology and Chemistry. Lecture Notes in Biomathematics, Vol. 66. Springer-Verlag, New York (1986) 19–33 11. Wilson, C.J., Kawaguchi, Y.: The Origins of Two-State Spontaneous Membrane Potential Fluctuations of Neostriatal Spiny Neurons. J. Neurosci. 16(7) (1996) 2397–2410

A Neural Network Model for Non-smooth Optimization over a Compact Convex Subset Guocheng Li1, Shiji Song1, Cheng Wu1, and Zifang Du2 1

Department of Automation, Tsinghua University, Beijing, 100084, China [email protected] 2 School of Statistics, Renmin University, 100872, China

Abstract. A neural network model is introduced which is aimed to solve nonsmooth optimization problem on a nonempty compact convex subset of R n . By using the subgradient, this neural network model is shown to obey a gradient system of differential inclusion. It is proved that the compact convex subset is a positive invariant and is a attractive to the neural network system, and that all the network trajectories starting from the inside of the compact convex subset converge to the set of equilibrium points of the neural network. The above every equilibrium point of the neural network is an optimal solution of the primal problem. A numerical simulation example is also given to illustrate the qualitative properties of the proposed neural network model.

1 Introduction Optimization problems arise in many areas of scientific and engineering applications. Most of the practice engineering applications problems are required to solve in real-time. One possible and very promising approach for solving in real-time optimization problem is to apply artificial neural networks. Hopfield and Tank [1-2] first proposed a neural network to deal with linear programming problems. Their works have motivated many researchers to investigate alternative neural networks for solving linear and nonlinear programming problems. And many models of neural networks used to optimization problems have been established in literatures [3-7]. In this paper, we extend Liang [6] for the recurrent neural network of nonlinear continuously differentiable optimization to the case of the non-smooth convex optimization. And another neural network model is introduced, which is aimed to solve in real time a class of non-smooth convex optimization over a compact convex subset. By using the subgradient, this network is shown to obey a subgradient system described by differential inclusion, and its dynamical behavior and optimization capabilities are rigorously analyzed in the framework of convex analysis and the theory of differential inclusions. Finally, the numerical simulation experiment is illustrated to substantiate our theoretical results.

2 Neural Network Model We consider the minimization problem formulated as the following: minimize E ( x) subject to x ∈ Ω J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 344 – 349, 2006. © Springer-Verlag Berlin Heidelberg 2006

(1)

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where E : R n → R is a convex function on R n and Ω is a compact convex subset of

R n , when the objective function E ( x) is smooth, and ∂i E ( x) = ∂i ( x) / ∂xi is singlevalued. However, for non-smooth convex E ( x) , the subdifferential ∂i E ( x) is a setvalued mapping. For example, if E : R → R is given by E ( x) =| x | , then we have

x>0 ⎧ 1 ⎪ ∂E ( x) = co[sgn( x)] = ⎨[−1,1] x = 0 ⎪ −1 x < 0. ⎩ The previous discussion makes it clear that neural network for optimization problem (1) is no longer described by a standard differential equation where the velocity vector field dx / dt is a single-valued mapping. Rather, the neural network for (1) obeys a differential inclusion where dx / dt becomes to be a set-valued mapping. Indeed, by explicitly expressing the neural network, we get that it should satisfy the system of differential inclusions: τ x& (t ) ∈ Fα ( x) = − x + PΩ ( x − α∂E ( x)) , (2) where PΩ (⋅) is a projection from

R n onto Ω , and

PΩ ( x − α∂E ( x )) = {PΩ ( x − α v ) : v ∈ ∂E ( x )} , where τ > 0 and α > 0 are the time constant and step-size parameter of the network model respectively. First, we need to explain what is meaning by a solution of a Cauchy problem associated with the system of differential inclusion (2). Since Fα is a set-valued mapping whose values are nonempty compact convex subset, in order to develop our discussion, we now present a definition which will be adopted in this paper (ref. Filippov [8]). Definition 1. We say that a function x (⋅) on [t1 , t2 ](t2 > t1 ) is a solution of (2), if the

x(⋅) is an absolutely continuous function and satisfies x(t1 ) = x0 and for almost all t ∈ [t1 , t2 ] and τ x& (t ) ∈ Fα ( x(t )) . The importance for the engineering applications of the concept of solutions in the sense of Filippov is due to the fact that they are good approximations of solutions of actual systems with very high slope nonlinearities (ref. [9-10]). Theorem 1. (i). For any x0 ∈ R , there exists at lest one solution of (2) with initial n

condition x(0) = x0 in the global time interval [0, ∞) . (ii) Ω is a positive invariant and is attractive set of (2). Proof. Since the projection operator PΩ (⋅) is continuous and ∂E (⋅) is an upper semi-

continuous set-valued mapping, so, Fα (⋅) is also an upper semi-continuous set-valued

Fα (x) is a compact convex subset for any x ∈ R n . Therefore, from Theorem 1 of [11], it follows that there exists at least one local solution x (⋅) on mapping and

[0, t1 ](t1 > 0) which satisfies (2) with x(0) = x0 .Similar to the processing discussed in

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Theorem 2 of [6], for the set-valued mapping rather than single-valued one, we have following argumentation: 1 Let H ( x) = d 2 ( x, Ω) . For almost all t ∈ [0, t1 ] , we have 2 dH ( x(t )) dx(t ) ) τ = ( x(t ) − PΩ ( x(t ))T (τ dt dt ≤ sup ( x(t ) − PΩ ( x(t ))T [−( x(t ) − PΩ ( x(t ))) + ( PΩ ( x(t ) − α v) − PΩ ( x(t ))] v∈∂E ( x ( t ))

≤ − || x(t ) − PΩ ( x(t )) ||2 = −2 H ( x(t )) .

(3)

By the comparison principle, from the differential inequality (3), it follows that t n d ( x(t ), Ω) ≤ d ( x0 , Ω) exp(− ) , x 0 ∈ R , ∀t ≥ 0 . (4)

τ

This means that the solution x(t ) of (2) with initial condition x(0) = x0 is bounded and hence well defined on [0, ∞) . From (4) we know that if x0 ∈ Ω , then d ( x(t ), Ω) = 0 , x(t ) ∈ Ω and if x0 ∉ Ω , then d ( x(t ), Ω) → 0 at an exponential rate

1

τ

as t → ∞ , which means that x(t )

□

converges to the set Ω exponentially.

An equilibrium point x ∈ R of (2) is a stationary solution. Clearly, x is an equilibrium point of (2) if and only if 0 ∈ Fα ( x e ) . The set of equilibrium point of (2) is thus given by e

n

e

Ωαe = {x ∈ R n : 0 ∈ Fα ( x)} = {x ∈ R n : x ∈ PΩ ( x − α∂E ( x))}. Theorem 2. For any α > 0 , we have that Ωαe ≠ φ . Proof. Since Fα is an upper semi-continuous set-valued mapping whose values are

nonempty compact convex subset, and Fα ( x) I TΩ ( x) ≠ φ for any x ∈ Ω , by Theo-

□

rem 1 in [11, pp228], (2) has at least one equilibrium point.

Now we consider the optimization problem (1). Let M = arg min E ( x) be set of x∈Ω

global minimum of problem (1), then the next basic result holds. Theorem 3. For any α > 0 , we have that Ωαe ⊆ M . Proof. If x ∈ Ωαe , then there exists an v ∈ ∂E ( x) , such that x = PΩ ( x − α v) .For any y ∈ Ω , we have

[ PΩ ( x − α v) − ( x − α v)]T [ PΩ ( x − α v) − y ] ≤ 0 , that is, vT ( y − x) ≥ 0 .Since E is a convex function, by the definition of subdifferential, for any y ∈ Ω and v ∈ ∂E ( x) , we have E ( y ) − E ( x) ≥ vT ( y − x) ≥ 0 . Therefore, the x is a global minimum of problem (1), i.e. x ∈ M .

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Next, we well recall the concept of limit point of a function to present the main result on trajectory convergence and optimization capabilities of network model (2). Definition 2. We say that

x * is a limit point of x(t ) when t → ∞ if there exists

sequence {t n } such that x (t n ) → x as *

t n → ∞.

Theorem 4. For any α > 0 , any limit point of trajectory of (2) starting from the inside of Ω belongs to Ωαe ,that is lim d ( x(t ), Ωαe ) = 0 . t →∞

Proof. Let

x(⋅) is any solution of (2) with initial condition x(0) = x0 . From (ii) of

Theorem 1, if x0 ∈ Ω , then the solution x(t ) ∈ Ω for t ≥ 0 . Note that τ x& ∈ Fα ( x) = − x + PΩ ( x − α∂E ( x)) ,

1 || x& (t ) ||≤ [|| x(t ) || + max || x ||] ≤ M τ

then

τ

Mτ =

2

τ

x∈Ω

for

almost

all

t≥0

,

where

max || x || . Let V ( x(t )) = τα E ( x(t )) and y (t , ω ) = x(t ) − αω , from Property 1 x∈Ω

in [5] and the projection theorem, differentiating V ( x(t )) along the solution x(t ) of (2), we have dV ( x(t )) = ταω T x& (t ) dt ≤ sup (αω )T (− x(t ) + PΩ ( x(t ) − αω )) ω ∈∂E ( x ( t ))

= sup [( x(t ) − PΩ ( y(t, ω ))) + ( PΩ ( y(t , ω )) − y(t , ω ))]T [ PΩ ( y(t , ω )) − x(t )] ω∈∂E ( x ( t ))

≤ sup − || x(t ) − PΩ ( y (t , ω )) ||2 ω ∈∂E ( x ( t ))

= sup − || x(t ) − PΩ ( x(t ) − αω ) ||2 ω ∈∂E ( x ( t ))

= −dist 2 ( x(t ), PΩ ( x(t ) − α∂E ( x(t ))) ≤ 0 .

(5)

Let Wα ( x) = dist 2 ( x, PΩ ( x − α∂E ( x))) . Since x → PΩ ( x − α∂E ( x)) is upper semicontinuous, then x → Wα ( x) is a lower semi-continuous nonnegative valued function n

defined on R . Clearly, p ∈ Ωαe if and only if Wα ( p) = 0 . For any

α > 0, let ωα ( x(⋅)) be the set of the limit points of a solution

x(⋅) of (2),

that is ωα ( x(⋅)) = I {x(t ) : t ≥ t ′} . Since x(t ) ∈ Ω , t ≥ 0 , it is sufficient to prove that t ′∈[0, ∞ )

ωα ( x(⋅)) ⊆ Ωα . For each p ∈ ωα ( x(⋅)) , there exists a increasing sequence {tn } such e

that lim tn = ∞ and lim x(tn ) = p . n →∞

n →∞

By (5), V ( x(tn )) is non-increasing and bounded from below. Therefore, there exists a constant V0 such that lim V ( x(tn )) = V0 . Since V ( x(tn )) is non-increasing, n →∞

V ( x(t )) ≥ V0 and lim V ( x(t )) = V0 . from (5) we further have t →∞

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t

t

0

0

∫ Wα ( x(s))ds ≤ − ∫

d [V ( x( s ))]ds = V ( x(0)) − V ( x(t )) . ds

Hence, we have

∫

∞

0

Wα ( x ( s ))ds ≤ V ( x0 ) − V0 .

(6)

Next, we prove that p ∈ Ωαe . If this is no true, that is Wα ( p ) > 0 . Since x → Wα ( x) is lower semi-continuous, then there exists a constant η > 0 and a small neighborhood of p , B2ε ( p) = {x ∈ Ω :|| x − p ||< 2ε } such that Wα ( x) > η , for all x ∈ B2ε ( p ) . We now claim that for all t ∈ U [tn − n ≥ n0

ε Mτ

, tn +

ε Mτ

], x(t ) ∈ B2ε ( p ) , where n0 is a

positive integer such that for all n ≥ n0 , || x(tn ) − p ||< ε . In fact, for t ∈ [t n −

ε Mτ

, tn +

ε Mτ

] , n ≥ n0 , we have

|| x(t ) − p ||≤|| x(t ) − x(tn ) || + || x(tn ) − p ||≤ M τ || t − tn || +ε . Therefore, for all t ∈ U [tn − n ≥ n0

ε Mτ

, tn +

ε Mτ

] we have Wα ( x) > η > 0 . Since lim tn = ∞ n →∞

and since the Lebesgue measure of the set t ∈ U [tn − n ≥ n0

ε Mτ

, tn +

ε Mτ

] is infinite, it

follows that

∫

∞

0

Wα ( x(t ))dt = ∞ .

(7)

The contradiction between (6) and (7) implies that Wα ( p) = 0 , i.e., p ∈ Ωα , this e

e mean that lim d ( x(t ), Ωα ) = 0 . t →∞

□

From Theorem 3 and Theorem 4, we know that every limit point of the trajectory of (2) starting from the inside of Ω is the optimal solution of Problem (1). In the following, we give a numerical simulation example to illustrate the qualitative behaviors of model (2) for solving an optimization problem over different compact convex subsets. Example 1. We consider the following optimization problem min | x12 − x2 | , x = ( x1 , x2 ) ∈ Ω .

The two cases of Ω being of Ω1 = {x ∈ R 2 :| x1 | + | x2 |≤ 1} and Ω 2 = {x ∈ R 2 : x12 + x 22 ≤ 1}. We select, respectively, 120 random points which are uniformly distributed over [−1,1] × [−1,1] as the initial ones of the solution trajectories of the model (2). Their geometric representations of these trajectories in the cases of Ω1 and Ω 2 are shown in Fig.1 and Fig. 2 respectively, which demonstrate that all the network trajectories are convergent to the minimum set of the optimization problem respectively.

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Fig. 1

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Fig. 2

Acknowledgment The authors of this paper would like to thanks for the support of 973 project (No.2002CB312205) and the National Natural Science Foundation of China (No.60574077).

References 1. Hopfield, J.J., Tank, D.W.: Neural Computation of Decisions in Optimization Problems. Biol. Cybern. 52 (1985) 141-152 2. Tank, D.W., Hopfield, J.J.: Simple ‘Neural’ Optimization Networks: an A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit. IEEE Trans. Circuits Syst. 33 (1986) 533-541 3. Kennedy, M.P., Chua, L.O.: Neural Networks for Nonlinear Programming. IEEE Trans. Circuits Syst. 35 (1988) 554-562 4. Rodriguez-Vazquez, A., Dominguez-Castro, R., Rueda, A., Huertas, J.L., Sachez-Sinencio E.: Nonlinear Switched-Capacitor Neural Networks for Optimization Problems. IEEE Trans. Circuits Syst. 37 (1990) 384-397 5. Forti, M., Nistri, P., Quincampoix, M.: Generalized Neural Network for Non-smooth Nonlinear Programming Problems. IEEE Trans. Circuits Syst. 51 (2004) 1741-1754 6. Liang, X.B.: A Recurrent Neural Network for Nonlinear Continuously Differentiable Optimization over a Compact Convex Subset. IEEE Trans. Neural Networks 12 (2001) 1487-1490 7. Wang, J.: A Deterministic Annealing Neural Network for Convex Programming. Neural Networks 7 (1994) 629-641 8. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Side. Kluwer Academic, Dordrecht (1988) 9. Utkin, V.I.: Sliding Modes and Their Application in Variable Structure Systems. U.S.S.R., MIR, Moscow (1978) 10. Paden, B.E., Sastry, S.S.: Calculus for Computing Filippov’s Differential Inclusion with Application to the Variable Structure Control of Robot Manipulator. IEEE Trans. Circuits Syst. 34 (1987) 73-82 11. Aubin, J.P., Cellina, A.: Differential Inclusion. Springer-Verlag, Berlin (1984)

Differential Inclusions-Based Neural Networks for Nonsmooth Convex Optimization on a Closed Convex Subset Shiji Song, Guocheng Li, and Xiaohong Guan Department of Automation, Tsinghua University, Beijing 100084, China [email protected]

Abstract. Differential inclusions-based dynamic feedback neural network models are introduced to solve in real time nonsmooth convex optimization problems restricted on a closed convex subset of Rn . First,a differential inclusion-based dynamic feedback neural network model for solving unconstrained optimization problem is established, and its stability and convergence are investigated, then based on the preceding results and the method of successive approximation, differential inclusions-based dynamic feedback neural network models for solving in real time nonsmooth optimization problem on a closed convex subset are successively constructed, and its dynamical behavior and optimization capabilities are analyzed rigorously.

1

Introduction

Most of existing feedback neural network optimization models for solving in real time optimization problems are commonly constructed with a concept of gradient,the object functions are required to be smooth and the network models are described with differential equations(ref. [1 − 7]).Recently, the neural networks with discontinuous neuron activation functions are also presented in Forti[8] and Lu[9],the dynamical behaviors and the global convergence in finite time are investigated for the respective networks. In addition, by using the Clarkes generalized gradient of the involved functions, generalized nonlinear programming circuit is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in Forti[10]. This paper is devoted to solve in real time nonsmooth convex optimization problems restricted on a closed convex subset of Rn . At first, by using the nonsmooth analysis theory instead of the Lyapunov stability theory and the LaSall invariance principle, we investigated the stability and convergence of network constructed by differential inclusion for solving unconstrained optimization problem. And for optimization problem on a closed convex subset, we construct successively an iterative energy functions sequence and the corresponding dynamic subnetworks which are described to obey differential inclusion systems respectively, we prove that the trajectory of J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 350–358, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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the every subnetwork converges to one minimum point of the its energy function which also correspond to an equilibrium point of the subnetwork, and the minimum points sequence corresponding to these energy functions converges to one minimum point of the primal optimization problem, and the convergence theorem is then obtained.Finally, the conclusions of this paper are summarized.

2

Uuconstrained Optimization Problem

In this paper,we firstly suppose that V is a convex function from Rn to R, then from Aubin-Cellina [11], the following properties are well known. Lemma 1. If V is a convex function from Rn to R, the following propositions hold: (i) V (x) − V (x − v) ≤ DV (x)(v) ≤ V (x + v) − V (x); (ii) ∂V (x) = {p ∈ Rn : V (x) − V (y) ≤ (p, x − y), ∀y ∈ Rn }. Lemma 2. Let V be a convex function from Rn to R. The following statements are equivalent: (i) x∗ minimizes V on Rn ; (ii) 0 ∈ ∂V (x∗ ); (iii) ∀v ∈ Rn , 0 ≤ DV (x∗ )(v). Moreover, we always assume that V (x) is a coercive function, that is, V (x) → . +∞, as ||x|| → +∞. Let x0 ∈ Rn , the set L(x0 ) = {x ∈ Rn : V (x) ≤ V (x0 )} is called the level set. Then the following result holds(ref. [12]). Lemma 3. Let V (x) be a continuously convex function , then any level set of V (x) is bounded if and only if V (x) is a coercive function. It is easy to verify that if V (x) is a convex coercive function, the V (x) achieves its minimum at some point. We consider the following general unconstrained nonsmooth convex optimization problem min V (x), x ∈ Rn

(1)

with the subgradient differential inclusion as follows x (t) ∈ ∂V (x(t)), x(0) = x0

(2)

where V : Rn → R is a convex function. A function x(·) defined on [t0 , ∞) is said to be a solution of (2), if the x(·) is a absolutely continuous function and satisfies x(t0 ) = x0 and x (t) ∈ ∂V (x(t) for almost all t ∈ [t0 , ∞). The concept of solution here is in the sense of Filippov and the derivative of solution means the Gˆ ateaux derivation(ref. [11] or [13]). An equilibrium point x∗ ∈ Rn of (2) means a constant solution of (2), that is, x(t) ≡ x∗ , t ∈ [0, ∞).

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It is clearly that x∗ is an equilibrium point of (2) if and only if 0 ∈ ∂V (x∗ ) and x∗ = x0 . The following Theorem 1 provides the existence, uniqueness and many of better properties of solutions for differential inclusion (2)(ref. Aubin-Cellina [11]). Theorem 1. Let V : Rn → R be a convex function. Then for any x0 ∈ Rn , there exists a unique solution x(·) defined on [0, ∞) to the differential inclusion (2). Moreover (i) t → ||x (t)|| is non-increasing function; (ii) Let x(·), y(·) be two solutions corresponding the initial points x0 and y0 respectively. Then for t ≥ 0, ||x(t) − y(t)|| ≤ ||x0 − y0 ||. (iii) V (x(t)) is a convex non-increasing function satisfying d V (x(t)) + ||x (t)||2 = 0. dt

(3)

From (ii) of Theorem 1, we know that the solutions of (2) have Lyapunov stability. In order to establish convergence theorem, we show the following definition and lemma(ref. Aubin-Cellina [11]). Definition 1. We say that x∗ is a limiting point of x(t) when t → ∞ if there exists sequence {tn } such that limtn →∞ x(tn ) = x∗ . Lemma 4. Let V : Rn → R be a convex coercive function and x(·) be a solution of (2). If all the limiting point x∗ of x(t) when t → ∞ achieve the minimum of V , then x(t) converges to one minimum point of V when t → ∞. From Lemma 4, we further have the following convergence theorem. Theorem 2. Let V : Rn → R be a convex coercive function. Then for any x0 ∈ Rn , the corresponding trajectory of (2) converges to one minimum point of V. Proof. For ∀x0 ∈ Rn , let x(t) = x(t, x0 ) be the unique solution of (2). By Lemma 4, we only need to check that the every limiting point x∗ of x(t) when t → ∞ achieve the minimum point of V . Integrating equality (3), we deduce that  t V (x(t)) − V (x(s)) + ||x (τ )||2 dτ = 0, s

since V (x(t)) is convex coercive,this implies that there exists the minimum point of V (x(t)), so, V (x(t)) is bounded from below and non-increasing by Theorem 1.Further we have  t lim ||x (τ )||2 dτ = lim (V (x(s)) − V (x(t))) = 0. t, s→∞

s

t, s→∞

Differential Inclusions-Based Neural Networks

Hence the Cauchy criterion yields that  ∞  t  2 ||x (τ )|| dτ = lim ||x (τ )||2 dτ < ∞. t→∞

0

353

(4)

0

From (i) of Theorem 1 and equation (4), we have limt→∞ ||x (t)|| = 0. Taking a minimum point x ¯ of V , then by Lemma 2, x ¯ is equilibrium point of (2), and by (ii) of Theorem 1 we have ||x(t) − x ¯|| ≤ ||x0 − x ¯|| for all t ≥ 0. Therefore, the set {x(t) : t ≥ 0} is bounded. By (ii) of Lemma 1, we deduce from the inclusion x (t) ∈ −∂V (x(t)) that for all y ∈ Rn , inf t≥0 V (x(t)) ≤ V (x(t)) ≤ V (y) + (−x (t), x(t) − y) ≤ V (y) + ||x (t)||||x(t) − y|| ≤ V (y) + My ||x (t)||, where My = supt≥0 ||x(t) − y|| is finite because {x(t) : t ≥ 0} is bounded. Therefore, by letting t → ∞ and taking the infimum for all y ∈ Rn , we obtain inf V (x(t)) = infn V (y).

t≥0

y∈R

Because the set {x(t) : t ≥ 0} is bounded, so there exists the limiting points x(t) when t → ∞, noting that t → V (x(t)) is a non-increasing function, by any limiting point x∗ of x(t), it follows that V (x∗ ) = lim V (x(tn )) = inf V (x(t)) = infn V (y). tn →∞

t≥0

y∈R

Lemma 4 guarantee that x(t) converges to one minimum point of V which is also the equilibrium point of (2) when t → ∞. Thus we complete the proof of this theorem. By the theorems described previously, two results are obtained immediately. (i) when V (x) has unique minimum point x∗ , then x∗ is globally, uniformly, and asymptotically stable. (ii) when V (x) has infinitely many minimum points, then given an arbitrary initial point x0 ∈ Rn , the trajectory of neural network (2) converges to one of its minimum points.

3

Optimization Restricted on a Closed Convex Subset

In this section, based on the preceding results and procedures of successive approximation,we construct the energy function and the corresponding neural subnetwork described by differential inclusions to solve in real time the nonsmooth convex optimization problem on a closed convex subset. Now we Consider the nonsmooth convex optimization problem: min V (x), x ∈ Ω

(5)

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where x ∈ Rn , V : Rn → R is a convex coercive function, Ω is a closed convex subset in Rn . By the weierstrass’ Theorem(ref. Appendix in [12]), (5) achieves at least one optimal solution on ∈ Ω. Taking M1 as a lower bound of the optimal value of (5), i.e., M1 ≤ V (x∗ ), where x∗ is an optimal solution of (5). Set V1 (x) = V (x) − M1 and F (x, M1 ) = 1 4 V1 (x)[V1 (x) + |V1 (x)|]. It is easy to verify that F (x, M1 ) is a nonnegative, continuous convex coercive function, and  (V (x) − M1 )∂V (x), if V(x) ≥ M1 ∂F (x, M1 ) = 0, if V(x) < M1 We construct the energy function on Rn E(x, M1 ) = F (x, M1 ) +

1 min ||x − y||2 . 2 y∈Ω

(6)

Obviously E(x, M1 ) is also a nonnegative, continuous convex coercive function. Its subdifferential is ∂E(x, M1 ) = ∂F (x, M1 ) + x − PΩ (x), where PΩ (x) is the projection of x onto Ω. For energy function E(x, M1 ), it is easy to see that the following two theorems hold, and their proofs are omitted here. Theorem 3. Let x∗ be an optimal solution of (5) and M1 ≤ V (x∗ ), then E(x, M1 ) has a minimum point on Rn and min E(x, M1 ) ≤ E(x∗ , M1 ).

x∈Rn

Theorem 4. 0 ∈ ∂E(x∗ , M1 )) if and only if x∗ is a minimum point of E(x, M1 ), that is, the set of equilibrium point of E(x, M1 ) is equal to that of its minimum points on Rn . From the preceding results, a neural network for solving the minimum point of E(x, M1 ) on Rn can be constructed as x (t) ∈ ∂E(x(t), M1 ).

(7)

The network described by differential inclusion (7) is a subnetwork of problem (5). From Theorem 1 and Theorem 2, we have following theorem about the subnetwork. Theorem 5. (1) ∀x0 ∈ Rn , the network (7) has a unique trajectory x(t, x0 ) on [0, +∞) with x(0) = x0 ; (2) Any minimum point of E(x, M1 ) is Lyapunov stable; (3) ∀x0 ∈ Rn , the corresponding trajectory of neural network (7) converges to a minimum point of E(x, M1 ) on Rn .

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Now, we construct energy function sequence and give the convergence theorem. suppose that x∗ is an optimal solution of problem (5) and M1 is an estimated value of the lower bounded of its optimal value, i.e., M1 ≤ V (x∗ ). We construct the energy function sequence as E(x, Mk ) = F (x, Mk ) +

1 min ||x − y||2 , k = 1, 2, ... 2 y∈Ω

(8)

where F (x, Mk ) = 14 Vk (x)[Vk (x) + |Vk (x)|], Vk (x) = V (x) − Mk , Mk+1 = Mk +  2E(xk , Mk ), and xk is the minimum point of E(x, Mk ) on Rn . Note that E(x, Mk ) is a convex coercive function, then from Theorem 2, xk is well defined. By the preceding analysis, we now establish the convergent result as follows: Theorem 6. (Convergence Theorem): Suppose that x∗ is an optimal solution of problem (5), then for above constructed energy function sequence E(x, Mk ), k = 1, 2, ..., the following results hold (1) for k = 1, 2, ..., Mk ≤ Mk+1 , Mk ≤ V (x∗ )and V (xk ) ≤ V (x∗ ); (2) limk→∞ Mk = M ∗ ; (3) the sequence {xk } has limiting point x ¯; and any limiting point x ¯ of {xk } satisfies V (¯ x) = V (x∗ ) = M ∗ .  Proof. (1) Since Mk+1 = Mk + 2E(xk , Mk ) ≥ Mk , k = 1, 2, ..., we have Mk ≤ Mk+1 (k = 1, 2, ...). By hypothesis M1 ≤ V (x∗ ), we suppose that for k, Mk ≤ V (x∗ ). Since xk is the minimum point of E(x, Mk )(k = 1, 2, ...), we have E(xk , Mk ) ≤ E(x∗ , Mk ) = F (x∗ , Mk ) + 12 miny∈Ω ||x∗ − y||2 = F (x∗ , Mk ) = 12 [V (x∗ ) − Mk ]2 .  Hence, we have Mk+1 = Mk + 2E(xk , Mk ) ≤ V (x∗ ). By the induction principle, Mk ≤ V (x∗ ) holds for k = 1, 2, ...  Moreover,  2E(xk , Mk ) ≥ 2F (xk , Mk ) ≥ V (xk ) − Mk , k = 1, 2, ..., that is, V (xk ) ≤ Mk + 2E(xk , Mk ) = Mk+1 ≤ V (x∗ ), so V (xk ) ≤ V (x∗ )(k = 1, 2, ...). Since the sequence {Mk } is monotone nondecreasing and it has an upper bound V (x∗ ), then (2) of Theorem 6 is immediately obtained and M ∗ ≤ V (x∗ ). (3) Since V (x) is a coercive function and V (xk ) ≤ V (x∗ ), then by Lemma 3 the sequence {xk } is bounded. Therefore, there exists subsequence {xki } ⊆ {xk } (ki < ki+1 ) such that limi→+∞ xki = x ¯. By the continuity of V (x), we have lim V (xki ) = V (¯ x) ≤ V (x∗ ).

i→+∞

 By Mki +1 = Mki + 2E(xki , Mki )(i = 1, 2, ...) and the continuity of E(x, Mk ), and taking the limit as i → +∞, we have 2E(¯ x, M ∗ ) = 0. From this fact, it follows that  F (¯ x, M ∗ ) = 0 ⇒ V (¯ x) ≤ M ∗ , 1 2 x − y|| = 0 ⇒ x¯ ∈ Ω. 2 miny∈Ω ||¯

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So, x ¯ is a feasible solution of problem(5), and M ∗ ≤ V (x∗ ) ≤ V (¯ x). Therefore, combining the above proof, we have V (x∗ ) = V (¯ x) = M ∗ . The proof of this theorem is thus completed. Since problem (5) may have many optimal solutions and so x ¯ may not equal x∗ , the previous equation shows that any limiting point x ¯ of {xk } is the optimal solution of problem (5), and the corresponding limit value V (¯ x) of {V (xk )} is the optimal value of problem (5). Based on the above convergence theorem, we can construct a feedback neural network for solving optimization problem (5). By Theorem 5, the trajectory of neural network (7) converges to a minimum point x1 of E(x, M1 ). Let M2 = M1 + 2E(x1 , M1 ), then by (1) of Theorem 6, M2 ≤ V (x∗ ). Construct the energy function on Rn E(x, M2 ) = F (x, M2 ) +

1 min ||x − y||2 , 2 y∈Ω

(9)

and the corresponding neural network for obtaining its minimum point becomes x (t) ∈ ∂F (x, M2 ) + x − PΩ (x).

(10)

Replacing M1 with M2 in Theorem 4-5 above, the trajectory of the network n (10) by the same token,  converges to a minimum point x2 of E(x, M2 ) on∗ R . Let M3 = M2 + 2E(x2 , M2 ), and by (1) of Theorem 6, M3 ≤ V (x ). We can also construct the energy function E(x, M3 ) and the corresponding neural network for solving its minimum point on Rn . The remainder can be deduced by analogy.  In general, let Mk = Mk−1 + 2E(xk−1 , Mk−1 ), then Mk ≤ V (x∗ ). We construct the energy function on Rn as E(x, Mk ) = F (x, Mk ) +

1 min ||x − y||2 , 2 y∈Ω

(11)

and the corresponding neural networks for obtaining its minimum points are x (t) ∈ ∂F (x, Mk ) + x − PΩ (x).

(12)

Similarly, Replacing M1 with Mk in Theorem 3-5 above, it follows that the trajectory of the network (12) converges to a minimum point xk of E(x, Mk ). Hence, the neural network, which is a feedback network, for solving problem (5) is constructed as follows   x (t) ∈ ∂F (x, Mk ) + x − PΩ (x), (13) x(0) = xk , k = 1, 2, ...,

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 where the parameter M1 ≤ V (x∗ ) and Mk+1 = Mk + 2E(xk , Mk ) in this network must be determined by the equilibrium point xk and the parameter Mk in the above network. By Theorem 6, every limiting point of the sequence {xk } produced by the feedback neural network (13) is the the optimal solution of (5) and limiting values of the sequence {V (xk )} and {Mk } are its optimal value.

4

Conclusions

This paper aim to solve in real time non-differentiable convex optimization problems on a closed convex subset of Rn , the dynamic feedback neural network models based on differential inclusions are introduced,through constructing successively an iterative energy functions sequence and the corresponding dynamic subnetworks which are described to obey differential inclusion systems, the dynamical behavior and optimization capabilities of the proposed network models are analyzed rigorously, and the convergence theorem for solving in real time the primal problem is then obtained. About the implementation technique relating with this paper,the reader can refer to authors’ paper[14]where some simulation algorithms and numerical simulation experiments are given.

Acknowledgment The authors of this paper would like to thanks for the support of 973 project (No.2002cb312205) and the National Science Foundation of China (No. 60574077).

References 1. Hopfield, J.J. and Tank, D.W.: Neural Computation of Decisions in Optimization Problems. Biol. Cybern.52(1985) 141-152 2. Tank, D.W. and Hopfield, J.J.: Simple Neural Optimization Network: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit. IEEE Trans. Circuits Syst.33(1986) 533-541 3. Kennedy, M.P. and Chua, L.O.: Neural Networks for Nonlinear Programming.IEEE Trans. Circuits Syst.35(1988) 554-562 4. Rodr´ıguez-V´ aquez, A.,Rueda, R., Huertas, J.L. and S´ anchez-Sinencio, E.: Nonlinear Switched-Capacitor Neural Networks for Optimization Problems. IEEE Trans. Circuits Syst. 37(1990) 384-397 5. Bouzerdoum, A. and Pattison, T.R.: Neural Network for Quadratic Optimization with Bound Constraints. IEEE Trans. Neural Networks 4(1993) 293-304 6. Sudharsanan, S. and Sundareshan, M.: Exponential Stability and a Systematic Synthesis of a Neural Network for Quadratic Minimization. Neural Networks 4(1991) 599-613 7. Leung, Y., Chen, K.Z. and Gao, X.B.: A High-Performance Feedback Neural Network for Solving Convex Nonlinear Programming Problems. IEEE Trans. Neural Networks 14(2003) 1469-1477

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8. Forti, M. and Nistri, P.: Global Convergence of Neural Networks with Discontinuous Neuron Activations.IEEE Trans.Circuits and Systems-I 50(2003) 1421-1435 9. Lu, W. and Chen, T.: Dynamical Behaviors of Cohen-Grossberg Neural Networks with Discontinuous Activation Functions. Neural Networks 18(2005) 231-242 10. Forti, M., Nistri, P. and Quincampoix, M.: Generalized Neural Network for Nonsmooth Nonlinear Programming Problems.IEEE Trans.Circuits and Systems-I 51(2004) 1741-1754 11. Aubin, J.P. and Cellina, A.: Differential Inclusions. Springer-Verlag, Berlin (1984) 12. Dimitri, P.B.: Nonlinear Programming, Second Edition. Athena Scientific,Belmont (1999) 13. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic,Dordrecht (1988) 14. Li, G., Song, S. and Wu, C.: Subgradient-Based Feedback Neural Networks for Nondifferentiable Convex Optimization Problems. Science in China, Series F 49(2006) 91-106

A Recurrent Neural Network for Linear Fractional Programming with Bound Constraints Fuye Feng, Yong Xia, and Quanju Zhang Dongguan University of Technology, Dongguan, Guangdong, China {fengye408, xiay}@dgut.edu.cn, [email protected]

Abstract. This paper presents a novel recurrent time continuous neural network model which performs linear fractional optimization subject to bound constraints on each of the optimization variables. The network is proved to be complete in the sense that the set of optima of the objective function to be minimized with bound constraints coincides with the set of equilibria of the neural network. It is also shown that the network is primal and globally convergent in the sense that its trajectory cannot escape from the feasible region and will converge to an exact optimal solution for any initial point chosen in the feasible bound region. Simulation results are given to demonstrate further the global convergence and the good performance of the proposed neural network for linear fractional programming problems with bound constraints.

1

Introduction

Although linear programming arising in various branches of human activity, especially in economics, has become well-known, fractional programming has known increasing exposure recently and its importance in solving concrete problems is steadily increasing. Economic problems described by fractional programming models can be found in Charnes, Cooper and Rhodes [1], Patkar [2], and Mjelde [3]. Besides the economic applications, it was found that the fractional programming problems also appeared in other domains, such as Physics, Information theory, Game theory and the others. Among all kinds of fractional programming problems, the linear fractional programming is the most important one for its much widely applications, see Stancu-Minasian [4] in details. Unlike most conventional algorithms which are time-consuming in solving optimization problems with large-scale variables, neural network approach can handle, as described in Hopfield’s seminal work [5]-[6], optimization process in real-time on line, and hence to be the top-choice. As it is known, artificial neural networks (RNN) governed by a system of differential equations can be implemented physically by designated hardware with integrated circuits and an optimization process with different specific-purposes,see Cichocki and Unbehauen [7], could be conducted in a truly parallel way. More intensive neural network models were investigated by Wang and other researchers, see [8]-[14], including for convex programming [8] and linear programming [9]-[10], which, proved to J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 359–368, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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be globally convergent to the problem’s exact solutions. Kennedy and Chua [11] developed a neural network model for solving nonlinear programming problems where a penalty parameter needed to tune in the optimization process and hence only approximate solutions were generated. Xia and Wang [12] gave a general neural network model designing methodology which put together many gradientbased network models under their framework. Neural networks for the quadratic optimization and nonlinear optimization with bound constraints were developed by Bouzerdorm, Pattison [13] and Liang, Wang [14], respectively. All these neural networks can be classified into the following three types: 1) The gradient-based models [8]-[10] and its extension [12]; 2) The penalty function based model [11]; 3) The projection based models [13]-[14]. Among them the first was proved to have the global convergence [12] and the third quasi-convergence [8]-[10] only when the optimization problems are convex programming problems. The second [11] could only be demonstrated to have local convergence and it might fail to find exact solutions and hence has little applications then. As it is known, linear fractional programming does not belong to convex optimization problems [4] and hence need to investigate further then. Motivated by this idea, we are going to propose a promising recurrent continues-time neural network model which has the following features. 1) The set of optima of the linear fractional programming coincides with the set of equilibria of the proposing model. 2) The model is invariant with respect to the problem’s feasible set and any trajectory from the feasible set converges to an exact solution of the problem. Remains of the paper are organized as follows. Section II formulates the optimization problem and describes the construction of the proposing RNN model. Complete and global convergence properties are proven in Section III. Illustrative simulation results are reported in Section IV as the test of good performance for the proposing model by numerical example. Finally, Section V is a conclusion remark which summarizes the whole paper.

2

Problem Formulation and the Neural Network Model

The linear fractional programming with bound constraints can be formulated as min{F (x) : a ≤ x ≤ b},

(1)

where (i) F (x) = cT x + c0 /dT x + d0 , (ii) c, d are n dimensional column vectors, (iii) c0 , d0 are scalars, (iv) superscript T denotes the transpose operator, (v) x = (x1 , x2 , · · · , xn )T ∈ Rn is the decision vector, (vi) a = (a1 , a2 , · · · , an )T ∈ Rn , b = (b1 , b2 , · · · , bn )T ∈ Rn are constant vectors with ai ≤ bi (i = 1, 2, · · · , n). It is assumed that denominator of the objective function F (x) maintains a constant sign on an open set O which contains the bound constraints W = {x :

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a ≤ x ≤ b}, say positive, i.e., dT x + d0 > 0, ∀x ∈ W and that the function F (x) does not reduce to a linear function, i.e., dT x + d0  = constant on W and c, d are linearly independent. If x∗ ∈ W and F (x) ≥ F (x∗ ) for any x ∈ W , then x∗ is called an optimal solution to the problem (1). The set of all solutions to problem (1) is denoted by Ω∗ , i.e., Ω∗ = {x∗ ∈ W |F (x) ≥ F (x∗ ), ∀x ∈ W }. Consider the following single-layered recurrent neural network whose state variable x is described by the differential equation dx = −x + fW (x − ∇F (x)), dt

(2)

where ∇ is the gradient operator and fW : Rn → W is the projection operator defined by fW (x) = arg min x − w . (3) w∈W

For the bound constrained feasible set W , the operator fW can be expressed explicitly as fW (x) = (fW 1 (x), · · · , fW n (x)) whose ith component is ⎧ xi < ai , ⎨ ai , fW i (x) ≡ fW i (xi ) = xi , ai ≤ xi ≤ bi , ⎩ bi , xi > bi . We can also reformulate (2) in the following component form dxi ∂F (x) = −xi + fW i (xi − ), i = 1, 2, · · · , n. dt ∂xi

(4)

The block functional diagram of the RNN model (4) is depicted in Fig.1. Accordingly, architecture of the proposed model (4) is composed of n integrators, n processors for F (x), 2n piece-wise linear activation functions and 2n summers. Let the equilibrium state of the RNN model (2) be Ωe which is defined by e Ω = {xe ∈ Rn |xe = fW (xe − ∇F (xe ))} ⊆ W. Its relationship with problem (1)’s minimizer set Ω∗ is explored by the coming section now.

3

The Complete Property of the RNN Model (2)

As proposed in [15], a neural network is said to be regular or normal if the set of minimizers of an energy function is a subset or superset of the set of the stable states of the neural network, respectively. If the two sets are exact the same, we say the network to be complete. Regular property implies the neural network’s reliability and normal effectiveness, respectively, while complete both. Here, for model (2), it is said to be regular, normal, and complete if three cases of Ω∗ ⊆ Ωe , Ωe ⊆ Ω∗ , and Ω∗ = Ωe occur respectively. Theorem 1. The RNN model (2) is complete, i.e., Ω∗ = Ωe . In order to prove Theorem 1, we need the following three lemmas.

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r .. .

.. .

∂F (x) ∂x1

∂F (x) ∂x2

+ ?  −  6 r

@ @

+ ?   −   r

+ ?  −  6

@ @

.. . .. .

r

.. . .. .

.. . .. .

.. . .. .

+ ?   −   .. . .. .

.. . .. .

r .. .

∂F (x) ∂xn

@ @

.. . .. .

6

xb01

r b x1 (t)

6 -

xb02

r b x2 (t) .. . .. .

.. . .. .

r

+ ?  −  6

+ ?   −  

6

xb0n

r b xn (t)

Fig. 1. Functional block diagram of the neural network model (2)

Lemma 1. Suppose that x∗ is a solution to problem (1), that is, F (x∗ ) = min F (y), y∈W

(5)

then x∗ is a solution to the variational inequality x ∈ W : (∇F (x), y − x) ≥ 0, ∀y ∈ W.

(6)

Proof: See [16] Proposition 5.1. Lemma 2. Function F (x) = cT x + c0 /dT x + d0 defined in (1) is both pseudoconvex and pseudoconcave over W . Proof: See [17] Lemma 11.4.1. Lemma 3. Let F (x) : Rn → R be a differentiable pseudoconvex function on an open set Y ⊆ Rn , and W ⊆ Y any given nonempty and convex set. Then x∗ is an optimal solution to the problem of minimizing F (x) subject to x ∈ W if and only if (x − x∗ )T ∇F (x∗ ) ≥ 0 for all x ∈ W . Proof: See [4] Theorem 2.3.1 (b). Now, we turn to the proof of Theorem 1: Let x∗ = (x∗1 , x∗2 , · · · , x∗n )T ∈ Ω∗ , then F (x) ≥ F (x∗ ) for any x ∈ W and hence, Lemma 1 means that x∗ solves (6), that is

A Recurrent Neural Network for Linear Fractional Programming

x∗ ∈ W : (y − x∗ )T ∇F (x∗ ) ≥ 0, ∀y ∈ W,

363

(7)

which is equivalent to, see [18], x∗ = fW (x∗ − ∇F (x∗ )), so, x∗ ∈ Ωe . Thus, Ω∗ ⊆ Ωe . Conversely, let xe = (xe1 , xe2 , · · · , xen )T ∈ Ωe , that is xe = fW (xe − ∇F (xe )), which, also see [18], means xe ∈ W : (y − xe )T ∇F (xe ) ≥ 0, ∀y ∈ W.

(8)

Since the function F (x) is pseudoconvex over W , see Lemma 2, it can be obtained by Lemma 3 that F (x) ≥ F (xe ), ∀x ∈ W, so, xe ∈ Ω∗ . Thus, Ωe ⊆ Ω∗ . Therefore, the obtained results Ωe ⊆ Ω∗ and Ω∗ ⊆ Ωe show the result of Theorem 1, Ω∗ = Ωe , comes to be true then.

4

The Global Convergence Problem

First, it can be shown that the RNN model (2) has a solution trajectory which is global in the sense that the existence interval can be extended to ∞ on the right hand for any initial point in W. The continuity of the right hand of (2) means, by Peano’s local existence theorem, see [19], that there exists a solution x(t; x0 ) for t ∈ [0, tmax ) with any initial point x0 ∈ W , here tmax is the maximal right hand point of the existence interval. The following lemma states that this tmax to be ∞. Lemma 4. The solution x(t; x0 ) of RNN model (2) with any initial point x(0; x0 ) = x0 ∈ W is bounded and so, it can be extended to ∞. Proof: It is easy to check that the solution x(t) = x(t; x0 ) for t ∈ [0, tmax ) with initial condition x(0; x0 ) = x0 is given by −t 0

x(t) = e x + e

−t



t

es fW [x(s) − ∇F (x(s))]ds.

(9)

0

Obviously, mapping fW is bounded, that is fW ≤ K for some positive number K > 0, where . is the Euclidean 2-norm. It follows from (9) that −t

−t



x(t) ≤ e x + e K 0

t

es ds,

(10)

0

≤ e−t x0 + K(1 − e−t ),

(11)

≤ max{ x , K}.

(12)

0

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Thus, solution x(t) is bounded. By the extension theorem for ODEs, see [19], it can be concluded that tmax = ∞. Another vital dynamical property about the RNN model (2) says the set W is positive invariant. That is, any solution x(t) starting from a point in W , e.g. x0 ∈ W , will stay in W for all time t elapsing. Theorem 2. W is a positive invariant set of the RNN model (2). Proof: Let W i = {xi ∈ R|ai ≤ xi ≤ bi } and x0i = xi (0; x0 ) ∈ W i . We first prove that, for all i = 1, 2, · · · , n, if x0i = x(0; x0 ) ∈ W i , then the ith component xi (t) = xi (t; x0 ) belongs to W i , that is, xi (t) ∈ W i for all t ≥ 0. Let t∗i = sup{t˜ |xi (t) ∈ W i , ∀t ∈ [0, t˜ ]} ≥ 0. It can be shown by a contradiction that t∗i = +∞. Suppose t∗i < ∞, then xi (t) ∈ W i for t ∈ [0, t∗i ] and xi (t) ∈ / W i for t ∈ (t∗i , t∗i + δ), here δ is a positive number. With no loss of generality, we assume that xi (t) < ai , ∀t ∈ (t∗i , t∗i + δ).

(13)

The proof for xi (t) < ai , ∀t ∈ (t∗i , t∗i + δ) is similar. By the definition of fWi , the RNN model (4) and the assumption (13), it follows that dxi (t)/dt ≥ −xi (t) + ai > 0, ∀t ∈ (t∗i , t∗i + δ). So, xi (t) is strictly increasing in t ∈ (t∗i , t∗i + δ) and hence xi (t) > xi (t∗i ), ∀t ∈ (t∗i , t∗i + δ).

(14)

Noting that xi (t) ∈ W for t ∈ [0, t∗i ] and assumption (13) means xi (t∗i ) = ai . So, by (14), we get xi (t) > ai , ∀t ∈ (t∗i , t∗i + δ). This is in contradiction with the assumption (13). Therefore, t∗i = +∞. So, W i is positive invariant. We can now explore the global convergence of the neural network model (2). To proceed, we need an inequality result about the projection operator fW and the definition of convergence for a neural network. Definition 1. Let x(t) be a solution of system x˙ = F (x). The system is said to be globally convergent to a set X with respect to set W if every solution x(t) starting at W satisfies ρ(x(t), X) → 0,

as t → ∞,

here ρ(x(t), X) = inf x − y and x(0) = x0 ∈ W. y∈X

(15)

A Recurrent Neural Network for Linear Fractional Programming

365

Definition 2. The neural network (2) is said to be globally convergent to a set X with respect to set W if the corresponding dynamical system is so. Lemma 5. For all v ∈ Rn and all u ∈ W (v − fW (v))T (fW (v) − u) ≥ 0. Proof: See [16] pp. 9-10. Theorem 3. The neural network (2) is globally convergent to the solution set Ω∗ with respect to set W . Proof: From Lemma 5, we know that (v − fW (v))T (fW (v) − u) ≥ 0, v ∈ Rn , u ∈ W. Let v = x − ∇F (x) and u = x, then (x − ∇F (x) − fW (x − ∇F (x)))T (fW (x − ∇F (x)) − x) ≥ 0, i.e., (∇F (x))T {fW (x − ∇F (x)) − x} ≤ − fW (x − ∇F (x)) − x .

(16)

Define an energy function F (x), then, differentiating this function along the solution x(t) of (2) gives us dF (x(t)) dx = (∇F (x))T = (∇F (x))T {fW (x − ∇F (x)) − x}. dt dt

(17)

According to (16), it follows that dF (x(t)) ≤ − fW (x − ∇F (x)) − x ≤ 0. dt

(18)

It means the energy of F (x) is decreasing along any trajectory of (2). By Lemma 4, we know the solution x(t) is bounded. So, F (x) is a Liapunov function to system (2). Therefore, by LaSalle’s invariant principle [20], we know that all trajectories of (2) starting at W will converge to the largest invariant subset Σ of set E like E = {x |

dF = 0}. dt

(19)

However, it can be guaranteed from (18) that dF dt = 0 only if fW (x − ∇F (x)) − x = 0, which means that x must be an equilibrium of (2) or, x ∈ Ω. Thus, Ω is the convergent set for all trajectories of neural network (2) starting at W. Noting that Theorem 1 tells us that Ω∗ = Ω and hence, Theorem 3 is proved to be true then.

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Up to now, we have demonstrated that the proposed neural network (2) is a promising neural network model both in implementable construction sense and in theoretic convergence sense for solving linear fractional programming problems. Certainly, it is also important to simulate a network’s effectiveness by numerical experiment to test its performance in practice. In next section, we will focus our attention on handling illustrative example to reach this goal.

5

Illustrative Example

We give a computational example as simulation experiment to show the proposed network’s good performance. Example. Consider the following linear fractional programming x1 + x2 + 1 , 2x1 − x2 + 3 0 ≤ x1 ≤ 2, 0 ≤ x2 ≤ 2.

min F (x) = s.t.

(20) (21) (22)

This problem has an exact solution x∗ = [0, 0]T with the optimal value F (x∗ ) = 1/3 and the gradient of F (x) can be expressed as  −3x2 + 1/(2x1 − x2 + 3)2 ∇F (x) = . (23) 3x1 + 4/(2x1 − x2 + 3)2 1

0.9

0.8

0.7

x1, x2

0.6

0.5

x

1

0.4

0.3

0.2

x

2

0.1

0

0

5

10 t

Fig. 2. Transient behaviors of neural trajectories x1 , x2

15

A Recurrent Neural Network for Linear Fractional Programming

367

0.5 x

1

x1, x2

0.4

0.3

0.2

x

0.1

0

0

2

2

4

6 t

8

10

12

Fig. 3. Neural trajectories x1 , x2 with starting point (0.5, 0.5)

We use neural network (2) to solve this problem. Simulation result is carried out by ODE 23 solver conducted on MATLAB 7.0. and the transient behaviors of the neural trajectories x1 , x2 starting at x0 = [0.4, 1]T are shown in Fig. 2. It can be seen visibly from the figure that the proposed neural network converges to the exact solution very soon. Also, Fig. 3 presents how the solution of this problem is located by the proposed neural trajectories from a different initial point, here x0 = [0.5, 0.5]T , in a more clearly visible way.

6

Conclusion

In this paper, we have proposed a neural network model for solving linear fractional programming problems with bound constraints. The network is governed by a system of differential equations with a projection method. The stability of the proposed neural network has been demonstrated to have global convergence with respect to the problem’s feasible set. As it is known, the existing neural network models with penalty function method for solving nonlinear programming problems may fail to find the exact solution of the problems. The new model has overcome this stability defect which appears in all penalty function based models. Certainly, the network presented here can perform well in the sense of real time computation which, in the time elapsing sense, is also superior to the classical algorithms. Finally, numerical simulation results demonstrate further that the new model can act both effectively and reliably on the purpose of locating the involved problem’s solutions.

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References 1. Charnes, A., Cooper, W. W., Rhodes, E.: Measuring the Efficiency of Decision Making Units. European J. Oper. Res. 2 (2) (1978) 429-444 2. Patkar, V. N.: Fractional Programming Models for Sharing of Urban Development Responsabilities. Nagarlok. 22 (1) (1990) 88-94 3. Mjelde, K.M.: Fractional Resource Allocation with S-shaped Return Functions. J. Oper. Res. Soc. 34 (2) (1983) 627-632 4. Stancu-Minasian, I. M.: Fractional Programming, Theory, Methods and Applications. Kluwer Academic Publishers, Netherlands (1992) 5. Hopfield, J. J.: Neurons with Graded Response Have Collective Computational Properties Like Those of Two-state Neurons. Proc. Natl. Acad. Sci. 81 (10) (1984) 3088-3092 6. Hopfield, J. J., Tank, D. W.: Neural Computation of Decisions in Optimization Problems. Biolog. Cybernetics. 52 (1) (1985) 141-152 7. Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. John Wiley & Sons, New York (1993) 8. Wang, J.: A Deterministic Annealing Neural Network for Convex Programming. Neural Networks. 7 (2) (1994) 629-641 9. Wang, J., Chankong, V.: Recurrent Neural Networks for Linear Programming: Analysis and Design Principles. Computers and Operations Research. 19 (1) (1992) 297-311 10. Wang, J.: Analysis and Design of a Recurrent Neural Network for Linear Programming. IEEE Transactions on Circuits and Systems. 40 (5) (1993) 613-618 11. Kennedy, M. P., Chua, L. O.: Neural Networks for Nonlinear Programming. IEEE Transaction on Circuits and Systems. 35 (5) (1988) 554-562 12. Xia, Y. S., Wang, J.: A General Methodology for Designing Globally Convergent Optimization Neural Networks. IEEE Transaction on Neural Networks. 9 (12) (1998) 1311-1343 13. Bouzerdorm, A., Pattison, T. R.: Neural Network for Quadratic Optimization with Bound Constraints. IEEE Transaction on Neural Networks. 4 (2) (1993) 293-304 14. Liang, X. B., Wang, J.: A Recurrent Neural Network for Nonlinear Optimization with a Continuously Differentiable Objective Function and Bound Constraints. IEEE Transaction on Neural Networks. 11 (11) (2000) 1251-1262 15. Xu, Z.B., Hu, G.Q., Kwong, C.P.: Asymmetric-Hopfield-Type Networks: Theory and Applications. Neural Networks. 9 (2) (2000) 483-501 16. Kinderlehrer, D., Stampcchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, New York (1980) 17. Bazaraa, M. S., Shetty, C. M.: Nonlinear Programming, Theory and Algorithms. John Wiley and Sons, New York (1979) 18. Eaves, B.C.: On the Basic Theorem of Complementarity. Mathematical Pragramming. 1 (1) (1970) 68-75 19. Hale, J.K.: Ordinary Diffential Equations. Wiley, New York (1993) 20. LaSalle, J.: The Stability Theory for Ordinary Differential Equations. J. Differential Equations 4 (1) (1983) 57-65

A Delayed Lagrangian Network for Solving Quadratic Programming Problems with Equality Constraints Qingshan Liu1 , Jun Wang1 , and Jinde Cao2 1

Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong {qsliu, jwang}@acae.cuhk.edu.hk 2 Department of Mathematics, Southeast University, Nanjing 210096, China [email protected]

Abstract. In this paper, a delayed Lagrangian network is presented for solving quadratic programming problems. Based on some known results, the delay interval is determined to guarantee the asymptotic stability of the delayed neural network at the optimal solution. One simulation example is provided to show the effectiveness of the approach.

1

Introduction

Neural networks based on circuit implementation can provide real-time solutions, and can be applied to solve many engineering problems, such as optimal control, structure design, signal and image processing. In recent years, many recurrent neural networks were proposed for solving linear and nonlinear optimization problems [1]-[12]. Tank and Hopfield [1] first proposed a neural network for solving linear programming problems in 1986. Based on penalty function and gradient methods, Kennedy and Chua [2] extended and improved the TankHopfield network for solving nonlinear programming problems. Bouzerdoum and Pattison [3] presented a neural network for solving quadratic convex optimization problems with bounded constraints. Zhang and Constantinides [4] proposed the Lagrangian network and Wang et al. [7] and Xia [8] studied the global convergence of the network. By primal-dual approach and projection approach, the primal-dual neural network and projection neural network were presented for solving linear and nonlinear convex programming problems [9]-[11]. However, these neural networks are based on the assumption that neurons communicate and respond instantaneously without any time delay. In fact, the switching delay exists in some hardware implementation, and the time delay is hard to predicted to guarantee the stability of the neural network theoretically. This is our first motivation to introduce the delayed neural network. Chen 

This work was supported by the Hong Kong Research Grants Council under Grant CUHK4165/03E, and the National Natural Science Foundation of China under Grant 60574043.

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 369–378, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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and Fang [12] proposed a delayed neural network for solving convex quadratic programming problems by using the penalty function approach. However, this network can not converge an exact optimal solution and has an implementation problem when the penalty parameter is very large. To avoid using finite penalty parameters, it is necessary to construct delayed neural network by using other approach. This is our second motivation to study the delayed neural network. For some quadratic programming problems, the convergence of some neural networks without time delays can not be guaranteed due to the asymmetry of the Jacobian matrix [8]. However, if we add some delays to the networks, but not change the equilibrium points, the topology structure of the solutions of the networks is altered and the convergence of the network can be obtained by selecting proper delays. This is our third motivation to introduce the delayed neural network and study its convergence.

2

Problem Formulation and Delayed Neural Network

In this paper, we consider the following quadratic convex programming problem: QP

minimize subject to

1 T 2 x Qx

+ cT x,

Ax = b,

(1)

where Q ∈ Rn×n is symmetric and positive semidefinite, c ∈ Rn , A ∈ Rm×n , and b ∈ Rm . Throughout this paper, we always assume that feasible domain F = {x ∈ Rn×n |Ax − b = 0} is not empty. In order to adopt the neural network method for solving problem (1), it is necessary to convert this problem into some equalities. It is well known that the Lagrangian function of (1) is defined as L(x, y) =

1 T x Qx + cT x − y T (Ax − b), 2

where y ∈ Rm is the Lagrange multiplier. By Karush-Kuhn-Tucker conditions [15], x∗ is a solution to (1) if and only if there exists y ∗ ∈ Rm such that (x∗ , y ∗ ) satisfies the following Lagrange condition:  ∇Lx (x, y) = Qx + c − AT y = 0 (2) ∇Ly (x, y) = Ax − b = 0 where ∇L is the gradient of L. Letting     Q −AT c W = , J= , A 0 −b

  x u= , y

then (2) can be rewritten as W u + J = 0. In [4], the following Lagrangian network was proposed to solve problem (1): du = −(W u + J). dt

(3)

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371

In [7] and [8], the global convergence of the Lagrangian network (3) has been investigated in case of the matrix Q being positive definite. However, if Q is positive semidefinite but not positive definite, the convergence of system (3) can not be guaranteed for some quadratic programming problems, which can be seen from the following example. Example. Consider the following convex quadratic programming problem: minimize f (x) =

1 T x 2



0.1 0.1 0.1 0.1



  x+ −1, 1 x, subject to (0.5, −0.5)x = −0.5,

where x = (x1 , x2 )T . This problem has a unique solution x∗ = (−0.5, 0.5)T . Then, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0.1 0.1 −0.5 −1 x1 W = ⎝ 0.1 0.1 0.5 ⎠ , J = ⎝ 1 ⎠ , u = ⎝ x2 ⎠ . 0.5 −0.5 0 0.5 y In the Lagrangian network (3), since the coefficient matrix −W has three eigenvalues λ1 = −0.2 and λ2,3 = ±0.7071i, this system has periodic solutions. The equilibrium point of this system is not asymptotically stable. From the above example, we know that the Lagrangian network (3) can not converge to its equilibrium point for some quadratic programming problems. In this case, our method is to change the topology structure of the solutions by adding some delay state items. In this paper, we propose the following delayed Lagrangian network for solving problem (1): du = −(D + W )u(t) + Du(t − τ ) − J, dt

(4)

where τ ≥ 0 denotes the transmission delay, D ∈ R(n+m)×(n+m) , u(t) with the initial value function u(s) = φ(s) s ∈ [−τ, 0]. It is easily to see that Lagrangian network (3) is a special case of the delayed Lagrangian network (4) when τ = 0. Obviously, the Lagrangian network (3) and the delayed Lagrangian network (4) have the same equilibrium point and we can get the exact solution of problem (1) by using network (4). From the results in [7], the following lemma holds. Lemma 1. x∗ is an optimal solution to problem (1) if and only if there exists y ∗ ∈ Rm such that u∗ = (x∗ , y ∗ ) is an equilibrium point of the Lagrangian network (3) or the delayed Lagrangian network (4). Assume that u∗ is an equilibrium point of (4), then −(D + W )u∗ + Du∗ − J = 0. By means of the coordinate translation z = u − u∗ , we have dz = −(D + W )z(t) + Dz(t − τ ). dt

(5)

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Time-Delay Analysis

In this section, we give the stability of an equilibrium of the following system based on the known results in [14]: dz = M z(t) + Mτ z(t − τ ), dt

(6)

where z(t) ∈ Rn , M, Mτ ∈ Rn×n , τ ≥ 0. For x ∈ Cn , the vector norms  · p are defined as

n n 

 x1 = |xi |, x2 =  xi x∗i , x∞ = max |xi |, i=1

1≤i≤n

i=1

where Cn denotes the n-dimensional complex vector space, x∗i is the conjugation of xi . The induced matrix norms  · p are displayed as follows A1 = max j

n 

|aij |,

A2 =

 λmax (A∗ A),

A∞ = max i

i=1

n 

|aij |,

j=1

where A = {aij }n×n , A∗ denotes the conjugate transpose matrix of A, λmax (·) is the maximum eigenvalue of corresponding matrix. Definition 1. [16] Let A = {aij }n×n is a matrix on Cn×n , then the corresponding matrix measure is the function μp : Cn×n → R defined by μp (A) = lim

ε→0+

I + εAp − 1 , ε

where  · p is an induced matrix norm on Cn×n , I is identity matrix and p = 1, 2, ∞. The induced matrix measure μp (·) has the following forms μ1 (A) = max{Re(ajj ) + j



|aij |},

μ2 (A) =

i =j

μ∞ (A) = max{Re(aii ) + i



1 [λmax (A∗ + A)], 2

|aij |},

j =i

where Re(·) represents the real part of the complex number. Lemma 2. [16] The matrix measure μp (·) defined in Definition 1 has the following properties: (i) Reλi (A) ≤ μp (A); (ii) μp (A + B) ≤ μp (A) + μp (B), ∀A, B ∈ Cn×n , where λi (·) represents an eigenvalue (i = 1, 2, . . .).

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373

Define the characteristic equation of system (6) as det(λI − M − Mτ exp(−τ λ)) = 0.

(7)

Equivalently, the characteristic function can be written as λ = λi (M + Mτ exp(−τ λ)).

(8)

In order to study the stability condition of system (6), let us define l1  μp (M ) + Mτ p ,

(9)

l2  μp (−ιM ) + Mτ p ,

(10)

and where ι2 = −1, p = 1, 2 and ∞. The first stability condition is given by Mori et al. [13] as the following theorem. Theorem 1. If l1 < 0, then every equilibrium point of the dynamic system defined by (6) is asymptotically stable. This is a delay-independent stability condition, and the other delay-depend stability condition is given by Mori and Kokame [14] in the following theorem. Theorem 2. When l1 ≥ 0, if Reλi (M + Mτ exp(−τ λ)) < 0,

(i = 1, 2, . . .),

(11)

then every equilibrium point of the dynamic system defined by (6) is asymptotically stable, where λ takes the values of ιω, l1 + ιω and r + ιl2 , with 0 ≤ ω ≤ l2 and 0 ≤ r ≤ l1 . In Theorem 2, it is very hard to check the stability conditions, since the eigenvalues in (7) are infinite. However, from Lemma 2 and Theorem 2, we can get several convenient way to guarantee to stability of system (6). We state them as the following corollaries. Corollary 1. When l1 ≥ 0, if μp (M + Mτ exp(−τ λ)) < 0,

(12)

μp (Mτ exp(−τ λ)) < −μp (M ),

(13)

or then every equilibrium point of the dynamic system defined by (6) is asymptotically stable, where λ takes the values of ιω, l1 + ιω and r + ιl2 , with 0 ≤ ω ≤ l2 and 0 ≤ r ≤ l1 . If l1 = 0, the stability conditions are more convenient to be checked.

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Corollary 2. When l1 = 0, if μp (M + Mτ exp(−τ λ)) < 0,

(14)

μp (Mτ exp(−τ λ)) < −μp (M ),

(15)

or then every equilibrium point of the dynamic system defined by (6) is asymptotically stable, where λ takes the values of ιω, with 0 ≤ ω ≤ l2 . If Mτ = kI, we have λi (M + Mτ exp(−τ λ)) = k exp(−τ λ) + λj (M ), then Reλi (M + Mτ exp(−τ λ)) = k exp(−τ Reλ) cos(τ Imλ) + Reλj (M ), where I is the identity matrix, k is a real scalar, Im(·) represents the imaginary part of complex number, i = 1, 2, . . . , and j = 1, 2, . . . , n. From Theorem 2, we can get the following useful corollaries. Corollary 3. When l1 ≥ 0, Mτ = kI and k ≥ 0, every equilibrium point of the dynamic system defined by (6) is asymptotically stable if the following conditions hold k exp(−τ r) cos(τ ω) + Reλj (M ) < 0, (16) where r, ω take the values of (a) r = 0, l1 , 0 ≤ ω ≤ l2 ; (b) 0 ≤ r ≤ l1 , ω = l2 , and j = 1, 2, . . . , n. By Lemma 2, a more convenient way for Corollary 3 is using k exp(−τ r) cos(τ ω) + μp (M ) < 0,

(17)

where r, ω take the same values as Corollary 3. Corollary 4. When l1 ≥ 0, Mτ = kI and k < 0, every equilibrium point of the dynamic system defined by (6) is asymptotically stable if one of the following conditions holds for 0 ≤ l2 ≤ for

π , 2τ

π π < l2 ≤ , 2τ τ for

for j = 1, 2, . . . , n.

k exp(−l1 τ ) cos(l2 τ ) + Reλj (M ) < 0,

π < l2 , τ

(18)

k cos(l2 τ ) + Reλj (M ) < 0,

(19)

−k + Reλj (M ) < 0,

(20)

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375

If l1 = 0, Mτ = kI, k > 0, and matrix M + Mτ is invertible, from (17), we have k exp(−τ r) cos(τ ω) + μp (M ) = k cos(τ ω) + μp (M ) < 0,

(21)

where 0 ≤ ω ≤ l2 . Since l1 = 0 and Mτ = kI, we have μp (M ) = −Mτ p = −k.

(22)

From (21) and (22), it follows that cos(τ ω) < 1. Since 0 ≤ ω ≤ l2 , we have 0 < τ ω < 2π, then, 0 < τ < 2π/l2 . Since matrix M + Mτ is invertible, equation (8) has no zero solution. Then λ can not take zero value. Therefore, we need only consider λ takes the values of ιω, with 0 < ω ≤ l2 in Corollary 3. Then the inverse procedure of above is also guaranteed. We can get the following results. Corollary 5. When l1 = 0, Mτ = kI, and matrix M + Mτ is invertible, every equilibrium point of the dynamic system defined by (6) is asymptotically stable if 0 < τ < 2π/l2 , where k > 0. Next, we apply the above results to solve the quadratic programming problem (1). We state it as the following theorem. Theorem 3. When matrix Q ∈ Rn×n is symmetric and positive semidefinite in problem (1), D = kI, and W is invertible in system (4), every equilibrium point u∗ = (x∗ , y ∗ ) of the delayed Lagrangian network (4) is asymptotically stable if 0 < τ < 2π/l2 , where x∗ corresponds to an optimal solution of problem (1), and k > 0. Proof. From system (4),    Q −AT M = −(D + W ) = − kI + , A 0    2Q 0 M ∗ + M = − 2kI + , 0 0

thus

then

   2Q 0 λi (M + M ) = −2k + λi − , 0 0 ∗

(i = 1, 2, . . . , n + m).

Since Q is symmetric and positive semidefinite, we have    2Q 0 max λi − = 0, 0 0 i

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then

1 max λi (M ∗ + M ) = −k. 2 i Since Mτ = D = kI, Mτ 2 = k, it follows that l1 = μ2 (M ) + Mτ 2 = 0. We have M + Mτ = −W is invertible. Therefore, by Corollary 5, every equilibrium point u∗ = (x∗ , y ∗ ) of the delayed Lagrangian network (4) is asymptotically stable if 0 < τ < 2π/l2 . From Lemma 1, x∗ corresponds to an optimal solution of problem (1). 

μ2 (M ) =

4

A Simulation Example

In this section, we use the delayed Lagrangian network to solve the problem in the example of Section 2. Let D = I3×3 identity matrix, then ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ −1.1 −0.1 0.5 100 −1 x1 M = ⎝ −0.1 −1.1 −0.5 ⎠ , Mτ = ⎝ 0 1 0 ⎠ , J = ⎝ 1 ⎠ , u = ⎝ x2 ⎠ . −0.5 0.5 −1 001 0.5 y √ By simple computation, we have l2 = μ2 (−ιM ) + Mτ 2 = 1 + 2/2. By Theorem 3, the equilibrium point of system (4) is asymptotically stable, if 3

3 x1 x2 y

2 1

1

0

0

−1

−1

−2

−2

−3

−3

−4

−4

−5

x1 x2 y

2

0

50

100

150 time

200

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−5

0

50

(a) τ = 0

150 time

200

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(b) τ = 0.5

3

3 x1 x2 y

2

x1 x2 y

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4

−4

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50

100

(c) τ = 2

150 time

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250

300

−5

0

50

100

150 time

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250

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(d) τ = 3.6

Fig. 1. State trajectories of the delayed Lagrangian network for different time delays

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0 < τ < 2π/l2 ≈ 3.68. Figure 1 gives the transient behavior with τ = 0, 0.5, 2 and 3.6. We can see that the solutions of system (4) converge to the equilibrium point (−0.5, 0.5, −2) when τ takes the values of 0.5, 2 and 3.6, where (x∗1 , x∗2 ) = (−0.5, 0.5) is the optimal solution of this problem and y ∗ = −2 is the Lagrangian multiplier.

5

Conclusions

In this paper, we have proposed a delayed Lagrangian network to solve convex quadratic programming problems with equality constrains. We give a delay interval for the asymptotic stability of the delayed Lagrangian network at the optimal solution. By choosing proper delays, we can get the exact optimal solutions. One simulation example is presented to show the effectiveness of the delayed Lagrangian network to quadratic programming problems.

References 1. Tank, D.W., Hopfield, J.J.: Simple Neural Optimization Networks: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit. IEEE Trans. Circuits and Systems 33 (1986) 533-541 2. Kennedy, M.P., Chua, L.O.: Neural Networks for Nonlinear Programming. IEEE Trans. Circuits and Systems 35 (1986) 554-562 3. Bouzerdoum, A., Pattison, T.R.: Neural Network for Quadratic Optimization with Bound Constraints. IEEE Trans. Neural Networks 4 (1993) 293-304 4. Zhang, S., Constantinides, A.G.: Lagrange Programming Neural Networks. IEEE Trans. Circuits and Systems II 39 (1992) 441-452 5. Zhang, S., Zhu, X., Zou, L.H.: Second-order Neural Nets for Constrained Optimization. IEEE Trans. Neural Networks 3 (1992) 1021-1024 6. Wang, J.: A Deterministic Annealing Neural Network for Convex Programming. Neural Networks 7 (1994) 629-641 7. Wang, J., Hu, Q., Jiang, D.: A Lagrangian Network for Kinematic Control of Redundant Robot Manipulators. IEEE Trans. Neural Networks 10 (1999) 11231132 8. Xia, Y.: Global Convergence Analysis of Lagrangian Networks. IEEE Trans. Circuits and Systems I 50 (2003) 818-822 9. Wang, J.: Primal and Dual Assignment Networks. IEEE Trans. Neural Networks, 8 (1997) 784-790 10. Wang, J., Xia, Y.: Analysis and Design of Primal-dual Assignment Networks. IEEE Trans. Neural Networks 9 (1998) 183-194 11. Xia, Y., Wang, J.: A General Projection Neural Network for Solving Monotone Variational Inequalities and Related Optimization Problems. IEEE Trans. Neural Networks 15 (2004) 318-328 12. Chen, Y.H., Fang, S.C.: Neurocomputing with Time Delay Analysis for Solving Convex Quadratic Programming Problems. IEEE Trans. Neural Networks 11 (2000) 230-240 13. Mori, T., Fukuma, N., Kuwahara, M.: On an Estimate of the Decay Rate for Stable Linear Delay Systems. Int. J. Control 36 (1982) 95-97

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14. Mori, T., Kokame, H.: Stability of x(t) ˙ = Ax(t)+Bx(t−τ ). IEEE Trans. Automat. Control 34 (1989) 460-462 15. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms (2nd Ed.). John Wiley, New York (1993) 16. Vidyasagar, M.: Nonlinear System Analysis. Englewood Cliffs, NJ: Prentice Hall (1993)

Wavelet Chaotic Neural Networks and Their Application to Optimization Problems Yao-qun Xu1,2, Ming Sun2, and Guangren Duan1 1

Center for Control Theory and Guidance Technology, Harbin Institute of Technology, 150001 Harbin, China [email protected] 2 Institute of Computer and Information Engineering, Harbin Commercial University, 150028 Harbin, China [email protected]

Abstract. In this paper, we first review Chen’s chaotic neural network model and then propose a novel wavelet chaotic neural network. Second, we apply them to search global minima of a continuous function, respectively. Meanwhile, the time evolution figures of the corresponding most positive Lyapunov exponent are given. Third, 10-city traveling salesman problem (TSP) is given to make a comparison between them. Finally we conclude that the novel wavelet chaotic neural network is more valid.

1 Introduction Many combinatorial optimization problems arising from science and technology are often difficult to solve entirely. Hopfield and Tank first applied the continuous-time, continuous-output Hopfield neural network (HNN) to solve TSP [1], thereby initiating a new approach to optimization problems [2, 3]. However, using the HNN to solve continuous-time nonlinear searching optimization and TSP suffers from several shortcomings. First, the network is often trapped at a local minimum in the complex energy terrain because of its gradient descent property. Second, HNN may converge to an infeasible solution. At last, sometimes, HNN does not converge at all within prescribed iteration. Chaotic neural networks have been proved to be powerful tools for escaping from local minima. From then on, there have been some researches on chaotic neural networks in the field. Chen and Aihara proposed chaotic simulated annealing (CSA) to illustrate the features and effectiveness of a transiently chaotic neural network (TCNN) in solving optimization problems [4]; and Wang proposed a stochastic noisy chaotic simulated annealing method (SCSA) [5] by combining stochastic simulated annealing (SSA) and chaotic simulated annealing (CSA). All above researches are based on simulated annealing methods; distinctly, now we do research on the activation function. In this paper, we first review the Chen’s chaotic neural network model. Second, we propose a novel chaotic neural network model. Third, we apply both of them to search global minima of a continuous nonlinear function and then the time evolution figures of J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 379 – 384, 2006. © Springer-Verlag Berlin Heidelberg 2006

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their most positive Lyapunov exponents are given. At last, we apply both of them to 10-city traveling salesman problem (TSP) in order to make a comparison. Finally we conclude the novel chaotic neural network we proposed is more valid.

2 Chaotic Neural Network Models In this section, two chaotic neural network models are given. And the first is proposed by Chen, the second is proposed by us. 2.1 Chaotic Simulated Annealing with Decaying Self-coupling Chen and Aihara’s transiently chaotic neural network [4] is described as follows:

x i (t ) = f ( y i (t )) =

1 1+ e

(1)

− yi ( t ) / ε

⎡ ⎤ y i (t + 1) = ky i (t ) + α ⎢ Wij x j + I i ⎥ − z i (t )( x i (t ) − I 0 ) ⎢ j ⎥ ⎣ ⎦ z i (t + 1) = (1 − β ) z i (t )

∑

(2) (3)

where x i (t ) is output of neuron i ; y i (t ) denotes internal state of neuron

i ; Wij describes connection weight from neuron j to neuron i , Wij = W ji ; I i is input bias of neuron i , a a positive scaling parameter for neural inputs, k damping factor of nerve membrane, 0≤ k ≤1, z i (t ) self-feedback connection weight (refractory strength) ≥0, β damping factor of z i (t ) , 0< β <1, I 0 a positive parameter, parameter of the output function, ε >0.

ε steepness

2.2 Morlet-Sigmoid Chaotic Neural Network (M-SCNN)

Morlet-Sigmoid chaotic neural network is a novel model proposed by us, described as follows: ⎡ y i (t + 1) = ky i (t ) + α ⎢ ⎢ ⎣

⎤ Wij x j + I i ⎥ − z i (t )( x i (t ) − I 0 ) ⎥ j ⎦ x i (t ) = f ( y i (t ))

∑

z i (t + 1) = (1 − β ) z i (t )

ηi (t + 1) = f ( yi (t )) = γe

−

( u1 yi ( t )(1+ηi ( t ))) 2 2

(5) (6)

η i (t ) ln(e + λ (1 − ηi (t )))

cos(5u1 yi (t )(1 + ηi (t ))) +

(4)

(7) 1 1+ e

− u 0 yi ( t )(1+η i ( t ))

(8)

where xi (t ) , y i (t ) , Wij , α , k , I i , β , z i (t ) , I 0 are the same with the above. And η i (t ) is the other simulated annealing factor, η i (t ) >0; λ is a positive parameter,

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which controls the speed of this annealing process; u 0 and u1 are important parameters of activation function which should be varied with kinds of special optimization problems.

3 Application to Search Optimization of Continuous Nonlinear Function In this section, we apply the two chaotic neural networks to search the minimum points of a famous Six-Hump Camel-Back Function [6] which can be described as follows: f ( x1 , x 2 ) = 4 x12 − 2.1x14 + x16 / 3 + x1 x 2 − 4 x 22 + 4 x 24

x i ≤5

(9)

Its minimum point is (-0.08983,0.7126) or (0.08983,-0.7126)， and its corresponding minimum value is -1.0316285. Moreover, the time evolution figures of the corresponding most positive Lyapunov exponent are given. 3.1 Chen’s Chaotic Neural Network

The parameters are set as follows:

α =0.5, k =1, β =0.004,I0=0.8, ε =1 ,z (0)=[17.5,17.5],y(0)=[0,0]. The time evolution figure of the corresponding most positive Lyapunov exponent is shown as Fig.1.

Fig. 1. Lyapunov exponent time evolution figure

We find out that when these parameters α 、 k 、 β 、 z (0) and y(0) are invariable, however we change the parameters I0 and ε , the minimum point computed by Chen’s model can not reach (-0.08983,0.7126) or (0.08983,-0.7126), and nor does the minimum energy -1.0316285. When these parameters are set as above, its minimum point is (1.3705e-131, 0.70717), and its corresponding minimum value is -1 within 2000 iterations. Now, these parameters α 、 k 、 β 、 z (0) and y(0) are fixed so as to make a comparison between Chen’s and our model.

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3.2 Morlet-Sigmoid Chaotic Neural Network (M-SCNN)

The parameters are set as follows:

α =0.5, k =1, β =0.004, I0=0.5,µ 0=0.05,µ 1=20,λ=0.002, z (0)=[17.5,17.5], y(0)=[0,0],η(0)=[0.05,0.05]. The time evolution figure of the corresponding most positive Lyapunov exponent is shown as Fig.5.

Fig. 2. Lyapunov exponent time evolution figure

Under these parameters, its minimum point is (-0.088431, 0.71251)， and its corresponding minimum value is -1.0316 within 2000 iterations. Seen from the above analysis, the result is more accurate than Chen’s, and the velocity of convergence is much faster than that of Chen’s. In order to verify the availability of our novel model, we apply it to the traveling salesman problem (TSP).

4 Application to Traveling Salesman Problem The coordinates of 10-city is as follows: (0.4, 0.4439)、 ( 0.2439, 0.1463)、 ( 0.1707, 0.2293)、 ( 0.2293, 0.716)、 ( 0.5171,0.9414)、 ( 0.8732, 0.6536)、 ( 0.6878, 0.5219)、 ( 0.8488, 0.3609)、 ( 0.6683, 0.2536)、 ( 0.6195, 0.2634). The shortest distance of the 10-city is 2.6776. Here are the results of the test about Chen’s and M-SCNN. The objective function we adopt is that provided in the reference [7].The parameters of the objective function are set as follows: A=2.5, D=1. The parameters of Chen’s are set as follows : α =0.5, k =1, I0=0.5, ε =1/20 ,z (0)=[0.5,0.5]. The parameters of M-SCNN are set as follows : α =0.5, k =1,u0=10,u1=0.8,I0=0.5,z(0)=[0.5,0.5],λ=0.001,η(0)=[0.8,0.8]. We make the test for 200 iterations in different β , as is shown in table 1. (VN= valid number; GN= global number; VP= valid percent; GP=global percent.)

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Table 1. Test result of two chaotic neural network

E 0.04 0.01 0.008

Reference M-SCNN Chen’s M-SCNN Chen’s M-SCNN Chen’s

VN 191 180 191 180 195 183

GN 188 177 188 177 191 182

VP 95.5% 90% 95.5% 90% 97.5% 91.5%

GP 94% 88.5% 94% 88.5% 95.5% 91%

The time evolution figures of the energy function of M-SCNN and Chen’s in solving TSP are respectively given in Fig.3 and Fig.4 when β =0.008.

Fig. 3. Energy time evolution figure of M-SCNN

Fig. 4. Energy time evolution figure of Chen’s

By comparison, it is concluded that M-SCNN is superior to Chen’s model. From the Fig.3, Fig.4, one can see that the velocity of convergence of M-SCNN is much faster than that of Chen’s in solving TSP.

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The superiority of M-SCNN contributes to several factors: First, because of the quality of Morlet wavelet function, the activation function of M-SCNN has a further performance in solving combinatorial optimization problems than Chen’s. Second, it is easier to produce chaotic phenomenon [8] in that the activation function is non-monotonic. Third, η i (t ) is varied with time, which denotes steepness parameter of M-SCNN.

5 Conclusions We have introduced two models of chaotic neural networks. To verify the availability of them, we have made comparison with Chen’s model in optimization problems. By comparison, one can conclude that M-SCNN is superior to Chen’s in searching global minima of continuous nonlinear function. Different from Chen’s model, the activation function of M-SCNN is composed by Morlet wavelet and Sigmoid. So, besides it has the nature of sigmoid activation, the activation function of M-SCNN has a higher nonlinear nature than Sigmoid, which is easier to produce chaotic phenomenon [8] because of its non-monotonic. Due to these factors, M-SCNN is superior to Chen’s in solving TSP.

Acknowledgement This work is supported by the National Science Foundation of China 70471087.

References 1. Hopfield, J., Tank, W.: Neural Computation of Decision in Optimization Problems. Biol. Cybern. 52(2) (1985)141-152 2. Hopfield, J.: Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc Natl. Acad. Sci. 79(4) (1982) 2554-2558 3. Hopfield, J.: Neural Networks and Physical Systems with Emergent Collective Computational Abilities. Proc. Natl. Acad. Sci. 81(4) (1984)3088-3092 4. Chen, L., Aihara, K.: Chaotic Simulated Annealing by a Neural Network Model with Transient Chaos. Neural Networks 8(6) (1995)915-930 5. Wang, L., Tian, F.: Noisy Chaotic Neural Networks for Solving Combinatorial Optimization Problems. International Joint Conference on Neural Networks. Italy: IEEE, (2000)37-40 6. Wang, L.: Intelligent Optimization Algorithms with Application. Tsinghua University & Springer Press, Beijing (2001) 7. Sun, S., Zheng, J.: A Kind of Improved Algorithm and Theory Testify of Solving TSP in Hopfield Neural Network. Acta Electronica Sinca 1(23) (1995)73-78 8. Potapove, A., Kali, M.: Robust Chaos in Neural Networks. Physics Letters A 277(6) (2000) 310-322

A New Optimization Algorithm Based on Ant Colony System with Density Control Strategy Ling Qin1,3, Yixin Chen2, Ling Chen1,3, and Yuan Yao1 1

Department of Computer Science, Nanjing University of Aeronautics and Astronautics, 210093, Nanjing, China [email protected] 2 Department of Computer Science and Engineering, Washington University, in St. Louis, St. Louis, MO 63130-4899, USA [email protected] 3 Department of Computer Science, Yangzhou University, 225009, Yangzhou, China [email protected] Abstract. A new optimization algorithm based on the ant colony system is presented by adopting the density control strategy to guarantee the performance of the algorithm. In each iteration of the algorithm, the solutions are selected to have mutation operations according to the quality and distribution of the solution. Experimental results on the traveling salesman problem show that our algorithm can not only get diversified solutions and higher convergence speed than the Neural Network Model and traditional ant colony algorithm, but also avoid the stagnation and premature problem.

1 Introduction Ant Colony algorithm (AC) was introduced by Dorigo, Maniezzo, and Colorni to solve the Traveling Salesman Problem (TSP) [1]. With the further study in this area, ant colony algorithm has been widely applied to complicated combinatorial optimization problems. However, the classical ant colony algorithm also has its defects, excessive positive feedback could cause premature solutions and local convergence. The major factor causes this two problems mentioned above is its lack of a diversity protection mechanism, which can keep the balance between the convergence speed and the quality of the solutions. In this paper, we present a new type of ant colony algorithm using the idea of density control strategy. In this algorithm, the individuals are selected to have crossover and mutation operations according to their fitness value and diversity. Experimental results on the traveling salesman problem show that our density controlled ant colony algorithm(DACA) can get diversified solutions and higher convergence speed than the Neural Network Model [2](NNM) and traditional ant colony algorithm[3](TA), but also avoid the stagnation and premature problem.

2 The Classical Ant Colony Algorithm Here we use TSP as an example to illustrate AC and its application. We denote the distance between city i and j as dij ( i,j =1,2,…, n). Let τij(t) be the intensity of trail J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 385 – 390, 2006. © Springer-Verlag Berlin Heidelberg 2006

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information on edge (i , j) at time t, and use τij(t) to simulate the pheromone of real ants. Suppose m is the total number of ants, at time t the kth ant selects from its current city i to city j according to the following probability distribution: ⎧ α (t )ηβ (t ) τij ij ⎪ ⎪ k pij (t ) = ⎨ β τα (t )ηir (t ) ⎪ ∑ r∈allowed k ir ⎪⎩ 0

j ∈ allowed k

(1) .

otherwise

where allowedk is a set of the cities can be chosen by the kth ant at city i for the next step, ηij is a heuristic function which is defined as the visibility of the path between cities i and j , for instance it can defined as 1/ dij . The relative influence of the trail information τij(t) and the visibility ηij are determined by the parameters α, β. The intensity of trail information should be changed by the updating formula, where 0<ρ<1, and (1-ρ) τij(t) represents the evaporation of τij(t) between time t and t+1, Δτij is the increment of τij(t). And Δτijk is the trail information laid by the kth ant on the path between city i and city j in step t, it takes different formula depending on the model used. in the most popularly used model called “ant circle system” it is given as Eq.(4), where Q is a constant, Lk is the total length of current tour traveled by the kth ant. τij(t 1) ρτij(t)+Δτij . (2) m (3) k Δτ = Δτ .

＋＝

∑ ij k =1 if the k t h ant pass es edge ( i , j ) in curren t tou r . otherwise ij

⎧Q / Lk Δ τ k (t ) = ⎨ ij ⎩ 0

(4)

3 The Density Controlled Ant Colony Algorithm Here, we present the density controlled ant colony system to improve the optimization capability of the ant colony algorithm. And we use it to solve different kinds of TSP to evaluate its performance. In fact, the experimental results have proved that our algorithm is suitable for solving not only TSP but also QoS routing problem, clustering problem and continuous optimization problems etc.[5] . 3.1 Framework of the Density Controlled Ant Colony Algorithm It’s obvious that the ant colony algorithm is mainly composed by an iterative course including searching and verification for the solutions. Although evolution of the solutions of each individual can be accelerated by positive feedback, sometime it may also be degenerated inevitably. Thus, we make full use of the characteristic of the problems to be solved and adopt the conception and theory of density control strategy [4] to restrain the degeneration in the iteration. The framework of the algorithm is given as follows: Begin 1. initialize 1.1 Generate Np initial solutions randomly, compute their fitness and set the initial trail information on edge(i,j) 2.While (t<=Nc)

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2.1for k=1 to N do //for n cities for i=1 to m do //for m ants 2.2.1The ant i select the next city j with the probability given by Eq.(5); 2.2.2Locally update the trail information on edge(i,j) according to Eq.(4); end for i end for k 2.2Select some solutions in current colony to have mutation operation according to Eq.(10) 2.3have global trail information updating operation, update the pheromone on the edges between each city according to Eq.(10); 2.4t++; if suffice the convergence condition then break the while loop; end while End In the line2.2.1, the next city j will be selected according to formula(5): arg max τij (t ), j ∈ allowedi q ≤ q0 ⎧ . j=⎨ ⎩select the next city j0 according to Eq.(1) otherwise

(5)

Here, q is stochastically generated number between (0,1),argmaxτij(t) is the available adjacent city of city i who has the highest trail information, q0 is a probability of selecting the optimal individual, for example, if q0 0.7, the city who has the highest trail information will be chosen with a probability over 0.7.

＝

3.2 Pheromone Updating Based on the Density Control Strategy The biological density control mechanism uses the individual density in its colony. The higher individual density the ant has, i.e. the more approximate individuals this ant has, the lower diversity the colony has. Suppose ant it =(ai1, ai2 ,…,aiN) is the solution of ith ant at time t, where aih is the hth vertex that the ith ant chooses. We let distance (antit,antjt) be the distance between the ith ant and the jth ant, density(i) denotes the the density factor of ant i at time t. Here dismax is a constant determined by the size of problem N. It was proved by large number of experimental results that N /5 dismax

∈[0, ].

N distance ( ant it , ant tj ) = ∑ ( aih ⊕ a jh ). h =1 m density ( i ) = ∑ ε ( i , j ). j =1 ⎧⎪ 1 distance ( ant it , ant tj ) < dis max here ε (i, j ) = ⎨ ⎪⎩ 0 otherwise

(6) (7) .

(8)

In the traditional ant colony algorithms, there are several methods to update the pheromone: increasing the pheromone on the tours of all the ants [1], or increasing the pheromone on the tours where the best ant passed, or updating the pheromone based on a certain rank [6], or increasing the pheromone on tours passed by the ants whose

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fitness are above the average fitness and decreasing the pheromone on other tours. All these methods only consider the fitness of solutions, but ignore their distribution. In our algorithm, we present an adaptive pheromone updating strategy based on density factor reflecting the quality and distribution of the solutions. Our strategy can adjust the pheromone on the tours dynamically, and avoid the pheromone being centralized on some individuals. ⎧ Q ⋅ density(i) ⎪ if ant i chooses city k on current iteration Δτi = ⎨ lengthi . lk ⎪⎩ 0 otherwise

(9)

Here lengthi is the length of the solution of the ith ant in the current iteration. The algorithm makes full use of the information of the solutions to update the pheromone. The b density factor reflects the distribution of solution. Compared with other ants, the larger the ant’s density is, the more similar its solution is. By the density factor, the tour of the ant with the high fitness and low-density solution will be greatly increased. This adaptive pheromone updating method not only can strengthen the pheromone on the tour of the best solution, but also can guarantee the diversity of solutions. 3.3 Density Controlled Mutation Operation In line 2.2, the algorithm have mutation operation with the mutation probability of : P mutation

= e

− (1 −

density m

(i )

)⋅

f (i, t ) sumf ( t ) .

(10)

Which is adaptively defined according to the density and fitness value of the solutions, here f(i,t) is the fitness of the ith ant at the tth iteration, sumf(t) is the total fitness value of the whole colony at time t i.e. the tth iteration, and m is the total number of the ants in current iteration. It can be seen that the ants that have high individual density and low fitness value will have high mutation probability.

4 Experimental Results The algorithm presented in this paper was applied to symmetric and asymmetric TSP benchmarks provided by TSPLIB. We also test these TSP benchmarks using the classical ant colony algorithm and neural network model to compare the performance with our algorithm. From our experience obtained by a large numbers of experiments [5], we let the parameters ρ=0.4,Q=5, α=1 and β=1. Table 1. The experimental results of symmetric TSP

Instance d198

att532

Algorithm NNM TA DACA NNM TA DACA

Best solution 15780 15780 15780 27686 27686 27686

Average solution 15780.4 15780.4 15780.2 27686.3 27686.3 27686.2

Number 22 20 23 23 22 24

）

Average time(s 112.4 117.3 107.5 521.5 533.0 472.6

allowed time(s) 300

1000

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Table 2. The experimental results of asymmetric TSP

Instance ry48p

ftv170

Algorithm NNM TA DACA NNM TA DACA

Best solution

Average solution

14422 14422 14422 2755 2755 2755

14424.7 14425.3 14422.0 2755.1 2755.0 2755.0

Number 20 19 25 25 25 25

） 76.4

Average time(s

71.1 35.9 114. 9 109.2 103.4

allowed time(s) 250

500

Here “number” is the number of trials obtaining the optimal solution “average solution ”is the average route length of the solution obtained ,“average time ”is the average time required to find the best solution of the trials,“allowed time”is the longest time allowed to be executed in each trial. The numbers included in the name of the problems denote the numbers of cities for its corresponding problem.

Fig. 1. It shows the evolutionary process of the best solution for d198 and ry48p using NNM, TA and DACA respectively. It is obvious that using DACA can not only get the same fitness valued solution, but also has a higher speed of reaching the best solution than NNM and TA.

From the tables we can see that the quality of our algorithm is much higher than that of NNM and TA, and it has much higher processing speed, stronger capability of finding optimal solutions, since DACA reaches the optimal solution in much less iterations. Fig.1 gives the evolutionary process of the best solution for symmetric and asymmetric TSP problems. In DACA, after reaching the best solution, the curve fluctuates within a very narrow scope around the best solution, this confirms the conclusion that DACA has higher stability and is suitable for solving large scaled TSP problems.

5 Conclusion In this paper we propose a new density controlled ant colony algorithm by adopting the density control strategy to guarantee the quality and diversity of the solutions. In this algorithm, the individuals are selected to have mutation operation according to the diversity and quality of the solutions. Experimental results on TSP benchmarks show that our algorithm has strong capability of optimization.

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Nevertheless, the ant colony algorithm is still a newly emerging optimization algorithm, though it is developed rapidly and has demonstrated its superiority of solving complex optimization problems. The ant colony algorithm lacks of a solid mathematic foundation, its selection of the parameters usually depends on experiments and experiences. All these indicate that there are lots of open problems of ant colony optimization worth of deep research in both practical and theoretical area.

Acknowledgements Supported in part by the Chinese National Natural Science Foundation under grant No. 60473012, Chinese National Foundation for Science and Technology Development under contract 2003BA614A-14, and Natural Science Foundation of Jiangsu Province under contract BK2005047.

References 1. Dorigo, M., Maniezzo, V., Colorni, A.: Ant System: Optimization By A Colony of Coorperating Agents. IEEE Transactions On Systems, Man, And Cybernetics-Part B 26(1) (1996) 29-41 2. Tan, K.C., Huajin, T., Ge, S.S.: On Parameter Settings of Hopfield Networks Applied to Traveling Salesman Problems. IEEE Transactions on Circuits and Systems - Regular Papers 52(8) (2005) 994 – 1002 3. Stützle, T., Hoos, H.H.: MAX-MIN Ant System. Future Generation Computer Systems Journal 16(8)(2000)889-914 4. Xiao, P., L., Wei, W.: A New Optimization Method on Immunogenetics. Acta Electronica Sinica 31(1)(2003) 59-62 5. Ling, C., Ling, Q., Hong, J. C., Xiao, H.X.: An Ant Colony Algorithm with Characteristic of Sensation and Consciousness. Journal of System Simulation 15(10) (2003)1418-1425 6. Bullnheimer, B., Hartl, R.F., Strauss, C.: A New Rank Based Version of the Ant System-A Computational Study. Central European Journal for Operations Research and Economics 7(1) (1999) 25-38

A New Neural Network Approach to the Traveling Salesman Problem Paulo Henrique Siqueira1, Sérgio Scheer2, and Maria Teresinha Arns Steiner3 1

Federal University of Paraná, Department of Drawing, Postfach 19081, 81531-990, Curitiba, Brazil [email protected] 2 Federal University of Paraná, Department of Civil Construction, Postfach 19011, 81531-980, Curitiba, Brazil [email protected] 3 Federal University of Paraná, Department of Mathematics, Postfach 19081, 81531-990, Curitiba, Brazil [email protected]

Abstract. This paper presents a technique that uses the Wang Recurrent Neural Network with the "Winner Takes All" principle to solve the Traveling Salesman Problem (TSP). When the Wang Neural Network presents solutions for the Assignment Problem with all constraints satisfied, the "Winner Takes All" principle is applied to the values in the Neural Network’s decision variables, with the additional constraint that the new solution must form a feasible route for the TSP. The results from this new technique are compared to other heuristics, with data from the TSPLIB (TSP Library). The 2-opt local search technique is applied to the final solutions of the proposed technique and shows a considerable improvement of the results.

1 Introduction The Traveling Salesman Problem is a classical problem of combinatory optimization of Operations Research’s area. The purpose is to find a minimum total cost Hamiltonian cycle [1]. There are several practical uses for this problem, such as Vehicle Routing [2] and Drilling Problems [3]. This problem has been used during the last years as a comparison basis for improving several optimization techniques, such as Genetic Algorithms [4], Simulated Annealing [5], Tabu Search [6], Local Search [7], Ant Colony [8] and Neural Networks [9], the latter used in this work. The principal types of Neural Network used to solve the TSP are: Hopfield’s Recurrent Networks [10] and Kohonen’s Self-Organizing Maps [9]. In a Hopfield’s Network, the main idea is to automatically find a solution for the TSP by means of a balanced state of the equation system defined for the TSP. By using Kohonen’s Maps for the TSP, the final route is determined through those neurons that have weights that are closer to the pair of coordinates ascribed to each city in the problem. The Recurrent Neural Network is able to solve others Operations Research’s classical problem, as the Shortest-Path Routing problem [11], [12] and the J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 391 – 398, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Assignment problem (AP) [13], [14]. The Wang Recurrent Neural Network (WRNN) with the "Winner Takes All" (WTA) principle [15] can be applied to solve the TSP, solving this problem as if it was an AP by means of the WRNN, and, furthermore, using the WTA principle on the solutions found with the WRNN, with the constraint that the solutions found must form a feasible route for the TSP. The parameters used for the WRNN are those that show the best solutions for the AP. The solutions found with the technique proposed in this work (WRNN+WTA) are compared with the solutions from the Self-Organizing Maps (SOM) and the Simulated Annealing (SA) for the symmetrical TSP, and with other heuristics for the asymmetrical TSP. The 2-opt Local Search technique [7] is used to improve the solutions found with the technique proposed in this work. The data used for the comparisons are from the TSPLIB database [16]. This essay is divided into 5 sections, including this introduction. In Section 2, the TSP is defined. In Section 3, the WRNN and the WTA principle are presented. In Section 4, the results of the proposed technique are presented and comparisons between this technique and other heuristics known in the literature are made. In Section 5, the work’s conclusions are presented.

2 Formulation of the Problem The TSP can be mathematically formulated as follows [1]: n

Minimize C =

n

∑∑ c

ij

xij

(1)

i =1 j =1

n

∑x

Subject to

ij

= 1,

j = 1, 2, ..., n

(2)

i =1

n

∑x

ij

= 1,

i = 1, 2, ..., n

(3)

j =1

xij ∈ {0, 1}, i, j = 1, 2, ..., n ~ x forms a Hamiltonian cycle

(4) (5)

where cij and xij are, respectively, the costs and the decision variables associated to the assignment of element i to position j. When xij = 1, the element i is assigned to the position j and this means that the route has a stretch that is the sequence from city i to city j. Vector ~ x has the whole sequence of the route that was found, i.e., the solution for the TSP. The AP has the same previous formulation, with exception of (5). The objective function (1) represents the total cost to be minimized. The constraints sets (2) and (3) assure that each city i will be assigned to exactly one city j. Set (4) represents the integrality constraints of variables zero-one xij, and can be replaced by constraints of the following type [1]: xij ≥ 0 , i, j = 1, 2, ..., n.

(6)

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Constraint (5) assures that in the final route each city will be visited once and that no sub-routes will be formed. The matrix form of the problem described in (1)-(5) is the following [14]: Minimize C = cTx

(7)

Subject to Ax = b x ij ≥ 0 , i, j = 1, 2, ..., n ~ x forms a Hamiltonian cycle

(8)

where vectors cT and x contain all the rows from the cost matrix and from the matrix with decision elements xij, this is, cT = (c11, c12, ..., c1n, c21, c22, ..., c2n, ..., cn1, cn2, ..., cnn) and x = (x11, x12, ..., x1n, x21, x22, ..., x2n, ..., xn1, xn2, ..., xnn). Vector b has the number 1 in all of its positions and matrix A has the following form:

⎡I A=⎢ ⎣ B1

... I ⎤ 2 ∈ ℜ 2 n×n ⎥ ... Bn ⎦

I B2

where I is the identity matrix, of order n × n, and each matrix Bi, for i = 1, 2, ..., n, has zeros, with the exception of the ith row, which has the number 1 in all of its positions.

3 WRNN and WTA to Solve the TSP The Recurrent Neural Network proposed by Wang, published in [13], [14] and [17], is characterized by the following differential equation: duij (t ) dt

= −η

n

∑x

ik

(t ) − η

k =1

n

∑ x (t ) + ηθ lj

ij

− λcij e

−

t

τ

,

(9)

l =1

where xij = g(uij(t)) and this Neural Network’s balance state is a solution for the AP. Function g is the sigmoid function, with a β parameter, i.e., g(u) =

1 . 1 + e −β u

(10)

The boundary vector is defined as θ = A b = (2, 2, ..., 2) ∈ ℜ n ×1 . Parameters η, λ and τ are constant and empirically chosen [14], thus affecting the Neural Network’s convergence. Parameter η is to penalize violations of the constraints set of the problem defined by (1)-(4). Parameters λ and τ control the AP’s objective function minimization (1). The matrix form for this Neural Network can be written as follows: 2

T

t

− du (t ) = −η (Wx(t ) − θ ) − λce τ dt

(11)

where x = g(u(t)) and W = ATA. The technique proposed in this work uses the WTA principle, accelerating the WRNN’s convergence and correcting eventual problems that may appear due to multiple optimal solutions or optimal solutions that are very close to each other [15].

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The parameters chosen for the WRNN are those that determine the best results for the AP [15]. Parameter η is considered equal to 1 in all cases tested within this work. Parameter λ is taken as a vector defined by: ⎛1

1 ⎞

1

λ = η ⎜⎜ , ,..., ⎟⎟ , dn ⎠ ⎝ d1 d 2

(12)

where di, for i = 1, 2, …, n, represents the standard deviation for each row in the cost matrix c. Each element of vector λ is used to update the corresponding row in decision matrix x. The best adjust for parameter τ uses the WRNN’s definition’s fourth term (9). When cij = cmax, the term λc exp(−t / τ ) = α must be such that g(α) =

φ ≅ 0. Isolating the value of α, we have: ⎛1 ⎞ − ln⎜⎜ − 1⎟⎟ ⎝φ ⎠ , α=

(13)

β

Using the value of α = λc exp(−t / τ ) and expression (13), the value of parameter τ is obtained.

τi =

−t ⎛ −α ln⎜⎜ ⎝ λcmax

⎞ ⎟ ⎟ ⎠

. (14)

In equation (11), Wx(t) − θ measures the AP’s constraints violation. After a certain number of iterations this term suffers no substantial changes of its value. This evidences the fact that the problem’s constraints are almost satisfied. At this moment, the WTA principle can be applied. When all the elements of x satisfy the condition Wx(t) − θ ≤ δ , where δ ∈ [0, 2], the proposed technique can be used and its algorithm is presented below: Step 1: Determine a maximum number of routes rmax. Find an AP’s x solution using the WRNN. If Wx(t) − θ ≤ δ , then go to Step 2. Otherwise, find another solution x. Step 2: Given the decision matrix, consider matrix x , where x = x, m = 1 and go to Step 3. Step 3: Choose a row k in decision matrix x . Do p = k, ~ x (m) = k and go to Step 4. Step 4: Find the biggest element of row k, x kl. This element’s value is given by the half of the sum of all elements of row k and of column l of matrix x, this is,

xkl =

1 ⎛⎜ 2 ⎜⎝

n

n

∑x + ∑x il

i =1

j =1

kj

⎞ ⎟ . ⎟ ⎠

(15)

The other elements of row k and column l become null. So that sub-routes are not formed, the other elements of column k must also be null. Do ~ x (m + 1) = l; to continue the TSP’s route, make k = l and go to Step 5.

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Step 5: If m < n, then make m = m + 1 and go to Step 4. Otherwise, do 1⎛ xkp = ⎜ 2 ⎜⎝

n

∑ i =1

xip +

n

∑x j =1

kj

⎞ ⎟ , ⎟ ⎠

(16)

~ x (n + 1) = p, determine the route’s cost, C, and go to Step 6. Step 6: If C < Cmin, then do Cmin = C and x = ~ x . Make r = r + 1. If r < rmax, then run the WRNN again and go to Step 2, otherwise Stop. The technique proposed in this work can be applied to symmetrical or asymmetrical TSPs. The next Section shows the results of applying this technique to some of the TSPLIB’s problems for symmetrical, as well as asymmetrical TSPs.

4 Results of Applying the Proposed Technique to Some of the TSPLIB’s Problems The results found with the technique proposed in this work for the TSPLIB’s TSP’s symmetrical cases are compared with SOM and SA results, and the asymmetrical cases are compared to remotion and insertion arc heuristic. For symmetrical TSPs, the following methods were used to compare with the technique presented in this work: the method that involves statistical methods between a SOM’s neurons’ weights [18] and has a global version (KniesG: Kohonen Network Incorporating Explicit Statistics Global), where all cities are used in the neuron dispersion process, and a local version (KniesL), where only some represented cities are used in the neuron dispersion step; the SA technique [5], using the 2-opt improvement technique; Budinich’s SOM, which consists of a traditional SOM applied to the TSP, presented in [5]; the expanded SOM (ESOM) [9], which, in each iteration, places the neurons close to their corresponding input data (cities) and, at the same time, places them at the convex contour determined by the cities; the efficient and integrated SOM (eISOM) [19], where the ESOM procedures are used and the winning neuron is placed at the mean point among its closest neighboring neurons; the efficient SOM technique (SETSP) [20], which defines the updating forms for parameters that use the TSP’s number of cities; and Kohonen’s cooperative adaptive network (CAN) [21] uses the idea of cooperation between the neurons’ close neighbors and uses a number of neurons that is larger than the number of cities in the problem. In Table 1 are the average errors/mistakes of the techniques mentioned above. Are also shown: the "pure" technique proposed in this work, the proposed technique with the 2-opt improvement algorithm, as well as the best (max) and worse (min) results of each problem considered. The solutions presented in bold characters show the best results for each problem, disregarding the results with the 2-opt technique. The results of the technique proposed in this work, together with the 2-opt improvement, are better than the results of the other Neural Networks, with the sole exception of problem lin105. The average errors range from 0 to 3.31%, as shows the last column in Table 1. Without the improvement technique, the results of problem eil76 are better than the results of the other Neural Networks.

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P.H. Siqueira, S. Scheer, and M.T.A. Steiner Table 1. Results of the experiments for the symmetrical TSP

TSP’s name

n

eil51 st70 eil76 gr96 rd100 eil101 lin105 pr107 pr124 bier127 pr136 pr152 rat195 kroa200 lin318 pcb442 att532

51 70 76 96 100 101 105 107 124 127 136 152 195 200 318 442 532

optimal solution

average error (%) for 7 algorithms KniesG KniesL SA

WRNN & WTA

BSom ESom EiSom Setsp CAN

430 2.86 2.86 2.33 3.10 678.6 2.33 1.51 2.14 1.70 545.4 5.48 4.98 5.54 5.32 514 NC NC 4.12 2.09 7,910 2.62 2.09 3.26 3.16 629 5.63 4.66 5.74 5.24 14,383 1.29 1.98 1.87 1.71 44,303 0.42 0.73 1.54 1.32 59,030 0.49 0.08 1.26 1.62 118,28 3.08 2.76 3.52 3.61 96,772 5.15 4.53 4.90 5.20 73,682 1.29 0.97 2.64 2.04 2,323 11.92 12.24 13.2 11.48 29,368 6.57 5.72 5.61 6.13 42,029 NC NC 7.56 8.19 50,784 10.45 11.07 9.15 8.43 27,686 6.8 6.74 5.38 5.67

2.10 2.09 3.89 1.03 1.96 3.43 0.25 1.48 0.67 1.70 4.31 0.89 7.13 2.91 4.11 7.43 4.95

max

min 2opt

2.56 2.22 0.94 1.16 1.16 0 NC 1.60 1.33 4.04 2.71 0 NC 4.23 2.04 2.49 1.03 0 NC NC NC 6.61 4.28 0 NC 2.60 1.23 7.17 6.83 0.08 3.59 NC 1.11 7.95 3.02 0.48 NC 1.30 0 5.94 4.33 0.20 NC 0.41 0.17 3.14 3.14 0 NC NC 2.36 2.63 0.33 0 NC 1.85 0.69 5.08 4.22 0.37 NC 4.40 3.94 6.86 5.99 1.21 NC 1.17 0.74 3.27 3.23 0 NC 11.19 5.27 8.82 5.55 3.31 1.64 3.12 0.92 12.25 8.95 0.62 2.05 NC 2.65 8.65 8.35 1.90 6.11 10.16 5.89 13.18 9.16 2.87 3.35 NC 3.32 15.43 14.58 1.28

Table 2. Results of the experiments for the asymmetrical TSP TSP’s name

n

Average error (%) For 5 algorithms

optimal solution GR

br17 ftv33 ftv35 ftv38 pr43 ftv44 ftv47 ry48p ft53 ftv55 ftv64 ft70 ftv70 kro124p ftv170 rbg323 rbg358 rbg403 rbg443

RI

KSP

GKS

RPC

COP

WRNN & WTA max

min

2opt

17 39 102.56 0 0 0 0 0 0 0 0 33 1,286 31.34 11.82 13.14 8.09 21.62 9.49 7.00 0 0 35 1,473 24.37 9.37 1.56 1.09 21.18 1.56 5.70 3.12 3.12 38 1,530 14.84 10.20 1.50 1.05 25.69 3.59 3.79 3.73 3.01 43 5,620 3.59 0.30 0.11 0.32 0.66 0.68 0.46 0.29 0.05 44 1,613 18.78 14.07 7.69 5.33 22.26 10.66 2.60 2.60 2.60 47 1,776 11.88 12.16 3.04 1.69 28.72 8.73 8.05 3.83 3.83 48 14,422 32.55 11.66 7.23 4.52 29.50 7.97 6.39 5.59 1.24 53 6,905 80.84 24.82 12.99 12.31 18.64 15.68 3.23 2.65 2.65 55 1,608 25.93 15.30 3.05 3.05 33.27 4.79 12.19 11.19 6.03 64 1,839 25.77 18.49 3.81 2.61 29.09 1.96 2.50 2.50 2.50 70 38,673 14.84 9.32 1.88 2.84 5.89 1.90 2.43 1.74 1.74 70 1,950 31.85 16.15 3.33 2.87 22.77 1.85 8.87 8.77 8.56 100 36,230 21.01 12.17 16.95 8.69 23.06 8.79 10.52 7.66 7.66 170 2,755 32.05 28.97 2.40 1.38 25.66 3.59 14.66 12.16 12.16 323 1,326 8.52 29.34 0 0.53 0 0 16.44 16.14 16.14 358 1,163 7.74 42.48 0 2.32 0.26 22.01 12.73 8.17 0 403 2,465 0.85 9.17 0.69 0.20 4.71 4.71 4.71 0 0 443 2,720 0.92 10.48 0 8.05 8.05 2.17 0 0 0

For the asymmetrical TSP, the techniques used to compare with the technique proposed in this work were [22]: the Karp-Steele path methods (KSP) and general Karp-Steele (GKS), which begin with one cycle and by removing arcs and placing new arcs, transform the initial cycle into a Hamiltonian one. The difference between these two techniques is that the GKS uses all of the cycle’s vertices for the changes in

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the cycle’s arcs. Were also used: the path recursive contraction (PRC) that consists in forming an initial cycle and transforming it into a Hamiltonian cycle by removing arcs from every sub-cycle; the contraction or path heuristic (COP), which is a combination of the GKS and RPC techniques; the “greedy” heuristic (GR) that chooses the smallest arc in the graph, contracts this arc creating a new graph, and keeps this procedure up to the last arc, thus creating a route; and the random insertion heuristic (RI) that initially chooses 2 vertices, inserts one vertex that had not been chosen, thus creating a cycle, and repeats this procedure until it creates a route including all vertices. Table 2 shows the average errors of the techniques described, as well as those of the "pure" technique presented in this work and of the proposed technique with the 2-opt technique. The results of the "pure" technique proposed in this work are better or equivalent to those of the other heuristics mentioned above, for problems br17, ftv33, ftv44, ft53, ft70 and kro124p, as shown in Table 2. By using the 2-opt technique on the proposed technique, the best results were found for problems br17, ftv33, pr43, ry48p, ftv44, ft53, ft70 and kro124p, with average errors ranging from 0 to 16.14%.

5 Conclusions This work presented the WRNN with the WTA principle to solve the TSP. By means of the WRNN, a solution for the AP is found and the WTA principle is applied to this solution, transforming it into a feasible route for the TSP. This technique’s solutions were considerably improved when the 2-opt technique was applied on the solutions presented by the technique proposed in this work. The data used for testing are from the TSPLIB and the comparisons that were made with other heuristics show that the technique proposed in this work obtains better results in several of the tested problems, with average errors below 16.14%. A great advantage of implementing the technique presented in this work is the possibility of using the same technique to solve symmetrical as well as asymmetrical TSPs.

References 1. Ahuja, R.K., Mangnanti, T.L., Orlin, J.B.: Network Flows. Prentice Hall, New Jersey (1993) 2. Laporte, G.: The Vehicle Roting Problem: An Overview of Exact and Approximate Algorithms. European Journal of Operational Research 59(2) (1992) 345-358 3. Onwubolu, G.C., Clerc, M.: Optimal Path for Automated Drilling Operations by a New Heuristic Approach Using Particle Swarm Optimization. International Journal of Production Research 42(3) (2004) 473-491 4. Affenzeller, M., Wanger, S.: A Self-Adaptive Model for Selective Pressure Handling within the Theory of Genetic Algorithms. In: Moreno-Díaz, R., Pichler, F. (eds.): Computer Aided Systems Theory – EUROCAST 2003. Lecture Notes in Computer Science, Vol. 2809. Springer-Verlag, Berlin Heidelberg New York (2003) 384-393 5. Budinich, M.: A Self-Organizing Neural Network for the Traveling Salesman Problem That Is Competitive with Simulated Annealing. Neural Computing 8 (1996) 416-424 6. Liu, G., He, Y., Fang, Y., Oiu, Y.: A Novel Adaptive Search Strategy of Intensification and Diversification in Tabu Search. In: Proceedings of the IEEE International Conference on Neural Networks and Signal Processing – ICNNSP’03, Vol. 1. IEEE (2003) 428-431

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7. Bianchi, L., Knowles, J., Bowler, J.: Local Search for the Probabilistic Traveling Salesman Problem: Correction to the 2-P-Opt and 1-Shift Algorithms. European Journal of Operational Research 162(1) (2005) 206-219 8. Chu, S.C., Roddick, J.F., Pan, J.S.: Ant Colony System with Communication Strategies. Information Sciences 167(1-4) (2004) 63-76 9. Leung, K.S., Jin, H.D., Xu, Z.B.: An Expanding Self-Organizing Neural Network for the Traveling Salesman Problem. Neurocomputing 62 (2004) 267-292 10. Wang, R.L., Tang, Z., Cao, Q.P.: A Learning Method iIn Hopfield Neural Network for Combinatorial Optimization Problem. Neurocomputing 48(4) (2002) 1021-1024 11. Wang, J.: Primal and Dual Neural Networks for Shortest-path Routing. IEEE Transactions on Systems: Man and Cybernetics - Part A: Systems and Humans 28(6) (1998) 864-869 12. Xia, Y., Wang, J.: A Discrete-time Recurrent Neural Network for Shortest-path Routing. IEEE Transactions on Automatic Control 45(11) (2000) 2129-2135 13. Wang, J.: Analog Neural Network for Solving the Assignment Problem. Electronic Letters 28(11) (1992) 1047-1050 14. Hung, D.L., Wang, J.: Digital Hardware Realization of a Recurrent Neural Network for Solving the Assignment Problem. Neurocomputing 51 (2003) 447-461 15. Siqueira, P.H., Scheer, S., Steiner, M.T.A.: Application of the "Winner Takes All" Principle in Wang's Recurrent Neural Network for the Assignment Problem. In: Wang, J., Liao, X., Yi, Z. (eds.): Advances in Neural Networks – ISNN 2005. Lecture Notes in Computer Science, Vol. 3496(1). Springer-Verlag, Berlin Heidelberg New York (2005) 731-738 16. Reinelt, G.: TSPLIB – A Traveling Salesman Problem Library. ORSA Journal on Computing 3(4) (1991) 376-384 17. Wang, J.: Primal and Dual Assignment Networks. IEEE Transactions on Neural Networks 8(3) (1997) 784-790 18. Aras, N., Oommen, B.J., Altinel, I.K.: The Kohonen Network Incorporating Explicit Statistics and Its Application to the Traveling Salesman Problem. Neural Networks 12(9) (1999) 1273-1284 19. Jin, H.D., Leung, K.S., Wong, M.L., Xu, Z.B.: An Efficient Self-Organizing Map Designed by Genetic Algorithms for the Traveling Salesman Problem. IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics 33(6) (2003) 877-887 20. Vieira, F.C., Doria Neto, A.D., Costa, J.A.: An Efficient Approach to the Travelling Salesman Problem Using Self-Organizing Maps. International Journal of Neural Systems 13(2) (2003) 59-66 21. Cochrane, E.M., Beasley, J.E.: The Co-Adaptive Neural Network Approach to the Euclidean Travelling Salesman Problem. Neural Networks 16(10) (2003) 1499-1525 22. Glover, F., Gutin, G., Yeo, A., Zverovich, A.: Construction Heuristics for the Asymmetric TSP. European Journal of Operational Research 129(3) (2001) 555-568

Dynamical System for Computing Largest Generalized Eigenvalue Lijun Liu and Wei Wu Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China Abstract. A dynamical system is proposed for generalized eigendecomposition problem. The stable points of the dynamical system are proved to be the eigenvectors corresponding to the largest generalized eigenvalue.

1

Introduction

Principal component analysis (PCA) [2, 3, 5, 9, 11] and linear discriminant analysis (LDA) [1] are widely used statistical techniques in various applications such as data compression, feature extraction and signal estimation. The PCA provides a set of adaptive algorithms for eigen-decomposition of a matrix A, which is the asymptotic expectation of a sequence of random matrices. LDA, on the other hand, provides adaptive algorithms for generalized eigen-decomposition of a matrix pair (A, B) which are the asymptotic expectations of two sequences of random matrices. According to the stochastic approximation theory [6], under some mild conditions, the asymptotic behavior of a PCA or LDA learning algorithm can be determined by its corresponding averaging differential equation [1–3, 5, 7, 9–13]. Therefore, to design and analyze PCA or LDA learning algorithms, we can design and analyze certain differential equations for solving the generalized eigenvalue problem. In this paper we are to find eigenpair (λ, x) ∈ R × Rn such that Ax = λBx

(1)

for given matrices A and B. When B is invertible, (1) is equivalent to the ordinary eigenvalue problem B −1 Ax = λx (2) Let us have a look at some existing results on the computation of eigenvalues by ordinary differential equation on which our work is based. Zhang and Yan [13] successfully use a neural network model for computing the ordinary eigenvalue problem Ax = λx for a symmetric matrix A as follows dx = xT xAx − xT Axx, dt

t≥0

(3)

We observe that generally B −1 A in (2) is no longer guaranteed to be symmetric. Thus model (3) cannot be extended directly to the generalized eigenvalue problem (1). J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 399–404, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Rao and Principe [10] propose an RLS type algorithm for generalized eigendecomposition, which can be summarized as the differential equation dx xT Bx −1 = T B Ax − x, t ≥ 0 (4) dt x Ax Their research is strongly related and successfully applied to signal processing theory. But their arguments and results can not be used directly to the general generalized eigenvalue problem (1). Zhang and Leung [12] propose a general framework for PCA algorithms based on generalized eigen-decomposition using the differential equation dx(t) = Ax − f (x)Bx, t ≥ 0 (5) dt This framework succeeds in covering many important algorithms. But in order to get a general result for the framework, they require that both A and B have the same eigenvector associated with the largest eigenvalue, which is hard to satisfy generally. Based on and inspired by the above mentioned works, in this paper, we propose the following dynamical system for the generalized eigen-decomposition of (1) dx = xT BxB −1 Ax − xT Axx, t ≥ 0 (6) dt We assume A is symmetric and B is symmetric positive. We see that, compared with (4), the term xT Ax has been moved out of the denominator in order to prevent the possible instability in case that xT Ax is small. The rest of this paper is organized as follows. In Section 2, some properties of the solutions to the dynamical system are studied as a preparation for the convergence results. In Section 3, the convergence of the system to the largest generalized eigenvalue is established. Numerical result is given in section 4.

2

Preliminary Results

Assumption 1. A and B are real, symmetric and positive definite. Remark 1. If B is positive definite but A is not, then we can simply replace A by A + cB, such that A + cB is positive definite with a suitable positive scalar constant c. It is easy to show that both A and A + cB have the same generalized eigenvector ξ with respect to B, with generalized eigenvalue λ and λ + c, respectively. T

Theorem 1. If x(t) is a solution to (6) for t ≥ 0, then x(t) Bx(t) keeps inT variant for t ≥ 0, i.e., x(t) Bx(t) ≡ x(0)T Bx(0). Proof. It is straightforward to check that d T dx (x Bx) = 2xT B = 2xT B(xT BxB −1 Ax − xT Axx) = 0 dt dt So xT Bx keeps constant for t ≥ 0.

(7)

Dynamical System for Computing Largest Generalized Eigenvalue

401

Remark 2. Luo [7] considered the invariant norm property of the principal component algorithm. Here we extend this property to generalized eigenvalue problem. Theorem 2. Let ω be any given vector in Rn . Then there exists a unique solution x(t) with x(0) = ω for the system (6). Proof. Denote the right hand side of (6) by f (x). Then f (x) is continuously differentiable in x. For a given point ω ∈ Rn , there exist a neighborhood N (ω) of ω and a positive constant K1 such that the local Lipschitz condition is valid, i.e., (8) f (u) − f (v) ≤ K1 u − v for all u, v ∈ N (ω), where  ·  is the usual Euclidean norm. Furthermore, from Theorem 1 and the positive definiteness of B, we have x(t)2 ≤ δ −1 x(0) Bx(0) T

(9)

where δ is the smallest positive eigenvalue of B. By the well-known PicardLinerlof theorem [8], there exists a unique solution x(t) with x(0) = ω for the system (6), which can be uniquely extended to the infinite time interval [0, ∞). Theorem 3. If ξ = 0 is an equilibrium point of the system (6), then ξ is an eigenvector associated with the eigenvalue λ = ξ T Aξ/ξ T Bξ of (1). Conversely, if ξ is an eigenvector corresponding to an eigenvalue of the generalized eigenvalue problem (1), then ξ is also an equilibrium point of (6). Proof. The proof is straightforward. For convenience of reference, we present the following well-known generalized eigen-decomposition theorem (cf. [4]). Theorem 4. Let Assumption 1 be valid, then generalized eigenvalue problem (1) has n linearly independent eigenvectors {Si }ni=1 associated with its real generalized eigenvalues {λi }ni=1 and λi > 0(i = 1, 2, · · · , n). Furthermore, denote S = (S1 , S2 , · · · , Sn ) ∈ Rn×n , then

3

S T AS = Λ = diag{λ1 , λ2 , · · · , λn }

(10)

S T BS = I = diag{1, 1, · · · , 1}

(11)

Convergence Analysis

Denote x(t) =

n 

θi (t)Si = SΘ, then according to Theorem 1 we have

i=1

Θ(t)2 = x(0)T Bx(0) = Θ(0)T S T BSΘ(0) = Θ(0)2

(12)

So there exists a uniform upper bound M > 0 such that |θi (t)| ≤ M,

(i = 1, 2, · · · n)

Now we give the main convergence theorem for the dynamics of (6).

(13)

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L. Liu and W. Wu

Theorem 5. If θ1 (0) = 0 and the largest generalized eigenvalue λ1 is unrepeated, then  lim x(t) = ± x(0)T Bx(0)S1 (14) t→∞

Proof. Under the transformation x = SΘ, (6) can be rewritten as S

dΘ = Θ(0)2 B −1 ASΘ − (ΘT S T ASΘ)SΘ dt

(15)

Clearly S −1 = S T B by Theorem 4. Then we have dΘ = Θ(0)2 B −1 ASΘ − (ΘT ΛΘ)Θ dt = Θ(0)2 S T ASΘ − (ΘT ΛΘ)Θ = Θ(0)2 ΛΘ − (ΘT ΛΘ)Θ Hence, (6) can be expressed in component form as ⎛ ⎞ n  dθi (t) ⎝ = Θ(0)2 λi − λj θj2 (t)⎠ θi (t), dt j=1 from which we have

⎛

θi (t) = θi (0) exp ⎝Θ(0)2 λi t −

t n

(16)

(i = 1, 2, · · · , n)

(17)

⎞ λj θj2 (s)ds⎠

(18)

0 j=1

Clearly if θi (0) = 0, then θi (t) = 0 for all t ≥ 0. Since θ1 (0) = 0, we can define θi (t) θ1 (t) for i = 1, 2, · · · , n. Then we have d dt



θi (t) θ1 (t)



⎛ ⎞ n  1 ⎝ = Θ(0)2 λi − λj θj2 (t)⎠ θi (t) θ1 (t) j=1 ⎛ ⎞ n  θi (t) ⎝ − 2 Θ(0)2 λ1 − λj θj2 (t)⎠ θ1 (t) θ1 (t) j=1

θi (t) Θ(0)2 (λi − λ1 ) θ1 (t)

(19)



θi (t) θi (0) = exp Θ(0)2 (λi − λ1 ) t θ1 (t) θ1 (0)

(20)

= This means that

So by λi − λ1 < 0 (i = 2, 3, · · · , n), we have lim

t→∞

θi (t) = 0. θ1 (t)

(21)

Dynamical System for Computing Largest Generalized Eigenvalue

403

From (12), (13) and (21), we have

  n   lim θ1 (t) = ± lim  θi2 (0) = ± x(0)T Bx(0)

t→∞

t→∞

and lim x(t) = lim

t→∞

n 

t→∞

(22)

i=1

θi (t)Si = ±

 x(0)T Bx(0)S1

(23)

i=1

Remark 3. If λ1 is repeated, say λ1 = · · · = λl > λl+1 ≥ · · · ≥ λn , then it is easy l  to show that lim x(t) = lim θi (t)S1 , which is still an eigenvector associated t→∞

t→∞ i=1

with λ1 . In fact, this limit can be expressed explicitly in terms of x(0) through the procedure adopted in [13].

4

Numerical Result

In this section, we randomly select symmetric matrix A and symmetric positive definite matrix B as follows ⎡ ⎤ 0.9660 0.2860 0.1939 −0.0699 −0.1404 ⎢ 0.2860 −0.2482 −0.0702 −0.3862 0.3025 ⎥ ⎢ ⎥ ⎥ A=⎢ ⎢ 0.1939 −0.0702 0.6894 0.2726 0.5524 ⎥ ⎣ −0.0699 −0.3862 0.2726 0.1384 −0.0330 ⎦ −0.1404 0.3025 0.5524 −0.0330 0.3111 ⎡

⎤ 1.7838 0.2825 0.2648 −0.4169 −0.1014 ⎢ 0.2825 2.6752 0.0882 0.1259 −0.2907 ⎥ ⎢ ⎥ ⎥ B=⎢ ⎢ 0.2648 0.0882 2.0226 0.7591 0.1542 ⎥ ⎣ −0.4169 0.1259 0.7591 2.6133 0.0689 ⎦ −0.1014 −0.2907 0.1542 0.0689 1.2801

and

1 0.8

x1(t) x4(t)

0.4

Actual Value λ

0.7

=0.6500

max

0.6

max

0.2

0.5

0

Estimated λ

value of each component

0.8

0.6

x2(t)

−0.2 −0.4 −0.6

x3(t)

−0.8

x5(t)

0.4 0.3 0.2 0.1 0

−1

−0.1

0

5

10

15

20

25

30

0≤ t ≤ 50

35

40

45

Fig. 1. Asymptotic stability of x(t)

50

−0.2

0

1

2

3

4

5

0≤ t ≤ 10

6

7

8

9

Fig. 2. Convergence of xT Ax/xT Bx

10

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The true largest generalized eigenvalue λmax = 0.6500. Using the network model, the estimated eigenvector corresponding to largest eigenvalue is computed as ξ = (0.4276, −0.2191, −0.6859, 0.2629, −0.9024)T with initial value x(0) taken as (0.1249, 0.2332, −0.7733, 0.7965, 0.5091)T . The result is shown in Figure 1 and Figure 2. The numerical result further confirms that the network model can be used to compute the largest generalized eigenvalue of real symmetrical matrix A associated with B.

Acknowledgement This work is partly supported by the National Natural Science Foundation of China (10471017). Corresponding author. [email protected]

References 1. Chatterjee, C., Roychowdhury, V. P., Zoltowski, M. D., Ramos, J.: Self-organizing and Adaptive Algorithms for Generalized Eigendecomposition. IEEE Trans. on Neural Networks 8 (1997) 1518–1530 2. Chen, T.P.: Modified Ojas Algorithms for Principal Subspace and Minor Subspace Extraction. Neural Processing Letters 5 (1997) 105–110 3. Diamantaras, K.J., Hornik, K., Strintzis, M.G.: Optimal Linear Compression under Unreliable Representation and Robust PCA Neural Models. IEEE Trans. on Neural Networks (1999) 1186–1195 4. Gloub, G.H., VanLoan, C.F.: Matrix Computations. Johns Hopkins Univ. Press (1983) 5. Hornik, K., Kuan, C.M.: Convergence Analysis of Local Feature Extraction Algorithms. Neural Networks (1992) 229–240 6. Ljung, L.: Analysis of Recursive Stochastic Algorithms. IEEE Trans. on Auto. Cont. AC-22(1977) 551–575 7. Luo, F., Unbehauen, R., Li, Y.D.: A Principal Component Analysis Algorithm with Invariant Norm. Neurocomputing (1995) 213–221 8. Miller, R.K., Michel, A.N.: Ordinary Differential Equations. New York Academic (1982) 9. Oja, E.: Principal Components, Minor Components, and Linear Neural Networks. Neural Networks 5 (1992) 927–935 10. Rao, Y.N., Principe, J.C.: An RLS Type Algorithm for Generalized Eigendecomposition. IEEE Workshop on Neural Networks for Signal Processing XI 7 (2001) 263–272 11. Xu, L., Oja, E., Suen, C.: Modified Hebbian Learning for Curve and Surface Fitting. Neural Networks 5 (1992) 441–457 12. Zhang, Q.F., Leung, Y.W.: A Class of Learning Algorithms for Principal Component Analysis and Minor Component Analysis. IEEE Trans. on Neural Networks 11 (2000) 200–204 13. Zhang, Y., Yan, F., Tang, H.J.: Neural Networks Based Approach for Computing Eigenvectors and Eigenvalues of Symmetric Matrix. Comp. and Math. with Appl. 47 (2004) 1155–1164

A Concise Functional Neural Network for Computing the Extremum Eigenpairs of Real Symmetric Matrices Yiguang Liu and Zhisheng You Institute of Image & Graphics, School of Computer Science and Engineering, Sichuan University, Chengdu, 610064, P.R. China [email protected]

Abstract. Quick extraction of the extremum eigenpairs of a real symmetric matrix is very important in engineering. Using neural networks to complete this operation is in a parallel manner and can achieve high performance. So, this paper proposes a very concise functional neural network (FNN) to compute the largest (or smallest) eigenvalue and one corresponding eigenvector. After transforming the FNN into a differential equation, and obtaining the analytic solution, the convergence properties are completely analyzed. By this FNN, the method that can compute the extremum eigenpairs whether the matrix is nondefinite, positive definite or negative definite is designed. Finally, three examples show the validity. Comparing with the other ones used in the same field, the proposed FNN is very simple and concise, so it is very easy to realize.

1 Introduction Using neural networks to compute eigenvalues and eigenvectors of a matrix has parallel and quick features, quick computation of eigenvalues and eigenvectors has many applications such as primary component analysis (PCA) for real time image compression, or adaptive signal processing etc. So, many research works about this field have been reported [1-14]. Recently, a recurrent neural network (RNN) is proposed to compute eigenvalues and eigenvectors of a real symmetric matrix [6]. It is as follow

[

)]

(

x& (t ) = − x(t ) + x T (t )x(t )A + 1 − x T (t )Ax(t ) I x(t )

(1)

where t ≥ 0 , x = ( x1 ,L , x n )T ∈ R n represents the states of neurons, I denotes an identity matrix, matrix A ∈ R n×n requiring eigenpair calculation is symmetric. Just now, Reference [11] proposed a neural network model dx(t ) dt = Ax (t ) − x T (t )x(t )x(t ) to execute similar computation. This paper gives an improved form of this model described as

[

]

x& (t ) = Ax(t ) − sgn x T (t ) x(t )x(t )

(2)

where definitions of A , t and x(t ) are same to those of eq. (1), sgn [x i (t )] returns 1 if xi (t ) > 0 , 0 if xi (t ) = 0 and -1 if xi (t ) < 0 . When A − I sgn[ x T (t )]x(t ) is looked as synaptic connection weights which obviously are functions of x(t ) , the neuron activation functions are assumed as pure linear functions, eq. (2) describes a functional J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 405 – 413, 2006. © Springer-Verlag Berlin Heidelberg 2006

406

neural

Y. Liu and Z. You

network

(FNN

(2)

).

Evidently,

FNN

(2)

is

simpler

than

dx(t ) dt = Ax (t ) − x T (t )x(t )x(t ) and RNN (1), this feature is important for electronic

design. Comparing with RNN (1), the connection count is much less, the electronic circuitry is more concise, so the realization is easier. Moreover, the performance is near to that of RNN (1), this can be seen in the “convergence analysis” section of this paper.

2 Some Preliminaries Since matrix A is real symmetrical, all eigenvalues are denoted as λ1 ≥ λ2 ≥ L ≥ λ n , corresponding eigenvectors and eigensubspaces are denoted as μ1 , L , μ n and V1 , L , V n . Assume matrix A has m ≤ n different eigenvalues, and denote them as σ 1 > σ 2 > ... > σ m , let K i (i = 1,..., m ) denote the sum of algebraic multiplicity of

σ 1 ,L, σ i , and let

K0 = 0 ,

λ0 = 0 , obviously K m = n . Let

U i = VK

i −1 +1

⊕ L ⊕ V Ki (i = 1,..., m )

(3)

where ‘ ⊕ ’ denotes the direct sum operator. Obviously U i is the eigensubspace corresponding to σ i . Lemma 1. λ1 + α ≥ λ 2 + α ≥ L ≥ λ n + α are eigenvalues of A + αI μ1 , L , μ n are the eigenvectors corresponding to λ1 + α , L , λ n + α .

(α ∈ R ) and

Proof: From Aμ i = λi μ i , i = 1,L , n .

It follows that Aμ i + αμ i = λi μ i + αμ i , i = 1, L , n .

i.e.

( A + αI )μ i = (λi + α )μ i , i = 1, L, n . Therefore, this lemma can be drawn. Let ξ denote the equilibrium vector of FNN (2). ξ ought to exist, because ξ = 0 is an invariable equilibrium vector. When ξ exists, there exists ξ = lim x(t ) . t →∞

and Aξ =

∑

n

i =1

ξi ξ ,

when ξ ≠ 0 , ξ is an eigenvector and the corresponding eigenvalue λ is

(4)

A Concise FNN for Computing the Extremum Eigenpairs of Real Symmetric Matrices

407

∑

(5)

λ=

n

i =1

ξi ,

3 Analytic Solution and Convergence Analysis Like the proving procedure of theorem 1 in ref. [11] or [12], the following theorems can be similarly proved. Theorem 1. Denote S i = μ i μ i , xi (t ) denotes the projection value of x(t ) onto S i ,

then xi (t ) can be presented as

⎛ xi (t ) = xi (0 ) exp(λi t ) ⎜1 + ⎝

∑

n j =1

x j (0 )

∫ exp(λ τ )dτ ⎟⎠ , ⎞

t

0

(6)

j

Using analytic solution (6) and the techniques used in ref. [11], we can prove the following theorems. Theorem 2. If x(0) ≠ 0 and x (0 ) ∈ Vi , then ξ = 0 or ξ ∈ Vi . If ξ ∈ Vi , then λi =

∑

n k =1

ξk .

Theorem 3. If x (0) ≠ 0 , x(0 ) ∉ Vi (i = 1, L , n ) , and ξ ≠ 0 , then ξ ∈ U 1 , λ1 =

∑

n k =1

ξk .

Theorem 4. When A is replaced with −A, If x (0) ≠ 0 , x(0 ) ∉ Vi (i = 1, L , n ) and ξ ≠ 0 , then ξ ∈ U m , λ n = −

∑

n k =1

ξk .

Theorem 5. Whether the sign of A is reversed, FNN (2) can not directly compute λn of positive definite matrix and λ1 of negative definite matrix. Theorem 6. Let ⎡λ1 ⎤ denote the minimal integer not less than λ1 . For positive definite matrix A , if x(0) ≠ 0 and x(0 ) ∉ Vi (i = 1, L , n ) , replacing A with − ( A − ⎡λ1 ⎤I ) makes ξ ≠ 0 and λn = −

∑

n

i =1

ξ i + ⎡λ1 ⎤ .

(7)

Theorem 7. Let ⎣λn ⎦ denote the maximal integer not more than λn . With negative definite matrix A , if x(0 ) ≠ 0 and x(0 ) ∉ Vi (i = 1, L , n ) , replacing A with ( A − ⎣λn ⎦I ) will make ξ ≠ 0 and λ1 =

∑

n

i =1

ξ i + ⎣λ n ⎦ .

(8)

Theorem 8. For FNN (2), there must have ξ = 0 or that ξ1 + ξ 2 + ... + ξ n equal to one positive eigenvalue.

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4 Steps to Compute λ1 and λn In this section, steps are designed to compute the largest eigenvalue and the smallest eigenvalue of a real symmetric matrix B ∈ R n×n with FNN (2), the steps are: 1. When A is replaced with B and ξ ≠ 0 , then λ1 = ξ1 + ξ 2 + ... + ξ n .

2. When A is replaced with − B and ξ ≠ 0 , we get λn = −(ξ1 + ξ 2 + ... + ξ n ) . If λ1 has been obtained, go to step 5. Otherwise, go to step 3. If ξ = 0 , go to step 4. 3. Through replacing A with (B − ⎣λ n ⎦I ) , we get λ1 = ξ1 + ξ 2 + ... + ξ n + ⎣λn ⎦ and go to step 5. 4. Through replacing A with − (B − ⎡λ1 ⎤I ) , we get λ n = −(ξ1 + ξ 2 + ... + ξ n ) + ⎡λ1 ⎤ and go to step 5. 5. End.

5 Simulation Results We give three examples to test the validity of the method for computing the smallest eignevalue and the largest eigenvalue of a real symmetric matrix. The simulating platform is MATLAB. Example 1. A real 7 × 7 symmetric matrix is randomly generated as ⎛ ⎜ ⎜ ⎜ ⎜ B=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0.5972 0.3858 0.8341 0.7884 0.5024 0.3249 0.5769 ⎞ ⎟ 0.3858 0.7015 0.2719 0.6883 0.4196 0.2061 0.6175 ⎟ , 0.8341 0.2719 0.9566 0.3936 0.7404 0.6784 0.3759 ⎟ ⎟ 0.7884 0.6883 0.3936 0.2473 0.2119 0.3470 0.4511 ⎟ ⎟ 0.5024 0.4196 0.7404 0.2119 0.3143 0.5366 0.2305 ⎟ 0.3249 0.2061 0.6784 0.3470 0.5366 0.5414 0.6682 ⎟ ⎟ 0.5769 0.6175 0.3759 0.4511 0.2305 0.6682 0.5030 ⎠

Through step 1, we get ξ = (0.5851 0.4572 0.6304 0.4575 0.4404 0.4783 0.4893)T . So, λ1 = ξ1 + L + ξ n = 3.5382 . The trajectories of xi (t ) (i = 1,2,L,7 ) and λ1 (t ) = x1 (t ) + L + x n (t ) are shown in Fig. 1 and Fig. 2. From Fig. 1 and 2, we can see that FNN (2) quickly reaches the equilibrium state and the largest eigenvalue is obtained soon. Through step 2, we get and ξ = (- 0.1301 - 0.0931 0.0269 0.1453 0.0858 - 0.0916 0.0848 )T (i = 1,2,L,7 ) and λn = −(ξ1 + L + ξ n ) = −0.6576 . The trajectories of xi (t )

λn (t ) = −( x1 (t ) + L + xn (t ) ) are shown in Fig. 3 and Fig. 4. Using Mathlab to di-

rectly compute the eigenvalues of and smallest eigenvalue

λ'n

B gives the true largest eigenvalue

= −0.6576 .

λ1' = 3.5379

A Concise FNN for Computing the Extremum Eigenpairs of Real Symmetric Matrices

Fig. 1. The trajectories of components of x when A is replaced with B

Fig.

2

409

The

trajectories of when A is

The

trajectories of when A is Re-

The

trajectories

λ1 (t ) = x1 (t ) + L + xn (t )

replaced with B

Fig. 3. The trajectories of components of x when A is replaced with -B

Fig.

4.

λn (t) = −( x1(t) +L+ xn (t) )

placed with –B

Fig. 5. The trajectories of components of x when A is replaced with C

Fig.

6.

x1 (t ) + L +

with C

x n (t )

of

when A is Replaced

410

Y. Liu and Z. You

Therefore, the results computed by the method are very close to the true values, the absolute difference values are and λ1 − λ1' = 3.5382 − 3.5379 = 0.0003 λ n − λ'n = − 0.6576 − (− 0.6576) = 0.0000 . This verifies that using this method to com-

pute the largest eigenvalue and the smallest eigenvalue of

B is successful.

Example 2: A positive matrix is ⎛ 0.6495 ⎜ ⎜ 0.5538 C=⎜ 0.5519 ⎜ ⎜ 0.5254 ⎝

0.5538 0.5519 0.5254 ⎞ ⎟ 0.8607 0.7991 0.5583 ⎟ . 0.7991 0.8863 0.4508 ⎟ ⎟ 0.5583 0.4508 0.6775⎟⎠

Fig. 7. The trajectories of components of x when A is replaced with -C

Fig.

8.

The

(

)

trajectories

− ( x1 (t ) + L + xn (t ) )

of

when A is Re-

placed with -C

Fig. 9. The trajectories of components of x when A is replaced with C’

Fig.

10.

The

trajectories

of

-

( x (t ) + L + x (t ) ) when A is Replaced 1

with C’

n

A Concise FNN for Computing the Extremum Eigenpairs of Real Symmetric Matrices

411

The true eigenvalues of computed by MATLAB are C λ '= (0.0420 0.1496 0.3664 2.5160 ) . From step 1, we can get λ1 = 2.5160 , the trajectories of the components of

x(t ) and

x1 (t ) + L + x n (t ) which will approach to

λ1 are shown in Fig. 5 and 6. Using step 2, the trajectories of components of x(t ) and λn (t ) = −( x1 (t ) + L + x n (t ) ) are shown in Fig. 7 and 8. The equilibrium vector

ξ = 1.0e - 5 * (0.1229 0.2827 - 0.2183 - 0.1342) T and λn = −(ξ1 + L + ξ n ) = -7.5811e - 6 .

Obviously, λn is very close to zero. So, step 4 is used to compute the smallest eigenvalue. Replacing A with

⎛ - 2.3505 0.5538 ⎜ ⎜ 0.5538 - 2.1393 C ' = −(C − ⎡λ1 ⎤I ) = −⎜ 0.5519 0.7991 ⎜ ⎜ 0.5254 0.5583 ⎝ the trajectories of components of

0.5519 0.7991 - 2.1137 0.4508

x(t ) and − ( x1 (t ) + L +

0.5254 ⎞ ⎟ 0.5583 ⎟ , 0.4508 ⎟ ⎟ - 2.3225 ⎟⎠

x n (t ) ) which will approach

to the smallest eigenvalue of C ' are shown in Fig. 9 and 10. In the end, we get the equilibrium vector ξ ∗ = (0.4168 1.0254 - 0.9303 - 0.5854 )T and the small eigenvalue of C λ∗n = − ξ1∗ + L + ξ n∗ + ⎡λ1 ⎤ = −2.9579 + 3 = 0.0421 .

(

)

λ1 = 2.5160 and λ∗n = 0.0421 are the results computed by FNN (2), comparing with the true values λ1' = 2.5160 and λ'n = 0.0420 , we can see that λ1 is very close to λ1' , the absolute difference is 0.0000 , and λ∗n is very near to λ'n , the absolute difference is 0.0001 . This verifies that using step 4 to compute the smallest eigenvalue of a real positive symmetric matrix is valid. Example 3. Matrix D = −C is used to test the validity of FNN (2). Obviously, D is negative definite and its true eigenvalues are λ '= (- 2.5160 - 0.3664 - 0.1496 - 0.0420 ) . Replacing A with D , from step 1, the obtained eigenvalue is 7.5811e - 6 , and we cannot decide whether this value is the true largest eigenvalue. Through step 2, the smallest eigenvalue is obtained λ4 = -2.5160 . Using step 3, we get λ1 = −0.0421 . Comparing λ4 and λ1 to λ'4 and

λ1' , respectively, it can be found that using step 3 to compute the largest eigenvalue is nice.

x(0 ) of example 1 is [0.2892 0.1175 0.0748 0.2304 0.2502 0.2001 0.2275]T , and x(0 )

of example 2 and 3 is [0.2892 0.1175 0.0748 0.2304]T . Repeating the three examples with the other initial values, the obtained results are all very close to the corresponding true values.

412

Y. Liu and Z. You

6 Conclusions This paper proposes a very concise functional neural network presented by a differential dynamical system, using the component analytic solution of this system gives the convergence behavior. This network is adaptive to extract the largest positive eigenvalue or the smallest negative eigenvalue. Detail steps are designed to compute the largest eigenvalue and smallest eigenvalue of a real symmetric matrix using this neural network. This method is adaptive to non-definite, positive definite, or negative definite matrix. Three examples show the validity of this method, B is a non-definite, C is positive definite and D is negative definite, the results obtained by this method are all very close to the corresponding true values. Comparing with other approach based on neural networks, this functional neural network is very concise and can be realized easily.

Acknowledgement This work is supported by the Youth Foundation of Sichuan University (No. 200574) and Youth Foundation of School of Computer Science & Engineering of Sichuan University.

References 1. Luo, F.-L., Unbehauen, R., Cichocki, A.: A Minor Component Analysis Algorithm. Neural Networks 10(2) (1997) 291-297 2. Luo, F.-L., Unbehauen, R., Li, Y.-D.: A Principal Component Analysis Algorithm with Invariant Norm. Neurocomputing 8(2) (1995) 213-221 3. Song, J., Yam, Y.: Complex Recurrent Neural Network for Computing the Inverse and Pseudo-inverse of the Complex Matrix. Applied Mathematics and Computing 93(2-3) (1998) 195-205. 4. Kakeya, H., Kindo, T.: Eigenspace Separation of Autocorrelation Memory Matrices for Capacity Expansion. Neural Networks 10(5) (1997) 833-843 5. Kobayashi, M., Dupret, G., King, O., Samukawa, H.: Estimation of Singular Values of Very Large Matrices Using Random Sampling. Computers and Mathematics with Applications 42(10-11) (2001) 1331-1352 6. Zhang, Y., Nan, F., Hua, J.: A Neural Networks Based Approach Computing Eigenvectors And Eigenvalues Of Symmetric Matrix. Computers and Mathematics with Applications 47(8-9) (2004) 1155-1164 7. Luo, F.-L., Li, Y.-D.: Real-time Neural Computation of the Eigenvector Corresponding to the Largest Eigenvalue of Positive Matrix. Neurocomputing 7(2) (1995) 145-157 8. Reddy, V. U., Mathew, G., Paulraj, A.: Some Algorithms for Eigensubspace Estimation. Digital Signal Processing 5(2) (1995) 97-115 9. Perfetti, R., Massarelli, E.: Training Spatially Homogeneous Fully Recurrent Neural Networks in Eigenvalue Space. Neural Networks 10(1) (1997) 125-137 10. Liu, Y., You, Z., Cao, L., Jiang, X.: A Neural Network Algorithm for Computing Matrix Eigenvalues and Eigenvectors. Journal of Software 16(6) (2005) 1064-1072

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11. Liu, Y., You, Z., Cao, L.: A Simple Functional Neural Network for Computing the Largest and Smallest Eigenvalues and Corresponding Eigenvectors of a Real Symmetric Matrix. Neurocomputing 67 (2005) 369-383 12. Liu, Y., You, Z., Cao, L.: A Functional Neural Network for Computing the Largest Modulus Eigenvalues and Their Corresponding Eigenvectors of an Anti-symmetric Matrix. Neurocomputing 67 (2005) 384-397. 13. Liu, Y., You, Z., Cao, L.: A Functional Neural Network Computing Some Eigenvalues and Eigenvectors of a Special Real Matrix. Neural Networks 18(10) (2005) 1293-1300 14. Wang, J.: Recurrent Neural Networks for Computing Pseudoinverses of Rank-deficient Matrices. SIAM Journal on Scientific Computing 18(5) (1997) 1479-1493

A Novel Stochastic Learning Rule for Neural Networks Frank Emmert-Streib Institut f¨ ur Theoretische Physik, Universit¨ at Bremen, Otto-Hahn-Allee 1, 28359 Bremen, Germany [email protected]

Abstract. The purpose of this article is the introduction of a novel stochastic Hebb-like learning rule for neural networks which combines features of unsupervised (Hebbian) and supervised (reinforcement) learning. This learning rule is stochastic with respect to the selection of the time points when a synaptic modification is induced by simultantious activation of the pre- and postsynaptic neuron. Moreover, the learning rule does not only affect the synapse between pre- and postsynaptic neuron which is called homosynaptic plasticity but effects also further remote synapses of the pre- and postsynaptic neuron. This more complex form of plasticity has recently come into the light of interest of experimental investigations in neurobiology and is called heterosynaptic plasticity. Our learning rule is motivated by these experimental findings and gives a qualitative explanation of this kind of synaptic plasticity. Additionally, we give some numerical results that demonstrate that our learning rule works well in training neural networks, even in the presence of noise.

1

Introduction

What are the mechanisms that modulate learning in the brain of an animal or a human? This question is up to now under debate, but the imagination one has for a biological learning rule is that the synaptic weights are changed according to a local rule. In the context of neural networks local means that only the adjacent neurons of a synapse contribute to modifications of the synaptic weight. Such a mechanism with respect to synaptic strengthening was proposed by Hebb [11] in 1949 and experimentally found by T. Bliss and T. Lomo [3]. In a biological terminus Hebbian learning is called long-term potentiation (LTP). Experimentally as well as theoretically there is a great body of investigations aiming to formulate precise conditions under which learning in neural networks takes place. For example, the influence of the precise timing of pre- and postsynaptic neuron firing [14, 17] or the duration of a synaptic change (for a review see [16]) have been studied extensively. All these studies share the locality condition proposed by Hebb [11]. However, there are also experimental findings which 

Present address: Stowers Institute for Medical Research, 1000 E. 50th Street, Kansas City, MO 64110, USA.

J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 414–423, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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extend the traditional view of synaptic plasticity. In this article we present a novel stochastic Hebb-like learning rule inspired by experimental findings about heterosynaptic plasticity [9]. This form of neural plasticity affects not only the synapse between pre- and postsynaptic neuron in which a synaptic modification was induced, but also further remote synapses of the pre- and postsynaptic neuron. Additionally, we demonstrate that this learning rule can be successfully applied in training multilayer neural networks. This paper is organized as follows. In section 2 we provide experimental results about heterosynaptic plasticity and learning in general. In section 3 we introduce our learning rule and in section 4 we present numerical results. This article finishes in section 5 with a discussion and conclusions

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Biological Learning

Recently, there is an increasing number of experimental results, which investigate a new form of synaptic modification, the so called heterosynaptic plasticity. In contrast to homosynaptic plasticity, where only the synapse between active pre- and postsynaptic neuron is changed, heterosynaptic plasticity concerns also further remote synapses of the pre- and postsynaptic neuron. This scenario is depicted in the left Fig. 1. Here we suppose, that the synapse w68 was changed either by LTP or LTD. Fitzsimonds et al. [9] found in cultured hippocampal neurons that the induction of LTD in w68 is also accompanied by back propagation of depression in the dendrite tree of the presynaptic neuron. Further more, depression also propagates laterally in the pre- and postsynaptic neuron. Similar results hold for the propagation of LTP, see [2] for a review. We emphasize all synapses, whose weights are changed, and all neurons, which enclose these synapses by drawing full lines respectively black circles. We want explicitly emphasize, that Fitzsimonds et al. found up to now no forward propagated postsynaptic plasticity. This would correspond to the synapses w811 , w812 of neuron n8 , which are drawn as dotted lines in the left Fig.1. A biological explanation for the cellular mechanisms of these findings are currently under investigation. Fitzsimonds et al. [9] suggest the existence of retrograde signaling from the postto the presynaptic neuron which could produce a secondary cytoplasmic factor for back-propagation and presynaptic lateral spread of LTD. On the postsynaptic side lateral spread of LTD could be explained similarly under the assumption that there is a blocking mechanism for the cytoplasmic factor which prevents forward propagated LTD. They are of the opinion that extracellular diffusible factors are of minor importance. The experiments of Fitzsimonds et al. [9] are certainly an extention of homosynaptic learning, which we denote briefly as Hebbian learning1 , but nevertheless both principles can be characterized as unsupervised learning because both learning types use exclusively local information available in the neural system. This 1

We are aware, that Hebbian learning is usually only used in the context of LTP as explained in the text above and want to emphasize by this paralance that homosynaptic LTP and LTD are more interrelated than homo- and heterosynaptic LTP.

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Fig. 1. Left: Visualization of heterosynaptic plasticity experimentally found by Fitzsimonds et al.. The neurons (synapses) which are affected by heterosynaptic plasticity induced in w68 are drawn as black circles (full lines). Right: Visualization of the symmetry of our learning rule. The cell character of a neuron is emphasized by a schematical contour around the whole neuron. Each neuron in this network has a new degree of freedom ci we call the neuron counter. Neuron counters can communicate which each other across synapses. The dynamics of the neuron counters is determined by the global reinforcement signal.

is in contrast to the famous back-propagation learning rule [21, 19] for artificial neural networks. The back-propagation algorithm is famous because until the 80’s there was no systematic method known to adjust the synaptic weights of an artificial multilayer (feed-forward) network to learn a mapping2 . Still, the problem with the back-propagation algorithm is that it is not biological plausible because it requires a back-propagation of an error in the network. We emphasize, that the problem is not the back-propagation process itself, because, e.g., heterosynaptic plasticity could provide such a mechanism, but the knowledge of the error, which can not be known explicitely to the neural network [6]. For this reason learning by back-propagation is classified as supervised learning or learning by a teacher [12]. However, there is a modified form of supervised learning namely reinforcement learning that is biologically plausible. Reinforcement learning reduces the information provided by a teacher to a binary reinforcement signal r that reflects the quality of the network’s performance. Interestingly, experimental observations from the hippocampus CA1 region have shown that there is a global signal in form of dopamine which is feedback to the neurons and causes thereby a modulation of LTD [18]. Schematically, this is depicted in the right Fig. 1. In this figure each neuron is connected with an additional edge which represents the feedback of dopamin in form of a reinforcement signal r. 2

The back-propagation algorithms was independently developed by Werbos [21] and Rumelhart et al. [19] but became aware to the broad research community after the article by Rumelhart et al.

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The working mechanism of the learning rule we suggest is based on the explanation of Fitzsimonds et al. [9] for heterosynaptic plasticity given above. To understand what kind of mathematical formulation is capable to describe ’a secondary cytoplasmic factor’ in a qualitative way we start our explanation with emphasizing that a neuron is from a biological point of view first of all a cell. The subdivision of a neuron in synapses, soma (cell body) and axon is a model and reflects already the direction of the information flow within the neuron namely from the synapses (input) to the soma (information processing) to the axon (output). Here, we do not question this model view with respect to the direction of signal processing, but learning. We see no biological reason why the model of a neuron for signal processing should be the same as the model of a neuron for learning. In the right Fig. 1 we emphasize the cell character of a neuron by underlying the contour of the whole neuron in gray. Now, our reason for drawing the synapses in an unusual way becomes clear, because it emphasizes automatically the cell character of a neuron. Suppose now, we assign to each neuron in the network one additional parameter ci as shown in the right Fig. 1. We call these parameters ci neuron counters. The neuron counters shall modulate the synaptic modification in a certain way defined in detail below. According to our cell view of the neuron, we assume further that the neuron counters of adjacent neurons, which are connected by synapses, can communicate with each other in an additive way. For example, in Fig. 1 the neuron counters c6 and c8 form a new value d68 = c6 + c8 in synapse w68 , which we call the approximated synapse counter. By this mechanism we obtain a star-like influence of ,e.g., the neuron counters c6 and c8 on all synapses connected with neuron 6 or 8, because either d6k = c6 + ck or dk8 = ck + c8 holds and regulates the synaptic update of the corresponding synaptic weight of the synapses w6k and wk8 respectively. This situation corresponds in a qualitative way to the learning behavior of heterosynaptic plasticity, however, with the difference, that we have a fully symmetrical learning rule. An interpretation of the communication between adjacent neuron counters can be given, if one views the neuron counters as cytoplasmic factors, which are allowed to freely move within the cytoplasm of the corresponding neuron (cell). Because, we introduced no blocking mechanism for the forward propagation of the postsynaptic neuron counter we result in a fully symmetric communication between adjacent neuron counters. Unfortunately, there are no experimental data available that would allow to specify the influence of dij on the corresponding synapse wij quantitatively. For this reason, we use an ansatz to close this gap and make it plausible [7].

3

Mathematical Definition of the Learning Rule

If one assumes, that the neuron counters shall modulate learning, then it is plausible to determine the values of ci in dependence on a reinforcement signal r reflecting the performance of the network qualitatively. In the most simple case, the dynamics of the neuron counters depends linearly on the reinforcement signal.

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⎧ ci − r > Θ ⎨ Θ, ci →c‘i = ci − r, Θ ≥ ci − r ≥ 0 ⎩ 0, 0 > ci − r.

(1)

Here, Θ ∈ IN is a threshold that restricts the possible values of the neuron counters ci to Θ + 1 possible values {0, . . . , Θ}. The value of ci reflects the network’s performance, but it has only relative and no absolute meaning with respect to the mean network error. This can be seen by the following example. Suppose c = 0, then it is clear that at least the last output of the network was right, r = 1. However, we know nothing about the outputs which occurred before the last one. E.g., the following two sequences of reinforcement signals can lead to the same value of the neuron counter c = 0. r1 = {1, −1, 1, −1, 1, −1, 1} and r2 = {1, 1, 1, 1, 1, 1, 1} if the start value is c = 1 for r1 and c = 7 for r2 . Obviously, the estimated mean error is different in both cases, if averaged over the last seven time steps. The crucial point is, that the start value of the neuron counter is not available for the neuron and, hence, the neuron can not directly calculate the mean error of the network. However, we can introduce a simple assumption, which allows an estimate of the mean network error. We claim that, if c1 < c2 for one neuron in a network, but trained by two different learning rules3 then the mean error of network one is lower then of network two. This may not hold for all cases, but it is certainly true in average. By this we couple the value of the neuron counter to the mean error of the network. Due to the fact, that this holds only statistically, we will introduce a stochastic rather than a deterministic update rule for the synapses that depends on the neuron counters. In the previous section we said, that adjacent neuron counter can communicate, if both neurons are connected by a synapse. This gives a new variable dij = ci + cj

(2)

we call the approximated synapse counter. We will use the approximated synapse counter as the driving parameter of our stochastic update rule, because its value reflects the performance of the synapse in the network which shall be updated, because the synapses are the adaptive part of a neural network. Hence, evaluating the value of an approximated synapse counter of a synapse will give us indirectly a decision for the update of this synapse. It is clear that, roughly speaking, the higher the approximated synapse counter of a synapse is the higher should be the probability the synapse is updated. This intuitively plausible assumption will now be quantified. Similar to [1, 5, 15] only active synapses wij which were involved in the last signal processing step can be updated, if the output of the network was wrong. This is plausible, because it prevents that already learned mappings in the neural network are destroyed possibly. If r = −1 the probability, that synapse wij is updated is given by PΔw (wij ) = P (pc < prdij ) (3) 3

The superscript indicates here not the neuron in the network, but the learning rule used.

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This probability has to be calculated for each synapse wij in the network. We want to emphasize, that this needs only local information besides the reinforcement signal. Hence, it is a biologically possible mechanism. If the synapse is actually chosen for update, the synaptic weight will be modified by  wij →wij = wij − δ,

(4)

Here, δ is a positive constant which determines the amount of the synaptic depression. To evaluate the stochastic update condition Eq. 3 the two auxiliary variables pc and prdij have to be identified. This is done in the following way: 1. Calculate the approximated synapse counter dij = ci + cj

(5)

2. Map the value of the approximated synapse counter dij to prdij by k = 2Θ + 3 − dij

(6)

Pkr ∝ k −τ , τ ∈ IR+

(7)

for k ∈ {1, . . . , 2Θ + 3}. We call Pkr rank ordering probability distribution 3. The random variable pc is drawn from the continuous coin distribution P c (x) ∝ x−α , α ∈ IR+

(8)

for x ∈ (0, 1]. We had three reasons to choose a power law in Eq. 8 for the coin distribution instead of an equal distribution, which would be the simplest choice. First, we see no evidence that a random number generator occurring in a neural system should favor a equal distribution. Second, it is highly probable that two different random number generators of the same biological system are not identical. Instead, they could have different parameters, in our case they could have different exponents. In this paper we will content ourself investigating the case of identical random number generators, but our framework can be directly applied to the described scenario. Third, by choosing α = 0, the coin distribution in Eq. 8 becomes the equal distribution. This allows us to investigate the influence of the distance of the coin distribution to an equal distribution on the learning behavior of a neural network by studying different parameters of α. We want to remark, that in this case the update probability Eq. 3 simplifies to PΔw (wij ) = prdij

(9)

Before we present our results in the next section, we want to visualize the stochastic update probability PΔw . Figure 2 shows the update probability PΔw as function of prdij . The different curves correspond to different values of the exponent α of the coin distribution. One can see, that the update probability follows the values of prdij . This holds for each curve in Fig. 2. That means, the higher the values of prdij are the higher is the update probability. This is the behavior one would intuitively expect, because high values of prdij correspond to

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high values of the approximated synapse counters dij indicating high values of the neuron counters, which correspond to a bad network performance. Moreover, one can see in Fig. 2 that the larger α the higher is the update probability for fixed prdij . In the limit α → ∞ the update probability equals one for all values of prdij . Hence, higher values of the exponent α of the coin distribution result in a higher update probability. That means, by α one can control the sensitivity the update probability depends on prdij . Another parameter our stochastic update rule depends on is the exponent of the rank ordering distribution τ . We display in Fig. 3 PΔw as function of τ and prdij to visualize its influence on the update probability. The values of the update probability are color-coded and blue corresponds to 0 and red to 1. For the left Fig. 3 we used α = 0.7 and for the right α = 1.5 as exponent for the coin distribution. If prdij = 0 no update takes place. For increasing values of the approximated synapse counter and fixed values of τ one obtains increasing values for the update probability. Moreover, higher values of α lead to higher values of PΔw . This can be seen by comparing the left and right Fig. 3. Increasing values of τ result in decreasing values of PΔw for fixed dij . To summarize, the stochastic update condition we introduced for a synaptic update depends on six parameters PΔw (wij ) = PΔw (r, ci , cj , Θ, α, τ ).

(10)

From the visualizations we gave in Fig. 2 and 3 we saw that increasing values of dij and α as well as decreasing values of τ lead to an increase in the update probability.

4

Numerical Results

Exemplarily, we demonstrate that our stochastic learning rule is capable to train multilayer neural networks by learning a XOR-function, given in table 1, in a

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three layer network. In the left Fig. 4 we show the mean learning time as function of the number of neurons H in the hidden layer. The curves are indexed by different values of the neuron counter Θ. In the right figure we demonstrate the robustness of these results in the presence of noise η by using a noisy winner-

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take-all mechanism as network dynamics which adds to the inner fields of the neurons noise η before the neuron with the highest inner field is selected. The noise was uniformly drawn from [0, η0 ]. From both figures one can see that the mean learning time decreases with an increasing number of neurons in the hidden layer as expected whereas the increase from 3 to 4 neurons has the biggest effect. This is due to the fact that the destructive path inference, which means that already correctly learned paths in the network are destroyed by a new synaptic modification, is strongly reduced by increasing the number of possible paths as a result of additional neurons in the hidden layer. Increasing the number of neurons beyond 19 has only marginal influence, because an additional increase of redundant paths has no affect. Even in the presence of noise our learning rule is capable of learning the XOR-function. One can nicely see how an increasing number of neurons in the hidden layer can efficiently reduce the amount of noise in the system.

5

Conclusions

In this article we presented a novel stochastic Hebb-like learning rule for neural networks and demonstrated its working capability exemplary in learning the XOR-function in a multilayer neural network even in the presence of noise. Our goal was a biologically inspired learning rule based on experimental findings about heterosynaptic plasticity rather than to learn functions in a multilayer network as fast a possible. Despite this fact, the results we obtained from numerical simulations are comparable to [1, 15] and indicate that the learning rule itself is useful for practical applications in training neural networks. In further investigations [8] we will demonstrate that our learning rule is not restricted to a feed-forward network topology but works also in recurrent networks. We hope, that our work stimulates further investigations in this exciting field and demonstrates how fruitful it is to create mathematical models based on experimental observations of biological phenomena.

Acknowledgments We would like to thank Tom Bielefeld, Rolf D. Henkel, Jens Otterpohl, Klaus Pawelzik, Roland Rothenstein, Peter Ryder, Heinz Georg Schuster and Helmut Schwegler for fruitful discussions.

References 1. Bak, P. and Chialvo, D.R.: Adaptive Learning by Extremal Dynamics and Negative Feedback. Phys. Rev. E 63 (2001) 031912-1–031912-12 2. Bi, G-g. and Poo, M-m.: Synaptic Modification by Correlated Activity: Hebb’s Postulate Revisited. Annual Review of Neuroscience 24 (2001) 139–166

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3. Bliss, T.V.P. and Lomo, T.: Long-lasting Potentiation of Synaptic Transmission in the Dentate Area of the Anaesthetized Rabbit Following Stimulation of the Perforant Path. J. Physiol. 232 (1973) 331–356 4. Bosman, R. J. C., van Leeuwen, W. A. and Wemmenhove, B.: Combining Hebbian and Reinforcement Learning in a Minibrain Model. Neural Networks 17:1 (2004) 29–39 5. Chialvo, D.R. and Bak, P.: Learning from Mistakes. Neuroscience 90 (1999) 1137– 1148 6. Crick, F.: The Recent Excitement about Neural Networks. Nature 337 (1989) 129– 132 7. Emmert-Streib, F.: Aktive Computation in Offenen Systemen. Lerndynamiken in biologischen Systemen: Vom Netzwerk zum Organismus. Dissertation, Universit¨ at Bremen, (2003), Mensch & Buch Verlag. 8. Emmert-Streib, F.: Active Learning in Recurrent Neural Networks Facilitated by a Hebb-like Learning Rule with Memory. Neural Information Processing - Letters and Reviews 9(2) (2005) 31–40 9. Fitzsimonds, R.M., Song, H-j. and Poo, M-m.: Propagation of Synaptic Modulation in Small Neural Networks. Nature 388 (1997) 439–448 10. Frey, U. and Morris, R.G.M.: Synaptic Tagging and Long-term Potentiation. Nature 385 (1997) 533–536 11. Hebb, D.O.: The Organization of Behavior. Wiley, New York (1949) 12. Hertz, J., Krogh, A. and Palmer, R.G.: Introduction to the Theory of Neural Computation. Addison-Wesley (1991) 13. Holland, J.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) 14. Kempter, R., Gerstner, W. and van Hemmen, J.L.: Hebbian Learning and Spiking Neurons. Phys. Rev. E 59 (1999) 4498–4514 15. Klemm, K., Bornholdt, S. and Schuster, H.G.: Beyond Hebb: Exclusive-Or and Biological Learning. Phys. Rev. Lett. 84 (2000) 3013–3016 16. Koch, C.: Biophysics of Computation, Oxford Press (1999) 17. Markram, H., L¨ ubke, J., Frotscher, M. and Sakmann, B.: Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs. Science 275 (1997) 213– 215 18. Otmakhova, N.A. and Lisman, J.E.: D1/D5 Dopamine Receptors Inhibit Depotentiation of CA1 Synapses via cAMP-dependent Mechanism. J. Neuroscience 18 (1998) 1270–1279 19. Rumelhart, D.E., Hinton, G.E. and Williams, R.J.: Learning Representations by Back-propagating Errors. Nature 323 (1986) 533–536 20. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P.: Optimization by Simulated Annealing. Science 220 (1983) 671–680 21. Werbos, P.: Beyond Regression: New tools for Prediction and Analysis in the Behavioral Sciences. Ph.D. Thesis, Harvard University (1974)

Learning with Single Quadratic Integrate-and-Fire Neuron Deepak Mishra1 , Abhishek Yadav2 , and Prem K. Kalra1 1

Department of Electrical Engineering, IIT Kanpur, India 2 Department of Electrical Engineering, G.B. Pant University of Agri. & Technology, Pant Nagar, India [email protected], [email protected], [email protected]

Abstract. In this paper, a learning algorithm for a single Quadratic Integrateand-Fire Neuron (QIFN) is proposed and tested for various applications in which a multilayer perceptron neural network is conventionally used. It is found that a single QIFN is sufficient for the applications that require a number of neurons in different layers of a conventional neural network. Several benchmark and real-life problems of classification and function-approximation have been illustrated.

1 Introduction Various Researchers have proposed many neuron models for artificial neural networks. Although all of these models were primarily inspired from the biological neuron, there is still a gap between the philosophies used in neuron models for neuroscience studies and neuron models used for artificial neural networks. Some of these models exhibit a close correspondence with their biological counterparts while others do not. Freeman [6] has pointed out that while brains and neural networks share certain structural features such as massive parallelism, biological networks solve complex problems easily and creatively, and existing neural networks do not. He discussed the issues related to the similarities and dissimilarities between biological and artificial neural systems of present days. The first artificial neuron model was proposed by McCulloch and Pitts [12] in 1943. They developed this neuron model based on the fact that the output of the neuron is 1 if the weighted sum of its inputs is greater than a threshold value and 0 otherwise. In 1949, Hebb [7] proposed a learning rule that became initiative for artificial neural networks. He postulated that the brain learns by changing its connectivity patterns. Widrow and Hoff [2] in 1960 presented the most analyzed and most applied learning rule. It was called the least mean square learning rule. Later in 1985, Widrow and Sterns [8] found that this rule converges in the mean square to the solution that corresponds to least mean square output error if all the input patterns are of the same length. Scholles [3] discussed a few biologically inspired artificial neurons. Feng and Li [16] introduced some neuron models with current as the inputs. Training of the integrate-and-fire model with the Informax principle was discussed in [14] and [15]. Learning with single integrate-and-fire neuron has been studied in [5]. In the same sequel, a learning with a more biologically realistic artificial neural system i.e. quadratic integrate and fire model is proposed and discussed in this paper. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 424–429, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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2 Quadratic Integrate-and-Fire Neuron (QIFN) Models Quadratic integrate-and-fire neuron is also known as the theta-neuron [18], or the Ermentrout-Kopell canonical model [19] (when it is written in a trigonometric form). We represent it here in terms of the following differential equation[10][20]. dv/dt = I + a(v − V1 )(v − V2 ); if v = V3 , then v ← V4

(1)

where V1 and V2 are the resting and threshold values of the membrane potential. V3 and V4 are the peak and reset values. This model is canonical in the sense that any Class 1 excitable system described by smooth ODEs can be transformed into this form by a continuous change of variables. Unlike its linear analogue, the quadratic integrate-andfire neuron has spike latencies, activity dependent threshold, and bistability of resting and tonic spiking modes. The solution of the first-order differential equation describing the dynamics of this model in sub-threshold region (eqn.(1)) can be found analytically. With v(0) = V1 , the solution is given by    V1 + V2 A At a(V1 − V2 ) −1 v(t) = + tan + tan (2) 2 a 2 A  A = 4aI + 2 a2 V1 V2 − a2 V2 2 − a2 V1 2 (3) Let us assume that t1 is the refractory period and v(t) hits V2 at t = t2 . Interspike interval, t0 = t1 + t2 , can be expressed as     4 w −1  t0 = t1 −  tan (4) (4aI − w2 ) (4aI − w2 ) where, w = a(V1 − V2 )

(5)

This equation can be written as t0 = t1 − 4x tan−1 (wx) 1 where, x =  (4aI − w2 )

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(8) (9)

Here the relationship between t0 and I is expressed in terms of the relationship between X and y in order to make it bounded. y is always between -1 and 1 for all real values of X. Therefore, besides its biological meaningfulness, this function is mathematically appropriate to serve as an activation function of a neuron.

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3 The Proposed Model Along with a biologically motivated and mathematically appropriate activation function, a suitable aggregation function is also required. In view of evidence in support of the presence of multiplicative-like operations in the nervous system, multiplication of net inputs to the activation function is considered [1][9]. In biological neural systems, this operation depends on the timings of various spikes. Aggregation of the exponential waveforms with different time-delays has been approximated by considering associated with weight and bias parameters. In the proposed model, these parameters are not measures of synaptic strength and resting potential as in conventional neuron models. They are measures of time delay and amplitude decay that represent the temporal summation as described in [11] for spiking neuron models. Equation (9), which is inspired from the relationship between injected current and interspike interval for quadratic integrate-and-fire neuron, can be modified to incorporate the temporal summation in case of multi-input neuron.  n   2 y = tan−1 (wi xi + bi ) (10) π i=1 where, n is the number of inputs. Equation (10), is used as the agregation function to calculate output y for input is x. wi and bi are connection weights and bias respectively. A simple steepest descent method is applied to minimize the following error function: e=

1 (y − t)2 2

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where, t is the target and y is the actual output of the neuron. e is the function of parameters bi & wi ; i = 1, 2, ...n. Therefore, the parameter update rule (weight update rule) is find out by calculating partial derivative of e w.r.t. the function of parameters.

4 Illustrative Examples 4.1 Classification Problem: XOR Data The XOR problem, as compared with other logic operations (NAND, NOR, AND and OR), is probably one of the best and most used nonlinearly separable pattern associator and consequently provides one of the most common examples of artificial neural systems for input remapping. We compared the performance of quadratic integrate-and-fire neuron (QIFN) with that of multilayer perceptron (MLP). For this purpose, we considered an MLP with 3 hidden units. Figure 1(a) shows the MSE vs. number of epochs curve with the proposed QIFN model while training it for XOR Problem. The proposed model takes only 3 iterations while MLP takes 523 iterations for training to achieve an MSE of the order of 0.0001. Figure 1(b) exhibit the comparison between MLP and QIFN in terms of the deviation of actual outputs from corresponding targets. It can be seen here that the performance of QIFN is almost same as compared with MLP. It means that a single QIFN is capable to learn XOR relationship significantly faster than an MLP

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Fig. 1. (a) Mean Square Error vs. Epochs for training for XOR problem, (b) Target vs. Actual Output with MLP and QIFN, for XOR-problem

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with 3 hidden units. Table 1 shows the comparison of training and testing performance with MLP and QIFN while solving the XOR-problem. 4.2 Function Approximation Problem: Electroencephalogram Data Electroencephalogram (EEG) data used in this work was taken from [21]. In this problem, four measurements y(t − 1), y(t − 2), y(t − 4) and y(t − 8) were used to predict y(t). We considered MLP with 6 hidden neurons for the comparison purposes. Figure 2 shows the comparison between MLP and QIFN in terms of the deviation of actual outputs from corresponding targets. Data till 100 sampling instants was used for training. It can be seen here that the performance of QIFN for training as well as testing data is much better than that of MLP. It means that a single QIFN is capable to learn this relationship faster than that in case of an MLP with 6 neurons and its performance on seen

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Fig. 2. Target and Actual Output with MLP and QIFN, for EEG Data

Table 2. Comparison of testing and training performance for EEG Data S. No. Parameter

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as well as unseen data is significantly better. Table 2 shows the comparison of training and testing performance with MLP and QIFN while applying for the EEG Data.

5 Conclusion The training and testing results with different benchmark and real-life problems show that the proposed artificial neural system with a single neuron inspired from the quadratic integrate-and-fire neuron model is capable of performing classification and function approximation tasks as efficiently as a multilayer perceptron with many neurons and in some cases its learning is even better. It is also observed that training and testing times in case of QIFN are significantly less as compared with MLP. Future scope of this work includes incorporation of these neurons in a network and analytical investigation of its learning capabilities (e.g., as universal function approximator).

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References 1. McKenna, T., Davis, J., Zornetzer, S.F. (ed.): Single Neuron Computation. Academic Press, San Diego, CA, USA (1992) 2. Anderson, J.A., Rosenfeld, E. (ed.): Neurocomputing: Foundations of Research. MIT Press, Cambridge (1960) 3. Scholles, M., Hosticka, B.J., Kesper, M., Richert, P., Schwarz, M.: Biologically Inspired Artificial Neurons: Modeling and Applications. Proceedings of International Joint Conference on Neural Networks, Vol. 3 (1993) 2300–2303 4. Iannella, N., Back, A.: A Spiking Neural Network Architecture for Nonlinear Function Approximation. Proceedings of the 1999 IEEE Signal Processing Society Workshop, (1999) 139–146 5. Yadav, A., Mishra, D., Ray, S., Yadav, R.N., Kalra, P.K.: Learning with Single Integrateand-Fire Neuron. Proceedings of International Joint Conference on Neural Network, (2005) 6. Freeman, W.J.: Why Neural Networks Dont Yet Fly: Inquiry into the Neurodynamics of Biological Intelligence. Proceedings of IEEE International Conference on Neural Networks, Vol. 2 (1988) 1–7 7. Hebb, D.: Organization of Behavior. John Weiley and Sons, New York (1949) 8. Widrow, B., Steams, S.: Adaptive Signal Processing. Prentice-Hall, Upper Saddle River, NJ USA (1985) 9. Koch, C.: Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, New York (1999) 10. Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. SpringerVerlag, New York (1997) 11. Gerstner, W., Kistler, W.M.: Spiking Neuron Models: An Introduction. Cambridge University Press, New York (2002) 12. McCulloch, W.S., Pitts, W.S.: A Logical Calculus of the Ideas Immanent in Nervous Activity. Mathematical Biophysics 5 (1943) 18–27 13. Sinha, M., Chaturvedi, D.K., Kalra, P.K.: Development of Flexible Neural Network. Journal of The Institution of Engineers (India) 83 (2002) 14. Feng, J., Buxton, H., Deng, Y.C.: Training the Integrate-and-Fire Model with the Informax Principle I. Journal of Physics–A 35 (2002) 2379-2394 15. Feng, J., Sun, Y., Buxton, H., Wei, G.: Training Integrate-and-Fire Neurons with the Informax Principle II. IEEE Trans. on Neural Networks 14(2) (2003) 326–336 16. Feng, J., Li, G.: Neuronal Models with Current Inputs. Journal of Physics– A 34 (2001) 1649-1664 17. Liu, S.C., Douglas, R.: Temporal Coding in a Silicon Network of Integrate-and-Fire Neurons. IEEE Trans. on Neural Networks 15(5) (2004) 1305–1314 18. Ermentrout, G.B.: Type I Membranes, Phase Resetting Curves, and Synchrony. Neural Computation 8(5) (1996) 979-1001 19. Ermentrout, G.B., Kopell, N.: Parabolic Bursting in an Excitable System Coupled with a Slow Oscillation. SIAM Journal of Applied Mathematics 46(2) (1986) 233-253 20. Latham, P.E., Richmond, B.J., Nelson, P.G., Nirenberg, S.: Intrinsic Dynamics in Neuronal Networks I. The Journal of Neurophysiology 83(2) (2000) 808-827 21. www.cs.colostate.edu/eeg/

Manifold Learning of Vector Fields Hongyu Li and I-Fan Shen Department of Computer Science and Engineering, Fudan University, Shanghai, China {hongyuli, yfshen}@fudan.edu.cn Abstract. In this paper, vector field learning is proposed as a new application of manifold learning to vector field. We also provide a learning framework to extract significant features from vector data. Vector data containing position, direction and magnitude information is different from common point data only containing position information. The algorithm of locally linear embedding (LLE) is extended to deal with vector data. The learning ability of the extended version has been tested on synthetic data sets and experimental results demonstrate that the method is very helpful and promising. Manifold features of vector data obtained by learning methods can be used for next work such as classification, clustering, visualization, or segmentation of vectors.

1 Introduction Whether at optical flow analysis in computer vision, surface reconstruction in computer graphics or flow simplification in computational fluid dynamics (CFD), vector data is the primary research object at all times. Vector data different from common point data includes direction and magnitude information other than position information. Just since much more information can be obtained from them, vector data is more worth studying and analyzing than common point data. In recent years, many efficient and feasible methods have been proposed to deal with vector data. Telea and Wijk ([1]) developed a bottom-up approach for clustering vector data. Heckel et al. ([2]) considered a top-down approach to segment vector fields using principal component analysis. Garcke et al. ([3]) proposed a continuous clustering method for vector field based on diffusion. A modified normalized cut algorithm was extended successfully for hierarchical vector field segmentation by Chen et al. in [4]. Li et al. ([5]) proposed a novel approach to partition 2D discrete vector fields based on discrete Hodge decomposition. The basic principle of these methods essentially is to learn or mine significant information from a set of vector data which is called as vector field learning(VFL) by us. The problem of vector field learning has become a significant concern for the exploratory study of vector data ([1, 4, 6, 7]). In this paper, we in detail explain the concept of vector field learning and propose a basic learning framework. In particular, we extend the classic nonlinear manifold learning method–locally linear embedding (LLE, [8]) to a modified version for vector data. The remainder of the paper is divided into the following parts. Section 2 introduces the concept of vector field learning and designs the extended version of LLE to deal with vector data. In section 3, manifold features of vector data are completely analyzed J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 430–435, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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with a synthetic data, and some experimental results are presented in section 4. Finally, section 5 ends with some conclusions.

2 Manifold Learning of Vector Fields In differential geometry, a vector field is defined as a map f : Rn → Rn that assigns each x in Rn a vector function f(x). For the purpose of application, discrete vector fields are considered more in general. For instance, the 2-D discrete vector field shown in Fig. 1 corresponds to a map function, f(x, y) = {−y, x}. Discrete points in the figure are sampled from boundaries of three circles with a common center, and the vector at each point represents the tangent vector at the current position. A discrete vector field is essentially composed of a set of vectors. Vector data is different from common point data in the ability of representation: given a set of common points, the geometrical meaning represented by them is usually indefinite; if a vector is additionally provided at each point, the implicit meaning of the set usually is totally definite. Just like in Fig. 1, if only considering 2-D Euclidean coordinates of points, it is difficult for us to demonstrate that those points are obtained from the boundaries of three circles. But, when vector data is given at each point, the meaning represented by the set of data becomes clear and definite. Vector fields can provide us more useful information to study. Vector filed learning aims to extract that significant information contained implicitly in vector data. In this paper, we use the classical manifold learning method–locally linear embedding (LLE)– as the method of feature extraction from vector field. Moreover, we need a similarity measure ([7]) to judge the affinity among vectors and a vector field space to extend the LLE algorithm. 2.1 Extended Locally Linear Embedding Traditional linear learning methods such as PCA and MDS ([9, 10]) usually work poor for data with nonlinear distribution. In last years, locally linear embedding (LLE) grad-

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ually became popular because of its simplicity and better performance for nonlinear manifolds. The original LLE method, however, is unable to deal with vector data. We need to modify it in the first step. In the modified version, determining the neighborhood of any vector is no longer based on Euclidean distance, but the similarity measure proposed in [7]. Given n points xi with vectors vi , the similarity between xi and xj can be described as follows, s(xi , xj ) = αe−d(xi ,xj ) + βe1−a(vi ,vj ) + γe−m(vi ,vj )

(1)

where d(xi , xj ) represents Euclidean distance between two points, a(vi , vj ) the angular distance between two vectors, m(vi , vj ) the difference between magnitudes of vectors, and α + β + γ = 1. As for the exact definition of parameters, please refer to the paper [7]. Those vectors with largest mutual similarity are considered as nearest neighbors. Besides, we replace Euclidean space with vector field space to extract features of vector data. In general, for a vector field constructed on a space Rn , its corresponding vector field space should be of (2 × n+ 1) dimensions which include n-dimensional Euclidean coordinate, n-dimensional cosine of angle between vectors and coordinate axes and 1dimensional magnitude of vectors. The modified LLE algorithm is briefly introduced as follows: 1. Given N points xi with vectors vi , constructing a vector field space τ ; 2. Finding k most similar neighbors for each vector data in terms of the similarity measure (1); 3. Calculating the weights wij on the vector field space τ by minimizing the reconstruction errors,   ε1 (W ) = |τi − wij τij |2 , i

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4. Finding coordinates ξi in a low-dimensional feature space by minimizing another error function,   ε2 (Ξ) = |ξi − wij ξij |2 . i

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3 Manifold Features of Vector Data For a 2-D vector field, we usually extend this 2-D space to a 5-D complex space where significant features about vector data are easier to be discovered. Applying the extended LLE to the vector field shown in Fig.1, we can mine some meaningful information and test the effectiveness and learning ability of our method. The learning results are shown in Fig.2 where the left and right panels respectively represent the first and second features extracted by our method. As one can see from the figure, the first feature only contains three values each of which corresponds to a set of vector data got from the same circle. In essence, the value on this feature is closely related with the radius of circle: when the radius of circle is changed, the feature value corresponding to the circle will also vary along. The second feature explains the periodicity of data: it actually implies the variation of central angle.

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If both features are simultaneously shown in a panel, we can get the result in Fig.3 where three data sets linearly distribute in order on such a feature space. The result is very helpful for next vector clustering or other applications.

4 Experimental Results Another example is provide in Fig.4. The vector data is composed of two sets which are randomly sampled on two circular loops respectively. On the 2-D Euclidean space, since

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data distributes on surfaces of nonlinear manifolds just like in Fig.4(a), it is difficult to find an appropriate measure to cluster vectors. But if a feature space where data is with linear distribution was discovered, the process of clustering could become simple and doable. When the extended LLE algorithm is applied to this set of data, two significant features can be extracted respectively shown in Figs.4(c) and 4(d): the first feature still corresponds to a variable related with the radius of circle, and the second one tells us the periodicity of variation of data. These features span a feature space shown in Fig.4(b), where vector data scatters in two linear regions. On such a space, we can easily find a decision boundary to separate two sets of data. Although vector data in Fig.4 is affected by noise which results that direction and magnitude of vectors are kind of changed, learning effects of our method obviously are not destroyed which proves that the method is robust to a little of noise. What’s more, from the figure, one can also find that the method is not related to the regularity of sampling. Even if in the situation of irregular sampling, the method can still work well and extract significant features.

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5 Conclusions In this paper, we propose the concept of vector field learning and provide a learning framework to extract significant features from vector data. Vector data different from common point data contains direction and magnitude information other than position information. We extend the locally linear embedding (LLE) algorithm to make it suitable for vector data. The learning ability of the extended method has been tested on synthetic data sets and experimental results demonstrate the merits of such approach. Manifold features of vector data extracted by learning methods can be used in the field of classification, clustering, visualization, or segmentation of vectors, which is the topic of our future studies.

Acknowledgments This research work is supported by the National Natural Science Foundation of China under Grant No. 60473104.

References 1. Telea, A.C., Wijk, J.J.V.: Simplified representation of vector fields. In Ebert, D., Gross, M., Hamann, B., eds.: Proc. of IEEE Visualization 1999. (1999) 35–42 2. Heckel, B., Weber, G., Hamann, B., Joy, K.I.: Construction of vector field hierarchies. In: Proc. of IEEE Visualization 1999. (1999) 19–25 3. Garcke, H., Preusser, T., Rumpf, M., Telea, A., Weikard, U., Wijk, J.J.V.: A phase field model for continuous clustering on vector fields. IEEE Trans. Visualization and Computer Graphics (2001) 230–241 4. Chen, J.L., Bai, Z., Hamann, B., Ligocki, T.J.: A normalized-cut algorithm for hierachical vector field data segmentation. In: Proc. of Visualization and Data Analysis 2003. (2003) 5. Li, H., Chen, W., Shen, I.F.: Segmentation of discrete vector fields. IEEE Transaction on Visualization and Computer Graphics (2006) (to appear). 6. Li, H., Chen, W., Shen, I.F.: Supervised learning for classification. In: FSKD (2). Volume 3614 of Lecture Notes in Computer Science. (2005) 49–57 7. Li, H., Shen, I.F.: Similarity measure for vector field learning. In: ISNN 2006. Lecture Notes in Computer Science (2006) (to appear). 8. Roweis, S., Saul, L.: Nonlinear dimension reduction by locally linear embedding. Science 290 (2000) 2323–2326 9. Borg, I., Groenen, P.: Modern Multidimensional Scaling. Springer-Verlag (1997) 10. Tipping, M.E., Bishop, C.: Mixtures of probabilistic principal component analyzers. Neural Computation 11 (1999) 443–482

Similarity Measure for Vector Field Learning Hongyu Li and I-Fan Shen Department of Computer Science and Engineering, Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai, China {hongyuli, yfshen}@fudan.edu.cn

Abstract. Vector data containing direction and magnitude information other than position information is different from common point data only containing position information. Those general similarity measures for point data such as Euclidean distance are not suitable for vector data. Thus, a novel measure must be proposed to estimate the similarity between vectors. The similarity measure defined in this paper combines Euclidean distance with angle and magnitude differences. Based on this measure, we construct a vector field space on which a modified locally linear embedding (LLE) algorithm is used for vector field learning. Our experimental results show that the proposed similarity measure works better than traditional Euclidean distance.

1 Introduction With the increase in the precision of computational simulation and Computational Fluid Dynamics (CFD), very large vector data sets become readily available. As a result, the problem of vector field learning became a significant concern for the exploratory study in visualizing vector data ([1]). The main task of vector filed learning is to extract significant features contained implicitly in vector fields. Once such features are found with learning methods such as principal component analysis (PCA, [2]) and locally linear embedding (LLE, [3]), we can easily cluster or classify vector data ([4, 5, 6, 7]), segment ([8]) and simplify ([9, 10]) vector fields on the feature space. Vector data different from common point data includes direction and magnitude information other than position information. Thus, methods for dealing with vector data should be different from those for common point data. The traditional method to compute the similarity between point data is based on Euclidean distance which is obviously inappropriate to vector data. It is necessary for us to define a new similarity measure to study the affinity between vector data. The measure proposed in the paper combines the direction and magnitude information with Euclidean distance in the form of linear combination which corresponds to a vector field space(VF space). On this space, significant features of vector data can be easily extracted using a modified LLE algorithm. The remainder of the paper is divided into the following parts. Section 2 introduces a similarity measure of vector data. In section 3, a VF space is constructed for vector field learning. Experimental results will be presented in section 4. Finally, section 5 ends with some conclusions. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 436–441, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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2 Similarity Measure of Vector Data For common point data (for example, Fig. 1(a)), the neighborhood selection is simply based on Euclidean distance in general. The measure is possibly inappropriate to vector data (for example, Fig. 1(b)) due to the appearance of other important information except spatial information. 1.5 1

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which is essentially a Gauss kernel. d(pi , pj ) represents Euclidean distance between pi and pj , the scaling parameter σ1 is a positive real constant. Under this measure, nearby points with smaller Euclidean distance are assigned higher similarity. Obviously the neighborhood of each point is completely determined by Euclidean distance. But a problem will appear in some cases: when data points are sampled from several different but close manifolds, those points with smaller distance are possibly got from different manifolds. That is, those data points in the local neighborhood of a certain point should actually belong to different sets. If such neighborhood is used for learning, the effects could be poor. For example, data points (p1 , . . . , p7 ) in Fig. 2 only containing spatial coordinates are sampled from two 1-d manifolds. Although the spatial distance between p1 and p7 is smaller than that between p1 and p2 , p1 and p2 actually belong to the same class and have higher similarity. Clearly it is not beneficial for extracting features of manifolds if purely using Euclidean distance to determine the similarity measure.

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Fig. 2. Angular difference between vectors to compute the similarity

2.2 Angular Difference If vector information at each point are provided like in Fig. 2, the most similar point to p1 is clearly p2 , not p7 since vectors at p1 and p2 have closer direction. Therefore, the angle information should also be included in the similarity measure. Given a set of points with vector information, the similarity between any two points pi and pj can be rewritten in the form of linear weighting, s(pi , pj ) = αe−d(pi ,pj ) + βe1−a(pi ,pj )

(2)

where α and β are positive real constants, and α + β = 1. a(pi , pj ) = vi · vj /(|vi ||vj |) represents the angular distance between pi and pj where vi is the vector at point pi and | · | denotes the norm of a vector. The smaller is the angle between vectors at two points, the larger is the similarity between two point data. In Fig. 2, points (p1 , · · · , p7 ) should be divided into two sets: (p1 , · · · , p4 ) belong to set 1; (p5 , · · · , p7 ) to set 2. According to the measure (2), there is clearly higher intrinsic similarity within points of either set, and lower similarity between points of different sets. 2.3 Magnitude Difference Besides Euclidean and angular distances, magnitude is an important factor influencing similarity between data. It should also be incorporated in the measure. So formula (2) can be reorganized as follows, s(pi , pj ) = αe−d(pi ,pj ) + βe1−a(pi ,pj ) + γe−m(pi ,pj )

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2.4 Analysis In this paper we will use the similarity measure (3). The measure contains a fact that the similarity becomes the largest at 1 when i = j. Furthermore, when α (β or γ) increases gradually from 0 to 1, the importance of the role Euclidean distance (angle or magnitude) plays in the measure is increasingly built up until the measure is completely dependent on the component when the parameter attains 1. Note the similarity measure (3) is specially designed for dealing with vector data. It is actually built on a vector field space different from traditional Euclidean space on which Euclidean distance is based.

3 Vector Field Learning 3.1 Construction of Vector Field Space From n points pi = {xi , yi } with vector information vi = {vix , viy }, we can construct a vector field space τ where τi = (αxi , αyi , βκxi , βκyi , γmi ). mi = |vi | represents the vy vx magnitude of vector, κxi = mii and κyi = mii respectively denote the cosine of the angle between vi and the coordinate axis, and the weights α, β and γ is equivalent to those in (3). Note if the measure (3) is used in a learning method, the learning process must work on a vector field space τ . 3.2 Nonlinear Learning Traditional linear learning methods such as PCA and MDS work poor for data with nonlinear distribution. In recent years, locally linear embedding (LLE) became more popular because of its simplicity and better performance for nonlinear manifolds. The original LLE method, however, is not suitable for dealing with vector data. We need to modify the method in the first step. In the modified version, the criterion of determining the neighborhood of any data is not Euclidean distance, but the similarity measure (3). Those points with largest mutual similarity are considered as nearest neighbors. Besides, we replace Euclidean space with vector field space to extract features of vector data. The modified LLE algorithm for vector field is discussed in [11] in detail.

4 Experimental Results The point data shown in Fig. 1(a) is actually sampled from a 1-d dimensional manifold. However, the reconstructed curve from such data set is not always with the similar shape to the original manifold if only the position information of points is provided. Fig. 3(a) shows one of possible curves reconstructed from the point data. If the tangent vector at each point is provided like in Fig. 1(b), it is clear that the geometrical structure of such a set of data is unique and definite as in Fig. 3(b). If the proposed similarity measure (3) is applied into a set of vector data (Fig. 1(b)) to estimate the neighborhood of each vector, the significant feature of the set of vector data can be effectively found using the modified LLE algorithm and the original manifold curve can be appropriately unfolded on the feature space. Fig. 4(b) shows the extracted

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feature along which vector data on the 1-d manifold is arranged in good order. However, if we use Euclidean distance as the measure and execute the original LLE algorithm on Euclidean space, the result will become disorderly. Fig.4(a) shows the result based on Euclidean distance, where vector data appears orderless.

5 Conclusions Vector data distributed on a low-dimensional manifold provides us more geometric structure information of data in comparison with common point data. To determine

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the neighborhood of each vector data appropriately, a novel measure different from Euclidean distance is proposed to estimate the similarity of vectors. Besides, we design a vector field space to incorporate vector information on which the modified LLE algorithm can effectively extract significant features of vector data. Furthermore, the proposed similarity measure is also applicable to the analysis of optical flow in the field of computer vision, and to the reconstruction of surface in the field of computer graphics.

Acknowledgments This research work is supported by the National Natural Science Foundation of China under Grant No. 60473104.

References 1. Scheuermann, G., Hamann, B., Joy, K.I., Kollmann, W.: Visualizing local vector field topology. SPIE Journal of Electronic Imaging 9 (2000) 356–367 2. Tipping, M.E., Bishop, C.: Mixtures of probabilistic principal component analyzers. Neural Computation 11 (1999) 443–482 3. Roweis, S., Saul, L.: Nonlinear dimension reduction by locally linear embedding. Science 290 (2000) 2323–2326 4. Li, H., Chen, W., Shen, I.F.: Supervised learning for classification. In: FSKD (2). Volume 3614 of Lecture Notes in Computer Science. (2005) 49–57 5. Chen, J.L., Bai, Z., Hamann, B., Ligocki, T.J.: A normalized-cut algorithm for hierachical vector field data segmentation. In: Proc. of Visualization and Data Analysis 2003. (2003) 6. Garcke, H., Preusser, T., Rumpf, M., Telea, A., Weikard, U., Wijk, J.J.V.: A continuous clustering method for vector fields. In Ertl, T., Hamann, B., Varshney, A., eds.: Proc. of IEEE Visualization 2000. (2000) 351–358 7. Heckel, B., Uva, A.E., Hamann, B.: Clustering-based generation of hierarchical surface models. In Wittenbrink, C., Varshney, A., eds.: Proc. of IEEE Visualization 1998 (Hot Topics). (1998) 50–55 8. Li, H., Chen, W., Shen, I.F.: Segmentation of discrete vector fields. IEEE Transaction on Visualization and Computer Graphics (2006) (to appear). 9. Tricoche, X., Scheuermann, G., Hagen, H.: A topology simplification method for 2d vector fields. In Ertl, T., Hamann, B., Varshney, A., eds.: Proc. of IEEE Visualization 2000. (2000) 359–366 10. Telea, A.C., Wijk, J.J.V.: Simplified representation of vector fields. In Ebert, D., Gross, M., Hamann, B., eds.: Proc. of IEEE Visualization 1999. (1999) 35–42 11. Li, H., Shen, I.F.: Manifold learning of vector fields. In: ISNN 2006. Lecture Notes in Computer Science (2006) (to appear).

The Mahalanobis Distance Based Rival Penalized Competitive Learning Algorithm Jinwen Ma and Bin Cao Department of Information Science, School of Mathematical Sciences and LMAM, Peking University, Beijing, 100871, China [email protected]

Abstract. The rival penalized competitive learning (RPCL) algorithm has been developed to make the clustering analysis on a set of sample data in which the number of clusters is unknown, and recent theoretical analysis shows that it can be constructed by minimizing a special kind of cost function on the sample data. In this paper, we use the Mahalanobis distance instead of the Euclidean distance in the cost function computation and propose the Mahalanobis distance based rival penalized competitive learning (MDRPCL) algorithm. It is demonstrated by the experiments that the MDRPCL algorithm can be successful to determine the number of elliptical clusters in a data set and lead to a good classification result.

1

Introduction

Clustering analysis is important not only in statistics but also in many aspects of practical applications. Analysis of clusters on a set of (unlabeled) sample data by means of mixture distribution, called mixture-model cluster analysis, is one of the most difficult problems in statistics, and its main task is to select a proper number of clusters in a sample data set. In the terminology of artificial neural networks, it is the problem of unsupervised classification on a sample data set where the number of clusters is unknown. Its importance and difficulty has been noted by many researchers (e.g., [1]-[2]). As an adaptive version of the classical k-means clustering, competitive learning (CL) has been developed from the field of neural networks and provided us a promising tool for unsupervised classification [3]. In the basic form of competitive learning, as each input sample is presented, the winning unit or neuron of the output layer of a CL network is activated and its corresponding weight vector is modified. While the rest of the units in the output layer are not activated and their corresponding weight vectors are not modified. This form of competitive learning is referred to as the simple or classical competitive learning (CCL) or winner-take-all (WTA) learning [4]. In the late of the 1980’s, it was found that this simple learning mechanism has the dead unit (or under-utilized) problem. In order to solve it, the Frequency Sensitive Competitive Learning (FSCL) algorithm was developed under the light of conscience mechanism [5]. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 442–447, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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However, these conventional competitive learning algorithms cannot be applied to solving the unsupervised classification problem on a set of sample data where the number of clusters is unknown. Xu, Krzyzak & Oja discussed this problem and showed that it is equivalent to the selection of an appropriate number of the units [6]. To tackle this problem, the Rival Penalized Competitive Learning (RPCL) algorithm was further proposed by adding a new mechanism into FSCL [6]. The basic idea is that for each input, not only the weight vector of the winner unit is modified (rewarded) to adapt to the input, but also the weight vector of its rival (the 2nd winner) is de-learned (penalized) by a smaller learning rate. When the learning and de-learning rates are appropriately selected, RPCL has the ability of automatically allocates an appropriate number of weight vectors for a sample data set, with the weight vectors of the extra units being pushed far away from the sample data. Actually, the RPCL algorithm is demonstrated well and has been applied to many fields such as clustering, vector quantization, and the training of RBF neural networks. Moreover, it has also been generalized to several versions for different types of sample data [7]. Via theoretical analysis, it has been recently proved that the RPCL algorithm can be considered as a special case of the distance sensitive RPCL (or DSRPCL) algorithms constructed by minimizing a cost function [8]. However, the distance between a sample point and a weight vector in the cost function is mainly analyzed and used in the form of the Euclidean distance, which actually limits the clusters to the spherical forms. In this paper, we use the Mahalanobis distance instead of the Euclidean distance in the cost function and propose the Mahalanobis distance based rival penalized competitive learning (MDRPCL) algorithm that can be successful to determine the number of elliptical clusters in a data set and lead to a good classification result.

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μ μ μ T μ d Given a set of sample data S = {X μ }N μ=1 with X = [x1 , x2 , · · · , xd ] ∈ R , the simple competitive learning or FSCL algorithm can be regarded as the adaptive versions of the well known k-mean algorithm, which essentially minimize the Mean Square Error (MSE) cost function as follows:

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where W = [W1 , W2 , · · · , Wk ], and each Wi = [wi1 , wi2 , · · · , wid ]T ∈ Rd is just the i-th weight vector in consideration. Moreover, we have  1 if i = c(μ) such that  X μ − Wc(μ) = minj  X μ − Wj , Miμ = 0 otherwise, with c(μ) denoting the index of the winner unit (or its weight vector) for the μth sample point.

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However, EMSE (W) cannot be used for detecting a correct number k because it decreases monotonically as the number of the units increases. Thus, it does not apply to the RPCL algorithm. Recently, Ma and Wang [8] constructed the following cost function to describe the mechanism of the RPCL algorithm: E(W) = E1 (W) + E2 (W) E1 (W)

1 1  X μ − Wc(μ) 2 , E2 (W) = 2 μ P



(2)  X μ − Wi −P ,

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where W = vec[W1 , W2 , · · · , Wn ] and P is a positive number. This new cost function is composed of the square mean error E1 and the model selection term E2 . Actually, the minimization of E2 can allocate the proper number of weight vectors to the sample data, while the minimization of E1 makes the classification of the sample data possible. It is shown by theoretical analysis and experiment in [8] that the minimization of this cost function can lead to a a correct number of weight vectors located around the centers of the clusters in the sample data, respectively, with the other weight vectors being driven away far from the sample data. However, the discussions in [8] are mainly based on the Euclidean distance for the cost function, which essentially assumes that the clusters in the sample data are spherical. To relax this constraint, we can use the Mahalanobis distance instead of the Euclidean one in the cost function. That is, the distance  Wi − X μ  is given by  X μ − Wi 2 = (X μ − Wi )T Σi−1 (X μ − Wi ),

(3)

where Σi is the covariance matrix of the cluster i assumed positive definite. Thus, the cost function given in Eq.(2) becomes a function of the parameters (W1 , Σ1 ), · · · , (Wn , Σn ). With the above preparations, we now construct the Mahalanobis distance based rival penalized competitive learning algorithm as follows. First we get the derivatives of E with respect to Wj :  j ∂E =− δc(μ) Σj−1 (X μ − Wj ) ∂Wj μ  j + (1 − δc(μ) )||X μ − Wj ||−P −2 Σj−1 (X μ − Wj ) μ,j

where δji is the Kronecker number. Then, the MDRPCL algorithm for the update of the weight vectors can be constructed as a gradient descent algorithm of the cost function as follows. (t)

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Instead of the batch mode, we can get the following adaptive algorithm for the update of the weight vectors: at current step t, we random choose a sample X μ , and have  ∂E1 −Σj−1 (X μ − Wj ), if j = c(μ); = (5) 0, otherwise, ∂Wj  ∂E2 0, if j = c(μ); = (6) μ −P −2 −1 μ ||X − W || Σ (X − W ), ∂Wj i j otherwise, j Meanwhile, we use the following update rule to modify the covariance matrix Σj at each step in the batch mode:   (t−1) (1 − η  )Σj + η  μ (X μ − Wc(μ) )(X μ − Wc(μ) )T , if j = c(μ); (t) Σj = (7) (t−1) Σj , otherwise, and in the adaptive mode:  (t−1) (1 − η  )Σj + η  (X μ − Wc(μ) )(X μ − Wc(μ) )T , if j = c(μ); (t) Σj = (t−1) Σj , otherwise.

(8)

where η  > 0 is a small positive constant number. This update rule was given by Xu [7] for the extension of the RPCL algorithm to the data sets of elliptical clusters. Although it does not follow the gradient learning rule of the cost function exactly, it can maintain a good stability on the convergence of the covariance matrices. Actually, the exact gradient learning rule often leads to a degenerate covariance matrix such that the algorithm cannot be convergent. Since the MDRPCL algorithm is a type of gradient descent algorithm, it may be possible to trap in a local minimum so that it may lead to a wrong clustering result. In order to overcome this problem, we can apply an simulated annealing mechanism to the MDRPCL algorithm in the same way as the ASRPCL algorithm given in [8]. We will use this simulated annealing method to make the classification of the wine data in the next section.

3

Experimental Results

In this section, some simulated experiments are carried out to demonstrate the performance of the MDRPCL. Moreover, the MDRPCL algorithm is applied to the classification of the wine data. 3.1

On Simulated Data Sets

We conducted two experiments on the set of samples drawn from a mixture of four bivariate Gaussians densities (i.e., d = 2), which can observed from the backgrounds of Fig.1(a)&(b), respectively. Clearly, all the four Gaussian distributions in each case were selected to be elliptical, with a certain degree

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of overlap. We implemented the batch MDRPCL algorithm on the data set given in Fig. 1(a) and the adaptive MDRPCL algorithm with the simulated annealing mechanism on the data set given in Fig. 1(b), which has a higher degree of overlap. In the both experiments, we let the parameters Wj being selected randomly and independently from the samples and took p = 0.02. The experimental results with n = 8 in the two cases are also given in Fig. 1(a)&(b), respectively. We can observe that four Gaussians based on the estimated weight vectors Wj and the covariance matrix Σj are finally located accurately at the actual Gaussians, respectively, while the other weight vectors are driven far away from the sample data. That is, the correct number of the elliptical clusters have been detected on each of these data sets. Moreover,the classification accuracies on the two data sets of Fig. 1(a)&(b) can reach at 98% and 93%, respectively. 3.2

On the Wine Data Set

Moreover, we applied the adaptive MDRPCL algorithm with the simulated annealing mechanism to the classification (or recognition) of the wine data set which is a typical real data set for testing a classification or clustering algorithm and can be downloaded from the web: http://www.ics.uci.edu/∼mlearn/MLRepository.html. Actually, it consists of 178 samples of three types of wine. Each datum is 13-dimensional and consists of chemical analysis of a sample from certain type of wine. We first regularized these wine data as into a interval of [0,3] and then apply the simulated annealing MDRPCL algorithm to solving the unsupervised classification problem of the wine data by setting k = 6 with the parameters wij being selected randomly from the interval [0,3] and let P = 0.2.

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It is shown by the experiments that the simulated annealing MDRPCL algorithm can detect the three classes in the wine data set with a classification accuracy of 100% or 99.64%(there is only one error) which is a rather good result for the unsupervised learning methods.

4

Conclusions

In this paper, we have investigated the function of the Mahalanobis distance in the special cost function associated with the RPCL algorithm and proposed a Mahalabis distance based RPCL algorithm. By relaxing the Euclidean distance into the Mahalanobis one, the proposed RPCL algorithm is able to be applied to a sample data set of elliptical clusters. It is demonstrated by the experiments that the MDRPCL algorithm can be successful to determine the number of elliptical clusters in a simulated or real data set and lead to a good classification result.

Acknowledgement This work was supported by the Natural Science Foundation of China for Project 60471054.

References 1. Galinski, R.B., Harabasz, J.A.: A Dendrite Method for Cluster Analysis. Communications in Statistics 3(1974) 1-27 2. Hartigan, J.A.: Distribution Problems in Clustering. In J. Van Ryzin (ed.): Classification and Clustering. Academic Press, New York (1977) 45-72 3. Nasrabadi, N., King, R.A.: Image Coding Using Vector Quantization: A Review. IEEE Trans. Commun. 36 (1988) 957-971 4. Hecht-Nielsen, R.: Neurocomputing. Addison-Wesley, Reading MA (1990) 5. Ahalt, S.C., Krishnamurty, A.K., Chen, P., Melton, D.E.: Competitive Learning Algorithms for Vector Quantization. Neural Networks 3 (1990) 277-291 6. Xu, L., Krzyzak, A., Oja, E.: Rival Penalized Competitive Learning for Clustering Analysis, RBF Net, and Curve Detection. IEEE Trans. Neural Networks 4 (1993) 636-649 7. Xu, L.: Rival Penalized Competitive Learning, Finite Mixture, and Multisets Clustering. In: Proc. of International Joint Conference on Neural Networks (IJCNN’98). vol.3. IEEE Press, Alaska (1998) 2525-2530 8. Ma, J., Wang, T.: A Cost Function Approach to Rival penalized competitive Learning (RPCL). IEEE Trans. Systems, Man, and Cybernetics B in Press.

Dynamic Competitive Learning Seongwon Cho1, Jaemin Kim1, and Sun-Tae Chung2 1

School of Electronic and Electrical Engineering, Hongik University, 72-1 Sangsu-dong, Mapo-gu, Seoul 121-791, Korea [email protected] 2 School of Electronic Engineering, Soongsil University, 511 Sangdo-dong, Dongjak-gu, Seoul 156-743, Korea [email protected]

Abstract. In this paper, a new competitive learning algorithm called Dynamic Competitive Learning (DCL) is presented. DCL is a supervised learning method that dynamically generates output neurons and initializes automatically the weight vectors from training patterns. It introduces a new parameter called LOG (Limit of Grade) to decide whether an output neuron is created or not. If the class of at least one among LOG number of nearest output neurons is the same as the class of the present training pattern, then DCL adjusts the weight vector associated with the output neuron to learn the pattern. If the classes of all the nearest output neurons are different from the class of the training pattern, a new output neuron is created and the given training pattern is used to initialize the weight vector of the created neuron. The proposed method is significantly different from the previous competitive learning algorithms in the point that the selected neuron for learning is not limited only to the winner and the output neurons are dynamically generated during the learning process. In addition, the proposed algorithm has a small number of parameters, which are easy to be determined and applied to real-world problems. Experimental results demonstrate the superiority of DCL in comparison to the conventional competitive learning methods.

1 Introduction Competitive learning has been used for rapid training of neural networks since it is of low computational complexity. Typical competitive learning models such as Simple Competitive Learning (SCL) [1], Frequency Sensitive Competitive Learning (FSCL) [2], [3], and Self-Organizing Feature Map (SOFM) [4], [5], [6], Learning Vector Quantization (LVQ) [7], [8], have the static structure of neural networks. Their learning performance extremely depends on the distribution of initial weight vectors and the number of output neurons, which has to be determined before learning. Contrariwise, as a typical model using a changeable structure of neural network, Adaptive Resonance Theory (ART) [9] holds the plasticity by consistently integrating the newly acquired knowledge to the whole knowledge base while keeping the previous knowledge. ART can simulate human learning behavior. However, its structure and learning algorithm is so complex and the adjustment of numerous learning parameters is so difficult, and thus it has some difficulties to apply to practical problems. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 448 – 455, 2006. © Springer-Verlag Berlin Heidelberg 2006

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Hence, in this paper, we propose a new supervised competitive learning neural network called dynamic competitive learning (DCL). In the proposed neural network, the output neurons are dynamically generated during the learning process and the initial weight vectors are automatically selected from the training patterns. The major differences between DCL and the conventional competitive learning models are the point that DCL introduces a new learning parameter, Limit of Grade (LOG), to decide the generation of output neurons and the selected neuron for learning is not limited only to the winner. In DCL, the output layer is changeable and there is a small number of parameters which have to be determined by users. The paper consists of 4 sections. In Section II, DCL is described. Experimental resu lts are presented in Section III. Section IV is conclusions.

2 Dynamic Competitive Learning 2.1 Basic Concept DCL neural network consists of 2 layers, input layer and output layer, as with the conventional competitive learning neural network. DCL is similar to SCL and FSCL in the point that only one weight vector is updated for the given training vector. However, in DCL the neurons of output layer are dynamically generated during learning and the next nearest output neurons are allowed to be updated if the class of the winning neuron with the minimum Euclidean distance is different from the class of the training vector incorrectly. The proposed method decides to generate output neurons during learning process by introducing a new learning parameter, LOG. DCL first checks whether the class of the winner is the same as the training vector or not. If the classes are identical, the winner learns. Otherwise the second nearest output neuron becomes a candidate for learning and its class is checked. The same procedure is repeated for the next nearest output neurons. LOG represents the number of the candidates for learning If all the candidates fail to learn, a new output neuron is generated and the given training vector is used as the initial weight vector for the output neuron. As learning proceeds, LOG of each class increases monotonically for preventing the number of generated output neurons to diverge. The terminologies used in this paper are as follows: t : the present learning time n : total number of the generated output neurons J : total number of classes j : index for class, 1≤ j ≤ J I : index for the output neuron, 1≤ i ≤ n x j : input vector belonging to class j

LOG j : LOG of class j ε : gain for LOG parameter n j : number of output neurons assigned to class j

w i : weight vector for output neuron i u i : labelled class of output neuron i u : vector consisting of the classes of all the output neurons, i.e.,

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u = [u1 , u 2 , L , u n ]T c r : the rth nearest output neuron (the winner is the first nearest output neuron) α(t) : learning rate at time t. 2.2 Learning Algorithm DCL algorithm is composed of parameter initialization, selection of candidates for learning, weight vector update, generation of a new output neuron, and parameter adjustment. The final weight vector and the label vector u can be obtained after all the learning and relabelling process. More detailed algorithm can be described as follows: Step 0. Initialize parameters: t = 0, n = 0 α(0) ∈ (0,1) n j = 0 for j = 1,2, L , J LOG j = 1 for j = 1,2,L , J Step 1. While stopping condition is false, do Steps 2-8. Step 2. For each training input vector xi , do Steps 3-7. Step 3. For r = 1, L , LOG j , a. Find C r such that

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b. If ucr = j , update the weight vector and go to Step 5: w cr (new) = w c (old) + α(t) ⋅ [x j − w cr (old)] c. Next r. Step 4. If u c r ≠ j for all r, then create a new output neuron and initialize the weight vector: n = n + 1, n j = n j + 1. w n (new) = x j , u n = j Step 5. Adjust limit of grade LOG j Step 6. Reduce learning rate alpha (t) Step 7. Next t ; Step 8. Test stopping condition : The condition may specify a fixed number of iterations. Step 9. Relabelling.

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In DCL the weight vector for an output neuron can be updated using several training vectors with different classes after initial labeling. Thus, after learning ends, relabeling is necessary for assigning the optimal class of the output neuron and removing dead neurons. DCL uses LOG as a parameter for output neuron generation. LOG takes the monotonically increasing form such as (1) or (2) to prevent the divergence of generated neurons. LOG j = 1 + ε ⋅ n j

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LOG parameter are associated with the number of output neurons and the gain ε is between 0 and 1 ( 0 < ε < 1 ). The equation of LOG parameter and its gain are predetermined before learning, and the number of generated neurons can be adjusted accordingly. As ε approximates to 1, the small number of output neurons is generated. In the contrary, as ε approximates to 0, relatively a large number of output neurons are generated. Especially, in case of ε = 1 in (2) the number of generated output neurons is the same with the number of classes. In case of ε = 0 , only the winner learns and DCL is similar to LVQ except that in DCL the weight vector is not updated when the training vector is classified incorrectly.

2.3 Characteristics of DCL Neural Network DCL dynamically generates output neurons during learning using the changeable structure of neural network such as in ART instead of static structure such as in SCL, FSCL, SOFM and LVQ. In this sense, it is not required to predetermine the number of output neurons and to initialize weight vectors. ART model is similar to DCL in the sense that it also has changeable structure of neural network, but ART model, which has both bottom-up and top-down vector, performs two-way learning and clusters input by using unsupervised competitive learning. Contrariwise, DCL performs only forward learning and its implementation is simple because it has only one parameter, LOG. Since DCL allows the weight vector for one of nearest output neurons to be updated or a new output neuron to be generated, it makes the generation probability of dead neurons low, and prevents the number of output neurons from diverging by introducing a monotonically increasing form of LOG. DCL constructs more sophisticated classifier by assigning many output neurons near the decision surfaces that misclassification occurs frequently.

3 Experimental Results 3.1 Recognition of Remote Sensing Data In this section, to check applicability and to demonstrate the superior performance of the DCL in the real problems, experiments with a remote sensing data were conducted in order to compare the proposed DCL with the other neural networks.

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A. Data Set The data set for the experiment is a multi-spectral earth observational remote sensing data called Flight Line C1 (FCL1) [10], [11]. The geographical location of the FCL1 is the southern part of Tippecanoe Country, Indiana, U.S.A. This multi-spectral image was collected with an airborne scanner in June 1966 at noon time. The FLC1 consists of 12 band signals. Each point of one spectral image represents one of 256 gray levels. In this experiment, only 8 spectral bands out of 12 bands are used. Each data vector is generated by concatenating eight corresponding band signal values together. 8 dominant farm products (alfalfa, corn, oats, red clover, soybean, wheat, bare soil, rye) are chosen to represent 8 particular classes. Training and testing with the networks were done with 200 signal samples per class and 375 other signal samples per class. Therefore, the total numbers of the training and the testing samples are 1600 and 3000, respectively. B. Simulation Results DCL generated 89 output neurons. In order to have a fair comparison, the number of output neurons used was 89 for FSCL and 10 x10 for SOFM. In the case of DCL, the learning rate coefficient was chosen as α(t) = 0.3 ⋅ (1 −

t ) Total number of iterations

and the gain for the LOG parameter was chosen as ε = 0.2 . In the case of FSCL and SOFM, the learning rate coefficients were chosen as α(t) = 0.9 ⋅ (1 −

t ) Total number of iterations

and the first 89 or 100 training patterns were used to initialize the weight vectors. Table 1 and Table 2 show the experimental results with the FCL1 data set. The training and testing accuracies of the DCL were 97.667% and 93.856%, respectively. The training and testing accuracies of FSCL were 94.125% and 92.217%, respectively. The training and testing accuracies of SOFM were 92.677% and 91.172%, respectively. Table 1. Recognition Rates of DCL with the FCL1 data set Epochs

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50 60 70 80 90 100 Average

82 84 84 82 89 84 84.1675

97.938 97.875 97.688 97.438 97.688 97.438 97.667

93.9 93.7 93.9 93.933 93.867 93.833 93.856

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Table 2. Recognition Rates of FSCL and SOFM with the FCL1 data set FSCL training(%) testing(%) 50 94 91.733 60 94.75 92.167 70 93.125 91.333 80 95.312 92.733 90 93.25 92.767 100 94.312 93.567 Average 94.125 92.217 Epoch

SOFM training(%) testing(%) 92.688 92.267 92.75 90.267 92.438 90.2 92.625 92 92.5 91.1 93.062 91.2 92.677 91.172

3.2 Handwritten Numeral Recognition Pattern recognition system is generally divided into 2 parts. First, feature extraction which computes the parameters that can provide the maximum information for the given pattern. Second, classifier which recognizes the object using extracted features. In the recognition of handwritten letters, we might want to recognize patterns without being influenced by distortions or transformations. For handwritten numeral recognition independent of the position or orientation of the pattern or its size, the Fourier descriptors were used as the features. A. Data Set Experimental data are 2000 handwritten Arabic numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) with various positions, sizes and orientations. They were made by 100 persons and acquired using a CCD camera and an image grabber. Each pattern has 128 x 128 pixels and 128 gray levels. 14 Fourier descriptors were computed for each pattern. Training and testing with the networks were done with 100 patterns per class and 100 other patterns per class. B. Simulation Results The number of output neurons, learning rate and LOG parameter are set to the same as experiments with remote sensing data. The initial learning rates of DCL, LVQ and SOFM were chosen as 0.3, 0.4 and 0.9, respectively. In the case of LVQ, the initialization of the weight vectors was carried out by choosing the training patterns belonging to each class. In the case of SOFM, the first 100 training patterns were used for the initial weight vectors. The recognition rates of DCL, LVQ and SOFM are shown in Tables 3 and 4. Table 3. Recognition Rate of DCL with the handwritten numeral data set Epochs 50 60 70 80 90 100 Average

No. of neurons 110 110 110 110 111 111 110.333

training(%) 97.4 97.6 97.8 97.8 97.8 97.8 97.7

testing(%) 84.4 84.6 84.6 84.4 84.6 84.6 84.533

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Table 4. Recognition Rate of LVQ and SOFM with the handwritten numeral data set LVQ SOFM training(%) testing(%) training(%) testing(%) 50 92.2 83.4 85.6 76.8 60 92 82.2 86.4 76 70 91.8 82.6 86.2 75.2 80 91.6 82.6 85.8 79.2 90 91.8 82.6 86.6 78.2 100 92 82.8 86.8 79.2 Average 91.9 82.7 86.233 77.433 Epoch

4 Conclusions We presented a new competitive learning neural network called DCL. It dynamically generates output neurons instead of determining before learning, and the initial weight vectors are selected from training patterns. DCL has the following features: 1) Learning algorithm itself includes the initialization process of weight vectors; 2) it has simple structure, learning algorithm and small number of parameters compared to the conventional neural networks with changeable structure; 3) the number of dead neurons can be reduced. For the best recognition, appropriate LOG gain needs to be chosen according to the distribution of given patterns. For the performance evaluation of the proposed method, we conducted the experiments using practical remote sensing data and handwritten numeral data, and demonstrated its superiority in comparison to FSCL, LVQ and SOFM.

Acknowledgement This work was supported by 2004 Hongik University Research Fund.

References 1. Rummelhart, D.E., McClelland, J.L.: Parallel Distributed Processing. In: Rummelhart, D.E., Zipser, D. (eds.): Feature Discovery by Competitive Learning. MA: MIT Press, Cambridge (1988) 2. Ahalt, S.C., Krishnamurthy, A.K., Chen, P., and Melton, D.E.: Competitive Learning Algorithms for Vector Quantization. Neural Networks 3 (1990) 277-290 3. DeSieno, D.: Adding a Conscience to Competitive Learning. Proc. IEEE International Conference on Neural Networks, San Diego, California (1988) I117-I124 4. Kohonen, T.: Self-Organization and Associative Memory. 3rd edn. Berlin, SpringerVerlag, Berlin Heidelberg New York (1989) 5. Kangas, J.A., Kohonen, T.K., and Laaksonen, J.T.: Variants of Self-Organizing Maps. IEEE Trans. Neural Networks l(1) (1990) 93-99

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6. Yin, F., Wang, J.: Advances in Neural Networks. Springer-Verlag, Berlin Heidelberg New York (2005) 7. Kohonen, T.: Learning Vector Quantization. Neural Networks 1(1) (1988) 303 8. Fausett, L.: Fundamentals of Neural Networks. Prentice Hall (1994) 9. Carpenter, G.A., Grossberg, S.: The ART of Adaptive Pattern Recognition by SelfOrganization Neural Network. Computer (1988) 77-88 10. Cho, S., Ersoy, O.K., Lehto, M.: Parallel, Self-Organizing, Hierarchical Neural Networks with Competitive Learning and Safe Rejection Schemes. IEEE Trans. Circuits and Systems, 40(9) (1993) 556-566 11. Cho, S.: Parallel, Self-Organizing, Hierarchical Neural Networks with Competitive Learning and Safe Rejection Scheme. Ph.D. Thesis, Purdue University (1992)

Hyperbolic Quotient Feature Map for Competitive Learning Neural Networks Jinwuk Seok1 , Seongwon Cho2 , and Jaemin Kim2 1

2

Embedded S/W Research Division, ETRI, 161, Gajeong-dong, Yoosung-gu, Daejun 305-700, Korea [email protected] School of Electronic and Electrical Engineering, Hongik University 72-1 Sangsu-dong, Mapo-gu, Seoul 121-791, Korea [email protected]

Abstract. In this paper, we present a new learning method called hyperbolic quotient feature map for competitive learning neural networks. The previous neural network learning algorithms didn’t consider their topological properties, and thus their dynamics were not clearly defined. We show that the weight vectors obtained by competitive learning decompose the input vector space and map it to the quotient space X/R. In addition, we define a quotient function which maps [1, ∞) ⊂ Rn to [0, 1) and induce the proposed algorithm from the performance measure with the quotient function. Experimental results for pattern recognition of remote sensing data indicate the superiority of the proposed algorithm in comparison to the conventional competitive learning methods.

1

Introduction

In the case of Backpropagation (BP), its object function depends on the ”Kullback Distance” obtained by ”Kullback Information”, and the stability, the convergence, and the convergence rate have been researched [1], [2]. The Hopfield model (or Chua-Lin-Kennedy Model) for the object function with some equality constraints has Lyapunov stability [3]. Genetic Algorithm (GA), which has been researched more vigorously nowadays, is proven a Markov chain. Ritter and Schuten induced the Fokker-Plank equation for Self-Organizing Feature Map (SOFM), and analyzed the convergence condition and dynamic properties with the equation [4], [5]. However, in the previous neural network algorithm analysis, no topological analysis has not been done yet except SOFM. In this paper, we analyze the topological structure of the simple competitive learning (SCL) and its performance measure. In addition, we construct a new performance measure with canonical function on the quotient space, and induce a learning equation for the hyperbolic quotient feature map neural network. The paper is organized as follows. Section 2 presents the performance measure of SCL. In Section 3, the topological information of input vector space is described. In Section 4, hyperbolic quotient competitive learning algorithm is induced. Section 5 discusses experimental results obtained with the proposed algorithm in comparison to SCL. Section 6 is conclusions J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 456–463, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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457

Performance Measure of SCL

The performance measure of SCL at time t ∈ R+ is as follows : 1  u ||x − wtr ||2 2 u r U

Jt =

W

(1)

In the above equation, xu is the u-th input vector such that xu ∈ (Rn , d), and wtr is the r-th weight vector such that wtr ∈ (Rn , d) at time t, respectively. Index u, r are the elements of the index set Λ ⊆ R+ := [0, ∞) and represent the indices of the input vector and the weight vector, respectively and satisfy 0 ≤ u ≤ U, 0 ≤ r ≤ W . Let the directional derivation ht be the gradient of Jt for the performance measure of SCL then, we obtain ht = ∇Jt = −(xu − wtr )

(2)

In equation (11), if we choose the step size (t) as a monotone decreasing function which depends on time t, then we obtain the following learning equation : r wt+1 = wtr − (t)ht = wtr + (t)(xu − wtr )

(3)

Suppose that in the above equation, the stochastic nonlinear programming criterion is used. Then the step size (t) satisfies the following condition:  t lim (t) = 0 lim (t)dt = ∞ (4) t→∞

t→∞

0

Considering the level set Lα of Jt , and suppose a separated level set Lsα ⊆ Lα satisfies the following conditions for an open -neighborhood N (p, ) = {x ∈ Rn |d(x, p) < }: ∃M < ∞ s.t. ∀y ∈ Lsα , ||y|| < M ∀x ∈

Lsα

s.t. ∀ >

0Lsα

∩ N (p, ) − {x} = ∅

(5) (6)

In addition, suppose that the performance measure Jt is locally Lipschitz and twice continuously differentiable, and for the local minima x ˆ, the Hessian of Jt , H(x) satisfies the following condition : ∃ 0 < m ≤ M < ∞ s.t. ∀ y ∈ Rn , m||y||2 ≤ y, H(ˆ x)y ≤ M ||y||2

(7)

The boundary set ∂Lsα of the level set Lsα with the limit point x ˆ has a quadratic function form. Thus, the noise term effects largely in the boundary set ∂Lsα . If Lsα is globally convex set, the norm of transfer vector ||xu − wtr || become larger and the sequence {wtr }∞ ˆ. However, the performance t=0 uniformly converges to x W measure Jt consists of the sum with the quadratic function r ||xu − wtr ||2 for each u. Thus the local minima of Jt become increasing as the index u is increasing. The competitive learning is basically one of the stochastic optimal strategies, and the weight vector may become stuck in the local minima. However, it reaches global minima with some probability.

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The Topological Information of Input Vector Space

Supposing that the finite number of the input vector exist in the input vector space (Rn , d), the hyper plane made by weight vectors wtr constructs the finite k(r) simplexes σwtr in (Rn , d) where k(r) is a function of the weight vector index r such that k(r) ∈ {x ∈ Z ++ |x ≥ 2} [6]. From the above induction, the input p vector space (Rn , d) is homopic with |K| = ∪σ∈K σw ˆr of the r . The limit point w t r ∞ sequence {wt }t=0 generated by competitive learning is one of the limit points in k(r) the simplex σwtr . Considering the topology of the input space, let the subbasis k(r)

k(r)

be σwtr − ∪∂σwtr and the open set(or space) (X, d) such that (X, d) ⊂ (Rn , d). We denote the space (X, d) as X abbreviated. Then we construct the topology for the input vector space such that k(r)

k(r)

k(r)

T = {X, ∅, ∪r (σwtr − ∪∂σwtr ), ∪∂σwtr }

(8)

On the above topology, suppose that x, x ∈ σwtr − ∪∂σwtr . Then, there exists a relation R such that R : x ∼ x . Hence, we can define a R an equivalence class k(r) k(r) R[x] = {y ∈ X|yRx} which makes σwtr − ∪∂σwtr a partial subspace denoted as X/R = {R[x]|x ∈ X} by R[x]. In addition, the input space X is a topological space by the above definition, and thus we can introduce a canonical function q(x) which maps X to X/R. We consider a function which maps X/R to level set Lα , as a new performance measure JtN combined with a canonical function q(x), then we obtain a surjective, continuous open (or closed) function. k(r)

4 4.1

k(r)

The Induction of Quotient Canonical Competitive Learning Algorithm Sub-alexandrov Compactification

In the above results, the performance measure Jt for competitive learning neural  network maps r ||xu − wtr ||2 to level set Lα defined on real line R for each u. Hence, a large noise affects on the difference of the performance measure Jt . However, if the difference of Jt is affected by the partition of X/R, i.e. if the input vector which has a large noise should not act on the performance measure and the input vector which has a small noise should do the performance measure largely. Consequently, if an algorithm has the above property, there are more simplexes in the partition X/R located in the dense cluster of input vectors. In order to locate each simplex in X/R more closely in the dense cluster of input vectors, we introduce a new performance measure JtQ which has large differential value in small JtQ and in contrast, has small differential value in large JtQ In addition, because the partition X/R is constructed by the whole weight vectors and each weight vector should locate in the center of the dense cluster of input vectors, it needs that we let the function only with the term of summation for weight vectors in performance measure. We denote the above

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 function as N et1 = r ||xu − wtr ||2 For the function N et1 defined on the [0, ∞), we want N et1 to be defined on [1, ∞). For this, let N et2 such that N et2 =  exp(N et1 /2σ 2 ) = exp( r ||xu − wtr ||2 /2σ 2 ). For the compactification, let C denote the sub-real line R[1, ∞) in Euclidean 2-space R2 , and let S denote the sphere with center (0, 0) and radius 1. The line passing through the ”north pole” pˆ = (0, 1) ∈ S and any point p ∈ C intersects the sphere S in exactly one point p distinct from pˆ. Let f : C → S be defined by f (p) = p . Then f is, in fact, a homeomorphism from the plane C which is not compact, onto the subset S − {ˆ p} of the sphere S, and S is compact. Hence S is a compactification of C. 4.2

The Induction of a New Performance Measure Using Sub-alexandrov Compactification

Let an equation for a line through a ”north pole” : pˆ and x = N etx in Figure 1 such that x y + =1 (9) N etx pˆ Since ”north pole” : pˆ = (0, 1), the equation of sphere S is the following that : x2 + y 2 = 1

(10)

Substituting (9) into (10), we obtain an equation for y which denote the compact measure JtQ for an input vector xu . y=

N et2x − 1 N et2x + 1

(11)

 Since N etx in (11) is exp( r ||xu − wtr ||2 /2σ 2 ), we obtain a new performance measure as follows :  exp( ||xu − wr ||2 /σ 2 ) − 1   t r u JtQ (x, w) = = tanh( ||xu − wtr ||2 /2σ 2 ) r 2 2 exp( ||x − w || /σ ) + 1 t r u u r (12) In equation (12), substituting σ 2 to σ 2 (t) such that σ 2 (t) ↓ 0, the Lebesgue measure of the gradient of σ 2 (t)JtQ (x, w) converges to 0 Theorem 1. The Lebesgue measure of the gradient of σ 2 (t)JtQ (x, w)converge to 0 Proof. The gradient of σ 2 (t)JtQ (x, w) is as follows  ||xu − wr ||2 t ∇(σ 2 (t)JtQ (x, w)) = −sech2 ( )(xu − wtr ) 2σ(t) r  u r 2 Let N et = r ||x −wt || . For  - neighborhood N (0, ) = {N et ∈ R+ |N et < }, Lebesuge–integrate ∇(σ 2 (t)JtQ (x, w)) as follows 

m(∇(σ 2 (t)JtQ (x, w))) ≤ m(||∇(σ 2 (t)JtQ (x, w))||) = ||

−sech2 ( N(0,)

≤ σ 2 (t)(tanh(

 ) ≤ σ 2 (t)||sup tanh(·)|| σ 2 (t)

N et )dN et|| σ 2 (t)

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Since σ 2 (t) ↓ 0, we can pick a tf ∈ R+ for an arbitrary positive number δ satisfying a following proposition. ∃tf > t ∈ R[0, t], s.t. σ(tf ) < δ Hence, the gradient of σ 2 (t)JtQ (x, w)weakly converge to Lebesuge measure with 0. (Q.E.D). 4.3

The Induction of the Quotient Canonical Feature Map Algorithm

Considering the corresponding performance measure, let the directional derivation ht be the gradient of the performance measure,  ||xu − wr ||2 t ht = ∇Jt = −sech2 ( )(xu − wtr ) 2 (t) 2σ r

(13)

Then, we obtain the following learning equation by substituting  to σ 2 (t) r wt+1 = wtr − ht  r wt+1 = wtr + sech2 ( r

||xu −wtr ||2 u 2σ2 (t) )(x

− wtr )

(14)

It can be easily verified that the sequence {wtr } converges to a limit point weakly from Theorem 1. The hyperbolic function for the gradient of the performance et measure sech2 ( σN2 (t) ) defined on the R[0, 1], it is the candidate of the canonical function for X → X/R. Therefore, we construct a canonical function as follows :   ||xu −wr ||2 xsech2 ( r 2σ2 (t)t )dx X ∀x ∈ X, q(x) =  (15)  ||xu −wtr ||2 2( sech )dx 2 r 2σ (t) X Since the function q(x) is a surjective, continuous and open(or closed) function, it is the canonical function which maps X → X/R. The proposed learning algorithm is as follows : step 0: Initialize the weight vector step 1: Find the weight vector satisfying the following for an input vector xu . wtr = minr¯||xu − wtr¯|| step 2: Compute the search direction for an input vector xu .  ||xu −wr ||2 ht = ∇Jt = −sech2 ( r 2σ2 (t)t )(xu − wtr ) step 3: Update wtr as follows :  ||xu −wr ||2 r wt+1 = wtr + sech2 ( r 2σ2 (t)t )(xu − wtr ) step 4: Replace t by t + 1. step 5: Go to step 1

Hyperbolic Quotient Feature Map for Competitive Learning

4.4

461

The Local Convergence Properties of the Proposed Algorithm

An important issue in analysis of learning algorithms for neural networks is the well-behaveness in a suitably defined sense. From the practical viewpoint, this issue is very important because disturbances such as noise or error are always present in a real system to knock it out of equilibrium. Therefore, in this section, we discuss the stability of the proposed algorithm, and prove that the proposed algorithm has the well-behavedness property. Definition 1. Directional Derivation ht is consisted of the as following: ht = λt (xμt − wtr ) λt = ∇(σ (t)J 2

Q

t)

·

(xμt

−

wtr )−1

R = sech ( r 2

(16) r 2 ||xμ t −wt || 2σ(t)2 )

(17)

Theorem 2. The sequence {wtr }∞ t=0 produced by the proposed algorithm is the Cauchy sequence in a measure. Proof. For an arbitrary positive n satisfying 0 < n < ∞, r r r r ||wtr − wt−n || = ||wt−1 + λt−1 (xμt−1 − wt−1 )χ(xut−1 , B o ) − wt−n || r r r r r r = ||wt−1 − wt−2 + wt−2 · · · −wt−n−1 + wt−n−1 − wt−n μ r u o +λt−1 (xt−1 − wt−1 )χ(xt−1 , B )||  μ r u o = || n−1 k=0 λt−k (xt−k − wt−k )χ(x t−k , B )||

(18)

r Hence, we obtain m{||wtr − wt−n || ≥ } as follows: r r r m{||wtr − wt−n || ≥ } = m{wt−n |wt−n ∈ / B o (wtr , )}

(19)

Therefore,  r r r r m{wt−n |wt−n ∈ / B o (wtr , )} ≤ Bo (wr ,) ||wtr − wt−n ||dwt−k t  n−1 r u o r ≤ Bo (wr ,)C k=0 ||λt−k (xμ t−k − wt−k )|| · ||χ(x t−k , B )||dwt−k n−1t  μ r r ≤ k=0 Bo (wr ,)C ||λt−k (xt−k − wt−k )|| dwt−k t   n−1  Q 2 r 2 Net = n−1 k=0 B o (wr ,)C ||∇w σ (t)Jt−k || dwt−k = k=0 B o (wr ,)C ||sech ( σ 2 (t−k) )||dN et t

t

(20) By Theorem 1, the above Lebesgue integral value has the following range.  n−1  N et N et 2 2 k=0 B o (wtr ,)C ||sech ( σ2 (t−k) )||dN et ≤ Rn [0,∞) ||sech ( σ2 (t−k) )||dN et n−1 2 ≤ k=0 σ (t − k) ≤ nσ 2 (t − k) (21) Since limt→∞ σ(t) = 0 and σ(t) ↓ 0 and n is a finite value, there exist a suitable  r n0 ≤ t − k such that 0 < σ(n0 ) < n , we obtain m{||wtr − wt−n || ≥ } ≤ . r ∞ Hence, {wt }t=0 is the Cauchy sequence in a Lebesgue measure. (Q.E.D) In Theorem 2, the proposed algorithm generates the limit point w∗r of {wtr }∞ t=0 Moreover, we simply note that the limit point w∗r is a local minimizer of JtQ ∈ C ∞.

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Fig. 1. Sub-Alexandrov compactification Table 1. The experiment for FLC1 8 class data Epoch 100 200 300 400 500 600 700 800 900 1000 Average Variance

5

The conventional SCL (%) Trainig Testing 90.19 88.63 87.56 83.3 90.81 87.93 91.12 90.23 89.75 89.2 88.94 86.27 91.69 88.93 90.56 87.37 88.13 83.67 88.38 86.87 89.71 87.24 1.33 2.18

The proposed Training 91.38 90.25 90.69 91.69 90.06 89.38 90.63 90.69 90.38 90.81 90.59 0.62

Algorithm (%) Testing 88.27 87.43 88.53 89.8 88.53 86.1 87.97 88.7 89.03 87.8 88.22 0.95

Experimental Results

The data set for the experiment is multispectral earth observational remote sensing data called ”Flight Line C1” (FLC1). The Geometrical location of the FLC1 is the southern part of Tippecanoe country, Indiana, U.S.A. The FLC1 consists of 8 bands signals. Each band signal value represents 256 gray levels, requiring a minimum of 8 bit for binary representation. 8 farm products were chosen to represent 8 classes (alfalfa, corn, oats, red clover, soybean, wheat, bare soil, rye). Training and testing of the networks were done with 200 signal samples per class and 375 other signal samples per class. For the 4 class data, 20 weight vectors were used, and for the 8 classes data, 40 weight vectors respectively. The weight vector initialization were carried out by choosing 20 or 40 input vectors randomly. The learning rates (t) in the conventional SCL algorithm and the proposed algorithm σ 2 (t) are as follows : (t) = 0.9 · (1 − σ 2 (t) = 10.0 · (1

t T otal number of Iteration ) − T otal numbert of Iteration )

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Conclusions

In this paper, we presented a canonical function which maps X → X/R based on the analysis of the topological information in the input vector space. A new performance measure by our topological analysis was defined and the learning equation for the quotient canonical feature map neural network was induced. In addition, we proved the convergence of proposed algorithm by the analysis of the gradient of our induced performance measure that converges weakly to Lebesgue measure 0. Experimental results for the remote sensing data recognition showed that the proposed algorithm produced higher classification accuracy.

Acknowledgement This work was supported by 2004 Hongik University Research Fund.

References 1. Cox, D.R., Hinskley, D.V., Reid, N., Rubin, D.B., Silverman, B.W.: Networks and Chaos - Statistical and Probabilitic Aspects. Chapman & Hall, New York (1993) 2. Yin, F., Wang, J.: Advances in Neural Networks. Springer-Verlag, Berlin Heidelberg New York (2005) 3. Hopfield, J.J., Tank, D.W.: Neural Computation of Decisions in Optimization Problems. Biol. Cybern. 52 (1985) 141-152 4. Ritter, H., Schulten, K.: The Convergence Properties of Kohonen’s Topology Conserving Maps : Fluctuations, Stability, and Dimension Selection. Biol. Cybern. 60 (1988) 59-71 5. Gihman, I.I., Skorohod, A.V.:The Theory of Stochastic Process. 2nd edn. SpringerVerlag, Berlin Heidelberg New York (1980) 6. Luenberger, D.G.: Optimization by vector space methods. John-Wiley and Sons, New Jersey (1968)

A Gradient Entropy Regularized Likelihood Learning Algorithm on Gaussian Mixture with Automatic Model Selection Zhiwu Lu1 and Jinwen Ma2 1

2

Institute of Computer Science & Technology of Peking University, Beijing 100871, China [email protected] Department of Information Science, School of Mathematical Sciences, and LMAM, Peking University, Beijing 100871, China [email protected]

Abstract. In Gaussian mixture (GM) modeling, it is crucial to select the number of Gaussians for a sample data set. In this paper, we propose a gradient entropy regularized likelihood (ERL) algorithm on Gaussian mixture to solve this problem under regularization theory. It is demonstrated by the simulation experiments that the gradient ERL learning algorithm can select an appropriate number of Gaussians automatically during the parameter learning on a sample data set and lead to a good estimation of the parameters in the actual Gaussian mixture, even in the cases of two or more actual Gaussians overlapped strongly.

1

Introduction

As a typical statistical model, Gaussian mixture (GM) has been widely used in a variety of practical applications. Although there have been several statistical or unsupervised learning approaches to solving this problem, such as the k-means algorithm [1] and the EM algorithm [2], the number k of Gaussians in the data set is usually assumed to be pre-known. Unfortunately, in many instances, this key information is not available, and then we have to select the number of Gaussians or mixture model before or during parameter estimation. The traditional approach to solve this kind of model selection problem is to choose the optimal number k ∗ of Gaussians with some statistical criteria such as the Akaike’s information criterion [3] and the Bayesian inference criterion [4]. However, since we need to repeat the entire parameter estimation at a number of different values of k, the process of evaluating these criteria incurs a large computational cost. Some more efficient approaches have also been developed on Gaussian mixture with automatic model selection using a mechanism that an appropriate number of Gaussians can be automatically selected during parameter learning, with the mixing proportions of the extra mixture components being reduced to zero. Under the Bayesian Ying-Yang (BYY) harmony learning theory [5, 6], a gradient BYY harmony learning algorithm [7] was proposed for Gaussian mixture modeling, which can make model selection automatically with parameter estimation. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 464–469, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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However, the number of Gaussians can not be detected accurately by this BYY harmony learning algorithm when two or more Gaussians are overlapped in a high degree. In this paper, we propose a gradient entropy regularized likelihood (ERL) learning algorithm to solve the model selection problem for Gaussian mixture in light of regularization theory [8]. It is demonstrated by the simulation experiments that the proposed gradient ERL learning algorithm can automatically determine the number of actual Gaussians in the sample data with a good estimation of the parameters in the original mixture in the mean time. Moreover, even when two or more actual Gaussians are overlapped in a relatively higher degree, the gradient ERL learning algorithm can still make model selection appropriately and outperforms the gradient BYY harmony learning algorithm.

2

Gradient Entropy Regularized Likelihood Learning Algorithm

We consider the following Gaussian mixture model: p(x | Θ) =

k 

αl p(x | θl ),

l=1

p(x | θl ) =

k 

αl = 1,

αl ≥ 0,

(1)

l=1

1 exp(−(1/2)(x − ml )T Σl−1 (x − ml )), (2π)n/2 |Σl |1/2

(2)

where n is the dimensionality of x, k is the number of Gaussians, and p(x | θl ) are Gaussian densities with parameters of mean vectors and covariance matrices, that is, θl = (ml , Σl ) for l = 1, ..., k. Given a sample data set S = {xt }N t=1 generated from a Gaussian mixture model with k ∗ actual Gaussians and setting k ≥ k ∗ , the negative log-likelihood function on the Gaussian mixture model p(x | Θ) is given by L(Θ) = −

N k 1   ln( (p(xt | θl )αl )). N t=1

(3)

l=1

The well-known maximum likelihood (ML) learning is just an implementation of minimizing L(Θ). With the posterior probability that xt arises from the l-th Gaussian component P (l | xt ) = p(xt | θl )αl /

k 

p(xt | θj )αj ,

(4)

j=1

we have the discrete Shannon entropy of these posterior probabilities for the sample xt E(xt ) = −

k  l=1

P (l | xt ) ln P (l | xt ),

(5)

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which is globally minimized at P (l0 | xt ) = 1, P (l | xt ) = 0(l  = l0 ),

(6)

that is, the sample xt is totally classified into or subject to the l0 -th Gaussian component. We now consider the average or mean entropy over the sample set S: E(Θ) = −

N k 1  P (l | xt ) ln P (l | xt ), N t=1 l=1

(7)

and use it to regularize the log likelihood function by H(Θ) = L(Θ) + γE(Θ),

(8)

where γ > 0 is the regularization factor. That is, E(Θ) is a regularization term to reduce the model complexity such that the Gaussian mixture can be made as simple as possible by minimizing H(Θ). In order to solve the minimization problem of H(Θ) without constraint conditions, we utilize the following substitution: αl = exp(βl )/

k 

exp(βj ).

(9)

j=1

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(10)

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(11)

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(12)

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and η is the learning rate which is usually selected as a small positive constant. As compared with the EM algorithm for Gaussian mixture, this gradienttype learning algorithm implements a regularization or penalization mechanism on the mixing proportions during the parameter learning process, which thus leads to the automatic model selection. The regularization factor is generally selected by experience.

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Simulation Results

In order to test the performance of the gradient ERL learning algorithm, several simulation experiments are carried out on eight different sets of sample data generated from Gaussian mixtures. Moreover, some comparison is made between the gradient ERL learning algorithm and the BYY harmony learning algorithm, especially in the cases of two or more Gaussians overlapped strongly. The eight sets of sample data for the simulation experiments are drawn from a mixture of four or three bivariate Gaussian densities (i.e., n = 2). As shown in Fig.1, each sample data set is generated at different degree of overlap among the clusters(i.e., Gaussians) in the Gaussian mixture by controlling the mean vectors and covariance matrices of the Gaussian distributions, and with equal or unequal mixing proportions of the clusters in the mixture by controlling the number of samples from each Gaussian density. For a sample data set with k ∗ clusters or Gaussians, we implement the gradient ERL algorithm with k ≥ k ∗ and η =0.1. Moreover, the other parameters are initialized randomly within certain intervals. In all the experiments, the algorithm is stopped if |ΔH| < 10−6 . Typically, we give the experimental results of the gradient ERL learning algorithm on the sample data set given by Fig.1(d) where k ∗ = 4. For two different regularization factors γ = 1.0 and γ = 0.5, the experimental results of the algorithm with k = 8 are shown in Fig.2(a) and Fig.2(b), respectively. We can observe that four Gaussian components are locally located accurately, while the mixing proportions of the other four Gaussian components are reduced to below 0.001, i.e, these Gaussian components are extra and can be discarded. Hence, the correct number of the Gaussians has been detected on this data set. Moreover, we can observe that the smaller the regularization factor γ is, the more slowly the ERL algorithm converges to a reasonable solution. We also make a comparison with the BYY harmony learning algorithm on the sample set from Fig.1(h), which contains three Gaussians (i.e., k ∗ = 3) with two ones overlapped in a high degree. Additionally, the three Gaussians have 4 3

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different numbers of samples, which makes it harder to learn on this sample set. Both the gradient ERL and BYY learning algorithms are implemented on this sample set with k = 8. As shown in Fig.3(a), the number of Gaussians is not correctly detected by the gradient BYY algorithm. However, the gradient ERL learning algorithm with γ = 0.3 detects the three Gaussian components correctly shown in Fig.3(b), with the five extra Gaussian components being canceled automatically once their mixing proportions are reduced to below 0.001. Therefore, the ERL learning algorithm can be considered to be more powerful than the BYY harmony learning algorithm on the model selection. The further experiments on other sample sets have also been made successfully for the correct number detection in the similar cases. Additionally, we also find that the ERL learning algorithm converges with a lower average error less than

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0.1 between the estimated parameters and the true parameters in the original mixture.

4

Conclusions

We have investigated the automatic model selection and parameter estimation on Gaussian mixture via a gradient entropy regularized likelihood (ERL) learning algorithm. The simulation experiments have demonstrated that our proposed gradient ERL learning algorithm can detect the number of Gaussians automatically during parameter learning, with a good estimation of the parameters in the original mixture in the mean time, even on the sample set with a high degree of overlap and then can outperform the BYY harmony learning algorithm on ceratin aspect.

References 1. Devijver, P. A., Kittter J.: Pattern Recognition: A Statistical Approach. Prentice Hall, Englewood Cliffs, N. J. (1982) 2. Render, R.A., Walker, H.F.: Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Review 26(2) (1984) 195–239 3. Akaike, H.: A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control 19(6) (1974) 716–723 4. Schwarz, G.: Estimating the Dimension of a Model. The Annals of Statistics 6(2) (1978) 461–464 5. Xu, L.: BYY Harmony Learning, Structural RPCL, and Topological Self-Organizing on Mixture Modes. Neural Networks 15(8–9) (2002) 1231–1237 6. Lu, Z., Cheng, Q., Ma, J.: A Gradient BYY Harmony Learning Algorithm on Mixture of Experts for Curve Detection. Lecture Notes in Computer Science 3578 (2005) 250–257 7. Ma, J., Wang, T., Xu, L.: A Gradient BYY Harmony Learning Rule on Gaussian Mixture with Automated Model Selection. Neurocomputing 56 (2004) 481–487 8. Dennis, D.C., Finbarr, O.S.: Asymptotic Analysis of Penalized Likelihood and Related Estimators. The Annals of Statistics 18(6) (1990) 1676–1695

Self-organizing Neural Architecture for Reinforcement Learning Ah-Hwee Tan Intelligent Systems Centre and School of Computer Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798 [email protected]

Abstract. Self-organizing neural networks are typically associated with unsupervised learning. This paper presents a self-organizing neural architecture, known as TD-FALCON, that learns cognitive codes across multi-modal pattern spaces, involving states, actions, and rewards, and is capable of adapting and functioning in a dynamic environment with external evaluative feedback signals. We present a case study of TDFALCON on a mine avoidance and navigation cognitive task, and illustrate its performance by comparing with a state-of-the-art reinforcement learning approach based on gradient descent backpropagation algorithm.

1

Introduction

Reinforcement learning [3] is a learning paradigm wherein an autonomous system learns to adjust its behaviour according to feedback from the environment. This paper describes a natural extension of Adaptive Resonance Theory (ART) [2] for developing an integrated reinforcement learner that adapts its behaviour through interacting with the environment. The proposed neural architecture, known as FALCON (Fusion Architecture for Learning, Cognition, and Navigation), learns multi-channel mappings simultaneously across multi-modal input patterns, involving states, actions, and rewards, in an online and incremental manner. Compared with prior systems, FALCON presents a truly integrated solution in the sense that there is no implementation of a separate reinforcement learning module or Q-value table. Using competitive coding as the underlying principle of computation, the network dynamics encompasses a myriad of learning paradigms, including unsupervised learning, supervised learning, as well as reinforcement learning. In this paper, we focus on a class of FALCON networks, known as TD-FALCON, that learns the value functions of the state-action space estimated through temporal difference (TD) methods. To evaluate the proposed system, experiments have been conducted based on a minefield navigation task, which involves an autonomous vehicle (AV) learning to navigate through obstacles to reach a stationary target (goal) within a specified number of steps. Experimental results have shown that TD-FALCON produces a much more stable performance and yet learns faster, by roughly two orders of magnitude, than a state-of-the-art reinforcement learning system using the gradient based backpropagation (BP) network. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 470–475, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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FALCON Architecture

FALCON employs a 3-channel architecture (Figure 1), comprising a cognitive field F2c and three input fields, namely a sensory field F1c1 for representing current states, a motor field F1c2 for representing actions, and a feedback field F1c3 for representing reward values. The generic network dynamics of FALCON, based on fuzzy ART operations [1], is described below.

Fig. 1. The FALCON architecture

Input vectors: Let S = (s1 , s2 , . . . , sn ) denote the state vector, where si ∈ [0, 1] indicates the sensory input i. Let A = (a1 , a2 , . . . , am ) denote the action vector, where ai ∈ [0, 1] indicates a possible action i. Let R = (r, r¯) denote the reward vector, where r ∈ [0, 1] is the reward signal value and r¯ (the complement of r) is given by r¯ = 1 − r. Activity vectors: Let xck denote the F1ck activity vector for k = 1, . . . , 3. Let yc denote the F2c activity vector. Weight vectors: Let wjck denote the weight vector associated with the jth node in F2c for learning the input patterns in F1ck for k = 1, . . . , 3. Initially, F2c contains only one uncommitted node and its weight vectors contain all 1’s. Parameters: The FALCON’s dynamics is determined by choice parameters αck > 0, learning rate parameters β ck ∈ [0, 1], contribution parameters γ ck ∈ [0, 1] and vigilance parameters ρck ∈ [0, 1] for k = 1, . . . , 3. 2.1

Predicting

In the predicting mode, FALCON receives input pattern(s) from one or more fields and predicts the pattern(s) in the remaining fields. The prediction process comprises three key steps, namely code activation, code competition, and activity readout. Code activation: Given the activity vectors xc1 , xc2 and xc3 , for each F2c node j, the choice function Tjc is computed as follows: Tjc =

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(1)

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where the fuzzy AND operation  ∧ is defined by (p ∧ q)i ≡ min(pi , qi ), and the norm |.| is defined by |p| ≡ i pi for vectors p and q. Code competition: A code competition process follows under which the F2c node with the highest choice function value is identified. The winner is indexed at J where TJc = max{Tjc : for all F2c node j}. (2) When a category choice is made at node J, yJc = 1; and yjc = 0 for all j  = J. This indicates a winner-take-all strategy. Activity readout: The chosen F2c node J performs a readout of its weight vectors to the input fields F1ck such that xc2(new) = xc2(old) ∧ wJc2 . 2.2

(3)

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In the learning mode, FALCON performs code activation and code competition (as described in the previous section) to select a winner J based on the activity vectors xc1 , xc2 , and xc3 . To complete the learning process, template matching and template learning are performed as described below. Template matching: Before code J can be used for learning, a template matching process checks that the weight templates of code J are sufficiently close to their respective input patterns. Specifically, resonance occurs if for each channel k, the match function mck J of the chosen code J meets its vigilance criterion: mck J =

|xck ∧ wJck | ≥ ρck . |xck |

(4)

If any of the vigilance constraints is violated, mismatch reset occurs in which the value of the choice function TJc is set to 0 for the duration of the input presentation. With a match tracking process, at the beginning of each input presentation, the vigilance parameter ρc1 equals a baseline vigilance ρ¯c1 . If a mismatch reset occurs, ρc1 is increased until it is slightly larger than the match c function mc1 J . The search process then selects another F2 node J under the revised vigilance criterion until a resonance is achieved. Template learning: Once a resonance occurs, for each channel k, the weight vector wJck is modified by the following learning rule: ck(new)

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(5)

When an uncommitted node is selecting for learning, it becomes committed and a new uncommitted node is added to the F2c field. FALCON thus expands its network architecture dynamically in response to the input patterns. The network dynamics of the FALCON architecture described above can be used to support a myriad of cognitive operations. We show how FALCON can be used to learn action policies and value functions for reinforcement learning in the next section.

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TD-FALCON

TD-FALCON incorporates Temporal Difference (TD) methods to estimate and learn value function Q(s, a), that indicates the goodness to take a certain action a in a given state s. The general sense-act-learn algorithm for TD-FALCON is summarized in Table 1. Given the current state s, the FALCON network is used to predict the value of performing each available action. The value functions are then processed by an action selection strategy (also known as policy) to select an action. Upon receiving a feedback, a TD formula is used to compute a new estimate of the Q value. The new Q value is then used as the teaching signal for TD-FALCON to learn the association of the current state and the chosen action to the estimated value. The details of the action selection strategy and the temporal difference equation are described as follows. Table 1. Generic flow of the TD−FALCON algorithm 1. Initialize the FALCON network. 2. Given the current state s, for each available action a in the action set A, predict the value of the action Q(s,a) by presenting the corresponding state and action vectors S and A to FALCON. 3. Based on the value functions computed, select an action a from A following an action selection policy. 4. Perform the action a, observe the next state s , and receive a reward r (if any). 5. Estimate the value function Q(s, a) following a temporal difference formula given by ΔQ(s, a) = αT Derr . 6. Present the corresponding state, action, and reward (Q-value) vectors, namely S, A, and R, to FALCON for learning. 7. Update the current state by s=s’. 8. Repeat from Step 2 until s is a terminal state.

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Action Selection Policy

Action selection policy refers to the strategy for selecting an action from the set of the available actions in a given state. The simplest action selection policy is to pick the action with the highest value predicted by the TD-FALCON network. However, if an agent keeps selecting the optimal action, it will not be able to explore and discover better alternative actions. There is thus a fundamental tradeoff between exploitation, i.e., sticking to the best actions believed, and exploration, i.e., trying out other seemingly inferior and less familiar actions. The -greedy policy selects the action with the highest value with a probability of 1 −  and takes a random action with probability . With a constant  value, the agent will always explore the environment with a fixed level of randomness. In practice, it may be beneficial to have a higher  value to encourage the exploration of paths in the initial stage and a lower  value to optimize the performance by exploiting familiar paths in the later stage. A decay -greedy policy is thus adopted to gradually reduce the value of  over time.

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Value Function Estimation

One key step of TD-FALCON is the temporal difference equation ΔQ(s, a) = αT Derr , where α ∈ [0, 1] is the learning parameter and T Derr is the error term. Using Q-learning [4], T Derr is computed by T Derr = r + γmaxa Q(s , a ) − Q(s, a)

(6)

where r is the immediate reward value, γ ∈ [0, 1] is the discount parameter, and maxa Q(s , a ) denotes the maximum estimated value of the next state s . With value iteration, the value function Q(s, a) is expected to converge to r + γmaxa Q(s , a ) over time. As ART and TD-FALCON typically assume all input values are bounded between 0 and 1, we develop a Bounded Q-Learning rule ΔQ(s, a) = αT Derr (1 − Q (s, a))

(7)

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The Minefield Navigation Task

The objective of the cognitive task is to navigate through a minefield to a randomly selected target position in a specified time frame without hitting a mine. In each trial, the autonomous vehicle (AV) starts from a random position in the field, and repeats the cycles of sense, act, and learn. A trial ends when the system reaches the target (success), hits a mine (failure), or exceeds 30 sense-act-learn cycles (out of time). The target and the mines remain stationary during the trial. For sensing, the AV has a coarse 180 degree forward view based on five sonar 1 sensors. In each direction i, the sonar signal is measured by si = 1+d , where i di is the distance to an obstacle in the i direction. Other sensory inputs include the current and target bearings. In each step, the system chooses one of the five possible actions, namely move left, move diagonally left, move straight ahead, move diagonally right, and move right. We conduct experiments with both immediate and delayed evaluative feedback. For both reward schemes, at the end of a trial, a reward of 1 is given when the AV reaches the target. A reward of 0 is given when the AV hits a mine. For the immediate reward scheme, a reward is estimated at each step of the trial by 1 computing a utility function utility = 1+rd , where rd is the remaining distance between the AV and the target position. To put the performance of TD-FALCON in perspective, we further conduct experiments using a reinforcement learning system, using the standard Q-learning rule and a gradient descent backpropagation (BP) algorithm as the function approximator. The BP learner employs a standard three-layer Perceptron architecture and has been shown to produce superior performance in many prior applications [5]. We experiment with a varying number of hidden nodes empirically and obtain the best results with 36 nodes. Figure 2 summarizes the performance of TD-FALCON and the BP learner in terms of success rate, hit-mine rate, and out-of-time rate in a 16 by 16 minefield

Self-organizing Neural Architecture for Reinforcement Learning

(a) FALCON with immediate reward

(b) FALCON with delayed reward

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Fig. 2. The performance of TD-FALCON and BP learner

containing 10 mines. The BP learner generally takes a very large number of (around 90, 000) trials to reach 90% success rates. In contrast, TD-FALCON consistently achieves the same level of performance within the first 1000 trials. In other words, TD-FALCON learns roughly two orders of magnitude faster than the BP learner. Considering network complexity, the BP learner has a highly compact network architecture. In terms of stability, however, TD-FALCON is clearly a much more reliable learner by consistently producing superior performance.

References 1. Carpenter, G.A., Grossberg, S., Rosen, D.B.: Fuzzy ART: Fast Stable Learning and Categorization of Analog Patterns by an Adaptive Resonance System. Neural Networks 4(6) (1991) 759-771 2. Carpenter, G.A., Grossberg, S.: Adaptive Resonance Theory. In: Arbib, M.A. (eds.): The Handbook of Brain Theory and Neural Networks. MIT Press, Cambridge (2003) 87-90 3. Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge (1998) 4. Watkins, C.J.C.H., Dayan, P.: Q-learning. Machine Learning 8(3) (1992) 279-292 5. Si, J., Barto, A.G., Powell, W.B., Wunsch, D. (eds.): Handbook of Learning and Approximate Dynamic Programming. Wiley-IEEE Press (2004)

On the Efficient Implementation Biologic Reinforcement Learning Using Eligibility Traces SeungGwan Lee School of Computer Science and Information Engineering, Catholic University, 43-1, Yeokgok 2-Dong, Wonmi-Gu, Bucheon-Si, Gyeonggi-Do, 420-743, Korea [email protected]

Abstract. The eligibility trace is one of the basic mechanisms in reinforcement learning to handle delayed reward. In this paper, we have used meta-heuristic method to solve hard combinatorial optimization problems. Our proposed solution introduce Ant-Q learning method to solve Traveling Salesman Problem (TSP). The approach is based on population that use positive feedback as well as greedy search and suggest ant reinforcement learning algorithms using eligibility traces which is called replace-trace methods(Ant-TD(λ)). Although replacing traces are only slightly, they can produce a significant improvement in learning rate. We could know through an experiment that proposed reinforcement learning method converges faster to optimal solution than ACS and Ant-Q.

1

Introduction

Reinforcement Learning is learning by interaction as agent achieves learning doing interaction by trial-and-error. Agent attempts an action that can be taken in the given state, and changes its state and receive reward value for the action. In this paper we introduce Ant-Q algorithm [7], [8] for combinatorial optimization has been introduced by Colorni, Dorigo and Maniezzo, and we suggest new ant reinforcement learning algorithms using eligibility traces are often called replacetrace methods(Ant-TD(λ)) to Ant-Q learning. Proposed Ant-TD(λ) reinforcement learning method searches goal using TDerror [3], [4], [5] and eligibility traces [13], there is characteristic that converge faster to optimal solution than original ACS [11], [12] and Ant-Q. The rest of the paper is organized as follows. In section2, we introduce Ant-Q and Ant-TD reinforcement learning. Section3 describes new ant reinforcement learning algorithms using eligibility traces are called replace-trace methods (AntTD(λ)). Section 4 presents experimental results. Finally, Section 5 concludes the paper and describes directions for future work.

2 2.1

Ant-Q Algorithm Ant-Q

Ant-Q learning method [7], [8] that is proposed by Coloni, Dorigo and Mauiezzo is an extension of Ant System(AS) [1], [2], [6], [9], [10], it is reinforcement learning reinterpreting in view of Q-learning. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 476–481, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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In Ant-Q, an agent(k ) situated in node(r ) moves to node(s) using the follow rule, called pseudo-random proportional action choice rule(or state transition rule):    arg max [AQ(r, u)]δ · [HE(r, u)]β if q ≤ q0 (exp loitation) u∈Jk (r) s= (1) S otherwise (exp loration) AQ(r, u) is Ant-Q value, be a positive real value associated to the edge(r,u), It is counterpart of Q-learning Q-values, and is intended to indicate how useful it is to move node(u) when in node(r ). AQ (r,u) is changed at run time. HE(r, u) is a heuristic value associated to edge(r,u) which allows an heuristic evaluation of which moves are better(in the TSP, the inverse of the distance). Let k be an agent making a tour. Jk (r) are nodes to be visited from the current node(r ). Let initial parameter is set to AQ(r, u)=AQ0 = 1/(average length of edges·n). Where δ and β is parameters which weigh the relative importance of the learned AQ-values and the heuristic values. q is a value chosen randomly with uniform probability in [0,1], q0 (0 ≤ q0 ≤ 1) is a parameter, and S is a random variable selected according to the distribution given by Eq.(2) which gives the probability with which an agent in node(r ) choose the node(s) to move to. ⎧ δ β ⎨ [AQ(r,s)] ·[HE(r,s)] if s ∈ Jk (r) [AQ(r,u)]δ ·[HE(r,u)]β pk (r, s) = u∈Jk (r) (2) ⎩ 0 otherwise The goal of Ant-Q is to learn AQ-values to find better solution as stochastic. AQ-values are updated by the following Eq.(3). AQ(r, s) ← (1 − α) · AQ(r, s) + α · (ΔAQ(r, s) + γ · M ax AQ(s, z))

(3)

z∈Jk (s)

α(0<α<1) is the pheromone decay parameter and γ is discount rate. M axAQ(s, z) is maximum reinforcement value that receive from external environment, global reinforcement is zero. Also, ΔAQ is reinforcement value, local reinforcement is always zero, while global reinforcement, which is given after all the agents have finished their tour, is computed by the following Eq.(4).  W if (r, s) ∈ tour done by the agent kib ΔAQ(r, s) = Lkib (4) 0 otherwise Where Lkib is the length of the tour done by the best agent which did the shortest tour in the current iteration, and W is a parameter, set to 10. 2.2

Ant-TD

To solve temporal-credit assignment problems, reinforcement learning that uses TD can be a method to learn prediction.

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Traditional learning methods such as supervised learning wait until final result happens to keep recording for prediction for all outputs that were calculated in intergrade. And then, as training error, we use difference between prediction value for output of each state and final result. But, the reinforcement learning using TD without waits final result. At each learning step, training error uses difference with prediction for output of present state and prediction for output of next state and this is known as TD-error(Temporal Difference error). The prediction for output of present state is updated to approximate with prediction for output of next state in TD-learning. Thus, it is known that reinforcement learning of simple form that uses difference of prediction values for output of each state is TD-learning. TD-learning that use TD-error calculates Q-function value of present state with Eq.(5) using difference with prediction for output of present state and prediction for output of next state. Q(st , at ) ← (1 − α) · Q(st , at ) + α · T D error

(5)

Here, α is learning rate, TD-error calculates with Eq.(6) as difference with prediction of present state and prediction of next state. T Derror = rt+1 + γ[ M ax Q(st+1 , at+1 ) − Q(st , at )] a∈A(st )

(6)

rt is reinforcement value and γ is discount rate. The goal of Ant-Q is to learn AQ-values to find better solution as stochastic. AQ-values are updated by the following Eq.(7). Appling Eq.(6) to Ant-Q, it is with Eq.(8). AQ(r, s) ← (1 − α) · AQ(r, s) + α · (ΔAQ(r, s) + γ · M ax AQ(s, z))

(7)

z∈Jk (s)

T Derror = ΔAQ(r, s) + γ[ M ax AQ(s, z) − AQ(r, s)]

(8)

z∈Jk (s)

Finally, an ant reinforcement learning model that apply TD-error in Ant-Q[14] calculates Q-function value for node(r,s) of present state with Eq.(9). α(0<α<1) is the pheromone decay parameter and γ is discount rate. M axAQ(s, z) is maximum reinforcement value that is received from external environment, global reinforcement is zero. Also, ΔAQ is reinforcement value, local reinforcement is always zero. AQ(r, s) ← (1 − α) · AQ(r, s) +α · (ΔAQ(r, s) + γ · [ M ax AQ(s, z) − AQ(r, s)]) z∈Jk (s)

(9) where

ΔAQ(r, s) = 0 M ax AQ(s, z) − AQ(r, s) = 0

z∈Jk (s)

if Local updating if Global updating

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Ant-TD(λ) : Ant-TD Using Eligibility Traces

An eligibility trace is one of the basic mechanisms of reinforcement learning. For example, in the popular TD(λ) algorithm, the reference to the use of an eligibility trace. Almost any temporal-difference (TD) method, such as Q-learning, can be combined with eligibility traces to obtain a more general method that may learn more efficiently [13]. An eligibility trace is a temporary record of the occurrence of an event, such as the visiting of a state or the taking of an action. The trace marks the memory parameters associated with the event as eligible for undergoing learning changes. When a TD error occurs, only the eligible states or actions are assigned credit for the error. Thus, eligibility traces help bridge the gap between events and training information. Like TD methods themselves, eligibility traces are a basic mechanism for temporal credit assignment.

Fig. 1. Feature of Replacing Eligibility Traces

In TD(λ), there is an additional memory variable associated with each state, its eligibility trace. The eligibility trace for edge(r,s) is denoted e(r, s). On each step, the eligibility trace for the maximum(M axAQ(s, z)) state visited on the step is incremented by 1 and the eligibility traces for another states decay by γλ.  1 if AQ(r, s) = M ax AQ(s, z) z∈Jk (s) e(r, s) = (10) γ · λ · e(rt−1 , st−1 ) otherwise Where γ is the discount rate and λ is the trace-decay parameter. This kind of eligibility trace is called a replacing trace because it replaces each time the state is visited, and then fades away gradually when the state is not maximum, as illustrated below: Finally, Ant reinforcement learning algorithms using eligibility traces are called replace-trace methods(Ant-TD(λ)) updated by the following Eq.(11). AQ(r, s) ← (1 − α) · AQ(r, s) +α · (ΔAQ(r, s) + γ · [ M ax AQ(s, z) − AQ(r, s)]) · e(r, s) z∈Jk (s)

(11) where

ΔAQ(r, s) = 0 M ax AQ(s, z) − AQ(r, s) = 0

z∈Jk (s)

if Local updating if Global updating

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Experimental Results

Prediction of performance of ant reinforcement learning algorithms using eligibility trace is called replace-trace methods(Ant-TD(λ)). We measure performance through comparison with original ant model(ACS and Ant-Q). Basis environment parameter for an experiment was decided as following, and optimum value decided by an experiment usually are β=2, α=0.1, q0 =0.9, γ=0.3, λ=0.7, m=n, W =10 and AQ0 = 1/(average length of edges·n). The initial position of agents assigned one agent in an each node at randomly, and the termination condition is that a fixed number of cycles or the value known as the optimum value was found. Table 1 shows optimal tour length and average tour length that are achieved by each algorithms(ACS, Ant-Q and Ant-TD(λ)) in case of repeated 1000 cycles in 10th trials about R×R grid problems, the performance of proposed method is excellent. Table 1. Performance evaluation of Ant-TD(λ)

Node 4×4 5×5 6×6 7×7 8×8 9×9 10×10

5

ACS Average Best length length 160 160 254 254 360 360 510 507 661 654 790 764 971 959

Ant-Q Average Best length length 160 160 254 254 362 360 509 502 654 648 782 753 963 941

Ant-TD(λ) Average Best length length 160 160 254 254 360 360 494 494 640 640 775 749 952 936

Conclusion and Future Work

In this paper, we suggested new ant reinforcement learning algorithms using eligibility traces are called replace-trace methods(Ant-TD(λ)). Proposed AntTD(λ) learning is method that is proposed newly to improve Ant-Q, this converged faster to optimal solution by solving temporal-credit assignment problems to use TD-error while agents accomplish tour cycle. Although replacing traces are only slightly, they can produce a significant improvement in learning rate. Proposed Ant-TD(λ) ant model using eligibility traces uses difference with prediction for output of present state and prediction for output of next state at each learning step, and updated to approximate with prediction for output of present state and prediction for output of next state in present state. Forward, we need study about several possible ways to generalize replacing eligibility traces in Ant-TD(λ).

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References 1. Colorni, A., Dorigo, M., Maniezzo, V.: An Investigation of Some Properties of an Ant Algorithm. In: Manner, R., Manderick, B.(eds.): Proceedings of the Parallel Problem Solving from Nature Conference. Elsevier Publishing (1992) 509-520 2. Colorni, A., Dorigo, M., Maniezzo, V.: Distributed Optimization by Ant Colonies. In: Varela, F., Bourgine, P.(eds.): Proceedings of the First European Conference of Artificial Life. Elsevier Publishing (1991) 134-144 3. Watkins, C.J.C.H.: Learning from Delayed Rewards. Ph.D. Thesis, King’s College, Cambridge, U.K (1989) 4. Fiecher, C.N.: Efficient Reinforcement Learning. Proceedings of the Seventh Annual ACM Conference on Computational Learning Theory (1994) 88-97 5. Barnald, E.: Temporal-Difference Methods and Markov Model. IEEE Trans. Systems, Man and Cybernetics 23 (1993) 357-365 6. Gambardella, L.M., Dorigo, M.: Solving Symmetric and Asymmetric TSPs by Ant Colonies. Proceedings of IEEE International Conference of Evolutionary Computation, IEEE Press (1996) 622-627 7. Gambardella, L.M., Dorigo, M.: Ant-Q: A Reinforcement Learning Approach to the Traveling Salesman Problem. In: Prieditis, A., Russell, S.(eds.): Proceedings of ML-95, Twelfth International Conference on Machine Learning. Morgan Kaufmann (1995) 252-260 8. Dorigo, M., Gambardella, L.M.: A Study of Some Properties of Ant-Q. In: Voigt, H.M., Ebeling, W., Rechenberg, I., Schwefel, H.S.(eds.): Proceedings of PPSN IV, Fourth International Conference on Parallel Problem Solving From Nature. Lecture Notes in Computer Science, Springer-Verlag, Berlin Heidelberg (1996) 656-665 9. Dorigo, M., Maniezzo, V., Colorni, A.: The Ant System: Optimization by a Colony of Cooperation Agents. IEEE Trans. Systems, Man and Cybernetics-Part B 26(1) (1996) 29-41 10. Stutzle, T., Hoos, H.: The Ant System and Local Search for the Traveling Salesman Problem. Proceedings of IEEE 4th International Conference of Evolutionary (1997) 11. Gambardella, L.M., Dorigo, M.: Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem. IEEE Trans. Evolutionary Computation 1(1) (1997) 12. Stutzle, T., Dorigo, M.: ACO Algorithms for the Traveling Salesman Problem. In: Miettinen, K., Makela, M., Neittaanmaki, P., Periaux, J.(eds.): Evolutionary Algorithms in Engineering and Computer Science, Wiley (1999) 13. Sutton, R., Barto, A.: Reinforcement Learning: An Introduction. MIT Press (1998) 14. Lee, S.G.: Multiagent Reinforcement Learning Algorithm Using Temporal Difference Error. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg (2005) 627-633

Combining Label Information and Neighborhood Graph for Semi-supervised Learning Lianwei Zhao1, Siwei Luo1, Mei Tian1, Chao Shao1, and Hongliang Ma2 1

School of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China [email protected] 2 Logistics Science Research Institute, General Logistics Department, Beijing 100071, China

Abstract. In this paper, we consider the problem of combining the labeled and unlabeled examples to boost the performance of semi-supervised learning. We first define the label information graph, and then incorporate it with neighborhood graph. We propose a new regularized semi-supervised classification algorithm, in which the regularization term is based on this modified Graph Laplacian. According to the properties of Reproducing Kernel Hilbert Space (RKHS), the representer theorem holds, so the solution can be expressed by the Mercer kernel of examples. Experimental results show that our algorithm can use unlabeled and labeled examples effectively.

1 Introduction The problem of learning from labeled and unlabeled examples has received significant attention in recent years. It is also known as semi-supervised learning, and a number of algorithms have been proposed for it, including Co-training [4], random field models [5,6] and graph based approaches [7,8]. However, learning from examples has been seen as an ill-posed inverse problem, and regularizing inverse problem means finding a meaning stable solution. So in this paper we focus on regularization approaches. Both measure based regularization[9] and information regularization [10] take density into consideration, and can get the decision boundary that lies in the region of low density in 2D example. However, it is difficult to apply them in high-dimensional real world data sets. Manifold regularization [1, 2] exploits the geometry of the marginal distribution to incorporate unlabeled examples within a geometrically motivated regularization term. One of the key assumptions of manifold based learning algorithm is that the data lies on the low dimensional manifold, and the data can be approximated by a weighted discrete graph. So how to use the labeled and the unlabeled examples construction an approximation graph is very important. In this paper, we first define the label information graph, and then incorporate it with neighborhood graph. Using Graph Laplacian regularizier, we propose regularized semi-supervised classification algorithm. According to the properties of Reproducing Kernel Hilbert Space (RKHS), the representer theorem holds, so the solution can be expressed by the Mercer kernel of examples. There is only one regularization parameter reflecting the tradeoff between the Graph Laplacian and the J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 482 – 488, 2006. © Springer-Verlag Berlin Heidelberg 2006

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complexity of solution. Experimental results show that our algorithm can use unlabeled and labeled examples effectively and are more robust than Transductive SVM and LapSVM. This paper is organized as follows. Section 2 briefly reviews Graph Laplacian. In section 3, we define label information graph with labeled examples, and propose the regularized semi-supervised classification algorithm. Experimental results on synthetic and real world data are shown in section 4, followed by conclusions in section 5.

2 Graph Laplacian Graph Laplacian [3] has played a crucial role in several recently developed algorithms [11] because it approximates the natural topology of data and is simple to compute for enumerable based classifiers. Let’s consider a neighborhood graph G = (V , E ) whose

vertices are example points V = {x1 , x 2 ,L , xl +u } and whose edge weights {Wij }li ,+ju=1 represent pairwise similarity relationships between examples. The neighborhood of x j can be defined as those examples which are closer than ε or the k nearest neighbors of x j . To ensure that the embedding function f is smooth, a natural choice is to get empirical estimate I (G ) , which measures how much f varies across the graph: I (G ) =

1 2∑ Wij

l +u

∑ ( f ( xi ) − f ( x j )) 2 Wij

i , j =1

(1)

i, j

where 2∑ Wij is normalizing factor, so that 0 ≤ I (G ) ≤ 1 . i, j

Defining fˆ = [ f ( x1 ,L , f ( xl +u )]T , and L = D − W as Graph Laplacian matrix, l+u

where D is diagonal matrix given by Dii = ∑ Wij , I (G ) can be rewritten as: j=1

I (G ) =

1 fˆ T Lfˆ 2∑ Wij

(2)

i, j

3 Regularized Semi-supervised Classification We assume that if two points x1 , x 2 ∈ X are close in the input space, then the conditional p( y | x1 ) and p( y | x 2 ) are near in intrinsic geometry of PX , as in [1]. However, this adjacency matrix of neighborhood graph doesn’t take into consideration the information carried by labeled examples. The regularization term

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I (G ) , especially for binary case classifiers, is proportional to the number of separated neighbors. Therefore for labeled examples, if x i and x j have the same label, they should not be separated by the decision boundary, so we can redefine the relationship between x i and x j by strengthening it.

Consider all the sample points {x1 , x 2 , L , xl +u } , the Least Squares algorithm solves with the squared loss function

l

l

i =1

i =1

∑V ( xi , yi , f ) = ∑ ( yi − f ( xi )) 2 , which is

based on the minimizing the error on the labeled examples. It is important to observe that: 2(l − 1)

l

l

i =1

i , j =1

∑ ( y i − f ( xi )) 2 ≥ ∑ (( f ( x i ) − f ( x j )) − ( y i − y j )) 2

(3)

l

If

∑ ( yi − f ( xi )) 2 → 0 , then i =1

l

∑ (( f ( xi ) − f ( x j )) − ( y i − y j )) 2 → 0

(4)

i , j =i

So if y i − y j < δ , then f ( xi ) − f ( x j ) < ε where δ , ε → 0, and δ , ε > 0 . Defining (l + u ) × (l + u ) matrix J as follows.

⎧1 or Wij , if i, j ≤ l , and y i − y j < δ ⎪ ⎪ J ij = ⎨0 or − Wij , if i, j ≤ l , and yi − y j ≥ δ ⎪ otherwise ⎪⎩0,

(5)

This can be seen as a label information graph G ′ = (V , E ′) , whose vertices are the labeled or unlabeled example points V = {x1 , x 2 ,L , xl +u } , and whose edge weights J ij represent appropriate pairwise label relationships between labeled examples i and j . So the right of equation 3 can be rewritten as: l +u

∑ ( f ( xi ) − f ( x j )) 2 J ij

(6)

i , j =1

This term can be seen as label information carried by labeled examples, and penalizes classifiers that separate the examples having the same labels. The neighborhood graph and the label information graph have the same vertices, and can be incorporated together. So the optimization problem has the following objective function:

Combining Label Information and Neighborhood Graph

min H [ f ] = γ f

2

=γ f

2

f ∈H

K

K

+

485

l +u

∑ ( f ( xi ) − f ( x j )) 2 (Wij + J ij )

i , j =1

(7)

+ fˆ T La fˆ l+ u

where La = D − (W + J ) , D is diagonal matrix given by Dii = ∑ (Wij + J ij ) and γ j=1

is a regularization parameter that controls the complexity of the clustering function. It has the same form as unsupervised regularization spectral clustering [1], and the Representer theorem holds and the solution of the equation the solution can be expressed by the kernel of examples as follows: l +u

f ( x ) = ∑ α i K ( x, x i )

(8)

i =1

where α can be solved by an eigenvalue problem, and the regularization parameter γ can be selected by the approach of L-curve. The complete semi-supervised learning algorithm consists of the following steps. Step 1. Construct adjacency graph G = (V , E ) with (l + u ) nodes using k nearest neighbors. Choose edge weights Wij with binary or heat kernel weights, construct label information graph G ′ = (V , E ′) , and then compute the Graph Laplacian La . Step 2. Regularized semi-supervised classification. At this step, we use equation 7 as objective function. Step 3. Label the unlabeled examples. Firstly, we select one labeled example from M = {xi , y i }li =1 . Without loss of generality, we select {x1 , y1} , so all the examples clustering with {x1 , y1} will have the same label y1 as {x1 , y1} , while the others will have the label different with y1 . So for every {xi , y i } ∈ M , we get a label yˆ i . l

Step 4. Compute

∑ yi − yˆ i

2

l

. Stop if

i =1

∑ yi − yˆ i

2

≤ threshold , otherwise, select

i =1

the i th labeled example where i = arg max y i − yˆ i . i

Step 5. Adjust the weights J ij . For the selected i th example, we can find the labeled examples j satisfying { y j − y i ≤ δ , y j − yˆ j ≤ ε ,1 ≤ j ≤ l} , and then adjust the weight J ij and re-compute the matrix La . Goto step 2.

4 Experimental Results We first conducted experiments on two moons dataset. The dataset contains 200 unlabeled sample points, and all the labeled points are sampled from the unlabeled points randomly. Figure 1 shows the results of semi-supervised classification using

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our algorithm on two moons dataset added Gaussian noise with 0 and 5 labeled points respectively. With 0 labeled points it can be regarded as unsupervised regularization clustering. From the figure, it is clear that unsupervised classification failed to label the points correctly. It is because after adding noise, the dataset has not the regular geometry structure. When using only 5 labeled points for each class with the proposed algorithm, we find the optimal solution as shown in Figure 1. We also show the experimental results on USPS dataset. Using the binary weight of the edge of the neighborhood graph, that is Wij = 0 or 1 , we show the results of 45 binary classification problems. We use the first 400 images for each handwritten digit, and processed using PCA to 100 dimensions as in [1]. As shown in Figure 2, we compare the error rate of Transductive SVM, LapSVM, and the proposed method at the precision-recall breakeven points in the ROC curves. Both experimental results show clearly that our proposed algorithm is more robust than Transductive SVM and LapSVM, as given in Figure 2. semi-supervised

clustering

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Fig. 1. Regularized Semi-supervised classification on two moons dataset added Guassian noise with 0, 5 labeled points respectively 8

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6 Error Rates

Error Rates

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10 20 30 40 45 Classification Problems

0 0

10 20 30 40 45 Classification Problems

Fig. 2. Comparing the error rate of ReguSCoM, Transductive SVM, and LapSVM at the precision-recall breakeven points

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5 Conclusions Semi-supervised learning is to benefit from a large number of unlabeled examples and few labeled examples. In this paper, by combining the labeled and unlabeled examples, we define the modified Graph Laplacian, and then propose a new regularized semi-supervised classification algorithm. Our method yields encouraging experimental results on both synthetic data and real world datasets. In future work, we will investigate other approaches of construction of graph using the labeled and unlabeled examples, further to improve performance of semi-supervised learning algorithm.

Acknowledgements This work is supported by National Natural Science Foundations of China (60373029) and the National Research Foundation for the Doctoral Program of Higher Education of China (20050004001). The author would like to thank Dr. M. Belkin for useful suggestion.

References 1. Belkin, M., Niyogi, P., Sindhwani, V.: Manifold Regularization: A Geometric Framework for Learning from Examples. Department of Computer Science, University of Chicago, TR-2004-06 2. Belkin, M., Niyogi, P., Sindhwani, V.: On Manifold Regularization. Department of Computer Science, University of Chicago, TR-2004-05 3. Belkin, M., Niyogi, P.: Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation 15(6) ( 2003)1373-1396 4. Blum, A., Mitchell, T.: Combining Labeled and Unlabeled Data with Co-training. In: Proceedings of the Eleventh Annual Conference on Computational Learning Theory. Madison, Wisconsin, USA. ACM (1998) 92-100 5. Szummer, M., Jaakkola, T.: Partially Labeled Classification with Markov Random Walks. In: Thomas G. Dietterich, Suzanna Becker, Zoubin Ghahramani (Eds.): Advances in Neural Information Processing Systems 14. MIT Press (2001) 945-952 6. Zhu, X., Ghahramani, Z., Lafferty J.: Semi-supervised Learning Using Gaussian Fields and Harmonic Functions. In: Tom Fawcett, Nina Mishra (Eds.): Proceedings of the Twentieth International Conference on Machine Learning. USA. AAAI Press (2003) 912919 7. Blum, A., Chawla, S.: Learning from Labeled and Unlabeled Data Using Graph Mincuts. In: Carla E. Brodley, Andrea Pohoreckyj Danyluk (Eds.): Proceedings of the Eighteenth International Conference on Machine Learning. USA, Morgan Kaufmann (2001)19-26 8. Zhou, D., Bousquet, O, Lal, T.N., Weston, J., Schoelkopf, B.: Learning with Local and Global Consistency, In: S. Thrun, L. Sauland B. Sch¨olkopf (Eds.): Advances in Neural Information Processing Systems 16. Cambridge, MA, MIT Press 2004

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9. Bousquet, O., Chapelle, O, Hein, M.: Measure Based Regularization, In: S. Thrun, L. Sauland B. Scholkopf (Eds.): Advances in Neural Information Processing Systems. Cambridge, MA, MIT Press 2004 10. Szummer, M., Jaakkola, T.: Information Regularization with Partially Labeled Data. In: Suzanna Becker, Sebastian Thrun, Klaus Obermayer (Eds.): Advances in Neural Information Processing Systems 15. Canada. MIT Press, (2003) 1025-1032 11. Krishnapuram, B., Williams, D., Xue, Ya, Hartemink, A., Carin L., Figueiredo M. A. T.: On Semi-Supervised Classification. In: L. K. Saul, Y.Weiss and L. Bottou (Eds.): Advances in Neural Information Processing Systems 17. Cambridge,MA, MIT Press (2005) 721-728

A Cerebellar Feedback Error Learning Scheme Based on Kalman Estimator for Tracing in Dynamic System* Liang Liu, Naigong Yu, Mingxiao Ding, and Xiaogang Ruan** School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100022, P.R. China {liuliang, mxdin}@emails.bjut.edu.cn {yunaigong, adrxg}@bjut.edu.cn

Abstract. Motivated by recent physiological and anatomical evidence, a new feedback error learning scheme is proposed for tracing in motor control system. In the scheme, the model of cerebellar cortex is regarded as the feedforward controller. Specifically, a neural network and an estimator are adopted in the cerebellar cortex model which can predict the future state and eliminate faults caused by time delay. Then the new scheme was used to control inverted pendulum. The simulation experimental results show that the new scheme can learn to control the inverted pendulum for tracing successfully.

1 Introduction “Brain’s engine of agility” is entitled to the cerebellum, since the cerebellum plays a key role in robust, adaptive neuromuscular control of multi-jointed movements, and helps to achieve smooth and dynamically efficient trajectories [1]. Presently, there is a highlight of researches in cerebellar neurobiology and control engineering [2]. Several early computational cerebellar models are derived from the regular structure of the cerebellum. In Albus’ perceptron model, inputs are summed in association units and connected via adjustable weights to an output unit. Marr’s view was that parallel fiber-Purkinje cell synapses should increase when a parallel fiber is active at about the time that a climbing fiber fires [3]. In this way, Purkinje cells become selectively to parallel fiber pattern. However, Albus proposed an error-correction rule: weight should decrease if the Purkinje cell was erroneously active. This is supported by Ito et al’s later experiment. The concept of internal model was first introduced into the field of neurophysiology by Ito [4], who suggested that there is internal model for motor learning and control in the central nervous system (CNS). Functional magnetic resonance imaging (fMRI) experiments on cerebellum activities showed that internal model does exist in the cerebellum [5]. Kawato proposed feedback error learning (FEL) model which comprises a fixed feedback controller that ensures the stability of the system and an adaptive feedforward controller that improves control performance [6]. The FEL scheme in the cerebellum is supported by the advance in neurophysiology [7]. * This work is supported by National Natural Science Foundation of China (60375017). ** Corresponding author. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 489 – 495, 2006. © Springer-Verlag Berlin Heidelberg 2006

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One landmark of the work in cerebellar neurobiology and control engineering is Paulin’s research on sensory prediction for dynamic state estimation and control [8]. In his view, the cerebellum can learn an implicit model of the dynamics between motor output and sensory input, from which to provide motor command structures with predictive information about relevant state variables. Inspired by his idea, this paper proposes a new feedback error learning scheme containing a cerebellar cortex model. The cerebellar cortex model can learn the dynamic inverse model of the controlled object according to the instructive information from the outputs of feedback controller and estimator to change the weights of the forward model. The inverted pendulum is adopted as a control object for the new scheme. This paper is organized as follows: in section 2, the present models about the cerebellar cortex are reviewed; in section 3, the principle of the Kalman estimator is described; in section 4, based on FEL model with the learning regulation, a new feedback error learning scheme to be implemented in motor control system is designed; in section 5, the simulation on the inverted pendulum was performed to testify the effectiveness of new scheme (this section is illustrated in two subsections: problem formulation; the analysis of simulation experimental results which are presented to validate the proposed new FEL scheme); finally, conclusions are stated in section 6.

2 Cerebellar Cortex Model According to Kawato’s FEL model, the engineering abstraction model of cerebellar cortex is shown in Fig. 1(a). In his view, the connection synapses between Parallel fiber (PF) and Purkinje cells (PC) can be adjusted by the teacher signal, which is provided from the channel of Climbing fiber (CF). The output of the model provides the feedforward motor command (FFMC). Cerebellar Cortex MF

GC

PF

SS

PF

PC

GO

PC CF

Actual Trajectory

FFMC

Desired Trajectory

GC

IO NUC

MF

CF Teacher Signal

(a) Kawato’s cerebellar cortex model

Excitatory Inhibitatory

Teacher signal

(b) Cerebellar microcomplex

Fig. 1. The models of cerebellum

A model of the adaptive control of saccades was developed by Schweighofer et al. [9]. This model inherits the idea of a structural and functional unit suggested by Ito [4] called a microcomplex as shown in Fig. 1(b). Input excites Granule cells (GC) and Nuclear cells (NUC), which provide the output of the unit via Mossy fibers (MF). The input of PC receive signal from PF, and NUC activity is modulated by inhibitory

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action from PC. Learning occurs at the Purkinje-parallel fiber synapses as a function of a second input system, the CF originating in the Inferior olive (IO). In most on-line learning cases, however, a problem is that the error signal is delayed and has to be temporally matched with the action that caused it. Therefore, a device which can eliminate the delay is indispensable, and an estimator is introduced.

3 Kalman Estimator The Kalman filter is taken as an estimator because of its powerful ability to support estimations of future states, and it can do so even when the precise nature of the modeled system is unknown [10]. Consisted of a set of mathematical equations that provide an efficient computational method to estimate the state of a process, the Kalman filter addresses the general problem of trying to estimate the state x ∈ ℜ n of a discrete-time controlled process.

x(t + 1) = Ax(t ) + Bu (t ) + w(t ) ,

(1)

where x(t ) is the state of the error signal and w(t ) is a random driving force of the system, having covariance Γ(t ) . The measurement data are y (t ) = Cx(t ) + v(t ) ,

(2)

where v(t ) is the measurement error term. It has covariance ∑ (t ) . The optimal estimate xˆ(t + 1) of system’s error state based on measurement up to time t is showed by

xˆ (t + 1) = [ A − K (t )C ]xˆ (t ) + K (t ) y (t ) .

(3)

The Kalman gain K determines the strength of coupling between the new data and the current state estimate. Its formulation shown as K (t ) = APC T [CPC T + ∑(t )]−1 ,

(4)

where P represents the covariance of current state estimate, and can derive from the matrix Riccatti equation, P(t ) = AP(t − 1) AT + Γ(t − 1) − AP(t − 1)C T . × [CPC T + ∑(t − 1)]−1 [ AP(t − 1)C T ]T

(5)

The equations (3)-(5) above are Kalman filter equations. They can be modified to optimally estimate the state at any time, based on current available information [10].

4 Feedback Error Learning Scheme The FEL model has been successfully applied to various control tasks, such as the control of a robot arm with pneumatic actuators. Here we aims to establish a new FEL scheme based on the FEL model.

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4.1 New FEL Scheme

In this paper, in order to use inverted pendulum for tracing task in the later experiments, an inverse model must be trained to generate the appropriate command for a desired movement of the plant. We proposed a new feedback error learning scheme for tracing which requires information in three ways: 1) from the input and output to the NNFC (Neural Network Feedforward Controller) that specify the current tracing requirement and previous performance, respectively, and 2) from the output of the CFC (Conventional Feedback Controller) that adjusts the NNFC as does in the original FEL and 3) from the output of the Kalman estimator that serves as an additional teacher signal as does in [11]. The framework of our new feedback error learning scheme for tracing is shown in Fig. 2.

Cerebellar Cortex NNFC

Kalman Estimator

r(t) Desired trajectory

+ -

e(t+1)

e(t)

CFC

uff +

+

u

ufb

Controlled object

y Actual trajectory

Fig. 2. The new FEL scheme with a cerebellar cortex model

4.2 The Learning Regulation for the New FEL Scheme

The new feedback error learning scheme for tracing is shown as follows: The controlled object’s dynamic equation is defined as: f (θ , θ&, x, x& ) = u

(6)

e = [θ d − θ , xd − x]T .

(7)

We define the error signal as:

The response of the CFC is formulated as: u fb = K d e& + K p e .

(8)

u ff = Φ(e, w)[α r + (1 − α ) y ] ,

(9)

NNFC is expressed as:

where w is the synaptic weight of NNFC, and | α |≤ 1 , the coefficient of the inputs. Thus the total dynamic formulation is given as: u = u fb + u ff = K d e& + K p e + Φ (e, w)[α r + (1 − α ) y ] .

(10)

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With respect to 0 < μ < 1 and k T ≥ 0 , the learning rule in the NNFC is dw ∂Φ T = −η ( ) ( μ u fb + (1 − μ ) k T e ) dt ∂w

(11)

5 Simulation and Experimental Results In this part, the inverted pendulum is used as the typical nonlinear plant to testify the new feedback error learning scheme. 5.1 Problem Formulation

A SIMO system in the real circumstance, the inverted pendulum balancing task involves a pendulum mounted on the top of a wheeled cart that travels along a track. This system was depicted by using the following equations of motion: 2 ( M + m) && x + ml cos θθ&& + dx& − ml sin θ (θ&) = u

(12)

2 (4ml / 3)θ&& + ml cos θ && x − mgl sin θ = 0

(13)

The goal of the inverted pendulum balancing and tracing task is to balance the upright pole of the inverted pendulum and follow the reference signal within the range of the rail in a short time, that is, the reference signal is r = [θ d , θ&d , xd , x&d ]T . The desired position of the cart can be set to any reasonable point on the rail, however, the rail origin is selected here as the desired position of the cart without losing generality. 5.2 Simulation Experimental Results Tracing. Firstly, the desired trajectory was given as r (t ) = 0 similar to the balancing problem. The proposed new FEL scheme is depicted in Fig. 3(b) prior to the FEL model shown in Fig. 3(a). The results indicate that the more information about future in the devised cerebellar cortex model, the more favorable trajectory can be got. 0.25

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Fig. 3. The tracing performances with zero reference signal

5

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Robustness. Then the reference signal is given as r (t ) = A sin(4π t ) and the track’s

length of the wheeled cart is limited by | x |< 0.6m . The results can be seen in Fig. 4(a). Then some disturbances are added to the controller’s output. When two increments are set as Δu = 350 ( t = 1s ) and Δu = 500 ( t = 2.5s ), the actual trajectory in Fig. 4(b) illustrates good curves for tracing. 0.3

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Fig. 4. The tracing performance of the new FEL scheme is presented by adopting a sine function as the input in the left column. In the right column, the robustness of the new scheme is shown by adding disturbances, at time 1s and 2.5s with the increments 350 and 500, respectively, to the summed controller output conforming to u = u ff + u fb .

6 Conclusions A novel feedback error learning scheme is described for tracing in motor control system. Inspired by the cerebellar circuitry, a cerebellar cortex model has been developed in the new scheme. A neural network can be used to simulate the synaptic plasticity resulting from correlations in activity between the Parallel fibers and Purkinje cells by the teacher signal via climbing fiber inputs, and a Kalman estimator is introduced to enhance the capacity for dynamic state estimation. In motor control system, the devised cerebellar cortex model can learn the dynamic inverse model of the controlled object according to the instructive information both from the output of the feedback controller and from the output of the estimator to change the weights of the forward model. The simulation results show that this new FEL scheme not only makes sure of the error convergence as conventional adaptive control but also acquires the exact inverse of the controlled object without loop delay. It is anticipated to build on this type of scheme for explaining cerebellar function and for using in robotic applications in the future.

References 1. Wickelgren, I.: The cerebellum: The Brain’s Engine of Agility. Science 281 (1998) 1588– 1590 2. Wolpert, D.M., Miall, R.C., and Kawato, M.: Internal Models in the Cerebellum. Trends Cognitive Sciences 2 (1998) 338–347

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3. Marr, D.: A Theory of Cerebellar Cortex. Journal of physiology. 202 (1969) 437-470 4. Ito, M.: Cerebellum and Neural Control. Raven Press: New York (1984) 5. Tamada, T., Miyauchi, S., Imamizu, H., Yoshioka, T., Kawato, M.: Activation of the Cerebellum in Grip Force and Load Force Coordination: an fMRI Study. Neuroimage 6 (1999) S492 6. Kawato, M., Gomi, H.: A Computational Model of Four Regions of the Cerebellum Based on Feedback Error Learning. Biological Cybernetics 69 (1992) 95-103 7. Doya, K., Kimura, H., Kawato, M.: Neural Mechanisms of Learning and Control. Control Systems Magazine. IEEE 4 (2001) 42-54 8. Paulin, M.G.: The Role of the Cerebellum in Motor Control and Perception. Brain Behavior and Evolution 41(1) (1993) 39–50 9. Schweighofer, N., Arib, M.A., and Dominey, P. F.: A Model of Cerebellum in Adaptive Control of Saccadic Gain. . The Model and Its Biological Substrate. Biological Cybernetics 75 (1996) 19-28 10. Davis, M.H.A., and Vinter, R.B.: Stochastic Modelling and Control. Chapman and Hall. London New York (1985) 11. Luo, Z.W., Fujii, S., Saitoh, Y., Muramatsu, E., Watanabe, K.: Feedback-error Learning for Explicit Force Control of a Robot Manipulator Interacting with Unknown Dynamic Environment. Proceedings of the IEEE International Conference on Robotics and Biomimetics 2004, 448, August 2004

Ⅰ

An Optimal Iterative Learning Scheme for Dynamic Neural Network Modelling Lei Guo1,2 and Hong Wang2 1 Research Institute of Automation, Southeast University, Nanjing 210096, China [email protected] 2 Control Systems Centre, The University of Manchester, Manchester, M60 1QD, UK [email protected]

Abstract. In this paper, iterative learning control is presented to a dynamic neural network modelling process for nonlinear (stochastic) dynamical systems. When (B-spline) neural networks are used to model a nonlinear dynamical functional such as the output stochastic distributions in the repetitive processes or the batch processes, the parameters in the basis functions will be tuned using the generalized iterative learning control (GILC) scheme. The GILC scheme differs from the classical model-based ILC laws, with no dynamical inputoutput differential or difference equations given a prior. For the “model free” GILC problem, we propose an optimal design algorithm for the learning operators by introducing an augmented difference model in state space. A sufficient and necessary solvable condition can be given where a robust optimization solution with an LMI-based design algorithm can be provided.

1 Introduction Neural network (NN) based approximations have been effectively used in many modelling and control problems for complex dynamic systems [1,2]. In [1], the genetic algorithm has been used in optimization of the parameters of radial basis function (RBF) networks. In [2] the orthogonal least square algorithm with basis pursuit and D-optimality has been studied. The repetitive processes exist in many industrial cases (e.g. the batch processes) where the NN models can be adjusted based on the obtained information. How to use and learn the information in the last batch to perform and improve a new nonlinear model dynamically is a significant and challenging problem. One typical example motivated from the stochastic systems, where the so-called stochastic distribution control (SDC) problem has been effectively applied to in chemical and mechatronics engineering is [3]. In this paper we will use the B-spline expansions in SDC problems to illustrate the iterative learning strategies for the parameter optimization in the NN-based modelling. It is noted that the proposed approaches are not limited to the SDC problems. In SDC problems, it is supposed that the probability density functions (PDFs) of system output have been measured or estimated. To transform the infinite dimensional control problem into a finite dimensional one, B-spline NNs have been adopted to J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 496 – 501, 2006. © Springer-Verlag Berlin Heidelberg 2006

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approximate the PDFs so that the PDF tracking and the fault detection problems can be performed [4-6]. However, basis functions in the B-spline expansions have been supposed to be fixed in a deterministic domain. One possible drawback is that dynamical changes may lead to serious approximation errors. Otherwise, one may unnecessarily select redundant basis functions so that the weighting system may have a high dimension. The basic idea of ILC is to improve the control signal for the present operation cycle by feeding back the control error in the previous cycle. ILC has been shown its applications in many processes, as an established control method in situations where the purpose of the system is to perform a particular operation repeatedly [8-10]. However, for such a learning process with respect to the parameters of a group of B-spline basis functions for a nonlinear distribution, most of the classical ILC laws are unavailable any more. Most mature ILC schemes are designed for the plants with given linear models, where the measurement for optimization or learning is a functional rather than the value of some signals. In [7], an iterative learning control (ILC) framework has been discussed for the SDC problem, where at each batch process the parameters of the basic functions can be tuned via an approximate ILC law. In [7], the primary idea is at each step to make the tracking errors between the practical PDF and the target one is smaller than the last step, where linearization has to be used to provide a convergence criterion. However, it is difficult for such a criterion to provide a constructive design algorithm with the required constraints of the weights. In this paper, we consider the real time optimization of the B-spline expansions based on the PDFs by using the ILC framework. The iterative learning process is used to tune the parameters that decide the shape and location of the basis functions. The proposed results can be used to the general neural network modelling processes for nonlinear dynamics. It also can be regarded a generalized dynamical identification problem if the parameters of the B-spline neural network are considered as the system parameters. By introducing an augmented vector related to the performance index function, the original generalized ILC (GILC) problem can be formulated to a classical nonlinear performance control problem, for which some nonlinear or robust design methodologies can be applied. It is noted that this framework also has potential significance in other ILC problems with nonlinear (non-quadratic) performance functions.

2 Formulation of the GILC Problem The following B-spline model for the SDC problems can be found in [3-6]. To simplify the mathematical description, we only consider one batch where only the selection of the basis functions is considered. For a dynamic stochastic system with the input

uk ,i ( t ) ∈ R m and the output yk ,i ( t ) ∈ [ a, b ] , the conditional PDF of output

yk ,i ( t ) is denoted by γ ( uk ,i ( t ) , z ) . The index k represents the step of batch while i stands for the step of control within the k th batch. In the previous works, usually the basis functions are supposed to be fixed while the weights can be controlled (see [3-6]). In this paper the linear B-spline model is adopted as follows

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γ (θ k , uk ,i , z ) = B (θ k , z ) V ( uk ,i , t ) + h (V ( uk ,i , t ) ) bn (θ k , z )

(1)

where h(V (t )) is a known nonlinear function as an irrelevant term in this paper (see [4]). In the tracking problem of SDC, the main task is to find the control law and the B-spline expansions such that the measured output PDFs can follow a target one, denoted by

γ g ( z) .

For a group of B-spline basis functions, the parameters to be

tuned in the k th batch is θ k . For example, θ k can be defined as the height and/or width of the quadratic polynomials (see [7]), or the mean and variance of a Gaussian distribution. In this paper, only the learning law of the θ k will be considered. The classical model-based ILC scheme needs a well-established plant model. The purpose of ILC is to find the learning rate such that the closed loop system is convergent and satisfies the stability and other performances. Simple comparison shows that two main differences exist between the classical model-based ILC formulations and the considered problem. Firstly, equality (3) as the relationship connecting the weights and parameters in the basis functions with the input is not a classical model in the conventional context, where a differential or difference model has been provided for learning. To some extent, such a problem is a type of “model free” problem since no conventional models can be used for learning. Secondly, the “model free” situation requires a generalization of the traditional output error since neither

yk − yk −1 nor

yk − yd can be measured for the learning procedures. Instead, the measured infor-

mation is the stochastic distribution of the system output. Noting the objective of the learning B-spline modelling is to find converge to a target PDF

γ g ( z)

yk − yd , we select

θk

such that the B-spline expansion (3) can

, as an analogue of the conventional tracking error

J k ,i (θ k ) = ∫ ( γ (θ k , uk ,i , z ) − γ g ( z ) ) dz b

a

2

(2)

as the “error distance” for the learning procedure. Denote the cumulative distance m

J k (θ k ) = ∑ J k ,i (θ k ) as the error for learning. The learning law for θ k can be i =1

given by

θ k = ϒ1θ k −1 + ϒ 2 J k −1 (θ k −1 )

(3)

ϒ1 and ϒ 2 are two learning operators to be determined. The objective is reduced to find ϒ1 and ϒ 2 such that J k (θ k ) can be minimized in the k th batch where

process. For the above problem, few available ILC design algorithms are applicable. It is shown that not only the measurement for learning, but also the performance functions

An Optimal Iterative Learning Scheme for Dynamic Neural Network Modelling

for optimization will be nonlinear functions of

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θ k . It is noted that even for the classi-

cal ILC problems, most feasible methods are dependent on linearization [8]. To this end, we denote an auxiliary vector as ξ k = J k (θ k ) and introduce a new quadratic performance index for

θk

can be formulated as

cost function

ξk

2 2

ξk .

Thus, the above iterative learning problem for

⎧θ k = ϒ1θ k −1 + ϒ 2ξ k −1 subject to the performance ⎨ ⎩ξ k = J k ( ϒ1θ k −1 + ϒ 2ξ k −1 )

, for which the classical control methods can be performed. In

order to obtain a feasible algorithm with a satisfactory convergent rate, we will perform a global optimization procedure by using an H ∞ formulation.

3 A Solution of the Transformed GILC Problem It is shown that the equation

ξ k = J k (θ k ) = J k ( ϒ1θ k −1 + ϒ 2ξ k −1 )

is highly nonlin-

θ k , which depends on the form of the basis functions. Within the k th batch, one can obtain J k (θ k ) = Fθ k + G1r + G2δ k −1 to represent the linearization of J k (θ k ) with respect to θ k , where F , G1 and G2 are known matrices, r is a known input related to the target PDF, and δ k −1 is the modelling error and unear with respect to

modelling dynamics which can be supposed

δk

2

< 1 by selecting the proper G2 .

As such, the original system can be further described as

⎧θ k = ϒ1θ k −1 + ϒ 2ξ k −1 ⎨ ⎩ξ k = F ϒ1θ k −1 + F ϒ 2ξ k −1 + G1r + G2δ k −1

(4)

It is shown that the transformed formulation (4) can be considered as a model-based ILC problem [9,10]. In this paper, in order to guarantee the convergent performance and the robust performance, a robust optimization problem will be studied. Define ρ k = W1θ k + W2ξ k as the reference output, where Wi (i = 1, 2) are two weighting matrices. Then we can formulate the following to find

H ∞ optimization problem:

ϒi (i = 1, 2) such that (4) is asymptotically stable in the absence of the ex-

ogenous inputs and satisfies

ρk

2

< ν , where ν is a given parameter. To simplify

the description, in this paper we suppose that the initial condition is zero. The following result gives a sufficient and necessary condition for the above optimization problem based on LMIs. Denote

ϒ = [ ϒ1

ϒ 2 ] , W = [W1 W2 ] , A = F ϒ , and

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⎡I ⎤ ⎡0 0⎤ ⎡θ k ⎤ ⎡r⎤ F =⎢ ⎥, G=⎢ , ϕk = ⎢ ⎥ , d k = ⎢ ⎥ ⎥ ⎣ F1 ⎦ ⎣G1 G2 ⎦ ⎣ξ k ⎦ ⎣δ k ⎦ Theorem. System (4) is asymptotically stable in the absence of

ρk

2

< ν if and only if there are matrices Q > 0 and R satisfying ⎡ −Q + W T W ⎢ 0 ⎢ ⎢ FR ⎣

where

d k and satisfies

γ =

ν2

2

r 2 +1 2

0 −γ 2 I G

RT F T ⎤ ⎥ GT ⎥ < 0 −Q ⎥⎦

(5)

. In this case, the learning rates can be calculated by

ϒ = RQ −1 . Proof: (4) can be rewritten as

ϕ k +1 = Aϕ k + Gd k , for which it has been shown that

it is asymptotically stable in the absence of only if there exist a matrix

d k and satisfies ρ k

P > 0 satisfying

⎡ AT PA − P + W TW ⎢ GT PA ⎣

2 2

< γ 2 dk

2 2

if and

AT PG ⎤ ⎥<0 −γ 2 I ⎦

Q = P −1 it can be verified that (5) is equivalent to the above inequality with R = ϒQ . In this case, it can be seen that Using Schur complement and

ρk

2 2

< γ 2 dk

2 2

(

)

≤ γ 2 r 2 + 1 . Q.E.D 2

Remark: The above result also provided an available design algorithm for the learning rates, where (5) is a linear matrix inequality (LMI) with respect to Q > 0 and R . PI-type GILC is expected to eliminate the static error, which will be considered in the future research [8]. Using the proper B-spline basis functions and the generalized distance, it is possible that J k (θ k ) can be selected as a linear function of θ k so that the linearization procedure is unnecessary. In some cases, constraints are required in learning procedures. These topics will be further discussed in the future.

4 Conclusions For the NN modelling procedure for a nonlinear dynamical output function used in SDC problems, we present a new type of ILC law to adjust the parameters in the basis functions. One feature is no classical models are required for the ILC process. Sec-

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ondly, the information for learning is the generalized distance between the measured PDFs and the target PDF. An augmented vector is firstly introduced such that the GILC problem can be transformed into an H ∞ optimization one, where an LMIbased design method can be obtained for the two learning rates. It is noted that the GILC can be applied to some general parameter learning problems in neural network modelling. Acknowledgement. This work is supported by the NSF of China (No. 60474050 and 60472065), the program of NCET of China and the Short Visiting Fund of the Royal Society of UK.

References 1. Chen, S., Wu, Y., Luk, B. L.: Combined Genetic Algorithm Optimization and Regularized Orthogonal Least Squares Learning for RBF Networks, IEEE Trans. on Neural Networks 10 (1999) 1239-1244 2. Hong, X., Brown, M., Chen, S., Harris, C. J.: Sparse Model Identification Using Orthogonal Forward Regression with Basis Pursuit and D-Optimality, IEE Proc.-Control Theory Appl. 151 (2004) 491-498 3. Wang, H.: Bounded Dynamic Stochastic Systems: Modelling and Control. SpringerVerlag, London (2000) 4. Guo, L., Wang, H.: PID Controller Design for Output PDFs of Stochastic Systems using Linear Matrix Inequalities, IEEE Trans on Systems, Man and Cybernetics, Part-B 35 (2005) 65-71 5. Guo, L., Wang, H.: Applying Constrained Nonlinear Generalized PI Strategy to PDF Tracking Control through Square Root B-Spline Models, International Journal of Control 77 (2004) 1481–1492 6. Guo, L., Wang, H.: Fault Detection and Diagnosis for General Stochastic Systems Using B-Spline Expansions and Nonlinear Filters, IEEE Trans. on Circuits and Systems-I: Regular Papers 52 (2005) 1644-1652 7. Wang, H., Zhang, J. F., Yue, H.: Iterative Learning Control of Output Shaping in Stochastic Systems, Proceedings of the 2005 IEEE International Symposium on Intelligent Control, Cyprus (2005) 1225-1230 8. Xu, J. X., Tan, Y.: Linear and Nonlinear Iterative Control. Springer-Verlag, London (2003) 9. Norrlof, M., Gunnarsson, S.: Experimental Comparison of Some Classical Iterative Learning Control Algorithms, Technical report from the Control & Communication group in Linkoping. http://www.control.isy.liu.se/publications (2002) 10. Amann, N., Owens, D. H., Rogers, E.: Predictive Optimal Iterative Learning Control. International Journal of Control 69 (1998) 203-226

Delayed Learning on Internal Memory Network and Organizing Internal States Toshinori Deguchi1 and Naohiro Ishii2 1

Gifu National College of Technology, Gifu, 501–0495, Japan [email protected] 2 Aichi Institute of Technology, Aichi, 470–0392, Japan [email protected]

Abstract. Elman presented a network with a context layer for the timeseries processing. The context layer is connected to the hidden layer for the next calculation of the time series, which keeps the output of the hidden layer. In this paper, the context layer is reformed to the internal memory layer, which is connected from the hidden layer with the connection weights to make the internal memory. Then, the internal memory plays an important role of the learning of the time series. We developed a new learning algorithm, called the time-delayed back-propagation learning, for the internal memory. The ability of the network with the internal memory layer is demonstrated by applying the simple sinusoidal time-series.

1

Introduction

For time-series learning, Elman’s network[1, 2] has been used[3, 4, 5]. Elman’s network is a simple recurrent network which has a context layer as a copy of the hidden layer, and is able to use the information of the context layer for the next calculation. Since the context layer is a simple copy of the hidden layer, the outputs of the hidden layer are sent directly to the context layer. In this paper, the connection weights are introduced to the connections from the hidden layer to the context layer.[6] With these connections, the context layer is no longer a simple copy of the hidden layer but the internal memory of the network. Therefore, this layer is named the internal memory layer in this paper. It is expected that the network with the internal memory layer is more powerful for learning in time series. For learning of this network, we developed a new learning algorithm, called here, the time-delayed back-propagation learning.[6] In this paper, the internal memory network and the time-delayed backpropagation learning are summarized, and the advantage of this network is presented with a simple example of time-series. J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 502–508, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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Network and Learning Internal Memory Network

Elman introduced the context layer to the 3-layer back-propagation network. In this paper, the connection weights from the hidden layer to the context layer are introduced to Elman’s network as shown in Fig.1. While the context layer in Elman’s holds the outputs of the hidden layer, this layer keeps the internal state calculated from the outputs of the hidden layer. Therefore, this layer is named the internal memory layer, and the 3-layer network with the internal memory layer is called the internal memory network. Input Layer

Output Layer Hidden Layer

Internal Memory Layer

Fig. 1. Internal memory network

Because the network has the internal memory calculated from the output of the hidden layer, the learning ability of this network is expected to be higher than Elman’s. The equations of the internal memory network are defined as follows. N  Nm i   h ih i mh m h xj (t) = f wkj xk (t) + wkj xk (t − 1) − θj , j = 1, · · · , Nh (1) k=0

xm j (t)

k=0

=f

N h 

 hm h wkj xk (t)

−

θjm

−

θjo

k=0

xoj (t)

=f

N h 

,

j = 1, · · · , Nm

(2)

 ho h wkj xk (t)

,

j = 1, · · · , No

(3)

k=0 o where xij (t), xhj (t), xm j (t), and xj (t) are the output of the j-th neuron at step t, in the input layer, in the hidden layer, in the internal memory layer, and in

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the output layer, respectively. θjh , θjm , and θjo are the threshold value of the j-th neuron in the hidden layer, in the internal memory layer, and in the output layer, ih mh ho hm respectively. wkj , wkj , wkj , and wkj are the connection weight from the k-th neuron in the input layer to the j-th neuron in the hidden layer, from the k-th neuron in the internal memory layer to the j-th neuron in the hidden layer, from the k-th neuron in the hidden layer to the j-th neuron in the internal memory layer, and from the k-th neuron in the hidden layer to the j-th neuron in the output layer, respectively. Ni , Nh , Nm , and No are the number of neurons in the input layer, in the hidden layer, in the internal memory layer, and in the output layer, respectively, and f is the output sigmoid function described below in Equation (4), f (x) = 2.2

1 1 + exp(−x)

(4)

Delayed Learning Algorithm

Back-propagation learning[7] is not able to apply directly to the internal memory network, because the internal memory layer does not have the desired output, while the output layer does. For the internal memory networks, time-delayed back-propagation learning — or simply ‘delayed learning’ — is developed. The basic idea of the delayed learning is the expansion of the network in time as shown in Fig.2. The outputs of the internal memory layer at step t − 1 are the inputs of the hidden layer at step t. The errors are to be back-propagated from the hidden layer at step t to the outputs of the internal memory layer at step t − 1. It is not enough when there are few neurons in the output layer, because the errors from the output layer are sometimes so small that the hidden layer is not able to learn. In the delayed learning, the hidden layer at step t − 1 receives the errors from both the output layer at step t − 1 and the internal memory layer at step t − 1. Let yi (t) be the desired value of i-th neuron in the output layer at step t. The error signal of i-th neuron in the output layer is shown as follows: δio = xoi (t)(1 − xoi (t))(yi (t) − xoi (t))

(5)

As same as the usual back-propagation, the error signal of i-th neuron in the hidden layer is No  ho δih = xhi (t)(1 − xhi (t)) δjo wij (6) j=0

Then, calculate the error signal of the internal memory layer at step t − 1. m δ  i = xm i (t − 1)(1 − xi (t − 1)) m

Nh 

mh δjh wij

(7)

j=0

Again, the errors of the output layer and of hidden layer at step t − 1 are calculated by the next equations. δ  i = xoi (t − 1)(1 − xoi (t − 1))(yi (t − 1) − xoi (t − 1)) o

(8)

Delayed Learning on Internal Memory Network supervised signal at t

supervised signal at t-1

step t-1

505

step t

Fig. 2. Time-delayed back-propagation learning

δ  i = xhi (t − 1)(1 − xhi (t − 1)) h

No 

ho δ  j wij + o

j=0

Nm 

hm δ  j wij m

(9)

j=0

Then, at step t − 1, the connection weights are changed as follows: pq pq wij (t + 1) = wij (t) + η δ  j xpi (t − 1), q

(p, q) = (i, h), (m, h), (h, m), (h, o) (10)

When you adopt a momentum term, pq pq pq pq wij (t + 1) = wij (t) + η δ  j xpi (t − 1) + α(wij (t) − wij (t − 1)), q

(p, q) = (i, h), (m, h), (h, m), (h, o)

(11)

If you need another back-step, you can calculate the error signals at t − 2. If you need more, you can get another back-step in the same way. Therefore, the delayed learning provides 1-delay, 2-delay, . . . , n-delay learning.

3

Simulations on Simple Time-Series

To demonstrate the internal memory network and the delayed learning, the sinusoidal values of sin(x) sampled at every π/10 are used. Therefore, this time-series is the cyclic values that come again after 20 steps. Each value was transformed by sin(x)+1 to fit to the range of the neuron from 0 to 1 as in 2 Equation (12).   sin πt 10 + 1 a(t) = (12) 2 In these simulations, 5 types of network are used. One is Elman’s network with 1 neuron for the input, 30 for hidden units, 30 in context layer, and 1

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for the output. The others are the internal memory networks with 1 neuron for input, 30 neurons for the hidden units, 30 neurons in the internal memory layer, and 1 for the output. The internal memory networks have variations of 1-delay, 2-delay, 4-delay, and 8-delay learnings. In the simulations, the input of the network at step t is a(t) and the desired output is a(t + 1). If the output is put to the input continuously, the network will generate the time-series of a(t). In this series, the value 0 comes twice a cycle, and the next values are different. Therefore, the network has to make the internal state which can tell the difference. In these learning, the parameter of the learning is set as α = 0.0005, η = 0.05. One cycle of sampled values of sin(t) by π/10 is counted as one learning time. The networks learned for 100,000,000 learning times. The maximum errors during one cycle at every 1,000,000 learning times are shown in Fig.3. 0.7

Max error in a cycle

0.6

Elman’s Net 1-delay 2-delay 4-delay 8-delay

0.5 0.4 0.3 0.2 0.1 0

1×107 2×107 3×107 4×107 5×107 6×107 7×107 8×107 9×107 1×108 Learning times

Fig. 3. Maximum errors during one cycle at every 1,000,000 learning times

Elman’s network and the networks with 1-delay and 2-delay delayed learning did not lower the errors. At the end of the learning in the 4-delay delayed learning, the maximum error has descended to 0.017. The 8-delay delayed learning shows almost the same results as the 4-delay case. This suggests the lower bound of the delay time in the learning.

Delayed Learning on Internal Memory Network

4

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Organizing Structure

After 100,000,000 learning of 4-delay delayed learning, the outputs of the network are shown in Fig.4.

1 Output I.M. #8 I.M. #14 I.M. #22

0.9 Outputs of neurons

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10 Steps

15

20

Fig. 4. Outputs and the internal state of neuron #8, #14, and #22 in 4-delay learning

The outputs are lined in sinusoidal curve. This means that the network has learned the time-series and that the network has made proper internal states in the internal memory layer. For viewing the internal memory, the outputs of the neurons in the internal memory layer are also shown in Fig.4. First, the output of #22 neuron is a similar shape of cos(t) function which is the derivation of sin(). Next, the output of #14 neuron is a opposite phase of cos(t). The output of #8 neuron is out of phase with sin(t) shape by 2 steps. During the delayed learning, the network organized these internal states.

5

Conclusion

In this paper, the connection weights are introduced to the connection from the hidden layer to the context layer of Elman’s. With this connection weights, the context layer is no longer a simple copy of the hidden layer but the internal memory of the network. With the time-delayed back-propagation learning, the internal memory network has the more ability to learn time series. Then, the advantage of this network is presented with a simple example of time-series, and the part of the organized internal states are reported.

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References 1. Elman, J.L.: Finding Structure in Time. Cognitive science 14 (1990) 179–211 2. Elman, J.L.: Learning and Development in Neural Networks: The Importance of Starting Small. Cognition 48 (1993) 71–99 3. Koskela, T., Lehtokangas, M., Saarinen, J. and Kaski, K.: Time Series Prediction with Multilayer Perceptron, FIR and Elman Neural Networks. Proc. of the World Congress on Neural Networks, INNS Press, San Diego, USA (1996) 491–496 4. Cholewo, T.J., and Zurada, J.M.: Sequential Network Construction for Time Series Prediction. Proc. of the IEEE Intl. Joint Conf. on Neural Networks, Houston, Texas, USA (1997) 2034–2039 5. Giles, C.L., Lawrence,S and Tsoi, A,C.: Noisy Time Series Prediction Using a Recurrent Neural Network and Grammatical Inference. Machine learning, Vol.44, No.1/2. Springer Science+Business Media B.V., Formerly Kluwer Academic Publishers B.V. (2001) 161–183 6. Iwasa, K., Deguchi, T., and Ishii, N.: Acquisition of the Time-Series Information in the Network with Internal Memory (in Japanese). IEICE Technical Report NC2001– 71 (2001) 7–12 7. Rumelhart, D.E., Hinton, G.E., and Williams, R.J.: Learning internal representations by error propagation. In: D.E.Rumelhart & J.L.McClelland (eds.): Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol.1. Cambridge, MA: MIT Press (1986) 318–362

A Novel Learning Algorithm for Feedforward Neural Networks Huawei Chen and Fan Jin School of Information Science & Technology, Southwest Jiaotong University, P.O. Box 406, Chengdu, Sichuan 610031, China [email protected]

Abstract. A novel learning algorithm called BPWA for feedforward neural networks is presented, which adjusts the weights during both forward phase and backward phase. It calculates the minimum norm square solution as the weights between the hidden layer and output layer in the forward pass, while the backward pass adjusts the weights connecting the input layer to hidden layer according to error gradient descent algorithm. The algorithm is compared with Extreme learning Machine, BP algorithm and LMBP algorithm on function approximation and classification tasks. The experiments’ results demonstrate that the proposed algorithm performs well.

1 Introduction The theory of neural network shows that a three layer feedforward net with sufficient number of hidden units of sigmoidal activation type can approximate any continuous or other kinds of function to any desired accuracy[1,2].Although neural network has ability to approximate any function in theory, how to assign appropriate weights of neural network is still a hard problem[3]. Backpropagation (BP) algorithm is the most widely used training algorithm for finding out the weights for feedforward neural networks[4]. The error information is propagated backwards and the weights are adjusted according to gradient descent within the solution’s vector space. Although BP algorithm is a first successful training algorithm for multi-layer feedforward neural networks, it easily converges to local minimums and converges slowly in practice Many variations have been proposed with the aim of making it run faster and making it less likely to become stuck in local minima, such as conjugate gradient backpropagation[5] and LevenbergMarquardt(LMBP) algorithm[6], etc. Both BP algorithm and its variants modify the weights only in the backward pass. Recently, a new learning scheme for single hidden layer feedforward neural networks (SLFNs) called extreme learning machine (ELM) [7] was proposed by Huang. In ELM, the output weights (connecting the hidden layer to the output layer) are directly calculated by using Moore–Penrose generalized inverse, while the input weights (connecting the input layer to the hidden layer) and hidden biases can be randomly chosen. It’s interesting and surprising that ELM can learn very fast as well as achieve good generalization performance. However, ELM may require more J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 509 – 514, 2006. © Springer-Verlag Berlin Heidelberg 2006

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hidden neurons in many cases[8] and ELM can’t make its error small enough because it only calculates the weights once and doesn’t make use of the input weights. In this paper, combining gradient-descent learning with idea of ELM, a novel hybrid algorithm taking advantages of both is proposed. The proposed algorithm can adjust the weights during both forward phase and backward phase according to generalized inverse and error gradient-descent principle respectively.

2 Model and Learning Problem of SLFNs Huang pointed out difference between universal approximation on compact input sets and approximation in a finite set[7]. Huang and Babri also shows that a SLFN with at most N hidden neurons and with almost any nonlinear activation function can learn N distinct observations with zero error[9]. For N arbitrary distinct samples (xi,ti), where xi = [ xi1, xi 2 ,...xin ]T ∈ R n ~ and ti = [ti1, ti 2 ,...tim ]T ∈ R m , standard SLFNs with N hidden neurons and activation function g(x) are mathematically modeled as a linear system ~ N

∑ β g (w • x i

i

j

+ bi ) = o j

j = 1,..., N

(1)

i =1

where wi = [ wi1 , wi 2 ,...win ]T is the weight vector linking the ith hidden neuron and the input neurons, β i = [ β i1, β i 2 ,...β im ]T is the weight vector linking the ith hidden neuron and output neurons, and bi is the threshold of the ith hidden neuron. wi ∗ x j denotes the inner product of wi and xj. If the SLFN with N hidden neurons with activation function g(x) can approximate N distinct samples(xi,ti) with zero error means that Hβ = T

(2)

H is the hidden layer output matrix of the neural network. So for fixed arbitrary input weights wi and the hidden layer biases bi, to train a SLFN is simply equivalent to finding a least-squares solution βˆ of the linear system Hβ = T . βˆ = H +T is the best weights[8,10], where H+ is the Moore-Penrose generalized inverse.

3 Bi-phases Weights Adjusting(BPWA) Algorithm Based on Moore-Penrose generalized inverse and the structure of SLFN, Huang proposed ELM. To be pointed out, although βˆ is the minimum norm least square solution, it can’t assure that output error is smaller than error threshold or make the output error small enough for lack of making use of the input weights. Another problem is that ELM always requires more hidden neurons than other learning algorithm. Incorporating generalized inverse with backpropagation, a hybrid scheme is proposed here. In this algorithm activation functions of hidden neurons are assumed as

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sigmoidal function. The algorithm caculates the minimum norm square solution as the weights between the hidden layer and output layer in the forward pass, while the backward pass, on the other hand, adjusts the weights connecting the input layer to hidden layer according to error gradient descent algorithm. Since it adjusts the weight during both phases, it’s called Bi-Phases’ Weights Adjust learning algorithm. The main thought of BPWA is illustrated in Fig. 1. From Fig.1, BPWA divides the process of weight adjusting into two processes during which it calculates or modifies different parts of weight vectors. Learning algorithm of BPWA is described as follows. Forward Pass: Adjusting the output weights

Backward Pass: Adjusting the input weights Fig. 1. The diagram of Bi-Phases Weights Adjusting algorithm

3.1 Output Weights Calculation During Forward Pass

For given any fixed arbitrary input weights and the hidden layer biases, 1. Calculate the hidden layer output matrix H, 2. According to H, compute βˆ = H +T that minimizes the error between the output and target, 3. Calculate the error between neural network’s output Hβˆ and target T. If error is less than given error threshold or the learning epoch has reached the maximum epoch, it stops else the learning process continues into backward pass. 3.2 Input Weights Adjusting Based on Gradient Descent During Backward Pass

Backward pass modifies the weights connecting input layer to hidden layer including hidden layer biases. To be noted, the output weights connecting hidden layer to output layer don’t change during this pass, only input weights can be adjusted according to error gradient descent. Forward pass calculates the optimal output weights that minimize the error, backward pass adjusts the input weights when the error is larger than error threshold. So the converge speed of this algorithm is fast due to adjusting weight during both phases. Different from traditional training algorithm, BPWA can train the parameters of neural network during both passes. Due to the minimum norm least square solution, the output weights can get small learning error as well as small generalization error after BPWA succeeded in training the neural network.

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4 Performance Evaluation In this section, the performance of the proposed algorithm is tested and compared with the BP algorithm, LMBP algorithm and ELM on some problems in the function approximation and classification areas. The simulations for these learning algorithms are carried out in Matlab 6.0 running in an AMD Athlon XP 1800+ CPU with 384MB memory. All the results shown in this paper are the mean values of 100 trails. 4.1 Two-Dimension Function Approximation

Target function is shown as follows[11] f ( x1 , x 2 ) = 3(1 −

2 2 x12 )e − x1 − ( x 2 +1)

− 10(

，

x1 5

−

x13

−

2 2 x 25 )e − x1 − x 2

e − ( x1 +1) − 3

2

− x 22

(3)

where x1 ∈ [−2,2] x2 ∈ [−2,2] For training purpose, the outputs are normalized to the interval (−1, 1). For each trial, the number of maximum epochs was fixed at 1000 epochs of exemplar presentation. An experiment was deemed to be successful, if it achieved a mean squared error (mse) value of 0.04. The results conducted are presented in the table 1. The architecture of neural network used here is 2-7-1. MSE is the mean of the mse achieved in all experiments for the task, SEMSE is the average of the mse in successful experiments, and SucRate is the mean of the ratio of the number of experiments succeeded to achieve the mse of 0.04 to total number of experiments. Table 1. Performance of BPWA, ELM, BP, LMBP on approximating f(x1,x2)

Algorithm BPWA ELM BP LMBP

MSE 0.027227 0.028618 0.10816 0.030121

SEMSE 0.027227 0.02755

－

0.030121

SucRate(%) 100 95 0 100

Training Time(s) 0.02312 0.02172 10.7438 0.67641

In the experiment, BP can’t achieve the error threshold when maximal learning epochs arrived, failing in this experiment. ELM really has fast learning speed, but it can’t finish the task every time. Although both BPWA and LMBP work well, it’s obvious that BPWA outperforms LMBP. 4.2 Benchmarking with Iris Problem

Iris database is a famous database in the pattern recognition field and always used for benchmarking for different methods. The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. Every instance has 4 attributes. We used 4-6-3 neural network for classifying the iris types. The whole dataset divided into two sets: training set and testing set. The two sets both have 75 instances randomly selected from the iris database and no intersect set exists between them.

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For one instance, if three output values satisfy the relation below yid − yi < θ

(4)

θ

The classification on this instance is considered as correct, else, false. is threshold ,which is set to The correct rate is defined as (8), which is one performance parameter evaluating the classification.

θ=0.5.

E=

n × 100% N

(5)

where n is the number of correct classification instances, N is size of training set or testing set. Training SucRate means the mean correct classification ratio over training set, Testing SucRate is the mean correct classification ratio over test set. The mean results over 100 trials are summarized in Table 2. Table 2. Performance of the BPWA,ELM,BP and LMBP over Iris database

Algorithm BPWA ELM BP LMBP

Training SucRate(%) 98.227 85.493 22.36 99.88

Testing SucRate(%) 94.56 80.747 20.987 93.533

Training Time(s) 0.06297 0.00953 0.47875 0.48562

Form Table above, ELM trained fastest, while its correct classification ratios are below 90 percent whether in training process or testing process. BP is the worst again. BPWA and LMBP yield similar correct classification ratio for training, while the correct ratio for testing of BPWA is slightly better than LMBP and the training time of BPWA is one eighth of LMBP. From experiments above, we can see that BPWA did well in the simulations. Since LMBP is one of the best gradient-based learning algorithms and ELM is fastest learning algorithm, we can draw a conclusion that BPWA performs well on both learning speed and generalization ability.

5 Conclusion This paper proposed a new learning algorithm for feedforward neural networks– BPWA, which can adjust the weights during both forward phase and backward phase. The algorithm calculates the minimum norm square solution as the weights between the hidden layer and output layer in the forward pass, while the backward pass, on the other hand, adjusts the weights connecting the input layer to hidden layer according to error gradient descent algorithm. Experimental results show that the proposed algorithm performs well comparing with BP algorithm, LM-BP algorithm and ELM Although BPWA is discussed based on single hidden layer feedforward neural networks, it’s obvious that this algorithm can also be applied to training multilayer feedforward neural networks.

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References 1. Hornik, K., Stinchcombe, M., White, H.: Multilayer Feedforward Networks are Universal Approximators. Neural Networks 2(5) (1989) 359-366 2. Chen, T, Chen, H.: Universal Approximation to Nonlinear Operators by Neural Networks with Arbitrary Activation Functions and Its Application to Dynamical Systems. IEEE Trans. Neural Networks 6(4) (1995) 911-917 3. Judd, J. S.: Learning in Networks is Hard. In: Proc. the 1st IEEE International Conference on Neural Networks. Vol. 2. New York (1987) 685-692 4. Rumelhart, D. E., Hinton, G. E., Williams, R. J.: Learning Internal Representations by Error Propagation. In: Parallel Distributed Processing, Vol.1 MIT Press, Cambridge (1986) 5. Hagan, M. T., Menhaj, M. B.: Training Feedforward Networks with the LevenbergMarquardt Algorithm. IEEE Trans. on Neural Networks 5(6) (1994) 989-993 6. Huang, G.-B., Zhu, Q.-Y., Siew, C.-K.: Extreme Learning Machine: a New Learning Scheme of Feedforward Neural Networks. In: Proceedings 2004 International Joint Conference on Neural Networks Vol. 2. (2004) 985-990 7. Huang, G.-B., Zhu, Q.-Y., Siew, C.-K.: Saratchandran P., Sundararajan N.: Can Threshold Networks Be Trained Directly?. IEEE Transactions on Circuits and Systems II: Express Briefs : Accepted for future publication 8. Huang, G.-B., Babri, H. A.: Upper Bounds on the Number of Hidden Neurons in Feedforward Networks with Arbitrary Bounded Nonlinear Activation Functions. IEEE Trans. on Neural Networks 9(1) (1998) 224-229. 9. Bartlett, P. L.: The Sample Complexity of Pattern Classification with Neural Networks: the Size of the Weights Is More Important than the Size of the Network. IEEE Trans. on Information Theory 44(2) (1998) 525-536 10. Chandra, P., Singh, Y.: An Activation Function Adapting Training Algorithm for Sigmoidal Feedforward Networks. Neurocomputing 61(10) (2004) 429-437

On H∞ Filtering in Feedforward Neural Networks Training and Pruning He-Sheng Tang1, Song-Tao Xue2, and Rong Chen1 1

Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, 200092, China [email protected] 2 Department of Architecture, School of Science and Engineering, Kinki University, Kowakae3-4-1, Higashi Osaka City, 577-0056, Japan

Abstract. An efficient training and pruning method based on H∞ filtering algorithm is proposed for Feedforward neural networks (FNNs). A FNNs’ weight importance measure linking up prediction error sensitivity obtained from H∞ filtering training and a weight salience based pruning technique are derived. The results of extensive experimentation indicate that the proposed method provides better pruning results during the training process of the network without losing its generalization capacity, also provides a robust global optimization training algorithm for given arbitrary network structures.

1 Introduction For neural networks (NNs) design, there are two crucial problems: one is the choice of a ‘fast’ and ‘robust’ training algorithm, and the other is the choice of a suitable or, ideally, minimal NNs topology to be adopted. In neural network training, the most well-known online training method is the backpropagation algorithm (BPA) [1], which is virtually a first-order stochastic gradient descent method and shows slow learning speed [2]. To overcome the slowness, many modified schemes based on the classical nonlinear programming technique have been suggested to speed up the training [3][4]. Recently, a class of second-order descent methods inspired by the theory of system identification and nonlinear filtering [5] has been introduced to estimate the weights of a neural network. Extended Kalman filter (EKF) [6][7][8] and Recursive least square (RLS) method have been applied to multilayer perceptron [9][10][11]. In the above mentioned EKF algorithm, although the learning speed is improved, the method requires the knowledge of the noise statistics. Convergence of this algorithm as well as the final values depends, to great extent, on this initial guess. The authors have presented suboptimal H∞ filtering to train feedforward multilayer network which is independent on noise statistics [12]. Besides the training algorithms, another concern encountered in the practical application of the NNs is the choice of suitable model architecture. Since an unsuitable topology will increase the training time or even cause non-convergence, it usually decrease the generalization capability of the network. If there are too few weights, the network may not be trained to learn the training data for the system mapping. On the J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 515 – 523, 2006. © Springer-Verlag Berlin Heidelberg 2006

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other hand, if the network size is too large, weights overfitting problems may usually occur and thus lead to worse generalization capacity[13][14]. Thus, in order to eliminate unnecessary weights, the pruning algorithm is applied. There are different pruning or model selection methods, such as Akaike Information Criterion (AIC) and cross-validation techniques [15][16] which require tens of networks to be exhaustively trained before the correct network size is determined, or simple weight decay method [17[18], or error sensitivity-based Optimal Brain Damage (OBD) [19] and Optimal Brain Surgeon (OBS) [20] methods, or OBD-like pruning methods [21][22], or growing methods [4] which may be sensitive to initial conditions and become trapped in local minima[23] . As the H∞ filtering was shown to be more efficient and robust than the Kalman filter[24][25], it would be interesting to inquire if there is any possibility of applying H∞ filtering training method together with network pruning. The objective of the present study is to develop a FNNs training and pruning method based on H∞ filtering algorithm for identification of nonlinear systems. The presented new method is able to reduce the complexity of the network during the training without diminishing the network’s estimation capacity. Also, independent of the statistics of the disturbances of the network’s inputs and outputs, the presented method provides a natural global optimization training algorithm for given arbitrary network structures. Examples of nonlinear system identification are given to verify the usefulness and effectiveness of the proposed method.

2 H∞ Filtering Algorithm in Neural Network Training Let y k = f (w k , u k ) be the transfer function of a single-layer FNNs where y k is the output, u k is the input and w k is its parameter vector that is combined by the weight 1 2 3 matrices w , w and w . Given a set of training data, the training of a neural network can be formulated as a filtering problem [6][8]. In this case, a discrete-time FNNs’ behavior can be described by the following nonlinear state-space model: w k +1 = w k + v k

(1)

y k = f (w k , u k ) + n k

(2)

Eq. (1) is known as process equation, where v k is process noise, the state of system is given by the network’s weight parameters values w k . Eq. (2) is the observation or measurement equation, represents the desired network response vector y k as a nonlinear function f (•) of the input vector u k and the weight parameter vector w k ; this equation is augmented by random measurement noise n k . To apply the optimal H∞ filtering algorithm, linear Taylor approximation of the ˆ −k −1 (prediction of w k ), u k is considered here, that is f (w k , u k ) at w ˆ −k −1 , u k ) + Ck (w k − w ˆ k−−1 ) f ( w k , u k ) ≈ f (w

where C k = ∂f ∂w | u =u ,w = wˆ − . A new quantity is introduced as follows: k k −1

(3)

On H∞ Filtering in Feedforward Neural Networks Training and Pruning ˆ −k −1 , u k ) + Ck w ˆ −k −1 ηk = y k − f ( w

517

(4)

The entries in the term ηk are all known at time k, and, therefore, ηk can be regarded as an observation vector at time k. Hence, the nonlinear model (Eq. (2)) is approximated by the linear model ηk = C k w k + n k

(5)

The problem addressed by the H∞ filtering is to find an estimate wˆ k of w k given u j , η j ( j = 0,1,L k ) . The suboptimal H∞ estimation [25][26] is interested not necessarily

in the estimation of w k but in the estimation of some arbitrary linear combination of w k using the noise-corrupted observations η j ( j = 0,1,L k ) i.e., zk = Lk wk

(6)

where L k ∈ R q×n . Different from that of the modified Wiener/Kalman filter which minimizes the variance of the estimation error, the design criterion of the H∞ filter is to provide a uniformly small estimation error, z k − zˆ k , for any n k ∈ l2 and w 0 ∈ R n . The H∞ filtering will search zˆ k such that the optimal estimate of z k among all possible zˆ k in the sense that the supremum of the performance measure should be less than a positive pre-chosen noise attenuation factor γ 2 , i.e., the worse-case performance measure

∑

sup x 0 ,{n k , v k }

ˆ0 w0 − w

2 P0−1

N −1 k =0

z k − zˆ k

{

2

+ ∑k =0 n k N −1

2

+ vk

2

}

<γ2

(7)

where wˆ 0 is an a priori estimate of w 0 and w 0 − wˆ 0 represents unknown initial condition error, P0−1 > 0 is weighting matrix. P0−1 > 0 denotes a positive definite matrix that reflects a priori knowledge on how the initial guess wˆ 0 close to w 0 is. Let γ > 0 be a prescribed level of noise attenuation. If this is the case, an optimized H∞ filtering algorithm for neural network training can be derived: ˆk =w ˆ −k −1 + K k ( ηk − Ck w ˆ k−−1 ) w ˆ −k −1 + K k (y k − f (w ˆ k−−1 , u k )) , w ˆ k− = w ˆk ,w ˆ −−1 = w ˆ0 =w

(8)

where wˆ k is an a posteriori estimate of the state at step k , the gain K k of the filter is given by K k = Pk C Tk (I + C k Pk C Tk ) −1

(9)

Pk +1 = Pk (I + C Tk C k Pk − γ −2 LTk L k Pk ) −1

(10)

where the attenuation factor γ must be tuned so as to satisfy the Pk positive definite.

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3 H∞ Filtering in Neural Network Pruning In this section, the conjunction of network training and pruning with the H∞ filtering algorithm will be illustrated. Without loss of generality, the network employed here is considered as a feedforward architecture with nI input units, nH hidden sigmoid unites and a single linear output unit. The initial network is fully connected between layers and implements a nonlinear mapping from input space u k to target output space yˆ k = f k (u k , w k ) , where f (•) is the actual output mapping function, w is the network parameters and yˆ k is the prediction of the target output yk . Then, for a given training set, the cost function can be expressed as: Ε(w ) =

1 2N

∑ ( yk − f k ) N

2

(11)

k =1

where N is the number of training examples. Under the assumption that the network is fully trained, that is, the cost function Ε has adjusted to a local or global minimum on the error surface, the second derivative of Ε with respect to w or the Hessian matrix [19] can be approximated as: Η( N ) ≈

1 N

⎛ ∂f ⎞⎛ ∂f ⎞ ∑ ⎜ ∂wk ⎟⎜ ∂wk ⎟ k =1 ⎝ ⎠⎝ ⎠ N

T

(12)

To illustrate the connection between the matrix Pk and the Hessian matrix Η of the cost function, the Riccati Eq. (10) can be rewritten in an alternative form that will be more convenient for analysis. By employing the following matrix inversion lemma (MIL), the following update for Pk−1 is obtained Pk−+11 = Pk−1 + CTk Ck − γ −2LTk L k

(13)

Suppose that the weight parameter and the ‘error covariance matrix’ Pk are both converge. Without loss of generality, it is convenient to select the matrix L k equal to Ck . The Riccati recursion Eq. (13) with initial condition P0 can be rewritten in the form of a recursion as: k

Pk−+11 = P0−1 + (1 − γ −2 )∑ CTi Ci

(14)

i =0

From Eq. (12) and Eq. (14), the inversion of Hessian matrix of the cost function is approximately expressed as:

[

Η −1 ≈ N (1 − γ −2 )Pk +1 I − (Pk +1 − P0 )−1 Pk +1

]

(15)

The pruning procedure simplifies the computations by making a further assumption of the Hessian matrix Η being a diagonal matrix [19]. Thus the saliencies for each parameter are as follows:

On H∞ Filtering in Feedforward Neural Networks Training and Pruning

Si =

1 [Η]i ,i w[2i ] 2

519

(16)

where [Η ]i,i is the i-th diagonal element of the Η and w[i ] is the i-th weight. The pruning strategy is to find the low-saliency or smallest saliency parameters which are then selected for deletion. The weights are pruned according to this pruning order until the change in the estimated training error is greater than the tolerance limit.

4 Illustrative Numerical Examples A non-linear hysteretic system [27] is selected for example analysis to illustrate the applicability of the proposed method. The non-linear system is described by the following second order difference equation: d 2u (t ) du (t ) + z( , u (t ), z (t )) = p(t ) 2 dt dt

(17)

2 3 dz (t ) du (t ) du (t ) du (t ) = 24.5 −2 z (t ) z (t ) − 0 . 5 z (t ) dt dt dt dt

(18)

where u (t ) and p(t ) are system’s output and input, respectively. The FNNs is taken into account to model the dynamic behaviors of the system, namely, the FNNs is trained to identify z (t ) . An FNNs with 3 input neurons ( du (t ) dt , u (t ) and z (t − 1) ), 30 hidden neurons (with hyperbolic tangent activation function), and one output neuron ( z (t ) ,with linear activation function) is trained to capture the unknown system. In this case, the total number of weights is 120. In this study, the effectiveness of the noise injection training is also investigated; noises are artificially added to the training data. The noise level is defined as the value of the standard deviation. For instance, if the standard deviation is 0.05, the noise level in the data can be referred as 5% in the root-mean-square (RMS) level. Training and test samples for the non-linear system identification problem are shown in Fig.1. The training set contains 1000 samples with 3% noise in RMS level and test set contains 500 samples. Numbers of training iterations of some parts of estimated network’s weights are shown in Fig.2. Some weights converge to stable values very quickly in first 500 iterations. This fast convergence demonstrated that the H∞ filtering is a very fast training algorithm. Last step values of the Pk (120×120 matrix, Fig. 3) of the Riccati recursion show that the matrix Pk is almost diagonal dominant. Fig.4 shows the estimated training error against the number of weights in the pruned network. It is clear that nearly only 70 weights are enough to capture the unknown system without increasing the training error dramatically. After training, another 500 pairs of testing data are passed to the pruned network. Fig. 5 depicts a segment of the actual and the desired network output for the test set. It shows that output of the pruned network is quite close to the desired output.

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H.-S. Tang, S.-T. Xue, and R. Chen p(t)

20 0 -20

(a)

10 u(t) 0 -1 -2 40 2 du(t)/dt 0 -2 -4 50 z(t) 0 -5

500

1000

1500

1000

1500

(b)

500 (c)

500 Training set

1000

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Test set

(d)

0

500

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Fig. 1. Training and test samples for the system identification problem 0.2 0 -0.2 0.5 0 -0.5 0.5 0 -0.5 0.5 0 -0.5 0.2 0 -0.2

w1 w2 w3 w4 w5 0

100

200

300

400

500

600

700

800

900

1000

Training iteration

Fig. 2. Convergence of the estimate weights

Pk

Mean square training

3

2

1

0

0 20 40 60 80 100 120 No. of weights in the pruned network

Fig. 3. Values of the Pk after trained H∞-network

Fig. 4. Number of weights in the pruned network versus training errors

On H∞ Filtering in Feedforward Neural Networks Training and Pruning 2 0 -2

521

z(t) (a)

1000 1050 2 z(t) 0 -2

1100

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1200

1250

1300

1350

1400

1450

1500

1300

1350

1400

1450

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(b)

1150

1200

1250

Discrete time

Fig. 5. A segment of the actual and the desired network output for the test set in the system identification example. Solid line represents the desired output; the dotted line corresponds to the actual output of: (a) the unpruned network; (b) the network pruned by our approach

To demonstrate that the generalization capability is not affected much if some weights in the network are pruned, another set of input signal (the test set) is fed for the pruned network. A system input p(t ) (Fig.6 (a)) is selected for the validation set. According to this input, corresponding desired output for this validation set are obtained. Fig. 6(b) depicts a segment of the actual and the desired network output corresponding to the test input. The instantaneous error is shown in Fig. 6 (c). The figure shows that the generalization capability of the pruned networks does not change much. 20 0 -20

p(t) (a)

40 2 0 -2 -4 20

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Fig. 6. A segment of the actual and the desired network output for the validation set in the system identification example. (a) Validation input of system; (b) comparison between actual output (solid line) and the pruned H∞-network output (dashed line); (c) instantaneous error

5 Conclusions In this paper, we have derived an efficient FNNs training and pruning method using the H∞ filtering algorithm. FNNs model pruned with H∞ filtering provides a good architecture design for generalization capacity and requires a smaller number of connection weights than a totally connected net. Examples of nonlinear system identifica-

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tion are carried out to evaluate the performance of the network. The results show the effectiveness of the neural network pruning and training by H∞ filtering algorithm.

References 1. Rumelhart, D., Hinton, G., and Williams, G.: Learning Internal Representations by Error Propagation. Parallel Distributed Processing, Vol.1, Cambridge, MA: MIT Press (1986) 318-362 2. Robbins, H. and Monro, S.: A Stochastic Approximation Method. Ann. Math. Stat. 22 (1951) 400-407 3. Bojarczak, O. S. P., and Stodolski, M.: Fast Second-order Learning Algorithm for Feedforward Multilayer Neural Networks and Its Application. Neural Networks 9(9) (1996) 1583–1596 4. Haykin, S.: Neural Networks: A Comprehensive Foundation, Prentice Hall, Inc. (1999) 5. Anderson, B. D. O., and Moore, J. B.: Optimal Filtering. Englewood Cliffs, NJ: PrenticeHall (1979) 6. Singhal, S., and Wu, L.: Training Multiplayer Perceptrons with the Extended Kalman Algorithm. In: Touretzky,D.S.(eds.): Advances in Neural Information Processing Systems, Vol.1, Morgan Kaufmann, San Mateo, CA (1989) 133-140 7. Fukuda, W. K. T., and Tzafestas, S. G.: Learning Algorithms of Layered Neural Networks via Extended Kalman Filters. Int. J. Syst. Sci. 22(4) (1991) 753–768 8. Iiguni, Y., Sakai, H., and Tokumaru, H.: A Real-time Learning Algorithm for a Multilayered Neural Network Based on the Extended Kalman Filter. IEEE Trans. Signal Processing 40(4) (1992) 959–966 9. Chen, S., Cowan, C. F. N., Billings, S. A., and Grant, P. M.: Parallel Recursive Prediction Error Algorithm for Training Layered Neural Network. Int. J. Contr. 51(6) (1990) 1215–1228 10. Kollias, S., and Anastassiou, D.: An Adaptive Least Squares Algorithm for the Efficient Training of Artificial Neural Networks. IEEE Trans. Circuits Syst. 36(8) (1989) 1092–1101 11. Leung, C. S., Wong, K. W., Sum, J., and Chan., L. W.: On-line Training and Pruning for RLS Algorithms. Electron. Lett. 32(23) (1996) 2152–2153 12. Tang, H., and Sato, T.: Structural Damage Detection Using the Neural Network and H∞ Algorithm. In: Kundu, T. (eds.): Health Monitoring and Smart Nondestructive Evaluation of Structural and Biological Systems III, Proceedings of SPIE, Vol. 5394, San Diego, CA (2004) 454-463 13. Abrahart, R.J., See, L., and Kneal, P.E.: Investigating the Role of Saliency Analysis with Neural Network Rainfall-runoff Model. Computer & Geosciences 27(8) (2001) 921-928 14. Makarynskyy, O., Pires-Silva, A.A., Makarynska, D., and Ventura-Soares, C.: Artificial Neural Networks in Wave Predictions at the West Coast of Portual. Computers & Geosciences 31(4) (2005) 415-424 15. Akaike, H.: A New Look at the Statistical Model Identification. IEEE Trans. Automat. Control AC-19 (6) (1974) 716-723 16. Stone, M.: An Asymptotic Equivalence of Choice of Model by Cross-validation and Akaike's criterion. Roy. Stat. Soc. Ser. 39(1) (1977) 44-47 17. Krogh, A., and Hertz, J.: A Simple Weight Decay Can Improve Generalization. In: Touretzky, D.S. (ed.): Advances in Neural Information-Processing Systems, Vol.4, Morgan Kaufmann, San Mateo, CA (1992) 950–957

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18. William, P.M.: Bayesian Regularization and Pruning Use a Laplace Prior, Neural Comput. 7(1) (1995) 117– 143 19. LeCun, Y., Denker, J. S., and Solla, S. A.: Optimal Brain Damage. In: Touretzky, D. S. (ed.): Advances in neural information processing, Vol.1, Morgan Kaufman, San Mateo, CA (1990) 396–404 20. Hassibi, B., and Stork, D. G.: Second-order Derivatives for Network Pruning: Optimal Brain Surgeon. In: Hanson, S. J., Cowan, J. D., and Lee Giles, C. (eds.): Advances in neural information processing, Vol.4, Morgan Kaufman, CA (1993) 164-171 21. Sum, J., Leung, C., Young, G. H., and Kan, W.: On the Kalman Filtering Method in Neural-Network Training and Pruning. IEEE Trans. on Neural Networks 10(1) (1999)161-166 22. Leung, C., Wong, K., Sum, P., and Chan, L., A Pruning Method for the Recursive Least Squared Algorithm. Neural Networks 14(2) (2001) 147-174 23. Prechelt, L.: A quantitative Study of Experimental Evaluations of Neural Network Learning Algorithms: Current Research Practice. Neural Networks 9 (3) (1996)457-462 24. Sato, T., and Qi, K.: Adaptive H∞ fillter: Its Application to Structural Identification. Journal of Engineering Mechanics 124(11) (1998) 1233-1240 25. Hassibi, B., Sayed, A.H., and Kailath, T.: Indefinite-Quadratic Estimation and Control: A Unified Approach to H2 and H∞Theories, SIAM (1998). 26. Didinsky, G., Pan, Z., and Basar, T.: Parameter Identification of Uncertain Plants Using H∞ Methods. Automatica 31(9) (1995) 1227-1250 27. Wen, Yi-K.: Method for Random Vibration of Hysteretic Systems. Journal of Engineering Mechanics 102(EM2) (1976) 249-263

A Node Pruning Algorithm Based on Optimal Brain Surgeon for Feedforward Neural Networks Jinhua Xu1 and Daniel W.C. Ho2 1

Department of Computer Sciences, East China Normal University [email protected] 2 Department of Mathematics, City University of Hong Kong [email protected]

Abstract. In this paper, a node pruning algorithm based on optimal brain surgeon is proposed for feedforward neural networks. First, the neural network is trained to an acceptable solution using the standard training algorithm. After the training process, the orthogonal factorization is applied to the output of the nodes in the same hidden layer to identify and prune the dependant nodes. Then, a unit-based optimal brain surgeon(UB-OBS) pruning algorithm is proposed to prune the insensitive hidden units to further reduce the size of the neural network, and no retraining is needed. Simulations are presented to demonstrate the effectiveness of the proposed approach.

1

Introduction

The feedforward neural network (NN) is one of the most popular neural network topologies. One of the difficulties of using the feedforward neural network is to determine the network architecture. In general, when a network is too small, it may not learn the problem well. However, a network which is too large may lead to overfitting and poor generalization performance. Algorithms that can find an appropriate network architecture automatically are thus highly desirable. There are three major approaches to tackle this problem. They are pruning algorithms [1], constructive algorithms [2] and regularization [3], respectively. Recent interest has been growing on pruning algorithms that start with an oversized network and remove unnecessary network parameters, either during training or after convergence to a local minimum. Network parameters that are considered for removal are individual weights, hidden units and input units. Sensitivity analysis has been used successfully to prune irrelevant parameters from feedforward neural networks. Sensitivity analysis techniques quantify the relevance of a network parameter as the influence that small parameter perturbations have on a performance function. The sensitivity measures could fail to identify possible correlations among nodes [4]. Other post-training pruning algorithms based on the correlations among nodes are proposed in [5, 6, 7]. In this paper, a new post-training pruning algorithm is proposed for feedforward neural networks. After the training process, the orthogonal factorization is applied to the output of the nodes in the same hidden layer to identify J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 524–529, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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and prune the linear dependant nodes. Then, a new sensitivity measure based on optimal brain surgeon(UB-OBS) is proposed to prune the insensitive hidden units to further reduce the size of the neural network, and no retraining is needed.

2

Problem Formulation

The neural networks considered in this paper are feedforward neural networks. For simplicity, the algorithm is derived for feedforward neural networks with one hidden layer. It is straightforward to extend the algorithm to multiple hidden layer networks. For one hidden layer feedforward network, the input-output relationship can be represented as ˆ = W2 σ(W1 x + B1 ) + B2 , y (1) where x ∈ Rn , y ∈ Rp are the network input and output, respectively. n is the input dimension and p is the output dimension. W1 ∈ Rm×n and W2 ∈ Rp×m are the weight matrices from the input layer to the hidden layer and from the hidden layer to the output layer, respectively. B1 ∈ Rm and B2 ∈ Rp are the bias vectors of the hidden neurons and output neurons respectively, where m is the number of the hidden neurons. σ is an activation function, such as sigmoid functions. With batch learning, a set of training patterns {x(t), y(t), t = 1, · · · , N } is given. The output of the ith hidden node for the tth training pattern is denoted as σi (t) = σ(Wi1 x(t) + b1i ). Define the matrix Φ := [φ1 , φ2 , · · · , φm , φm+1 ],

(2)

where φi = [σi (1), σi (2), · · · , σi (N )]T , i = 1, · · · , m, and φm+1 (j) = 1 for j = 1, · · · , N is the output of a bias unit for all training patterns. The output of the network is denoted as Yˆ := [ˆ y (1), yˆ(2), · · · , yˆ(N )]T and let W := [W 2 , B 2 ] to obtain ΦW T = Yˆ .

(3)

then the training problem can be formulated as an optimization problem as follows: min V (θ) = tr{(Y − ΦW T )T (Y − ΦW T )} (4) θ

where Y = [y(1), y(2), · · · , y(N )]T , and θ represents the weight parameters of the NN. The standard NN training algorithms such as backpropagation (BP) or Newton’s method can be used to train the NN to a local minimum.

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Dependent Nodes Pruning Algorithm Using QR Factorization

Consider a network trained by the standard training algorithm to a local minimum of the error function V in equation (4). Next, QR factorization with column pivoting is applied to matrix Φ to detect the independent columns, that is Φ · P = Q · A,

(5)

where Q is a N ∗ N unitary matrix, A is an upper triangular matrix of the same dimensions as Φ, P is a permutation matrix with PT P = PPT = I, which is chosen to make the absolute value of the diagonal elements of A in decreasing order. If |arr |/|a11 | < ρ, ρ is a very small positive number, then |ajj |/|a11 | < ρ for j = r + 1, · · · , min(m + 1, N ), this means there are r nonzero diagonal elements in A, and therefore there are r independent columns in matrix Φ. Decompose A in appropriate block matrix forms as   A11 A12 A= , (6) 0 0 where A11 is an r ∗r upper triangular matrix, and r is the number of the nonzero ¯ := WP, and it diagonal elements in A. Substitute equation (5) into (3), let W implies that ¯ T = QT Y. ˆ AW (7) ¯ Correspondingly, decompose Q, W in appropriate block matrix forms as Q = [Q1 , Q2 ],

¯ = [Wr , Wp ], W

(8)

where Wr and Wp are p × r and p × (m + 1 − r) matrices, representing the output weight of independent nodes and dependent nodes, respectively.   T   T  Wr A11 A12 Q1 ˆ = Y (9) 0 0 WTp QT2 ˆ = 0. Setting Wp = 0, the independent weight Wr can One can verify that QT2 Y be easily obtained by solving the following linear equation: ˆ A11 (Wr )T = QT1 Y.

(10)

From the above analysis, the output weights of the m + 1 − r dependent nodes can be set as zeros and the output weights of the independent nodes can be recalculated using equation (10) without any effect on the network’s input-output behavior.

4

Unit-Based Optimal Brain Surgeon Pruning

In this section, a novel sensitivity measure is proposed to remove the insensitive nodes based on optimal brain surgeon(UB-OBS) developed by Hassibi et al. [4]. For notation simplicity, the pruned network is described as

A Node Pruning Algorithm Based on Optimal Brain Surgeon

ΦW T = Y,

527

(11)

here Φ is a N × r full column rank matrix, and the remaining weight W can be easily obtained as W = Y T Φ(ΦT Φ)−1 . (12) Pruning the j th hidden node is equivalent to setting the corresponding weights from the j th node to all the output nodes to be zeros, that is Wij = 0, i = 1, · · · , p. The new weight W after pruning needs to be updated such that the output of ΦW T remains as close as possible to Y . The functional Taylor series of the error function in (4) with respect to the output weights is δV = (

∂V 1 )δW + δW T · H · δW + O(δW 3 ), ∂W 2

(13)

where H = ∂ 2 V /∂ 2 W = ΦT Φ is the hessian matrix. The first term vanishes for a network trained to a local minimum, and the third term is zero. The updating process can be formulated as an optimization problem: subject to δW ej = −Wj ,

min δV δW

(14)

where ej and Wj are the j th column of the identity matrix and W respectively, that is Wj = W ej . The constraint in (14) means that the j th node is assumed to be pruned. Solving the constrained linear programming problem in (14) using the Lagrangian method, and δW and δV can be obtained as follows: δW = −Wj eTj (ΦT Φ)−1 /(ΦT Φ)−1 jj , δV =

WjT Wj /(ΦT Φ)−1 jj .

(15) (16)

Therefore the sensitivity measure of the minimal effect after pruning the j th node and readjusting the remaining weights can be defined as: mesj := WjT Wj /(ΦT Φ)−1 jj .

(17)

The lower the value of mesj , the less effect on the output. Hence an insensitive ˆ can be node j ∗ th to be pruned can be chosen according to equation (17) and W calculated using equation (15).

5

Simulation Results

In this section, a regression problem is used to demonstrate the effectiveness of the proposed node pruning algorithm. The performance of the network is measured by the so-called metric fraction of variance unexplained (FVU). For details about the meaning of FVU, see [8]. Note that the FVU is proportional to the mean square error (MSE).

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Table 1. Comparisons between the mean FVUs of the training and the generalization for the HAS regression problem (SNR = -3dB, 40 runs) sensitivity measure Number of Training Generalization hidden nodes FVU FVU measure in [5] 10 0.7764 0.3864 proposed measure 10 0.7368 0.2758

1

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Fig. 1. Comparison of the FVU for training data using different sensitivity measure (solid line: proposed measure; dashed line: measure in [5])

Consider the following five-dimensional function (HAS) [8] that is given by g(x1 , x2 , x3 , x4 , x5 ) = 0.0647(12+3x1−3.5x22 +7.2x33 )(1+cos4πx4 )(1+0.8sin3πx5 ) Three thousands random samples with additive noise were generated for network training, the SNR is -3dB. Ten thousand random samples without additive noise were generated for verifying the network generalization capabilities. Forty independent runs were performed (where for each run the network is initialized to a different set of weight values). The neural networks with 20 hidden nodes are trained to an acceptable solution, that is the FVU for training data is less than 0.71. Then at first stage, dependent nodes are pruned using QR factorization, and at second stage, insensitive nodes are selected and pruned until ten hidden nodes are remained, and the FVUs for training and generalization are checked. Means of FVUs for the training and the generalization performance of networks for the HAS function are summarized in Table 1. The sensitive measure in [5] was defined as follows: V˜j = (φTj φj )(WjT Wj ).

(18)

For comparison, this measure is also used at second stage to select the insensitive nodes, and output weight of the remaining nodes are readjusted by computing

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the pseudo-inverse. The result is also shown in Table 1. It can be seen that the training and generalization performance is much better using our proposed sensitive measure. In Figure 1, means of training FVUs are also displayed after each hidden nodes is pruned for different measures. It follows that the proposed algorithm has less effect on the neural network performance after each hidden nodes is pruned.

6

Conclusion

In this paper, a node pruning algorithm based on optimal brain surgeon is proposed for feedforward neural networks. At the first stage, the orthogonal factorization is applied to the output of the nodes in the same hidden layer to identify and prune the linear dependant nodes. At second stage, a unit-based optimal brain surgeon(UB-OBS) pruning algorithm is used to prune the insensitive hidden units to further reduce the size of the neural network. No retraining is needed.

References 1. Reed, R.: Pruning Algorithm-A Survey. IEEE Trans. Neural Networks 4(5) (1993) 740-747 2. Kwok, T., Yeung, D.: Constructive Algorithms for Structure Learning in Feedforward Neural Networks for Regression Problems. IEEE Trans. Neural Networks 8(3) (1997) 630-645 3. Girosi, F., Jones, M., Poggio, T.: Regularization Theory and Neural Network Architectures. Neural Comput. 7(3) (1995) 219-269 4. Hassibi, B., Stork, D.G.: Second Order Derivatives for Network Pruning: Optimal Brain Surgeon. In: S.J. Hanson, (Ed.): Advances in Neural Information Processing, Vol.5. CA: Morgan Kaufmann (1993) 164-171 5. Castellano, G., Fanelli, A.M., Pelillo, M.: An Iterative Pruning Algorithm for Feedforward Neural Networks. IEEE Trans. on Neural Networks 8(3) (1997) 519-531 6. Chandrasekaran, H., Chen, H., Manry, M.T.: Pruning of Basis Functions in Nonlinear Approximators. Neurocomputing 34(1) (2000) 29-53 7. Fletcher, L., Katkovnik, V.,Steffens, F.E., Engelbrecht, A.P.: Optimizing the Number of Hidden Nodes of a Feedforward Artificial Neural Network. In: IEEE World Congress on Computational Intelligence, Anchorage, Alaska (1998) 1608-1612 8. Ma, L., Khorasani, K.: New Training Strategies for Constructive Neural Networks with Application to Regression Problems. Neural Networks 17(4) (2004) 589-609

A Fast Learning Algorithm Based on Layered Hessian Approximations and the Pseudoinverse E.J. Teoh, C. Xiang, and K.C. Tan Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576 [email protected] Abstract. In this article, we present a simple, effective method to learning for an MLP that is based on approximating the Hessian using only local information, specifically, the correlations of output activations from previous layers of hidden neurons. This approach of training the hidden layer weights with the Hessian approximation combined with the training of the final output layer of weights using the pseudoinverse [1] yields improved performance at a fraction of the computational and structural complexity of conventional learning algorithms.

1

Introduction

Typical learning algorithms for training feedforward neural architectures for a variety of applications such as classification, regression and forecasting utilize well-known optimization techniques. These numerical optimization methods usually exploit the use of first-order (Jacobian) and second-order (Hessian) methods. Typically, the Jacobian or gradient of the cost function can be computed quite readily and conveniently; however, the same cannot be said of the Hessian, particularly for larger-sized networks as the number of free parameters (synaptic weights) increase. With this in mind, approximation of the Hessian matrix, commonly found in numerous quasi-Newton approaches for optimization, is of great interest. Although more efficient methods in computing the Hessian exactly have been suggested [2, 3], exact computation of the Hessian is costly - moreover, in many practical applications, such accurate computation is not required. Some of the more well-known approximation techniques include only using the diagonal entries of the Hessian, omission of certain terms (such as the second partial derivatives) [4] or through numerical differentiation [3]. While various approximations to the Hessian have been proposed, computation of the Hessian matrix based solely on local information, such as (input and output) signals available to weights of a particular layer, is often desirable, as this may allow the partitioning of the training of a multi-layered neural architecture into layers, each of which can be trained separately from each other. On the other hand, non-gradient based approaches such as evolutionary algorithms (EAs), and the ELM paradigm [1] which trains only the output layer weights while leaving the hidden layer weights initialized randomly with a uniform probability distribution provide interesting alternative methods of training a neural J. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 530–536, 2006. c Springer-Verlag Berlin Heidelberg 2006 

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network. Non-gradient based techniques are attractive particularly because they are less likely to converge to a local minima, which holds true for increasingly complex problems. However, EAs are largely stochastic optimizers and are hence slow in discovering solutions; the ELM approach is fast in obtaining solutions but at the cost of the need to use an architecturally more complex (as measured by the number of hidden layer neurons used) structures. In [1], the Moore-Penrose generalized inverse (or more commonly known as the pseudoinverse) was used to train the output weights of the feedforward network (for either a conventional MLP or an RBF network) while leaving the hidden layer weights (initialized randomly using a uniform probability distribution) untrained. While this method is extremely fast in the training stage, relatively larger number of hidden layer neurons are needed – some form of training if utilized, to learn the weights of the hidden weights, is always able to obtain a smaller, or minimal network architecture using fewer hidden neurons. The number of iterations (or epochs) that is additionally used to train the hidden layer weights need not be exhaustive, as we will susequently demonstrate.

2

The Proposed Approach

Given a cost function E(w) : n → , such as the sum-of-squares cost function, we can construct E(w) as a localized model using the Taylor series approximation by constructing an expansion about a nominal point w0 and subsequently analyze the effect of perturbing the parameter vector (w) wbout w0 (the local minimum). Thus, we obtain E(w) = E(w0 ) + J(w0 )(w − w0 ) + 12 (w − w0 )T H(w0 )(w − w0 ) + O(w − wo 3 ), where J(w) and H(w) are the Jacobian (gradient) and Hessian of the cost function respectively. The essential idea that underlies the proposed approach is the layer-wise (iterative) estimation of the Hessian, which together with the gradient (Jacobian) information forms a weight update rule for hidden layer weights. This, coupled with the least-squares or pseudoinverse approach for the weights of the final (output) layer is the basis of our proposed training algorithm which offers faster convergence and better generalizability. We manipulate the weights indirectly through the modification of the desired activations of each layer, and is derived by noticing the separability of the layers of the network into linear (given by the affine form of the locally induced field) and a nonlinear (given by the activation function of the nodes) blocks. Consider the activation of a layer k at time step (iteration) t as Yk (t), where Yk (t) = Xk (t)Wk (t) Xk+1 (t) = [f (Yk (t))

1]

(1) (2)

The new estimate for Yk at the next time step is Yk (t + 1) = Yk (t) + Dk (t) where Dk (t) is the estimate for the direction in which the new activation for layer k should proceed. Dk (t) can be derived using a variety of methods, but for simplicity, is found using the gradient descent on the cost function with respect to the

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output activations of the hidden layers (Yk (t)), i.e. Dk (t) = −η(t)∇Yk (t) E(w, t). Thus, Yk (t + 1) = Yk (t) − η(t)∇Yk (t) E(w, t) Yk (t + 1) = Xk (t)Wk (t + 1)

(3) (4)

Xk (t)Wk (t + 1) = Xk (t)Wk (t) − ∇Yk (t) E(w, t)

(5)

†

†

Wk (t + 1) = Xk (t) Xk (t)Wk (t) − η(t)Xk (t) ∇Yk (t) E(w, t) †

Wk (t + 1) = Wk (t) − η(t)Xk (t) ∇Yk (t) E(w, t)

(6) (7)

Clearly, ∇Wk (t) E(w, t) = Xk (t)T ∇Yk (t) E(w, t) ⇒ ∇Yk (t) E(w, t) and thus ∇Wk (t) E(w, t) = (Xk (t)T )−1 ∇Wk (t) E(w, t). By virtue of the fact that Xk (t)† = (Xk (t)T Xk (t))−1 Xk (t)T , this then leads to Wk (t + 1) = Wk (t) − η(t)Xk (t)† ∇Yk (t) E(w, t) −1

Wk (t + 1) = Wk (t) − η(t)(Xk (t) Xk (t))

(8)

Xk (t) ∇Yk (t) E(w, t)

(9)

Wk (t + 1) = Wk (t) − η(t)(Xk (t)T Xk (t) + λ1 I)−1 ∇Wk (t) E(w, t)

(10)

T

T

−1

Wk (t + 1) = Wk (t) − η(t)(Φk (t) + λ1 I)

∇Wk (t) E(w, t)

(11)

From this, Φk (t) represents the correlation matrix of the activation outputs of the preceding layer and λ1 being the regularization parameter for the hidden layer weights. On the other hand, the gradient or the Jacobian of the cost function with respect to the weights in layer k, ∇Wk (t) E(w, t) can be recursively computed layer-by-layer. This approach was previously suggested by [5, 6] (without involving the use of the pseudoinverse), which offers a simpler method to building an approximation of the Hessian in a second-order approach. The use of the pseudoinverse in the layer-by-layer computation of the weights of each layer affords us the ability to introduce regularization terms to ‘control’ the resulting size of the weights in each layer as measured by its L2 -norm. This prevents overtraining of the network, and is similar in idea to that of ‘earlystopping’. As was rigorously discussed by Bartlett, the size of the weights of the network is important for the generalization performance of the network [7]. The computational advantage obtained from the use of the proposed approach however, is dependent on the number of neurons that is used in the hidden layer(s). Another point worthy to note is that this layered estimation of the Hessian assumes the independency of weights on different layers – such as assumption is generally acceptable in the case when the learning is close to convergence [6]. The convergence rate using only the pseudoinverse as a learning algorithm is extremely fast - however the use of the pseudoinverse in determining the weights of a layer in a multi-layer feedforward network is only optimal with respect to the weights of the other layers. This is to say that global optimality is only guaranteed for that layer. In a global context, the weights found using the pseudoinverse is locally optimal. Based on Cover’s 1965 function counting theorem [8], we know that projecting samples or observations from a low dimensional input space to a higher

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dimensional feature space would make these transformed features more likely to be linearly separable. Placed in our context, we could then use a single hidden layer with a large number of neurons (with randomly initialized hidden layer weights). To find the output layer weights, we use the pseudoinverse. Note that the output nodes need not be a linear activation function, for if f (y) = d, then y = f −1 (d) where d is the desired signal, f −1 (·) is the functional inverse. The pseudoinverse is then used with y as the desired signal. Like other numerical minimization algorithms, the method being suggested here is an iterative procedure. The gradient (Jacobian) of the cost function (sum-of-squares) with respect to the weights in the different layers can be found quite readily. The estimate of the Hessian is based on a layer-wise approximation that only uses local information (the correlation of the activation signals from the previous layer). Clearly, this Hessian approximation is a function of the layer index (k) and the iteration step (t). The output layer, using a linear activation function is computed using the pseudoinverse. For the final (output) layer (k = L), with the superscript † denoting the pseudoinverse and d the desired signals, XL−1 (t)wL (t + 1) = d †

wL (t + 1) = XL−1 (t) d  −1 T wL (t + 1) = XL−1 (t) XL−1 (t)λ2 I XL−1 (t)T d

(12) (13) (14)

XL−1 (t) is the set of activation (output) values from the previous layer in the current time step. The system is usually overdetermined as we have more samples than hidden layer neurons in layer L − 1. The use of this training process, in a way, allows a layer-by-layer training approach in determining the appropriate set of weights for each hidden layer – by utilizing different learning algorithms for learning the weights on different layers, we are able to exploit the properties and characteristics of each layer.

3

Experimental Results

The datasets that were used to demonstrate the effectiveness and efficiency of the proposed approach are taken from the UCI Machine Leaning Repository. We compare the proposed approach with the ELM paradigm [1] and a MLP-based neural network (using the NETLAB toolbox). All initial simulation parameters (such as the weights, ∈ [−1, 1]) of the network used were randomly initialized but are the same for the different approaches. The comparison of the proposed algorithm with the back-propagation algorithm (scaled conjugate gradient variant) is based on the CPU time. The ratio of the improvement of the proposed algorithm over the BP algorithm (the speed-up) is approximately 3-5 times faster – notably it is difficult to compare the speeds of the algorithms based solely on the CPU time as different implementation and coding techniques may bias the results and hence the run times. Moreover, the convergence of the algorithms are different, for example the ELM approach converges in a single step (‘one-shot’

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Table 1. Performance comparisons – Mean accuracies and standard deviations Dataset

MLP MLP training testing

Diabetes

0.7517 (0.0238) 0.9659 (0.0106) 0.8861 (0.0329) 0.7395 (0.0492) 0.8357 (0.0340) 0.8088 (0.0357) 0.9184 (0.0275) 0.6736 (0.0369) 0.8039 (0.1232) 0.9450 (0.0342) 0.8695 (0.0432) 0.9373 (0.0388)

0.7777 (0.0146) Breast cancer 0.9680 (0.0072) Biomedical 0.9166 (0.0199) Sonar 0.8695 (0.0409) Ionosphere 0.8884 (0.0237) Heart 0.8646 (0.0211) Glass 0.9617 (0.0127) Liver 0.7351 (0.0282) E.Coli 0.9503 (0.0165) Iris 0.9498 (0.0269) IMOX 0.9214 (0.0217) Wine 0.9498 (0.0298)

Fast Fast training testing (λ = 0.001) (λ = 0.001) 0.8184 0.7403 (0.0178) (0.0245) 0.9789 0.9668 (0.0110) (0.0072) 0.9285 0.8671 (0.0333) (0.0380) 0.9924 0.7388 (0.0091) (0.0392) 0.9886 0.8736 (0.0072) (0.0270) 0.9404 0.7837 (0.0174) (0.0350) 0.9913 0.8635 (0.0097) (0.0948) 0.7836 0.6581 (0.0406) (0.0341) 0.9656 0.9250 (0.0123) (0.0318) 0.9682 0.9420 (0.0318) (0.0461) 0.9674 0.8976 (0.0186) (0.0322) 0.9687 0.9320 (0.0322) (0.0457)

Fast training (λ = 1) 0.7945 (0.0132) 0.9790 (0.0054) 0.9178 (0.0157) 0.9729 (0.0126) 0.9744 (0.0097) 0.8981 (0.0163) 0.9710 (0.0117) 0.7539 (0.0199) 0.9670 (0.0084) 0.9587 (0.0149) 0.9552 (0.0158) 0.9598 (0.0145)

Fast testing (λ = 1) 0.7629 (0.0193) 0.9666 (0.0078) 0.8821 (0.0298) 0.7507 (0.0358) 0.8664 (0.0245) 0.8295 (0.0251) 0.9219 (0.0224) 0.7032 (0.0377) 0.9467 (0.0173) 0.9423 (0.0327) 0.9126 (0.0283) 0.9417 (0.0309)

ELM ELM training testing 0.7690 (0.0188) 0.9666 (0.0087) 0.8963 (0.0170) 0.7270 (0.0451) 0.8227 (0.0383) 0.8087 (0.0369) 0.9340 (0.0182) 0.7069 (0.0252) 0.9543 (0.0125) 0.9431 (0.0269) 0.8864 (0.0249) 0.9491 (0.0365)

0.7530 (0.0256) 0.9640 (0.0110) 0.8871 (0.0309) 0.6776 (0.0626) 0.7967 (0.0388) 0.7963 (0.0462) 0.9073 (0.0272) 0.6706 (0.0404) 0.8150 (0.1677) 0.9307 (0.0421) 0.8453 (0.0414) 0.9263 (0.0427)

learning) as no iterative correction of the weights are required since the training on the linear output layer of weights is achieved via the use of the pseudoinverse. The MLP on the other hand takes a longer time to converge while the proposed algorithm’s convergence speed is between the two, depending on the problem complexity and network size. The tansig nonlinearity is used in the hidden layers. Results obtained, comparing the performances of the MLP, the proposed approach (using 2 different regularization parameters) and the ELM are as in Table 1. 60% of available data was used as the training data while the remaining 40% was used as the testing data – these were randomly selected for each of the 50 runs, trained for 10 epochs, with 10 hidden neurons used.

4

Discussion

The ‘size’ of the weights, as measured by the norm of the weights at each layer is smaller using the proposed approach than that obtained from using the backpropagation. For the output layer weights this can be attributed to the operation of the pseudoinverse as the pseudoinverse obtains the minimum norm of possible weights that solves an overdetermined problem in a least-squares sense. For better generalization performance, clearly, fitting the network to the data of

A Fast Learning Algorithm

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the training set is not desirable. Some form of regularization is advantageous in preventing the network from being overtrained. However, determining the appropriate amount of regularization is an art in itself. Regularization is introduced explicitly through the use of the regularization parameter λ, and through the approximation of the Hessian. Approximations made during the weight update in the training phase is akin to regularization, since little attempt is made to fit the samples in the training data in an exact manner. In the approach presented here, the Hessian, is approximated on a layer-by-layer basis using the correlation matrix of the output activations from the preceding layer. Moreover, exact computation of the Hessian is not only computationally expensive but may also result in an ill-conditioned and even singular matrix. The regularization term that is introduced in the computation of the pseudoinverse has an effect on the classification performance of the resulting feedforward network – generally a larger regularization term leads to poorer training performance but better testing performance. Geometrically, this is seen as the construction of increasingly smoother separating hyperplanes in hidden layer space as larger regularization is used. Clearly, the magnitude of the regularization term assists in the generalization ability of the resulting network only up to a certain value, beyond which any increase would be detrimental to both the training and testing performance. While the ELM is fast in the trainin