Neuroelectrodynamics: Understanding the Brain Language - Volume 74 Biomedical and Health Research

NEUROELECTRODYNAMICS

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Neuroelectrodynamics Understanding the Brain Language

Dorian Aur Barrow Neurological Institute, Phoenix, Arizona, USA

and

Mandar S. Jog University of Western Ontario, London, Ontario, Canada

Amsterdam • Berlin • Tokyo • Washington, DC

© 2010 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-60750-091-9 (print) ISBN 978-1-60750-473-3 (online) Library of Congress Control Number: 2009943663 doi: 10.3233/978-1-60750-473-3-i Publisher IOS Press BV Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the USA and Canada IOS Press, Inc. 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: [email protected]

LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Neuroelectrodynamics D. Aur and M.S. Jog IOS Press, 2010 © 2010 The authors and IOS Press. All rights reserved.

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Preface The brain is an incredibly powerful computing machine shaped by an evolutionary process that has developed over millions of years. The essence of brain function consists in how information is processed, transferred and stored. Understanding how our brain computes can explain why our abilities of seeing and moving in space are substantially more powerful as compared to any artificially built system. Without a clear model of how computation is performed in the brain, we are not able to understand the relationship between perception learning, memory and molecular changes at DNA, gene or protein level. Many models in computational neuroscience claim the existence of parallel, distributed processing system in the brain. Similarly many advances in the structural, microscopic and genetic fields have shed light on the molecular machinery within the brain. Additionally, we know that the brain processes and communicates information through complex electrochemical processes. Any theoretical and computational model must be able to bring all of these aspects together and build a generally true model of brain computation. However, this has not happened. From a neurophysiological point of view, current doctrine remains focused within a spike timing paradigm. This paradigm has limited capability for advancing the understanding of how the brain works. Additional hypotheses such spike stereotypy (a visual assumption), temporal coding (which does not explain information processing) and cable function (which cannot explain memory storage and simplifies a complex, dynamic axonal system into a transmission cable) are required, due to which the reach of the neural computational model is limited. Assumptions made in any theoretical construct that are unproven make the theory limited and thus inherently weak. Taken from this perspective any subsequent result and further hypotheses are invalid. These reasons were the beginning of our dissatisfaction with the state of the current trend in neuroscience and neural computation and the main reason for writing this book. Our view is that information processing in the brain transcends the usual notion of action potential occurrence with millisecond precision. The experimental evidence and the models presented in this book extensively demonstrate, for the first time to our knowledge, that spike timing theory considerably restricts the richness of how the brain carries out computations. The ability to record voltage extracellulary directly implies the existence of significant changes in local electric fields. Indeed action potentials themselves are shaped by the electric field and the presence, distribution and dynamics of charges that locally change molecular dynamics and electrical excitability of voltage-gated ion channels in active neurons. We therefore set out to put forward a new model from a ground-up set of universal principles on the basis established by the laws of physics. These laws such as those of classical mechanics, thermodynamics and quantum physics can be used as guiding principles based upon which a general theoretical construct of brain computational model can be built. Such model attempts to incorporate the neurobiology of the cells, the molecular machinery itself along with the electrical activity in neurons to explain experimental results and in addition predict organization of the system. In this book we show that spike timing models are special cases of a broader theoretical construct – NeuroElectroDynamics (NED) neuro- from neurons, electrofrom electric field and -dynamics meaning movement. The main idea of NED is that under the influence of electric fields, charges that interact perform computations and

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are capable to read, write and store information in their spatial distribution at molecular level within active neurons. In this domain, the universal physical laws from classical mechanics to thermodynamics and quantum mechanics are applied in order to give mathematical and theoretical support to charges as the operators of the brain computing machine. Experimental data is used where necessary to corroborate, support and extend these observations. This novel and substantial approach of the NeuroElectroDynamic model extends the previous approach to a different level where the physical aspect of information processing is fundamental and can be directly related to macromolecular changes, thermodynamic properties and intrinsic quantum phenomena. This new model represents the crucial paradigm shift which opens the door to a new subfield of neuroscience that bridges neural computational models and research in molecular computation. Any model requires certain assumptions and the proposed theory is no different. However, the assumptions that are made in this theory are those based upon already validated laws and physical principles that should and indeed must be transferable between the fields of physics and neuroscience. No simplifications to these laws are made in the NED construct. The very novelty and strength of the model is in such an application of basic physical principles which are used to explain how the brain computes. This makes the theoretical framework used to generate NED “general” and by default very amenable to expansion, with clear possibility for experimental verification. The book is organized into five large chapters. The first chapter outlines and provides an overview of the deficiencies of the current approaches to neural computation. The current development of neuronal computation based upon the assumption of spike stereotypy, known as spike timing theory is fraught with several gaps and controversies. Many of these issues are then discussed in detail in individual chapters along with a parallel development of the new theoretical architecture alluded to in this first chapter. The chapter is also a gentle introduction to laws and principles required to build a new model that addresses the issues raised in the chapter. This description highlights a systemic approach where universal laws should shape the nature of computations. Since the link between electrophysiological properties of neurons and theory of neural computation, is critical, Chapter 2 describes such experiments and their results. The analyses performed in the in-vivo behaving animals are presented with a very fresh approach to reflect the essential characteristics of neurons and neuronal ensembles and disqualify spike timing as a unique modality which can fully describe brain computations. Spike timing analyzed in local or larger areas does not seem to be rich enough and does not tell us the whole story. Fundamental processing is hidden, observables such the timing of spikes (firing rate, ISI), oscillations partially include information regarding these hidden processes. Importantly, new methods and techniques, designed to characterize neuronal activity well beyond the spike timing approach are revealed in this chapter. This new description provides the required link between computation and semantics analyzed during behavioral experiments. Chapter 3 highlights some important concepts which relate the theory of information with computation and electrophysiological properties of neurons. The presentation differentiates between popular view of ciphering (i.e. performing encryption and decryption) and fundamental mathematical principles of coding and decoding information. Previous theories of neural coding are discussed in this context

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and new concepts are presented. This novel view relates information processing and computation with cognitive models. Information flow within the nervous system is driven by electrical phenomena with full support from the biological and neurochemical infrastructure of the neurons. Chapter 4 is the heart of the presentation of the new model of computation and forms the raison d’être for the book. The chapter builds a systemic approach of brain computation in terms of interactions, dynamics of charges (NeuroElectroDynamics) from the nanoscale level to macroscale phenomena using a theoretical design, the computational cube, which can be regarded as a compartmental model. This novel model proves that complex, grained charge level of information processing is the subtle phenomenology behind the commonly observed spiking activity and temporal pattern occurrence. Such processes have implicit quantum mechanisms that can be demonstrated as intrinsic results of experimental data analysis. A unified mathematical view of computation expressed in terms of charge dynamics as successive coding and decoding phases as developed in the NED theory is able to characterize the activity within single cells or neuronal ensembles. It is clear that computation depends on intrinsic physical laws (the principle of least action, second law of thermodynamics) and anatomical substrate of neurons and is reflected in their electrophysiological properties. The concepts of computation based on the dynamics and interaction of charges, is no different and is the grounding upon which NED model has been developed. Additionally, the new approach is able to explain “coding and decoding” mechanisms, information retrieval, perception, memory formation, and other cognitive processes. We show as an example that temporal organization/temporal patterns can be obtained as a result of ‘learning’ starting with a charge movement model. The description of these state-of-the-art techniques of information processing (coding and decoding) are gradually developed in this book, using a simple mathematical formalism. This approach will allow readers from different backgrounds to implement and experiment new models and perform computer simulations of brain computation. There is no real equivalent for such development in the models that consider temporal aspects of spike occurrence (firing rate, ISI). The theoretical approach of NED builds a mathematically general model of brain computation that incorporates the current spike timing theory as a particular case. This ability of generalization is sorely missing from the current models of brain function. As the NED theory is accepted and developed in the future, it is likely that it can itself be incorporated within another even more generalized approach. Chapter 5 establishes and expounds the new NED model and several examples of its application are shown in order to reveal and characterize brain computations. These applications are only partially developed here in this book and will form the basis of our next work. Several processes including memory and cognition can be explained if spike timing models are replaced with the new systemic approach of dynamics and interaction of charges. The emergent topology of charges as a result of their intrinsic dynamics during action potentials and synaptic spikes are at the origin of distancebased semantics, a phenomenon previously experimentally evidenced in several brain recordings. NeuroElectroDynamics is a completely different model where the intrinsic computational processes are described by the dynamics and interaction of charges. Such a description very naturally includes and incorporates the internal neuronal molecular processes, the effect of neurotransmitters and genetic regulatory mechanisms. At fundamental molecular level, information carried by electric field and

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charges can be selectively transferred and can perform intricate operations on regulatory mechanisms at the gene level which are required to build appropriate proteins. Importantly, NeuroElectroDynamics shows that computational properties of neurons are built and change at the quantum level. The subtle changes that occur at a molecular level determined by the interactions of charges and electric fields can be related with qualitative information processing. Most important to the concept of human behavior, it is possible to relate these interactions to the emergent property of semantics of behavior, which has been elusive to date except in the realms of psychology and psychophysics. As presented this multimodal aspect of information transfer and processing in the brain incorporates and indeed requires interactions with intracellular neuronal and nonneuronal complex molecular regulatory mechanisms from gene selection/expression, DNA computations to membrane properties that spatially shape the dynamics and interaction of charges. Spike timing including spike patterns can be regarded as an epiphenomenon or a result of these hidden processes. The often observed heterogeneity in activation as response to a given stimulus reflects strongly processes that occur at molecular level within selected neurons. Therefore, NeuroElectroDynamics provides the structure upon which one can bridge molecular mechanisms and the spike timing framework. This systemic view unifies neural computation and may provide fresh guidance to research in artificial intelligence using newly developed models of computation. Instead of revolving around spike timing code the new model highlights several different levels where information is processed, emphasizes and indeed necessitates within the theory the role of fine sensitive structure of cells, the role of ion channels, proteins, genes and electric charges in information processing. Additionally, this chapter alludes to the role of afferent brain rhythms including sleep in learning and memory storage. Also, specific neuropathological situations where the brain is diseased such as in schizophrenia or Parkinson are briefly developed in the context of NED. The analysis of brain processes in terms of dynamic changes and resulting electric perturbations is fundamentally necessary to allow us to consider a true brain-computer interface or to perform what we feel is true neuromodulation. The future of these concepts and how we should achieve them are also discussed here. We explain why currently built artificial systems based on ultra-simplified neurons using spike timing paradigms are inconsistent with brain computations and their future is limited by this reductionist approach. Built on principles supported by established pillars of physical laws in describing brain computation NeuroElectroDynamics is a step forward, required to develop the next generation of artificial intelligent systems and understand the mind in computational terms. We anticipate that the development in nanotechnology will allow this progress to be made soon. The NED model highlights the importance of charge dynamics, electric interactions and electric field in information communication at a nanoscale level of protein synthesis. Traditional molecular biological approaches are unable to link the electrophysiological components of neural function to protein and genes and indeed most importantly with information models. Although NED does not provide a complete solution, we feel that it at least gives a path along which further research could be carried out in a way that emphasizes the full relationship between genes, proteins and complex regulatory mechanisms of information flow.

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We started this journey more than five years ago by demonstrating that spikes are not stereotyped events by computing spike directivity and providing electrical images of recorded spikes. Since the fundamentals of spike timing theory of computation in the brain are based on the assumptions of ‘spike stereotypy’, representation of axons as cables and others, the resulting temporal coding theory and its conclusions are restrictive, limited and possibly inaccurate. Indeed, they are certainly incomplete in terms of explaining the complexity of function and behavior that we see in the real world. After we began our effort of writing this book in February 2008, terms that we have defined such as – spike timing code as “a myth” – have already been borrowed from our proposal to publishers, by other experts in the field. We are delighted to see that some distinguished members of the academic community are beginning to incorporate these ideas into their own work. We are fully aware of how slow progress is in science and how hard it is to induce a paradigm change (see Planck, 1949; Kuhn, 1962). We hope that this book will appeal greatly to many students. The models hinted at in chapter 5 can serve as research pathways for both experimental and theoretical development. Those that study physics and mathematics will know that ideas put forward by giants in their fields such as those of Schrödinger and Sir Roger Penrose can indeed be applied in neuroscience. New tools and mathematical concepts will need to be developed and applied. The students that are interested in systems neuroscience can revel in the fact that complex systems dynamics can be made to work and deeply explain experimental data. To the neurophysiologist there are now clear testable conjectures that are predicted and will result in a new line of experiments well beyond the current spike timing measurements. To the computer science and AI group, a true and detailed framework for brain function will allow much better algorithmic development of computations that can be translated into applications. From philosophical, psychological or psychoanalytical point of view, mind and brain are inseparable and this model of computation will appear as a naturally significant change that resonates strongly with previous views and new concepts of the science of the mind. Finally, to biologists that are struggling to find connections between the myriads of proteins, genes, their properties and behavior, NeuroElectroDynamics can be way of making sense of their findings in a comprehensive way. The list of scientists that pioneered this work of in-depth investigation of intricate phenomena in brain is long. We can mention here all the Nobel laureates cited for their contribution in this field along other names such as Wilder Penfield, Mircea Steriade, Alwyn Scott, Roger Penrose, Stuart Hameroff and others that did not receive this honorary recognition but made significant observations in analyzing how the brain works. By contrast there is an even larger list of scientists that have simplified their observations to the temporal domain by building spike timing models, emphasizing their role and creating the current fashion in computational neuroscience. We hope that for a new generation of students, researchers and scientists an important legacy stands in novel concepts initiated and developed in this book. We anticipate that their further development would establish a new field of research to emerge that will determine a paradigm shift required to significantly change current insights in computational neuroscience and connected fields which finally will provide a better understanding of the human brain and mind.

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Acknowledgements We have succeeded in publishing this book in this environment where every model that does not display the fashion of spike timing is highly marginalized in the mainstream publications. We have to thank those people that had the vision and made this publication possible especially to reviewers and publisher Mark Eligh from IOS Press. Our foremost acknowledgement is to Shrikrishna Jog for carefully correcting the manuscript for valuable insights and useful comments. Also, the authors would like to express many thanks to Dr. Radu Balan, Dr. Neil Lambert and Liviu Aur for several suggestions to improve this book. Finally, we would like to thank our parents and families especially Monica, Teodora and our daughters, Alexandra, Rachna and Mihika for their patience. The Authors

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A Note on the Illustrations The online edition of this book has a free supplement with all the available colour illustrations of this publication. Please visit www.booksonline.iospress.nl and navigate to Book Series Biomedical and Health Research (Volume 74) or use the doi:10.3233/978-1-60750-473-3-i.

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CONTENTS Preface ........................................................................................................................ ... v A Note on the Illustrations .......................................................................................... xi CHAPTER 1. Understanding the Brain Language.................................................... 1 1.1.1. The importance of asking the right questions........................................... 1 1.1.2. Does neuroscience ask the right questions?.............................................. 1 1.1.3. Cognition, Computation and Dynamical Systems.................................... 2 1.2. Myths, Controversies and Current Challenges ............................................. 4 1.2.1. The inadequacies of temporal coding....................................................... 4 1.2.2. The cable myth ......................................................................................... 5 1.2.3. The dilemma of synapses ......................................................................... 6 1.2.4. The Hebbian approach of connectivity..................................................... 6 1.2.5. The myth of neurotransmitters ................................................................. 7 1.2.6. The myth of molecular biology ................................................................ 7 1.3. Spike Timing – Current Challenges ............................................................... 8 1.3.1. Is the neuron an integrator of incoming inputs – Temporal integration or coincidence detection? ....................................................... 8 1.3.2. Firing rate or ISI? Which technique is better to quantify neuronal activity? .................................................................................................... 8 1.3.3. Do irregular spikes convey more or less information than the regular ones?......................................................................................................... 8 1.3.4. Are synchrony and oscillations important? .............................................. 9 1.3.5. Across-fiber pattern, grandmother cell, or labeled-line.......................... 10 1.3.6. Optimal efficiency or a high metabolic cost of information transmission? .......................................................................................... 11 1.4. Issues of Brain Computation......................................................................... 11 1.4.1. Brain Computations – Classical or Quantum?........................................ 11 1.4.2. Brain computation and AI ...................................................................... 12 1.4.3. Is computation in the brain closed to a reversible process?.................... 12 1.4.4. Is ‘noise’ useful for brain computations? ............................................... 12 1.4.5. What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? ....................................... 12 1.5. How can Information be Represented in the Brain..................................... 13 1.5.1. Is postsynaptic activity required for dendritic development? ................. 13 1.5.2. How and where are memories stored?.................................................... 13 1.5.3. The key of the topographic map ............................................................. 13 1.6. Interdisciplinary Formalism ......................................................................... 14 1.6.1. Sets and functions................................................................................... 14 1.6.2. Dynamical Systems ................................................................................ 15 1.6.3. The laws of thermodynamics.................................................................. 17 1.6.4. Conclusion.............................................................................................. 18 CHAPTER 2. Imaging Spikes – Challenges for Brain Computations ................... 25 2.1. Methods, Models and Techniques................................................................. 25 2.1.1. Tetrodes.................................................................................................. 27 2.1.2. The Necessity for a New Model ............................................................. 28 2.1.3. The Model of Charges in Movement...................................................... 28

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2.1.4. The Triangulation technique................................................................... 30 2.1.5. Charges and Electric potential................................................................ 30 2.1.6. The Projective Field: Spatial Directivity of Charges.............................. 31 2.1.7. What is “Spike Directivity”? .................................................................. 33 2.1.8. Triangulation and Independent Component Analysis Together ............. 34 2.2. From Spikes to Behavior ............................................................................... 38 2.2.1. Behavioral procedures............................................................................ 40 2.2.2. Spike sorting and unit classification....................................................... 40 2.2.3. Computing and Analyzing Spike Directivity ......................................... 41 2.2.4. Upcoming Choices, Decision and Spike Directivity .............................. 43 2.2.5. Learning: One Spike Seems Enough ...................................................... 48 CHAPTER 3. Information and Computation .......................................................... 55 3.1. What is Computation? ................................................................................... 55 3.1.1. Coding and decoding.............................................................................. 56 3.1.2. Are Turing Machines Universal Models of Computation? .................... 57 3.1.3. Von Neumann Architectures .................................................................. 59 3.1.4. Reversible Computation ......................................................................... 60 3.1.5. Models of Brain Computation ................................................................ 63 3.1.6. Building the Internal Model ................................................................... 66 3.2. Neurons as information engines.................................................................... 67 3.2.1. Maximum Entropy Principle .................................................................. 67 3.2.2. Mutual Information ................................................................................ 69 3.2.3. Information transfer................................................................................ 72 3.2.4. Efficiency of Information Transfer......................................................... 73 3.2.5. The Nature of Things – Functional Density ........................................... 76 3.2.6. Charge Distribution and Probability Density ......................................... 77 3.2.7. The Laws of Physics............................................................................... 79 3.2.8. Minimum Description Length ................................................................ 83 3.2.9. Is Hypercomputation a Myth? ................................................................ 85 CHAPTER 4. Models of Brain Computation ........................................................... 95 4.1. Dynamics Based Computation – The Power of Charges ............................ 95 4.1.1. Space and hidden dimensions................................................................. 96 4.1.2. Minimum Path Description – The Principle of Least Action ................. 97 4.1.3. Natural Computation – Abstract Physical Machines.............................. 99 4.1.4. Back to Building Brains ....................................................................... 104 4.1.5. Charge Movement Model – A Computational Approach..................... 109 4.1.6. Properties of Computation with Charges.............................................. 131 4.2. Quantum Model – Combining Many Worlds ............................................ 137 4.2.1. Many Worlds in a Single Spike............................................................ 138 4.2.2. Spike Models – A Quantum Formalism ............................................... 139 4.2.3. Quantum models................................................................................... 142 4.2.4. Brain the Physical Computer................................................................ 145 4.3. The Thermodynamic Model of Computation ............................................ 147 4.3.1. The Maxwell Daemon .......................................................................... 147 4.3.2. Computation in a Thermodynamic Engine........................................... 148 4.3.3. Entropy in Neuronal Spike ................................................................... 151 4.3.4. Thermodynamic Entropy and Information ........................................... 153 4.3.5. Synaptic spikes ..................................................................................... 155 4.3.6. Spikes as Thermodynamic Engines...................................................... 156

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CHAPTER 5. From Brain Language to Artificial Intelligence............................. 167 5.1. How are memories stored? .......................................................................... 167 5.1.1. Are proteins the key?............................................................................ 169 5.1.2. What sleep is for? ................................................................................. 183 5.2. Spike Timing – An Incomplete Description ............................................... 185 5.2.1. Models of Auditory Processing ............................................................ 187 5.2.2. Models of Schizophrenia...................................................................... 189 5.2.3. Models of Learning .............................................................................. 192 5.2.4. Models of Parkinson’s disease and Dyskinesia .................................... 195 5.2.5. Brain Computer Interfaces ................................................................... 196 5.2.6. Moving from Spike Timing to NeuroElectroDynamics ....................... 199 5.2.7. Computation, Cognition and Artificial intelligence ............................. 208 5.3. Instead of Discussion.................................................................................... 211

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Neuroelectrodynamics D. Aur and M.S. Jog IOS Press, 2010 © 2010 The authors and IOS Press. All rights reserved.

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CHAPTER 1. Understanding the Brain Language 1.1.1. The importance of asking the right questions It is believed that progress in science is determined by asking the right questions at the right moment in time. At the beginning of the twentieth century Hilbert asked if mathematics could be completely built based on an axiomatic formalism. The answer came three decades later. Gödel's incompleteness theorems (Gödel, 1931) showed that Hilbert's idea for a consistent set of axioms for mathematics was not possible. Maxwell addressed a similar and important question in physics regarding the second law of thermodynamics. Known today as Maxwell’s Daemon, the proposed thought experiment regarding controversy between thermodynamic and information entropy and was tackled later by many scientists. A remarkable advance made by Leo Szilard was followed by Brillouin and Landauer who showed that the Daemon can be defined in terms of information (Brillouin, 1956, Landauer, 1961). In the artificial intelligence field, the right question was asked by Searle (Searle, 1980). Searle questioned the current developments in AI where the main issue is with regard to acquiring semantics using an algorithmic approach. The thought experiment imagined by Searle assumes that a programmed computer is able to behave similarly to a human who understands Chinese and passes the Turing test. The computer uses Chinese characters as the input and syntactic rules to generate other Chinese characters. Searle’s thought experiment asked an important question; Is a computer which runs the above algorithm able to “understand” Chinese language? In other words, is the manipulation of symbols using syntactic rules performed by a computer capable of providing the required semantics in a manner similar to a person that speaks Chinese? With an exemplar philosophical clarity Searle showed that computers are only able to follow the syntactic rules and they do not understand the meaning of the Chinese language (Searle, 1980). Since the Chinese speaker has acquired in advance the semantics of words or symbols, he already knows what the symbol means. However, this process does not occur for computers, which manipulate symbols and follow syntactic rules. Many critics do not understand the gist of Searle’s philosophical argument. The problem of acquiring meaning within a system that manipulates symbols is termed symbol grounding. It is commonly believed that grounding can be achieved during learning or evolution. Despite strong opposition and criticisms (Sprevak, 2007) there isn’t a complete answer and yet no final algorithmic solution to grounded semantics. However, there is no question that Searle plays the same role in artificial intelligence as Maxwell played in physics or Hilbert in mathematics in terms of asking the right questions. 1.1.2. Does neuroscience ask the right questions? In neuroscience, we are at a critical cross-roads similar to the problem that the above three examples have demonstrated. The issue of recognizing the meaning within the neural code is closely represented by the problem in Searle’s thought experiment. Understanding the brain code using neuronal recordings faces a similary enormous problem of semantic representation within a symbolic framework. What is the language that the brain really uses to represent meaning? Is this language in terms of firing rate,

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Chapter 1: Understanding the Brain Language

spike timing, recorded oscillations or a hitherto undiscovered rhythm? Truly looking deeply at this question and attempting to understand it is akin to figuring out the semantics of a different language that we do not speak (Chinese as in Searle example). Any method that attempts to explain function of the nervous system must therefore intrinsicially be able to reveal how meaning or sematincs of behavior may be represented. The issue of representing neural function in spike timing faces the same crucial problem of having a symbolic representation without revealing how meaning is represented. Increasing or decreasing the firing rate may not have any meaning for us or at least not the same meaning that it has for neurons. In other words we are not able to understand neuronal language, since spiking seems to be a symbolic way of communication as the Chinese language is for a computer or for a non-Chinese speaker. Therefore, the main point that Searle made about digital computers can be extended to brain analyses based on neuronal recordings. Additionally, although the binary revolution has brought immense benefits, it has created a state of mind that makes us see just digital bits 0 and 1 almost everywhere. For many scientists the theory of computation is reduced to bit level operations. This has tremendously influenced the interpretation of neuronal activity. Action potentials are currently approximated almost everywhere as all or none events (stereotyped “one” for an action potential and “zero” values for the absence of an action potential). However, there is no experimental or theoretical evidence that such approximation correctly describes neuronal activity. Furthermore, if neuronal computation is reduced to symbolic manipulation then indeed it seems that there is no way to understand the intricacies of neuronal electrophysiology as a language representing meaning. A series of stereotyped action potentials and their timing can be simulated and programmed on almost any digital computer. However such an approach in the time domain does not seem to provide the richness required to build a human like information processing system and the required semantic level. 1.1.3. Cognition, Computation and Dynamical Systems Cognitive awareness and the resulting actions required to interact with the environment are a behavioral manifestation of an intrinsic semantic representation in brain. For several decades cognitive representation has been understood as a straightforward manipulation of internal symbols. Despite Searle’s demonstration, symbolic computational paradigms still dominate cognitive science. Regarding this issue, there are several claims that the brain computes differently than actual digital computers, though there is no clear understanding of how such computations are carried out. Additionally, dynamical systems theory plays an important role in emergent cognitive processes. This theory is a mathematical framework which describes how systems change over time. The early idea of the dynamical hypothesis in cognition came from Ashby's “design for a brain” (Ashby, 1952) where for the first time dynamical systems theory was used to assess cognitive behavior. The fact that intelligence is a real time embedded characteristic was also hypothesized by Ashby. More recently this idea was fully developed in a book by Port and Van Gelder (Port and Van Gelder, 1995). As presented in both books, cognition is indeed a dynamical process, which necessarily has a representation in neuronal activity. The critical point here is that the theoretical analysis of cognition requires a separation between computation and system dynamics. However, in general, such theoretical development

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pictures the theory of cognition in two distinct and opposed views one that reflects the theory of computation and another one with models of dynamic systems. Moreover, the dynamical system hypothesis presents a general theoretical approach that can be applied to several types of systems in different areas of science. This mathematical framework alone cannot explain how cognition arises or develops. A deeper understanding of the dynamics can come only from an intrinsic analysis of brain processes correlated with changes in developed semantics. Since mental processes are supposed to express the emerging behavior, the understanding of semantics should come from studying brain processes in relation to emerging changes in behavior. These interdependent, symbiotic issues that are intrinsically interconnected have not been presented anywhere yet to our knowledge. David Hume expressed this view of “mental mechanics” more than two hundred years ago (Hume, 1978). Inspired by Newtonian mechanics he developed an analogy between mental processes and the laws of motion. At that time Hume didn’t have any idea about electrophysiology, ion channels, movement of charges. This view of “mental mechanics” requires a deep understanding of brain processes related to intricacies of neuronal function. However, such data were not available two centuries ago. Extending Hume’s analogy we will show that mental processes are not only described by physical laws, they are intrinsically built using principles of motion. Hence, there should not be any contrast between a computational hypothesis and dynamic hypothesis regarding the emergence of behavioral semantics. The new development presented in this book not only ties the link between dynamical systems and brain computation, it shows that the dynamic framework is a distinct way of performing computation which shapes all cognitive processes in brain. A fundamental axiom is that the measurement of brain activity using behavioral experiments can be used to prove that animals acquire semantics during behavioral learning phases. Acquisition of semantics occurs naturally and can be expressed as changes in the emergent dynamics of electrical charges (Aur and Jog 2007a). We have demonstrated, for the first time to our knowledge, that computations performed and shaped by the dynamics of charges are radically different than computations in digital computers. Therefore, instead of a list of symbol manipulations without semantics, the representation in memory requires a physical determined structure fashioned by the intrinsic dynamical laws of charge movement which should be physically and directly associated to the represented behavior. This way of writing into memory facilitates the later retrieval of semantically connected concepts or events. Importantly, the critique that computational models are different than dynamic models is based upon only the current computational techniques algorithmized in von Neumann type machines. Since any coupled dynamic system could be used to perform computations (Ditto et al., 1993) the issue of symbolic manipulation in terms of Searle’s thought experiments should not affect the entire computational theory. Every physical realization of a computable system indeed involves a certain dynamics, whether or not the proposed framework develops a cognitive system. As an example even the implementation of von Neumann architecture of digital computers requires a dynamics for electron charges. Additionally, not all dynamical systems are able to embed semantics. This issue is now well perceived by the artificial intelligence community which is striving to build new principles for AI development. The important point to make here is that building a complex cognitive system requires particular dynamic structures and intrinsic interactions specific to neuronal activity.

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Therefore, this book proposes a broader view of the dynamical system hypothesis which directly connecting the theory of systems and physical laws of computation. The book relates this to neural mechanisms. Fundamentally, chapters 2 through 5 attempt to show that there is indeed an answer to Searle’s question posed above with the answer coming directly from an understanding of brain computation. Further, since cognition cannot be achieved as a simple manipulation of symbols, the understanding of the semantics carried by neuronal activity is highlighted as an important prerequisite. This analysis may provide a solution to build new types of intelligent systems using current digital computers that solve the grounding problem in terms of Searle’s thoughts. Each of the topics that are presented below is a subject that requires extensive discussion. This overview chapter does not make any attempt to provide a comprehensive look at the many years of work and thought that have gone into the development of these ideas. The purpose here is to state the problems and issues that neuroscience faces today. Many of these concepts will become the focal points of the chapters within the book. Neuroscience is presently at a stage where the development of classical physics found itself over a century ago. The rapidity of collection of intricate, factual data such as gene and protein identification in molecular biology and timing/rate of coding in electrophysiology and neural computation, has not answered the crucial issues alluded to above. The very focus of this book is to tackle this impasse and propose an important yet intuitive approach through a series of provocative chapters that will handle many of the issues which will be raised in the text below. 1.2. Myths, Controversies and Current Challenges The fact that neural code and information is transmitted by neuronal firing rate goes back to the very start of experimental recordings. Neurophysiologists believe that information is a function of spike time occurrence which when related to an event can be completely described by a Post-Stimulus Time Histogram (PSTH) (Winter and Palmer, 1990). Even today, almost all existing models of neural systems take for granted that information is transmitted by neurons in the form of interspike interval (ISI) or firing rate. Such myths have shaped the current theory of computation. Ignoring the fact that an AP is not a stereotyped phenomenon, several controversies and challenges are tied to this spike time analysis. Current advances in brain computations are then intrinsically related to the following four myths that are supposed to explain computation and memory storage. 1.2.1. The inadequacies of temporal coding There are several speculations regarding the nature of the neuronal code. Earlier analyses that started with electrical recordings proposed the firing rate as a coding method in neural networks (Adrian 1926, Adrian et al., 1932). The firing rate is determined as the average over time of the number of spikes generated by a nerve cell. After more than half a century of research in this field there are numerous views regarding the spike timing code. The coincidences of presynaptic events are considered to determine accurately the time of postsynaptic spikes and therefore the spike trains are supposed to represent the precise timing of presynaptic events (Abeles, 1991; Mainen and Sejnowski, 1995). Another view is that presynaptic input may alter the probability of the postsynaptic spike and therefore claims of “precise timing” are due to random occurrences (Shadlen and Newsome, 1994). Hence, Shadlen and Newsome

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rejected the idea that spike time patterns and intervals convey information. In their view only the firing rate is a relevant measure of neuronal code. The ‘synfire’ approach suggests the existence of cell ensembles that fire synchronously (Abeles 1982; Abeles, 1991). However, the synchrony mechanism cannot explain coding at low firing rates in the striatum or hippocampus and currently this mechanism is considered to be limited to certain brain areas. Additionally, other studies (Dinse, 2002) seriously called this view into question. They suggest that the “synfire” approach can explain information propagation in neural ensembles but it cannot explain information processing. Assessing these theories Thorpe, et al., (Thorpe, et al., 2001) proposed another scheme where information is encoded in the relative timing of spikes across ensembles of cells determined by the order in which the neurons fire. This mechanism of coding has also been rejected since to reach such neurons, information about the stimuli would need to travel several layers of neuronal connectivity. Then, the time of processing becomes critical for the proposed rank-order coding algorithm. However, recently, criticizing firing rate as a “true” measure of neural code several scientists claim that they had instead found neural coding in the interspike interval (ISI) (Rieke et al., 1997, Gerstner and Kistler 2002a, Maass, 1997). Their argument is that the nervous system does not have enough time to perform temporal averages since transmission times in several systems are too short, of the order of tens of milliseconds, to result in goal achievement (Rieke et al., 1997). Even though their arguments seem plausible, this issue regarding firing rate does not necessarily imply that the model based on ISI can accurately describe the incoming stimuli or that the timing of action potential (AP) is the “natural” code used by the neuron. Together, all of the above controversies and challenges demonstrate that ultimately the temporal aspect surrounding the occurrence of spikes may be an inadequate or at least an incomplete or partial description of neuronal coding. 1.2.2. The cable myth The axon and dendrites are frequently simulated as electrically conductive cables using the Hodgkin-Huxley parameterization. Based on available electrophysiological data several scientists were eager to estimate the passive cable parameters such as internal resistance, which is the membrane resistance at resting voltage using frequency-domain analysis (Koch and Poggio 1985; Meyer et al., 1997). However, reducing neuronal activity to electrical propagation in cables is an over simplification which cannot work as a general model of computation by neurons. Moving charges in cables (Figure 1) cannot account for complex processes that occur within the neuronal membrane, dendrite, soma or axons. It is unlikely that these structures act simply as conductors without involving additional mechanisms to allow information processing. While these initial studies enhanced the role of dendritic arborization, they spread confusion regarding the possible computational mechanisms.

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Aa

J

Ab

Δl

Figure 1 Schematic representation of a cable

Additionally, the cell membrane which separates the interior of the cell from the extracellular environment is most likely performing complex functions beyond the well claimed “capacitor approximation”. Treating the axons and dendrites as cables ignores all these possibilities. 1.2.3. The dilemma of synapses Synapses are considered to play a key role in information processing. Changes in the transmission properties of synapses are supposed to be directly connected with learning and memory. Another hypothesis that results from the spike timing theory is that synapses are capable of supporting computations based on a highly structured temporal code. This view of spike timing dependent plasticity (STDP) has been encouraged by existent temporal correlations on a millisecond level time scale between neuronal spikes. The synaptic weight is assumed to depend on the “relative timing between spikes” and is presented as a new Hebbian learning paradigm (Gerstner and Kistler, 2002b). The origins of STDP can be found in the early work of Grossberg (Grossberg, 1969), which was adapted to describe synaptic depression by Markram and colleagues (Markram et al., 1997). Current dogma hypothesizes that memories are stored through long-lasting modifications of synaptic strengths (structural modifications of synaptic connections). On the other hand several experiments show a dynamic change of synaptic strengths which, contradicts the synaptic long-term memory storage assumption (Khanbabaie et al, 2007; Banitt et al., 2005; Lewis and Maler, 2004). Therefore, this model using work done on the memory system should display only a short lifetime instead of the hypothesized lifetime memory preservation (Arshavsky, 2006). The link with molecular mechanisms starting from a timing description (see Clopath et al., 2009) needs a long series of additional hypotheses which are likely to make the timing model irrelevant in this case. Thus, alteration in synapse formation/elimination and changes in their ‘strength’ alone cannot fully explain the richness of information processing and especially that of memory storage. 1.2.4. The Hebbian approach of connectivity The connectionist view started with the historical remark of Hebb: “When an axon of cell A is near enough to excite cell B or repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased” (Hebb, 1949). Connectionist models assume that computation and representation is carried by cognitive units (neurons) organized in networks. Changes in their connectivity are

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supposed to provide an adaptive transformation between input and output of the system. Since this mechanism was translated into a mathematical formalism several attempts have been made to apply this formalism to mimic brain computations. This view was clearly opposed to symbolic models where computation is determined by symbol handling using inference rules. Both models have been developed separately. Unfortunately, this progress generates a huge gap between so called “high level semantics” acquired by symbolic development and lower-level connectionist representations in neural networks. 1.2.5. The myth of neurotransmitters The neuron doctrine defines the nerve cell as an embryological, metabolically distinct unit (Cajal, 1909, 1910) where the law of ‘dynamic polarization’ requires that information flows from dendrites to axon and its terminal branches. Since neurons are physically separated how any information is passed between two neurons? It is now known that synapses allow the transfer of information. In this context, the dynamics of neurotransmitters and their release in a stochastic manner was felt to describe this process of information transmission between neurons. Therefore the control of behavior was reduced to neurotransmitters’ role to alter the properties of neurons. Based solely on these observations regarding neurotransmitters many followers of spike timing theory rejected the importance of electric interactions claiming that “neurons do not interact through extracellular fields”. Even though these interactions are letely being reconsidered (see Milstein and Koch, 2008), computation cannot be linked/connected to intracellular processes. Most of the information processing takes place intracellularly and at the most fundamental level, a local sudden increase in charge density occurs within neurons which can be demonstrated within every spike delivered by a neuron (see Chapter 2). This description of computation and information processing in neurons would indeed be incomplete if it is reduced to AP occurrence or chemical synapse. Information transfer at the gap junction level is another important component that is directly carried by ions over a shorter distance, about 3.5 nm (Kandel et al. 2000, Bennett and Zukin, 2004). Additionally, the ionic flux is well recognized to have an important, active role in information processing at electrochemical synapse level (Abbott and Regehr, 2004) even though in this case the situation is somewhat more complex. If the presynaptic neuron develops an action potential, then a dynamic flow of charges occurs within the nerve cell which changes the internal and external environment without delay. 1.2.6. The myth of molecular biology The gene-centred view of evolution, seen as a sort of general philosophy capable of explaining almost every aspect of biological life is also a matter of concern. The full relationship between genes and the organism is missing and the genetic reductionism is not able to explain “how whole creatures work” (Ford, 1999). Known as the central dogma of molecular biology this model states that the flow of information is directed from genes by transcription translation processes to protein structures. Once proteins are synthesized information cannot be transferred back from protein to either protein or nucleic acid (Crick, 1958). This ‘dogma’ assigned a unique direction for information flow. Despite this earlier scheme, the reverse flow from RNA to DNA was later demonstrated by Baltimore (Baltimore, 1970) and Crick had to include this new aspect into another scheme (Crick, 1970). The intricacy of processes

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that operate at gene and protein level indicates a more complex regulatory mechanism of information flow than earlier proposed. 1.3. Spike Timing– Current Challenges 1.3.1. Is the neuron an integrator of incoming inputs - Temporal integration or coincidence detection? It had long been assumed that neurons behave as temporal integrators. Put simplistically, measures of neuronal activity firing rates have to be proportional to the time integral of stimulus inputs (Aksay et al., 2001; Loewenstein and Sompolinsky, 2003). Therefore, the probability of AP occurrence depends on the synchronization in neuronal ensembles (Roy and Alloway, 2001) and additionally neuronal synchronization should play a critical role especially in sensory information transmission. Konig challenged this view and considered coincidence detection as the main operational mode of cortical neurons (Konig et al., 1996). They used simulations to prove that neurons act more as a detector for dendritic activation with millisecond resolution than as temporal integrators (Softky and Koch, 1993). Also, they showed that simple models such as the leaky integrate-and-fire models are not able to explain the variability of cell firing in vivo. This issue poses a serious problem and weakens the hypothesis of the neuron as an integrator of incoming inputs. 1.3.2. Firing rate or ISI? Which technique is better to quantify neuronal activity? Typically, neurons receive thousands of synaptic inputs and the probability of the postsynaptic neuronal firing depends on the arrival of spikes from different inputs. The time between the resulting successive action potentials is known as the interspike interval (ISI). The relative spike timing and ISI duration is supposed to be the coding mechanism for sensory input neurons such as the visual neurons that “are well equipped to decode stimulus-related information” (Reich et al., 2000). On the other hand neurophysiologists generally assume that spike generation is a Poisson process and this approximation can provide a realistic description of information transmission. In the cortex the passive membrane time constant is about 15 ms, therefore the spiking rate should be a relevant measure (Shadlen and Newsome, 1994).This issue regarding firing rate or ISI as the more appropriate technique to quantify neuronal activity has generated a strong debate. 1.3.3. Do irregular spikes convey more or less information than the regular ones? If external events are represented by neuronal spiking activity then there should be a relationship between input stimuli and spikes. However, each neuron receives large numbers of synaptic inputs and displays a broad variability in temporal distribution of spikes. It is assumed that alterations in synaptic strength as a measure of connectivity should generate plasticity phenomena in neural circuits. Even if this connection between input stimuli, alteration in synaptic weights as proposed by Hebbian hypothesis and summation of all of these stimuli by the neuron (as an integrator) is truly robust and rich, a fundamental question still remains. How are these characteristics transformed and represented within the temporal sequences of spikes?

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The main issue here is that we don’t know how neurons integrate these synaptic inputs to produce the spike output. After presentation, neurons usually respond selectively to complex stimuli in only 100-150 ms (Thorpe and Imbert, 1989). If an adequate length of the stimulus sequence is specified, then spike trains are highly reproducible for adequate lengths of input stimuli and their standard deviation is of the order of 1-2 ms (Liu et al., 2001). However, similar stimuli may generate a variable number of AP (Britten et al 1993, Vogels et al., 1989). This source of irregularity has serious repercussions for neural information processing which seem to be unknown and have, to a large extent, been ignored. Several hypotheses ascribed this variability to NMDA conductance nonlinearity (Durstewitz and Gabriel 2007) while others hypothesized that spike-timing dependent plasticity alters the synaptic connections (Song, 2000) but no intrinsic explanation of the mechanism has been offered. Similarly, experiments in vivo and in vitro provide confirmation of varying responses of neurons to similar stimuli. If a neuron in vitro is stimulated, it fires a regular train of action potentials. However, by contrast in vivo recordings show that neurons from many areas will generate irregular trains of spikes to the same sensory stimulus. Given this property, Steveninck et al. showed that the patterns of action potentials have to be variable in order to provide information about dynamic sensory stimuli (Steveninck et al, 1997). Specifically as described by Mainen and Sejnowski, “constant stimuli led to imprecise spike trains, whereas stimuli with fluctuations produced spike trains with timing reproducible to less than 1 millisecond” (Mainen and Sejnowski, 1995). This view was opposed to the Shadlen and Newsome hypothesis that insists on the fact that an irregular ISI may reflect little information about the inputs (Shadlen and Newsome, 1994; van Vreeswijk and Sompolinsky, 1996). There are further contradictory results about information transfer and regularity or irregularity of spiking time. Davies et al. found that irregularity in time patterns reflects “the rich bandwidth” of information transfer (Davies et al., 2006). By contrast Hoebeek et al., demonstrate that irregularity is not needed and regular firing is important in Purkinje cells for information processing. As was shown by Hoebeek et al., the variability in firing rate of Purkinje cells is associated with severe motor deficits (Hoebeek et al., 2005). Also, spontaneous, irregular spikes can occur even in the absence of external inputs or extraneous sources of noise (van Vreeswijk and Sompolinsky, 1996; Brunel and Hakim, 1999). Trying to keep a balance between these two views Manwani showed that the variability of spike occurrence could either enhance or impede the decoding performance (Manwani, 2002). 1.3.4. Are synchrony and oscillations important? Moving from single neurons to network level theories, Edelman hypothesized that synchrony and oscillations are crucial for perceptual categorization, memory and consciousness (Edelman, 1989). Firing rates and their response profiles change under the influence of attention, memory activation or behavioral context. If synchrony or oscillations are so important then their temporal representation in the firing rate also is clearly an important phenomenon. Therefore, synchrony of firing is considered to be a way to encode signals with high temporal fidelity (Stevens and Zador, 1998) and is assumed to be dependent on the millisecond precision of spiking as shown by Mainen and Sejnowski (Mainen and Sejnowski, 1995). Taking the typical example of the visual system, “the binding model” suggests that neural synchrony is required for object recognition (Malsburg, 1981). This hypothesis,

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if confirmed, would imply that neurons should respond to the same sensory object and fire in temporal synchrony with precision in the millisecond range. Brunel and Hakim showed that such synchrony may be controlled by synaptic times and depend on the characteristics of the external input (Brunel and Hakim 1999). However, synchrony occurrence seems to be intrinsically generated in neuronal ensembles and can sometimes be independent of external stimuli. Also, high-frequency oscillations do not depend on synaptic transmission; they seem to be more related to direct electrotonic coupling between neurons (Draguhn et al., 1998). At this level several gap junctions assist chemical synaptic signaling in determining oscillatory synchronization with high frequency. Additionally, an increase in slow wave oscillations occurs usually in states associated with the loss of consciousness (Buzsáki and Draguhn, 2004) and these oscillatory patterns can be found also during sleep periods (Wilson and McNaughton, 2004). 1.3.5. Across-fiber pattern, grandmother cell, or labeled-line The nervous system is able to perceive the stimulus input in terms of intensity and quality. Over a half century of research in neuronal computation and brain coding has been largely focused on understanding how information about stimulus quality is processed by neurons by analyzing the spike-timing domain of single neurons or neuronal circuitry. There is not yet a unanimous answer to this coding problem and the issue has generated a long-standing controversy between several types of coding. The “labeled-line” model supports the idea that a specific characteristic such as taste, colour etc have a particular set of neurons in the brain (Hellekant et al., 1998; Dacey, 1996). Additionally, some experiments identified visual single units that respond selectively to a specific image of a familiar object. Jerzy Konorski called such cells “gnostic” units (Konorski, 1948). These neurons are better known as “grandmother” cells since the analyzed neuron is activated only by a specific meaningful stimulus (e.g. grand mother image). Many scientists have been critical regarding this grand mother hypothesis since one to one correspondence is not a plausible mechanism for neuronal coding, although, this correspondence between neurons and objects continued to be theoretically speculated (Edelman and Poggio, 1992). Despite the criticisms against the Konorski view, however, recent Quiroga et al., experimental evidence has supported the existence of these neurons which respond as exemplified in their paper only to “Jennifer Aniston” image as exemplified in their paper (Quiroga et al., 2005). Fearing the ridicule of the “grandmother” story the authors tried to explain such characteristics as a resultant of a “sparse distribution coding” in the neuronal ensemble. However, they were unable to prove and identify such circuitry or the hypothesized coding mechanism. The "across-fiber patterning" was first hypothesized by Pfaffmann who was trying to identify how neurons respond to specific tastes (Pfaffmann, 1941). Since he observed that the same fiber responds to different chemical substances he hypothesized that for a specific taste several neurons are activated and the coding takes place as “across-fiber patterning”. This model of distributed computation was later confirmed by several studies (Erickson, 1967, Erickson, 1968, Erickson, 1982, Caicedo et al., 2002, Scott, 2004). However, since the same neuron might be activated by other characteristics, there seems to be no clear ability of neuronal ensembles to detect a certain specific taste or color.

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1.3.6. Optimal efficiency or a high metabolic cost of information transmission? Energy demands are recognized as an important factor in mediating neuronal computation. It is unanimously recognized that the rapid, sustained trains of action potentials and the high rates of synaptic activity require high levels of energy. The major source of cellular energy, ATP, is generated by mitochondria. An important role is played by neuron-astrocyte metabolic interactions where astrocytes regulate the flux of energy required by neurons (Escartin et al., 2006). Slight dysfunctions at the mitochondrial level may determine the occurrence of severe neurodegenerative disorders. Importantly, protein expression changes (determined with a proteomic analysis) may also underlie alterations in efficient energy production. These phenomena can involve several mechanisms such protein degradation, DNA damage and changes in intracellular signal transduction. In order to guarantee correct function calcium homeostasis dependent energy production is important for neurons. Since neurons are highly vulnerable to energy deficit several neuroprotective mechanisms are intrinsically present that ameliorate an eventual energy deprivation. The relationship between this high-energy demand of the brain and the neuronal function is not really understood. Laughlin et al., showed that the costs to transmit a bit at a chemical synapse is 104 ATP molecules, and 106 - 107 ATP for spike coding (Laughlin et al., 1998). However, neuroscience has not yet grasped how this energy and functional constraints translate into information processing. There is currently no explanation for the substantial differences in the metabolic costs required at the chemical synapse and spike level. Moreover, there is no explanation between the so called optimal efficiency of spike time coding in the fly’s visual system (Rieke et al., 1997) and the observed excessive energy demands. 1.4. Issues of Brain Computation 1.4.1. Brain Computations--Classical or Quantum? Several scientists have hypothesized that brain computation lies in a quantum framework as a result of effects inside cellular structures such as the microtubules (Penrose, 1989; Penrose, 1994; Hameroff and Schneiker, 1987) and cannot be fully explained in a classical framework. However, the existence of quantum effects was severely criticized by Tegmark who demonstrated that quantum behavior requires longer decoherence times than are supported by the laws of classical physics (Tegmark, 2000). The collisions with other ions, with water molecules and several Coulomb interactions may very quickly induce decoherence that would destroy any required superposition to perform quantum computation. Progress in quantum theory generated an open debate between classical and quantum concepts regarding cognitive processes including consciousness. Actual images of spikes show electrical activities with micron level precision. However, if one goes close to the nanometer-size dimensions the observed physical laws are different from the classical ones. Automatically, at this scale, laws and theories of quantum mechanics turn out to be relevant and important. There is an ongoing debate wether several phenomena in brain including computation can be classically explained or they require a quantum framework. The advocates for quantum theory, Penrose and Hameroff insist that brain and mind have to be explained in a quantum framework.

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1.4.2. Brain computation and AI One of the goals in understanding brain computations is to develop human-like intelligent machines. Artificial intelligence has been viewed as a computational model that traditionally included a simple network of neurons, symbolic systems based on inference and probabilistic rules. However, difficulties of finding solutions to several problems in a human-like fashion do not seem to be solved by designing algorithms on current computers. Several years ago many scientists claimed that the key algorithm that has humanoid characteristics had been developed and only required computational power to be able to run the program. Since there is currently an easy availability of such computing power, the common agreement is that all these attempts have been mere speculations. Our understanding is that the basic principles that govern human kind of intelligence have not yet been revealed. These principles still seem to be buried in this unknown description of brain computations. 1.4.3. Is computation in the brain closed to a reversible process? Neurons are biological entities that function in a milieu. During each spike the neuron processes information and communicates with other neurons and also with its milieu. Since physical computation needs to be explained by physical laws of electric field propagation, charge motion, ion fluxes seen as charges in motion describe spatial trajectories that can be perceived and recorded in each real spike (Aur and Jog 2006). This new view of computation highlights the possibility of applying reversibility principles developed by processes at microscopic scale (Day and Daggett, 2007). 1.4.4. Is ‘noise’ useful for brain computations? Noise is omnipresent and is expected to influence the computation in neurons. Seen as strong stochastic membrane potential fluctuations (Pare et al., 1998) to random input stimuli, the presence of noise may be responsible in modulating neuronal activity. Almost everywhere in our daily life, noise is considered to be an undesired phenomenon. However, recent results of Salinas, if confirmed, suggest that noise can be useful for neuronal computations since the addition of noise may allow neuronal ensembles to maintain an “optimal performance” bandwidth (Salinas, 2006). 1.4.5. What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? The presence of metabolic subunits within the neuron, defies the idea of a simple indivisible neuron. The current neuron doctrine assumes a passive role for dendrites in acquiring information. The role of passive dendrites has been contradicted by several patch-clamp experiments performed in dendrites. The dendrites and soma have voltage gated ion channels that generate electrical potentials at the origin of diverse recorded electrical activities. The existence of dendritic spikes based on opening of voltage-gated sodium and calcium channels show the dynamic role of dendrites in information processing. Their diversity in information processing is responsible for specialized functions of different neurons (Häusser et al.,2000) The existence of several forms of dendritic excitability and multi-compartmentalization in dendrites was explained as being necessary for multiple sites of integration of synaptic inputs that may enhance the computational power of single neurons (Poirazi and Mel, 2001). The importance of

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dendritic morphology for AP propagation has recently been demonstrated (Vetter et al., 2001). 1.5. How can Information be Represented in the Brain 1.5.1. Is postsynaptic activity required for dendritic development? A biological means by which information representation has been thought to occur is by changes in the synapses in “connected” neurons, as discussed above. This could also be seen as happening at the dendritic level. Dendritic growth is seen as an adaptive process to better respond to incoming signals (Konur and Ghosh, 2005; Wong and Wong, 2000). The role of excitatory neurotransmitters for such growth seems important since cells in cultures have less developed dendritic trees than in vivo, where, all kinds of inputs shape growth and arborization. However, in case of Purkinje cells dendritic development can be achieved by a predetermined growth program (Adcock et al., 2004). 1.5.2. How and where are memories stored? Declarative memory representation is supposed to have a distinct place in the hippocampus (Eichenbaum, 2004). Additional working memory is shown to be located in the prefrontal cortex (PFC) and somehow connected with the activity in the basal ganglia. However, there is a continuous debate whether memory access in PFC is related to a remembered stimulus or is preparatory for a decision (Boussaoud and Wise, 1993). Some studies show that neuronal responses correlate with the outcome of a perceptual decision (Shadlen, and Newsome, 1996). Other, electrophysiological recordings show an involvement of PFC by mediating working memory phenomena (Jonides et al.1993, Funahashi, et al., 1993). The short term memory capacity depends on the number of objects required to be memorized and on their complexity (Xu and Chun, 2006). However, it is expected that neural coding for stimuli is similar for the senses such as vision, hearing, and touch. 1.5.3. The key of the topographic map Brain-imaging measures suggest that a strong relationship exists between brain structure and function. It has been demonstrated that during a simple decision task functional connectivity occurs between several brain areas (Boorman et al., 2007). However, currently there is no plausible explanation for exactly how this topographic projection of information actually occurs within the brain. While topography can result from a wiring optimization process (Buzsaki, 2004) Hebb claimed that such topography is an artifact and in fact that connectivity (synaptic connectivity) can and should exclude the role of topography. The above discussion has only touched upon the substantial difficulties faced by neuroscience and a deeper understanding of neural computation remains out of our reach. These controversies and challenges show that the “brain language” is a complex and strange way of communication, which is hard to understand especially from the limited spike timing approach. The assumption of spike stereotypy and homogeneity between neurons remains an extremely important problem present everywhere in theoretical neuroscience. Since neuronal spikes are not stereotyped events (Aur et al.,

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2005, Aur and Jog 2006, Aur and Jog, 2007a, Aur and Jog, 2007b) they have richness and semantics, similar to words in a well-defined language. Electrophysiological recordings in behaving animals have revealed considerable information about the nature of coding. Recent new methods and techniques have for the first time provided a glimpse into the relationship between neuronal activity and emergent semantics (Aur and Jog, 2007a). These intrinsic neuronal activations seen as movement of charges could be the building blocks for higher order functions such as learning, memory and cognition. The movement of charges can play the role of letters in composing “words” in terms of brain language. With this new level of abstraction, which operates explicitly at neuron level and implicitly at network ensembles levels, we will be able to explore the full power of brain computation. Therefore, it is appropriate to settle these controversies by building a new model of neural computation using classical and quantum physical laws. However, understanding this complex mechanism of information processing requires knowledge from several disciplines. 1.6. Interdisciplinary Formalism The fundamentals of mathematical thought are often developed using several axioms accepted without proof. The challenge of building a theoretical framework for the brain computations in the same manner does not seem to be solved yet. Whether brain computations can be built on such fundamental notions which are similar to the approach taken by mathematical science, still remains an open question. However, since the approach of thought proposed in this book requires an interdisciplinary systemic approach, mathematics and the laws of physics are the essential tools for implementation. 1.6.1. Sets and functions Many important concepts in mathematics are based on the idea of a set. A set can be defined as a collection of distinct objects that may include numbers, letters, symbols, or other sets. Sets are listed as members inside parenthesis X = {4, 9, 3, 10} where the set X contains four elements 4, 9, 3 and 10. In order to show that 3 is an element of the set X one can write: 3∈ X (1) While for the element 5 that is not contained by X one can write: 5∉ X

(2)

If for two sets X and Y, all the members of X are also members of Y this can be represented as: X ⊂Y (3) or that X is a subset of Y. The union of two sets X ∪ Y is the set that includes all members of sets X and Y while the intersection X ∩ Y contains only the elements that are common for X and Y.

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The mathematical concept of a function shows an input-output relationship between two or several quantities. A function can be given by a graph, an algorithm or usually by a formula: y = f (x) (4) where f is the function with the output value y and the input x . The function f is called a transformation or mapping and can be written using the notation ƒ:X→Y where X is the domain and Y is the codomain of two given sets. Occasionally, these important concepts of set theory will be applied to characterize computation performed by charges. The required mathematical framework will be further developed in chapter 4 of this book. 1.6.2. Dynamical Systems The understanding of complex systems behaviour over time is the main focus of system theory. Ideas of system theory have grown in several scientific areas from engineering to sociology, biology and economics. The states of a general system are described as a state variable. All state variables and their combinations can be used to express the state space of the system. This formulation in state space allows us to represent almost every system in an n-dimensional space where n is the number of states S1 , S 2 ,...S n that describe the system. As an example, a single particle moving along the x axis can be considered to have two degrees of freedom namely position and velocity which can be represented in 2-dimensional space. Such state definitions may represent a particular combination of physical properties or can be more abstract. These states are thus the required set that describes the specific conditions for a particular system. The evolution of the system is seen then as a succession of stages from one state to another state. If the system does not satisfy the superposition principle then the representation has to be written in terms of nonlinear functions. In case the transition between states is perceived to be in continuous time then the system is considered to be a continuous time system and described as a continuous set of ordinary differential equations: dS1 = f1 ( S1 , S 2 ,...S n ) dt dS 2 = f 2 ( S1 , S 2 ,...S n ) (5) dt . dS n = f n ( S1 , S 2 ,...S n ) dt This description is well known and very common since almost any physical system can be modelled as a set of differential equations. In a similar fashion, every neuron can be regarded as a dynamical system with inputs and outputs and neuronal state (Figure 2)

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Inputs

Outputs Neuronal State

Figure 2 Schematic representation of a neuron as an input-output dynamical system

The dynamics of electric charges are determined by the laws that govern how the system evolves over time. Additionally, this evolution depends on selected state variables and their order. The inputs and the outputs of the system can be selected based upon any biochemical or electrical outcome that is relevant to neuronal function. This dynamical state dependent evolving system level approach is exceedingly important and has begun to be described in relation with information transfer in neurons (Aur et al., 2006). A detailed construct of this framework will be presented in subsequent chapters. A much simpler example can then be considered using the case of the particle (charge) dynamics. The state space description of the particle can be considered in the form: dX = f 1 ( X ,V ) dt (6) dV = f 2 ( X ,V ) dt where X is the position and V is the velocity of the particle. A plot of position as a function of time across the velocity variable for the particle is called a phase diagram (Figure 3). 0.8 0.6 0.4 V

0.2 0 -0.2 -0.4 -0.6 1

2

3 X

4

5

Figure 3: Phase diagram for a charged particle, which moves with velocity V over X-axis and describes damped amortized motion

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If the transition is seen in discrete steps then the description is in terms of a discrete time system as a set of difference equations: S1 (k + 1) = f 1 ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....) S 2 (k + 1) = f 2 ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....) .

(7)

S n (k + 1) = f n ( S1 (k ), S1 (k − 1)..., S 2 (k ), S 2 (k − 1),...,...S n (k ), S n (k − 1,....)

Such reduction of a continuous description to this discrete form is performed in von Neumann computers and has several consequences. In case of smaller sampling rates under the Nyquist rate, information is always lost from the original continuous description (Cover and Thomas, 1991; Sobczyk, 2001; Aur et al., 2006). There is a similar terminology regarding the state of a machine in computer science. The transition of states can describe the behavior of a finite state machine, such as a Turing machine. Since usually the state term defines a particular set of instructions that are executed the sense of state and the considered transitions can be restrictive and sometimes significantly different than the terminology used in dynamical systems. 1.6.3. The laws of thermodynamics A deeper understanding of natural processes including life, brain function and cognition should come from well recognized general principles. The law of entropy is a general principle known as the second law of thermodynamics, which can be applied from physics to biology and evolutionary theory. Viewed as a 'law of disorder' in a system the second law of thermodynamics makes the link between physics and statistical properties of systems. Boltzmann defines this concept in the terms of entropy, H B : H B = k B ln Ω (8) where k B is the Boltzmann constant and Ω is the number of microstates. The second law states that in an isolated system which is not in equilibrium the entropy tends to increase over time, approaching a maximum value at equilibrium (Halliday et al., 2004). Importantly, the second law also establishes the direction of heat transfer. The first law of thermodynamics states that the total energy of a system does not change. Different forms of energy, mechanical, electrical or thermal energy can be converted from one to another, however, their sum must remain constant. This may also demonstrate the equivalence between heat and other forms of energy. The system can change its internal energy ΔWi only by work or heat exchanges: ΔWi = Q − L (9) where L represents the work and Q is the heat exchange. Importantly, the second law establishes the direction of heat transfer.

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QC

QH

W

Cold Reservoir

Hot Reservoir

Figure 4 Heat engine scheme

Additionally it can be successfully shown that it is not possible to convert all thermal energy Q to do work L . This result was extended in engineering and thermodynamics to the theoretical concept of heat engines (see Figure 4). Physical implementation of such an input-output system is based on a specific thermodynamic cycle and transforms the heat input to an output, which usually is mechanical work (e.g. steam engines, diesel engines). The efficiency of the thermodynamic engine can be written: η=

QH − QC QH

( 10 )

where QH is the amount of heat from hot reservoir and QC is exhausted at the cold reservoir. Depending on their interaction with the environment these heat engines can have open or closed cycles. The major revolution in science was the establishment of the link between the "law of maximum entropy” statistical mechanics and information theory (Shannon, 1948; Jaynes, 1957). It can be successfully shown that information encoding can be better performed within a distribution, which maximizes the information entropy. 1.6.4. Conclusion An integrated application of such universal laws will help us to understand how the brain computes. Therefore, in order to understand the brain language, neuroscience has to ask the right questions. The main issue yet unsolved is to forge a link between cognition with information processing and theoretical computation. The cable myth, the dilemma of synapses, the Hebbian approach of connectivity and spike stereotypy reflect the actual myth that pointed to spike timing as a single path in explaining neural computation. The temporal coding approach seems to be inadequate to solve important issues integral to brain processing begging for questions like. Do irregular spikes convey more or less information than the regular ones? Is the neuron an integrator of incoming inputs? Which technique is better to quantify neuronal activity? Firing rate or ISI? Across-fiber pattern, grandmother cell, or labeled-line?

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Other questions should aim to tackle the inherent aspect of computation: Is noise useful for brain computations? Is computation in the brain a quantum process? What is the role of active dendrites and metabolic subunits within the neuron from a computational perspective? Information representation in brain is also a critical issue. How is information represented in the brain? How and where are memories stored? Is there any meaning behind the topographic map? Solving all these issues requires an interdisciplinary formalism where the laws of physics should play a greater role than spike timing model. The view proposed here is that, ultimately, spike timing is a natural outcome of a much richer process that defines the brain language. However, this mechanism of computation may never be understood if the entire analysis is restricted to spike time patterns observed in single neurons or neuronal ensembles. Since such formalism based study of the brain may allow finding the link between the semantics of computation and emergent behavior an analysis of both phenomena is desirable. However, before developing these principles it is important to have a construct of the current advances in experimental data analysis. Therefore, in order to study the computational power we have to switch back and carefully re-analyze behavioral experiments and the strength of interpretations of earlier recordings performed in-vivo. This is developed thoroughly in the next chapter. Summary of Principal Concepts • • • • • • •

• • •

Brain computes differently than current digital computers The problem of acquiring semantics using an algorithmic approach has no solution as yet in terms of a symbolic framework The hypothesis of spike stereotypy is at the origins of several controversies in neuroscience and neural computation Understanding the brain code using neuronal recordings faces the problem of semantics. If neural computation is reduced to symbols manipulation (stereotype action potentials) then semantic issues may arise in Searle’s terms. Spikes (action potentials) as ‘binary pulses’ cannot be considered to be fundamental units for computation in the brain. This view regarding spikes determines several controversies in terms of information transmission, integration of input stimuli, synchrony, memory storage and retrieval. Neuronal spike, seen as a binary pulse has historically been the cause of much scientific debate: if the neuron is an integrator of incoming inputs or if it performs coincidence detection, if irregular spikes convey more or less information than the regular ones or if firing rate is better than ISI to quantify neuronal activity. Spike timing theory is a radical view that attempts to explain brain computations in terms of an existent temporal code. The description of brain computation has to be made in terms of universal laws of nature Brain analysis requires an interdisciplinary formalism where computation can be described in terms of dynamical systems theory and thermodynamic laws

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Stevens C. F and Zador A.M., (1998) Input synchrony and the irregular firing of cortical neurons, Nature Neuroscience 1, 210 - 217. Steveninck De R.V., R. R. Lewen, G. D. Strong, S. P. Koberle, R. Bialek, W. (1997) Reproducibility and Variability in Neural Spike Trains, Science, , 5307, pp. 1805. Thorpe S. J., and Imbert, M., (1989) Biological constraints on connectionist models. In R. van Vreeswijk C. and Sompolinsky H. (1996) Chaos in Neuronal Networks with Balanced Excitatory and Inhibitory Activity, Science 6 ,Vol. 274. no. 5293, pp. 1724 – 1726. Vogels R., Spileers W., Orban G.A., (1989) The response variability of striate cortical neurons in the behaving monkey. Exp Brain Res 77:432-436. Von der Malsburg, C. (1981) The Correlation Theory of Brain Function, Internal Report 81-2, Max Planck Institute for Biophysical Chemistry, Gottingen, Germany Vetter P., Roth A. and Hausser M., (2001) Propagation of action potentials in dendrites depends on dendritic morphology, J Neurophysiol 85 pp. 926–937. Xu Y. and Chun M. M., (2006) Dissociable neural mechanisms supporting visual short-term memory for objects Nature 440, 91-95 Wilson M. A., McNaughton B. L., (1994) Reactivation of hippocampal ensemble memories during sleep Science 265, 676. Winter IM, Palmer AR. (1990) Responses of single units in the anteroventral cochlear nucleus of the guinea pig. Hear Res. Mar;44(2-3):161-78. Wong W. T and Wong R. O.L., (2000) Rapid dendritic movements during synapse formation and rearrangement, Current Opinion in Neurobiology, Volume 10, Issue 1, Pages 118-124.

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CHAPTER 2. Imaging Spikes – Challenges for Brain Computations The Copernicus model generated the greatest revolution in astronomy and questioned the beliefs of Aristotle and Ptolemy that the Earth is at the center of the Universe. The initial failure to accept Copernicus’s theory was due in part to an inadequate ability to see far away in space. “Seeing is believing”, and a revolution in thinking was possible only after several improvements to the telescope led to observations of Jupiter’s moons. This is a classic example where the progress in technology confirmed earlier theoretical ideas. We face the same challenge today in neuroscience. Imaging techniques such as magnetic resonance imaging (MRI), functional MRI (fMRI) along with electro-encephalography (EEG), play an important role in monitoring brain activity. From sensations to body movements, from memory to language in general, imaging techniques have significantly changed our perception about brain functions. Recording of electrical activity of individual and networks of neurons can be thought of as imaging at a much finer level. Activity patterns revealed in such recordings can be studied at levels of slice recordings up to multiple cells in vivo where the activity is elegantly related to observed behavior. The correlation of large scale imaging such as fMRI to the electrophysiological substrate is however sorely lacking and a significant gap exists between imaging techniques at larger scales and electrophysiological recordings in single cells. Action potentials were assumed to be uniform pulses of electricity and we believe that the origins of this view are buried in the early single electrode recordings. Since neurophysiologists could see only similar shapes of action potentials, this occurrence of electrical signals in nerve cells was considered to be a stereotyped uniform pulse signal. This revelation came about from hearing such electrical impulses from the optic nerve as described by Adrian (Adrian 1928): "The room was nearly dark and I was puzzled to hear repeated noises in the loudspeaker attached to the amplifier, noises indicating that a great deal of impulse activity was going on. It was not until I compared the noises with my own movements around the room that I realized I was in the field of vision of the toad's eye and that it was signaling what I was doing." These early recordings claimed that action potentials are uniform pulses of electricity and their variability and importance consist only in their time of occurrence. Interestingly, this viewpoint has essentially remained unchallenged for a long time. Newly developed techniques of multi-tip electrodes however provide a tool for powerful analysis of electrical propagation during single action potentials and shows that the earlier claims of action potential stereotypy are not valid. 2.1. Methods, Models and Techniques Initiated by Ramón y Cajal in the late 19th century, the neuron doctrine was developed based on single electrode recordings and anatomical tracing studies of nerve cells in the brain. This doctrine states that neurons are distinct structural and functional units. Neurons were considered to carry information through electrochemical processes having an intrinsic ability to integrate both aspects as a part of their signaling capability. From an electrical perspective, two important assumptions that have shaped

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

the field of neurophysiology were made: that an AP propagates predominantly in the axon and the spike can be considered a stereotyped phenomenon. Given the electrochemical nature of processing, the ability to integrate a rapid (electrical) and a much slower (chemical) transmission in order for neurons to communicate became an important problem, especially since neurons are spatially separated. This led to an important controversy that started in the 1930s regarding neural communication. The issue of whether nerve cells communicate with each other chemically or electrically generated a strong debate between scientists in the field. After a long dispute the prior theory of “continuous nature of nervous transmission” (Fulton, 1926) was replaced. A new theory for information coding in the nervous system was developed and the neuron was regarded as the basic information-processing unit. Additional theoretical work of McCulloch and Pitts contributed to formalize brain processes where the neuron was considered the key element for processing and communication in the brain (McCulloch and Pitts, 1943). Initial supporters of the electrical theory suddenly changed their viewpoint in favor of a new approach. This change in interpretation was determined by images of synaptic vesicles that contain neurotransmitters and theoretical approaches regarding chemical mediators (Eccles, 1957). Neurotransmitters such adrenaline, acetylcholine, norepinephrine, glutamate or serotonin were associated with the inhibition or initiation of AP while the synapse was felt to be the substrate of neuronal communication. Over time, learning and memory phenomena also became closely related to synaptic plasticity (Kandel et al., 2000). Accumulation of experimental data determined the development of theoretical models with the main goal of studying electrophysiological properties of neurons. An important step in understanding the electrical nature of the AP was made by the Hodgkin-Huxley model (Hodgkin andHuxley, 1952). This model showed that almost every portion of the cell membrane is far more complex than the switching device model proposed early by McCulloch and Pitts. A new theoretical framework able to explain brain activity was required. Barlow (Barlow, 1972) made the first theoretical attempt to define rigorously the field of neuroscience and neuronal computation. He proposed a set of five dogmas that are considered even today to be the foundation of research in neuroscience. The “first dogma” states that we can understand the relationship between the brain and mind by simply analyzing cell interactions. This was the primary goal of Hubel and Wiesel and their studies expanded our knowledge regarding the visual system. They showed for the first time that visual experience in early life influences neural connections (Hubel and Wiesel, 1962; Hubel and Wiesel, 1968). Recently, neuronal activity in individual neurons or network of neurons has been analyzed using information theoretical measures (Mainen and Sejnowski 1995, Rieke et al., 1996). Several scientists have tried to describe the mechanism of coding in neuronal ensembles and new theories regarding interspike interval (ISI) are considered the basis of the “third generation” of neural networks (Gerstner and Kistler, 2002, Maass, 1997). However, all these analyses and proposed theories have their origins in an important assumption that was stated above, namely the spike stereotypy hypothesis. Neuronal activity and thus neuronal coding of information is analyzed predominantly in the temporal domain. This viewpoint has recently been challenged by the demonstration of information transfer in a much smaller time window, within milliseconds during every AP. Using simulations of Hodgkin-Huxley model it can be shown that the values of mutual information between input stimulus and sodium fluxes

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are more than double compared to mutual information between the membrane voltage and stimulus input (Aur et al., 2005). Such theoretical postulation that information is physically transferred during spike occurrence by the movement and interaction of electrical charges has implications and may define a new model of computation. All these issues regarding information transfer are presented in detail in the next chapter. However, before elaborating this new model we have to verify this theoretical outcome using experimental data. 2.1.1. Tetrodes Since cell bodies of neurons are packed at distances of tens of microns, single extracellular electrodes are able to record signals simultaneously from several neurons. Electrical activity in neurons can be revealed using multi-tip recording devices followed by data analysis using several processing techniques. The needle-like electrodes are brought in the proximity of the cell. As a result the amplitude of the recorded signal is slightly larger than the amplitude of neighbour neurons. This separation of amplitudes of recorded spikes and their correspondence to a specific individual cell is an important step in analysing the activity of neuronal ensembles. However, despite sophisticated hardware and software methodologies, it is difficult to assign recorded spikes effectively to corresponding neurons. More recently McNaughton et al., utilized a new implantable device with multiple electrodes packed together named stereotrodes and later tetrodes (McNaughton et al., 1983, Wilson and McNaughton, 1993). Since an electrode receives signals from several neurons it is difficult to assign a certain spike to the corresponding neuron. Finding this correspondence between cells and spikes is important if one analyses how neurons work together in the network. A tetrode uses inexpensive material and can be made virtually in any laboratory manually by twisting four insulated wires into a bundle (Figure 1). This manual assembly can produce a usual spacing between the geometric locations of wires on average between 20 and 30 micrometers and can be in an ensemble, which can be used to record from several areas simultaneously (Jog et al., 1999). Therefore, with this device the firing pattern of neurons in a very small region of tissue of about 100micrometer radius can be recorded.

Figure 1: Ensemble of several tetrodes

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

A band-pass filter is then used to pre-process the recorded signal followed by spike threshold detection. This detection procedure can be implemented using a hardware device or a software program. Then the amplitudes of the spike recorded by each channel of a tetrode are considered to generate a feature vector that is used to "cluster" the spikes. Clustering techniques group spikes with similar feature vectors and assign each group to the corresponding neuron. Improvements in the spike sorting methods of a recorded population of neurons using multi-tip electrodes were carried out in the early 1990s and is at the origin of fabrication of multi-site electrodes (McNaughton et al., 1983, O'Keefe and Reece, 1993). If this process of spike sorting is done manually, it is time consuming, and may be affected by subjective factors. It has been proposed that automatic methods may significantly speed up the clustering process, and reduce the effect of subjective factors (Fee et al. 1996; Lewicki, 1998; Sahani, 1999; Jog et al., 1999). However, despite these advances clustering is done manually in many laboratories. Even though the tetrode recordings were designed to improve spike-sorting techniques they can ideally be used to analyze the pattern of electrical activities during every AP. The methodology that allows this analysis of electrical patterns has been recently developed as a straightforward computational approach (Aur et al., 2005, Aur and Jog 2006). Two distinct processing techniques have to be applied in order to allow the visualization of electrical patterns during APs. These methods are discussed in detail elsewhere (Aur and Jog 2006). In short, they require assignment of charges or group or charges with charge points and then to perform independent component analysis (ICA) and triangulation techniques to spatially localize the position of charges. 2.1.2. The Necessity for a New Model Current models of brain activity are highly spike-time oriented and they omit some other important aspects of neural activity, including the intrinsic electrical communication with electrical charges, gap junction transmission, active dendrites, ion fluxes etc. Action potentials are indeed carriers of information; however they are not uniform (stereotyped) pulses of electricity. At the nerve cell scale, for dimensions between 100 to 300 micrometers, every action potential (AP) can be well described by mapping electrical patterns of activities in a three dimensional space. Using new techniques we provide evidence that APs are not stereotyped events and their analysis even for mathematical modeling purpose cannot be reduced to spike timing occurrences without loosing the essence of what is contained within spiking activity. The representation of independent signal sources in three-dimensional space shows that charges are preferentially spread during every AP across certain areas of the nerve cell where ion channels are likely to be more active and generate electrical patterns of activations (Aur and Jog 2006). The new model of charges in movement highlights all these aspects and provides a better interpretation of computation at a smaller scale within every action potential. 2.1.3. The Model of Charges in Movement Electrophysiologists have known for a long time that the amplitude of extracellularly recorded AP attenuates strongly and inversely with distance. Therefore, several experiments suggested that a “point source model” would be able to approximate well enough the extracellularly recorded AP (Drake et al., 1988). The point approximation does not seem to perform well in real recordings due to several

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issues presented below including noise level. Additionally, since every spike is a complex electrical phenomenon (Aur et al., 2005) neurons cannot be considered as point sources during the action potential occurrence. An important implication of this aspect is that neurons cannot be practically separated based on their spatial location (Recce and O'Keefe, 1989). Lately, this issue has become important since several researchers have tried to determine AP propagation. They anticipated that this point approximation with a single charge does not seem to be an adequate description of an AP. Calling this view into question Holt and Koch approximated the occurrence of AP with a line of charges named “the line source model” (Holt and Koch, 1999). Supporting earlier studies including the view that an axon can be approximated with a cable; recently, this model was recently extended to extracellular recordings by Gold et al (Gold et al., 2006). They assumed that the extracellularly recorded AP can also be treated as a line of point sources and then the resulting voltage can be computed as the sum of potentials generated in this “line” of point sources. By contrast, another proposed model (Aur et al., 2005) shows that even this “line” of point sources or the “point source model” alone is a poor descriptor of electrical phenomena during an AP. Simply, if one considers tetrode recordings, where the distance between tips is about 20 microns, then the spatial occurrence of charge movement during every AP is about five to ten times wider (or higher) than this distance between tips. Therefore, it is unlikely that electrical activities within an active neuron could be well approximated with a single point source or even with a simple line of electrical sources. This demonstration of spatial geometry shows that we may need a different model to describe the complexity of the electrical phenomenon within a spike. Since biological neurons are complex three-dimensional structures with dendrites and axonal arborizations every spike can be considered a superposition of several charges in movement. The electric field and charges that move in space are the sources for generated signals in electrode tips. This simple view has important implications for modeling extracellular measurements and consequently the electrical events such as spikes can only be well described as results of dynamics of many charges (Aur et al., 2005, Aur and Jog, 2006). Due to several factors already mentioned above, action potentials are considered to be stereotypical events. This view of spike stereotypy has remained unaltered for a long time. The factor that has contributed to a slow progress in this field is the beautiful mathematical theory that can be developed assuming spike stereotypy. Spike timing theory has attracted some of the best minds and any suggestion that may show a completely different view by contrast to the stereotypical, all or none AP event is stopped in advance (Scott, 1999). There are also several practical reasons why this spike timing theory has remained so popular. First, the spike phenomenon occurs within a short time about one millisecond. Second, the powerful propagation of electrical wave in the axon buries other electric phenomena, which take place in dendrites. Hence, these activities remained unobservable for a long period of time and indeed, as mentioned by Gold et al., dendritic morphology has very little impact on the extracellularly recorded waveform (Gold et al., 2006). Despite the currently prevalent view, we have been able to show that the biological and architectural details of dendritic arbors and axonal branches slightly yet significantly modulate the extracellular AP. However, these minor changes in AP waveforms can have a strong impact in determining electrical ‘connectivity’. We will demonstrate that (a) this weak activity in dendrites is extremely important for neuronal function, (b) this small variability can tremendously change patterns of activations in a

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neuron and (c) it is likely that such changes will alter the function of neuronal ensembles. Additionally, even though well accepted, axons cannot be approximated with simple (linear) cables since they always have a certain degree of arborization and contain numerous spatially distributed voltage-gated channels (Debanne, 2004). Further, the membrane itself is a complex electrical structure that is hard to approximate as a linear cable. Modulation of proteins and changes in their characteristics at ion channels or receptors level may have a strong impact in electrical signal transmission and implicitly on the interaction with synapses, other dendrites, or axonal coupling. The fundamental postulates of this book are based upon providing a mathematical formulation and an experimental base to understand brain computations. The issues raised in chapter one and the beginning of chapter two will be directly tackled by applying physical principles to allow an integrative view from macromolecular alterations seen in proteins to modulations of APs. These subtle changes that occur during neuronal activity will then be related with information transfer and emergent behavior. 2.1.4. The Triangulation technique As a technique, triangulation was intensively applied during and after the Second World War for detecting submarines with noisy diesel engines. The system called SOSUS (Sound Ocean Surveillance System) used the property that sound spreads in the ocean uniformly in all directions, with known attenuation. Since sound detectors are distributed in space, then the position of each submarine can be obtained by processing the signal received from detectors using as a simple geometrical procedure called triangulation (Figure 2).

D

A

B

Figure 2: The distance D to the ship can be estimated if the spatial coordinates of sound detectors A and B are known

2.1.5. Charges and Electric potential The charge movement model (CMM) has been developed in order to address many of the concerns raised above (Aur et al., 2005). This model challenges current view which relates spiking activity to several myths and suggests a new development where the remaining formalism of spike timing can still be maintained and demonstrated as a special case with particular characteristics. This CMM formalism is critical for

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understanding how information is electrically communicated to downstream brain structures and ultimately how they are related to spatiotemporal spiking activities. The electric potential, V generated by every charge, q and recorded by tetrode tips depends on the distance r from the charge (Figure 3a): q V= (1) 4πε r and ε is the electrical permittivity of the medium. Additionally, for a line of charges, the electric potential can be computed by summing elementary potentials determined by single charges (Figure3, b): qi V =∑ ∑ (2) 4πε ri i In the proposed model the main hypothesis is that different charges or groups of charges under an electrical field have distinct movements. Therefore, in order to apply independent component analysis (ICA), the AP is assumed to be the output of several independent sources of signals generated by charges that move in space. Hence, for recorded AP signals, using ICA, a blind source separation can be performed. This technique generates a linear transformation which minimizes the statistical dependence between components (Comon, 1994). a

b -3

-3

x 10 Electric Potential [Vm-1 ]

Electric Potential [Vm-1 ]

x 10 10 8 6 4 2 5

5 0 Y [um]

0 -5

-5

X [um]

15 10

5

5 5 0 Y [um]

0 -5

-5

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Figure 3: The representation of electric potential a. b.

The electric potential representation for a single charge, The electric potential representation for a line of charges obtained by summation of elementary potentials

Consequently, the localization of charges or group of charges is obtained performing triangulation. Importantly, the distribution of charges can then be related to selective activation of ion channels during AP and spatial propagation of electrical signal (Aur and Jog, 2006). 2.1.6. The Projective Field: Spatial Directivity of Charges The activity of the sensory system has been a preferred area of investigation and several studies are concentrated in analyzing visual, auditory and tactile sensory properties. A classic example in this area is the early work of Hubel and Wiesel who studied the dependency of retinal receptive fields on the input stimuli characteristics (Hubel and Wiesel, 1962).

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Additionally, beyond the study of patterns in receptive fields it is important to understand the spatial path where these neurons project the electrical signal. The idea of a projective field is not new. In the 1930's Penfield, the famous Canadian neurosurgeon was able to show that specific muscles can be activated if particular areas from motor cortex are electrically stimulated. Therefore, we should be able to prove the existence of certain electrical paths where information is selectively transferred. Since the projective field characterizes in general the activity of neuronal ensembles, we anticipate that this type of information should somehow originate in neurons. However, to identify where in space specific nerve cell projects information is not a trivial analysis. Staining and labeling studies are unlikely to trace these electric pathways since these projections are rapidly altered within millisecond time range by the occurrences of spikes. For this purpose we have developed the new model that is able to analyze changes in electrical propagation at microscale level within recorded action potentials. The simplest model that we were able to simulate describes the movement of several charges in space during the action potential (AP) generation. If a frame of reference is considered in space as the origin of measurement, then the arrangement of these charges can be characterized globally by their “directivity” over the selected reference system (Figure 4 a, b, Figure 5 a, b).

a

b -3

x 10 15

-3

Electric Potential [V]

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x 10 15 10 5

5 0

5 0 Y [um]

-5

-5

X [um]

10 5

5 5

0 Y [um]

0 -5

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Figure 4: Representation of charge directivity as a measure of spatial distribution of charges over the reference .The frame of reference has the coordinate origin of X axis. Charges are represented in blue color and the directivity is represented in magenta. a Representation of a line of charges in blue and computed directivity in magenta b Since the position of charges over the reference is changed from a, then the computed directivity has an opposite direction

The method for computing the directivity of group of charges and their application to compute spike directivity was detailed in (Aur et al., 2005). Briefly, the method consists of singular value decomp osition (SVD) for the matrix that contains the coordinates of charges in space. The position of charges is determined from recorded voltages using the triangulation technique assuming the charge point approximation.

Chapter 2: Imaging Spikes – Challenges for Brain Computations

a

33

b x 10

-3

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Figure 5: Representative changes in computed “directivity” are determined by different arrangements of charges in space. Charges are represented in blue color and the directivity of spatially distributed charges is represented in magenta. a. An example of spatial distribution of charges in blue color and computed directivity b. A different spatial distribution of charges in blue color and computed directivity

2.1.7. What is “Spike Directivity”? Spike directivity is a new concept that describes electrical activity in a biological neuron. However, since the mathematical formalism is not always easily understood a physical interpretation of spike directivity measure is important. The computational approach underlying “charge movement model” offers such a physical representation in terms of a vectorial representation of every spike. Small differences in recorded voltage of APs from the same neuron (Figure 6, a,b) can be translated into specific spatiotemporal patterns of electrical activations (Figure 7 a,b). Additionally, spike directivity can be estimated for every spike.

-4

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Figure 6 Representation of two sequential spike waveforms recorded from the same nerve cell with the same tetrode. a, Four recorded waveforms and their corresponding electrical patterns of activation are represented in Figure 7a b, Four recorded waveforms and their corresponding electrical patterns of activation are represented in Figure 7b

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

Figure 7 Representative electric patterns of activation from the same neuron in red and computed directivity in blue. The computational processes occur along electrical spatial patterns of activation in dendrites, soma and axon. The axon appears to be strongly involved in electrical flux within each spike. a, Specific electric patterns of activation corresponding to waveforms from Figure 6a b, Specific electric patterns of activation corresponding to waveforms from Figure 6b

These spatio-temporal patterns during each AP are likely to be related with a preferential activation of ion channels in specific regions of nerve cell membrane, dendrites or axon. Indeed, Oesch et al., 2005, using patch-clamp recordings and two-photon calcium imaging demonstrated that dendrites are selective to AP directivity. Equally, the dendritic architecture has an influence on the cell response and these slight spatial changes in spike directivity in several spikes may be determined by a preferential activation of dendritic tree during each AP (Aur and Jog, 2006). In a simplified manner spike directivity can be seen as a reflection of the distribution of charges in space. Since the recordings are made with tetrodes, these spatial distribution of patterns describe electrical characteristics revealed over the frame of reference (tetrode position). The spike directivity term can be described by a preferred direction of propagation of electrical signal during each action potential (AP) and approximated with a vector (Aur et al., 2005). This characteristic can be obtained using tetrodes or any other type of multi-tip receiver (with number of tips n≥4) but cannot be measured based on single electrode recording techniques. Such analyses add an important level of subtlety to information processing by these neurons (Aur et al., 2006). Therefore, when simplified, every spike can be represented as a vector, which characterizes spatial distribution of charges in space. Following this rationale, the group of neurons that exhibits spiking activities can be described as a set of vectors spatially oriented at distinct moments in time. This model even in this simplified form offers a much richer description of neural activity than any current spike timing models. Therefore, beyond their time of occurrence, each spike can be characterized by a new feature called spike directivity. 2.1.8. Triangulation and Independent Component Analysis Together Now, imagine that instead of submarines one would like to determine the position of charges in space during AP generation. Switching back to the issue of AP recordings, the tips of a tetrode function like the sound detectors used by submarines. The movements of charges in space generate electrical currents, which determine the

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potential difference ΔV at the tip of a tetrode. As a result the corresponding current i (t ) produced by charge q can be computed: 1 q 1 q − = ΔV 4πε rB (t ) 4πε rA (t )

(3)

where ε is medium permittivity, and rA (t ) , rB (t ) are distances from the charges to conductor ends. The total drop of potential for several charges in movement can then be seen as a linear result determined by several currents generated by these charges in movement: ΔV (t ) = ∑ α k ik (t ) i =1, N

(4)

where α k are constant. Therefore, the voltage recorded during an AP by the tetrode tip (k) can be written in a discrete form:

V ( k ) = [V ( k ) (1) V ( k ) (2) . . . V ( k ) (n)]

(5)

where n represents the number of points recorded in time from a single spike. Since the recorded voltages are hypothesized to be some mixture of signal sources generated by an ionic flux, then each spike recording can be rewritten for each tip in the simplified form: V = Ms + N n

(6)

where V is data obtained from tetrode tips, M ∈ Rmxn mixing matrix, s is the signal matrix and N n models the noise. In order to provide an internal model of the data, an independent component analysis (ICA) technique is used where for simplicity we have omitted the noise term. Writing sˆ for the estimate of s , and Mˆ for the estimate of M, the estimated signal sources can be obtained after ICA is performed: sˆ = Mˆ −1V

(7)

If the number of observed linear mixtures is at least as large as the number of independent components then Eq. 6 has a solution. Even in the case where m
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Chapter 2: Imaging Spikes – Challenges for Brain Computations

t io ( x − x1 ) 2 +( y − y1 ) 2 + ( z − z1 ) 2 = t i1 x2 + y2 + z2

t io ( x − x 2 ) 2 +( y − y 2 ) 2 + ( z − z 2 ) 2 = ti 2 x2+ y2 + z2

(8)

t io ( x − x3 ) 2 +( y − y3 ) 2 + ( z − z 3 ) 2 = ti3 x2 + y2 + z2

where xi,yi,zi, i=0,1,2,3 are the coordinates of the four channel tips. Several details for this algorithm have already been published in (Aur et al., 2005). The position p k of electrical sources (i.e. where recorded signals are coming from) is being generated can be obtained with an iterative Newton-Raphson algorithm that solves the nonlinear system Eq (8): p k +1 = p k - J −1 F(p k ), k ∈ N

(9)

where matrix F is: ⎡ a − ti1 (( p11 − x1 ) 2 + ( p21 − y1 ) 2 + ( p31 − z1 ) 2 ) ⎤ ⎢ ⎥ F = ⎢a − ti 2 (( p11 − x2 ) 2 + ( p21 − y2 ) 2 + ( p31 − z 2 ) 2 )⎥ ⎢ a − ti 3 (( p11 − x3 ) 2 + ( p21 − y3 ) 2 + ( p31 − z3 ) 2 ) ⎥ ⎣ ⎦

( 10 )

and the value of a is computed: a = ti 0 ( p112 + p212 + p312 )

( 11 )

The value of the Jacobian, J has to be computed at each step: ⎡ ti 0 p11 − ti1 ( p11 − x1 ) ti 0 p21 − ti1 ( p21 − y1 ) ti 0 p31 − ti1 ( p31 − z1 ) ⎤ J = ⎢⎢ti 0 p11 − ti 2 ( p11 − x2 ) ti 0 p21 − ti 2 ( p21 − y2 ) ti 0 p31 − ti 2 ( p31 − z2 ) ⎥⎥ ⎢⎣ti 0 p11 − ti 3 ( p11 − x3 ) ti 0 p21 − ti 3 ( p21 − y3 ) ti 0 p31 − ti3 ( p31 − z31 )⎥⎦

( 12 )

The number of solutions provided by the Newton-Raphson algorithm is equal with the number of blue dots represented in Figure 9. Therefore, in order to improve image resolutions a higher sampling frequency could be used for recording the data. Currently employed low frequencies between 20 and 30 KHz generate low-resolution images for electrical activity. However, if the sampling frequency is increased by about 10 times the resolution would improve significantly since there will be about 320 data values recorded for each spike. Due to several nonlinearities the computed trajectories are bent. Therefore a “debending” technique needs to be performed by first constructing the symmetric plot and subsequently rotating the image to get a better image of electrical patterns as in Figure 8 (Aur et al., 2005). Since during every AP occurrence, voltage dependent channels are activated over specific areas along dendrites soma and the axon, the charge spread reveals parts of the neuronal architecture (Figure 9). Indeed, the spatial displacement of charges during an AP is a way of providing an electrical image of activity of a neuron.

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Z

β = -α α β

Y X Figure 8 The computed and schematic representation of trajectory for linear charge movement in the tetrode space. The “de-bending” is performed in two steps. First the symmetric plot schematically shown in the figure is constructed and then in the second step the image is rotated in order to perceive trajectory patterns from the axon.

Figure 9 A 2D-view of electrical potential during an AP estimated on spatial position of charges represented in blue color. These patterns of activation (micromaps) occur during an action potential due to spatial distributions of charges. The division on the X and Y-axes are in microns.

In fact, changes in AP amplitude and width are correlated with spatial distribution of densities of charges in specific areas. It is likely that this variability in charge distribution is a result of the dynamics of ion channels, which are selectively activated during the electrical event of an AP. The patterns may occur along ion channels within dendrites, soma and axon. During every spike, the axon always appears to be strongly activated which seems to be facilitated by the highly conductible nature of the axonal membrane. In summary, the steps required for imaging electrical activities during an action potential are (Figure 10) a) Perform the ICA algorithm for all four channels b) Construct the transformation T1, as the law of cosines ui=T1(s). d) Compute the position pk of electrical sources. e) De-bending by constructing a symmetric plot.

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Chapter 2: Imaging Spikes – Challenges for Brain Computations

T1

ICA

v

sˆ

p k+1 = pk - J− 1 F(pk )

u

p

Figure 10: A schematic representations of required phases for imaging electrical activities during AP

This picture along with other images of analyzed spikes of neurons suggests that the AP cannot be considered a static stereotypical event. Using this technique, spatial patterns determined by the movement of charges during several successive AP from the same neuron are slightly different. Therefore, viewed as flux of charges, spikes are clearly not the same. This is important because the variability that is clearly pointed out in charge flux from spike to spike is a window into the subtle and complex computations that are physically occurring in neuron. It is the first step in realizing that: 1. Simply considering the spike as a stereotypic, all-or-none phenomenon is inadequate 2. There are many intrinsic and deeper layers of information processing that are occurring within neuron that cannot be ignored. 3. Charge flux analysis is a start of opening this new window and is the first step in understanding neural information processing that is discussed further in subsequent chapters. The obvious questions then are: Why do these electric patterns of activity (micromaps) exist within action potentials? What is their role? Could it be that these specific electric patterns of activation within neuronal spikes were correlated with computational processes performed by this intrinsic dynamic and interaction of charges? Does the underlying spike propagation and morphology reflect the intracellular information processing? It is already known that charges that move in space in an electric field intrinsically perform computations (Moore, 1990). Therefore, at this microscopic level it is probable that computation can be linked to changes in electric field and this physical movement of charges. Based on these observations, we suggest that spatiotemporal activation of ion channels and interaction of electric charges can be used to account for fundamental computational aspects that are observed. The current models, based solely in terms of a temporal representation can be easily incorporated in the new general theoretical framework. Additionally, it is likely that this phenomenon of electrical patterns within AP can be related to other processes including neurotransmitters, synaptic transmission. These aspects will be discussed in detail in Chapter 4. 2.2. From Spikes to Behavior The idea that neurons communicate electrically, thereby causing the generation of spikes and the relationship of spikes to behavior has been the central theme of research for many years (Cajal, 1909; Eccles, 1982). Typically, spikes were perceived to have stereotyped waveforms that can be reduced to an all or none event (Bialek et al., 1993; Strong et al., 1998; Maass et al., 1999; Purwins et al., 2000; Gerstner and Kistler 2002;

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Brody and Hopfield, 2003; Koch et al., 2004). Such reduction allowed linking of neuronal activity to behavior on a temporal basis (Wilson and McNaughton, 1993; Jog et al., 1999, Barnes et al. 2005). Thus, temporal analysis of spike trains became a standard method to understand the relationship of spiking to behavior. However, outside of relating spikes to behavior, the idea of semantics or meaning within the neural activity, started to gain ground even earlier (O'Keefe and Dostrovsky, 1971). Spatial location of the rat was associated with changes in firing rates in so called "place cells" from hippocampus. However, recent investigations show that place cells in the hippocampus appear to alter their preferred firing even in the absence of any changes in the environment (Lee et al., 2006). The predictability of the animal’s location based on firing in the “place cells” may therefore not be as robust as once thought and such temporal analyses may be too restrictive in their ability to define the complexity of neuronal activity. Finally, the current view of behavioral representation is based on population coding and not based on the activity in single neurons (Passaglia et al., 1997). We have already demonstrated above that the spike is a complex observable phenomenon that is potentally rich in information (see chapter 4). In the next sections of this chapter we will continue to use the above model of charge dynamics and extend it to the concepts of semantics. Specifically, we will show that the activity in selected single neurons from the striatum reflects behavior and even more particularly that these neurons demonstrate semantic differences regarding the emerging behavior. Due to its critical importance in motor learning and the substantial amount of existing data, we will use the striatum in order to demonstrate the relationship between spike directivity and semantics of complex behavior during procedural learning. We feel that such techniques could be applied to analyze any other area of the nervous system. The role of the striatum in learning stimulus-response associations, action selection or habit formation is unquestionable and has been discussed and demonstrated in several papers (Alexander et al., 1986; Jog et al. 1999, Packard and Knowlton, 2002; Devan and White, 1999; Yin and Knowlton, 2004; Barnes et al. 2005). The dorsal striatum, which receives a substantial component of its input from the sensorimotor cortex, plays a central role in behavioral action selection studies (Jog et al., 1999; Matell et al., 2003; Barnes et al., 2005). It is for this reason that the dorso-lateral striatum was chosen in the current study to examine the relationship between observed behavior and spike directivity properties from neurons. The presence of the so-called “expert” neurons in the striatum has been advanced for some time (Jog et al., 1999, Apicella P., 2002, Kimura et al., 2003). The term “expert” neuron was used before by Barnes et al., relating spike timing of neurons from the striatum to meaningful events of the task on the T-maze (Barnes et al., 2005).We have already shown that using a charge movement model (CMM) it is possible to compute spike directivity for every action potential (Aur et al.,2005). Additionally, performing information theoretic analyses, we have been able to reveal that during rewarded T-maze learning tasks, spike directivities of “expert” neurons from the dorsolateral striatum become organized with behavioral learning (Aur and Jog, 2007a).The important question is what the newly described feature of spike directivity adds to our understanding of the relationship to behavior. The current discussion describes the value of this organization of spike directivity in terms of what it represents within behavior. In these recordings, every tetrode implanted in the dorsolateral striatum is considered to form a frame of reference over which the electrical flow of charges in space can be analyzed during each individual

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spike and trains of spikes. In this view, the directivity property of a spike is embedded in a vector that reflects the flow of charges over the frame of reference (tetrode) for every spike. Having already shown the organization of directivity as rats learnt a procedural task (Aur and Jog, 2006) we embarked on finding the semantic relevance of this organizational process by analyzing the T-maze data during the time that is of the highest value for action selection in order to obtain the reward. This is the time between the cue (tone) delivery and the beginning of turn where spiking activity of striatal neurons is reduced with learning (Jog et al., 1999, Barnes et al., 2005). We have shown (Aur and Jog, 2007a;Aur and Jog, 2007b) that in animals which have already acquired the task, after tone delivery, spatial modulation of spike directivity (during this time of highest value) predicts the correct turn that the animal would make in order to get the reward. The details of these experiments, the protocol and the techniques are summarized below. 2.2.1. Behavioral procedures The training procedure is summarized here. The T-maze learning sessions began by opening of the start gate. On the maze during each trial, one of two tones (2 and 6 kHz pure tones, ~70 dB) was presented. By their low or high tone these stimuli indicate the position of food (chocolate sprinkles), on the right or left arm of the maze and determined the direction of turn. After few training sessions rats are able to discriminate the food position by auditory instruction cues and turn appropriately. Tone-turn arm assignments were randomized. During each session approximately 40 trials separated by 1-2 minute inter-trial intervals were given. Each rat was required to reach the correct goal in at least 70% of trials in a session to attain the acquisition criterion for significant correct performance. The number of training sessions ranged from 15 to 24. Daily recordings were made for an average of 39 ± 3 trials for every session. Tetrodes were not moved within each session during the time of data acquisition. Tetrodes position was adjusted by few microns (no more than 20 to 40 microns) between sessions to improve data quality if necessary. 2.2.2. Spike sorting and unit classification We analyzed electrical activity of isolated units from the dorsolateral portion of the striatum of 3 Sprague-Dawley rats. Tetrode recordings were obtained with established methods (Wilson and McNaughton, 1993, Jog et al., 2002, Buzsáki, 2004) and data was captured at an acquisition rate of about 25 KHz per channel using a Neuralynx@ data acquisition system. On average, 6 tetrodes were available for analysis in each animal. Subsequent processing included clustering into putative neurons and de-noising of the data. The details of spike activity estimation and the spike directivity computation based on the charge movement model are briefly explained and presented below. The activity of neurons in the sensorimotor striatum was recorded chronically during behavioural training on a conditional reward-based T-maze task. All animals were maintained on feeding restriction of not less than 80% of baseline weight required for behavioral training. The animals were anesthetized, a burr hole was drilled for the purposes of tetrode penetration (for striatum: AP 9.2mm, DV 5.9mm, L 3.5mm) and dura was removed. The headstage drive was lowered such that the cannula holding the tetrodes just touched the surface of the brain (Jog et al., 2002). Upon the rat awakening

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postoperatively, the tetrodes were lowered out of the cannula. Tetrodes were advanced partially on each day so as to allow the brain to settle. The brain targets were reached by day 3 or 4 post-operatively. Recordings commenced after this was achieved. All procedures were approved by the animal care facilities at the University of Western Ontario, London, Ontario. After classifying spikes into putative cells it is necessary to determine whether the spikes are correctly assigned to the determined units (Jog et al., 2002).Measures of unit isolation quality, Lratio and isolation distance were used to evaluate the performance of the sorting technique (Schmitzer-Torbert et al., 2005). Within recording sessions the location of the cells relative to the tetrode was considered stable only for spike recordings where the AP amplitudes and their profiles did not change significantly. Only data from well-separated units were used in computing spike directivity. Additionally, the quality of sorting was tested by analyzing autocorrelograms and corresponding refractory period of single neurons. 2.2.3. Computing and Analyzing Spike Directivity The specific aspects of spike directivity computation based on the charge movement model are summarized above and presented in detail in (Aur et al., 2005). Briefly, the following steps were utilized. Using the triangulation method and the point charge model, the trajectory of electric field propagation (charge flow) was calculated for each spike. Then the obtained trajectory was used to estimate spike directivity. This trajectory was analyzed using singular value decomposition (SVD) for every spike in order to find the best linear approximation of the spike direction. In this scenario, the tetrode acts as a frame of reference. This SVD technique generates three cosines ( v1 , v 2 , v3 ). Higher singular values of the decomposition indicate dimensions with higher energy within the data. By analyzing SVD during each spike, it is clear that about 70-80% of the AP energy has a preferred direction for propagation. Therefore, each AP can be represented by a vector with directivity determined by three cosine angles v1 , v 2 , v3 obtained from SVD decomposition. In order to show the directivity orientation, a PCA analysis is performed for direction cosines in recorded spikes from selected neurons. Analyzing the distribution of direction cosines for the best linear approximation using principal component analysis (PCA) we found two high-density clusters (Aur and Jog, 2007b). These clusters are represented in specific colors for each type of turn (yellow and blue for right, red and magenta for left). Then, during a session, yellow and blue vectors can be plotted to represent these directivities for the right turn related trials on the maze. In a similar way red and magenta arrows are drawn for left turn related trials on the maze. To understand the variability of distributions of the cosine angles the probability distribution is estimated for each cosine angle v1 , v2 , v3 , separately within each of the clusters. This analysis is carried out on computed directivity for each and every neuron both before learning (free exploration) and after the learning stage is achieved. The analysis of spike directivity before and after learning in the selected neuron reveals substantial changes in the probability density function of cosine angles. The recordings used for this analysis have been selected for the same neuron prior to and after

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neuron after the animal had achieved a stable 75% correct turn response. The neuron performs better than random in choosing spiking directivity. a, An example of pdf for the

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rewarded learning on the T-maze task. Only the v3 cosine angle is shown in the Figure 11 and Figure 12. The blue color represents the probability density function (pdf) for the first cluster and the red color the second cluster. A clear difference between pdf shapes for both clustered spikes is visible in Figure 11 and Figure 12. The two graphs in Figure 11 (free exploration) show a higher level of randomness than those in Figure 12 (after learning). This difference in randomness can be quantified using Shannon information entropy that has almost double the value before learning than after T-maze learning. In the presented pdf examples before T-maze learning the mean estimated

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information entropy for v3 cosine angle is H S = 4.2 ± 0.13 bits, while after learning the information entropy decreases substantially around the mean of H S = 2.7 ± 0.13 bits for about the same number of points in the data. This important decrease in the Shannon information entropy although visible in the shape of the pdf reflects the random nature of spiking directivity before learning and substantial organization in “charge flow” after learning. More details about organization in spike directivity can be found in (Aur and Jog, 2006). Having represented AP’s as vectors, the angle between the directions or the semantic distance of two spike vectors by definition can be determined:

Δθ = arccos(

< v, v ' > ) | v || v ' |

( 13 )

In the selected “expert” neuron for several spikes in a session during all T-maze trials the average value of cosines angle v r is computed: vr = avg (vi ) , i ∈ T

( 14 ) L

To obtain the variations of spike directivity during a left trial θ ( j ) we used the computed average v r as the reference value: Δθ L ( j ) = arccos(

< v( j ), vr > ) j∈N | v( j ) || vr |

( 15 )

and in a similar manner for turning right on the maze: Δθ R ( j ) = arccos(

< v( j ), vr > ) j∈N | v( j ) || vr |

( 16 )

2.2.4. Upcoming Choices, Decision and Spike Directivity The recorded signal from one neuron is used to obtain s ( k ), k ∈ N which represents the occurrence or absence of a spike in a neuron during a trial. The occurrence of the spike is assigned a value of 1 ( s(k ) = 1 ) while for non-spiking activity a zero value is assigned ( s( k ) = 0 ). To obtain an estimation of spiking activity sB (k ), k ∈ N , a low pass Butterworth filter of the 2nd order with the cutoff frequency of 10Hz is used to process the spiking signal s (k ) . The result is a smooth curve that shows the variations of spiking activity during each trial. This estimation of spiking activity is performed only for demonstration purposes to compare it with changes in spike directivity shapes. The computed value is not used for analysis at any point afterward. During T-maze sessions (Figure 13, a) all three animals showed the expected improvement revealed in average percentages of correct responses during behavioral learning (Figure 13, b). We recorded a daily average of 30 separate units per animal with six tetrodes each. Events associated with hexadecimal values are recorded simultaneously with neuronal spikes during each trial and color coded for display purposes (Table1).

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Figure 13: Experimental T-maze and behavioral response. a, Schematic representation of the T maze experiment. Events are associated with colors as per Table 1. b, An example of behavioral response that shows average percentages of correct responses (APCR). The cyan color shows low levels of performance while the magenta color shows at least 70% correct responses in the three rats. The vertical bar shows the color scheme used to show the change in performance level across the number of training sessions.

The neurons showed an increase in spiking activity visibly correlated with behavioral events (tone cue, turning on T-maze, etc., Figure 14 a and b). The percentage of units that responded to events in the task increased from approximately 50% to a maximum of 85% of recorded units, corresponding to a χ2 value with P<0.001. The number of units that responded to more than one event rose from 40% to 60%. At the beginning of training, 27% of task-related units responded during turns while by the end of training only 14% responded (P<0.001). Between the events of tone delivery and turn beginning, with learning, the number of responsive units decreased as did the actual number of spikes per unit occurring in this time period confirming previously published results (Jog et al., 1999; Barnes et al., 2005) This decrease in activity has been recorded by others but there is no explanation for this decrease during what may be the most important phase of the task. It is during this phase that the animal has to make and commit itself to the decision of the turn direction. Yet, spike-timing analysis is unable to highlight the importance of this phase of behavioral performance. Directivity analysis of the remaining responsive neurons showed increased modulation in spatial directivity with learning as shown below. Table 1: Relationship between events and recorded values during left and right trials

Left

Right

Start Tone Turn Begins Turn Ends Goal

All data analyzed and shown below are after at least 70% correct response stage is achieved. An example of temporal representation of events as recorded from one of the many neurons is shown (Figure 14, a). The lower half of the frame shows event flags as a color-coded bar (per Table 1). The graph above represents actual spikes from this

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sB in red color.

b, An example of temporal representation of events (bottom) where high tone was associated with right arm reward. Occurrence of spikes for the expert neuron is shown in blue color and the estimation of spiking activity s B in red color.

neuron in a single trial, plotted in each behavioral period. Figure 14a shows neuronal activity from the same neuron for a left turn while Figure 14b shows the same activity during the right turn. The filtered version of unit activation sB for the selected neuron is plotted in red color. Neuronal activity occurs upon signal tone delivery, during turning movements on the right or on the left of the T-maze and before or during attaining the goal. It is clear that the same expert neuron shows activity for both turns. At the end of the learning period, all neurons that are linked to the learning activity of cued turn showed this dual turn related responsivity. Other neurons showed a diversity of activities including response to start, tone or goal (analyzed but not shown). Spiketiming analysis seems not rich enough to effectively show this fact nor can it explain the phenomenon. Spatial directivity is now calculated for neurons spiking during different phases of the T-maze task. Every spike yields only one vector (Aur et al., 2005). It is important to note that spike directivity analysis is carried out on neurons after the animal has already acquired the task and organization in spike directivity has occurred. The fact that directivity is reorganized during learning has already been demonstrated above by computing Shannon information entropy. High values of Shannon information entropy reflect the random nature of spiking directivity before learning and reduced values show substantial organization of spike directivity after learning (Aur and Jog, 2006). The change of direction cosines during the whole trial is first computed in selected neurons for left and right turns. An example of these shapes for cosines angle variations is presented in Figure 15a, b for the left and for the right turn trials on T-maze. At least two distinctive parts can be easily seen to exist in these plots for both the right and left turn trials. In each trace of the change in cosines angle during the task, the

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a

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Figure 15: Changes in cosines angle in an expert neuron from striatum as related to events (bottom) over time during the whole trial. Event flags are represented as a bar, color coded as in Table 1. a, An example of temporal representation of events (bottom) with low tone for turning left and variation of directivity angle Δθ in blue color. b, Temporal representation of events (bottom) with high tone for turning right and variation of L

directivity angle

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first change occurs in Δθ L or Δθ R angle during or just after tone cue event. The second distinctive change in Δθ L or Δθ R angle appears just before or during left or right turns. The change of angle during movement shows similar variability during several trials. This characteristic profile is akin to a signature. In order to highlight the powerful and unique aspect of spike directivity not detectable by using spike timing alone, the directivity vector of every spike occurring between start and tone (Figure 16, a) and tone and beginning of turn (Figure 16, b) is plotted for about 20 trials and between beginning of turn and goal (Figure 16, c) for a single trial. Sequential spikes from the same expert neuron during a specified turn show variability in directivity as shown in Figure 15. Rotation of the three obtained 3D images showed a separation for spike directivity associated with the left versus the right turn trials on the T-maze only in Figure 16b. This separation between left and right turn related spikes cannot be seen in Figure 16a or Figure 16c in any of the analyzed neurons. It is important to note that for Figure 16b directivity is calculated for spikes between tone delivery and before the turn has actually begun and not calculated for those spikes that occur during the actual turn. This analysis demonstrates that the portion of the trial that has the lowest firing rate (after the learning is achieved), is the period of time that is most representative. It is in this period of time that the animal is deciding what turn it is required to make. It is crucial to note that this feature of predictability of an upcoming event is not demonstrable by using the spike counts (firing rate analysis) or timing of spike occurrence. Once learning progresses, neurons from the striatum significantly reduced their number of spikes between when the gate is open and turn is performed while the number of spikes between the turn and goal phase show an increase. This temporal reorganization phenomenon of neuronal activity was previously described in detail (Jog et al., 2007, Barnes et al., 2005) and can be directly observed in the number of spikes represented in Figure 16c and Figure 15a, b. While spike directivity for the data between when the gate opening and turn can be computed for every spike and represented as vectors (Figure 16, b), a reliable statistical analysis of these data is considerably limited due to the low numbers of spikes.

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Figure 16: Representation of spike directivity vectors for spikes in an expert neuron. Left turn trial is represented by yellow and blue arrows while the right turn trial is represented by red and magenta arrows. a, Spike directivity vectors of all spikes between gate opens and tone in about 20 trials b, Spike directivity vectors of all spikes between tone and turn starts in about 20 trials c, Representative spike directivity vectors of all spikes between beginning of turn and goal. Only one trial is shown here for better visualization due to larger number of spikes.

Figure 17 a, b show the cosines angle changes from the same neuron in three successive trials for both right and left turns while Figure 17c and d show corresponding spikes (for which the directivity analysis was performed) in blue color and spike activity estimates in red color. An important observation here is that the profiles for left versus right turns are visibly similar with slight changes from trial to trial, both in the temporal and the spatial directivity plots. Finally, these changes in the cosines angle can be compared to what happens in the traditional, spike timing view during a trial. This comparison allows us to reveal powerful similarities in dynamic variations in spike directivity and temporal activity. A correspondence between changes in the firing rate and modulation of spike directivity appears to exist as seen in Figure 17a-d. Thus when a significant change is seen in the timing of spike activity during a trial, there is a corresponding change in cosines angle. plotted for spikes (Figure 17: a versus c; b versus d). Three such trials are represented for the left turn and right turn trials respectively. The variation in cosines angle for every trial is similar to the corresponding changes in timing of spike occurrence.

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2.2.5. Learning: One Spike Seems Enough In the absence of any reward or punishment the decrease in behavioral responsiveness follows a repetitive presentation of a stimulus. After several repetitions of a stimulus there is a general tendency to become familiar with the stimulus. This phenomenon of learning was observed from Aplysia (Bailey and Chen, 1983) to cats (Groves et al., 1969) and was called habituation. Thompson and Spencer tried to characterize habituation using their experimental analyses (Thompson and Spencer, 1966). They showed that the repeated application of a stimulus leads to a decreased response. This is like the animal that learns how to ignore an input that does not show any new information. In case the stimulus is withdrawn, the response improves over time. Many neurons habituate during a repeated stimulation. If the stimulus presentation is more rapid then the habituation is faster. Potentiation or faster habituation occurs also if training and recovery phases are used. Usually habituation has not been associated with synaptic accommodation determined by an exhaustion of neurotransmitter at the synapse, which also may result

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in a severe decrease in behavioral responsiveness. McFadden and Koshland showed that alteration of the habituation pattern is caused by activators of the protein kinase pathway and changes in the activity of voltage-gated calcium channels (McFadden and Koshland, 1990). These observations link habituation patterns at temporal scales of spike trains with significant changes that occur at a molecular level. However, longterm potentiation (LTP), a relatively long-lived increase in synaptic strength, remains the most popular model for the cellular process that may underlie information storage within neural systems. All these experiments showed reduced activity of neuronal firing. Different biophysical explanations that range from protein folding to neurotransmitter and ion fluxes modification have been hypothesized to explain this phenomenon. Classical explanation of changes in wiring or synaptic plasticity does not seem to address the intrinsic physical aspect and the real meaning in terms of a computational model for such mechanism is still in debate. The answer is simple and comes from an analysis of electrical flow in specific neurons that fire before and during the learning process. The reduction in firing rate is directly related to organization of spike directivity at the choice point as presented above and in (Aur and Jog 2006). Using the tetrode system as a frame of reference, we have shown that an analysis of the newly described feature of spike directivity has important implications to understanding neuronal activity (Aur and Jog, 2006; Aur and Jog, 2007a). Based on electrical recordings and computational methods we are able to show that in certain neurons, spike directivity changes occur in a determined, behaviorally dependent fashion with events on the T-maze. Spike directivity is a hidden characteristic. Unlike spike timing, spike directivity cannot be directly perceived by experimental observation; it has to be computed in advance to allow further analysis. Additionally, as spike counts may decrease with learning, firing rate dependent analyses and ISI statistics cannot be performed. Since spike directivity is computed for every spike, only few spikes are required to exploit the significance of this feature. Coupled with spike timing analysis, our work uses spike directivity to reveal a hitherto unrecognized yet important aspect of how behavior is expressed by neurons. The first result is that “expert” spiking units responded to more than one meaningful event. This suggests that spiking behavior of each cell is likely to represent more than one class of events/features and does not imply the existence of individual neurons coding uniquely for certain discrete events. Since neurons receive inputs from a variety of sources the response pattern will vary significantly depending upon the nature of the input. This provides a much larger range of response sensitivity for the ‘expert’ neuron which can be seen in the temporal patterns of response (Figure 14a and b) and can be equally well shown in the directivity analysis of spikes from the same neurons (Figure 15a and b). Such results confirm the published observations that neurons do respond by changing spiking activity in specific periods of the T-maze behavior. The changes in the cosines angle in spike directivity have similar inflections as spike activity plots, corresponding to the same phases in the same neuron. Therefore, alterations in spatial directivity occur simultaneously to changes in the spike timing. This could imply that spike directivity characteristics are inclusive of what is shown by spike timing analysis techniques. It is likely that immediately after delivery of the auditory tone, action selection of which turn is to be taken occurs, which is then followed by turn execution. However, this had never actually been demonstrated before as has been with spike directivity. Neurons that respond to meaningful events of the task appear to have robust firing rate

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changes relating to periods corresponding to “before” (seen as the “decision” period) and “during” the actual turn (seen as an “execution” period). This implies that action selection and execution can be represented in the same expert neuron as shown in both the temporal and spike directivity analyses. Such neurons can then be seen to combine spatial directivity of spiking with time contingent properties. Yet a crucial difference exists between what we can gain from spike time versus directivity analysis. If merely the spiking activity is considered, one can see clearly the occurrence of turn selection and execution in a successive manner. However, based on estimated spike timing activity (Figure 14 a and b, Figure 17c and d), an observer would be unlikely to determine the direction of the turn that the animal may select since there is no significant difference between any specific temporal spike activity corresponding to either direction of turn. This crucial piece of information that recognizes which turn selection is valuable, i.e. has the potential to result in reward is however elegantly revealed in the spike directivity characteristic. The current result demonstrates that spike directivity during the period between tone and beginning of turn highlights the arm to be selected, information which is of great value to the animal in order to obtain reward. Most significantly, this turn oriented separation in spike directivity does not occur in the previous phase, between gate opening and tone delivery (Figure 16a) and is also not present after the selection of turn between the turn start and when goal is achieved (Figure 16c). The conclusion is then obvious that upon tone delivery, the spike directivity distinguishes the left versus the right turn represents the selection phase. Since correct selection increases the chances of reward, this intriguing result implies that the spike directivity predicts which turn is subsequently to be executed. Summarizing then, we show that the same “expert” neurons respond to multiple events of the T-maze task, that they show response modulations in spiking frequency (temporal domain) and in spike directivity (spatial domain). This temporal and spatial response sensitivity occurs to similar events in the task. In addition, we show that spike directivity can be correlated with meaning in terms of which turn selection would have the highest likelihood of obtaining reward. We anticipate that these changes in spike directivity are a result and a manifestation of a form of neuronal plasticity. Since electrical flux of charges carry information (Aur et al., 2006) “expert” neurons can be seen as projecting information related to behavior, temporally and spatially, to different targets at certain moments in time. The directionality of charge flow as observed by the tetrodes is a reflection of intracellular and dendritic events (Figure 7 a,b) that we have been able to obtain using ICA techniques and these electrical patterns may to some extent be determined by the received inputs (Aur and Jog, 2006). Finally and most importantly, our work has shown that predictability of the upcoming behavior is an inherent property of analyzed neurons as demonstrated by their spatio-temporal dynamics. Based on measured electrical activity from tetrode recordings, the demonstration of such a subtle yet necessary feature is in essence equivalent to decoding the neural code. Current investigations of neuronal coding are restricted to time domain analyses where computation of firing rate or interspike interval assumes the stereotypy of action potential (AP). However, viewed with multi-tip receivers (tetrode) the sequential APs from one neuron are not alike. The present study reveals the existence of an intrinsic spatial code within neuronal spikes that predicts behavior. As rats learnt a T-maze procedural task, simultaneous changes in temporal occurrence of spikes and spike directivity are evidenced in “expert” neurons. While the number of spikes between the

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tone delivery and the beginning of turn phase reduced with learning, the generated spikes between these two events acquired behavioral meaning that is of highest value for action selection. If the tetrode is considered a frame of reference, spike directivity characterizes the propagation of charges in every recorded action potential. This hidden feature can be computed and then graphically represented as a vector pointing in space to the center of mass of charges, which move during an AP. Since this electric propagation in every spike transfers information, spike directivity provides a vectorial representation of information propagation. Thus the analysis of spike directivity reveals the semantics of each spike and in the current experiment, predicts the correct turn that the animal would subsequently make to obtain reward. Semantic representation of behavior can then be revealed as spatial modulations of spike directivity. The predictability of observed behavior based on these subtle changes in spike directivity represents an important step towards decoding and understanding the underlying neural code. Even though temporal aspects including synchrony or oscillations seem to be important often we are not able to obtain information regarding the intrinsic mechanism of neural coding. In this book we propose to demonstrate that for the first time it is now possible to relate the neuronal activities with animal behavior and importantly with behavioral semantics. These results are of value in determining representations of distributed spike computations. During the action selection phase, the number of spikes decreases with learning. However, the spike directivity analysis reveals inherent additional decision-making value in indicating the turn arm being selected. It is intriguing to suggest that the occurring spikes are now more efficient at providing information about the task. After learning is achieved by the rat, analyzing electrical characteristics of only one spike recorded from a selected expert neuron would be enough to know in advance the arm where the rat will move to get reward. One spike seems to be enough to know in advance the decision that the rat may make. In this context the cosine between two spike directivity vectors from the same neuron or just their scalar product can be considered to measure the semantic distance between neuronal spikes. The computed semantic distance of spikes shows the similarity of meaning between two spikes from the same neuron implying that the more separated the directivity between spikes, the more distinct the spikes are semantically. • Spike directivity and its modulations challenge the traditionally held fundamental idea of ‘binary pulses’ or spike stereotypy of action potentials Indeed one may predict the animals’ turn direction on the T-maze simply based upon the directivity of spikes in the selection period. Once organized, at a point when the animal has acquired the task, the rat “understands” the meaning of the low or high tone and selects the turning arm to maximize the way of getting the reward. In our work, the use of tetrodes provides a local frame of reference that allows the visualization of these subtle changes in time and space on a small single spike-level scale. Communication in the brain appears to be represented by such spatiotemporal electrical flux on a small scale, which contains behaviorally meaningful data that is exchanged between anatomically interconnected structures. Such electrical waves can be effectively measured in the brain in recordings such as field potentials or electroencephalography and identified as rhythms of the brain (Buzsáki, 2006).

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Summary of Principal Concepts •

• • • • • • • • • • •

The origins of several gaps in neuroscience are buried in early single electrode recordings (Adrian 1929; Eccles, 1957) where action potentials have been considered to be similar uniform pulses of electricity and only their temporal variability of occurrence was perceived. New technological ideas and developed techniques based on tetrode recordings prove that action potentials recorded from the same neuron display changes in electrical patterns not just temporal variability The action potentials as electric events can be spatially modulated and therefore “point source” models or the “line source” models are inadequate to describe neuronal activity A new approach, the model of charges in movement (charge movement model) is introduced as a description of electrical activity during AP generation in neurons The directivity of electrical propagation modulated by dendritic activity (axonal branches) within every spike (spike directivity) can be computed if the charge movement model is combined with a triangulation technique Additionally, using independent component analysis of recorded signals and triangulation techniques with this new model of charges in movement the electric patterns of activation within neuronal spikes can be represented. Such electrical patterns that occur during AP propagation are likely to be a result and reflect intra-cellular information processing Spike directivity can be used as an effective characteristic that can explain changes in neural in activity during learning a T-maze procedural task Spike directivity organizes with learning. The generated spikes show responsiveness to the acquired behavioral meaning that is of the highest value for action selection and decision mechanism. Spikes are not alike, the reduction in firing rate during learning is correlated with an increase of significance (value) of few delivered action potentials With learning spatiotemporal electrical patterns occurring within analyzed spikes from expert neurons reflect behaviorally meaningful data Since two different spikes from the same neuron can carry different semantic we cannot add spikes, we cannot average them or measure ISI without loosing important information regarding the semantics of the stimulus or behavior.

References Alexander, G. E., DeLong, M. R. and Strick, P. L. (1986) in Annu Rev Neurosci (eds. Cowan, W. M., Shooter, E. M., Stevens, C. F. & Thompson, R. F.) 357-381, Annual Reviews, Palo Alto. Amari, S. Chen T.P, Cichocki A., (1997) Stability analysis of learning algorithms for blind source separation, Neural Networks, Vol.10, No8, 1345-1351, Apicella P., (2002), Tonically active neurons in the primate striatum and their role in the processing of information about motivationally relevant events. Eur. J. Neurosci. Dec; 16(11):2017-26. Aur D., Connolly C.I., and Jog M.S., (2005) Computing spike directivity with tetrodes. J. Neurosci. Vol. 149, Issue 1, pp. 57-63. Aur D., Connolly C.I. and Jog M.S., (2006) Computing Information in Neuronal Spikes, Neural Processing Letters, 23:183-199. Aur D., Jog MS (2006), Building Spike Representation in Tetrodes, Journal of Neuroscience Methods, vol. 157, Issue 2, 364-373.

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Aur D., Jog, M., (2007a) Reading the Neural Code: What do Spikes Mean for Behavior? Nature Precedings, Aur D, Jog MS (2007b), Neuronal spatial learning, Neural Processing Letters, vol. 25, No 1, pp 31-47. Bailey C.H., Chen M. (1983) Morphological basis of long-term habituation and sensitization in Aplysia. Science 220:91–93 Barlow, H. B. (1972) Single units and sensation: a neuron doctrine for perceptual psychology. Perception, 1:371-394 Barnes T.D., Kubota Y., Hu D., Jin D.Z.and Graybiel A.M., (2005) Activity of striatal neurons reflects dynamic encoding and recoding of procedural memories, vol 437, Nr. 7062 pp. 1158-1161. Bialek W., DeWeese M., Rieke F. and Warland D., (1993) Bits and brains: Information flow in the nervous system Physica A: Statistical and Theoretical Physics, Volume 200, 1-4, 581-593 Brody C, D. and Hopfield J. J., (2003) Simple Networks for Spike-Timing-Based Computation, with Application to Olfactory Processing ,Neuron, Volume 37, Issue 5, 843-852. Buzsáki G., (2006) Rhythms of the Brain, Oxford University Press. Buzsáki G., (2004) Large-scale recording of neuronal ensembles. Nat. Neurosci., 7:5. Cajal S. R., (1909) Histology of the Nervous System of Man and Vertebrates (trans. N. Swanson and L. W. Swanson 1995), New York: Oxford University Press. Cardoso J.F., 1991.Super-symmetric decomposition of the fourth-order cumulant tensor, blind identification of more sources than sensors. In Proc. ICASSP'91, pp. 3109-3112, Comon P, (1994) Independent Component Analysis: a new concept? Signal Processing, Elsevier, 36(3):287-314 Debanne D., (2004) Information processing in the axon, Nature Reviews Neuroscience 5 (4), pp. 304-316. Devan, B. D. and White, N. M., (1999) Parallel information processing in the dorsal striatum: relation to hippocampal function. J Neurosci 19, 2789-2798. Drake KL, Wise KD, Farraye J, Anderson DJ, BeMent SL. 1988 Performance of planar multisite microprobes in recording extracellular single-unit intracortical activity. IEEE Trans Biomed. Eng.; 35: 719. Eccles JC. (1982) The synapse: from electrical to chemical transmission. Annu. Rev. Neurosci. 5:32539. Eccles J. C. (1957) The Physiology of Nerve Cells. Baltimore: Johns Hopkins Press. Fee M.S., Mitra P.P. and Kleinfeld D., (1996) Automatic sorting of multiple unit neuronal signals in the presence of anisotropic and non-Gaussian variability. J. Neurosci. Methods 69 (1996), pp. 175–188. Fulton, J. F., (1938) Physiology of the nervous system. (Oxford Med. Publ,) New York, OxfordUniv. Press. Gold C, Henze DA, Koch C, Buzsáki G., (2006) On the origin of the extracellular action potential waveform: amodeling study. J Neurophysiol 95: 3113-3128. Groves, P. M., Lee, D., and Thompson, R. F. (1969). Effects of stimulus frequency and intensityon habituation and sensitization in acute spinal cat. Physiology and Behavior, 4, 383-388. Hodgkin A., and Huxley, A., (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117:500–544. Hubel D. and Wiesel, T. (1962) Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. Journal of Physiology of London, 160:106—154. Hubel D and Wiesel, T., (1968). Receptive fields and functional architecture of monkey striate cortex. J. Physiol 195, 215–243. Gerstner W and Kistler W. M., (2002) Spiking Neuron Models. Single Neurons, Populations, Plasticity. Cambridge University Press. Holt G.R. and Koch, C. (1999) Electrical Interactions via the Extracellular Potential Near Cell Bodies. Journal of Computational Neuroscience, 6 (2). pp. 169-184. Jog M.S., Aur, D., Connolly C.I., (2007) Is there a Tipping Point in Neuronal Ensembles during Learning? Neuroscience Letters Volume 412, Issue1, pp. 39-44 Jog M.S., Kubota,Y., Connolly,C.I., Hillegaart,V. and Graybiel, A.M., (1999) Building neural representations of habits. Science 286, 1745-1749. Jog M.S., Connolly C.I., Kubota Y., Iyengar D.R., Garrido L., Harlan R., Graybiel A.M., (2002). Tetrode technology: advances in implantable hardware, neuroimaging, and data analysis techniques. Journal of Neuroscience Methods 117,141-152. Jourjine A., Rickard S., and Yilmaz O., (2000) Blind separation of disjoint orthogonal signals: Demixing n sources from 2 mixtures, in Proc. Int. Conf.Acoust. Speech Signal Process. (ICASSP’00), Istanbul, Turkey, vol. 5, pp. 2985–2988. Kandel ER, Schwartz JH, Jessell TM 2000. Principles of Neural Science, 4th ed. McGraw-Hill, New York. Koch K., McLean J., Berry M., Sterling P., Balasubramanian V. and Freed M. A., (2004) Efficiency of Information Transmission by Retinal Ganglion Cells Current Biology, Vol. 14, Issue 17, pp 1523-1530. Kimura M., Matsumoto, N., Okahashi, K., Ueda, Y., Satoh, T., (2003) Goal-directed, serial and synchronous activation of neurons in the primate striatum. Neuroreport. 14(6):799-802.

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Lee I., Griffin A. L., Zilli E. A., Eichenbaum H. and Hasselmo M. E., (2006) Gradual Translocation of Spatial Correlates of Neuronal Firing in the Hippocampus toward Prospective Reward Locations, Issue 5, pp. 639-650. Lee T.W., Girolami M., Bell A.J., Sejnowski T., (2000) A unifying information-theoretic framework for independent component analysis, Int. J. Comput. Math. Appl. 39 ,11, 1–21, De Lathauwer L., De Moor B., Vandewalle J., (1999) ESAT, ICA techniques for more sources than sensors, Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics: 121-124. Lewicki M.S. (1998) A review of methods for spike sorting: the detection and classification of neural action potentials. Network 9 (1998), pp. R53–R78 Lewicki M.S., and Sejnowski T.J., (2000) Learning overcomplete representations. Neural Computation, Vol 12; Part 2, pages 337-366 Maass W., (1997) Networks of Spiking Neurons: The Third Generation of Neural Network Models, Neural Networks -Oxford- Vol 10; Number 9, pages 1659-1671 Maass W., Bishop C.M., Sejnowski T.J., (1999) Pulsed neural networks, Cambridge, Mass, MIT Press. Mainen ZF, Sejnowski T.J., (1995) Reliability of spike timing in neocortical neurons, Science, Vol 268, Issue 5216, 1503-1506. Matell M.S., Meck W.H., Nicolelis M.A., (2003) Interval timing and the encoding of signal duration by ensembles of cortical and striatal neurons, Behav. Neurosci, Vol. 117, No. 4, 760–773. McFadden PN, Koshland DE., Jr (1990), Parallel pathways for habituation in repetitively stimulated PC12 cells. Neuron. Apr; 4(4):615–621 McNaughton BL, Barnes CA, O’Keefe (1983) The contributions of position, direction, and velocity to single unit activity in the hippocampus Exp. Brain Res. 52:41-49. McCulloch W. and Pitts W., (1943) A Logical Calculus of Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics 5:115-133. Moore C., (1990) Unpredictability and undecidability in dynamical systems, Physical Review Letters, 64(20):2354–2357. Oesch, N., Euler, T., and Taylor, W. R., (2005) Direction-selective dendritic action potentials in rabbit retina, Neuron 47(5) 739–750. O'Keefe J. and Dostrovsky, J., (1971) The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat, Brain Res. 34 (1971), pp. 171–175. O'Keefe J., Reece, M. L. (1993) Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3: 317-330, Packard, M.G. and Knowlton, B. J., (2002) Learning and memory functions of the basal ganglia) Annual Review of Neuroscience, Vol. 25: 563-593 Passaglia C, Dodge F., Herzog E., Jackson S.and Barlow R., (1997) Deciphering a neural code for vision, Proc. Natl. Acad. Sci. USA, Vol. 94, pp. 12649-12654, November Neurobiology Purwins H., Blankertz B., Obermayer K., (2000) Computing auditory perception, Organized Sound 5(3): 159–171. Recce and O'Keefe, (1989) The tetrode a new technique for multi-unit extracellular recording. Soc. Neurosci. Abstr 19,1250. Sahani M., (1999) Latent Variable Models for Neural Data Analysis. Ph.D. Dissertation. Computation and Neural Systems. California Institute of Technology, 1999. Schmitzer-Torbert N., Jackson, N., J., Henze, D. Harris K. and Redish A.D., (2005) Quantitative measures of cluster quality for use in extracellular recordings, Neuroscience Volume 131, Issue 1, Pages 1-11. Strong S.P., Koberle, R., Steveninck, R. R., Bialek, W., (1998), Entropy and information in neural spike trains. Phys. Rev. Let., 80, 197–200. Thompson R.F. and Spencer W.A, (1966) Habituation: A model phenomenon for the study of neuronal substrates of behavior, Psychology Review 73 (1966), pp. 16–43 Wilson M.A. and McNaughton, B.L. (1993). Dynamics of the hippocampal ensemble code for space. Science 261, 1055-1058. Yin H. H. and Knowlton, B. J. (2004) Contributions of striatal subregions to place and response learning. Learn Mem 11, 459-63.

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CHAPTER 3. Information and Computation Ever since technological advances have become intergral to science, brain computations have gradually been imagined as switches or nonlinear functions and lately as parallel digital computations. To represent high-level abstractions or complex biological functions (e.g. in vision, language, and other AI-level tasks), one first needs to gain a deeper understanding of the “brain structures”. A key concept of brain function is its ability to encode information. However, many past attempts to describe brain processing have ignored the subtle relationship between information processing and computation. Such ideas of subtle relationship have been prevalent in physics to link the geometry of space, energy and time with information (Bekenstein, 1981, Bekenstein and Schiffer, 1990). It is clear that this progress in an understanding of information is closely related to the existent laws of physics that govern all natural phenomena. We are still far from understanding the link between information and computation in the brain. It is still unclear whether there is a difference between living (neurons) and nonliving (silicon) matter in terms of information processing capability or if the brain intelligence involves more than mere computation (Pfeiffer, 1999). One limitation appears to be the current idea of computation. Chapter three will begin by exploring this vital concept compared to what we understand by it today. Currently our understanding of computational processes seems to be limited to theoretical principles developed by Turing which are supposed to have been built on as an “universal” model (Turing, 1939). 3.1. What is Computation? The study of computation started with the early work of Alan Turing and Kurt Gödel before the Second World War. Initially, this work was seen as a step to advance the field of cryptography. This initial development has evolved into a new area of research known as complexity analysis. However, it can be demonstrated that there is a close link between system computability and the study of complexity. Recent theoretical breakthroughs in both these areas are stepping stones that may help us to understand subtle processes that define brain computations. Usually, when one asks about the limits of computers or computation, such a question is with respect to computations performed by digital computers. This popular view shows that we are unable to picture different types of computation, other than those built within binary, digital machines. However, there are varieties of natural processes that perform complex computations. In fact one can view all processes in the universe as carrying out “computations” (Lloyd, 2006). Even though we are far from the Lloyd extension, the brain is a naturally shaped system where all processes have to be entirely governed by biophysical laws. Seen as a distinct “universe” determined by several steps of natural evolution, the brain is able to perform complex computations. The main issue then is whether we can envision the complexity of brain computations in terms of an extension of the accepted machine models of computation. As mentioned above, Turing machines are universally accepted as a general model that formalizes several types of computations. Can brain computations be assimilated with Turing Machines? Can such

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computational processes be simulated using the von Neumann architectures? These are important questions that need to be answered. The current chapter will begin to address these issues. 3.1.1. Coding and decoding The easiest way to see computation in terms of information processing is simply as the classic example of the “Caesar cipher”. This ciphering technique was used by Caesar to protect several transmitted messages during the war campaigns. A "cipher" line is obtained by shifting the letters of the alphabet. Such operations can be formalized as mathematical transformations. The techniques of encryption (coding) by replacing letters from the original message with letters from a "cipher" line can be formalized and described mathematically. However, even though the letters from the original message can still be read one cannot associate meaning to the new “words” due to the change of letters. This example should remind us of Searle’s experiment where messages written in Chinese are interpreted by the computer. One cannot interpret the semantics hidden in such an encrypted message. However, in this particular case of Caesar cipher the process of decoding (decryption) can be simply performed by obtaining the frequency distribution of the letters in the message. Then, by evaluating the differences between the usual frequency of appearance of letters in a regular text and in the coded text the shift in letters can be assessed. Therefore, after this technique is performed the text can be decoded and interpreted. Importantly, as in the Searle example, the intrinsic semantics can be understood only by the human interpreter after reading the message. Understanding of the coding and decoding processes performed in the brain is essential to model brain computation. In case of the Caesar cipher the decoding procedure can be performed very fast using current computers. However, there is a particular aspect that current computers cannot perform. Reading and interpreting the decoded message or, in other words extracting the semantics behind the text seems to be a unique capability of our brain. Searle showed that this characteristic couldn’t be achieved by any algorithmic processing in current computers. This simple example of ciphering shows that our brain is capable of performing additional tasks beyond simple coding or decoding of messages. While almost all theoretical frameworks in building models of the brain analyze the coding and decoding strategies, the topic of semantics is not extensively covered by a theoretical implementable approach. Is this saying that the interpreter has to be human? No, not necessarily; however it shows that the computational system may require that information processing/storage function needs to be performed differently than that being done in current computers in order to fulfill the criteria of the required “intelligence”. A simple approach to coding and decoding shows the importance of probability distributions (frequency of letters) as a way of addressing a certain code. An extension of this description in terms of frequency of occurrence of events or items representing X

T

Y

Figure 1 Schematic representation of coding as a set of transformations T required to achieve the representation Y from external input X.

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or carrying information was originally developed by Shannon and later expanded into the theory of information (Shannon, 1948). Coding/decoding can be seen as a set of transformations to achieve a representation Y when an external input X is applied (Figure 1).Since coding/decoding is seen essentially as mathematical transformations, functions and operations between functions, can formalize this approach. Therefore coding/ decoding phases can be related to functions that operate on other functions. This basic model of computation shapes a formal description of cognition where the “representation law” is described in terms of coding and decoding functions (Newell, 1990) (1) decode[encode(T)(encode(X))] = T(X)

where X is the external situation or oncoming stimuli and T defines the set of transformations required to achieve the representation. Using the system theory formalism this law can be extended further to a set of internal brain states that match an external input X or which may be considered to convert the arbitrary input X to an observed/measured output Y= T (X). In this example, computation can be theoretically described as information processing or simply as a formal process of data manipulation. When this is applied to the brain, any type of computation that requires and involves physical processes implies that well known physical laws have to be involved in explaining brain computations. Therefore, as for any other processes in the brain, computations start within an arbitrary physical state and then end usually in a different “particular” state. Since the brain computations are physically driven, then it is logical that any theoretical framework applicable to machines can equally become a valid method required to clarify some important issues regarding brain computations. Initially, they can be formulated as a series of specific queries: How is information encoded and decoded in brain? What are the specific processes required in coding and decoding mechanisms? Does the brain function change in a way so as to determine the semantics of input stimuli or behavioral situations and if so, how is this represented in computational terms? Is there a computational formulation that allows us to extend brain coding into the realm of semantics and meaning? We will begin by first providing a brief grounding in computation theory which may help summarize the current knowledge in this field before we highlight specific details of brain and neural function. 3.1.2. Are Turing Machines Universal Models of Computation? As discussed in the introduction, current theoretical computational examples are generally restricted to examples of Turing machines considered to be the universal model of computation. Every Turing machine (TM) can be seen as a simple theoretical computer which transforms input strings into output strings using a sequence of state transitions. Physical aspects of implementation as well as energy requirements are not relevant for theoretical Turing machines (Figure 2). The TM has a finite set of internal states s (k ) ∈ Q , k ∈ N and a fixed design. It reads one binary symbol at a time

i (k ) ∈ Γ , supplied on a tape. Each machine’s action O(k + 1) and D(k + 1) depends only on the input symbol i(k) and internal state s(k) and moves the tape one place in direction D(k + 1) (left L or right R). This sequence is resumed by the next equations: s (k + 1) = G ( s (k ), i (k ))

(2)

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O( k + 1) = F ( s( k ), i ( k ))

D(k + 1) = D( s (k ), i (k ) )

1

0

1

0

(3) (4)

1

Read/Write Head

Figure 2 The Turing machine reads a symbol from the tape and moves the head one cell to the left L or to the right R depending on the internal states s (k ) and received input i(k). While the computations are carried out the Turing machine does not accept any inputs from the external environment. Each output O (k + 1) is determined only by inputs and internal machine states

The Turing machine can read and write on a tape by moving the read-write head to the left and to the right and the transition function μ which describes this phenomenon is: μ : QxΓ → QxΓx{L, R} (5) where Q is the set of states, Γ is the tape alphabet and {L, R} defines the transition to left or to right. The Turing machine follows the rules imposed by the transition function, may accept a certain state or reject a certain state and then the machine halts. For a program which follows the Turing model, halting can be seen as a decision problem and can be described as a sequence of mathematical objects where the algorithm may halt if a certain input i is presented: H = {( M , i ) | the machine M will halt if the input is i

(6)

Importantly, the halting issue is considered to be the first computational problem which has been proven to be undecidable since there is no universal method (or algorithm) that works for all programs and inputs to determine if an algorithm halts. However, there are cases when instead of halting, the machine repeats succesively a set of states and follows a specific loop (oscillations). Halting does not guarantee that the correct answer for a problem is found since sometimes the algorithm may not return any answer. The importance of Turing machines for computation theory derives from the fact that it can simulate any program on any type of von Neumann computers. This simulation provides the solution within a certain amount of time; usually, such computations are much slower. Some problems are easy to solve by TM, some of them are hard while others are unsolvable. Regarding this aspect in terms of complexity theory, several decision problems can be solved using a deterministic Turing machine in a polynomial computation time known as complexity class P. However, a more inclusive class is known to be the non-deterministic polynomial time decision class NP. Such problems can be solved in polynomial time by a

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nondeterministic Turing machine. Different types of algorithms can be simulated by Turing machines and in general, there is an equivalence in terms between recursive algorithms and Turing Machines . One important question is whether Turing machine models are capable of completely describing all forms of computation. The fact that several problems do not have an algorithmic solution was clear even for Turing. However, for several decades after Turing presented this formalism it was believed that TM can describe almost every type of computation. Additionally, these beliefs were strengthened by the Church-Turing thesis, which assumes that what is computable by an algorithm is also computable using a Turing machine. However, lately, it has been realized that this is not the case and several scientists claim that Turing machines are able to model only a small subset of computational possibilities. It is important to remember that TMs do not accept any external inputs while they compute.This considerably limits the capability of machine interaction with the environment and its history dependent behavior (Goldin, 2000). Additionally, in order to perform an operation a computing machine has to transfer information between its internal components. Such transfer has to be physically performed and requires discrete speed limits applicable for information transfer. However, computation time or energy required to perform certain computations are irrelevant for Turing machines. Therefore, it was projected that TM framework does not capture all forms of computations (Wegner, 1997). Since Turing machines compute via a series of discrete steps they provide only a discrete model of computation which cannot fully capture all physical aspects of computation. Recently, new models of TM have been proposed trying to break the limits of the conventional Turing Machine. These versions of Turing machines include fuzzy Turing machines (FTMs), superrecursive or inductive Turing Machines (Burgin and Klinger, 2004) and even quantum Turing machines. It is claimed that such machines can be very efficient and they might have more computational power than classical Turing machines (Wiedermann, 2002; Wiedermann, 2004; Wegner and Eberbach, 2004). Similar to physics at the beginning of the nineteenth century, the theory of computation appears to be evolving and currently there is a healthy debate regarding several forms of computation. However, some new proposed forms of computation are still strongly rejected by the mainstream of science (Cockshott and Michaelson, 2007). By analogy, it seems that a major issue is to limit brain processes to discrete Turing models of computation since such models constrain the analyses to be intrinsically discrete. Experimental evidence from neuroscience(Aur and Jog 2007, Aur et al., 2006) shows that computation can be performed continuously by a physical movement of charges in the brain which interact and intrinsically can carry and store information. This new understanding of computation in the brain challenges the classical assumption a conventional Turing model of computability by the brain. Understanding how the brain computes may also change our perception regarding TM and their supposed “universality”. 3.1.3. Von Neumann Architectures Current electronic computers that sequentially perform tasks are von Neumann architectures (VAN) which store data and instructions as binary values. In essence any such computer architecture has four important components: the memory, the arithmetic logic unit (ALU), the control unit (CPU) and the input-output devices (Figure 6). Von

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Neumann computations are defined by algorithms or programs that use a series of instructions. Every algorithm that runs on a von Neumann machine can compute the same class of functions as a universal Turing machine. The data are read or written from or to the memory in a sequential order. The memory stores data, intermediate results of computation and also the instructions. Therefore computation involves a frequent transfer of information between memory and control processing unit (CPU).

Input/Output

Memory

Control Unit Figure 3: Schematic representation of von Neumann architectures

ALU

This process of data transfer needs time to be performed and generates severe limitations regarding the speed of computation. However, since current computers do not use a tape as memory support unlike the classic Turing implementation, the machine can read and write at any location in memory. With higher data lengths, the time required to perform the task significantly increases since the memory has to be sequentially accessed. Such centralized control and sequential execution of instructions limits the speed of processing and is the main issue with von Neumann style architectures. Additionally, power dissipation and information storage capabilities are other critical issues. The finding of new alternatives for computation that use different principles may provide in the near future solutions for these issues. Inspired by biological life, in the late 1940s von Neumann's work was concentrated on developing novel computational principles that can adaptively change and self-replicate. This approach may challenge current theoretical view and generate in the future some useful, real-world models that may defy current computer architecture. 3.1.4. Reversible Computation Understanding the thermodynamic nature of computation took a long time and generated a strong debate. Landauer showed that the erasure of information in any computational device requires an increase in the thermodynamic entropy (Landauer, 1961). Another important aspect stressed by physicists is that certain computations can be theoretically performed without a required thermodynamic cost (Bennett, 2003). They are called logical reversible computations. Any logical transformation can be logically reversible if and only if it defines a one-to-one mapping. After a reversible

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logical transformation takes place the input state can be uniquely reconstructed based on information provided by the output state. There is a difference between a logical transformation which is a mathematical operation and the physical process required to perform this operation. The associated physical system has to go from an arbitrary physical state to another particular physical state. This physical process can be assimilated under certain conditions to become thermodynamically reversible without loss or dissipation energy. Landauer presented the theoretical solution regarding computation reversibility. If the computation is carried out slowly, then computers can dissipate arbitrarily small amounts of energy per step and therefore reversible computations may be carried out at each step without discarding information (Landauer, 1961; Landauer, 1988). An important aspect is that the laws of physics at microscopic scale are invariant under the reversal of time; therefore one should imagine that such processes could in principle be reversible. However, the transport of information in almost all devices where

0

a

0

0

b

a b

a+b

Output

Input Figure 4: A logically irreversible operations, an adder can be implemented in a mechanically ‘reversible’ way (adapted from Denker, 1994)

dissipative processes occur is an irreversible process. It is expected that if such computation can be carried out slowly enough the process can be physically considered to be close to an adiabatic process (Landauer, 1961; Landauer, 1988). If the speed of electrical propagation is higher then dissipation and energy requirement increases. However, the relationship between logical reversibility and thermodynamic reversibility is still unclear. Mathematically, logical reversibility has to map a given input pattern to a unique output pattern. This behavior cannot be easily determined just based on a thermodynamic analysis and this difference is explained in Figure 4. An adder even though it is not logically reversible can be made thermodynamically reversible (mechanically reversible) in the usual sense. This means that in this operation logical reversibility is not required and does not necessarily imply thermodynamic reversibility. One can see that “a” and “b” are not deleted during this operation and they are preserved (Denker, 1994). This relationship of reversibility between thermodynamic and logical operations is of great importance in the biological systems. The development of processes that are based upon the principles of reversibility of logical and thermodynamic operations have gradually evolved over time and have been shaped to improve process efficiency. These processes have resulted in higher order, yet efficient biological systems that presumably started off as individual

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systems and have now become much more complex. The nervous system can also be expected to have followed a similar process. As was discussed in the previous chapter and will be elaborated in detail in this chapter, electric charges can be envisioned to be the carriers upon which such logical and thermodynamic operations can be performed. From an information processing perspective, a trade off between these two properties results with evolution to maximize efficiency and minimize energy expenditure so as to produce the most informationally rich state. Therefore, one can envision the flow of electrical charges within neurons as an optimal trade off, achieved biologically, between required power for computation and limited sources of energy to provide such power. Mathematically, the processing of information can be seen as a transformation of the incoming signals. These transformations of the input signal may preserve Ndimensionality of the input in a possible reversible framework. Depending on design or in case of a biological system, connectivity, this transformation can map from the Ndimensional space to N-m a lower-dimensional space. Such an operation of dimensionality reduction results in an irreversible transformation and information is lost. This problem would be substantially compounded if the numbers of carriers which perform these operations are constrained. With a small number of charges computation would be unattainable and information transfer would be limited. However, within a given neuron and its environment, an extremely large number of charges exist that make possible to preserve the dimensionality of the process. Therefore, within the neuronal system, a high number of electrical charges are involved in computation. An interesting property is seen in terms of information entropy when a computation occurs. Computation can be seen as being closely related with changes in physical entropy. Although a computation may not intrinsically add ‘new’ information about the input, physical entropy must increase when information is communicated along with the movement of charges. Therefore, computations that require a dynamic transfer of information are commonly dissipative, these processes requiring deletion of information in one part while creating it in another, different part (Parker, and Walker 2007). To understand this computation performed by charges requires an integration of the knowledge gained from information theory with the physics of charge movement and electric interaction. A simple theoretical experiment can reveal the link between spatial movement of electrical charges and statistical properties of generated signals recorded by an electrode (Aur and Jog 2006). If several charges q1 , q2 ,,, qn are →

considered to be distributed in an electric field E then the force acting on each charge is dependent on the electrical field : r r Fi = qi E (7)

r

and can also be written as a product between the mass of charge ai is the acceleration: r r Fi = mi ai (8) →

→

→

→

→

→

Given the initial spatial position r01 , r02 ,…, rn3 and initial velocities v01 , v02 ,.., vn3 the equations of motion can be integrated numerically: r r i r i qi E (9) v = v0 + dt m

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r r r r i = r0i + v dt

63

( 10 )

and the trajectories can be plotted. Electrical signals generated by moving several charges in an electric field can be used to relate changes in information with changes in charge trajectories. These signals generated in the conductor can be statistically analyzed using mutual information (MI) measures (Aur, et al., 2006). This simple theoretical experiment above shows that if charges have similar trajectories in space they provide similar information while if their dynamics include trajectories that are far apart then the transmitted information may be different. In other words charges increase their statistical independence if they move far apart. Therefore, during every AP each charge or more accurately several group of charges that move independently in space could be treated as satisfying a certain degree of statistical independence. Additionally, the transfer of information between two charges is maximized if their trajectory is close enough. It is likely that charges that move in the same region in neuron carry similar information. Since writing down any information is a physical process a neuron or a group of neurons that is topologically located in certain location in space would have to theoretically share similar specific information. o Charges are able to read, carry and transfer information during their movement in space. In neurons the transferred information by a charge or a group of charges is directly related to their spatial trajectory. o Importantly, this discussion illustrates the dual role of a charge - as a sensor that can “perceive” the presence of other charges and as a carrier of information during their movement in space and electric/magnetic field generation. Additionally, two or more charges would provide more information about the electric field or the presence of other charges than one single charge. Theoretically, this aspect can be understood by introducing a new measure for information, namely multiinformation (Studeny, 1989; Studeny and Vejnarova, 1998). 3.1.5. Models of Brain Computation Currently a number of theories have been proposed to describe how computation is performed in the brain. The recent history of neural computation includes an abundance of theoretical models that started with a simple description of a neuron developed by McCulloch and Pitts (McCulloch and Pitts, 1943). This theoretical model was followed by the book of Donald Hebb (Hebb, 1949) that described the principle of plastic changes and connectivity between neurons. The first simulations of neuronal networks were developed by Frank Rosenblatt (Rosenblatt , 1958) building the well known perceptron model. Later, Bernard Widrow and Marcian Hoff introduced several adaptation algorithms in their paper “Adaptive Switching Circuits” (Widrow and Hoff, 1960) while Marvin Minsky and Seymour Papert have made important theoretical contributions to the development of neural learning and object recognition (Minsky and Papert, 1969). Other significant theories regarding network architecture include Adaptive Resonance Theory (ART) developed by Stephen Grossberg (Grossberg, 1976; Grossberg, 1987). The neocognitron model built by Kunihiko Fukushima (Fukushima, 1980) achieved recognition for simple visual patterns based on learning. Shun-Ichi Amari introduced the concept of information geometry and related it with the theory of the natural gradient (Amari, 1977; Amari, 1985). Inspired by the Ising model, John Hopfield developed, with binary

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threshold units, models of content addressable memory (Hopfield, 1984; Hopfield and Tank 1985, Hopfield and Tank 1986). Teuvo Kohonen has demonstrated that unsupervised learning is able to preserve the topological properties of the input signal and this novel theoretical approach was at the origin of self-organization algorithms (Kohonen, 1974; Kohonen, 1982; Kohonen, 1984). Mclelland and Rumelhart (McClelland and Rumelhart, 1988) challenged the dominant symbolic paradigm, which claimed “mental processes can be modeled as programs running on a digital computer". Cognitive abilities become in this context result of lower level processes, rather than the application of “higher level rules”. This is the most important hypothesis that shaped the theory of neural networks. Information is represented by patterns of activation over neuron-like "units" that perform parallel distributed processing where synapses provide the required connectivity. Since every neuron can process information, computation is distributed in the network and does not require the existence of a central processing unit as in current digital computers. Behavior needs to be described by adaptive neural networks where the connection weights are changed using the generalized delta rule as a result of interaction with stimuli from the environment. This is an important premise that shaped several theoretical developments and their view remained valid for a considerable length of time. Therefore one can surmise that: o “Many of the constructs of macrolevel descriptions such as schemata, prototypes, rules, productions, can be viewed as emerging out of interactions of the microstructure of distributed models" As mentioned above, the Adaptive Resonance Theory (ART) was developed by Stephen Grossberg (Grossberg, 1976; Grossberg, 1987). There are two general issues that Grossberg and collaborators themselves raised regarding connectionist implementations. First, the stability-plasticity dilemma enquires how new knowledge can be incorporated within an already trained network, essentially asking how can learning continue without resulting in catastrophic forgetting. This is an important problem encountered in building artificial neural networks where information is preserved in the set of weights. The second issue called the noise saturation dilemma questions how features are processed in noisy cells with finite input sites without being contaminated by either noise or saturation. Despite their effort and some partial solutions both problems seem to remain critical issues in the development of connectionist theory and artificial neural networks. Other approaches follow the Mclelland –Rumelhart development and are heavily rooted in ‘parallel distributed processing (PDP) framework in building neural networks (Anderson 1995; Gardner, 1993; O'Reilly and Munakata 2000). As an example O'Reilly identifies three categories of PDP principles (O'Reilly, 1998): o “The first principle, biological realism, is in a category by itself, providing a general overriding constraint on the framework. The next three principles, distributed representations, inhibitory competition, and bidirectional activation propagation (interactivity), are concerned with the architecture of the network and the general behavior of the neuron-like processing units within it. The final two principles, error driven task learning and Hebbian model learning, govern the way that learning occurs in the network.” Churchland and Sejnowski (Churchland and Sejnowski, 1992) following the same principles insisted on computational models of the dynamics of neural networks

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extending PDP models to a new level. Their work stressed the role of inhibitory/ excitatory connections and made distinction between stored (weights of connection) and non-stored representations (neural activations). These models represented a new attempt to understand distributed representations and similarities to mind-brain models. In a more recent attempt using the cable theory, Koch described the basics of ionic currents (Koch, 1999) including dendritic attenuation, interaction of synaptic inputs . However, this description missed some important phenomena such as learning. Koch’s vision regarding information processing by neurons can be summarized by the following paragraph: o “Individual neurons convert the incoming streams of binary pulses into analog, spatially distributed variables, the postsynaptic voltage and calcium distribution throughout the dendritic tree, soma, and axon. These appear to be the two primary variables regulating and controlling information flow, each with a distinct spatio-temporal signature and dynamic bandwidth. The transformation between the input and the postsynaptic Vm and [Ca2+], involves highly dynamic synapses that adapt to different time scales, from milliseconds to seconds and longer. Information is processed in the analog domain, using a menu of linear and nonlinear operations. Second-order and higher multiplication, amplification, sharpening and thresholding, saturation, temporal and spatial filtering, and coincidence detection are the major operations readily available in the dendritic cable structure augmented by voltage-dependent membrane and synaptic conductances” The importance of spikes beyond the standard firing rate model is also the main topic of the book “Spikes: Exploring the Neural Code”. The power of computation with spikes reveals the fact that “the individual spike, so often averaged in within neighbors, deserves more respect” (Rieke et al., 1997). Using information theoretic techniques the authors envisioned that information processing in the brain approaches the limits imposed by “physical limits”. This new path approach where the timing of occurrence of individual spikes plays a key role, is developed further by Gerstner and Kistler (Gerstner and Kistler, 2002). The fundamental assumption of the current development of spike timing theory is that an action potential is stereotyped (Maass, 1997; Maass and Bishop 1998; Mainen and Sejnowski, 1995). In this treatment, the spike is simply an event, either present or absent, and the interspike interval is considered important, performing better than the average firing rate in coding input signals. o “The exact timing of spikes can, in principle, convey a lot of information which is lost if only the temporally averaged mean firing rate is considered.” Instead of adopting an ”input-output unit with a nonlinear transfer function” as previously described, the neuron is therefore mathematically built to approximate the temporal activity of spiking activity simplifying previous conductance models (Hodgkin and Huxley, 1952). Importantly, the spike response model can incorporate temporal properties including the refractoriness (Jolivet et al., 2008). This view was earlier developed by (Eliasmith and Anderson, 2003) and summarized by Eliasmith based on the following three principles: o Neural representations are defined by the combination of nonlinear encoding (exemplified by neuron tuning curves, and neural spiking) and weighted linear decoding (over populations of neurons and over time).

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o

Transformations of neural representations are functions of the variables represented by neural populations. Transformations are determined using an alternately weighted linear decoding. o Neural dynamics are characterized by considering neural representations as control theoretic state variables. Thus, the dynamics of neurobiological systems can be analyzed using control theory. There are two main contributions that Eliasmith and Anderson stress in their approach. Firstly, they formalize computation as an input/output model where encoding and decoding phases are successively performed. The second aspect is that “neural representations are modeled as control theoretic state variables”. On the neurophysiological front, advances in electrophysiological recordings during behavior in rodents and primates allowed correlative analyses between the electrical properties and functions in the brain. The adaptation mechanism called Hebbian plasticity became applicable to these systems and was recently applied to spike-timing-dependent plasticity, which has become an active area of research for many neuroscientists. As already discussed previously, a stereotyped action potential formed the basis of current development of spike timing theory (Maass, 1997; Maass and Bishop 1998; Mainen and Sejnowski 1995). The interspike interval is postulated to perform better than the average firing rate in coding input signals. The dynamics of spiking models have lately been analyzed by Eugene Izhikevich (Izhikevich, 2007), bringing new theoretical analysis in the field. Other recent contributions may include the confabulation theory of Hecht-Nielsen as an attempt of reconciliation between connectionist view and symbolic paradigms (Hecht-Nielsen, 2007). All these models are examples of original research. Many of these computational models have theorized and algorithmized the adaptation process. Seen as plastic changes of synaptic strength or modulations in firing rate as an effect of neurotransmitters, adaptation has embraced a theoretical mathematical formalism. The common issue for all these contributions is that they mainly reflect and extend the scientific vision based on early electrical recordings performed by Adrian (Adrian, 1932) where the timing of spikes, either as to their firing rate or the associated interspike interval, was felt to fully describe neural processes. All of these theoretical constructs assume yet fail to describe an important, unforgettable fact: for any computation and transfer of information, underlying, physical information carriers are needed to perform the classical computation. As an example, electrons play the role of information carriers in order to physically perform simple computations in von Neumann machines. Temporal description of computation in the brain ignores or limits this vital requirement of a physical carrier for information. The predominant issue is that the temporal unit so valued to date, is not a rich physical quantity compared to complex dynamics and interactions between electric charges. Therefore, even though many things in nature occur in time, there is always a physical mechanism that is acting within this time domain. Spike timing theory limits this vital fact to the time where an action potential event occurs. 3.1.6. Building the Internal Model Developing conceptual models of complex real world entities and simulation of dynamic agent-based systems can help us to understand some basic phenomena and the development of cognition can be easily exemplified using the theory of cognitive agents (Lee and Loia, 2007). If one assumes that the environment E can be described

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using the state variables (e1,e2,…,en) then the agent function is to build an accurate model of the environment. Therefore, the agent A which is represented as a dynamic system with internal states (a1,a2,…an) can build during time an equivalent model E’ with state variables (e1’,e2’,…en’). Since the agent is sensorially coupled (S’) then it will provide motor commands M’ to the internal model exactly in the same manner as it receives sensorial inputs S and gives motor commands M when it interacts with the environment (Figure 5). In this case the internal model E’ acts as an emulator of the environment and its design depends on dynamical interaction with the environment. Therefore, the executive part, the controller C is directly dependent on the accuracy of the internal model M’. The study of cognitive agents shows that building a

M

M' E' S'

S

C

Environment Figure 5: The representation of information and cognitive system development determined by the dynamic interaction of agent –environment .See related text for symbol description.

representative model of the environment is not an easy task. An increase in complexity of the environment or in its interaction with the agent requires an increase in complexity of the internal model. This way of building cognitive agents can be extrapolated to our work of understanding how knowledge is embedded in our brain. Therefore, it is expected that in order to describe the dynamics of the external world, the brain should be able to provide a similar complex dynamic mechanism that acts as an emulator or an internal model of the external world. Understanding how these dynamic processes occur and how they are physically built is the essence of building every system that underlies “intelligent” computations. 3.2. Neurons as information engines It is generally established that neurons play an important role and are involved in brain information processing. The accepted view assumes that information is transferred from the axon of the presynaptic neuron to the dendrites of the postsynaptic neuron. Such transfer of information is generally biochemically modulated. However, the basic physical principles that govern this information transfer still remain unrevealed. 3.2.1. Maximum Entropy Principle Individual cells fire many action potentials and this firing activity can be measured using both extracellular and intracellular recording electrodes. As discussed above,

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neural activity cannot be completely described by or reduced to AP occurrence alone. Additionally, neural architecture is extremely complex with large numbers of spatially distributed synaptic inputs on every neuron and such bioelectrical activity determined by synaptic inputs is almost continuous and spatially spread out. The complexity of the relationship between information transfer and electrical properties of neurons can be seen to occur at various scales. At the highest level a moderate analysis has been developed for “wave coding” as a large-scale feature of a group of neurons that encode behavior in human fMRI studies (Cerf-Ducastel and Murphy, 2003; Poellinger et al., 2001). The second level at the cell scale involves spike coding usually viewed in the time domain, as discussed above. Known examples are well studied involving rate coding or spike coding (Maass, 1997; Gerstner and Kistler, 2002) that have been the premise of the strong development of artificial neural networks and artificial intelligence. Such processing can occur at the ensemble level where groups of neurons can be seen to encode for complex behavior (Wilson and McNaughton, 1993; Barnes et al., 2005). However, it is likely that information processing and transfer does also occur at an even more subtle scale, one that is based upon the dynamics of different ionic fluxes within neurons. This topic, along with experimental evidence has been discussed in detail in chapter 2. Such level of information processing may be the most fundamental and possibly the richest level of information processing for the entire nervous system. Our work has tackled the problem of information coding by neurons at this level. We have shown that neuronal spiking and the resulting charge flux is characterized by spatial directivity. These changes in spike directivity could result in important effects during every AP and may alter the neuron’s ability to communicate information (Aur et al., 2005; Aur et al., 2006; Aur and Jog 2006; Aur and Jog, 2007). When a neuron spikes, it considers all received inputs. Many of these inputs are synaptic while several other inputs are non-synaptic. At this time scale of a spike, these inputs come predominantly in the form of ionic flow and the spike results in its own ionic flux that can be measured electrophysiologically. During synaptic communication there is a substantial involvement of ion channels. As the AP invades the presynaptic terminal it generates a depolarization of the presynaptic neuronal membrane. Depolarization operates on several voltage-gated ion channels, such as Na +, Ca2+, K+ , Cl- channels. The probability of opening of the channels usually increases during depolarization (Meir et al., 1999). In the neostriatal cholinergic interneurons, spikes are produced spontaneously in the absence of synaptic inputs (Bennett et al., 2000). Another typical example can be found in the cerebellar Purkinje neurons where spontaneous firing appears in the absence of synaptic input (Bell and Grimm, 1969; Latham, 1971). Spontaneous firing can be maintained also in cerebellar slice preparations when synaptic activity is blocked (Häusser and Clark, 1997). Such spontaneous firing implies that in the absence of synaptic input, the extracellular milieu may serve as the medium through which Purkinje cells can be depolarized. Hence, in terms of information communication, the activity of the ionic channels on neuronal membranes may be modulated by synaptic and non-synaptic electrical events and result in generating the action potential. The work of Boltzmann, Gibbs, Shannon has led to the Jaynes Maximum Entropy Principle (MEP) (Jaynes, 1957). This principle applies to a large variety of nonequilibrium systems including large-scale climate changes to small scales of crystal growth or bacterial metabolism. Within every neuron there is a large number of components such as atoms, electric charges, molecules, which can be categorized into

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a certain number of classes. Such an ensemble of several components allows us to see each neuron as a self-contained and complete system. Boltzmann’s statistical physics approach assumes that all microstates of a system are equally probable. His definition of entropy: H B = k ln Ω ( 11 ) correlates elementary disorder in a given system with entropy. Low entropy is specific to an ordered system which should have only a few microscopic states while a less ordered system should increase the number of macrostates and also the entropy. The postulate of maximum entropy states that the entropy of a closed system never decreases in any process ( ΔH B ≥ 0 ). Additionally, Boltzmann’s formula can be obtained as a result of minimizing the “average index of probability of phase” ∑ pi log pi with respect to the microstate probabilities (Gibbs, 1902). In a similar i

fashion but with respect to information entropy, Shannon introduced the equation for information entropy, (Shannon, 1948) ∑ pi log pi as a measure of uncertainty i

associated with the probability distribution pi . Therefore, Shannon’s information entropy measures the level of our ignorance about the state of the system. Jaynes has further formalized this concept in terms of probability distribution related to information. “Information theory provides a constructive criterion for setting up probability distributions on the basis of partial knowledge, and leads to a type of statistical inference which is called the maximum entropy estimate. It is the least biased estimate possible on the given information” (Jaynes, 1957). This development of partial knowledge of a process can be related to Planck’s work in statistical thermodynamics where he hypothesized the equality of a priori probabilities for the states of a system (Planck, 1900). Based on this theoretical approach one has to select from among all possible distributions the initial states that show maximum entropy values (Tolman, 1979). It is quite likely that from such a broad spectrum of every possible outcome, Nature selects the most probable one which is the state that obeys the Maximum Entropy Principle. 3.2.2. Mutual Information It is therefore possible that with information entropy fluctuations, a neuron can transfer information at multiple levels during AP propagation and diffusion processes (Goychuk and Hanggi, 2000; Aur and Jog 2007).

X

H(X)

∫

Y

H(X|Y)

Figure 6 Schematic representation of dynamic model of information processing in the brain where H(X) is the entropy of the input variable X and H(X|Y) is the conditional entropy. Input and output variables can physically describe the flow of charges in the system.

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Voltage sensitive channels under the influence of electric fields and concentration gradients that allow charge transfer are responsible for this phenomenon. The voltagegated ion channels are distributed along dendrites, soma and axons and the ionic flow is greatly enriched almost everywhere during each spike. Even the process of diffusion in liquid–liquid systems “can be compared to information transfer in information systems of multiple components” (Ceruti and Rubin, 2007). Artificially we can generate signals that share large volumes of information, however in the real world in order to obtain large values of mutual information, information has to be transferred. Therefore the process of diffusion of ions during AP occurrence can be shown to process and transfer information within every spike. Information can thus be accumulated or removed from the neuron by electrical ionic flow during each spike. Therefore, the analysis of information transferred within AP’s by the electric field and flow of charges is critical. Information entropy of any continuous variable contains more information entropy than its discrete version. However, for high sampling rates, the information entropy of a discrete variable comes close to the continuous value (Sobczyk, 2001). This characteristic allows us to see the mutual information MI between two continuous variables as the limit of computed MI values for the discrete versions using the Hodgkin Huxley (HH) model. Since the mutual information measures arbitrary dependencies between random variables such measure is far superior to the autocorrelation function (Fraser and Swinney 1986). Additionally this measure can be used to analyze dynamical systems (Figure 6). Mutual information measures information shared by two selected variables X1 and X2 (Cover and Thomas, Gray, 2000): I(X1 ; X2 ) = H(X1 ) + H(X2 ) - H(X1 , X2 ) ( 12 ) where H(X1,X2) is the joint information entropy. Substituting I (Na+, K+) in Eq. 12, we can measure the reduction in uncertainty about the Na+ flux due to the knowledge of K+ flux. Considering the two main ionic fluxes Na+ and K+ the information transferred between them is: I spike = I ( Na + ; K + ) = H(Na + ) + H(K + ) - H(Na + , K + ) ( 13 ) spike represents mutual information between these two fluxes. By using values of where I ionic INa+ and IK+ fluxes from HH model it is possible to estimate the joint probability distribution and mutual information between Na+ and K+. o The total amount of correlations is given by computing mutual information, large correlations imply large mutual information in classical or quantum systems (see Horodecki et al., 2005) The concepts and the methods for computing distributions, entropy and mutual information have been presented elsewhere in detail (Aur et al., 2006). Since the analysis of ionic fluxes in vivo during AP activity is not an easy task we have used for this study extensive simulation of the Hodgkin-Huxley (HH) model. This model is close enough to biological cell activity to allow us to model the dynamics of ionic currents during AP (Appendix 1). Based on simulated input-output relationships the quantity of information transferred between input stimuli and ion fluxes can be estimated using general principles from signal processing and information theory. The interdependencies between input signal and the output are analyzed in the context of mutual information. Theoretically, mutual information measure does not provide the direction of information transfer (Kaiser and Schreiber, 2002). However, for physical processes the directionality in most cases is predictable.

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Briefly, to perform this analysis the following steps are developed. First, a RungeKutta technique is used to compute ion currents in the HH model that occur when the input stimuli with variable frequency are applied. The accuracy for integration was tested for the 4th order Runge-Kutta method and we have found it to be accurate enough to perform the integration of the HH model. Since the HH model is described by known equations, the sampling rate of the discrete model is adjusted to keep certain dimensions for analyzed data. For computing MI we kept these dimensions between 330x330 and 2000x2000. The estimated average error is under 0.2 bits for all range of data between 330 and 2000 points and the standard deviation is below 0.15 bits. For these data dimensions the maximum error is less than 0.23 bits. The errors and standard deviation between estimated values with the histogram method and analytical estimated values are computed (see Appendix 2, and also Aur et al., 2006). Simulation of the HH model and mutual information estimation were implemented using a PC computer (Pentium 4, 2.8 GHz, 512 MB RAM) and Matlab - MathWorks, Inc. All the routines were custom developed or are freely available on the world-wideweb. Mutual information I IstimNa was estimated between the stimuli current I stim and sodium flux I Na : I IstimNa ( I stim ; I Na ) = H( I stim ) + H(I Na ) - H( I stim , I Na ) ( 14 ) and Ispike between I stim and membrane potential V: I spike ( I stim ; V) = H( I stim ) + H(V) - H( I stim , V)

( 15 ) and potassium

Additionally, mutual information I IstimK was estimated between I stim flux I K : I IstimK ( I stim ; I K ) = H( I stim ) + H(I K ) - H( I stim , I K ) ( 16 ) and mutual information I NaK between I Na and I K : I NaK ( I Na ; I K ) = H( I Na ) + H(I K ) - H( I Na , I K ) ( 17 ) During each spike, ions keep in contact through space and time and these correlations can be quantified based on MI estimations. Therefore, to find the information “encoded in spikes one has to analyze the dependencies in information entropies of several ionic fluxes. Different ionic fluxes convey information mutually because they are correlated. As discussed above, I(Na+, K+) measures the reduction in uncertainty about the Na+ flux due to the knowledge K+ flux: I(Na+; K+)=H(Na+)- H (Na+/ K+) ( 18 ) where H (Na+/ K+) is the conditional entropy. Considering only these two main ionic fluxes Na+ and K+ the “encoded information” is: Ispike = I (Na+ ; K + ) = H(Na+ ) + H(K+ ) - H(Na+ , K+ ) 2

( 19 ) By using values of ionic fluxes from HH model it is possible to find their joint probability distribution as presented in (Aur et al. 2006). Since mutual information is a positive quantity, it is expected that many fluxes together tell us more than two fluxes. Indeed neuronal processes include the activity of many ionic fluxes. Stochastic dependence between several variables can be characterized by the multi-information measure. Multi-information always increases

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by adding a new variable (Appendix1). Therefore, introducing Cl − flux, Eq.( 19 ) becomes: I3spike= I(Na+ ; K+ ;Cl− ) = I(Na+ ; K+ ) + I((Na+ , K+ );Cl− ) ( 20 ) The increase in multi-information is based on mutual information between Cl − and previously considered Na+ and K+ ionic fluxes. From Eq. 22 and 23 the inequality is: I 3spike ≥ I 2spike ( 21 ) Inequality 24 can be generalized by adding the transport of different ionic fluxes such as Ca2+, Zn2+, Mn2+,Co2+, Cd2+, Cu2+, Ni2+ (Colvin, 1998). Adding enumerated ionic fluxes inequalities can be written as: spike I nspike ≥,... ≥ I 3spike ≥ I 2spike ( 22 ) +1 ≥ I n showing mainly that considering several ionic fluxes in the spike model, multiinformation measure increases. Therefore, from informational point of view, it is clear that assemblies of ionic fluxes display more information within the system than each pair of ionic fluxes. These correlations between several ions can be quantified based on multiinformation estimations. The concepts and the methods for computing distributions, entropy and mutual information have been presented in several papers (Silverman 1986, Gray, 1990, N. Tishby et al, 1999, Moddemeijer, R. 2000, Dinh-Tuan Pham, 2004). The computing and assessment of the quality of MI estimations have been explained in detail (Aur et al., 2006). The following section describes the application of theoretical concepts discussed above. 3.2.3. Information transfer In the real world large correlations between a set of variables in any system are unlikely to occur unless information is physically transferred. A variety of results from HH model simulations and calculations of mutual information have already been presented in Aur and Jog, 2006. Here we would like to illustrate the importance of input phase modulation in information transfer. We are able to show this phenomenon since the values of MI transferred in sodium fluxes can be even higher than previously estimated at low frequencies. The values of mutual information depend on the input phase (Figure 7 a-d). This dependence on phase increases at low frequency. The lower the frequency of stimulus, the stronger is the dependence on phase. Therefore, to show this dependence we lowered the input frequency to 50 Hz and plotted the obtained values. Previously the estimated MI per spike was Ispike= 3.56 ± 0.2 bits and the maximum value for INa fluxes was 7.05 ± 0.2 bits at 100Hz. (Aur et al., 2006). If the frequency of Istim is lowered to 50 Hz the maximum value for IstimNa =8.7840 ±0.2 bits while for Ispike is 3.3264 ±0.2 bits. Since phase differences can be translated in time, this may explain why, by analyzing spike timing, the values of information are comparable with those obtained from Ispike estimations. This is an extremely important point as it is the beginning of our ability to show that the analysis that we have presented above can incorporate subtle concepts related to spike timing theory. Sodium and potassium currents are highly correlated. This is an indication that there is a strong information transfer between incoming Na+ and outgoing K+. Therefore, in terms of mutual information represented by Na+ and K+ components, one can estimate relevant information of Na+ ionic flux with respect to K+ flux and vice versa during each AP. Using Na+ influx with respect to K+ efflux during AP, the values

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Figure 7: The dependence between estimated values of mutual information IIstimNa and Ispike plotted across the phase of sinusoidal input at 50 Hz Istim(µA/cm2) during 22 ms. a, The 3D plot of IstimNa over amplitude of input stimuli and phase of sinusoidal input. b, The estimated mean of IstimNa over the phase of sinusoidal input. c, The 3D plot of Ispike over amplitude of input stimuli and the phase of sinusoidal input d, The estimated mean of Ispike over the phase of sinusoidal input

of mutual information are computed and represented in (Figure 7a) for different frequencies and amplitudes of the input stimuli. Mutual information encoded in Na+ and K+ fluxes have comparable values to information transferred from sinusoidal input signal to Na+ fluxes. o Information is transferred synaptically and non-synapticaly within ion fluxes and this electric interaction is able to transfer on average about 3.5 bits/spike and maximum values of 8.7840 ±0.2 bits o Temporal encoding (spike timing) can be seen as a natural outcome generated by changes in information transfer modulated by the phase of the stimulus input The dependence between estimated values of mutual information I NaK and frequencies for a sinusoidal stimulus Istim computed over 30 ms give values between 3 and 3.5 bits. 3.2.4. Efficiency of Information Transfer In order to compute the values of information transfer that occur from ionic fluxes and their dependence on input stimuli, we have used simulations of signal input and output in the Hodgkin-Huxley (HH) giant squid axon model (Aur et al., 2006). The

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applied inward current I stim could come from another neuron or can be artificially generated. Indeed, I stim from HH simulations can be viewed as a dendritic current, a synaptic current, or a current injected through a microelectrode. Before using the method to compute mutual information (MI) between ionic currents during a spike we have extensively analyzed the histogram method to compute MI for Gaussian noise signals (mean μ=0 and standard deviation σ=1) (see Appendix 2). In comparison, the estimated values of MI between ionic currents are at least ten times higher, > 2 bits. To compare information by spikes or by ion fluxes we use the same method therefore we expect robust results. One way to see efficiency within each spike is to consider the mutual information I stimNa between I stim and sodium flux as the input MI and mutual information

I spike between I stim and membrane potential as an output MI. Then efficiency can be computed: η1 =

I spike I stimNa

( 23 )

One can view this in terms of ionic fluxes alone. If I Na is seen as an intra-

neuronal carrier of information then the K+ outflow can be seen as an export of information. One can then compute the efficiency as the ratio between MI transfer from I stim to K+ and I stim to Na+: η2 =

I istimK I istimNa

( 24 )

Interestingly, in both cases the values of efficiency of information transfer in spike seem to have the same shape over the frequency of the input stimuli: η 2 ( f ) ≅ η1 ( f ) = η ( f )

( 25 )

where f is the frequency of the sinusoidal input. The upper bound in efficiency can attain values of η ≅ 0.7 in both cases for frequencies lower than 200 Hz. The maximum and mean values of efficiency decrease for frequencies higher than 200 Hz (Figure 8 ad). The dependence between efficiency values η1 , η 2 and frequencies of Istim is presented in Figure 9. The maximum and mean values of efficiency have been computed. One can see clearly that the efficiency of information transfer within a spike depends on input frequency. For lower frequencies less than 200Hz, neurons efficiently transfer information from input stimuli during each spike while this transfer is reduced significantly for higher input stimuli frequencies. Additionally, at lower frequencies less than 200Hz the MI value and computed efficiency values for information transfer present a nonlinear complex dependency over Istim phase (Figure 9 a-d) (Aur et al., 2006 ). Since the phase between Istim and the occurrence of AP can be translated into a temporal domain, this variability of MI values over phase shift is confirmed by the experimental observations of current spike timing theory. While the current spike time theory was built from observation of MI between stimuli and spikes occurrence our model reveals this dependence from a different theoretical approach by analyzing ion fluxes in HH model. Indeed, our approach provides a firm framework upon which the

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observations of spike timing theory can be justified. The broader platform is also capable of providing much more expandability as we show in this book. It is also clear that the notions of spike timing are limited in what they can offer to understanding neuronal function. Thus, unlike the spike-timing model, the new model we have developed is far richer and complex and yields a picture that is closer to the nature of neuronal computation than the spike-timing model can ever expect to achieve. An extremely powerful consequence of experimental results presented in Chapter 2 and the above development is that if sequential APs from neurons can now be treated as being distinct implying that (see Aur and Jog 2006) every spike may transfer slightly different amounts of information during their occurrence.

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3.2.5. The Nature of Things - Functional Density The major goal in cognitive neuroscience is to determine how information is processed in the brain. Mathematical formalism has been applied to mimic neuronal computation and provide basic tools for developing various theories of computation and neural coding. Although Putnam claimed that the brain could be adequately modeled by a classical Turing representation (Putnam, 1960) many other scientists have rejected this idea (see Eliasmith, 2002). However, since the current theories of neural computation fail to mimic the power of the brain, new concepts regarding computation need to be developed. Understanding the basic principles of brain computation can be a result of direct observation and analysis of several aspects that shape all natural processes. At a large scale, a view from an airplane may only show several cities and some roads connecting them. At a smaller scale in every city we are able to see details such as conglomerations of houses, traffic lights, cars and finally the people. The city has high density of buildings, streets where each component certainly plays a functional role. This increase in density is a natural way of increasing interaction that seems to work everywhere as a universal principle. Urban conglomeration plays a role since energy resources can be

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effectively used. This may include energy spent for transportation, human resources, labor force. The development of certain companies in the region may affect long term, production from another that has effects upon nearby economic activity. These observations regarding increased efficiency of resource utilization based upon increased density can be extended to several other natural phenomena and can be used to understand brain processing. Our universe contains several things that repeat in space or in time with or without obviously visible patterns. One can observe leaves, trees, or even neurons in the brain that are spread in space. This analysis can go to a microscopic scale where water molecules, atoms or electrons are evenly or randomly distributed in space. The same phenomenon can occur and can also be repeated in time as in the heart or in brain rhythms (Buzsáki, 2006). This property is called recursivity. The case of trees, leaves, migration and arborization of neurons are forms of spatial recursivity while brain and heart rhythms are typical cases of temporal recursivity. o Once derived as an appropriate structure or a suitable mechanism, the structure/mechanism is repeated in space or in time with small variability to complete certain functions. Nature itself, and not mathematics, found this way of building complex systems. Mathematical models described by equations, functions or systems of equations can in some measure represent these real-life phenomena. In theory, recursion as an important property occurs when a function calls itself (Odifreddi, 1989). Typical examples in mathematics are computing factorials, Fibonacci series or graphing Sierpinski triangles (Rich, 2007). Beyond nice-looking images spatial recursion is a principle that shows us how to put together constructs improving understanding of many complex phenomena. In quantum physics changes in density at atomic scale are described by so called density functional theory that replaced the many-body electron-based wave function with the electronic density. This replacement provides a way to develop a new theoretical framework that can be applied more easily than a many body wave function approach to study microscopic phenomena. 3.2.6. Charge Distribution and Probability Density It is well known that density estimation is a powerful technique for modeling distributions in case of sampling different types of data. The basic formulation specifies that the probability of a given variable in the interval between any two points a and b can be computed: b

P( x ∈ [a, b]) = ∫ p( x)dx

( 26 )

a

where p(x) is the probability density of a normalized function. Since information transfer is dependent on the movement of charges it is important to find a way to characterize their spatial distribution (Figure 10). The amount of uncertainty about a distribution of charges P =( p1 , p2 ,... pn ) can be computed using Shannon information entropy H: n

H ( p1 , p2 ,... pn ) = −∑ pk log 2 pk k

( 27 )

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aa

b

Figure 10: Schematic representation of two different distributions of charges across X axis. a, The charges are uniformly distributed b,The charges are non-uniformly distributed

If one considers charges evenly distributed in space (Figure 10,a) with the given probability distribution P1 ={1/3, 1/3, 1/3}then information entropy value is: n 1 1 H 1 = −∑ log 2 = 1.585 bits ( 28 ) 3 k 3 If the distribution of charges follows the schematic representation (Figure 10,b) with given P2 ={0.1,0.8,0.1} then the computed entropy is: H 2 = −2(0.1 log 2 0.1) − 0.8 log 2 0.8 = 0.921 bits

( 29 )

This simple example shows the relationship between spatial distribution of charges and information entropy values. Once the charges are non-uniformly distributed Shannon information entropy computed for this system composed by charges is reduced. This implies that (Figure 10, b) there is also a reduction in uncertainty since charge particles can be better located across the X axis. In other words for an external observer this distribution provides more “information” regarding the spatial location of charges across the X axis. This problem of linking information with spatial distribution of charges becomes complex if these charges move in space. These charges can be seen as points in a threedimensional space (Figure 11). Then, the observer has to further consider the time as an additional dimension when the measurement or the observation of charges in motion has to be performed. However, an accurate measurement of charge density in a certain region of space for such a dynamical system composed of several charges is not a trivial issue. In fact solving this issue provides the answer of how the brain decrypts information carried by electric field waves and charges. The information carried and delivered by these charges is likely the information which is read by biochemically modulated synapses. Therefore, neural activity can be artificially separated into two main phases. The first phase consists of generating the movement of charges or an uneven spatial distribution of charges and the second phase is the above problem of reading information embedded in such a distribution. Both phases should be physically realizable and must obey the laws of physics.

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Looking at the brain from small to large scales, the occurrence of changes in charge densities plays an important role.

S1

S2 S3

Figure 11: Representation of spatially distributed charges in blue and a set of solid spheres S1, S2….Sn with radius R. These spheres can cover with a certain approximation the regions with high density of charges

A certain arrangement of spheres or in a general case of ellipsoids is able to pick up the regions with high densities of charges (Figure 11). Choosing an optimal number of spheres/ellipsoids, their radius and/or location in space to minimize the number of charges left out is a delicate problem. The theoretical model can be reduced to the Khachiyan problem (Khachiyan,1996). Since the charge points should be inside the ellipsoids/spheres then one can write: (x i − c)T M ( xi − c) < 1, i = 1, N

( 30 )

where x i are the coordinates of the N charges, M and c are variables that need to be determined. Finding the optimal ellipsoid/sphere with the minimum volume that covers the position of charge points can be obtained by solving the Khachiyan problem (Khachiyan, 1996). If instead of ellipsoids we use solid spheres with similar radius R the Khachiyan problem can be simplified (Figure 11). This problem becomes more complex when these charges move and interact in space which in fact may be the role played by the active synapses if computations are performed by electric charges at synaptic cleft level. Additionally, considering charges as basic elements, “neural computation” becomes in fact computation by all elements that make up the brain, since several other mechanisms including glia and other buffering systems dynamically regulate the distribution/redistribution of charges in space (Kofuji and Newman, 2004). 3.2.7. The Laws of Physics The laws of motion are determined by the existence of fields and forces and several other factors involved in the physical process. The key to solving this subtle problem of dynamics provides the answer of how computation is built from nanoscale level as a process of diffusion. This also shows why neurons have several types of synapses

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spatially distributed all over. Furthermore when and where the observation (measurement) is performed provides insight about reading certain specific “information” processed further in the nervous system. Starting from this simple presentation several models of computation in brain can be imagined. The simplest model can start with a charge density model expressed via the Dirac function:

ρ ( r ) = ∑ q iδ ( r − ri ) i

( 31 )

where ri gives the position of the ith charge. This simple description of a group of charges allows us to relate the movement and interaction of these charges with signal processing techniques and the theory of dynamical systems. Importantly the main difference between charges and their counterpart as simple particles is that charges are r always at the origin of the generated electric field E that mediates electrical interactions and information transfer. As a classic example the electric potential of a group of charges distributed in space can be computed using the Poisson law: ∂2 ∂2 ∂2 ρ ( x, y , z ) ( 2 + 2 + 2 )Φ ( x , y , z ) = − ( 32 ) ∂x ∂y ∂z ε The equations of classical mechanics can be used to describe the motion of charges in a multidimensional potential energy surface. The force that acts on charge qi

r

depends on the intensity of the electric field E : r r Fi = qi E

( 33 )

where for each charge within a certain volume V the intensity of electric field is:

r 1 E (r ) = 4πε

∫

V

ρ (r ) r r dV r2

( 34 )

r Additionally the force Fi can be written as a product between the mass of charge r and the acceleration. ai : r r Fi = mi ai ( 35 )

Naturally these charges will redistribute themselves by moving away one from another or attracting each other until they achieve the lowest state of electrostatic potential energy. Classical description provides the evolution and location of the charges as points in three dimensional space (Figure 12). The attraction or repulsion force depends inversely on the square of distance between charges: r r N r q (r − r ) Fi = ∑ j i j 3 ( 36 ) 4πε (r − r ) j =1 j ≠i

i

j

In case of N charges their kinetic energy can then be written: 1 N K = ∑ mi vi2 ( 37 ) 2 i=1 and their potential energy P in the electrostatic Φ i potential can be described: 1 N P = ∑ qi Φ i ( 38 ) 2 i =1

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Xb, Vb

Xa, Va

Figure 12 : The movement of charge from point A to point B deletes information in point A and transfers information to point B. Energy is required to perform such movement to transfer information in space.

where the electrostatic potential Φ i for charges q j ,j=1,2,…N is: N

Φi = ∑ j =1 j ≠i

qj 4πε | ri − rj |

( 39 )

The effective potential includes the effects of the Coulomb interactions between charges that can be represented by different types of ions, electrons or even polarized molecules. The easiest way to find the distribution of charges in space is to minimize their electrostatic potential energy P . The process of obtaining the position of charges in space can then be viewed as a mathematical multidimensional optimization procedure. However, if the number of charges increases the modeling and the simulation of charge movement on current computers becomes problematic since it is not easy to describe all interactions that occur between such charges in movement. At microscopic scales the attractions between individual atoms, molecules etc can be expressed as van der Waals forces. Since the electrons are not evenly distributed in atoms and molecules they may determine polarizations. The resulting forces display an attractive or a repulsive force determined by an electrostatic interaction between charges, polarization or dispersion. Sometimes van der Waals forces are assumed to be only attractive. However, the repulsive forces are equally important components since they prevent molecules from collapsing. Once the distance between constituents becomes smaller the interaction sharply increases (see Figure 13) and the trajectory of charges can be described in a probabilistic fashion using quantum theory (Schrödinger, 1926). The charged particle can be seen as a wave ψ that moves in a potential well, known as Schrödinger Equation (Schrödinger, 1926): ∂ψ h 2 ∂ 2ψ ih =− + Vψ ( 40 ) ∂t 2m ∂x 2 where h is the reduced Planck constant: h h= ( 41 ) 2π -34 2 and h =6.626068 × 10 m kg / s.

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For a plane wave, ψ ( x, t ) : ψ ( x, t ) = ψ 0 e i ( kx−ω t )

( 42 )

one can write: −

h ∂ψ = hωψ i ∂t

( 43 )

Since:

K = hω ( 44 ) then the Schrödinger wave equation can be rewritten as: h2 2 Kψ = − Δ ψ + Pψ ( 45 ) 2m where ψ is the wave function, P is the wave potential and K is the kinetic energy. At this microscopic level it can be shown that information communication can be seen also to be intrinsic to the treatment of charges behaving either as ‘particles’ or with a wave like behavior. Since: 2 ( 46 ) ∫ |ψ ( x) | dx = 1 R3

then information that is transferred from one location to another spatial location can be seen in terms of a normalized wave function: dH S = ∫ | ψ (r ) |2 log 2 | ψ (r ) |2 dr ( 47 ) R3

(Parker and Walker, 2007) At atomic scales the strong nuclear force that occurs cannot be modeled in terms of a Coulomb force law and such interactions depend on potential energy and internuclear distance. There is no straightforward formula to describe such dependence (Lippincott and Schroeder, 1955). As represented in (see Figure 13) interaction depends on the distance between particles and varies in a nonlinear manner from weak interaction to strong interaction (Landau and Lifshitz, 1960). Since computation is related to the movement of charges it is expected that such changes in interaction would alter the movement of charges and information transfer. o Information is communicated by electric interaction; any charge or group of charges is able to alter the movement/dynamics of closed charges due to their electric field. In this case the electric field becomes the physical entity involved in information transfer. The applicable laws of physics also change with quantum mechanics and probabilistic laws of movement replacing at the microscopic level classical dependence 1 where interaction depends inversely on the square of distance 2 . Since movement and r interaction are different depending on distance it is likely that computational laws are applicable differently depending on the scale where they are described. Many of these models have to include subtle phenomena at atomic scales described by quantum computation (Nielsen and Chuang 2000). Such description requires a quantum mechanics formalism where the probability for a charge to follow a certain path depends on the difference between kinetic energy and potential energy. Based on the Schrödinger equation this probability can be mathematically expressed in terms of the wave state function | ψ (t ) > (Feynman and Hibbs, 1965):

| ψ ( t ) >= all

∑

paths

e

−

i h/

∫[

m x&

2

2

(τ )

− P ( x ( τ ))] d τ

( 48 )

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where m is the mass of the charge, x is the position and P is the electric potential. Such description relates charge path trajectory to charge energy. In the same time the electric potential can be translated into the required work to obtain a certain spatial position for a certain charge. This wavelike behavior at microscopic level is at the origin of the uncertainty principle (Heisenberg, 1930) and represents a direct consequence of unmodeled strong interactions at microscopic level.

Interaction energy

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Figure 13: Representation of weak interaction and strong interaction energy depending on van der Waals interaction, electrostatic attraction forces and London dispersion forces.

For an external observer this principle provides theoretical limits in predicting particle trajectory in contrast with classical approach of Newtonian principles where the real trajectory is determined with enough confidence. Science faces an ill posed problem trying to find which model is better suited to model neural computations since both the quantum and classical models coexist together cannot be separated (see Figure 13). Their coexistence reflects our limits to find a unique model to describe natural phenomena at different scales rather than the superiority of one model over another. 3.2.8. Minimum Description Length Statistical models are generally used to find regularities in observed data. The main inconvenience is that the same set of data can generate several different descriptions and therefore several models. Comparing these models is not an easy task and one may use complexity analyses to discriminate between many competing models (Li and Vitanyi, 1997). Kolmogorov has defined complexity of a sequence as the length of the shortest program that prints the sequence and then halts. Finding the right model which provides the shortest data description or “minimum description length” (MDL) is often required in science. Ray Solomonoff suggested the use of computer code as a complexity measure (Solomonoff, 1964). Such description of complexity has been extensively studied and described by Rissanen:

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“The number of digits it takes to write down an observed sequence x1, …, xN of a time series depends on the model with its parameters that one assumes to have generated the observed data. Accordingly, by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters” (Rissanen, 1978). Seen as regularity in a given set of data MDL, allows the largest compression of the data (length in bits). Minimum description length is also an information theoretic measure. In other words the MDL principle provides the solution for selecting the model in the model class which achieves the shortest code length for the data and the model. Since computations in brain are performed by a continuous dynamic of charges which builds a required model, then the main question is if we can find a theoretical principle similar to MDL that can depict the process. It has been shown that charges that move are also able to perform universal computations (Moore, 1990). Therefore, it is likely correct to assume that there should be a ‘minimum description’ for such computations. Achieving a certain goal or reaching a certain point in space can be described at a microscopic scale as “a many path description” which can be a model built in a quantum mechanics framework. This approach is consistent with the Occam’s razor principle since among all hypotheses/paths consistent with the observations we have to select the simplest one which can be a straight line or a curve depending on external fields. Therefore, it is expected that the selection of the optimal path for charge movement is consistent with a minimum path description (MPD) principle. The charge displacement should result in the shortest dynamics which may generate the more accurate representation or model of the input: t r r Δr = arg min ∫ ( F (r , E ( Stimulus, ε M ) + n(t ))dt ( 49 ) f

εM

t0

r where Δr is the infinitesimal displacement of the charged particle, and the force F depends on the electric field. The noise source n(t ) and the error between the input and

the model ε M is :

ε M = StimulusData − Model

( 50 ) Additionally the MPD principle asserts that the best model of given input data is the one that minimizes the combined cost of describing the model in terms of number of charges by minimizing the error between the model and input stimuli data. Therefore MPD should also minimize the number N of charges involved in such description by considering the misfit between input stimulus data and the internal model built by the distribution of electric charges : N * = arg min ( StimulusData − Model ( N )) ( 51 ) N

This approach when applied to biological systems reveals a very crucial and important feature. In the brain computation this natural phenomenon can be seen as a self-organization process or learning. In a dynamic model the shortest ‘program’ has to be equivalent to producing a charge trajectory that reflects the “minimum path description”. It is important to show that once these dynamic patterns occur in the movement of charges within action potential, synaptic spikes they may be perceived at

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a macroscopic level and recorded as resulting temporal patterns of AP or synaptic activities. Therefore, temporal patterns or dynamics of AP in neurons or networks of neurons as described in the literature may be a direct outcome of electric field propagation and spatial charge displacement. It is therefore likely that the analyses of such patterns provide less information than one may obtain by analyzing the movement and interaction of charges. Given a current distribution of electric charges and a performance index that measures the error between the ‘model’ and stimuli inputs it is likely that the MPD principle can be reduced to find the minimum displacement of charges able to provide an accurate model of the input. 3.2.9. Is Hypercomputation a Myth? The theory of computation founded by the work of Church, and Alain Turing is still in early stages of development. Theoretical models and algorithmic techniques have been developed mathematically in great detail. At their heart the ideas that information and computation are related have their origins from analyses in physics and computer science. These developments are mainly related to the work of Feynman on quantum physics and new progress in quantum information processing (Nielsen and Chuang 2000; Marinescu and Marinescu, 2005; Angelakis, 2006). The early development of computation related Turing Machines with brain computation capabilities (Putnam, 1960). Computation was generally perceived as a function-based transformation of an input to an output. However, recently it has been suggested that such models cannot describe all forms of computation (Goldin and Wegner, 2008). They show that the definition of Turing computability is highly restrictive since it constrains computation to discrete sequences that generate values in a finite number of steps. Indeed, the Turing model was not constructed to be a general model for computation (Cleland, 2004) and “there is nothing in mathematics or physics to prevent the implementation of a hypercomputational systems”(Stannett, 2006). Even Turing himself suggested that human intelligence “requires more than the standard version of computation described by his machines”(Stannett, 2006). There are several examples where computation cannot be expressed in terms of Turing machines. The behavior of a quantum system cannot be reduced to a Turing model (Komar, 1994; Copeland, 2002; Bringsjord and Zenzen, 2003). Additionally, it has been suggested that “membrane computing” an inspired bio-computational model developed by Paun, (Paun, 2002) can be a new solution for hypercomputation. Using classical physics examples, several nonrecursive computational phenomena can be demonstrated (Scapellini, 2003). Recursive functions are a class of computable functions. Such functions can be expressed in term of themselves and this process is known as "recurrence" or "recursion". As presented above, many natural processes are recursive either in time or space. However the existence of several physical systems whose time evolution is not describable by recursive functions (e.g. wave equation of classical physics) are known to exist (Siegelmann and Sontag, 1992). Scapellini suggested “that the brain may rely on analogue processes for certain types of computation and decision-making”. Possible candidates which may give rise to such processes are the axons of nerve cells.” Even though Scapellini does not describe issues that define new computational paradigms, the proposed hypercomputational model in brain encourages the idea that our brains’ capabilities are beyond those of simple Turing models.

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The new model of computation that includes charges, their movement and interaction show why we cannot reduce brain computation to Turing models. 1. First, the charges move and interact continuously rather than in “isolated jumps” as computation is pictured in the Turing model. Such continuous behavior is clearly a non-Turing behavior. 2. Second the number of charges involved in computation is high enough to provide high degrees of freedom and high dimensionality of processing that can be hardly emulated by single TMs. 3. Third, such movement of charges determines changes in electric field, intrinsic interactions and high levels of parallelism which cannot be easily modeled by TM. Accumulation of information in different forms and the resultant history dependency cannot be also be naturally included in Turing models. Therefore, physical computation in each spike (action potential/synaptic spike) seen as a propagation of electric wave and diffusion process involves the interaction and dynamics of various types of charges. The laws of physics including electric field propagation, charge motion and interaction would then define the general rules of computation. Since charge fluxes are information carriers in neurons, information transfer occurs during an action potential (AP) via the voltage gated ion channels in membranes and the same physical mechanism can be extended to various types of synapses. The action potential can be regarded as providing the required inputoutput dynamics to perform computation by a neuron (Aur et al., 2006). Sodium ions that enter during the depolarization phase can be considered as an input variable while potassium current can be considered an output variable. By computing mutual information between these to variables one can measure the interaction between these two components and the efficiency in information transfer as shown above in section 3.2 of this chapter. This reductionist scheme of charges (ions) as carriers of information represents the required physical machinery for a dynamic process of information transfer (Figure 14) that is substantially limited in the current spike-timing description. Yn

∫ Y2

X

∫ ∫

Y1

Figure 14: Schematic representation of coding as a feedback mechanism that uses the dynamics of charges to compute and build a model during an action potential. Multidimensional input X is dynamically transformed in a spatial distribution Y1, Y2, … Yn of outputs via ion channels

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As a powerful example the visual system has been presented as a highly efficient system that processes information from the environment. Photoreceptors transduce information from received photons into electrical signals that propagate in the network of neurons in the retina, then to the primary visual cortex. The representation of information can be evidenced in distributed electrical activity of neurons and generates models of perceived objects. The intrinsic processing phase is performed by a continuous movement of charges, which corresponds to a permanent information flow. As presented above such processes are governed by the laws of motion and electrodynamics. Charges become the common denominator into which every sensorial input (sound, vision etc) is transformed and processed. Therefore, an adequate representation of information in the brain is shaped by the density of charges and their continuous interaction. At the microscopic level, the laws of quantum mechanics can describe these interactions between charges. However, a detailed analysis of interactions and effective potential energies of the system cannot always be achieved due to high degrees of freedom of charges.

Figure 15 Schematic representation of decoding activity. Electrical synapses in blue and chemical synapses in red as solid spheres allow the transfer of information between neurons.

Information is distributed spatially during every action potential within ion channel activities in axons, membranes and dendrites which represents the coding phase. The mechanism is repeated at larger scales where activation spreads from sensory input areas to “output” areas due to a continuous flow of charges. Between cells this information is carried by electrical synapses (gap junctions) and chemical synapses (Figure 15) and the neuron is able to “read” the input which represents the decoding phase. Therefore, the role of spheres with radius R in (Figure 15) can be seen as a tool to measure the concentration/density of charges. Such activity is repeatedly performed in the brain by synaptic connections. In chemical synapses, considered to be either excitatory or inhibitory, ion channel kinetics are controlled by several neurotransmitters and are considered to be either excitatory or inhibitory. The neurotransmitter release and the activation of ionic conductances in synapses depend on ion concentration

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(Zenisek and Matthews, 2000). As a general mechanism excitation and inhibition is determined at small scales (synapses) and large scales (group of neurons) by the dynamic changes in the densities of charges and their interaction mediated by several neurotransmitters which regulate the computation performed by charges. We can reinforce the fact that once derived this appropriate structure /mechanism that uses the movement and interaction of charges to compute (code/ decode). The structure/mechanism is repeated in space with small variability to design neural networks and in time to complete certain brain functions perceived or measured as brain rhythms. Summary of Principal Concepts •

• • • •

• • • • • • • •

The movement of charges from one point in space to another point in space deletes information in the first point and transfers information to the second point in space. Every charge can get information regarding other charges through the electric field. The erasure of information in any computational device requires an increase in the thermodynamic entropy (Landauer, 1961). Moving charges across the nerve cell membrane determines changes of thermodynamic entropy. These variations of thermodynamic entropy can be directly related with changes in Shannon information entropy (Brioullin,1956; Zurek, 1989) Therefore, using the HH model one can demonstrate that high levels of information can be transferred between the input signals and electrical charges that move (ionic fluxes) within a millisecond-level time domain (Aur et al., 2006, Aur and Jog, 2006) Since these charges are carriers of information then the movement and interaction of electrical charges can be seen as natural forms of physical computation. Therefore, computation in the brain is performed by a dynamical movement and interaction of electrical charges. The optimal path for charge movement is consistent with a Minimum Path Description (MPD) principle. The charge displacement should result in the shortest dynamics which may generate the most accurate representation or model of the input. Learning (self-organization) processes can be seen as an optimization phase which minimizes the number of charges that describe or build a certain model for a set of given inputs. The role of interaction between electrical charges during computation is critical since information is shared between several electric fluxes. Information is distributed spatially during every action potential with ion channel activations in axons, membrane and dendrites which can be described as a process of coding. The distribution of several types of synapses in space is used to “read” and decode the spatial distribution of charges. This approach provides a unifying description of neuronal activity in computational terms at a more fundamental level unachieved by current spiketiming theory.

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• • • •

89

At the microscopic scale quantum probabilistic laws govern the movement of charges their interaction and information processing and communication Turing computability constrains computation to discrete sequences and seems to be inadequate to describe highly interactive systems Brain computations display a non-Turing behavior, the continuous interaction and dynamics of charges provide a high degree of freedom and high dimensionality of information processing Classical and quantum laws and models based on either laws do not compete together, they both coexist and provide a description of computation at different scales

Appendix 1: The Hodgkin-Huxley model describes the action potential in a giant squid axon:

Cm

dV = I leak + I Na + I K + I stim dt

( 52 )

where Cm is the specific membrane capacitance, INa is the voltage-dependent fast-inactivating sodium, IK is a delayed rectifier potassium current, and Ileak is the leakage current. The inward current coming to cell body Istim could be viewed as a dendritic current, a synaptic current, or a current injected through a microelectrode. Then, I leak ,

I Na and I K can be written:

I leak = g leak (V − Eleak )

( 53 )

I Na = g Na m 3 h(V − ENa )

( 54 )

I K = g K n 4 (V − EK )

( 55 )

g K , g Na , g leak are the maximum conductances for currents of the K+, Na+ and respectively of leak currents, while E K , E Na , Eleak are the reversal potentials determined by the Nernst equation. where

A probabilistic model of changes in ionic conductances was presented by Hodgkin and Huxley (Hodgkin and Huxley, 1952) is outlined by the following equations:

dm = α m (1 − m) − β m dt

( 56 )

dn = α n (1 − n) − β n n dt

( 57 )

dh = α h (1 − h) − β h h dt

( 58 )

The conductance variables m, n, and h take values between 0 and 1 and they approach steady state values m∞ (V ) ,

n∞ (V )

and

h∞ (V )

with time constants τ m (V ) ,

τ n (V )

and τ h (V ) :

dm = m∞ (V ) − m dt

( 59 )

τ n (V )

dn = n∞ (V ) − n dt

( 60 )

τ h (V )

dh = h∞ (V ) − h dt

( 61 )

τ m (V )

90

Chapter 3: Information and Computation Appendix 2: Consider a linear input -output model:

y = as + n ( 62 ) where n the noise has a Gaussian distribution (mean μ=0 and standard deviation σ=1). To compute MI for a Gaussian channel we assume that the signal s has also a Gaussian distribution. From Rieke et al., A12, (Rieke et al., 1997 [27]) mutual information can be computed: I = 0.5 log 2 (1 +

a2 < s2 > ) < n2 >

( 63 )

This model is used to asses the errors of the histogram method in computing MI for a Gaussian input. Appendix 3 For three variables X1, X2, X3 by definition multi-information is the difference between marginal distribution and joint distribution: I(X1 ; X 2 ; X 3 ) = H(X 1 ) + H(X 2 ) + H(X 3 ) - H(X 1 , X 2 , X 3 )

( 64 )

Then from Eq.16 and 36 multi-information is: I(X 1 ; X 2 ; X 3 ) = I(X 1 ; X 2 ) + H(X 1 , X 2 ) + H(X 3 ) - H(X 1 , X 2 , X 3 )

( 65 )

Eq 37 can be written then:

I(X 1 ; X 2 ; X 3 ) = I(X1 ; X 2 ) + I((X 1 , X 2 ); X 3 ) ( 66 ) Since mutual information is always positive, multi-information always increases by introducing a new variable. In addition Eq. 37 can be generalized by induction for several variables. References Adrian E., (1932) The Mechanism of Nervous Action: Electrical Studies of the Neuron. Philadelphia: University of Pennsylvania Press. Amari S. (1977) Neural Theory of Association and Concept Formation, Biological Cybernetics, Vol. 26, pp. 175–185. Amari S., (1985) Differential-geometrical methods in statistics, Lecture notes in statistics, SpringerVerlag, Berlin. Anderson J.A., (1995) An Introduction to Neural Networks, The MIT Press. Angelakis D.G (2006) Quantum information processing: from theory to experiment. Proceedings of the NATO Advanced Study Institute on Quantum Computation and Quantum Information, Chania, Crete, Greece, 2-13 May 2005 IOS Press. Aur D., Connolly C.I., and Jog M.S., (2005) Computing spike directivity with tetrodes. J. Neurosci. Vol. 149, Issue 1, pp. 57-63. Aur D., Connolly C.I. and Jog M.S., (2006) Computing Information in Neuronal Spikes, Neural Processing Letters, 23:183-199. Aur D., Jog MS (2006) Building Spike Representation in Tetrodes, Journal of Neuroscience Methods, vol. 157, Issue 2, 364-373. Aur D, Jog MS (2007) Neuronal spatial learning, Neural Processing Letters, vol. 25, No 1, pp 31-47. Barnes T.D., Kubota Y., Hu D., Jin D.Z. and Graybiel A.M., (2005) Activity of striatal neurons reflects dynamic encoding and recoding of procedural memories, Nature, vol 437, Nr. 7062 pp. 1158-1161. Bell C.C., Grimm R.J., (1969) Discharge properties of Purkinje cells recorded on single and double microelectrodes. J Neurophysiol, 32:1044-1055. Bennett C.H., (1973), Logical Reversibility of Computation,IBM J. Research and Develop. 17 525-532 Bennett C.H., (2003) Notes on Landauer's principle, reversible computation, and Maxwell's demon, Studies in History and Philosophy of Modern Physics, Volume: 34, (2003), pp. 501—510. Bennett B. D., Callaway J.C., and Wilson C. J., (2000) Intrinsic Membrane Properties Underlying Spontaneous Tonic Firing in Neostriatal Cholinergic Interneurons, The Journal of Neuroscience, November 15, 20(22):8493-8503. Bekenstein, Jacob D. (1981) Universal upper bound on the entropy-to-energy ratio for bounded systems. Physical Review DD 23 (215).

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Bekenstein J. D. and Schiffer M., (1990) Quantum Limitations on the Storage and Transmission of Information, Int. J. of Modern Physics 1:355-422. Bringsjord S., Zenzen M., (2003) Superminds: People Harness Hypercomputation, and More, Kluwer Academic Publishers, Dordrecht. Burgin, M., Klinger, A., (2004) Three aspects of super-recursive algorithms and hypercomputation or finding black swans, Theoretical Computer Science 317 (1-3), pp. 1-11. Buzsáki G., (2006) Rhythms of the Brain, Oxford Press. Cerf-Ducastel B., and Murphy C., (2003) FMRI brain activation in response to odors is reduced in primary olfactory areas of elderly subjects, Brain Res. 986, pp. 39–53 Colvin R. A., (1998) Characterization of a plasma membrane zinc transporter in rat brain, Neuroscience Letters, 247 147–150. Cover T.M., Thomas J.A., (1991) Elements of Information Theory, John Wiley & Sons, Inc. Copeland, B.J., (2002) Hypercomputation , Minds and Machines 12 (4), pp. 461-502. Jaynes E.T., (1957) Information Theory and Statistical Mechanics, in Physical Review, Vol.106, pp. 620-630. Cleland C., (2004) The concept of computability, Theor. Comput. Sci., Volume: 317, pp. 209—225. Ceruti M.G., Rubin S.H., (2007) Infodynamics: Analogical analysis of states of matter and information Information Sciences, 177 (4), pp. 969-987. Cockshott P. and Michaelson G., (2007) Are there new models of computation? Reply to Wegner and Eberbach , The computer journal, vol:50 iss:2 pg:232 -247. Churchland P.S. and Sejnowski T. J., (1992) The Computational Brain, MIT Press. Denker, J.S., (1994) A review of adiabatic computing, Low Power Electronics, Digest of Technical Papers., IEEE Symposium 10-12, Page(s):94 – 97. Eliasmith C., (2002) The myth of the Turing machine: the failings of functionalism and related theses,Journal of Experimental & Theoretical Artificial Intelligence, Volume 14, Issue 1 , pages 1 – 8. Eliasmith C. and Anderson C. H., (2003) Neural Engineering: Computation, MIT Press. Feynman R. P., and Hibbs, A. R., (1965) Quantum Physics and Path Integrals, New York: McGrawHill. Fukushima K., (1980) Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position, Biological Cybernetics, 36[4], pp. 193-202. Fraser AM., Swinney H.L., (1986) Independent coordinates for strange attractors from mutual information Phys. Rev. A 33, 1134 – 1140. Gardner D., (1993)The Neurobiology of Neural Networks., MIT Press. Gerstner W., and Kistler W.M., (2002) Spiking Neuron Models Single Neurons, Populations, Plasticity Cambridge University Press. Gibbs J. W., (1902): Elementary Principles in Statistical Mechanics, Yale University Press. Goldin, D. Q., (2000) Persistent Turing Machines as a Model of Interactive Computation, Lecture Notes In Computer Science, ISSU 1762, pages 116-135. Goldin D., Wegner, P., (2008) The interactive nature of computing: Refuting the strong Church-Turing Thesis , Minds and Machines 18 (1), pp. 17-38. Gray R.M., (2000) Entropy and Information Theory, Springer-Verlag New York. Grossberg, S., (1976) Adaptive Pattern Classification and Universal Pattern Recoding: I. Parallel Development and Coding of Neural Feature Detectors, Biological Cybernetics, Vol. 23, pp. 121–134. Grossberg S., (1987), Competitive learning: From interactive activation to adaptive resonance, Cognitive Science (Publication), 11, 23-63. Goychuk I Hanggi P., (2000) Stochastic resonance in ion channels characterized by information theoryPhys. Rev. E 61, 4272 - 4280. Häusser M., Clark B.A. (1997)Tonic synaptic inhibition modulates neuronal output pattern and spatiotemporal synaptic integration. Neuron, 19:665-678. Hebb D.O., (1949) The organization of behavior, New York: Wiley Hecht-Nielsen R., (2007) Confabulation Theory the Mechanism of Thought, .Springer Verlag. Heisenberg W., (1930) Die Physikalischen Prinzipien der Quantenmechanik (Leipzig: Hirzel). English translation The Physical Principles of Quantum Theory University of Chicago Press. Horodecki M., Horodecki P., Horodecki R., Oppenheim J., Sen A., Sen U., Synak-Radtke B., (2005) Local versus nonlocal information in quantum-information theory: Formalism and phenomena Physical Review A - Atomic, Molecular, and Optical Physics 71 (6), art. no. 062307, pp. 1-25 Hodgkin A. L. and Huxley, A.: (1952) Quantitative description of ion currents and its applications to conduction and excitation in nerve membranes, J. Physiol. (Lond.), 117.500–544. Hopfield J., and D. Tank (1985) Neural Computations of Decisions in Optimization Problems, Biological Cybernetics, Vol. 52, pp. 141–152.

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Hopfield J., (1984) Neurons with Graded Response Have Collective Computational Properties Like those of Two-State Neurons, Proceedings of the Proc Natl Acad Sci U S A. 1984 May, 81(10): 3088–3092. Hopfield, J., and Tank D., (1986) Computing with Neural Circuits, Science, Vol. 233, pp. 625–633. Izhikevich E. M., (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Jolivet R, Schürmann F., T. Berger, R. Naud, W. Gerstner, A. Roth (2008) The quantitative singleneuron modeling competition, Biological Cybernetics, Vol. 99, No. 4, pp. 417-426. Kaiser A., Schreiber, T., (2002) Information transfer in continuous processes, Physica D: Nonlinear Phenomena 166 (1-2), pp. 43-62. Koch C., (1999) Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press: New York, New York. Kohonen T., (1972) Correlation Matrix Memories, IEEE Transactions on Computers, Vol. C-21, pp. 353–359. Kohonen T., (1982) Self-Organized Formation of Topologically Correct Feature Maps, Biological Cybernetics, Vol. 43, pp. 59–69. Kohonen T., (1984) Self-Organization and Associative Memory, Springer Verlag, Berlin. Komar A., (1964) Undecidability of Macroscopically Distinguishable States in Quantum Field, Theory, Physical Review, second series, 133B, pp. 542–544. Kofuji P., and Newman E.A., (2004) Potassium buffering in the central nervous system, Neuroscience Volume 129, Issue 4, Pages 1043-1054. Khachiyan L. G., (1996) Rounding of polytopes in the real number model of computation,Mathematics of Operations Research 21 307, 320. Landau L. D. and Lifshitz E. M., (1960) Electrodynamics of Continuous Media, Pergamon, Oxford, pp. 368–376. Landauer R., (1961) Irreversibility and heat generation in the computing process, IBM Journal of Research and Development, Volume: 5, (1961), pp. 183--191 (Reprinted in Leff & Rex (1990). Lee R.S. T., Loia V., (2007) Computational Intelligence for Agent-Based Systems, Springer. Landauer, R., (1961) Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 5, 183-191. Landauer R., (1988) Dissipation and noise immunity in computation and communication, Nature 335 (6193), pp. 779-784. Latham A., Paul D.H., (1971) Spontaneous activity of cerebellar Purkinje cells and their responses to impulses in climbing fibres. J Physiol ,(Lond) 213:135-156. Li M. and Vitanyi P. M B., (2007) An introduction to Kolmogorov complexity and its applications. Springer, New York, 2nd edition 1997, and 3rd edition. Lippincott E R. and Schroeder R., (1955) General Relation between Potential Energy and Internuclear Distance for Diatomic and Polyatomic Molecules. I J. Chem. Phys. 23, 1131. Lloyd S, (2006) Programming the Universe: A Quantum Computer Scientist Takes on the Cosmos. Marinescu D. C and Marinescu G. M. (2005) Approaching Quantum Computing, Prentice Hall. Maass W., (1997) Networks of spiking neurons: the third generation of neural network models. Neural Networks, 10:1659-1671. Maass W. and Bishop C.M., (1998) Pulsed Neural Networks, MIT Press. Mainen ZF, Sejnowski TJ., (1995) Reliability of spike timing in neocortical neurons, Science, Vol 268, Issue 5216, 1503-1506. McClelland J.L. and Rumelhart D.E., (1988) Exploration in Parallel Distributing Processing, Brandford Books, MIT Press, Cambridge, MA. McCulloch W. and Pitts W., (1943) A Logical Calculus of Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics 5:115-133. Meir A., Ginsburg S., Butkevich A., Kachalsky S. G., Kaiserman I., Ahdut R., Mesulam, M., (1998) From sensation to cognition, Brain 121 (6), pp. 1013-1052. Minsky, M. and Papert S., (1969) Perceptrons: An introduction to Computational Geometry, MIT Press. Moore C., (1996) Recursion theory on the reals and continuous-time computation, Theor. Comput. Sci., Volume: 162, pp. 23-44. Moore C., (1990) Predictability and Undecidability in Dynamical Systems, Physical Review Letters, Vol. 64, pp. 2354-2357. Nielsen M. A., I Chuang L.I., (2000) Quantum computation and quantum information, Cambridge University Press. Newell A., (1990) Unified Theories of Cognition. Cambridge, Massachusetts: Harvard University Press. O'Reilly C. R. and Munakata Y., (2002) Computational Explorations in Cognitive Neuroscience Paun G., (2002) Membrane Computing. An Introduction. Springer-Verlag, Berlin.

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CHAPTER 4. Models of Brain Computation Transposing biological information into algorithms that mimic biological computations in neurons has proved to be a difficult task and current work in neuronal computation is still influenced by the McCulloch-Pitts model considered to be the first model in neural computation. The firing rate and sigmoid neural activation function are at the origin of implementation of the second generation of neural networks. More recently the spike timing approach is considered the third generation of models (Maass, 1997, Rieke et al., 1997; Gerstner and Kistler, 2002; Mainen and Sejnowski 1995). However, it is already accepted that, as a whole, the brain can be regarded as a physical computer. This is not necessarily a new theoretical concept since many developments in physics and mathematics showed that complex computations can be achieved by moving billiard balls when they are modeled as particles within a dynamical system (Bunimovich and Sinai, 1980). As presented by Moore (Moore, 1990), a classical example shows that a single particle which moves in a threedimensional potential is able to perform computation and is theoretically equivalent to a Turing machine. Computation is, therefore, performed in this case following the Newtonian laws of motion where the state-space system that describes the dynamics of single particles has definite inputs and can generate outputs. It is believed that coding and decoding of information in the brain is governed by classical physics; however, at microscopic level many interacting particles within molecular machines, namely the proteins, may require a different model. Quantum properties seem to occur almost anywhere at microscopic scales (e.g. Brooks, 2005) and sometimes such characteristics can be macroscopically perceived (Julsgaard et al, 2001). Does the brain exploit these quantum features? Undoubtedly the macroscopic outcome recorded as a train of spikes cannot be generated without this infrastructure that obeys the quantum laws of physics. However, when we talk about models, the problem of classic or quantum model seems to be ill posed since no matter how complex, models that are mathematical descriptions cannot fully characterize the real world phenomena. 4.1. Dynamics Based Computation – The Power of Charges The details of electrical patterns within a spike and the relationship with emerging behavior have been observed during T-maze procedural experiments (Aur and Jog, 2007b). Each spike can be reduced to a vector representation (see spike directivity) that is spatiotemporally modulated during learning. This view provides a natural interpretation for vector representation or vector coding (see Georgopoulos et al. 1993); however this model although very useful is an over simplification for computation with charges. Experimental data, neurophysiological recordings, the model of information transfer based on Hodgkin-Huxley description (Aur et al., 2006), prediction about semantics (Aur and Jog 2007a) are consistent with and support the idea that computation and information transfer is carried out by electrical charges.

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4.1.1. Space and hidden dimensions Our common view of space is shaped by a 3D view of the world. Frequently, in physics, time can be added as the fourth dimension. In quantum mechanics or string theory extra dimensions are used to describe new concepts since there is no demonstration which limits any space analysis to three dimensions. The general theory in mathematics is based on the concept of an n-dimensional Euclidian space E n. This ndimensional presentation where n>3 can be extended to every incoming stimulus s(t) or its discretized version s(k), (k ∈ N) which can be seen as a point in an n-dimensional

Figure 1 Schematic representation of a three-dimensional sphere and a three-dimensional cube. If these three- dimensional objects are projected in a single dimension both become lines

space. The point Sk={s(k), s(k+1), ..., s(k+n)} corresponds to n-discrete samples of the signal. Since, usually the incoming stimulus is not constant the variation in signal amplitude can be seen as a movement or a trajectory of the point in space (Figure 2). This system where particles may operate under Newtonian laws can well describe complex computational processes (Beggs and Tucker, 2007). In this case the dimension of processing space becomes critical. A very simple example shows the importance of space dimension selection. If the space dimension is lowered from 3D to 2D and subsequently to 1D, some relevant characteristics required for object perception are lost. It is likely that with dimension reduction, the information required for object recognition is lost. A cube or a sphere projected onto 1D space cannot be distinguished since both objects become lines in the single dimension. If information is lost, the cube or the sphere cannot be reconstructed. This simple example shows that projecting onto higher dimensional space may be a better alternative than reducing dimensionality for solving object recognition problems. Theoretically, the Support Vector Machine is a good abstract example of increasing dimensionality for learning classifiers (Vapnik and Chervonenkis, 1974; Vapnik et al., 1997). o The dimension of processing space is critical in terms of information resources required to obtain/find solutions for certain problems. In the brain it involves electric charges as a way of building a higher dimensional space. This simple thought experiment reveals the fact that the brain seen as a collection of electrical charges that move and interact may hide information since a human observer used to a three dimensional view of the world cannot perceive the depth and richness of higher dimensional processing with charges. These characteristics seem to

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be essential for several processes that occur simultaneously in the brain. In this case the principles of brain computation can be related to an application of fundamental physical principles such as the principle of least action or second law of thermodynamics. Therefore, the physical mechanism of coding should map the incoming signals/features into the dynamics of charges or group of charges in a multidimensional space. The X

S(t) Voltage

Xi

Time

Figure 2: The incoming signal properties can be translated to a distribution of charges in space. An n-dimensional input signal has an equivalent representation in brain as a distribution of electrical charges in space. A multi-dimensional input signal s(t) can be represented in the brain using a spatial distribution of electrical charges

process can be experimentally evidenced by measuring the spatial distribution of electrical activities within certain regions of the brain starting with smaller microscopic scales during spike occurrences at the level of ion channels. These characteristics of many charges that have a certain dynamics show the importance played by unperceived hidden dimensions in the brain for processing since there is no natural constraint to limit any space analysis to three dimensions (Randall, 2007). Additionally, the assumption of a Euclidian space is restrictive and in contrast to the realm of space curvature determined by the presence of charges and electric fields that are likely to build a non-Euclidean universe in the brain. 4.1.2. Minimum Path Description - The Principle of Least Action The minimum description length concept has been discussed in Chapter 3. This principle is used to find an optimal way to describe data or models (Rissanen, 1984; Barron et al., 1998). As shown in the previous chapter this model that describes the dynamics of charges may require a similar mathematical formalism since the physics of charge movement and interaction can be described in terms of universal physical laws. The application of physical principles should provide answers to several questions. What is the trajectory of charged particles? Can we predict electric propagation of charges knowing some constraints? To find a way to express the movement of a charge in physical space we have to exploit the laws that govern energy constraints. For at least two hundred years this law is known as the principle of least action (Maupertuis, 1774; Hanc et al., 2005). The analysis of this dynamic system can be made therefore in terms of a Lagrangian L : L=K −P

(1)

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where P is the potential energy and K is the kinetic energy of the system. If one considers that each charged particle i has the dimensional coordinates ri , then, to express the movement of charges we can introduce the generalized coordinates q1 , q2 ,..., qm : ri = ri (q1 , q2 ,..., qm , t ) (2) Therefore, the trajectory of charges can be obtained by writing the Lagrange equations: d ∂L ∂L , i=1,2…m ( )− dt ∂q&i ∂qi and the evolution in time of the system can be summarized: 0=

(3)

tf

S = ∫ L(q1 , q2 ,...., qm , q&1 , q& 2 ,...., q& m , t )dt

(4)

ti

where S is the action. Another way to express the evolution in time is to write the Hamiltonian as the Legendre transform of the Lagrangian: H (qi , p i , t ) = ∑ q& j p j − L( qi , pi , t ) j

(5)

where qi , pi are the canonical coordinates in phase space (Marsden and Ratiu,1999) The theory known as the principle of least action states that the charged particle r should follow the “worldline” r for which the average kinetic energy minus the average potential energy is minimum and can be written in the form where the optimal *

action S : tf

r r S * = arg min ∫ ( K (r ) − P(r ))dt r

(6)

ti

The “path description” formalism represents, in this case, the physical law that r requires a certain path for the charged particle that has to follow the “worldline” r representing the “least action”. Therefore, using this principle of least action (de Maupertuis 1744, Hanc et al., 2005) we are able to predict the trajectory of the particle/charge. Since the energy is not a vectorial quantity, the easiest way to figure out the trajectory is to consider a simple example where the kinetic energy is zero. In this case the charged particle will move along the force direction: r F = − grad (P) (7) The sharper the gradient slope, the faster the charge movement in the steepest direction since nature selects the lowest possible value of the kinetic energy which is required for the movement between the two selected points. The displacement of the particle or group of particles can indeed be represented as a vector in 3D space (see Chapter2) and the minimum path description highlights the principle where every charge moves only along the selected “worldline”. o The Minimum Path Description states that the electric propagation (charge propagation) follows the principle of least action in selecting the trajectory where the average kinetic energy minus the average potential energy is minimum as long as these charges follow a given topology

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This basic principle can be directly related to changes that occur in dendritic topology, development of synapses, ion channel distribution and neuronal topology along with selective genetic changes at molecular level. Such modifications are results of emergent adaptations required to follow the principle of least action and to represent with a minimum cost certain inputs and stimulus features. Since topological properties of the electric fields created by point charges require a non-Euclidean description, in a Riemannian manifold the curves that correspond to extremal paths (geodesics) are locally of minimal length and then the shortest possible path is connected along the geodesics. o In a Riemannian space the Minimum Path Description follows the curve of minimal length which minimizes the energy function (see do Carmo, 1992). This can be interpreted as a natural extension of the Minimum Path Description to the framework of general Riemannian manifolds and it is likely to be related with morphological changes at neuronal level that allow locally the shortest possible paths for electric charges. Other physical constraints may derive from the concept of free energy (Gibbs, 1873) which reflects the thermodynamic perspective. 4.1.3. Natural Computation –Abstract Physical Machines The idea of computation by a physical system is not new. Recent research accredited the idea that physical systems can compute more than a machine which performs computation based on algorithms (Beggs et al., 2008). Examples of such systems include the abstract physical machines that may compute noncomputable/ computable Turing functions using experimental computational models. Basically any physical or biological entity as a part of nature can be seen as a system which can be transformed, reduced and simulated using algorithmic procedures. These forms of natural computation may include variants of the analogue-digital scatter machine, DNA computing (Adleman, 1994; Kari and Rozenberg, 2008), membrane computation which may display some hypercomputational powers (Calude and Păun, 2004; Beggs et al., 2008). •

Membrane computing

Membranes represent physical barriers for charges and molecules to move and interact. Every charge can escape from this trap due to certain potential configurations. The result is that charges/molecules cross the membrane and determine the nature of neuronal activity in that region. Theoretical membrane computation combines some abstract computing ideas which formalize the biological substrate of (excitable) polarized membranes (Păun 2002; Păun 2000). The power of computation of the membranes consists of using rewriting rules analogous to reactions taking place in the cell that encode local transitions in cells with specific mechanisms (e.g. division mechanism). Called P-systems (from Păun) they can be used to solve problems in an algorithmic manner where computations are sequences of synchronous transitions. These models are equal in power with Turing machines and can be generated by a deterministic Turing machine in polynomial time. The membrane can be seen as an abstract machine: M = ( A, μ , ω1 ,..., ω m , E , R1 ,..., Rm ) (8) that includes an alphabet of objects A , a membrane structure μ , ω1 ,..., ω m the strings that represent the set of objects E present in the environment and a finite set of rules

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R1 ,..., Rm that define the membrane transport properties. Importantly, these rules apply to transport phenomena when an “object” enters or leaves the membrane and can be labeled as ‘uniport’ rules (a, in), (a, out), ‘symport’ rules where both objects ab enter together or leave the membrane (ab,in),(ab, out) or antiport rules when the object a enters and object b leaves the membrane (a, in; b out). Such mathematical rules can formalize changes that occur during polarization and depolarization phases in neurons where ions Na+ and K+, Ca2+ cross the neuronal membrane. Some new membrane models include dynamic membrane structures with rules that define membrane “creation”. Additionally, some new advances in this field incorporate distributed computing models, neural like systems based on membrane development (Ionescu et al., 2006; Chen et al., 2008)

•

Particle Computation

The model proposed by Eberhart and Kennedy in 1995 was inspired from models of social behaviors in birds (Eberhart and Kennedy, 1995). In optimization problems this model seems to perform better than evolutionary algorithms (genetic algorithms). Given a multivariable optimization problem: Minimize f ( x1 , x2 ,...., xn ) (9) x1 , x2 ,...., xn

the solution is found by the moving particles in the search space to find the best solution, i.e. the moving particles perform the searching. The theoretical model assumes an n-dimensional search space S, and the j-th, particle position is an ndimensional vector: x j = ( x j1 , x j 2 ,....x jn )T ( 10 ) where the velocity of this particle is also an n-dimensional vector v j = (v j1 , v j 2 ,....v jn )T

( 11 ) and j=1,2,….N. The matrix of velocities of particles can be updated at each discrete step k: Vk +1 = WVk + c1 A( Pb − X k ) + c2 B( Pg − X k ) ( 12 ) where Pb is the memory of the best solution and Pg is the best position group of neighborhood particles, W is the inertia weight matrix, c1 , c2 two positive coefficients, and A and B two random matrices. The position of particles is obtained: X k +1 = Vk + X k ( 13 ) based on their previous position X k . Besides the standard version proposed by Eberhart and Kennedy in 1995 this technique can be used to find solutions for several optimization problems and since it has received constant interest from scientists, numerous versions and improvements have been recently developed (Parsopoulos and Vrahatis, 2004; Li and Liu, 2002; Coello et al., 2004; Jie et al., 2006) •

Computation with Charges –An Example

As seen above, computation can be seen as an intrinsic physical property of a natural system to solve an optimization problem by searching in the space for the best solutions using the movement of several particles/charges. A simple example considers

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a quadratic function V which represents an electric potential that takes the values of a given function: V ( x, y ) = ( x − 0.5) 2 + ( y − 0.7) 2 − 9 ( 14 ) The optimization problem can be seen as a mathematical problem to find the values x * , y * for which the potential V reaches the minimum value. There are several possible paths that one charge may follow to reach the goal. Here we are interested to reveal a possible physical way to implement such a procedure given the fact that V is an electric potential. In order to find the minimum value for the function V one can apply several theoretical techniques. However, the best answer will be provided by the charge or group of charges with the path that provides the correct solution (the optimal path) following the principle of least action. The electric field across each axis can then be computed: ∂V = 2( x − 0.5) Ex = − ( 15 ) ∂x ∂V Ey = − = 2( y − 1.5) ( 16 ) ∂y ∂V =0 Ez = − ( 17 ) ∂z As an example, when we consider the conservation of energy for every charge qi with mass mi over the x axis we can then write: Wi =

1 2 mi vi + qiV ( x) 2

( 18 )

where the velocity of the charge is: 2 (Wi − qiV ( x) ) mi If there is no dissipation, in terms of conservation energy then dW=0 and: .. dV mi x = −qi = − qi E x dx vi =

( 19 )

( 20 )

In this particular case the electric field generates an electric force that can be assimilated with an elastic force determining periodic movements of the charges. The total energy W can be rewritten: 1 1 2 mv x + k E x 2 2 2 where the oscillatory solution has the form: W=

x=±

2W sin ω x t k

( 21 )

( 22 )

and ω x = k E / m . The above technique can be applied similarly for the y and z axes. In the presence of a dissipation force, the charge will always move to the minimum potential energy and will remain in that position which in fact provides the expected solution of the proposed problem to find the coordinates x* , y * , z * where the potential

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reaches the minimum value V * . Since almost all processes are dissipative then this natural way of searching and finding solutions should work. The time required to find/search the optimal solution and the robustness of this solution are two other important issues. Under dissipative constraints a rapid, stable response can be obtained if several charges are involved in the process. However, if dissipation is weak, once these charges have reached their spatial position for the minimum value, they come close enough to generate an electrostatic force. Since all charges have the same sign, they will move far apart and the process of finding the minimum potential is repeated indefinitely. This simple example shows a natural way of computing with charges which follows the “least action” principle. One can find a similitude between the movement of charges during action potential and this description where several charges move to reach the solution (minimum value for electric potential). There are several important ideas which can be drawn from this theoretical experiment: First, under certain conditions the electric force can be compared (in a parabolic field) with an elastic force. Second, if the number of charges is increased then the space exploration is dramatically improved and the solution can be found faster (parallel search=the power of many) (see Figure 8, a). Additionally, many charges can provide a richer representation and the error in representing the feature/stimulus tends to decrease exponentially with the number of charges involved in the process (Figure 8 b). Third, the random motion of charges usually seen as a noisy process in some cases may help to achieve a better solution (Figure 7) since the charge ci can be anywhere in the threedimensional space. For every individual charge ci or a group of charges ci we can define a function V ( xci , y ci , z ci ) which characterizes the quality of solution reached by the charge or the group of charges ci in providing the minimum electric potential V . If the charge cj is better placed than ci then V ( xci , yci , z ci ) > V ( xcj , ycj , zcj ) One can see that this issue in case of charge computation gives another perspective for noise processes and challenges the current view regarding “noise” in brain processes (see Eliasmith and Anderson, 2003) Importantly, this natural process can be modeled on a computer performing the following algorithmic steps: Step 1: Charges are distributed randomly in space, they are characterized by a mass center position xc, yc, zc, and their velocity vx , vy, vz Step 2: Their velocity and position are computed using a Eq ( 11 ) and Eq.( 13 ) respectively Step 3: Groups of charges are found from the population of charges which are better placed than other charges in the group ( V * ( xcg , y cg , z cg )
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received. Indeed, the solution can be found faster if in time these charges are constrained to follow certain paths. One way to constrain a charge or a group of charges to pursue a certain trajectory is to develop electrical or mechanical barriers with certain configurations. Or the electric field should be configured in such a way so as to constrain the movement of charges. Importantly, the existence of neurons with membranes, receptors, dendrites, axons and ion channels plays this role since channel gating

Figure 3: Schematic representation of electric field lines and the required direction in red for the group of charges in blue color to reach the point in three- dimensional space where the potential has the minimum value V

*

( x, y, z ) .

processes can be considered to be part of this mechanism of controlling the movement of electric charges in space. Other cellular structures including glia also regulate this extracellular flux (MacVicar, 1984; Volterra and Steinhäuser, 2004). As an example the astrocytes provide potassium homeostasis by removing the excess of ions (Nielsen et al., 1997). Since glial cells outnumber neurons (Steinhaeuser and Gallo 1996) their role in charge motion (or flux), homeostasis and computation cannot be ignored. The existence of such refined regulatory mechanisms at the brain scale proves that the dynamics of electrical charges is an essential component in this process and is not solely the attribute of neurons. o Rhythmic processes (e.g. action potentials, synaptic spikes) can be seen as a way of periodically restarting (reiterating) the computation performed by moving electrical charges. o Computation with charges may use the “noise” as a better way to improve the finding/searching strategy in the space. o Neurons with spatially distributed synapses/ ion channels restrict/ constrain the diffusion of charges to certain spatial topologies. o Since the homeostatic processes including ion regulation in the brain is provided by glial cells, a deeper understanding of their role in assisting and maintaining normal ionic environment is essential and cannot be ignored. Therefore, the term ‘neural computation’ doesn’t seem to fully describe the complexity and dynamics of electric charges in brain. This is the main reason why we

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prefer to avoid the term “neural computation” and we think that ‘brain computations’ or ‘brain language’ as a metaphoric expression are more appropriate to describe computational processes in the brain. 4.1.4. Back to Building Brains Seen as movement and interaction of charges, brain computations can be regarded as a set of transformations that map input signals acquired from different sensory systems (audio, visual, tactile) into spatio-temporal distribution of charges at neuron level. Due to their multidimensional intrinsic behavior and parallelism charges are likely to achieve a strong computational power. Such computations could provide the required capability to process complex input signals. These signals/input stimuli tend to be fully reconstructed (decoded) if the inverse set of transformations T-1 is performed. This inverse set of transformations can be carried out even in the absence of the external input stimuli. Theoretically, as a result of bijectivity, this set of inverse transformations carried out by charges could be seen as the basic phenomenon that helps us to remember/recreate previous events, places, names or imagine data, objects. Additionally, it is likely that this distribution of charges and their intrinsic interaction can provide a rich representation of the solution space. While these transformations are physically performed, we would like to further analyze how such transformations can be modeled at the micro scale level. It is already known that every action potential is a result of the high number of charges that move in space with trajectories controlled by the cell machinery, spatial architecture of neurons, T CODE

INPUT SPACE

CHARGE SPACE

DECODE

T- 1

Figure 4: Computation in the brain can be reduced to a set of transformations T between input signals and spatio-temporal distribution of charges. Such transformations may define a morphism between the signal space Sn and the charge space Cm. Brain mapping can be described as set of transformations between the input signal space and spatial distribution of charges which can be homeomorphic if the map is a bijection. The input signals/stimuli tend to be fully recovered after the inverse set of transformations T-1 is performed.

distribution of ion channels and several external constraints including electric fields. Therefore, instead of defining only two distinct states for a neuron (presence or absence of action potential, “Up and Down states”), computation can be described by the

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dynamics and interaction of electrical charges. A systemic approach of the phenomenon should identify the following data: 1. The set of inputs with initial conditions r ∈ R (stimuli, intensity of electric field) 2. The set of outputs obtained during measurements at a certain time (e.g. coordinates of selected charged particles, x ∈ X ); 3. The states of the system that define the properties of the smallest set of variables which characterize the dynamics of the system; in this case the set of positions xi and velocities vi of electric charges; 4. The state transition is determined by physical laws of movement (Newtonian, quantum electrodynamics, Poisson-Nernst-Planck equations) The motion of charges in an electric potential can be simply modeled using the Langevin equation: r r r r dvi mi = −miγ i vi + Fi R (t ) + qi Ei ( 23 ) dt where mi is the mass of the qi charge. The average frictional force that acts on the charge qi is dependent on the velocity v of the charge: r r Fi D = miγ i vi ( 24 ) where γ a coefficient is a function of the viscosity of the fluid. The random collision

term Fi R (t) is added and can be assimilated within as white noise. At a given position r r where the charge qi is located, the force exerted by the electric field is Fi e = qi Ei . Working with large numbers of charges may generate severe computational difficulties, therefore, to simulate the movement of charges and their interaction we have to select r r only a few charges. For every two selected charges at distances r1 and r2 respectively in space the Coulomb interaction depends on the value of charges q1 and q2 and the distance between them: r r r (r − r ) F = kq1q2 r1 r 2 3 ( 25 ) | r1 − r2 | where k= 4πε0 εr where ε0 =8.854x10-12 F/m, εr=80 . If the two charges have the same sign, they cannot come very close since the repulsive force will drive them apart. This feedback model tries to resume the effect of the electric field where charges are mainly driven by the dynamics of the input signals generated by stimuli and/or spatial distribution of other charges within macromolecular formations such as protein structures/polarization. The electric force that acts on each charge (Figure 5) depends on its position xi in the field. Therefore, under certain conditions (electric field generated by other charges) the electric interaction between charges can be seen as described in the above example as an elastic force that drives the charge to a certain r position ri in space. The previous equation can be rewritten: mi

dvi = −miγ i vi + Fi R (t ) + k E (ri − xi ) dt

( 26 )

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FR r

+

kE

+

dv/dt

∫

-

-

V

X

∫

mγ

Figure 5: Schematic representation of

the feedback system which drives the charge qi to

position xi in space. This plot formalizes the effect of a three dimensional potential where computation can be seen to be performed by the movement of charges.

Xb, Vb

Xa, Va

Fe

Figure 6 Schematic representation of an elastic force which occurs as a result of electrical interactions. This force can be repulsive or attractive depending on the sign of charges. For two positive ions the electrostatic force is repulsive

where the elastic force depends on the error in displacement ri − xi and the spring constant k E (Figure 6). If the velocity of the charge is known, then the position xi of the charge qi can be obtained: t

xi = ∫ vi dt

( 27 )

0

In order to simulate this model the simplest way is to use the Euler integration method (Ascher and Petzold, 1998) and the dynamics of charges can be seen as a map: X k +1 = f ( X k , Rk )

( 28 ) where the function f specifies the dynamics of the system, X k and Rk are the set of positions, respectively the set of references at k step ( k ∈ N ). In this context the physical description required to reach the final position in a minimum number of steps

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is related to minimum action principle and can be expressed and easily translated into an objective function: J = arg min ∑ ( Rk − X k ) 2 ( 29 ) k k The number of steps required in simulation to achieve the goal depends on several factors that should include initial position of charges, their initial velocities, and the existence of collisions. Importantly, the analysis of the number of steps shows that collision dependent noise can play an important role and sometimes these random events may allow the goal to be reached faster (see Figure 7) b

a

2.5

2

Z

Z

2 1.5

1.5 1 2

1 2

3

3 1.5

1.5

2 1

1 0

Y

2 1 0

Y

X

1 X

Figure 7 : The representation of a single charge in movement where the probability to reach the goal in a certain number of steps depends on initial position, velocity and the existence of collision generated noise. a. Without noise the charge reaches the goal in 33 steps ( J <0.01) b. With added white noise the charge reaches the goal just in 28 steps

Another factor that may determine how to reach the goal faster is the number of charges involved in this dynamic movement. If the number of charges is increased, the time required to find a certain solution may be significantly reduced since many charges can cover and explore a larger area/volume better. These simulations indicate an exponential decrease of required time to find a certain solution with an increase in the number of charges. However, these charges will require far more energy to move and interact in space. -3

2.2

a

x 10

0.3

b

2

0.29

1.8

0.28

Time[ms]

Error

1.6 1.4 1.2 1

0.26 0.25 0.24

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700

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Figure 8: Searching /finding solutions with particles/charges depends on the number of charges involved in computation. a, The error in feature representation decreases exponentially with the number of particles/ charges b, If the error ε is kept constant the time required to find the solution decreases with the number of particles/ charges that perform the task.

1000

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o

The solution can be found faster if the number of charges involved in computation increases o The increase in the number of charges that perform computation requires a growth in energy resources. o The increase in complexity of problems will always require an increase in computational complexity (Ding-Zhu and Ker-I, 2000) and an activation of larger volume areas in the brain (neuronal circuits). Therefore, for new problems a direct correspondence is expected to exist between computational complexity and activation of volume/area in brain. The trajectory of a charge can be also plotted in the phase space if all possible states of the system are represented. If the system is a charged particle in one dimension, then its position in phase space can be given by the pair ( xk , v k ) where xk is the position of the particle and vk is its velocity at discrete moment k ∈ N . Given enough time, some set of states can be repeated and may define a subset of the phase space, called an attracting set A. The neighborhood of A for the dynamical system is called the basin of attraction B(A). If the pair ( xk , vk ) is an element of the phase space, then ( xk , vk ) totally specifies the state of the system at some instant, k ∈ N . 1.5

a

b

1

1

0.5

0

V

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0.5

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-0.5

-0.5 -1 -1.5 -2

0

2

4

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-1 0

1

X

2

3 X

4

5

6

Figure 9: Phase space representation of a charge movement a, The solution is found in the absence of random input variables b, A slightly better solution can be found if a random, Brownian motion is considered

The subset of phase space, the attracting set A, can be a point, a group of points, a curve, a manifold or a complex shape. The “point attractor” is the simplest type of attractor which may direct the charges to a single stable point (Figure 10). Since the patterns of electrical activity occur in every action potential, they can be mathematically modeled as temporal sequences of point attractors (see Figure 10, chapter 2). It is likely that electrical fluctuations generated by this movement of charges can be reflected in fluctuations of firing rate or spike timing. As presented above (Figure 10) such fluctuations and oscillations can be perceived as a direct way of exploring/searching in the solution space within the volume/area where the solutions for some class of problems can be found while the motion of particles/charges, the attractor model can be seen to be the physical root of electrical patterns.

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4

Z

3 2 1 4 6 2 Y

4 0 0

2 X

Figure 10 Representation of a “point attractor” in three-dimensional space in red. This amortized oscillatory trajectory in space of a selected charge may play an important role in finding/searching solutions

4.1.5. Charge Movement Model - A Computational Approach Brain activity shows complex patterns either in simple spikes or in neuronal ensembles and we have had difficulty understanding what these activities represent. Our ability to interpret or make decisions is generally diminished by the increase in complexity (Zadeh, 1968). Therefore, before moving to a complex framework it is important to analyze fundamental characteristics that define brain computations which may help us refine our understanding of information transmission and computation in brain. Since we cannot assume that brain processing can be simply performed by “uniform pulses of electricity” (Bains, 2003) this subchapter is an attempt to develop new theoretical ideas and to investigate some new models of computation that can better mimic some processes in brain. Within each collection of neurons, cells or their subdivisions one can define a processing space where the movement/trajectories of charges characterize/represent different computational levels carried on in order to find solutions. It is clear for us that modeling with current generation of computers cannot account for the required richness provided by the natural system. However, these models may allow us to better understand the outcomes observed in several experiments regarding dynamics and plasticity of the neural processes including the occurrence of the highly studied spike timing patterns. Our previous approach highlighted the important role played by ion fluxes as main carriers of information. However, it is expected that the number of positive and negative charges within the brain (e.g. electrons) is equal since the brain has to be electrically neutral. Their local distribution in space can be significantly changed for short periods of time and this process of changing spatial charge distribution defines the essence of computation, since as shown above the movement of charges in an electric field is able to provide natural solutions. Therefore, grouping or increasing the density of charges for short periods of time in a specific spatial region defines the coding phase and this step can be considered to be the first phase of computation where information is projected in space by an intrinsic movement of charges as a result of electric field propagation. Too often several assumptions of spike timing theory reject the importance of charges or electric interactions claiming that “neurons do not interact through extracellular fields”. However, the main issue regarding computation is not

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intercellular interaction since most of this processing occurs within every cell. At a fundamental level, local increase in charge density occurs within neurons and can be demonstrated within every spike delivered by a neuron (see Chapter 2). Additionally reaction-diffusion phenomena lead to significant changes in charge concentration. Even in case of chemical synapses information transfer and the interaction between neurons is greatly determined by ion fluxes. Calcium (Ca2+) regulates the process of neurotransmitter release that furthermore allows ions to move in and out determining the occurrence of the postsynaptic potential. Several other phenomena highlight the importance of electrical charges in brain computations: • Important quantities of information are transferred by ionic fluxes during every spike within a millisecond time scale (maximum average values over 89 bits/spike). Using the HH model, this phenomenon has been proved by computing mutual information between input stimulus and sodium fluxes (Aur et al., 2006). These values are more than double compared to mutual information between the membrane voltage and stimulus input. A similar computational model of mutual information was previously used to claim that spike timing and ISI embed information regarding stimuli/behavior. This mainly demonstrates that information regarding stimuli/behavior is distributed in several observable/measurable quantities; however, the significant values of mutual information provide a higher status in this hierarchy for sodium/potassium fluxes. • The study of atypical alterations that occur in terms of changes in local electrical properties as well as information transfer within brain areas can be strongly connected with the occurrence of symptoms of schizophrenia / Parkinson’s disease in animal models (see Chapter 5- Parkinson’s and schizophrenia). • Electrical charges are traditional carriers of information in many biophysical systems (see electron transport chain). The processing of extracellular recordings (e.g. spike directivity) demonstrates that the striatum uses this dynamics of charges to represent semantics during procedural learning in Tmaze tests (Aur and Jog 2007a). • Charge conservation principle states that electric charges cannot be created or destroyed; therefore these charges under a set of conditions represent a valuable resource for computation. Even though in the brain the number of positive charges should equal the number of negative charges their topology/spatial distribution is continuously changing. In its resting state each neuron contains a small excess of negative charges, therefore, an inherent change of density of positive ions (e.g. Na+) close to the membrane may determine voltage gated sodium channels to open and neuron to generate an action potential. We have shown that electrical patterns of activation occur within every spike as a result of diffusion and charge dynamics at micro-scale level within neurons (Aur and Jog, 2006). • Since within the electric field inherent electrostatic interactions play a crucial role at a micro-scale level by directly influencing protein folding and aggregation, the changes in spatial distribution of charges can be directly related with alterations in the internal structure of the cell. Therefore computation with charges is able to resolve the current gap between molecular

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description, neuronal function and spike timing patterns analysed in several experiments. • The electric/electrostatic interactions between charges are extremely important in providing a triple role for charges in brain computations. They are able to read information, transfer information by their movement and store information within a given spatial topology of charges within macromolecular formations (e.g. proteins). • As a consequence of learning, the organization of spatial charge distribution at micro-scale level determines several observable effects at neuronal scale: o The antagonistic surround in the receptive fields observed in the activity of cells of the lateral geniculate nucleus becomes stronger with learning shaping the Mexican hat form in terms of firing rate measure (Ohshiro and Weliky, 2006) o The millisecond occurrence of spikes within neurons (Mainen and Sejnowski, 1995) o The organization of spike timing within neuronal ensembles (Jog et al., 2007) o Overrepresentation, with learning larger areas/volumes are assigned to familiar stimuli inputs (Penfield and Rasmussen, 1950). All these observables can be demonstrated to occur in simulation as outcomes of reorganization in charge distribution: • Seen as particles that have a certain dynamics, these charges during their movement from an initial state to a later state can carry out complex mathematical transformation and serve as direct means for computation. • Several properties attributed to brain/neural computations such as temporal patterns, multidimensionality, parallelism, fuzziness, topological fractality can be naturally seen as direct effects of computation with charges. • Importantly the laws of physics are well known for electrical charges, and there is no need to discover other physical principles in order to build a basic model of computation. All these aspects summarized above show why we need to develop a new, different model beyond the current limited view provided by spike timing framework in order to reflect the inherent natural richness of capacity for complex computations within the brain. The essence and richness of brain computations in terms of charge dynamics and their interaction can be structured within six fundamental principles that form the basis of brain computation. 1. The Principle of Information: To understand brain computations one needs to look at the brain’s fine grained structure that includes electrical charges. Information processing and transfer occur naturally by moving these charges in space (e.g. Na +, K+, Ca2+) intra and extracellularly as a result of electrical wave propagation, selective diffusion process during action potentials, synaptic spikes. Intracellularly, where information processing predominantly occurs, the generated electric field is the fundamental component that allows a fast transfer of information and mediates the interaction between electrically charged particles at a molecular level. Importantly, the information carried by charges does not relate solely to stimuli or behavior. At a fundamental molecular level information carried by electric field and charges can be selectively transferred, and this information operates at gene level to determine intricate regulatory mechanisms required to build appropriate proteins intracellulary. Several

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other intrinsic physiological regulatory mechanisms require information transfer (e.g. homeostasis). 2. Brain Computation Principles: Since information travels with the propagation of electric field and movement of charges, neurons and other cells (e.g. astrocytes) can be considered essential units in this complex process that spatially modulates the flow and interaction between charges. During every AP/synaptic spikes the spatial trajectories of electric field and charges are determined by a selective activation of ion channels and interaction with molecular structures. Computations can be reduced to a set of transformations that define a morphism between the input signal/stimuli space and electric patterns represented by the ‘charge space’. The inverse set of transformations carried out in the presence/absence of the input stimuli may help us to identify or remember/recreate previous events, place names. 2.1 Semantic Acquisition Principle: Computation with electric charges is mediated by many neurotransmitters and neuromodulators which can alter the ion channel function and directly relate neuronal propagation of charges with behavioural or stimulus input semantics (e.g. see changes in spike directivity). This principle provides several answers regarding non-deterministic computations and intrinsic properties of electric field propagation and charge movement. An increased spatial charge density within molecular formations, their emergent spatial topology and interaction with charges in movement is at the origin of distance-based semantics. 2.2 Rhythm Dependency Principle: Specific phases of computation require different brain rhythms. As an example, physical phenomenon of ‘writing’ in memory during sleep time is frequency dependent since changing polarization at molecular level and also conformational changes at protein scale are voltage and frequency dependent events that may determine a preferential expression level of certain genes. 2.3 Fundamental Thermodynamic Principle of Computation: The understanding of thermodynamic nature of computations relies on the analysis of thermodynamic entropy. The transmembrane movement of charges is correlated with changes in thermodynamic entropy and information transfer during spike activity (action potentials, synaptic spikes). Reading and writing or deleting information at molecular level has to follow the fundamental thermodynamic principle (e.g. second law of thermodynamics, minimum entropy production principle, Landauer principle). Computation has a thermodynamic cost that has to be paid when information is erased. 2.3 The Principle of Coding: A charge or a group of charges that move during an AP/synaptic spike are able to read (decode) and write (code) information at or in the same time. The role of the “Turing tape” is played by molecular structures including ion channels where the redistributions of charges within macromolecular formations determine changes in polarizations, conformational changes in proteins and selective gene expression. These physical processes briefly described above are carried out by several repetitive electrical phenomena (synaptic spikes, action potentials). These reiterating electrical phenomena are essentially modulated by changes that occur at the molecular level. 2.4 The Principle of Decoding: The patterns of external events, stimuli are directly reflected in changes of electric patterns that occur in neurons determined by the dynamics of electric charges. This “reading” of information (about stimulus, or behaviour) is altered by a continuous interaction with already existent data stored in memory (in terms of conformational states, polarizations). This view can simply describe the basic mechanism of perception where the input information from stimuli is modulated by previously accumulated and stored information in memory.

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2.5 The Principle of Parallel Processing and Multidimensionality: The richness of computation in the brain relies on parallel processing performed by the movement and interaction of many types of charges. Automatically, parallelism and multidimensionality occur directly in terms of physical laws that govern the motion of charges. Synchronous neuronal firing can be explained as a requirement to increase interaction and the number of charges involved in information processing in a particular region which directly increases the power of computation and decreases the required time to find the solution (see above section 4.1.4). Thus, local environmental charge enrichment occurs in order to increase computational efficiency. 2.6 Global Computational Principle: The richness of computation with charges is lost if the inputs/stimuli/other processing are seen as local disconnected events, analyzed in vitro in brain slices. Computationally, the dynamics and interaction of charges involve the whole brain as an entire machine tuned to unravel the emerging behavioral aspect or the semantics of incoming stimuli. 2.7 Biomolecular computation: At the molecular scale biological information follows a direct path from DNA to RNA through to protein synthesis. It is likely that a reverse path in information flow is possible through various types of interactions at the molecular and electric level required by an appropriate gene selection for protein synthesis. This reverse path involves electric interactions and provides a feedback mechanism critical for maintaining, consolidating and generating new memories. 3. The Principle of Representation: “Constructs at ‘macrolevel scale’ (image representations, movement control) are results of emergent dynamics and interactions between electric charges that alter and build charge configuration at molecular scale. From motor output to visual processing, mental representations are essentially built from smaller to larger scales by a spatial distribution of charges which temporarily increases their density in specific regions and can be seen then as natural, physical solutions (electrical patterns within action potentials, synaptic spikes). At a molecular level spatial distribution of charges is modulated by changes that occur in gene selection for new protein synthesis which contribute to an intrinsic memory formation. 3.1 The Principle of Optimality: During learning the computational process in the brain is organised to achieve the required representation of a certain feature with the minimum number of electrical patterns (charges) in minimum time. The “minimum path description” follows the physical principle of least action and has in this case to be adapted to the cell’s and ion channel’s morphology. This process can be evidenced in terms of spatio-temporal dynamics of charges within neurons and neuronal ensembles. Spikes (action potentials, synaptic spikes) can be considered as ‘discrete’ steps that carry out in time this optimization process based on an active fundamental loop of molecular computations (changes in polarization, folding mechanism, changes in gene expression) able to provide the required dynamic adaptation. 3.2. The Learning Principle: Learning represents a natural solution for reducing energy expenditure required to solve repeatable, complex problems. The rhythmic occurrence of an increase in spatial distributions of charges determines in time alterations/organization of charges within macromolecular structures (e.g. proteins) which in the long term tend to preserve information regarding the metric properties of represented scenes, objects or events. This process is deeply rooted within specific bimolecular mechanisms of ‘computation’ that include gene selection, regulation and expression related to DNA-RNA transcription, protein regeneration and new protein synthesis in response to a wide variety of extracellular signals and electrical interactions. As a result learning changes occur at the microscopic scale and determine

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effective changes in molecular-genetic mechanisms of protein biosynthesis, proteinprotein associations mediated by electrical interactions and then determine organization/re-organization of spike patterns observed/measured by common electrophysiological recording techniques and revealed as forms of learning (habituation, sensitization, classical conditioning). With learning the electrical maps related to frequently used inputs become overrepresented (see the hand and the mouth in the Penfield 'homunculus'). 4. Scalability Principle: The model of the dynamics of charges has the property to function and effectively accommodate at different scale/sizes from macromolecules/ proteins to neurons and networks of neurons where these models can be built and analyzed. (See the computational cube, Chapter 4) 5. Quantum Computing Principle: Computation with charges occurs naturally at the microscopic scale within macromolecular formations. When one goes deeper at the nano-scale level, various quantum phenomena such as time-dependent tunneling, Coulomb blockade, stochastic resonance can be considered to physically drive computations and information exchange. This quantum scale is possibly the most fundamental level of computation. 6. The Unity Principle: The electrodynamic model of computation unifies current observables at macro scale level (spike timing patterns, firing rate) with macromolecular changes in terms of spatial distribution of charges that occur at ion channel and protein level (within neuronal membranes, inside synapses) and their dynamics. Based on these six principles we have to build a new model of computation which should better reflect what happens in brain. In order to be simulated on current computers we are forced to use a reductionist approach and these models are simplified and discretized. Therefore, only some of the relevant aspects of computation with charges can be exposed in the development below. o The coding/writing phase is a process where information regarding an input feature/stimulus/behavior is embedded in a spatial distribution of charges that can be seen at different scale levels from AP to macromolecular structures, proteins. An example of this new model of computation at the neuronal level provides the solution in terms of charge distribution that is “read” by the dynamics of other charges in certain spatial regions defined as synaptic boundaries, gap junctions and ion channels. Due to synaptic and non-synaptic connectivity and high impedance in these structures, neurons are able to “read” this spatial distribution of charges. The phase of receiving information regarding spatial distribution of charges can be seen as the “reading” or decoding phase, a phase where information is received by neuronal structures (e.g. synapses) that spatially lie or present certain type of connectivity in a definite specific region. o The decoding/reading phase is a process of acquiring information regarding the spatial distribution of charges The increased number of charges involved in this process and their spatial trajectories reflect the multidimensional aspect of computation. In fact McClelland and Rumelhart in their seminal work (McClelland and Rumelhart 1988) stepped up to the next level of analyzing firing rate and missed this fundamental “lower level” where parallel distributed processes occur as results of dynamics and interaction between electric charges.

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Despite common thought regarding noise, the random movement of charges during the process of diffusion (Brownian motion) can drive charged particles to appropriate and sometimes faster solutions which may increase the performance of the computing system. Incorporating all these observations into an algorithm should allow us to describe mathematically and simulate the main characteristics of computations in brain on current computers. This separation in two main phases (coding and decoding) is presented here only for modeling purposes. In reality the process of computation is unlikely to be strictly separable in phases since reading and writing information are continous and simultaneous processes driven by the movement and interaction of charges. There are some important differences between the current algorithm and how computation emerges in biological neurons. The movement of charges during AP/synaptic spikes and their interaction has two main purposes; to read and to write information and both activities are simultaneously achieved. Therefore, there is no real separation between the coding/writing information and reading/decoding information as presented above when one considers a biological system. Another issue is to artificially introduce the input features by another group of charges. Such separation between charges that “model” the input and charges that “compute” does not exist in the brain and is introduced just for demonstrative purposes. This model of computation with charges has to incorporate the most essential characteristics of the system; it should include the spatio-temporal distribution of charges, spatial distribution of synapses/neurons and importantly the dynamical aspects of computation. The computational model presented below is in two forms, a static mapping model and a dynamic mapping model representing a general framework where the main goal is to build a theoretical approach that can explain brain computations starting from spike occurrence including synaptic role and learning. This model can be modified in several ways to fit data from brain recordings and other experiments; however it remains just a model which cannot replace the real brain functions. In other words quoting Stevan Harnard, “the virtual brain is no more able to think than a virtual plane is able to fly” (Harnard, 2001) •

The Static Mapping Model

The recent progress in machine vision faces several challenges. The ability of current computers to recognize classes of objects based on their shape is as yet unsolved. Standard optical devices provide just a partial view of objects; therefore solving the problem using partial information is difficult due to large variations in shapes and uncertainty. The more intuitive example is to design a model to process and recognize visual stimuli; therefore, we have chosen the object recognition problem as an example. Since every electrical phenomenon (action potential, synaptic spike) can be reduced to movement of electric charges then the model should use as an input feature the distribution in space of electric charges. At this micro-scale we have to consider just one component of a larger input pattern of activity. For simplification we consider a computational cube where the input features I 0 = (i1 , i2 ,...., i p ) , ( ik = ( xk , yk , z k ), k ∈ N , (e.g. letter E, Figure 11) are shaped by an arrangement of charges.

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Decoding – Reading Phase In this model the decoding phase is a process of reading the distribution of charges that can be performed by electric activity within macromolecular structures (e.g. proteins) or at “synaptic” level, “gap junction. This phase is a simplified model of the role played by synapses, the macromolecular structures in computation. Therefore, it is considered in a reductionist approach that “reading” or decoding at neuronal level preferentially occurs where synapses are located in space by the movement and interaction of electric charges mediated by neurotransmitter release. In this case, the synaptic activity is initially modeled by the existence of a charge/group of charges spatially positioned within the computational cube. The main idea is first to find the ‘correct’ spatial location of synapses where these charges occur and diffuse and second to model the ‘synapse-specific plasticity’. These characteristics can be analyzed by using an algorithm following the steps:

S1

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Figure 11: Schematic representation of a computational cube. The input feature is presented at input as a distribution of electric charges that shapes the letter E. Initially, the synapses from a neuron/group of neurons s1 , s2 ,...., sm are modeled as being randomly positioned in the three-dimensional space.

Step 1: Guess the initial spatial geometry of synapses S 0 = ( s1 , s2 ,...., sn ), n ∈ N where their position is defined by Cartesian coordinates x, y, z within the computational cube s1=(x1,y1,z1), s2=(x2,y2,z2)…sn=(xn,yn,zn). Step 2: Compute the n-dimensional feature vector R = [r1 , r2 ,...., rn ] received by these synapses: 1 rj = ω j ∑ , j = 1,2,..., n ( 30 ) 2 k ( s j − ik ) where the above formula is a way to account for interactions between synapses and charges, ω j ∈ Ω , where Ω = [ω1 , ω 2 ,...., ω n ] is the synaptic vector, ω j ∈ [-1,1] are

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initially arbitrary random weighting coefficients and the input vector I i represented by the coordinates of charges as the input feature. These coefficients (weights) that compose the synaptic vector Ω as well as the spatial location of “synapses” are subject to plastic/elastic changes during the learning process. Importantly, one can observe that the obtained input feature vector R characterizes the spatial distribution of charges and incorporates nonlinearity as a charge interaction effect (see Eq. 36, Ch 3). o Nonlinearities often introduced in connectionist models can be seen as natural dependencies of physical interactions between electric charges. If the input charge distribution is close to the synapses then the flux of charges can be triggered and mediated by neurotransmitters. Mathematically this behavior can be modeled by selecting a certain threshold θ and the synaptic activation occurs when the threshold is reached ( norm(R) ≥ θ ). Since one neuron often “connects” synoptically to more than 10,000 postsynaptic neurons this huge n -dimensionality becomes an issue 1 is neglected then the model is reduced for computer simulations. If the term ( s j − ik ) 2 to a connectionist framework and several properties of computing with charges are lost. The positive/negative coefficients ωi in Eq. ( 30 ) can be naturally seen as a result of physical interaction between groups of charges distributed in space with different polarities . •

Coding Phase

The schematic representation of the coding phase uses the feedback system described above which drives the charges to the required position in an n-dimensional space using an optimization algorithm. A genetic algorithm (Goldberg, 1989; Koza, 1992; Fogel, 2006) or a particle swarm algorithm as presented in the example below can be used as an optimization step to find the optimal position for charges that are able to represent the input feature R .This coding phase generates a spatial distribution of charges/particles and builds a certain representation for the input feature vector R . The coding phase can be summarized into the following steps: Step 3: Guess the initial spatial charge geometry and initial charge velocities. Define an initial random position for the charge i as n-dimensional vector X i = ( x1 , x2 ,..., xn ) and an initial random velocity of charge i as Vi = (v1 , v2 ,..., vn ) in an n-dimensional space Step 4: The velocity of N charges at step k + 1 can then be rewritten in the matrix form: Vk +1 = ΑVk + Fk + K E ( R − X k ) ( 31 ) where Vk represents the velocity matrix at step k , Α is the inertial matrix that in this case also models the interaction between charges, Fk is the random noise matrix and the difference R − X k behaves as an elastic force that projects charges in an ndimensional space. Step 5: Since these charges are projected in an n-dimensional space ( n ≥ 3 ), it is necessary to integrate the equation to find the new position of charges in space: X k +1 = ΕX k + Vk ( 32 )

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where Ε is an inertial/interaction matrix and X k is the matrix of previous position at step k of charges. Step 6: Project the n-dimensional vectors of charge position within the three dimensional cube using an iterative process across dimensions (Normand et al., 2005). High density of charges in a certain region is analogous to high density of opened ion channels. Once projected in a three – dimensional space the increase of charge density in a certain spatial region as a result of the coding phase can be synaptically decoded/ read by other “neurons”. Selecting initial velocities for charges/particles may adjust the time scale for finding the solution. However, the essence of building this simple model is to demonstrate how input feature representation can be spatially generated by increasing the density of charges/particles. o Feature Representation: Specific input features are represented by an increase in spatial density of charges that occupy different position in space within the computational cube (see Figure 12) Interestingly, this simple model may also show that information regarding a feature/stimulus can be stored within a certain spatial distribution of charges and this specific aspect will be analyzed in detail in the next chapter and related to memory •

The Basics of Effective Learning

To reduce the time needed to find the solution in simulation/brain will always require more charges to be involved in computation. Additionally, it is likely that complex problems always require more resources translated in more charges resulting in many neurons to be involved. o Learning represents a natural solution for optimizing energy expenditure required to solve repeatable complex problems. This is achieved in time by a reorganization of spatial distribution of charges at micro-scale level where the trajectories of charges are controlled to provide rapid solutions. o Intracellular processes and extracellular milieu determine a dynamic distribution of electric charges, therefore learning can be modeled as a method by which the intracellular distribution of charges and synapse location play the game together. The dynamics and the plasticity of processes developed in neurons during learning can be algorithmically resumed to error driven task learning which can be performed in two main phases that combine fast and slow-learning steps. •

Fast Learning Phase

Given a synaptic vector Ω that models synaptic strengths and a spatial positions of synapses S 0 , the fast learning phase can be interpreted as an energy minimization process performed by a the group of charges within a neuron or group of neurons during a train of successive p action potentials that minimize the mismatch error between the input feature R and spatial position of charges X : * X = arg min ∑ ( Rk − X k ) T Q( Rk − X k ), k ≤ p ( 33 ) Xk

k

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where Q is a positive definite matrix. Even though here we explicitly refer to action potentials, this model can be extended to synaptic spikes given the scalability property of the charge movement model. Step 7: Check if the spatial distribution of charges provides an accurate model for the input. In mathematical terms this can be translated to check if the matrix norm satisfies the inequality: norm( Rk − X k )T Q( Rk − X k ) < ε ( 34 ) where ε is a small positive constant. Repeat steps 2 to 6 until the condition of Eq. ( 34 ) is fulfilled. If the minimum * X is a global minimum then the position of charges represents an optimal solution for the given input. The speed of finding the optimal value can be improved if instead of the norm, the algorithm uses the memory of a best solution Pb and the best performing particle in the neighborhood Pg following the particle swarm optimization algorithm (Eberhart and Kennedy, 1995). The final position of charges depends on the outcome determined by the fast learning LF algorithm that minimizes the difference between the input feature vector R and spatial position of charges in an n-dimensional space: ΔX = LF (|| X − R ||) ( 35 ) During this fast learning process, the information about the characteristic features of the input data is therefore transferred and stored within a spatial distribution of charges which are at the origin of distance-based semantics. Physically this phenomenon should be further related to the process of writing this information within macromolecular configurations and new synthesized proteins that are related to memory formation. Besides this fast optimization stage, there are plastic phenomena that can change the “strength” of synapses by adjusting the synaptic matrix Ω and the spatial position of synapses S 0 . •

Slow Learning Phase

In the brain, there is an almost continuous process of synaptogenesis. This process includes an explosion of synapse formation which is more active during early stages of brain development. The slow learning phase includes plastic phenomena at synapse level including changing their strength Ω and spatial positioning. Often these changes at synapse level are experimentally observed to depend on the relative timing of preand postsynaptic action potentials and are modeled as forms of Hebbian learning. Related to depression or synaptic potentiation, these models are known as spike timing dependent plasticity STDP (Gerstner and Kistler, 2002; Song et al, 2000). However, such rules in “strengthening or weakening” of synapses and their relationship to temporal windows are not general and are expected to depend on several factors including changes in polarizations, dynamics of charges and their interaction at molecular level (Bi and Poo, 2001). Synapse formation and synapse elimination in the brain allows us to transpose this mechanism into an algorithmic phase where learning is seen as a process of changing the spatial position of synapses and their ‘strength’ as a result of changes in polarizations within neuronal systems at the macromolecular level that may occur on a millisecond time scale.

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Step 8: Therefore, changes in position of charges can be followed by alterations in the strength Ω of “synapses” and changes in spatial position S 0 of synapses which should provide optimal values for a selected performance index J : J Ω , S = arg max J *

*

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( 36 )

(see the performance index as a measure of discrimination of input features Eq. ( 39 ) below). Finding the optimal values for S * and Ω* can be performed using genetic algorithms (Goldberg, 1989; Koza, 1992; Fogel, 2006). The algorithm provides the best values for S * and Ω* that is tuned for a certain class Ck of inputs. This step can be performed in the context of a certain task such as recognizing a class of objects or performing specific movements. The changing of values in the synaptic matrix works as a typical learning algorithm. This is driven by error signals as the difference between the distribution of charges after the coding phase (neuronal output), synapse position in space and synaptic strength. The input feature vector R depends on the outcome determined by the slow learning algorithm Ls that adjusts the spatial position of synapses S and their strength Ω : ΔR = LS ( S , I , Ω) ( 37 ) Information about the characteristic features of input data is therefore transferred during the learning process and stored in the ‘synaptic weights’, spatial distribution of synapses and in this case distributed charges in space are at the origin of distance-based semantics. •

Computational Cube – An Example

The power of computation with charges can be demonstrated using the steps provided above within a bounded cube space following a pattern recognition task. The image is acquired considering as the input the position of several charges placed in the feature space - the top surface which generates a letter or a number (see Figure 12). The concept behind the computational cube can be included in the larger group of earlier developed compartmental models (see Hines and Carnevale, 1997). o Computational cube: The computational cube is a theoretical design that can be used to explore the dynamics and interaction of electric charges and can be regarded as a compartmental model. While compartmental models try to relate experimental data with quantitative modeling the computational cube reveals some relevant qualitative aspects of computation as presented below. The computational cube as a theoretical approach can be seen at different levels. At the most fundamental level the computational cube can describe electric processes within proteins. At a macro-scale level, the computational cube can describe processes within a neuron where charges move and interact during the action potential/synaptic spikes, depending on spatial distribution of ion channels at the level of the soma, within dendrites or axonal branches. This model represents a viable alternative to the well described compartmental models (Koch and Segev, 1998) by avoiding some detailed compartmental modeling. o Scalability Principle: The computational cube can be implemented at different scale/sizes in order to model computations performed by electric charges; o At the most fundamental level the computational cube can describe processes of coding and decoding of information at micro-scale level within

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macromolecules. The model can be extended at macro-scale level to describe computation within neurons or ensemble of neurons; o Within a computational cube, the high density of charges that occurs as a result of the learning process carries information regarding specific features of stimuli or behavior. At macro-scale level this information regarding the dynamics of charges is “read” by neighboring neurons synaptically and non-synaptically. This algorithm requires first a normalization of the coordinates and input data. As described above the entire procedure is a two stage computational process of coding and decoding. Two groups of charges are considered. The groups of charges that generate the input features (stimuli/memory) are positioned in space (top of the cube) with shapes that for simplification in this example generates letters (e.g. E or C) which are sequentially presented (Figure 12). The main group of charges then performs “searching” for a spatial distribution that best describes the input feature. During the fast learning phase, the charges tend to cluster to a certain spatial position that better maps the input features and therefore only certain ‘synapses’ that are spatially placed better are likely to be selected to transfer this information. In this model a good description is obtained when the error between the input feature and spatial charge distribution is at a minimum. If the origin of spatial measurements is considered to be one corner of the cube, once distributed in space, the position of charges can be represented by a vector that points to the location where the center of mass of the charges lies. In this example only the shape of the letters is considered as input feature, however, other different characteristics of selected objects (color, texture, position) can be considered and mapped in a distribution of charges. Importantly, if the model is defined at a neural scale (hundreds of micrometers) then the distribution of charges describes the outcome of intracellular processing and is similar to the one we have previously computed in chapter 2 for the spike directivity vector. Therefore, in this case r the input letter E, can be described by the vector VE . After the fast learning phase is completed, in a similar manner for letter C, the position of charges can be characterized r by a different “spike vector” VC . The angle between these two spike vectors α EC is computed as the inner product: α

EC

r r VE VC r = ac cos r || VE || || VC ||

( 38 )

as a form of semantic distance. o During object presentation information regarding different input features is dynamically micromapped in different areas of neurons in an explicit topographic manner. o Semantic distance: Topological similarity is provided by a metric attribute (angle, Euclidian distance) that characterizes the information regarding emerging behaviour or stimulus input based on the resulting spatial distribution of charges (see also the feature representation) o Computation is mediated by many neurotransmitters and neuromodulators which can alter the ion channel function and directly relate the propagation of electrical waveform and charges with behavioural or stimulus input semantics (e.g. changes in spike directivity).

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The above principle provides several answers regarding intrinsic properties of charge dynamics and their required spatial distribution. Intracellularly, increased charge density and an emergent topology around synapses is at the origin of semantic representation and expressive power of non-deterministic computations. After several classes of inputs Ci , i = 1,2,...h , including letters, numbers are presented; the coding phase drives the charges within the computational cube in such a way that they uniquely describe different input features if the inner product of spike vectors is maximized during learning. This performance can be better achieved if the strength and spatial position of synapses are adjusted during the “slow” learning phase. Therefore, the learning algorithm built in two phases maximizes the performance index J 1 and allows a higher discrimination of input features based on spatial distribution of charges. If several distinct letters are considered as input features (E, C, I ….) then the position of synapses S 0 and their “strength” are adjusted to maximize the sum of several angles α IE , α EC , α IC between several “spike vectors”

VC VE αC

αE

Figure 12 Representation of the computational cube. The input features are considered to be letters or numbers which can be represented as stationary charges that shape the letter or the number. In this case letters E and C are successively presented. During learning another group of charges moves (appears) inside the computational cube driven by input features onto spatial positions that should uniquely describe the characteristics of the input.

J 1* = arg max( | α 1IE | + | α 1EC | + | α 1IC | ...+ | α 1MN | ...) Ω,S

( 39 )

where α XY is the angle between spike vectors when letters X and Y are presented at input. It is likely that the accuracy of separating classes of inputs depends on the dimensionality of the input data and the number of charges involved in the processing phase. Results This algorithm that is able to discriminate several input features may help us to understand the outcome generated by learning processes including spatio-temporal

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observables recorded from the brain. When we talk about observables we understand several categories of data. These data should not be limited to spike timing properties in single neurons or ensemble of neurons/ networks. They should include spatial distribution of synapses, spatial distribution of charges, and finally, temporal properties of “spikes”. The discrimination of input features increases with learning (Figure 13). This increase is almost ten times higher if both inhibitory and excitatory synapses are considered (Figure 13, b) instead of when only excitatory synapses are introduced in the algorithm (Figure 13, a). If the same input patterns are repeatedly presented, information regarding the input features is embedded during learning as expected within the strength of synapses Ω* and in their spatial position S * (see: Figure 14). Additionally, we have computed the probability distribution function for the charges ∑ ( X 0 − X i ) 2 where X 0 is the mass center of points and we found that the i

distribution of charges in space, after learning, embeds information regarding the input feature (Figure 15). The analyses of spatial distribution of charges over the mass center, seen as charge points, show that after learning charges are not uniformly distributed in space anymore. Therefore, input information is embedded during learning into a resulting spatial distribution of charges (Figure 15) that can determine the occurrence of ‘spike directivity’. Importantly, this process is likely to determine changes that occur intracellularly at the most fundamental level everywhere within macromolecular structures (e.g. proteins) including changes in synapses. Similar features are micromapped together, topographically distributed in computational cube and display cluster configuration alike to ones observed in neurons. o Spike directivity changes with spatial distribution of micromapped electric features in selected neuron and reveals the process of computation. The main effect of spatial organization of charge distribution occurs at macro-scale levels as electrical patterns observed within recorded action potentials (see Fig.9, Chapter 2).

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This observation of intracellular spatial distribution of electrical charges is critical in the context of information reading by other neurons that are synaptically connected (Figure 16). o Regularities in spike timing occurrences with millisecond time precision are a result of spatial organization/re-organization of charge distributions at microscale level within macromolecular formations during the learning process. This observation shows that spike patterns can be obtained as a result of a model where computation is performed by charges/particles and their dynamics. Regularity in spike timing occurrences with millisecond time precision is one of many properties of computation with charges. Therefore, such regularities in spike timing are unlikely to express the essence and richness of brain computations. 0.03 0.025

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VE αE Figure 16 During every spike information propagates by moving electrical charges (e.g ions Na+,K+). The distribution of charges in the extracellular and especially within intracellular space has important consequences of how information is received “synaptically” and non-synaptically by other neurons located in the region.

o

The level of abstraction may be extended from single neurons to neuronal ensembles activated by stimuli or behaviour. Therefore, simultaneous occurrence of electric patterns within two or more different topological brain areas can be the result of concurrent/parallel processing of dissimilar stimulus feature inputs or behavioural outcomes. o Cognition occurs as a result of emerging categorization (Harnad, 2005) seen as an intrinsic evolving process developed during learning as an organization of spatial distribution of charges at molecular scale within neurons. o Importantly, in the brain this representation is not limited to physical concrete entities it includes semantic categories, abstract concepts related to input stimuli or behaviour. The algorithm of coding and reading information has to be repeated several times in a similar way as it occurs during APs/synaptic spikes where the electric flux is rhythmically generated within a neuron or in an ensemble of neurons. o Firing rate (mean instantaneous frequency) gives an estimate of the average frequency of coding/decoding certain set of features in a neuron. o With learning, information regarding frequent stimuli or behaviour is spatially distributed and overwritten within selected neurons High density of similar charges/ions may naturally generate a fluctuating outcome at microscopic scale since electric forces tend to disperse similar charges that are spatially located very close to each other. The Dynamic Mapping Model Why do we need a dynamic mapping model? Input signals are continuously received and there is no pause between input presentations as we have artificially generated in the above simulations. Therefore, the computational model has to provide a “continuous integration” of input signals allowing a constant processing of stimulus or input features. Since the coding stage determines a concentration of charges in a

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certain volume then to follow the physics of charges we have to include a repulsive 1 force in the above model which can be expressed by Γ = − . Therefore, norm( X T X ) Eq.( 31 ) can be re-written in the form: Vk +1 = ΑVk + Fk + Β( Rk − X k ) + ΓVk ( 40 ) The electrodynamic system is schematically represented in (Figure 17) and the result of the coding phase is again a non-even distribution of charged particles in space that can be computed using Eq. ( 32 ). Due to repulsive forces these charges are redistributed in space and the process of coding/decoding is automatically reiterated. This cyclical behavior allows rhythmic coding and decoding of the input signal. Therefore, computation performed by electric charges consists in repeated successive steps of codingdecodingcodingdecoding….. and so on. If an electrode is closely placed in this computational cube it will record an electrical activity that is similar to a train of action potentials followed by pauses or “interspike intervals” (see Figure 19). o Computations carried out by the dynamics and interaction of electric charges in synapses within neurons or group of neurons can be seen to perform successive repeatable cycles of coding and decoding phases. Since energy is required for any physical process, this rhythmic behavior of charges can then be seen in terms of fluctuations and conversions between potential and kinetic energy. The potential energy can be computed: 1 1 P= ∑ ( 41 ) n i ( X i − X )2 where X is: X =

1 ∑ Xi n i

( 42 )

The kinetic energy is computed based on their velocity Vk : T

K =|| Vk Vk || ( 43 ) The conversion between kinetic energy and potential energy follows a cyclical behavior and therefore under some energy requirements this entire physical computational process can be rhythmically repeated.

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This dynamic model however raises some new issues. In this case the main issue is with regard to the decoding process or more specifically the timing when information has to be “synaptically” read. This is an important problem which algorithmically can be related to the occurrence of a certain density of charges in a certain volume and this issue also requires a dynamic process for information reading. Since high density of charges can be translated to low values of potential energy, the algorithm can be adapted to perform a “dynamic” reading at the critical moment when information of input is embedded within a certain spatial distribution of charges. o An efficient reading of information requires that ‘synaptic values’ Ω that model the intrinsic conductances change dynamically. The advantage of a dynamic model is that computation can be cyclically performed. Therefore, if the input stimulus changes in time this new model is able to follow and “read” the changes of the input (Figure 18).Theoretically an optimization algorithm can provide required parameters for a selected performance index. Genetic algorithms can be used for optimization purposes since they provide ways of solving optimization problems by mimicking natural evolution as a mutation and selection of genes (Fogel, 1994). The selection of a certain set of “gene string” allows finding the optimal values for the synaptic strength and their spatial position that minimizes the performance index J (see Eq. ( 36 )). Ω,So

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Figure 18: Schematic representation of the dynamic system where during slow learning phase a genetic algorithm changes the values of synaptic strengths Ω and their spatial position So

However, when a similar feature is presented at input, the same information is again reprocessed. This problem can be solved subsequently since this issue is related to information storage and memory mechanisms (see Chapter 5). As was experimentally demonstrated the strength of a synapse diminishes if the train of APs arrive in rapid succession in a so-called frequency dependence of synapses (Deisz and Prince, 1989; Galarreta and Hestrin, 1998; Galarreta and Hestrin, 1999) The main issue in this case is to estimate the moment when it is required to read information since coding also occurs cyclically. The increase in the density of charges in a specific region is a transitory phenomenon which for very short time provides “the solution”. “Reading” information at the neural level is played by synaptic structures capable of supporting computations based on “highly structured temporal codes”. To

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model this aspect we have investigated changes in kinetic energy and potential energy of charges. It is likely that a solution occurs in terms of spatial distribution of charges when the kinetic energy reaches a minimum value and the potential energy attains maximum values. Such analysis allows knowing the information synaptically delivered by other neurons by “reading” the spatial distribution of charges, rhythmically. a

b

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Figure 19: Spike like behavior within a computational cube a. Representation of cyclical behavior of electrical activity determined by charges in the computational cube b. The occurrence of “spikes” becomes organized with learning and the time of spike delivery achieves millisecond precision.

Introducing a dynamic model has important consequences regarding some temporal observables since spike timing occurrences organize also with learning and the neuron/network generates temporal patterns with millisecond time precision (Figure 19) which have been previously analyzed in many recordings (Mainen and Sejnowski, 1995). o During learning the information regarding stimulus input/behavioral outcome is transferred by electric charges and determines regularities /organization of spatio-temporal spikes. Spike timing models (see Gerstner and Kistler, 2002) have considered this temporal information and added a new dimension to the previous parallel distributed processing (PDP) models approach developed as a connectionist framework by (Mclelland and Rumelhart 1988; O'Reilly and Munakata 2000). However, the richness of physical computation is once again lost since spike timing models do not consider computations performed by electric charges, their interactions. Mediated by neurotransmitters, these computations performed by electric charges are at the origins of temporal patterns. Seeing the cube presented above as an abstract model of computation at neuron level, the dynamic model may incorporate at each synapse, a subcube level which follows the same principles of rhythmic coding and decoding of information as described above. Besides the ‘remote’ input (shaped by the E charges) at each subcube level which in this case may have little impact (electric interactions depend inversely on the distance where charges are located) the flow of charges within the sub-cube is determined by nearby concentration/distribution of charges inside the cube (see Figure 20) as a result of previous computations. The set of subcubes positioned in space at S 0 = ( s1 , s2 ,...., sn ), define in this case the activity at synapse level. This new model can be further extended since every subcube may include several sub-subcubes distributed in space and this third level can model the dynamics of electric charges at ion channel level within macromolecular formations in proteins. The spatial distribution of subcubes and sub-subcubes inside the computational cube can

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V αE

V αE

V αE

V α

E

Figure 20 Representation of successive parallel steps of coding and decoding at multiple level scales. Within every computational cube/subcube the coding phase involves the movement and interaction of electric charges in space while the decoding phase ‘reads’ this spatial distribution

define hierarchical structures that function using coding/decoding techniques that can model the activity in neurons and ‘synaptic connections’ and can be seen as natural extensions of multi-compartmental models of neurons (Hines and Carnevale 1997, Hines and Carnevale, 2001; Gunay et al., 2008). Importantly, these new ‘compartmental models’ as coding-decoding structures can be built or simulated using current computers and do not require supplementary assumptions regarding spatial isotropy or electrical homogeneity as needed in current cable models. Ensemble of Neurons within Computational Cube This description of computation can be limited to a few cells or ensembles of many neurons to implement complex processes that involve composition of primitives, hierarchical structures. Such behavior can be naturally developed by several units/neurons that share the same computational cube and adapt to an explosive growth of dimensionality in the input data. Each neuron in the computational cube performs coding based on the received “synaptic” inputs which accounts for input data. As in the previous example, for every neuron in the cube, the input is determined by the presented letter/number and results after learning in a slightly different distribution of charges within the cube. If N units/neurons are considered, the slow learning phase can be seen to be built in order to adjust the strength of synapses Ω1 , Ω 2 ,…. Ω N and their spatial position S1 , S 2 ,…, S N and maximize the performance index J N : *

JN =

N

arg max( ∑ | α

Ω 2 , Ω 2 ,...Ω N , S1 , S 2 ,.... S N

k =1

XY k

|)

( 44 )

where α 2XY is the change in the angle between the “spike” vectors generated by two units when two different features X an Y are presented at the input. Finding the minimum values for this performance index builds the required set of transformations

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that map the input stimuli within a spatial distribution of charges and determine certain temporal patterns of activation which commonly organize during the learning stage. Additionally, some unit/neurons may consider as inputs the distribution of charges determined by coding output PX, PY since these charges are distributed within the computational cube and the performance index can be rewritten: λ

JN =

N

arg max( ∑ | α

Ω 2 , Ω 2 ,...Ω N , S1 , S 2 ,.... S N k =1

XY k

| + λ | α kP P |) X

Y

( 45 )

where λ is the fraction of unit/neurons that consider the distribution of charges that results after coding. The presence of several cells in the cube allows for an increased number of charges to receive slightly different inputs and to perform computations that can be useful to code by spatially redistributing many charges. Besides this spatial coding, the results of simulation show that information regarding the shape of the input is incorporated within the temporal pattern of “spikes”. Representation of probability distribution of “spikes” shows that a part of information regarding the input is indeed presented in the temporal occurrences of spikes. If one analyses the distribution of ISIs for these computational units one may find several distributions that resemble the theoretical random Poisson point processes that are alike with the ones obtained from spike trains analysis from brain recordings (Nowak et al., 1997; Song et al., 2000). Additionally, central to a recognition process, the composition of primitives in this case can be seen in terms of unions, intersections of spatial regions where the density of charges increases within the computational cube as a result of the coding phase in “neurons” (Figure 21). It is likely that this spatio-temporal distribution of charges and their resulting interactions is the fundamental phenomenon that shapes the spatiotemporal spike occurrences in real neurons and ensembles of neurons and their highly heterogeneous response to input stimuli observed in multi-neuron recordings (Yen et al., 2007; Tolhurst,et al., 2009).

Figure 21 : Separation between letter E and C during the learning process

Therefore, the spike timing model (ISI, firing rate) currently used to explain temporal aspects of brain computations can be shown to be a particular case of the

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proposed model where charges perform dynamic computations during their spatialtemporal movement. o The entire computational theory of spike timing can be seen as an approximation of a more general model of computation with charges. Importantly, an input image (e.g. I E for the letter E) can be decoded or remembered as long as information regarding this input image is shared by an ensemble of N neurons and is electrically generated. Therefore, the image I E can be retrieved from electrical patterns of action potentials in a given time interval Δt : t + Δt

I E = T −1 ( ∫ t

N

mi

i

j =1

∑∑ ω

ji

Π V dt )

( 46 )

E ji

where mi represents the number of selected electric features Π V registered in the r neuron i (for corresponding spike directivity VE ) when the subject first view the Eji

ij

-1

input image E, ω ji is the synaptic matrix and T represents a set of transformations from which the input signals/stimuli are recovered from spatial distribution of charges (see Figure 4). This phenomenon of remembering images has been recently experimentally investigated. The free recall always selectively reactivates the same ensemble of neurons that were active before when the subjects first view the presented images (Gelbard-Sagiv et al., 2008). 4.1.6. Properties of Computation with Charges Computing based on charge dynamics/interaction is a natural way of finding a solution for complex problems. Inspired from AP driven events, computation with charges is performed within every cell/neuron and this new framework provides several answers versus classical neural models. The biophysical mechanism of synaptic transmission where ion fluxes are extensively involved extends this model of computation with charges at neural ensemble level. Semantics occur naturally as a temporal conglomeration of charges that can be theoretically explored using already developed formalism of fuzzy approach and granular computing (Lin,1999; Pedrycz et al., 2008). •

Fuzziness and Granularity

Previous analyses considered the chain of chemical processing (Signal Transduction Pathways) as the basis of fuzzy operations in the brain (Freitas and Pereira, 2005). Extending this earlier attempt to explain fuzziness in processing information in the brain, the fuzziness property occurs naturally as an intricate information processing developed with charges (Figure 22). The probability that a charge or a group of charges occupies a certain spatial position at a given moment can be described in terms of numerical uncertainty and can be naturally seen and well modeled as a fuzzy property. o While probability defines a subjective state of knowledge (Jaynes, 2003) fuzziness describes uncertainty related to event ambiguity. Both measures characterize the uncertainty numerically with values between 0 and 1. The ambiguity or vagueness is specific to any physical phenomenon and becomes more evident within a quantum description. Using probability measures in the real world proved to be a difficult way to account for uncertainty.

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The probabilistic approach should satisfy the requirements of a probabilistic model which associates with the values of probabilities pi . This issue is solved differently in a fuzzy description. For any set X of parts, a membership function on X is any function f from X to the real unit interval [0, 1] ( f : X → [0,1] ). The membership functions μ (X ) on X can be defined as a fuzzy subset of X ( μ ( X ) ⊂ X ) and therefore the membership functions are context dependent. Let X = (x1,…, xn) be a set of elements. A classical probability measure P on X is a mapping P : 2 X → [0,1] . The values of probabilities pi are used to uniquely determine P(A). Specifically, P(A) is given: P ( A) = ∑ pi ( 47 ) i

0.06 0.04

Z

0.02 0 -0.02 -0.04 0.1 0 Y

-0.1 -0.2

0

-0.1

0.1

0.2

X

Figure 22: Fuzziness in the feature/stimuli discrimination. The charges/particles that represent three different inputs share some common spatial areas/volumes

The concept of entropy can then be used to calculate the uncertainty associated with such a probability distribution (Shannon, 1948). The probability measure was extended by Zadeh (1968) to assign probabilities to the fuzzy subsets of X: P( A) = ∑ μ ( xi ) pi ( 48 ) i

where μ ( xi ) is defined as the membership grade of xi . Furthermore, for the special case where μ = xi , then P(A) = pi . As an example if two fuzzy sets are described as a set of geometric points (charge points in the above example- Figure 21) within two unit cubes A and B then the standard fuzzy set operations such as fuzzy union can be written: A ∪ B = max( A, B) ( 49 ) where the result is the volume in space of the two cubes. Similarly for fuzzy intersection: A ∩ B = min( A, B) ( 50 ) the outcome is the intersection of the two cubes. o Fuzzy logic allows high dimensional exploration space for computation that is in most cases a critical requirement for information processing.

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This standard set of fuzzy operations makes sense for current description of brain computations carried out by charges. Operations such as contraction μ 2 or dilation μ 0.5 can be naturally seen to occur as emerging properties of computation with charges as a result of increasing or decreasing the density of charged particles in certain spatial regions. In a fuzzy description, neurons or networks of neurons within a computational cube can be seen as “granules”. Generally, the density of charges in specific regions can serve as a physical model of granularity. Therefore, the computational universe can be seen as a set of n cube grains Ki. By measuring the percentage of particles that belongs to a certain cube Ki, the membership function value μK(x) can be obtained and is called the membership degree of x in the fuzzy set K. A combination between the charge movement model, fuzzy and granular computing provides a solution for how membership functions can acquire meaning. Such solutions can be obtained automatically and can be modified to acquire new knowledge, concepts during learning. o The granulation process (Zadeh, 1997) can be seen in the brain as an intrinsic development of computation with charges from micro-scale level to larger ensembles scales As a property, granulation allows different representations of the same problem to have different levels of details. Granulation assumes the separation of this computational universe into subsets by grouping individual components/ clusters. On the other hand organization involves the integration of these parts into whole that is performed by physical movement interaction and agglomeration of charges. One may select some typical “cells” from a granule as being representative. Instead of searching for a single optimal solution, this framework where charges are seen as points provides just a good approximation of the solution. Switching among different granularity levels, between the coarse grain and the fine grain may allow us to analyze various hierarchical structures and emerging related semantics. Similarity or distance functions can then be used to define the relationships between abstract “objects”. As an example the distance between two input features F1 , and F2 (e.g. letters E and C) can be written as sets of points in an n-dimensional space: d ( F1 , F2 ) = Max

Min || x

x1∈F1 , x2 ∈F2

1

− x2 ||

( 51 )

where ||.|| is the Euclidian distance. The distance d can be regarded as an alternative of the semantic distance computed above. Shannon entropy measure can be additionally used as a measure of the granularity of the partition. Based on computation with charges it is possible to develop a conceptual model of information processing of the human brain starting from this granular approach. Importantly granular computation is a growing research field for expressing semantics using rich rules (Zadeh, 1997; Lin 1999). Separate pieces of information can be combined based on belief functions and plausible reasoning. It is likely that this combination of algorithmic and non-algorithmic information processing of granular computing is able to mimic human intelligent response (Bargiela and Pedrycz, 2006). Since the dynamics of charges, their interaction and clustering is a reflection of computation in brain, this fuzzy approach makes the required step to reduce the gap between the connectionist view and symbolic framework. Additionally, this approach makes a general computation theory from granular computing that mimics algorithmic

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and non-algorithmic processing in brain. Certainly understanding how brain computes using the dynamics and interaction of charges should serve as a positive feedback on developing this new field. •

Fractal Properties

Spike timing models of computation merely ignore spatial or morphological structure of neurons. Usually spatial architecture of dendrites, axonal branches, is considered to be irrelevant for many computational models (e.g. HH model). Additionally even the computational cube pays little attention to these structural details. However, spatial dendritic and axonal morphology is less random than once believed. Statistical analyses show the possibility of an intrinsic homeostatic control and regulation during development (Samsonovich and Ascoli, 2006) that may influence specific information processing and possible pathophysiological processes in neurons (Buckmaster et al., 2004). Electrical patterns observed at small scales within neurons are likely to be the result of using ion channel activation as a method to move and allow an interaction of charges during every AP. These patterns that occur within neurons resemble patterns seen at large scales in neuronal ensembles. Several anatomical examples show the presence of a large number of stereotypical neurons and brain modules grouped in microcircuits with highly fractal structure (Liu et al., 2005) and suggest that brain signals have fractal properties (Popivanov et al., 2006) . This fractal structure observed at several scales in the brain shows that once derived as an appropriate structure or a suitable mechanism, the structure/mechanism is repeated several times with small variability to complete certain brain functions (Figure 23, Figure 24). Artificial neural networks are typical examples of fractal principle application. Each artificial neuron is considered to be the basic unit in designing the artificial neural network. However in this case the computation is reduced to some simple sets of mathematical functions for implementation. The idea of McCulloch and Pitts (McCulloch and Pitts, 1940) was that an artificial brain may be assembled by connecting similar computational modules operating at a very abstract level. However, to obtain a powerful system such connectivity should be built based on physical principles that allow natural interactions between components. Computer vision aims to understand how images are perceived, processed and recognized in different conditions such as noise, distortions or changes of luminance (Poggio et al., 1985).

Figure 23: A fractal structure made by repeating squares and triangles. They show similar shapes of trees or branches of trees which can be similar to neuronal dendrites and axonal branches.

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All observed images in nature are subject to geometric distortion depending on their relative position to the observer. Additionally, the apparent dimensions of the real object geometry are automatically scaled or rotated. Any distorted image can be obtained by using a set of complex nonlinear transformations.

Figure 24: Fractal occurrence in a selected simulation using the charge movement model. The electric potential displays fractal properties (a similar image at a different scale is in the blue rectangle).

A broad variety of geometric transformation such as contraction, expansion, dilation, rotation can be obtained within a linear transformation: x = Ax + B ( 52 ) In this simple case the matrix A can describe pure rotations which can be combined with scaling properties while matrix B can carry pure translations. As an example, a triangle can be transformed into any other geometric shape by an affine transformation. This shows that a simple linear transformation can carry a combination of translation, rotation, scaling and/or shearing. As an example building neuronal architecture can be easily described in terms of fractal topology. These shapes can be directly generated using simple linear transformations and the results are several different shapes of artificially generated neurons (Figure 25). This simple observation may have a deep impact in understanding the propagation/occurrence of charges and their interaction in neurons, especially within dendritic and axonal branches. o Neuronal morphogenesis is a critical process that shapes dendritic and axonal growth and arborizations. Local changes in neuron architecture reflect requirements for the minimum path description principle. The existence of complex three-dimensional shapes allows complex fractal transformations during computation with charges. In brain the axonal, dendritic growth in a defined direction, the formation of branches and dendritic spines to form connections with presynaptic axons is likely to be additionally regulated by the densities of charges and concentrations of diffusible ligands (Tessier-Lavigne and Goodman, 1996). Therefore, once a point of interest that represents a certain feature is determined and reflected by an increase in the density of charges, the propagation of electrical charges follows the selected morphology of the neuron and can carry all kinds of transformations as described above (Figure 26). As an example information regarding scaling and rotating properties of the red diamond is carried out between the two points where the densities of charges increase during an AP.

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Figure 25: Schematic representation of several fractal structures that model arborescent shapes of neurons

Figure 26: As a result of electric propagation the action potential can be seen as a succession of micromaps of high density of charges (blue-grey areas) that dynamically map a set of features within dendrites, soma and axon and carry information regarding fractal transformations (e.g. rotation combined with scaling properties for the representation of the red diamond).

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o

Fractal property: The sequential increase of the density of charges within neuronal structures such as dendrites, soma and axons during every action potential can be seen to carry out information regarding fractal transformations. This example shows that some computational characteristics can be intrinsically developed based on spatial shapes of dendrites, axonal branches. One can imagine the resulting computational power if this simple linear transformation as described above Eq.( 37 ) is replaced with complex fractal shapes determined by real neuronal architecture. o The shape of neurons including dendritic arborizations and axonal projections affects information processing in neurons and networks of neurons This fractal model may be further related with experimental mappings of dendritic morphology and axonal projections obtained from neuroanatomical analyses (Ascoli, et al., 2001) in order to evaluate the connotation determined by neuronal morphology variability. To achieve such propagation these electric charges under the influence of electric field and due to intrinsic interactions perform series of complex maps/transformations that are likely to provide greater computational capabilities than the modulations in “firing rate” or spike timing. •

Attractor Property

Several scientists proposed memory formation in terms of attractor states or continuous attractors at network level using spike timing approaches (Amit, 1989; Hopfield, 1982; Skarda and Freeman 1987; Sompolinsky et al., 1988, Wills et al., 2005). Such interpretation should indeed be related to memories of series of complex features or series of events if they can be interpreted as a reflection of charge dynamics with their distribution at a micro/macro scale level. Repulsion and attraction are two main characteristics of charges that have the same sign and respectively the opposite sign. o The attractor description is a natural, physical model in terms of spatial distribution and interaction of charges. Therefore, ions or group of ions of Na+, K+, can be considered as point attractors for negative charges in physical space. If such physical attractors can be naturally defined at this level as an effect of interaction between electrical charges, then the movement and spatial distribution of electrical charges and their interaction can indeed be directly connected with the required model of memory implementation. 4.2. Quantum Model – Combining Many Worlds One of the most delicate problems in science is to model the relation between the macroscopic and the microscopic world. Quantum physics is an extremely successful model at the microscopic scale. Several phenomena in nature such as atomic spectra, solids’ behavior, properties of superfluids have explanations within quantum mechanics theory (QM). Without a quantum model, molecular bonding, the structure and function of DNA would not have received any coherent explanation (Home and Chattopadhyaya, 1996). Newtonian physics assumes that velocity of objects and their precise location can be always determined. Despite this strong belief, Heisenberg proved that at atomic level

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“physical reality” is different and there is always incertitude in a precise determination of position and velocity of particles. Additionally, De Broglie showed that particles under certain conditions behave as waves. A corresponding wavelength λ can be associated to any particle in movement. At macroscopic scale this wavelength is imperceptible. For example, a ball with a mass of 2 kg with a velocity of 10 m/s has only a wavelength of 3.3 x10-35 m. If the mass is reduced to a microscopic level, this wave effect is observable in particularly designed experiments (e.g. diffraction, two silt experiment). The macroscopic world is described by classical physics while the microscopic one is governed by the laws of quantum mechanics. 4.2.1. Many Worlds in a Single Spike The role and the importance of quantum phenomena in explaining brain processes have been pointed out recently by (Kurita, 2005). If and how the “classical world” emerges from quantum mechanics is given by analyzing the decoherence process. The decoherence converts quantum entanglement into classical correlations (Zurek, 1991). Tegmark showed that ionic coherence during AP is destroyed by a collision with other ions, collision with water molecules or by Coulomb interactions with more distant ions (Tegmark, 2000). However, in Tegmark’s description only the axon is viewed to be involved in AP delivery, cell membrane and dendritic tree are considered passive elements. Additionally, if the quantum phenomenon is reduced to ions being inside or outside the membrane (electrons are not considered), then this description indeed is highly altered by a fast decoherence process and there is no question regarding physical approach provided by Tegmark. However, extending physical laws to biological domain has to be carefully made. Results from recent observations may show “two different types of decoherence”: one type that may occur in living things and another one specific for the non-living entities (Thaheld, 2005). In physics this subtle frontier between the quantum and the classical world was revealed by Ueda and Leggett (Ueda and Leggett 1998). Julsgaard showed that a macroscopic population of caesium atoms can be quantum entangled for periods of up 0.5 milliseconds, at room temperature (Julsgaard et al., 2001). Several processes within neurons develop at mesoscopic scale. If one considers the flow of ions and electrons at this scale their movements are between few nanometers to hundreds of microns. At this scale where ionic masses of Na+, K+ are of order 10-26 kg the interaction between charges can be stronger and all processes could be highly interdependent. However, this Coulombian interaction is approximately 80 times lower in the water than in the air due to a higher water permittivity. Therefore, it is expected that an experimental setup and a careful analysis of recorded data during neuronal spikes may be able to provide some details regarding the presence of quantum phenomena. Despite the estimated values of decoherence rates (Tegmark, 2000) several theoretical approaches indicate a quantum specific behavior. The gating of ion channels has been intensively investigated over the last 10 to 15 years and a number of models have been proposed for gating based on electrons controlling the ion channel gates (Ralston 2005). Single-electron tunneling with long time-constant inactivation may also provide a simple mechanism for producing burst-mode oscillations in L-type calcium channels. Therefore, the explanation of physical phenomena may involve a quantum formalism that could include electron tunneling as a simple mechanism for controlling ion channel gates.

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Analyzing spatial propagation of spikes with a charge movement model we found evidence that electrical spatial patterns of activation during each action potential AP may hide the existent quantum phenomena. For a classical analysis with the charge movement model (Aur et al., 2005) the quantum outcome is less visible. However, using a filtering ICA approach (Aur and Jog, 2006) we are able to show that the peaks in off-diagonal elements of the density matrix may reflect a phenomenon of coherence that can be modeled using a quantum formalism. 4.2.2. Spike Models –A Quantum Formalism During each AP charges including several ions (e.g. Na+, K+, Ca2+) and electrons move in space under the influence of electric fields and concentration gradients. Classical dynamical quantities can be expressed as a function M of position r, moment p and time t (Figure 27). The probability of a charge particle to be located within a certain region τ ⊂ R at time t is equal to: 3

∫ | Ψ(r, t) | τ

3

2

d 3r

( 53 )

and in all R space is:

∫

R3

| Ψ (r, t) |2 d 3 r = 1

( 54 )

The state of the charge particle can be represented by the normalized vector | Ψ > in the Hilbert space. Then the expectation value of the Hamiltonian H (r,p,t) can be associated with a quantum mechanical operator Hˆ . The expectation value of the observable quantity H is given by: < H > = ∫ Ψ (r , t ) * Hˆ Ψ (r , t )

( 55 )

where Hˆ should be hermitian (Griffits, 1995) and can be written as an inner product: < H > = < Ψ | Hˆ | Ψ > ( 56 ) since the observable quantities in quantum mechanics are described by hermitian operators. A measurement of an observable O of several electrical charges in state | Ψ > will have to return an eigenvalue λ. For several N charges the wave function can be written in the form Ψ (r1 , r2 ,..., rN , t ) where r1 , r2 ,..., rN represents the coordinates of each charge. The correct interpretation of Ψ is based on statistical meaning. For N charges in movement| Ψ ( r1 , r2 ,..., rN ,t)|2 represents the probability density for charge 1 to be at the position r1, charge 2 at r2 . . . and charge N at rN. Therefore, the charge flux in each spike could be described as a quantum ensemble, which consists of large number of charges N. If a measurement/observation O is made on every one of these charges, approximately piN will yield a value λi where pi defines the probability to be in a certain state. For a sufficiently large N the deviation from p i will be small. In a quantum framework the states are given by the eigenvalues Λ of the observable O : O | Ψ >=Λ | Ψ > ( 57 ) If one considers the computational approach from (Aur et al., 2005) where X matrix represents the trajectory coordinates of the spike, then O =XTX is hermitian and then the quantum formalism can be further developed for each recorded spike.

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Y

Z

dτ

X

Figure 27: Statistical interpretation of the wave function requires that the squared modulus of the wave function is interpreted as the probability density, p(r,t), for the position of the charged particle. Therefore, the probability of a charged particle to be located in a small volume element, d τ =dxdydz, at position r at time t, is p(r,t). The region in which the particle can be found becomes larger with time if the measured momentum has a precise value.

The density matrix for a source of quantum mechanical states is accessible by measurement and is supposed to contain all information about the quantum mechanical state of the system. A quantum system defined by a superposition of several states | Ψ1 >, | Ψ2 >,..., | ΨN > with probabilities p1 , p2 ,..., pn can be characterized by the density matrix of the system in its pure state realization given by: ρ=

∑p

i =1, N

i

| Ψi >< Ψi |

( 58 )

The result of any measurement of a quantum system described by the state vector ψ is always one of the eigenvalues of the operator. The computational details of spike directivity calculations for each recorded spike are performed using the charge movement model (Aur et al., 2005). Using the charge movement model, for each spike “measurement”, we computed three eigenvalues λi and three state vectors ψi. One can predict for each spike the probability of obtaining each of the possible results using: pi ( Spike in state Ψi ) =

λi2 ∑ λi2

( 59 )

i

The algorithm for computing the density matrix for a selected spike based on charge movement model is (Figure 28): a. Determine the trajectory coordinates of the spike X, b. Compose the observation matrix O =XTX, c. Determine the eigenvalues λi and state vectors ψi for O , d. Determine the probability pi of the spike to be in state ψi , (Eq.( 59 )), e. Compute the density matrix ρ (Eq. ( 58 )).

Chapter 4: Models of Brain Computation

X

O=XTX

λi

ψi

pi

ψ

141

ρ

i

Figure 28 A: reference scheme for computing density matrix

Figure 29: Two schematic representations of action potentials. A deep analysis of spikes at microscopic scale reveals an intrinsic quantum nature (adapted from the Journal of Neuroscience Methods). a. The representation of spike based on the movement of charge model. The spike trajectory in black and computed directivities of the corresponding three charges (red, blue and magenta arrows). b. Representation of an action potential after applying ICA technique, where each spike is perceived as a flux of N charges. Data processing in this case shows an electrical pattern which may reveal an underlying quantum phenomenon.

For a selected spike sk the values of probabilities and states are given in Appendix1 and the corresponding density matrix is: ⎛ 0.0456 - 0.0122 0.0920 ⎞ ⎟ ⎜ ρ CMM = ⎜ - 0.0122 0.0150 - 0.0218 ⎟ ⎜ 0.0920 - 0.0218 0.9394 ⎟ ⎠ ⎝

( 60 )

Due to several phenomena including the decoherence process this measurement does not reveal a possible quantum phenomenon and the spike can be seen as charges in movement that obey classical laws of physics (Figure 29, a). However, if a filtering ICA algorithm is performed as in (Aur and Jog, 2006) the position of electrical spatial pattern of activation X N _ ch arg e is obtained. After following the same steps for the spike sk where ON _ ch arg e = X NT _ ch arg e X N _ ch arg e the computed density matrix is: ⎛ 0.1822 0.0224 0.3593 ⎞ ⎜ ⎟ ρ N _ ch arg e = ⎜ 0.0224 0.0257 0.0796 ⎟ ⎜ 0.3593 0.0796 0.7921 ⎟ ⎝ ⎠

(61 )

This model corresponds to an N-charge movement model where the spike is seen as a superposition of N moving charges (Figure 29, b). The size of the off-diagonal elements shows that particles can be seen in a coherent superposition of states.

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a

b 1.2 0.8

1 0.8

0.6

0.6 0.4

0.4 0.2 3 0

3

0.2 2

2 1

-0.2 3

2

1

0

1 3

2

1

Figure 30: Spatial representation of the density matrix for a given spike represented above in Figure 29 a. b.

For “three charge movement model” the off diagonal elements are small showing the presence of decoherence and the movement of charges can be approximated with a classical model. For N charge model the peaks away from main diagonal may be attributed to a quantum coherence process where charges can be seen in a superposition of states.

The environment induces decoherence which is a fast process and therefore with a ‘three charge model’ we are able to see just a decoherent state (Figure 30a). Since the spike is a result of charge interactions at microscopic levels, after the ICA technique is applied in case of the ‘N-charge model’ it is likely that we are able to perceive the intrinsic quantum nature of the event since the off-diagonal elements support the existence of subtle phenomena of coherence within the analyzed spike (Figure 30b). Since coherence seems to be hidden one needs special processing tools to reveal the subtle frontier between quantum and classical worlds. The common perception of the surrounding universe is a classical one. This above example may also shed light on why in our daily life we are not able to filter and observe quantum phenomena. However, this shows that phenomena in Nature cannot be separated and once we explore the microscopic levels these quantum phenomena may indeed occur. It is likely that at the microscopic level during spike activity, communications within gap junction, synaptic connections may not always induce decoherence or not in the same degree as it is induced by the measurements with recording electrodes. Based on this analysis at microscopic scale we claim that these phenomena can be described using quantum models where spikes may act like memory registers for quantum computations driven by charge fluxes. This interesting application of ICA filtering shows that the quantum phenomenon may be hidden to our observation. However, such quantum occurrence may be responsible for some intrinsic characteristics of brain computations where processed information is not lost in the “heat bath”. Such subtle observations cannot be revealed within a classical framework and therefore nothing should stop us from modeling these processes as emerging quantum events. 4.2.3. Quantum models In a quantum formalism the charged particle will explore all possible paths between initial position and final position. In this case the description of a selected path has a probabilistic interpretation. Every state is described by a quantity called wavefunction or probability amplitude. A particle starting at a certain point A at time t A will arrive at point B at time tB with the square of the probability amplitude (Feynman, 1964). The total amplitude can then be formally written as the sum of the amplitudes for each possible path:

Chapter 4: Models of Brain Computation

∫e

−

143

i S h

( 62 )

all paths

where h is the reduced Planck's constant, S is the action for a specific path: S = ∫ Ldt

( 63 )

where L is the Lagrangian of the system. In case of classical systems one can show that amplitudes for all paths which have very different phases cancel out by taking the sum, except for those that are extremely close to the path with minimal action. A quantum system can be modeled by Schrödinger equation: ∂ψ (t , x) i = H ( E )ψ (t , x ) ( 64 ) ∂t h where x is the coordinate, ψ is the wave function and H is the Hamiltonian of the system: ( 65 ) H =K +P+M where K is the kinetic energy, P is the potential energy and M is the dipole moment. As shown above in a classical approach, finding of specific paths for charges to move can be interpreted as an optimization procedure or as a learning mechanism. During learning, changes in the path of charges occur as a way to select the “minimum path”

r

for charges. In order for the charge to follow a certain trajectory, the electric field E has to be controlled to minimize the function: J ( E ) = − ∫ψ * (t , x)Qψ (t , x)dA ( 66 )

where A is the boundary region and Q is a hermitian matrix. The cost J can be considered to be the sum of all path costs, which takes the form of an integral over time. A similar description can be found in the description of Pontryagin's minimum principle extensively used in optimal control theory (Iyanaga and Kawada, 1980) Since the charge or the group of charges have an initial position, the problem of “minimum” path description can be reduced in terms of computational geometry to connect two points in the Euclidean space eventually with energy constraints by the shortest path.

S1

B S2

A

S3

Figure 31: For a particle to move from point A to point B there are several possible paths. Learning can be regarded as an optimization process that drives the charge/particle over the “minimum” path. Particularly, in this schematic representation it is assumed that the selected “wordline” S2 satisfies the requirements (S2<S1 and S2<S3)

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o

Either in a quantum or a classical framework, learning can be seen in terms of wiring, re-wiring schemes where synapses are created in the network of neurons to optimize the path of charge flow. Such a view can be related to a Minimum Path Description principle that may explain and extend the mathematical framework developed earlier by connectionist models. This process of “path selection” clearly has to induce in time structural changes at molecular level that determine changes in synaptic plasticity and result in different ‘connectivity’ of dendritic and axonal branches of wired neurons. o The synaptic function allows the transfer of information between neurons carried by electric charges and modulated by several neurotransmitters The idea of building quantum models requires depicting phenomena associated with neuronal processing by adapting well-known models of quantum computing to describe changes of individual electric dipole moments (Penrose,1997; Hameroff and Penrose, 1996; Hameroff and Tuszynski, 2004). At molecular scale these processes are seen to occur at fundamental level of protein synthesis from DNA (Karafyllidis, 2008). The mesoscopic/macroscopic quantum states can be formally described by considering the state vectors for the two different “spikes” generated by two different input features χ1 , χ 2 for the neuron k. For the first spike, the state can be written in Dirac bracket notation | ψ kχ > as a superposition of basis states | ψ i > : 1

k

| ψ kχ >= ∑ α i | ψ i >

( 67 )

1

i =1

and for the second spike | ψ χ 2 > k

| ψ kχ >= ∑ β i | ψ i >

( 68 )

2

i =1

where:

∑ (α )

2

i

i

= 1, ∑ ( β i ) 2 = 1 i

( 69 )

To formalize the coding mechanism one can define propagators to describe how this wave propagates from their initial configuration. The decoding/reading phase can be modeled by a synaptic quantum reading that gives the measured wave function: | ψ m >= ∑ α i | ψ i >| si > ( 70 ) i

where si are the orthonormal states of synaptic quantum measurements (Beck, 2008). If two different features are presented at input the learning process should maximize the inner product between the resulting “spike” vectors: < ψ kχ | ψ kχ > = ∑ β * jα j ( 71 ) 2

1

j

where * denotes conjugation. If several “N” ‘’spikes’’ are involved in the process, learning tries to minimize the performance index: N

J = arg min ∑ < ψ kχ | ψ kχ > 2

α k ,βk

k

1

( 72 )

At a macroscopic scale, the spike directivity vector can be considered to be an observable where the resulting spikes can be written using the formalism of a two state system: | Ψ (t ) > = a | ↑> +b | ↓> ( 73 )

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145

where the squares | a |2 and | b |2 give the probability to detect the “neuron qubit” (quantum bit) in the state | ↑> for upward spike directivity and | ↓> for downward spike directivity where: | a |2 + | b |2 = 1 ( 74 ) If the measurement is performed (e.g. tetrode recordings) then the system decoheres and will be in the classical state | ↑> with probability | a |2 and in the state | ↓> with probability | b |2 (Figure 32).

a

b z

2

ϑ

z

1.5 1

x

0.5 0 -1

ϕ

ψ

y

-1 0

0 1

1 2 2 y

x

Figure 32: Representation of spike directivity for several consecutive spikes recorded from a unit/neuron and the corresponding Bloch sphere. a. Representation of several spikes recorded from the same neuron. Initially, before learning, spike directivity points randomly in space. The upward spike directivity in red and downward spike directivity is represented in blue. b. Bloch sphere representation of a qubit.

| Ψ (t ) > may be written : θ θ | Ψ (t ) >= cos | ↑> +e iϕ sin | ↓> ( 75 ) 2 2 where θ and ϕ define a point on the Bloch sphere and the formalism of quantum computation can be used to develop quantum neuronal networks. A variety of models that provide the algorithmic formalism to implement superposition, interference, entanglement or qubit registers are given in the literature (Ezhov and Ventura, 2000; Narayanan and Menneer, 2000; Gupta and Zia 2001).

4.2.4. Brain the Physical Computer Recently analyzed experiments have shown that encoding of certain mental state/semantics can be described by a directed electric flux which locally within a spike can be represented by a vector. The electrical output of a neuron in terms of spike directivity reflects a certain semantics that can be demonstrated using the T-maze procedural learning tasks. This view of the brain as a physical computer that regulates the dynamics of charges which follows physical principles can provide tremendous computational power. Further, this view also extends our understanding beyond the

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realm of the physical aspect according to previous theories regarding semantic compositional rules as developed by Smolensky, (Smolensky, 1990). Since computation involves the physical movement and interaction of charges, electrical phenomena including electric fields and polarizations are important components at the micro-scale level (Hameroff and Tuszynski, 2004). In terms of polarizability, the nonlinear dependence upon the electrical field can be described as a power series: r P ( E ) = k ( χ1 E + χ 2 E 2 + ....,+ χ n E n ) ( 76 ) The existence of charge clusters, changes in their density and their spatial distribution determines polarizations of molecular formations that can be related to the direction of r electric field E : r r P = kχE ( 77 ) Such expression can be written in a three dimensional space where χ can play the role of the dielectric tensor: ⎛ χ xx (t ) χ xy (t ) χ xz (t ) ⎞⎛ E x (t ) ⎞ ⎛ Px (t ) ⎞ ⎜ ⎟⎜ ⎜ ⎟ ⎟ ( 78 ) ⎜ Py (t ) ⎟ = k ⎜ χ yx (t ) χ yy (t ) χ yz (t ) ⎟⎜ E y (t ) ⎟ ⎜ (t ) ⎟⎜ E (t ) ⎟ ⎜ P (t ) ⎟ ( ) ( ) t t χ χ χ zy zz ⎝ z ⎠ ⎝ zx ⎠⎝ z ⎠

where with time the values of χ change. Since the polarization changes with time it determines slight changes in the spatial flow of charges which carries information as shown before: r r I = ∫ ∇JdV ( 79 ) V

where the gradient of the current density can be computed and depends on the density ρ ' of charges: ∂ρ ' ∇J = − ( 80 ) ∂t Since individual electric dipole moments at molecular level and resulting vectors at macroscopic scale (e.g. spike directivity) can exhibit certain semantics, any resulting t t tensor χ can be seen as an outer product of vectors. The second order tensor χ can be r r expressed based on two vectors a1 and a2 : t r r a1 ⊗ a2 = χ ( 81 ) t which may both exhibit semantics. If χ is a third order tensor then it can be seen as an outcome of three vectors: t r r r a1 ⊗ a2 ⊗ a3 = χ ( 82 ) This result can be generalized for an Nth order tensor: t r r r a1 ⊗ a2 ⊗,...,⊗a N = χ ( 83 ) t t The rank of the tensor χ is the number p of decomposable states such that χ can t be written as a linear combination of components v1 , v2 ,..., v p = χ . This formalism gives a natural way of composing semantics of individual elements and provides semantic interpretations for complex expressions as tensor products (see Smolensky, 1990). A natural extension from vectors to tensors can be directly derived from an electric/charge model description and does not necessarily require a spike timing approach Such description may be seen as a granulation phenomenon from micro-scale level

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147

determined by polarization of macromolecules (e.g. proteins) within individual neurons to electric phenomena that occur at larger scales of networks of neurons. Extending these compositional rules in the network of neurons depicts coding as tensor composition where input stimuli seen as vectors may activate the value in the tensor product node. The decoding phase can be performed by computing the inner product of an input cue and the trace (Smolensky, 1990). The coding phase requires an exponential growth of the number of binding units representing the number of processed items (Rachkovskij and Kussul, 2001) and such increase in number would need physical support within every neuron at the biomolecular level. It is likely for such representations to occur naturally at microscopic scales in terms of electric patterns within spikes (see Aur and Jog, 2006). The physical aspect of information processing related to this dynamics and interaction of charges, seen as simple particles, can be additionally explored within a thermodynamic framework. 4.3. The Thermodynamic Model of Computation The neuron with its cellular machinery, with inputs and outputs which change in a complex interdependent fashion, can be regarded as a system that processes information where the intrinsic mechanism of membrane transport has to obey the established physical laws. Therefore, since the laws of physics govern neuronal processes, the thermodynamic model can be seen as a unified conceptual model of information and computation in the brain. In this context, a thermodynamic framework related to charge movement and their interaction during synaptic and non-synaptic activities can be inferred. Since this charge flow is viewed as a mechanism of communication, applying a thermodynamic framework to it may allow the representation of this flow in terms of information transfer. Thus the dependency between thermodynamic characteristics and information changes in the analyzed system can be understood in terms of charge mass transfer. Simulations using the Hodgkin-Huxley (HH) model and mutual information analyses are used to demonstrate this relationship between ionic flow and efficiency in information transfer. The results can be then extrapolated to multiple ionic fluxes. Such application of information theory to neuronal models is novel and has not been discussed in the literature. 4.3.1. The Maxwell Daemon The Webster’s dictionary defines Maxwell’s Daemon to be Quote: “a hypothetical agent or device of arbitrarily small mass that is considered selectively to admit or block the passage of individual molecules from one compartment to another according to their speed, constituting a violation of the second law of thermodynamics”. The Maxwell experiment (Maxwell, 1871) describes a simple thought experiment of separating molecules/particles from two different containers based on their velocity (Figure 33). The supposed “daemon” is able using a gate to separate the molecules that have low velocity from the molecules that have high velocity. Since the higher speed corresponds to a higher temperature, then the temperature will have to increase in one container while the other one should decrease its temperature.

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a

b

Figure 33: Schematic representation of the Maxwell Daemon. After a certain time the Daemon succeeds to separate the two types of molecules by sliding the gate.

The Maxwell experiment seems to be contrary to the second law of thermodynamics since without doing any work the entropy of the system apparently decreases. This phenomenon intrinsically provides a subtle clarification which can be explained in terms of information. Leo Szilárd showed that acquiring information or information storage may require some form of energy (Szilárd, 1929). Measurements of the molecular speed by the daemon is a process that may require under certain conditions energy expenditure. Lately, the work of Landauer (Landauer, 1961) and Bennett (Bennett, 2003) showed that some measurements may be performed without any thermodynamic cost and only information erasure requires energy. The minimum increase in entropy that should be associated with the erasure of one bit is: ΔH B = k B ln 2 ( 84 ) where k B is the Boltzmann’s constant. Their work demonstrates that erasing information cannot be performed in a thermodynamically reversible manner and this problem has lately become an issue for classical computation since writing in memory in current computers has to delete previously stored information. o Landauer's principle: Erasing information requires a thermodynamic cost of k B ln 2 for every deleted bit of information This thermodynamic analysis shows why in current computers the erasure of information increases the temperature. From this theoretical analysis one question naturally arises above all others. How is this process performed in the brain? For several decades Maxwell’s daemon proved to divide scientists. Here, we show that this thought experiment is able to reveal the subtle mechanism of computation and information processing in the brain. 4.3.2. Computation in a Thermodynamic Engine It is already known that thermodynamic analysis plays an important role in understanding computation. A simplified model that includes a bath and two separate compartments is considered (Figure 34). This theoretical model is used to illustrate the computational phenomenon and the thermodynamic implications derived from physical laws. Suppose that several particles are free to move in this bath and there is communication between the bath and each chamber. For simplification both chambers have the same volume V. The particles could enter or leave each chamber by a diffusive process and the thermodynamic entropy can be computed initially for particles that are in the left chamber using the Boltzmann-Maxwell formula:

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149

H B = k B log Ω

( 85 ) where k B = 1.3806503 × 10 m kg s K is the Boltzmann constant and Ω is the number of microstates (see Appendix 2). The non-equilibrium state evolves to an equilibrium state through a thermodynamic entropy increase. This equilibrium corresponds to an equal distribution of particles with the highest level of computed entropy (Table 1) -23

State

Chamber1

2

-2

-1

Bath

Chamber 2

Entropy

1

9

0

0

-19.7750 k B

2

8

1

0

-16.6355 k B

3

7

1

1

-13.6214 k B

4

6

2

1

-12.1369 k B

5

5

2

2

-10.8198 k B

6

4

3

2

-10.2273 k B

7

3

3

3

-9.8875 k B

Table 1 The entropy in a closed system increases monotonically until it reaches its maximum at the state of thermodynamic equilibrium.

After the equilibrium occurs, there is no force able to drive this thermodynamic engine unless a special mechanism is built to separate these particles (see Maxwell Daemon).

a

b

-8

b

Boltzmann Entropy

-10 -12 -14 -16 -18 -20 1

2

3

4 States

5

6

7

Figure 34: Schematic representations of a thermodynamic engine and entropy variation in a multicompartment model a, Representation of two separate chambers, the left one (L) in blue color and the right chamber (R) in red color within the bath (B) b, The entropy of the closed system increases toward equilibrium. According to the second law of thermodynamics, the closed system always maximizes its entropy. If the system is not in an equilibrium state, it moves towards it.

150

Chapter 4: Models of Brain Computation -21

a

3

x 10

-21

b

a

3 2.5

2

2

HB

2.5

1.5

1

1

0.5 0 0

b

dHB

1.5

x 10

0.5 0.2

0.4

0.6 0.8 t [ms]

1

1.2

1.4

0 -1

-0.5

0 dHB/dt

0.5

1 -22

x 10

Figure 35 A computation cycle for the spike “engine” shows variations in thermodynamic entropy a, The entropy variations in a cell/neuron during particle/charge influx and particle/charge efflux. b, The plot of entropy variation across entropy time derivative.

Can these mechanisms be formalised using charges that are moved as a computational process? The extended Church thesis states that every physically realizable Turing machine is able to perform any type of computation with at most a polynomial slowdown. Therefore, in a reductionist approach, the proces of particle transport can be algorithmically formalised as below: S1. Observe the particle’s chamber [B] (read/measure the state of the bath) S2. If L>B (read the input L) send particle to [B] (output) O(t + 1) = F ( S (t ), i (t )) S3. If L B(read the input R and B), send particle to [B] (output) S5. If R
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151

o

If one imagines charges as particles, neurons can be reduced to these chambers where the bath may represent the extracellular space. Glial cells /astrocytes are responsible for homeostatic processes and regulate the required concentration of charges needed to restart and perform the computational process. Analysing transport phenomena in neurons is important since these homeostatic processes are intrinsicaly related to computational aspects developed at membrane level within neurons. Additionally, we would like to know if this specific biophysical principle of information processing in brain may allow it to perform computation with a lower cost than in currently manufactured elctronic devices. 4.3.3. Entropy in Neuronal Spike It is already known that during the interspike interval (equilibrium state), for each three Na+ ionic efflux there occurs an influx of two K+ through the Na+-K+ ATP dependent pump (Kandel et al., 2000). From statistical point of view more information is conveyed or transferred when an event is less likely to happen than when it actually occurs. This means that low levels of information are coded within ionic fluxes during the neuron’s interspike interval. It is therefore possible that, only the transient state described by an AP occurrence transfers high levels of information. Since an AP is generated by ionic fluxes, it is expected that a relation between ionic flux and information transfer could be established. This estimation of the ionic flow can be provided during every AP using Hodgkin-Huxley equations that give this quantitative description. Therefore, an analysis of relative information determined by ionic fluxes mainly of Na+, K+ and Cl- during the AP would then provide insights into information communication due to the AP event. Estimating a probability model for activities of ionic fluxes during AP, the relation with information is interpreted in terms of information entropy as a measure of uncertainty of random variables. Indeed using “mutual information” we have obtained an estimation of information that one random variable, for example sodium flux contains about stimulus input during the action potential (Aur et al., 2006). Switching back to thermodynamic entropy, according to the second law of thermodynamics, thermodynamic entropy never decreases in closed systems. In an open system, thermodynamic entropy can be modified by matter flow and temperature exchange. One can imagine neurons as open systems that share matter, energy and information with their environment. Considering the work from Nicolis and Prigogine (Nicolis and Prigogine, 1977) the change of thermodynamic entropy, ∆H, during each spike for one neuron is composed by two terms: ΔH = ΔH e + ΔH i ( 86 ) The external thermodynamic entropy ∆He is due to exchange of matter and energy with the exterior while the internal entropy variation ∆Hi is due to internal irreversible processes inside the neuron. The external variation of entropy ΔH e can be accessible to measurement. Internal entropy production ΔH i can be obtained by measuring the “metabolism” of the neuron as a result of diverse physical phenomena generated primarily by biochemical reactions and diffusion. During each spike the neuron is an open system where a metabolic exchange takes place. In isolation, internal entropy ΔH i always increases reflecting the second law of

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thermodynamics. As the neuron tries to maintain its thermodynamic state, the effect is that the internally produced entropy ( ΔH i ) has to be exported to the exterior in the form of matter or heat ( ΔH e ). Since the heat transfer during the AP is very small (time is very short, 1ms, for heat transfer) the exchange is mainly in the form of matter. When an ionic mass mi enters into the nerve cell, the thermodynamic entropy is increased by the amount mihi where hi is the specific entropy of the mass mi. As the ionic mass ∆mo leaves the system, the thermodynamic entropy is decreased by the amount ∆moho. In neurons during the AP, the ionic influx or the efflux consisting mainly of Na+ and K+ is primarily responsible for the thermodynamic entropy variation (Figure 36). This variation of thermodynamic entropy during the AP can be computed by: ΔH e = ∑

Qk d d + (mNa − mKr hKr ) hNa Tk

( 87 )

Outside of nerve cell K+ Na+ Cl-

Membrane

Figure 36: During an AP the permeability to sodium increases by opening of sodium channels. Sodium ions enter inside the cell and the membrane is depolarized. Then, with a delay the voltage-gated K+ channels open and K+ ions rush out of the cell almost instantaneously during repolarizing phase. This exchange of ions across the neuronal membrane is responsible for Inside of nerve cell thermodynamic entropy variations.

where Qk represents the heat of ionic mass entering and leaving and Tk their temperature, with the indices d and r indicating the depolarizing or repolarizing phase of the AP. The total variation of external entropy during each AP is: d d ΔH e = mNa hNa − mKr hKr

( 88 )

In the steady state (interspike interval), the variation of external thermodynamic entropy is net negative due to mass leaving the neuron. Simultaneously, the internal entropy is net positive and represents the metabolic processes within the neuron. This balance between the external and internal entropy in the steady state represents an equilibrium state and the variation of thermodynamic entropy is close to 0 ( ΔH ≈0). During an action potential, this equilibrium changes such that the entropy of the neuron

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153

may increase or decrease depending on the balance between internal and external entropy: ΔH <0 or ΔH >0 ( 89 ) This mechanism of altering the thermodynamic entropy within nerve cell by mass transfer during every action potential can underlie information transfer between the neuron, neuronal milieu and connected nerve cells. 4.3.4. Thermodynamic Entropy and Information Having discussed variations in thermodynamic entropy during AP in terms of mass exchange, it is now important to introduce the concept of Shannon information entropy and its link to thermodynamic entropy. In addition, we will also build upon this relationship of thermodynamic entropy and Shannon information entropy and discuss how these measures can be viewed in terms of bits of information exchanged. In 1948, Shannon introduced the notion of information entropy for a random discreet variable X, (Shannon, 1948) as a function of the probability distribution p: H SE ( p) = −∑ p( xi ) log p( xi ) ( 90 ) i

Shannon information entropy H SE measures the uncertainty about the outcome of a random variable and does not depend on the state values x1, x2…xn. If the probability distribution of these variables is spread across many states, then H SE value is low, so H SE can be considered as a measure of the compactness of a certain distribution. Importantly, the amount of information in bits obtained about a system by an external observer is commensurate with the reduction in the Shannon information entropy of the studied system (Shannon, 1948). Therefore, the removal or addition of bits of information can be explained in terms of variations in Shannon information entropy. In this case, the extracellular milieu including connected cells can take the place of the external observer. Moving back to thermodynamic entropy, the Boltzmann-Maxwell thermodynamic entropy, in a classical framework, gives an explicit expression for the relation between information that can be acquired and Shannon information entropy (Brillouin, 1956). As discussed above, in equations ( 86 ) and ( 87 ), the thermodynamic entropy changes within neuron are equal to the sum of entropy transfer by heat, by mass flow and internal entropy variation that can be written as: ⎛ Q ⎞ ΔH = ⎜ ∑ k + ∑ mi hi − ∑ mo ho ⎟ + ΔH i ( 91 ) Tk ⎠ ⎝ where the term in parenthesis represents ΔH e . According to the second law of thermodynamics, internal thermodynamic entropy always increases in a closed system. However, since neurons are open systems, they can export entropy. Thus, in a hypothetical steady state of the neuron, a part of the external entropy ΔH e , accessible to measurement as ionic flux, is negative and compensates for the internal thermodynamic entropy production in the neurons. Since the internal thermodynamic entropy always increases, the corresponding reduction of external thermodynamic entropy has to compensate for this increase and at a certain moment the equality below must hold true: ΔH e + ΔH i ≅ 0 ( 92 )

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Otherwise, a continuous increase in the internal thermodynamic entropy would inevitably destroy the cell. The internal thermodynamic entropy ΔH i of the neuron at this point in time can then be computed using a general statistical definition of Boltzmann-Maxwell thermodynamic entropy: N

H B = -k B ∑ p i log pi i =1

( 93 )

where kB is the Boltzmann constant (1.380658×10-23 Joules/Kelvin) and pi is the probability distribution to be in the ith configuration. If all N configurations are equiprobable, then Eq.( 93 ) leads to well known definition of entropy in terms of the number of microstates associated with the macrostate (HB=kBlogN). Combining equations ( 91 ) and Eq.( 93 ), one can write: Qk

∑ Tk + ∑ m h − ∑ m h i

i

o

o

+ k B (−∑ pi log pi ) −k B (−∑ pi* log pi* ) = 0

( 94 )

where pi and pi* are the probability distributions of two adjacent thermodynamic states of the neuron. Assuming that this phenomenon is close to an adiabatic process Q ( ∑ k ≅ 0 ), the relation between Shannon information entropy and biophysics of Tk neurons becomes clear. Equation ( 94 ) shows that Shannon information entropy represented by the term − ∑ p( xi ) log p( xi ) gives the amount of information in terms of bits. If one assumes that the difference between these two adjacent states provides 1 bit of information (i.e. ∑ pi* log pi* − ∑ pi log pi = 1 bit ), then Eq.( 94 )can be rewritten:

∑m h − ∑m h i

i

o

kB

where

∑m h i

i

o

=1

represents thermodynamic entropy of incoming mass and

( 95 )

∑m h o

o

represents thermodynamic entropy of outgoing mass. The analysis of equation Eq.( 95 ) then shows that for adding 1bit of Shannon information entropy as in the ∑ mi hi is necessary. In the depolarization phase in each neuron, an ionic influx of kB + depolarization phase the influx of Na is predominant and as a consequence the Shannon information entropy increases. For removing 1bit of Shannon information ∑ mo ho is entropy as in the repolarization phase in each neuron, an ionic efflux of kB + required. In the repolarization phase the efflux of K results in a decrease of Shannon information entropy. This idealization is necessary only to show the relationship between thermodynamic processes and information entropy. In reality, the biophysical process in the neuron is not adiabatic for obvious reasons due to dissipative factors (e.g. friction, diffusion, thermal loss) that occur during ionic movements. During an AP, the equilibrium between internal and external thermodynamic entropy depends on the movement of charges across the cell membrane. In this way,

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155

changes in thermodynamic entropy can be effectively tied to Shannon information entropy and then extended to information transfer during an AP event by computing mutual information (see Chapter 3). As stated above, the amount of information obtained about a system for an external observer is commensurate with the reduction in the Shannon information entropy (Shannon, 1948). Therefore, an external observer such as the extracellular milieu or even us may acquire (increase) information if Shannon information entropy is reduced. Practically speaking then, ion movements are responsible for information flow within the connected cells and neuronal milieu. The fact that ion fluxes carry information about stimuli input was demonstrated earlier in (Aur et al., 2006) by performing MI analyses between input signal and ionic fluxes using the HH model. During the repolarization phase, information is acquired by extracellular milieu and cells in the proximity by removing Shannon information ∑ mo ho ). Thus a change in the thermodynamic entropy of a entropy as an ionic efflux ( kB neuron is reflected in information alteration of the surrounding milieu which includes both cellular and extracellular elements. Explicitly, the decrease in thermodynamic entropy from mass outflow results in an increase in the information in these elements surrounding the nerve cell. Conversely, an increase in thermodynamic entropy in a neuron from mass influx results in an overall information transfer from the surrounding elements. These observations can be seen as an extension of Landauer principle applied to biological cells (Landauer, 1961) which implies that information transfer during AP has physical representation expressed in ionic matter flow. 4.3.5. Synaptic spikes This description of computation and information processing in neurons would be incomplete if it is reduced merely to AP occurrence. At the gap junction level, there cannot be any possible confusion regarding the carrier of information since the electrical current is directly generated by ions on a shorter distance, about 3.5 nm (Kandel et al. 2000, Bennett and Zukin, 2004). The situation is somewhat more complex at the electrochemical synapse. The ionic mechanism presented above occurs during synaptic activity. The ionic flux is well recognized to have an important, active role in information processing at synapses (Abbott, and Wade, 2004). If the presynaptic neuron develops an action potential, then a dynamic flow of charges occurs within the nerve cell that changes the internal and external environment without delay. The presynaptic membrane then through calcium influx and other mechanisms causes the neurotransmitter vesicle to dock and release its contents across the synaptic cleft over a distance of 20 to 40 nm that separates the cells. This chemical binds to the receptors on the post synaptic neuron. The binding of the neurotransmitters to the receptors generates an electrical response and ions are transferred depending on gsyn(t) synaptic conductance and the difference between the voltage of the postsynaptic neuron V(t) and the membrane reversal potential Vsyn(t) of the presynaptic nerve cell: I syn (t ) = g syn (t )(Vsyn − V (t )) ( 96 ) From a thermodynamic perspective these biochemically modulated processes, at a smaller scale, are similar to exchanges of ions during AP when ions move in or out of the cell. Even though synaptic processes develop at this small scale they can be

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recorded with electrodes and since the number of synapses is of order of thousands for a nerve cell these thermodynamic variations of entropy are not negligible. These changes in thermodynamic entropy determined by ionic mass transfer are directly linked to Shannon information entropy as stated above. (see Appendix 1). o Information communication is a continuous process where electric charges and electric field are the preferred carriers and neurotransmitters modulate this process The chemical neurotransmitter now can be viewed as changing synaptic conductance from Eq. ( 96 ) and the variability of g syn modulates both ionic and information transfer.

Figure 37: While the cars are similar to ‘information carriers’, the neurotransmitters play the role of ‘traffic lights’. Additionally, the intricate function of ‘traffic lights’ directs the movement of cars on the road. This effect is similar to neurotransmitter release as a result of calcium (Ca2+) flux.

This basic description in theoretical terms demonstrates that charge carriers and electric field communicate information during synaptic spikes. In the literature these processes are known as electric synaptic computations (Abbott, and Wade, 2004). The importance of electric charges and ion fluxes can be exemplified by a suggestive analogy to a traffic scenario where cars become ‘information carriers’ and the neurotransmitters play the modulatory role of traffic lights (Figure 37). 4.3.6. Spikes as Thermodynamic Engines The brain “bath” model with chambers that mimic neurons and their environment presented above reduces computation to transport phenomena where ions/charges play the role of particles. In this case the thermodynamic engine is powered in each neuron by already known sodium/potassium exchange pumps that maintain a high concentration of K+ inside the neuron and a high concentration of Na+ outside the neuron. These concentrations are required to reach the resting potential that drives the AP engine. It is likely that spike propagation properties depend on the input features and during each spike this electric comunication transfers information preferentialy in space. A formal description of the neuronal process can be modelled using a thermodynamic framework (Aur and Jog, 2007c). The thermodynamic entropy changes

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occurring within a neuron are equal to the sum of entropy transfer by heat, by mass flow and the entropy generated and can be directly related to Boltzman-Maxwell entropy (Figure 38). The fluctuations in sodium current are given by the influx of sodium charges: dN Na iNa = q ( 97 ) dt where q=1.60217646 × 10-19 C. In a similar way for iCl and iK can be written: +

K+

−

dN dN Cl , iK = q iCl = q dt dt

( 98 )

If all configurations are equiprobable then the fluctuation of Boltzmann entropy due to ionic flow within each spike can be approximated: ΔH B (t ) ≅

t t t k k kB log ∫ i Na dt + B log ∫ (−iCl )dt − B log ∫ i K dt q q q 0 0 o

-15

a

8

-15

b

x 10

a

8

b

x 10

6

6 Entropy [J/K]

( 99 )

4 HB

4

2

2 0

0

-2 0

-2 -0.5

5

10

15

20

25

0

0.5

1 1.5 dHB/dt

2

2.5

3 -16

x 10

Time[ms]

Figure 38: The variation of external entropy during a spike computed based on Hodgkin-Huxley model (Appendix2). a. Entropy variation for a spike across time. The influx of Na+ increases the entropy during the depolarization phase while the outflow of K+ during repolarization phase decreases the entropy of neuronal cell and equilibrium appears as a ‘cooling’ process. b Plot of entropy variation within a spike across the entropy derivative. The computation does not come to a halt since the entropy variation is a cyclic process.

Removing the heat from a hot object, decreases the entropy while adding heat to a cold object increases the entropy. Within this paradigm each spike and the adjacent interspike interval can be viewed as a thermodynamic engine (Figure 39). “Overheating” can occur only in case of an abnormal function that may physically destroy the neuron.

SH

SL

TL

TH

Figure 39: A schematic view of the thermodynamic spike engine. The increasing of entropy during depolarization phase is followed by a decrease during repolarization and interspike interval. In the depolarization phase the neuron is the cool reservoir where Na+ ions enter. During the repolarising phase the neuron becomes the “hot reservoir” and K+ export reduces the entropy that may remain still high. Therefore, the cooling process continues during interspike interval (blue color) and is driven by ATP pump.

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The changes in each neuron during AP, are reflected in the variation of energy, work and heat content. These entropy changes may be associated with energy loss which leaves few resources to do useful work. For a given exchange of energy dU, the maximal work at constant temperanture T and volume of a system is given by: dWmax = dU - TdH B ( 100 ) Since some extra heat is transferred to the surroundings as entropy increases during each spike, the electrical work dWel is: dWel = μ Na dmNa + μ K dmK + μCl dmCl ( 101 ) +

where μ Na , μ K , μ Cl +

+

−

+

+

+

−

−

define electrochemical potential as the mechanical work required

to bring 1 mole of ions from a standard state to a specified concentration and electrical potential. This electrical work generated during each AP is always less than the maximal work: dWel ≤ dWmax ( 102 ) Chemical energy in the form of ATP powers nearly all this work. The ion pump uses the chemical energy stored in ATP to actively pump charges or molecules across the membrane against a gradient of concentration or electrical charge. The large number of distinct observational states determined by charges seen as particles may be thought of as a large memory capacity where information can be transferred and stored. The entropy variation governs the amount of information that each AP can maximally register. Therefore, theoretically, particles/charges within a spike can register a number of bits where the maximum value can be computed (Lloyd, 2000): ΔH Bspike ΔI spike = ( 103 ) k B ln(2) HH

This maximum value ΔI spike can be estimated from Hodgkin-Huxley simulations to HH

be about 108 bits of information within a single spike. o It is unlikely that this entire information carried by ion fluxes is stimuli or behaviorally dependent. Mutual information values can be computed in order to find how much information is transferred about the input stimuli or behavioral states within ion fluxes (see Aur et al., 2006) If one interprets the ion movement within ionic channels as a potential well then there is a physical model of memory that may indeed sustain some high values (Levitt, 1986) that may show an ‘unlimited’ capacity of information transfer and possible storage in brain. Theoretically, the large number of distinct observational states is given by the number of charges/particles during every AP. However, we do not expect that every spike will be able to register this amount of bits in memory and the registered value ΔI Rspike is expected to be much lower: ΔI Rspike < ΔI spike

( 104 ) It is likely that high information quantities are required to create, maintain and consolidate memories at a fundamental molecular level. This and several additional aspects regarding the process of building memory will be further discussed in the next chapter. Biological processes are by definition metabolically driven, they utilize energy to process information that changes the thermodynamics of the cell. Therefore, the laws of thermodynamics are universally applicable to these physical systems and as shown HH

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above they can be intrinsically related to information processing, deleting and storage mechanisms. This shows that changes in information transfer by charges that cross neuronal membranes can be incorporated in a thermodynamic outline. This natural form of physical computation within a spike seems to be very efficient. A simple analysis regarding energy consumption per transferred bit shows that spikes are more efficient than actual processors. The energy associated with a single ATP molecule is about 10−19 Joules. It is estimated that for a single spike, 10+6 ATP molecules are required (Aiello and Bach-y-Rita, 2000). If one spike transfers on average 3 bits of information, the energy required is equivalent to about 10-13 Joules per bit. In the current hardware technology for processors, the dissipation per transferred bit is

10 −8 Joules (Rusu et al., 2004), which is more than ten thousand times higher. Additionally if indeed one can demonstrate a thermodynamic construct and relate it to neuronal function, it is intriguing to suggest that this framework may be linked to several phenomena connected with the processes of neurodegeneration in brain. Summary of Principal Concepts • •

• • •

•

• • •

Models are able to characterize only a few properties of real physical phenomena. Many scientists believe that coding and decoding information in the brain is governed by classical physics. However, at the microscopic level many interacting particles, molecular machines, proteins, genes and specific regulatory mechanisms may actually naturally display quantum properties. A single particle which moves in a three-dimensional potential is able to perform “universal” computation and can be theoretically reduced to a Turing machine (Moore, 1990). During physical computation a charged particle should follow the principle of least action for which the average kinetic energy minus the average potential energy is minimum. During movement of charged particles, selected set of states can be repeated and may define attracting sets that can be directly related with revealed electrical patterns within spikes and memory representation of simple input features. The computation with charges in the brain builds an extremely powerful machine. The multidimensional intrinsic behavior, intrinsic interactions and parallelism of computation with charges display outstanding capabilities and they may be able to compute more than a machine that performs computation based on algorithms (Beggs et al., 2008). With charge dynamics/interaction model, noise driven solution, attractors, synapse dependent plasticity can be better explained in terms of computation with charges. The computational cube defines a theoretical, abstract model for computation by charges during successive series of coding decoding phases. This reductionist scheme can be regarded as a compartmental model. The coding phase is a process where information regarding an input feature/stimulus/behavior is embedded in a spatial distribution of charges. This phase can be seen at different scale levels from AP to macromolecular structures, proteins.

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• • • • • • •

• • • • •

•

• • • •

The decoding phase is a process of reading the spatial distribution of charges that mimic electric activities at “synaptic” level, “gap junction, protein level. The model of learning can be resumed in two main phases which combine fast and slow-learning steps. The fast learning phase can be simulated within a computational cube as an energy minimization procedure by a genetic algorithm/ swarm algorithm. The slow learning phase tries to mimic synapse level processes, plastic changes in synaptic strength, synapse formation, synapse elimination that depend on complex processes that occur at molecular scale. The learning algorithm tries to maximize a function to allow a higher discrimination of input features based on spatial distribution of charges. A performance index is introduced in order to optimize the inner product of “spike” vectors and generate a higher discrimination of input features within the computational cube. During learning the information about the characteristic/features of input data are transferred and stored in the ‘synaptic weights’ which may have a correspondence in electric interactions at micro-scale level (macromolecules, proteins), spatial distribution of synapses. In a reductionist approach brain mapping can be described as a set of transformations between the input signal space and spatial distribution of charges in brain which can be homeomorphic if the map is a bijection. Information regarding input features is represented (mapped) in neurons and can be evidenced in single spikes (see Chapter 2, micro-mapping). With learning larger areas/volumes are assigned to familiar stimuli inputs and this information becomes overrepresented Spatial distribution of charges are at the origin of distance-based semantics and determine temporal properties of spikes delivered with millisecond time precision Fractal properties occur naturally since once derived an appropriate structure or a suitable mechanism that computes uses the dynamics and interaction of charges. The structure/mechanism is repeated several times with small variability to complete certain brain functions. Once a point of interest is determined (that represents a certain feature by an increased density of charges) the propagation of electric field/charges follows the selected morphology of neurons and carries selected sets of transformations. Granular properties occur naturally as an agglomeration of charges from micro to macro-scale levels and semantics can be theoretically explored using the already developed formalism of granular computing. The macroscopic world of spikes is described by classical physics while the microscopic one that includes charges, ions is governed by the laws of quantum mechanics. A particle (charge) starting at a certain point A at time tA will arrive at point B at time tB with the square of probability amplitude (Feynman, 1964). If a “three charge movement model” is used for AP representation then the off diagonal elements of the density matrix are small showing the presence of decoherence and the spike can be well approximated with a classical model. However, if an ICA filtering technique is used then the occurrence of peaks of

Chapter 4: Models of Brain Computation

• •

• • • • •

161

the density matrix away from the main diagonal can be attributed to a quantum coherence process where electric charges can be seen in a superposition of states. Experimental analysis shows that quantum phenomena are hidden from our observation; however, they exist and they influence information processing at a molecular scale. For several decades Maxwell’s Daemon (MD) succeeded to divide scientists from different fields. Here, we show that the MD experiment is able to reveal the subtle mechanism of computation and information processing that occurs in the brain following a thermodynamic approach. In the depolarization phase the influx of Na+ is predominant and as a consequence the values of Shannon information entropy increase. In the repolarization phase the efflux of K+ results in a decrease of Shannon information entropy. Spikes can be seen as thermodynamic engines able to register a certain number of bits that the unit/neuron can hold within macromolecular structures. Erasing information in these structures requires a thermodynamic cost of k B ln 2 as heat for every deleted bit of information. The efficiency of information transfer in the brain is higher than current limits allowed in the hardware technology for processors. These limits are reflected in the dissipation values per transferred bit of information; in brain the limit is 10-13 Joules per bit while in current processors the limit is about

10 −8 Joules/bit. Appendix 1

p1= 0.000125

Classic system. Spike can be viewed as three charges in movement 0.3611 ⎞ | Ψ1 > = ⎛⎜ ⎟

p2= 0.0018

p3= 0.9981

⎜ 0.9324 ⎟ ⎜ - 0.0138 ⎟ ⎝ ⎠ ⎛ - 0.9270 ⎞ = | Ψ2 > ⎜ ⎟ ⎜ 0.3606 ⎟ ⎜ 0.1035 ⎟ ⎝ ⎠

⎛ 0.1015 ⎞ ⎟ ⎜ - 0.0246 ⎟ ⎜ 0.9945 ⎟ ⎝ ⎠

| Ψ3 > = ⎜

The same spike can be viewed in a superposition of three states that are obtained after an ICA algorithm embedded in N-charge model. Quantum three-level system. 0.6498 ⎞ p1= 1.7947e-005 | Ψ > = ⎛⎜ ⎟ 1

⎜ 0.6675 ⎟ ⎜ - 0.3636 ⎟ ⎝ ⎠

⎛ 0.6353 ⎞ ⎟ ⎜ - 0.7396 ⎟ ⎜ - 0.2224 ⎟ ⎝ ⎠

p2= 9.9185e-004

| Ψ2 > = ⎜

p3= 0.9990

| Ψ3 > = ⎛⎜ 0.4174 ⎞⎟

⎜ 0.0865 ⎟ ⎜ 0.9046 ⎟ ⎝ ⎠

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In order to compute the value of Boltzmann entropy for a particular macrostate of the system, one has to compute the number of microstates. The number of microstates can be deduced from a combinatorial argument. For N particles there are N! number of ways in which they can be placed in all energetic levels. For each energy level there are Nj ways of rearranging the ions within this energetic level. Therefore from this combinatorial scheme

Ω

becomes: Ω=

N! N1! N 2!...N m

( 105 )

where N + N + ... + N = N . For large integers N, Stirling’s approximation gives good 1 2 m approximations for factorial: N != N N e − N 2πN

( 106 )

For high N values one may perform the approximation log(N!) ≅ N ln( N!) − N

( 107 )

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Pedrycz W., Skowron A., Kreinovich V., (2008) Handbook of Granular Computing, Wiley. Maass, W., (1997) Networks of spiking neurons: The third generation of neural network models. Neural Networks, 10(9), 1659–1671. Nowak, L.G., Sanchez-Vives, M.V., McCormick, D.A., (1997) Influence of low and high frequency inputs on spike timing in visual cortical neurons, Cerebral Cortex 7 (6), pp. 487-501, 145. Ohshiro, T. Weliky, M., (2006) Simple fall-off pattern of correlated neural activity in the developing lateral geniculate nucleus Nature Neuroscience ,Vol 9; Number 12, pages 1541-1548. Pǎun, G., (2000) Computing with membranes,Journal of Computer and System Sciences 61 (1), pp. 108-143 Pǎun G., (2002) Membrane Computing Springer Verlag. Penfield W., Rasmussen T., (1950) The cerebral cortex of man. New York: Mac-Millan. Penrose, R. (1997) Physics and the mind. In M. Longair (Ed.), The large, the small and the human mind (pp.93–143). Cambridge, England: Cambridge University Press. Popivanov, D. Stomonyakov, V. Minchev, Z. Jivkova, S. Dojnov, P. Jivkov, S. Christova, E. Kosev, S., (2006) Multifractality of decomposed EEG during imaginary and real visual-motor tracking Biological Cybernetics ,, Vol 94; Number 2, pp.149-156. Poggio T., Torre, V., and Koch, C., (1985) Computational vision and regularization theory. Nature, 317:314-319. Ralston W. P., (2005) Electron-Gated Ion Channels, SciTeh Publishing Inc. O'Reilly, R.C. , Munakata, Y., (2000) Computational Explorations in Cognitive Neuroscience: Understanding the Mind by Simulating the Brain. Cambridge: MIT Press. Rieke F., Warland D., Steveninck R. and Bialek W., (1997) Spikes: Exploring the Neural Code, MIT Press, Cambridge. Rissanen, J., (1984) Universal coding, information, prediction, and esitmation. IEEE Transactions on Information Theory 29(4):629-636. Randall L., (2007) Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions, Barnes & Noble Rachkovskij, D. A., & Kussul, E. M., (2001) Binding and normalization of binary sparse distributed representations by context-dependent thinning. Neural Computation, 13, 411–452. Rusu S, Muljono H., Cherkauer B., (2004) Itanium 2 processor 6M: higher frequency and larger L3 cache, IEEE Micro, - ieeexplore.ieee.org. Samsonovich, A.V., Ascoli, G.A., (2006) Morphological homeostasis in cortical dendrites, PNAS 103 (5), pp. 1569-1574 Skarda C. A. and W. J. Freeman (1987) How brains make chaos in order to make sense of the world. Behavioral and Brain Sciences 10, 161–195. Smolensky P., (1990) Tensor product variable binding and the representation of symbolic structures in connectionist networks. Artificial Intelligence, 46, 159-216. [Reprinted in G. Sompolinsky H., A. Crisanti and H.J. Sommers (1988) Chaos in random neural networks. Phys. Rev. Lett. 61: 259–262. Song S. Miller, K. D. Abbott, L. F., (2000) Competitive Hebbian learning through spike-timing-dependent synaptic plasticity, Nature Neuroscience , Vol 3; No 9, pages 919-926. Steinhaeuser, C. Gallo, V., (1996) News on glutamate receptors in glial cells , Trends In Neurosciences, Vol 19; No 8, pages 339-345. Szilárd, L., (1929) On the Decrease in Entropy in a Thermodynamic System by the Intervention of Intelligent Beings, Zeitschrift fur Physik, 53, 840-56. Tegmark, M., (2000) Importance of quantum decoherence in brain processes, Physical Review -Series E- Vol 61; Part 4; Part B, pages 4194-4206 Thaheld FH., (2005) An interdisciplinary approach to certain fundamental issues in the fields of physics and biology: Biosystems Volume 80, Issue 1, Pages 41-56 Tolhurst, D.J., Smyth, D., Thompson, I.D., (2009) The sparseness of neuronal responses in ferret primary visual cortex Journal of Neuroscience 29 (8), pp. 2355-2370. Ueda, M. Leggett, A. J., (1998) Macroscopic Quantum Tunneling of a Bose-Einstein Condensate with Attractive Interaction ,Physical Review Letters , Vol. 80; Number 8, pages 1576-1579. Vapnik V., Chervonenkis, A., (1974) Theory of Pattern Recognition [in Russian]. Nauka, Moscow. (German Translation: Wapnik, W.; Tscherwonenkis, A., (1979) Theorie der Zeichenerkennung, AkademieVerlag, Berlin. Vapnik V., Golowich, S., Smola, A., (1997) Support Vector Method for Function Approximation, Regression Estimation, and Signal Processing. In: M. Mozer, M. Jordan, and T. Petsche (eds.): Neural Information Processing Systems, Vol. 9. MIT Press, Cambridge, MA. Volterra A., Steinhäuser C., (2004). Glial modulation of synaptic transmission in the hippocampus. Glia 47 (3): 249–57.

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CHAPTER 5. From Brain Language to Artificial Intelligence The original project of Artificial Intelligence (AI) has been abandoned by even the AI community due to the grounding issue which can be explained in terms of Searle’s argument of finding the "true meaning" within results of computations. The novel model of brain computations developed in this book may allow a new start in this field inspired by the concepts we propose as the theory of NeuroElectroDynamics. For the first time to our knowledge physical laws spanning across from classical to quantum level have been applied comprehensively to understand the theoretical framework that describes brain computation. In this case computation has to explain how information is acquired, transferred, processed, stored and read. Previous chapters elaborately describe all the above issues except the process of how information is stored. It is required that the stored information must be readable and retrievable in order to be further used. We believe that the new framework developed in this chapter will provide a step forward and suggest solutions in terms of computation to build a new generation of artificial intelligent systems. 5.1. How are memories stored? Starting with the first months of life you can see faces, hear voices; however, you cannot remember any details of this past experience. You can remember almost everything that happened yesterday, last year or two years ago but you won’t be able to remember anything from the first years of your life. Why one cannot remember is still a mystery and science cannot as yet provide any plausible answer. Where have I parked the car? Where are the keys? These are simple questions which show that for everyone, forgetting is a natural phenomenon. However, what is the mechanism that allows us to remember images, sounds or taste? The distortion or loss of memory occurs inevitably with the passage of time. Aging is another factor, memory becomes “funny” since you can remember perfectly a meeting fifty years ago but you cannot remember what you had yesterday at lunch (King, 2009). Also people can voluntarily forget information. This issue was observed by Freud more than one century ago (Freud, 1896). How are these memories pushed into the “unconscious” or how do we recover memories of our childhood? Current explanation is still based on the Atkinson-Shiffrin model and its variations. Rehearsal is considered to be the main mechanism which transfers information from the short term memory into stable long term memories. However, there is little or almost no effective explanation about the intrinsic phenomena at neuronal level that may reflect this model. What is the physical mechanism that leads to this transfer? If you ask any graduate student in neuroscience what is the role of sleep he/she will almost unanimously answer that “sleep consolidates the memory”. This cliché however, does not go further since none of them can actually explain how sleep is able to ‘consolidate’ the memory. The connectionist view in terms of wiring and re-wiring (Hebb, 1949) assumes that synaptic plasticity is mainly responsible for the memory phenomena. The classic

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Rehearsal

Stimulus Short Term Memory

Long Term Memory

Forgetting Figure 1: Schematic representation of the Atkinson-Shiffrin memory model

postulate of Hebb of increasing activity is reproduced as a universal response in many text books and is considered to be able to explain the wiring and re-wiring hypothesis: “When an axon of cell A is near enough to excite cell B or repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that ”A’s efficiency, as one of the cells firing B, is increased.” Current paradigms presume that memories are stored through long-lasting modifications of synaptic strengths (structural modifications of synaptic connections) (Kandel, 2000; Kandel, 2001) or so called “activity-dependent changes in synaptic strength”. Following the Hebb postulate and experimental data, long-term memory can be seen as a result of structural plasticity in the adult brain including changes in the strengths of connections between neurons. Supporting a Hebbian view several scientists suggested that memory depends on changes in the 'wiring diagram' induced by learning (Chklovskii et al., 2002; Chklovskii et al., 2004). Theoretical studies assume also a passive attenuation of the electrical signal in dendrites (see also Koch and Poggio, 1985). Therefore, the maximization of the density of synapses is viewed in terms of the length of “wires” that shape the axonal and dendritic branches. The primary question is if we can use this cable model of wiring exactly as is commonly used to build electrical circuits in electric/electronic devices. There are several issues with these hypotheses. The most important problem remains the assumption that dendrites are passive cables. We claim to show here that the axon, its axonal branches and dendrites cannot be modeled in any circumstance as passive cables. Such biological structures have an intrinsic composition which includes several types of macromolecular complexes and spatially distributed ion channels. By themselves, the formation/elimination of synapses, their plasticity known in terms of changes in conductivity and connectivity cannot directly explain information processing or memory storage. Moreover, the organization of synaptic connections and wiring can be partially determined by genetic programs (Benson et al., 2001) and become shaped afterwards by different supplementary processes. Additionally, a

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passive cable based structure fails to reveal a concrete physical description of memory formation and it seems that the wiring and re-wiring schemes offer an incomplete explanation of how long term memories are built. Several experiments performed by Penfield showed that electrical stimulation of certain areas evoked memories from long-term storage (see memory retrieval, Penfield, 1958). Where are these memories stored? It is expected that during registering information at least the acquisition of input information during learning should determine specific physical changes within the neuron. The idea that only synapses store memories and the hypothesis of synaptic plasticity are self-contradictory as a plastic system would not be expected to have static storage capability. Almost every experiment shows that synapses change their conductances dynamically and information is definitely “written and rewritten” at synaptic level and thus based on this model memories should exist for only a short time. The “consolidation” of acquired information is therefore unlikely to be based only on ‘synaptic strength’. The “synaptic strengths” seem not to be a stable support of all long-lasting storage and therefore longlife persistence of memory cannot be explained in the context of synaptic changes. The relative timing of activation of the presynaptic and postsynaptic neurons is considered to describe the process of adaptation that occurs at synapse level, the so called spike timing dependent plasticity (STDP) (Markram et al., 1997). Even though this analysis has recently received significant attention (Maass and Bishop 1999; Koch, 2004) this framework cannot model the long lasting storage within a temporally based approach. o Memory (information) storage cannot be explained in terms of spike timing or synaptic strengths alone; this process has to be related to a physical machinery able to keep information unaltered for longer time periods. Therefore, the response to this issue has to be searched for in another direction which may provide a better explanation than spike timing theory or ‘synaptic storage’. The genomic hypothesis of memory introduced by Arshavsky (Arshavsky, 2006) can be another possibility. In order to last a long time, memory must somehow be persistently stored within individual neurons through long lasting modifications of the DNA in proteins and other molecules within cell. Therefore, it would be important to investigate a different hypothesis where each neuron can store memory (information) within its intrinsic structure including soma, axon and dendrites and synaptic connections mediated by complex bioelectrical processes modeled in previous chapters. The increase in charge density (charge clustering) that was demonstrated to occur within every AP in different regions of analyzed neurons can be seen as an outcome of how memory storage is shaped at macromolecular level. The importance of synaptic communication and long-term potentiation (LTP) may also provide an important key to understanding this mechanism (Bliss 1993). It is well recognized for long time that LTP requires gene transcription and new protein synthesis (Teyler and DiScenna, 1987; Chardonnet et al., 2008). This behavior adds another hint of how memories are formed and stored. 5.1.1. Are proteins the key? Every neuron has a specific structure and shape determined by its cytoskeleton which includes the membrane and many macromolecular formations. The hydrophobic lipid cell membrane is a barrier for ions. They can cross the membrane through protein

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based ion channels. Biological macromolecules, such as proteins and nucleic acids are important constituents that influence the neurotransmitter release and recruitment of vesicle pools. Since their spatial position is not fixed, beyond this task they undergo considerable motion determined by changes in electric fields and polarizations. Their mobility and molecular processing as protein-based elements can be related to long lasting memory phenomena as they are distributed all over the soma, dendritic branches, axonal and synaptic formations. These proteins act as miniature molecular machines which fold into certain conformational states that may allow or block the flow of charges/ ions. Every functional protein has an ordered structure where amino acids that are linked into a peptide chain have folded in a strictly specified conformation (three-dimensional structure). Additionally, the passage of ions through the channel as shown above may change significantly the conformational state of the channel protein. Significant amount of electrical energy is exchanged between the ions and the channel wall and may change the conformational state of the protein, as was suggested by the model of Chinarov et al. (Chinarov et al. 1992). The alterations related to slight changes in dielectric properties of polar molecules are closely determined by electromagnetic interactions. The synthesis of proteins occurs in the cytoplasm and then they are distributed almost everywhere within the neuron into various organelles, cytoplasm and within membrane (e.g. ion channels). Proteins participate in almost every process within cells including signal and energy transduction and electron transport where electrostatic effects have major roles (Warshel, 1981; Sharp and Honig, 1990). maintaining the normal resting potential of cells. Abnormal changes in protein expression are associated with cortical hyperexcitability and the onset of epileptiform activity (Chu, et al., 2009). It is therefore expected that a modulated electric potential will affect proteins, their conformational behavior including enzymatic activity. Importantly, a molecule/protein does not necessarily have to contain an excess of charges to actively participate in this process, but such proteins need only to have a non-even charge distribution which commonly is referred to as polarization. We will show that changes in the existing polarity of the system of electric charges at a microscopic level within molecules and protein structures is the critical mechanism that in time builds our memory. Every protein is an assembly of amino acids that create a distribution of atoms and charges with a given set of spatial coordinates. Therefore, within a reductionist approach proteins can be viewed as simple distributions of charges (Laberge, 1998). These charges are not evenly distributed in the three-dimensional protein structure (Figure 2). Since these proteins have intrinsic conformational fluctuations, the coordinates of charges can be modified and therefore, the protein polarization is self-modulating responding in electrostatically important ways to different inputs within its environment (Brown, 1999). As shown above, ionic flow and thermal noise may determine conformational changes during synaptic spikes and series of action potentials. Building memory patterns within macromolecular formations should shape the spatial distribution of charges. These charge densities should appear in similitude to the annual rings in growing trees that keep within them an entire history of impacts from climate change to floods or insect-damage. It is likely that in neurons such memory patterns are a result of protein folding/ binding. The electric and hydrophobic interactions may make a difference between protein folding and binding in mediating interactions and proteinprotein associations (Xu et al, 1997). Since protein folding determines new spatial distribution of electric charges in the protein matrix, such transformations are related to

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memory alterations/changes, while protein binding could be seen as a fundamental dynamic process that allows an increase in memory content. Protein folding can be determined by several mechanisms that include neurotransmitter control, electrostatic interactions, hydrogen bonds, van der Waals interactions, hydrophobic interactions acting all together or alone. It can be expected that at this level there is a direct influence between changes in the conformation state of the channel protein and the dynamics and interaction of ions during their motion. Additionally, the dynamics of these interactions has to play an essential role in channel function (Chen et al., 1992; Liebovitch and Krekora. 2002). Changes in electrostatic energies can be calculated using the screened Coulomb potential approach (Mehler and Solmajer, 1991). A slight change in “spatial position” of negative charges may determine slight changes of the trajectory of ions as an effect of ion- electron interactions. As a result, this movement and interaction of charges intrinsically mediates elastic and plastic changes within the “semi-elastic structure” that affects the spatial distribution of charges in the protein or their orientation in space (Figure 2). It is clear that at this level quantum effects are not negligible and the potential energy at protein level can be seen to be composed of the following terms: P = Pclassical + Pquantum + Pquantum _ classical (1) where the classical Pclassical and the quantum mechanical Pquantum potential energies are considered with a the coupling term Pquantum _ classical . In this case the quantum mechanical potential energy can be computed using the quantum-mechanical consistent force field method that takes account of all the valence electrons (Warshel, and Levitt. 1976). K2 K4

K2

K3

K3

K1 K4

r p K5

K1

K5

r p

K6

K6

Figure 2: Schematic representation of a protein model as a distribution of electric charges in space connected by springs within two different conformational snapshots. The dipole moment r p represented in blue color reflects slight changes of spatial distribution of atoms/charges

Within every protein the dipole moment can be associated with a separation of two point charges, one positive (+q) and another one with a negative charge (−q) that can r then be characterized by a vector with the dipole moment p : r r p = ∑ qi ri (2) i r where ri is the displacement vector that points from some reference point to the charge qi . Importantly, the polarizability of a protein structure is a function of spatial

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distribution of atoms/charges and depends on frequency of the electric field. The r r induced dipole moment p is proportional with the field intensity E : r r r p = p p + αE (3) r where p p is the permanent polarization and α is the polarizability of the protein/molecule. Therefore, the protein/molecule shows a preferred direction for polarizability: px = p px + α xx E x + α xy E y + α xz E z p y = p py + α yx E x + α yy E y + α yz E z

(4)

pz = p pz + α zx E x + α zy E y + α zz E z

where α ij is the polarizability tensor for the protein. Writing in memory or coding is aprocess performed by electric interactions, repeating the movement of a charge or more likely of a group of charges (flux of ions) which are expected to determine in time the dipole strength and orientation. Since these spatially distributed charges generate an electric field then the polarized protein will r experience a torque T : r r r T = pxE (5) in the electric field. If the field is not uniform, the occurring force on the dipole can be computed: r r ΔF = qΔE (6) r where ΔE represents spatial variation of the electrical field. At this fundamental microscopic scale the decoding phase or “reading” can be performed by any charge that moves closely to these polarized proteins/molecules. In a classical interpretation the spatial “trajectory” of such charges depends on dipole orientations since the electric force which acts over every charge that moves closely is highly dependent on the orientation of these dipole moments. During each day sensory information is transferred between sensory areas by a continuous flux of electrical charges which reaches several neurons. This permanent bombardment with charges determined by a strong ionic flow during every action potential is able to alter the polarization and orientation of such dipoles. At the most important phase these changes occur at ion channel level within chemical synapses or gap junctions that are protein regulated structures (Eisenberg, 2003). There are several data that relate the polarization effect at protein scale with neuron function and as has recently been shown; charge selectivity in ion channels is determined by protein polarization (Boda et al., 2007). The channel protein is represented as a dielectric continuum and the electrostatic effects appear to be amplified by polarization at the dielectric boundary between channel protein and the pore of the channel. It is likely that these alterations in polarization of biological macromolecules within ion channels in dendrites soma and axon control not only the temporal patterns of action potentials, or synaptic plasticity, they also change local conductance and spatial directivity of electric propagation. The dipole moments induced in proteins by charge movements, their interaction (external electric fields) can display some physical properties that may include relaxation, hysteresis (Xu et al., 1996). Additionally, if an

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+

173

-

+

Figure 3 Biological macromolecules, such as proteins or nucleic acids become polarized and they may change their conformation and orientation of their dipole moments due to changes in the electric field.

external electric field is applied this may induce transitions between several protein configurations which can power the ionic ATPase pumps (Astumian, 2003). These subtle molecular processes provide additional reasons that highlight major differences from earlier cable model proposed in the literature. Thermal fluctuations play a critical role during ionic flow and it is likely that information transfer takes place at the molecular level. In this category of thermal fluctuations we can incorporate all movements including the ones determined by electric flow (e.g. ions) during every action potential, synaptic spikes. The thermodynamic approach can explain this phenomenon. For example for a protein the difference of free energy between folded and the unfolded (i.e. two different structural conformations) is of the order of tens of k BT , on average, ( U f − U uf ≅ 20k B T ) (Wagner and Kiefhaber, 1999) . At this scale any microscopic entity with some degree of freedom inevitably has a significant thermal energy. As an example the system temperature T at thermal equilibrium, can be obtained by averaging the kinetic energy of particles: N

T=

∑m v i =1

i

2 i

Nk B

(7)

Every sodium ion that moves with an average speed v=3 m/s within an ion channel may provide about 0.6090e-022 J (20 k BT for T=300K is 8.28e-020J) then hundreds of ions are theoretically required to act together to change the folding state. Therefore, to alter or create new memories the number of charges involved in the process becomes critical. Additionally, the energy of the electrostatic interaction between a charge in protein and the ion: Wi =

q2 4πεd

(8) −12

is close to k BT limit ( ε 0 =8.8541878176× 10 F/m, d=10-9m, q=1.6 × 10-19 C, Wthreshold=55kBT; Wi = 0.6947kBT ) (see Liebovitch and Krekora. 2002). This simple calculation shows that conformational change based on electric interactions is thermodynamically possible for a protein, however, it does not mean that it always occurs since many electric charges have to be involved in order to provide Wi ≥ Wthreshold and there are many proteins as well. The folding process can be enhanced by the presence of fast-acting neurotransmitters and slower acting neuromodulators (Halbach and Dermitzel 2006).

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For every charge embedded in the protein matrix the Langevin equation can be expressed as: d vip = −mip γ i vip + Fi p + Fi R (9) dt where, vip and mip represent the velocity and mass of the charge i, embedded in the r r matrix protein structure and γ i is the frictional coefficient. Fi p = qE is the electric mip

force on charge i, and

Fi R is a random force with mean 0 and variance < Fi R (t)

Fi R j(t) ≥ 2γ i k B m i T δ(t-t’); that derives from the effects of structure/solvent (Ramachandran and Schlick,1995). Since within a protein the viscous force is expected to be much greater than the inertial term, then the equation for the overdamped limit can be written: mip γ i

d ri = qip E (t ) + Fi R dt

( 10 )

Therefore, the displacement Δri p of the charge in the protein structure is: Δri p =

qip E (t ) Z Δt Δt + ip mipγ i mi γ i

( 11 )

where: Z i ≈ N (0,2γ i k B m i T) ( 12 ) represents the Gaussian noise. The expression in Eq.( 11 ) shows that the displacement Δri p of charge i in the protein matrix structure depends on the electric force generated by other charges (e.g. flux of ions) and the Gaussian noise like term N (0,2γ i k B m i T) with the variance expressed as the thermal energy. Therefore, under certain circumstances the kinetic energy provided by the ionic flux and electric field may determine the transition of proteins between the unfolded states to folded states. It is likely that this flux of charges is modulated by several neurotransmitters as presented in the literature (Kreiss et al., 1996; Schultz, 1998; Monti et al., 1988). o Protein folding is a process that significantly alters the spatial distribution of atoms/charges within proteins. Importantly, at this fundamental level during the writing/coding phase, information retrieved by charges within the protein structure I p cannot be greater than information

carried and transmitted by the movement of charges/flux of ions: I p ≤ IC

( 13 ) Since the information exchange is limited during this transfer, repetition or rehearsal of the process/activity, event etc is usually required for a permanent change in protein conformation. o Building long term memories (LTM) within macromolecular structures involves several biophysical processes, where the selection of information storage depends on neurotransmitter release, number of charges involved, their electric force/ kinetic energy and intrinsic thermal generated noise.

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The above presentation links molecular computing to observations and the earlier memory model provided by Atkinson and Shiffrin (Atkinson and Shiffrin, 1968). While reading and writing are affected by the thermal noise (see Eq.( 11 ))( 11 ) repetition determined by a rhythmic writing/reading process may ensure an acceptable level of signal-noise ratio since the noise may always present slightly different distributions while the signal tends to maintain almost the same patterns. At this level both elastic and plastic changes within macromolecular formations (proteins) can be seen to be determined by Coulomb forces. These infinitesimal ‘dislocations’ of charges and electrons within macromolecular structures can be written in terms of energy: E dis = E elastic + E plastic ( 14 ) Imagine these charges having a certain distribution in the protein structure. Then these particles are perturbed or slightly moved due to interaction with ballistic movement of ions during action potentials/ synaptic spikes determining a slight change in charge distribution, protein folding/binding. To understand these changes, a global picture of protein charge interactions within the cell/synapse is required. Such interactions that occur between charges in movement and electrical charges within proteins can be described by classical inter-atomic potentials. Once the movement of charges is repeated regularly the coding/writing of similar information in the same structures is reinforced. This model of charge distribution allows a natural link to the well known two stage model proposed by Atkinson and Shiffrin model (Figure 1). It is likely that the resultant of electric interaction determine a slight change in the spatial distribution of charges, protein folding. Since some proteins degenerate after a while some of these charges with density ρ e can be seen as short term elastic changes in the electrostatic potential while the others ρ p may be seen as stable plastic changes determined by protein folding. The resulting electrostatic potential Φ is dependent on this new spatial distribution of charges: r r ρ e (r ) ρ p (r ) ∂ 2Φ ∂ 2Φ ∂ 2Φ + + = − − ( 15 ) ∂x 2 ∂y 2 ∂z 2 ε ε This physical implementation allows reading of the electric field Φ (distribution of charges) by simple movements and interaction of other charges at different levels (synaptic spikes, neuronal spikes etc) since the electric field generated by this charge distribution within the infinitesimal volume dτ is: r r ρr 1 E= ( 16 ) ∫ r 2 dτ 4πε Volume

r

where ρ = ρ e + ρ p . The total force FQ acting on the charge Q that moves in space r r r r FQ = QE + QV × B ( 17 ) depends on this spatial distribution of charges within proteins and generated magnetic r r field B (Figure 4). Since usually the component B is negligible, then the dynamics of charges are predominantly changed by electric interactions. Therefore, a change or a new distribution of charges will affect the movement of the ion Q and determine a new trajectory in space. ‘Reading’ information is a process where a moving charge or a group of charges interacts and changes its trajectory depending on spatial distribution of charges embedded within macromolecular formations.

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Chapter 5: From Brain Language to Artificial Intelligence K2 K4

K3

K1

K5

Q

K6

Figure 4: The dynamics and trajectory of the mobile ion Q is determined by interaction with electric charges distributed within protein structure. As a result of changes in conformational state (e.g protein folding) these interactions and trajectory of the ion Q is altered (before folding in blue continuous arrow and after folding in blue discontinuous arrow)

It is likely that information ‘writing’ that simultaneously takes place is influenced by this ‘reading’ that changes the spatial trajectory of the charge. At this microscale level the classic description of interaction from Eq( 17 ) would be incomplete if quantum interactions are ignored. Reading/decoding is performed rhythmically by this charge Q or most likely by a group of charges that moves close by and interacts with the generated electric field. Due to dissipative processes, the rhythmic movement described by the Hamiltonian H : ( 18 ) H =K+P may become small and unobservable during time where K is the kinetic energy and P is the potential energy. The quantization of the energy of this charge can be demonstrated to occur even in case of a classical description, see for instance (Giné, 2005) where dissipative harmonic oscillations may be directly related to intrinsic quantum phenomena: ΨQ (r , t ) = ΨQ (r )e − iHt ( 19 ) In a quantum description reading/decoding is performed by the charge Q with the wavefunction ΨQ and can be seen as the result of interaction with charges within the protein macromolecular formation with the total wavefunction Ψnuclei+electrons : Ψnuclei+electrons = Ψnuclei + Ψelectrons ( 20 ) which may allow the writing of protein energy as being depending on atomic coordinates. In this case changes in memory dM can be seen to be related to changes in spatial distribution of charges embedded in chains of proteins within neurons: r r dM = M (dρ nuclei (r ) + dρ electrons (r )) ( 21 ) At the fundamental level memory is altered in time by even this slight change within spatial distribution of charges within the protein structures. Additionally, such changes in spatial distribution of charges can be maintained by a continuous generation and redistribution of proteins which is likely to be the critical mechanism for long term memory preservation (Li et al., 2004). o Cellular basis of memory: Local protein synthesis within neurons, correct protein folding and binding are critical processes for memory generation and long term preservation

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For simplification we have considered in simulation, regions with high density of positive charges that carry information about the stimulus input (Figure 5). The result presents groups of negative charges represented in blue color that are attracted to the regions with high densities of positive ions which become in this case physical attractors. a

b

Figure 5: Negative charges are attracted to the regions with high densities of positive ions which may act as physical attractors. a,Three-dimensional representation of electric potential for negative charges in blue and positive charges in red b, Bi-dimensional projection of the above representation

Therefore, it is expected that the movement of charges/ions during an action potential depends on their interaction with electric charges which are spatially distributed at molecular level within the protein structures. This spatial distribution thereby determines local polarizations in the proteins which may represent the physical embedding of our memories. As the increase in the density of charges (usually ions) occurs during an AP, pattern formation within a spike can be seen as a consequence or a result of already existent spatial distribution of electric charges within proteins. If spiking is repeatedly performed the flow of electric charges is expected to interact with the existent charges in the protein matrix determining slight changes/deviations in position, changes in polarization or even protein folding. These charges which are nonevenly spatially distributed become further physical attractors to electric charged particles (electrons/ions) and shape the physical memory design (Figure 5). As shown above, processes that occur within the cell related to protein machinery including protein interaction/ binding are likely to determine, during a certain period of time, an accumulation of information. Going back to the computational cube discussed in detail in chapter 4, the effect of these charges distributed within proteins can be modeled by another group of charges (Figure 6) that generates the input component F M for the feedback model (Figure 7). This interaction with charges embedded within the protein structure represented by the component F M in Figure 7 is important since it critically determines the trajectory of charges and their position X in space. This effect can be briefly modeled by returning to the previous dynamic model; the velocity of N charges at step k + 1 can be rewritten in the matrix form:

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Vk +1 = ΑVk + Fk + K E ( R new − X k ) + ΓVk

( 22 )

where Vk represents the matrix of the velocity of charges at step k , Α is the inertial/interaction matrix, Fk is the random noise matrix and R new = [ R | FkM ] includes the effect of charges embedded within macromolecular structures that behaves as an elastic force that projects charges in an n+m-dimensional space depending on stimuli input and ‘memory’ effects. Importantly, as presented above the interaction with charges embedded within macromolecular formations acts as a dimensional expansion phase. The new position of charges in the cube can be computed by integrating this equation: X k +1 = ΕX k + Vk

( 23 )

where Ε is an inertial/interaction matrix and X k is the matrix of previous position at step k of charges. Besides the role played by stimuli this new model incorporates the effect of memories seen as distribution of charges and provides an input component FkM for the dynamic system. There are several phases that can be observed analyzing the dynamics of the system from Figure 7.

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VE αE Figure 6: Representation of the computational cube and the interaction of several groups of electric charges. The input features are represented by charges that shape the letter E. After learning, the distribution of charges is considered to be fixed within the protein structure for a given time interval and activate the component F

M

. Computation occurs as an interaction process between the mobile

group of charges in blue color that moves inside the computational cube and previous memories stored within a charge distribution in proteins.

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Phase 1: The initial step is when memories are not shaped, the input F M ≅ 0 and the system is driven only by the input stimuli S , ( R new = [S ] ). This phase characterizes the early stage of biological life when the semantics of input stimuli are not built. Phase 2: With learning the organization of charges in macromolecular structures gradually provides the input F M ≠ 0 which combined with the input R from diverse stimuli S drives the dynamics of electric charges ( R new = [ S | F M ] . This phase characterizes the normal state of the system where information received from input stimuli is altered by previous accumulated information in memory. In addition, this interaction also changes the spatial distribution of macromolecular structures, their intrinsic conformational changes leading to changes in memory. In this phase, electric interactions shape the expression of electric potentials and internal representations of computation at molecular level. Alterations of memory storage gradually occur at smaller scales, fundamentally altering protein generation by regulating gene selection and expression (Klann and Sweatt, 2008). The RNAi can catalyze these reactions and new proteins that are made may store new and previous information under certain thermodynamic conditions (Tomari et al., 2004; Mückstein et al., 2006). The ability to sustain these processes at molecular level requires energy. Sometimes these reactions can be unsuccessful for example due to alterations in metabolism and as a consequence memory formation or retrieval or association to the semantics can fail. The process of interaction between several groups of charges (where the information received from input stimuli is modulated by previously accumulated and stored information) will determine in time the occurrence of semantics/meaning and is constitutive in developing the basic mechanism of perception that is an important step for the organism. Additionally, the dynamics of charges and their interaction with previous memory can distort information from stimuli and may generate illusory contours creating nonexisting features (such as in the retina or other sensorial areas). The illusion of presence can be explained in terms of dynamics of electric charges and interaction with previous accumulated information. This explanation seems to be more plausible than any other

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hypothesis in the field (Grossberg, Mingolla, 1985; Halko et al., 2008; Domijan et al., 2007). Phase 3: In the absence of input stimuli ( R ≈ 0 ) charges can be driven by information already stored in memory R new = F M . This computation can be seen analogous to processes that help us to remember/recreate previous events, places names carried out in the absence of the external stimuli (see Eq.(44), Chapter 4). The mechanism operates during the dream state as well during a playback of “memories”. Phase 4: Different changes in computational state may occur during our lifetime when widespread functional disturbance at protein level (Kang et al., 1987) affects memories and can be modeled by severe alterations in the input F M . During their movement, charges (e.g. ions) are able to ‘read’ information and their trajectory can be slightly changed by their interaction with existent charges within the protein structure. Importantly, alterations in gene selection, mutations or an incorrect protein folding may determine in this case severe neurodegenerative disease such as Alzheimer (e.g. amyloid accumulation) where information previously stored is distorted or cannot be read (Duff et al., 1996). As a result the altered state of computation can change significantly the outcome of the system and in some cases the acquired semantics are lost (Martin et al., 1985, Hodges et al., 1991). In this case, the model can be seen as a returning to the first phase where R new = [S ] . Besides Alzheimer disease the neuroelectrodynamic model can be extended to account for other disorders such schizophrenia and Parkinson’s disease. Additionally, seeing electric charges as information carriers offers a better understanding of long term potentiation through its intrinsic relationship with molecular mechanisms (Cooke and Bliss, 2006). Importantly, the continuous involvement of electric charges in computation and memory mechanism extends the “storage” role substantially beyond synaptic structures and in fact engages all levels of the cellular machinery. The schematic presentation below (Figure 8) shows that by enhancing synaptic transmission for path A instead of path B the flux of electric charges is able to read different information stored at macromolecular level of proteins within selected dendrites. The selection of a certain path increases the entire activity in the region determining significant morphological changes on the pre and postsynaptic cell (Malenka and Bear, 2004). Additionally, the information carried by these charges is ‘written’ shaping the distribution of charges within chains of proteins embedded in dendritic structures and all synapses involved in this process. We have already demonstrated in Chapter 4 that learning a simple feature (e.g. the letter E) shapes a certain distribution of charges within the computational cube. However, since computation is a dynamic process, this spatial distribution of moving charges is transitory and the time point of reading/decoding information is critical. o Spike timing dependent plasticity (STDP) is a resultant of reorganization of spatial distribution of electric charges within dendritic structures and synapses at micromolecular scale. This phenomenon allows changes in rhythmic information transfer between neurons. With learning the inherent mechanism of organization of charge distribution within macromolecular structures becomes the main process that contributes to an emerging organization of spike timing patterns with millisecond precision.

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Therefore, spike timing and STDP appear to be epiphenomenal results of computation with charges, their interaction and dynamics at molecular level.

A

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Figure 8: Schematic representation of path selection by enhancing synaptic transmission. Chemical synapses in red as solid spheres allow the transfer of information between neurons. The selection of path A instead of path B can be critical in processing, reading and writing different information within macromolecular structures embedded within involved dendritic/axonal branches.

This schematic presentation shows that beyond synapses it is likely that the entire cell structure that contains macromolecular formation must be involved in information processing, communication and storage. o Long term storage in the brain is not solely an attribute of synapses. Embedded macromolecular structures/proteins throughout the nerve cell within soma, dendrites, and axonal branches are involved in this process. o As a consequence these high concentrations of charges are intracellularly or synaptically built where macromolecular structures (proteins) are located in order to facilitate the interaction between incoming information (e.g. ionic flux) and pre-existing stored memories. At larger scales, the activity of cells is usually surrounded by an antagonistic field forming a Mexican hat shape confirmed by several experiments. The activity of cells in the region is commonly measured by analyzing the firing rate of neurons. In the immature ferret the lateral geniculate nucleus presents a weak antagonistic surround in the receptive fields (simple fall-off pattern) while in the adult animals there is a well shaped antagonistic surround (Mexican hat, see Ohshiro and Weliky, 2006). This form is considered to be a result of excitatory and inhibitory processes at network level. However, it is likely that the center-surround receptive field is a result of charge distribution that outlines the effect of learning in the respective area/volume. The example of a specific distribution of charges (Figure 5) may explain this effect (changes in the firing rate/ Mexican hat shape) of the center-surround receptive field.

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Therefore, theoretically the simplest computational model of a neuron should at least include a Turing tape as macromolecular like structures where reading and writing is electrically performed by the movement and interaction of charges (Figure 9).

Figure 9: The computational model of neuron should include protein/molecular structures where the distributions of atoms/ electric charges are formally equivalent to the Turing tape model.

o

Long term potentiation seen as a form of plasticity is determined by molecular changes at synaptic level that alter the propagation path of charges and their interaction o Neural computation: Information processing occurs within the neuron and is carried out by constrained movement and interaction of electrical charges. o Protein folding- Plasticity: The memory model of an event/stimuli is built during changes of macromolecules/proteins (conformational changes) that shape a certain spatial distribution of electric charges at micro-scale level where information is embedded (e.g. protein folding occurs in the milliseconds to seconds range, even minutes). o Protein stability –Stable LTM: The memory model of an event, stimuli is preserved as long as the folding states of involved proteins are kept stable. o Protein regeneration: The degenerated proteins are continuously replaced by new ones as a two-step process of transcription and translation from genes into proteins. First, the regenerative processes tend to preserve previous information incorporated within spatial distributions of electric charges. This process occurs as a consequence of a variety of mechanisms. Since these proteins are in a chain of many, and importantly not all of the proteins are degenerating at the same time, the new ones are naturally shaped by the local electric field and several other bioregulatory mechanisms in order to preserve old and then store new information (Kohen et al., 1990). Second, it is likely that over time, a fundamental and stable change affects the selection of genes at the level where proteins are synthesized and therefore it is likely that the stable, structural conformational state of new generated proteins may be gradually determined. Third, since the electric field that is being expressed itself provides only transient information, the new generated proteins will be shaped to “fit in” rapidly to preserve and store previous and new information, over successive action potentials and ‘charge

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flux iterations’. Therefore, the storage of information within a chain of proteins is locally modulated by complex bioregulatory processes and dynamically shaped by the generated electric field. 5.1.2. What sleep is for? We remember our anniversary or the wedding party by just seeing the label on a bottle of wine. This process of formation or recovery of collections of memories has not been modelled and has never been simulated on a computer to our knowledge. How are these memories built? What is the function of sleep? Why do we need to sleep more during childhood and less when we are adults? A wide range of experimental observations show that during the sleep phase, there is a rhythmic activity of neurons. What is the goal of this rhythmic neuronal activity in the absence of familiar external input stimuli (visual, auditory etc)? What are the reasons to maintain continuous nervous system activity during sleep? Following the Atkinson-Shiffrin model the rehearsal phase can be modeled as a repetitive directed flow of charges that is able to determine long lasting effects in the displacement or polarizations within macromolecular complexes. Action potentials/ synaptic spikes are able to perform changes at microscale level that may involve long lasting effects in neuronal ensembles. Several studies show that there are considerable individual differences in sleep duration which changes with age; however, almost everyone sleeps about one third portion of our lifespan (Aekerstedt, 1998). It is already known that the sleep period is critical for memory consolidation. There is a continuous decrease in the amount sleep over one's lifetime and from birth to adulthood the sleep time is reduced almost to a half (Mindell, 1999). In fact beside several functions such as brain thermoregulation, tissue "restoration", brain detoxification during sleep two other objectives are achieved. First, changes in electric rhythms during sleep phase provide memory maintenance/consolidation. Second, sleep phases are involved in providing the required bioelectrical rhythms mandatory for creating new long term memories based on gene selection processes and appropriate protein synthesis. It was observed that the rapid eye movements are characterised by low-amplitude and faster rhythms of EEG recordings and it is considered that these faster rhythms consolidate memory by playing back events during sleep. The brain seems to be faster at re-running these events during sleep state than performing them in the awake state (Euston et al., 2007; Mehta, 2007, Hahn et al., 2007). This process is similar to the fast forward mode versus a playback mode of a device such as a DVD. It is likely that short time ‘elastic’ changes are responsible for so called “short memory” phenomena while plastic/long term changes determined by new protein synthesis and their conformational changes are related to long term memories. We hypothesise here that the effect of rehearsal (sleep rehearsal) allows that previous “elastic” alterations in the protein structure to be selectively transferred into plastic irreversible conformational changes. These changes in memory can be physically seen as stable modifications in spatial distribution of charges embedded within the proteins or chains of proteins. This phase of new protein generation that occurs predominantly during sleep and incorporates “new memories” allows us to formally describe these changes in memory with a model that after each sleep phase increases its dimension. The current memory M n depends on sleep changes that include the experience from the previous day Dayn−1 within the previous memory M n−1 :

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M n = sleep[ M n−1 | Dayn−1 ] ( 24 ) The process of re-distributing electrical charges within proteins and spatial distribution of protein ensembles is at origin of the process commonly called selforganization. Therefore, self-organization occurs first at this microscopic level with changes in macromolecular formations and later their final effects are perceived at macro-scale level in the reorganization of temporal occurrences of spikes as largely covered by spike timing theory (Sejnowski and Destexhe, 2000). To read this memory other charges/ions have to be rhythmically moved close enough in this space. Synthesis of new proteins, writing information and maintaining memories require different electric rhythms than the ones the brain usually experiences during daily activities and these rhythms are provided during the sleep phases. Importantly, with sleep different categories of genes provide the required support for new protein synthesis (Cirelli, et al., 2004). Therefore, sleep has an important role in generating critical electrical rhythms and it is likely that the sleep phase provides a selection of genes involved in building/writing memories (Stickgold, 2007; Born et al., 2006). Additionaly, molecular and electrophysiological changes detected during homeostatic sleep in animal studies at synaptic level are likely to support this hypothesis (Mignot and Huguenard, 2009). This may explain why we need to sleep more during childhood when we build the memory “model” required to perceive the external world where we live. Sleep deprivation impairs the “maintenance” and further retrieval of stored memories generating so called false memories (Loftus et al., 1995; Schacter et al., 1998; Diekelmann et al., 2008). This explains how different types of memory are acquired at different stages on life. Memories in childhood may be those that are important for the internal structure of the system and help to actually build the system itself. Such memories are not necessarily event based and sometimes not really retrievable. As such, writing this information is energy intensive and initially the data is massive; children require a much higher amount of sleep as mentioned above. o Self-organization is a process of spatial rearrangement of electrical charges embedded in macromolecular formations, spatial distribution of new generated proteins. This process occurs during learning and involves other specific physiological adaptation mechanisms. o With learning the information regarding frequent input stimuli, motor inputs is overwritten at molecular level and some features for more frequent inputs are likely to become overrepresented (see the Penfield ‘homunculus’). o (Sleep vs memory) The process of sleep and internally generated rhythms during the phases of sleep determine the selection of relevant genes for the synthesis of new proteins. Such proteins could then maintain and subsequently consolidate the spatial distribution of electric charges that embed our memories within these macromolecular structures. The analysis of artificial cognitive agents shows that building artificial mental states requires a good model of the environment. Simulations of the agents’ behavior in case of different complex environmental conditions can be extrapolated to our work of understanding how knowledge is embedded in our brain (Sharpanskykh and Treur, 2008). o The model of the external world (MEW) has to be constructed in advance (during the first few years of life) and kept/preserved in memory. This model

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acts later as an initial reference for almost all emerging mental and cognitive processes. An increase in complexity of the environment or in its interaction with the agent requires an increase in complexity of the internal model and this observation can be extrapolated to processes that refine and build memories in brain. 5.2. Spike Timing- An Incomplete Description Starting with Adrian’s investigations firing rate was considered to be an important tool to study neuronal activity. Spike timing coding assumes that spikes from an anatomical source are identical and only the timing of their occurrence really counts. Therefore, more recently the millisecond range of spike time patterns is being considered to be a reliable measure of neural code. The measures of firing rate or interspike interval are hypothesized to accurately reflect neuronal activity and several papers pointed out the importance of this analysis (Mainen and Sejnowski, 1995; Rieke at al., 1997; Gerstner and Kistler, 2002). Determining the neural code from spike timing analyses proved to be a very difficult task which has remained unsolved despite considerable effort in the field. During the last decade intense research in interspike interval coding does not appear to have provided significant advances in brain understanding or useful applications beyond the standard model of firing rate. Several scientists have been lured by the apparent synchrony between spike occurrence and behavioral events in humans or studied animals. This is not surprising since behavioral outcome is often correlated to events at the micro and macromolecular scale. Therefore, in some fashion, indeed spike occurrence is related to analyzed behavioral events. However, there is no reason to believe that spike timing sequences at millisecond range can completely describe/encode the complexity of these events. The timing code is thus an incomplete description and several unanswered questions are left over by using a spike timing paradigm. What is the meaning of the time code in terms of semantics of behavior? What is the relationship with memory? Several experiments performed in behaving animals show that spikes cannot be characterized only by the timing of their occurrence. In vivo recordings performed in freely behaving rodents during a T-maze procedural learning task proved that there are other important features intrinsic to cell neurophysiology which are modulated during the learning phase beyond spike timing alterations (Aur and Jog, 2007a). Neurons showed an increase in spiking activity visibly correlated with behavioral events (tone cue, turning on T-maze). With learning, the number of spiking units decreased between tone delivery and the beginning of turn as did the number of spikes per unit. Spikes from the remaining responsive neurons showed increased modulation in spatial directivity. Importantly, the electrical flow within a spike demonstrated with a vector called spike directivity associates behavioral meaning to spike sequences as rats learnt a T-maze procedural task. As shown in Figure 10, spike directivity is a vector. While firing rate and ISI statistics require sufficient numbers of spikes, spike directivity analysis can be performed for every recorded spike. Spike directivity becomes an important feature able to relate neural activity with the semantics of behavior. The analyses of interspike interval or firing rate do not seem to show a clear relation between neuronal activity and behavioral meaning. At the same time by using spike timing analysis for the same neurons we are unable to demonstrate the relationship with the semantics of the task. Spike timing does not seem to relate this

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activity with underlying meaning of behavior, therefore does not seem to be a consistent measure of complex neuronal activity. Importantly, the semantic representation of behavior can be revealed as modulations in spike directivity within selected neurons during the time that is of highest value for action selection (Aur and Jog, 2007). Additionally, neuron simulations with HH models demonstrate that information is transferred by ionic fluxes within millisecond-level time domain. Mutual information between input signal and sodium flux is about two times that between input signal and output spikes during each spike within a millisecond-level time domain (Aur et al, 2006). Why is spike timing theory in trouble? It seems that by taking into account only the time of AP occurrence the richness of neuronal processing that occurs within the neuron is lost. In fact can anyone show that “information” in a wave can be completely described by the instant of time when the wave is crossing the zero value? Since the spike itself is a spatiotemporal event, information is definitively lost if only the time of spike occurrence is considered (Figure 10). This is similar to the loss of information from moving from a higher dimensional to a lower dimensional computational space. It is unlikely tha such transformation will preserve information since a vector (spike directivity) cannot be fully described by the corresponding time with ‘millisecond precision’. Therefore, time coding alone cannot completely represent neuronal activity and at least the observer should consider both characteristics, the time of spike occurrence and the vector property of the spike.

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Figure 10 : Schematic transformation from spatial coding (spike directivity) to temporal coding. This transformation cannot provide a one to one correspondence since a vector (spike directivity) cannot be fully described as a temporal event.

Based on these experimental results we claim that spike timing alone may not entirely reflect the richness of the neuronal code. Therefore, finding neural code only in terms of spike timing seems to be an “ill posed problem”. The continuous flux of electrical charges during synaptic spikes, action potentials, allows us to record with adequate tools and analyze several brain processes. Seen at smaller scales, recorded with electrodes, changes in electrical activities may be related to alterations determined

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by severe dysfunctions or diseases in the symptoms of Parkinson disease such as rigidity, tremor or even dyskinesia as well as illnesses such as schizophrenia. Others involve regular stimulus processing of sounds or image recognition. These electrical events can be recorded and analyzed and all models presented below are genuine proofs of complex computation in the brain. In the following paragraphs we provide a glimpse of how the neuroelectrodynamic model can be applied to diverse issues such as sensory auditory processing, normal phenomena of learning, schizophrenia and degenerative disorders such as Parkinson’s disease. The application is not meant to be exhaustive in any way and will be developed further in future publications. However, it will at least give the reader a concept of the applicability of neuroelectrodynamics to real neuroscientific problems, taking it from a theoretical to a more practical domain. 5.2.1. Models of Auditory Processing The brain is an active analyzer of input stimuli which changes local and global states dynamically. Eyes, ears, nose, skin and tongue can be claimed in terms of computational view to be our interface with the external world. Every received stimulus determines a flow of charges (ions/electrons) in the intra and extracellular space of neurons that generate fluctuating electric fields. As described above it is expected that this dynamics and interaction of charges is able to reflect emerging cognitive activities and motor processes. This aspect can be evidenced by analyzing electroencephalographic recordings during visual and auditory tasks. Auditory evoked potentials reflect the processing of acoustic stimuli in the cortex. Common interaction between visual and auditory processing in the brain makes it difficult to analyze these brain waves since the detection of auditory evoked potentials (AEP) are altered by motor or visual activity of the subject resulting in a very low signal-to-noise ratio. Our ability to integrate some sounds and dissociate others depends on this movement and interaction of charges that span neuronal ensembles and leads to a natural decomposition, integration and interpretation of input features. It is commonly recognized that the components of the input stimulus or their transformed characteristics are hidden in the EEG activity. There are two main theories that attempt to explain these evoked potentials. The additive model sees any recorded signal e(t ) as a superposition of response to stimuli s (t ) , electrical activities that are not correlated with stimulus a(t ) and the unavoidable noise n(t ) due to the recording process: e(t ) = s(t ) + a(t ) + n(t ) ( 25 ) In order to increase the amplitude of the signal relative to artifacts several approaches use averaged EEG recordings over trials. The second approach, the “phase resetting” model suggests that the “average model” is inappropriate to analyze the recorded signal. We can easily demonstrate using information theory that the averaging procedure masks information contained in individual trials and does not necessarily eliminate the sources of noise. Therefore, the recorded signal can be decomposed in single trials and the mutual information between the input signal s(t) (in magenta) and the recorded potential e(t ) in red can be computed (Figure 11). This type of analysis shows the dynamics of how the sounds are perceptually integrated and processed in brain.

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Figure 11: Representation of information processing in the auditory cortex. High amplitude of mutual information in blue color shows in time where specific information regarding the input stimulus is processed. The stimulus (5KHz, 50 ms pure tone, 75 dB SPL) is represented in magenta and the EEG values are in red.

There are three significant changes in mutual information values that follow the delivery of the signal processing in the auditory pathway. First, it involves the primary sensorial areas with a sharp rise, a flat aspect and a fast decrease in mutual information values that seems to preserve the duration length to be about the same as the stimulus’ temporal length. The second and third changes may represent different auditory areas and are characterized by a sharp increase in mutual information followed by a sharp decrease and then a slow decrease in value with temporal length greater than the stimulus length. High intensities of mutual information correspond to sensory memory process supposed to last about 200-300 ms after the occurrence of the stimulus. The sound is processed within the primary auditory cortex and the electrical waves in these auditory areas show opposite phases that propagate spatially and attenuate after several seconds. This analysis of mutual information in auditory event-related potentials is able to dissociate early cortical responses and late latency components related to late memory (Chao et al., 1995). The presence of stimulus processing can be seen in mutual information peaks that occur even after 0.5 s after stimulus. The analyses of evoked potentials in single trials using mutual information techniques illustrate several important aspects. First this demonstrates that information about the stimulus is processed in successive spatially defined areas in the brain. Additionally, it seems that this information about the input stimulus is rhythmically/cyclically processed. A better view of this phenomenon is expected to come from a spatial projection of electrical activity in a similar way as we have developed for action potential recordings. Importantly, all these observations reinforce the idea that changes in electric field and computation with charges could be essential to information processing. Since electrical charges are information carriers the new model can explain these fluctuations in MI as a physical requirement to write and read information by charges.

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o

The presentation of a stimulus input determines a successive, repetitive phenomenon of information processing and communication carried by the generated electric wave in the auditory pathway The model of computation with interacting charges (neuroelectrodynamic) and the role attributed to ‘memories’ in controlling the flux of charges is presented in (Figure 12). Schematically the rhythm of electrical flow is changed by modulating the term Γ (see yellow arrow) while the intensity of the charge flux is modulated and represented in this model by green arrow. Both controls depend on stored memories F M embedded within proteins in neurons/synaptic connectivity. As stated earlier, information written within the protein chains cannot be greater than the information carried by the movement of charges/flux of ions. Therefore, to read/write this information a certain number of iterations are required and this assumption is confirmed by the peaks in mutual information values (see Figure 11). FM

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Additionally, the above investigation shows that the analysis of evoked potentials can be performed in single trials and to obtain useful results, there is no need to average several trials as is currently done in many labs. In fact this analysis demonstrates that “average techniques” are inadequate to process evoked potentials since even in case of similar repeated stimulus inputs, the characteristic of EEG recordings in individual trials are different depending on emerging cortical processes and artifact occurrences. Additionally, averaging techniques do not guarantee that the noise or artifacts are completely removed. It is likely that these rhythmic processes are related to a rhythmical reading and writing in memory. Such observations can be linked to the resonance type models developed previously by Grossberg and colleagues (e.g. adaptive resonance theory, Carpenter and Grossberg 2003). 5.2.2. Models of Schizophrenia It is well known that schizophrenia develops over time typically in late adolescence (Harrop and Trower 2001). Schizophrenia is manifested by several symptoms including

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hallucinations, impairments in perception and selective attention, delusions, thought disorder, disorganized speech, cognitive impairments or disorganized behaviors. The study of the schizophrenic brain in humans and animals shows that once brain structures have been changed to manifest schizophrenia, there are significant alterations within the prefrontal cortex (PFC), nucleus accumbens, hippocampus and amygdale (Zmarowski et al., 2005). The main hypotheses regarding schizophrenia is related to observed changes in dopaminergic activity. Dopaminergic tone is defined by the properties of synaptic events, which include the amount of released dopamine, the sensitivity of dopamine receptors, and the length of time dopamine spends in the synaptic space. Gray showed that an excess in dopamine release causes a disruption in latent inhibition in rats (Gray, 1998). If the dopaminergic transmission is blocked a potentiation of latent inhibition occurs. We will show that these changes in neurotransmitter activity control the electric activity in neurons and network of neurons. However, neurons are electrochemical structures which communicate information electrically. Therefore it is expected that these changes can be evidenced in electrophysiological properties as recordings in single neuron or in groups of neurons (Middleton et al., 2002; Konradi et al., 2004; Mirnics et al., 2000). The main issue that remains to be resolved is to build a schizophrenia-like model which can be used to test the effects of different drugs. Recently, a number of animal models have been described showing considerable validity as models of schizophrenia. These models and method s previously described (Rajakumar et al., 2004) enable functional studies of schizophrenia at network level. This work shows the results obtained from the analyses of extracellular multi-tip recording in freely behaving animals. The effect of pharmacological administration of ampheta mine (subcutaneous amphetamine administration 1mg/kg) shows that electrical changes occur differen tly in these two models and there is a huge difference in electrical activity response between a normal rat and a rodent model of schizophrenia. Amplitudes of each incoming spike waveform were recorded individually on every tip of tetrodes in the PFC and in both the normal animal (Figure 13) and in the schizophrenic model animal (Figure 14). The analysis of the average spike amplitude reveals that in the normal animal at baseline there are no large amplitude spikes until amphetamine is given (Figure 13,b). However, schizophrenic animals already have large amplitude spikes even at baseline which disappear after amphetamine administration. In addition these changes in electrical activity from recorded neurons in the schizophrenia model animal occur gradually after amphetamine administration. The process of returning to the baseline values usually also takes about four hours in almost all recorded neurons. However, some neurons continued to present increased values over the baseline level even after 4 hours. Such differences in electrical response confirm known observations that neuronal activity can be further modulated by pharmacological interventions such as administering amphetamine or other drugs. Dopaminergic input is considered to selectively potentiate the 'memory fields' of prefrontal neurons (Williams and Goldman-Rakic, 1995). Maintaining normal probability of neurotransmitter release is controlled by specific proteins (Powell et al., 2004) and is likely to look similar to schizophrenia-like changes in an existing mouse mutant model (Rushlow et al 2005).

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However, this does not demonstrate that information in the technical sense is effectively conveyed by chemical modulations. The comparative study of the normal versus schizophrenia induced animal model shows differences in electrical amplitude patterns in neurons that are directly related to observed changes in the rat’s behavior (increase in sniffing activity etc). The visualization of change in the amplitudes of spikes takes about 20 to 30 min after amphetamine administration. Importantly, our observations show that the amplitudes of action potentials increase or decrease gradually after amphetamine administration. This mainly contradicts the hypothesis of an “all or none” spike event currently used as a model in spike timing theory and reinforces the model of charge movement and electric interaction as a form of information processing (computation) that can be modulated by several neurotransmitters. In the long term, permanent changes in electrical activity after amphetamine administration can be related to protein synthesis at dendritic spines and cytoskeleton

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associated proteins (Guidotti et al., 2000, Benítez-King et al.,2007) and can be schematically modeled using the neuroelectrodynamic model (Figure 15). The plot formalizes the interaction with the resultant electric potential determined by the input stimuli and charges embedded/distributed within macromolecular structures in protein formations. The increased flow of charges in the normal rat model can be seen as an amplification of input component F M and number of charges N involved in computation (Figure 15,a) as a result of amphetamine administration. However, for a schizophrenic model it determines a decrease in F M and number of charges N involved in computation which may be due to altered microtubulin protein function (Figure 15,b). Modeling this effect can be attributed to changes determined by the amplification matrix Π in the dynamics of charges: Vk +1 = ΑVk + Fk + Β( R new − X k ) + ΓVk ( 26 ) where R new = [ Rk | ΠFkM ] increases also the input dimension. Additionally, these alterations at protein level could have an impact on Γ matrix values which control the rhythms of electrical flow and may be seen as an effect in increasing or decreasing firing rate/interspike interval in analyzed neurons from prefrontal cortex. Importantly, the molecular mechanisms involved in the alteration of proteins seem to be strongly related with putative susceptibility genes (Egan et al., 2001; Harrison and Weinberger, 2005). 5.2.3. Models of Learning The organization of spike directivity presented as an example in Chapter 2 shows that after procedural T-maze training a clear electrical pathway for information propagation is displayed. This organization of electrical activity is a reflection of subtle phenomena that occur within neurons during learning. If initially the neuron displays a whole range of spike directivities the process of learning chooses just the preferred pathways for electrical flow and spike directivity points specifically to a certain preselected region after learning. Therefore, it is expected that a stimulus or a certain behavior may have a representation in terms of spatial distribution of charges within

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such a region. The successive action potentials in neurons and in the ensemble of neurons depend on synaptic alteration, changes in gap junction connectivity. However, in this case it is likely that all these activities have a final outcome in directing electric flux toward a specific region in brain. Once the agglomeration of charges has achieved the required density it is likely that this information is rhythmically “written” and “read”. Therefore as a result of this organization in charge distribution with learning, temporal changes in the occurrence of action potentials are determined. This process can be evidenced by computing ensemble mutual information (EMI) where changes and re-organization in spike timing that occur with learning in neural ensembles are analyzed (Jog et al., 2007).

Figure 16: Mean ensemble mutual information (EMI) plots for a rat during training. The X-axis shows training sessions while the Y axis shows the mean EMI values. (Adapted from Neuroscience letters, Jog et al., 2007)

It is likely that these changes that occur in charge distribution at the most fundamental level within macromolecular formations /proteins are altered during several training sessions and they make a sudden big difference at macro-level in pattern of spikes and in learning motor habits. This moment was termed to represent the “tipping point” in neuronal ensembles during learning and occurred after several sessions of training. Therefore, the observed “tipping point” may be a property that reflects the flow of charges within the network that occurs as a result of selforganization of charges within macromolecular structures. This is important because the intrinsic reorganization of charge flux depends on changes in interaction at macromolecular level resulting in the spatial and temporal reorganization of spikes in the network. Thus, temporal observables form a small and limited portion of changes determined by computations that have occurred and have enriched the outcome of the ensemble. o The reorganization of spatial densities of charges on a small scale, within macromolecular formations in neurons determines the occurrence of synchronous spike timing patterns in recorded neurons resulting in a sudden increase in ensemble mutual information. Once this organization of electric charges occurs within proteins, chain of proteins and their distribution within neurons (Figure 17) the “tipping point” is reached and the neuronal ensemble maintains a high level of mutual information seen in visible, correct behavioral responses. However, it is suggested that these temporal spike patterns are a poor representation of information richness that is being handled and stored intracellularly at charge configuration level within macromolecular formations.

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Starting from a persistent low level, a point of abrupt change in mutual information in the neuronal network occurs as the task is acquired. Even though building memories is a gradual process that shapes the charge distribution within macromolecular structures, the change in ensemble mutual information is an abrupt nonlinear development determined by the time scale where the analysis can be performed (sessions of recordings). Besides this increase in local interaction with learning, the flux of charges becomes spatially oriented and the directivity of electric flow can be determined by three cosines angles v = [v1 , v 2 , v 3 ]T . The dynamic model can be then extended to include this spatial modulation of charge flow: ⎡ X k +1 ⎤ ⎡ Ε O ⎤ ⎡ X k ⎤ ⎡Vk ⎤ ( 27 ) ⎢ν ⎥ = ⎢ ⎥⎢ ⎥ + ⎢ ⎥ ⎣ k +1 ⎦ ⎣ M J ⎦ ⎣ν k ⎦ ⎣ O ⎦ where the coefficients in matrix M and J characterize local changes in directivity. This model can be rewritten in the following form: X kE+1 = Ε E X kE + VkE ( 28 ) and schematically represented in the (Figure 17) . Other modulation may involve changes in spatial propagation of information which can be seen as alteration in matrix coefficients M and J that characterize local changes in directivity of charge flow, see Eq.( 27 ).

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5.2.4. Models of Parkinson’s disease and Dyskinesia Parkinson’s disease (PD) is a neurodegenerative disorder characterized by rest tremor, rigidity, difficulty in the initiation of movement, slowness in the execution of movement and postural reflex impairment. The motor changes in PD occur due to basal ganglia dysfunction, especially the death of the dopamine neurons of the substantia nigra pars compacta (SNc). Pharmacological therapy restores dopaminergic equilibrium, by using levodopa. However, these beneficial effects are not permanent and after a period of treatment are accompanied by fluctuations in the motor response in addition to development of levodopa-induced dyskinesia in a large majority of the patients. b 3.5

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To understand the differences between Parkinson patients and levodopa induced dyskinesia we used rat models where we recorded with multi-tip electrodes in the striatum. The data analysis shows that fundamental alterations occur within the interconnected neuronal network of the basal ganglia nuclei, measurable in terms of changes in local electrical properties as well as information flow within the network. Since levodopa induced dyskinesia develops over time, electrophysiological recordings during dyskinesia development present these modifications in electrical activity. The changes in mutual information can be differentiated between Parkinson and levodopa induced conditions. These changes in ensemble mutual information in the normal animals and dyskinetic animals have been observed (Figure 18a). The processed data from striatum showed a slight increase in the local information transfer after levodopa treatment (see the mean ensemble MI in striatum over time after levodopa treatment in red and before treatment in blue (Figure 18b). This change in ensemble MI corresponds to a shift in the power spectrum of the mean ensemble MI to a lower frequency domain (Figure 18b). While high frequencies are easily filtered by the body dynamics this shift in low frequency bandwidth has an outcome and becomes observable as dyskinesia. Seen as changes in the spectrum of information transfer, these observations reflect the model of computation with charges and the predominant role attributed to proteins in the pathogenesis of neurodegenerative disorders that can be altered by disease or

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current therapies as drug treatments (Buée et al., 2000). These systemic changes are often related with cognitive processes and memory problems that occur in Parkinsonian patients (see interrupted border in blue arrow, Figure 19). FM

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Importantly, such alterations at protein level cause changes in the rhythm of electrical flow and can be modeled by the term ΓFζ represented by the red arrow in the neuroelectrodynamic model (see Figure 19 and Eq. ( 26 )). 5.2.5. Brain Computer Interfaces It has been shown for several decades that electrical flow in the cortex encodes information regarding movement (Georgopoulos, 1982). The average firing depends on the preferred direction of movement for which the neuronal firing is intense. If movement directions are farther from the analysed preferred direction, then neuronal activity is gradually reduced shaping a cosine tuning curve. The cosine tuning curves, describe the average firing rate of neurons. Currently, there is no satisfactory explanation for this occurrence; however, the dynamic model of computation with charges is likely to display similar properties and suggests the basis of these observations. It is expected that two different motor movements with specific features would have characteristics coded as the flow of charges that point to two different motor cortex areas (Figure 20). It is likely that spatial distance between these two areas will increase if the movement patterns are different. Therefore, since the signal is recorded only in a specific region of the brain it is expected that this area receives an increase in the density of charge flow /ions which is naturally correlated with the observed increase in the firing rate. Therefore, it is expected that by controlling this electric field/flow of charges one can direct and construct different movements. Such control can in principle be achieved by silicon chips implanted in the brain or using an external processing or computational device. Brain computer interfaces (BCIs) had initially been seen as a science fiction idea however, they may become one day a way to restore human brain functions. Recent advances in computer science and technology made this idea feasible. Research

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on BCIs began early in the 1970s. However they had not very successfully been implanted in animals or humans until the mid-1990s.

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Brain-computer interfaces that use recordings from chronically implanted electrodes provide a direct communication between human brain and computers or an external device. Sometimes called direct neural interface or brain-machine interfaces, these devices are able to facilitate information transfer from the brain to a computer or reversely from a computer to brain. Even though currently there are many reports of success in the field, this success has not been based upon a well designed theoretical model and indeed may have been helped substantially by brain plasticity. There are two main categories of BCI. The first category includes prior subject training to build the neuroprothesis. The best example is a self-feeding task where the monkey (Macaca mulatta) uses a prosthetic device controlled by cortical signals (Velliste, 2008). The monkey was firstly trained to control the arm using a joystick. After this training period the monkey was implanted with microelectrode arrays in the primary motor cortex and the monkey used cortical signals to control the prosthetic arm to feed itself. The second category uses non-invasive EEG systems where recorded signals are processed using machine learning techniques and the need for subject training is considerably reduced. This system is used to help people who have experienced severe motor disorders or paralysis to modulate motor-sensory rhythms or to trigger the movements of the prosthesis (Blankertz, et al., 2004). The major issue for this method is the huge inter-subject variability with respect to signal patterns associated with recorded signals. Beside movement disorders, neurocognitive prostheses are a new class of devices able to repair and sustain attention, language and memory by means of implanted electrodes, transcranial magnetic stimulation or EEG signals (Serruya and Kahana, 2008). With recent advances in technology and knowledge, next generation of electrical interfaces to the central nervous system are expected also to supplement a broader range of human functions rather than simply repair them. The current treatment of neurological disorders including Parkinson’s disease, and other movement disorders also use electrical stimulation for direct therapeutics. Knowledge of cortico-basal ganglia-cortical circuit is essential for understanding these motor dysfunctions (Alexander et al., 1990; Albin et al., 1989).

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However, the outcome of manipulating the brain through electrical stimulation is not always predictable. As an example the application of high-frequency electric stimulation in thalamus seems to generate lesion-like effect while the patients treated with DBS of the subthalamic nucleus (STN) and globus pallidus internus (GPi) suggested a more complex mechanism of action.(Benazzouz and Hallett, 2000, Bellinger et al., 2008). The specific changes in stimulation waveforms, correlated with local degree of inhomogeneity and anisotropy seem directly to affect the neural response (Butson and McIntyre 2007; Sotiropoulos and Steinmetz, 2007). What exactly is neuromodulation? How well can people learn to control neural responses? How can these be used to control external devices? These are important questions that have not received yet a clear response. The most fundamental limitation in understanding the effects of electrical brain stimulation comes from the fact we are not able to understand the principles of how neurons function. The complexity of processing and computation has been simplified to spike timing approach. This view has had a strong impact on the outcome of the past research efforts and this is one reason why we have not been able to understand how deep brain stimulation (DBS) acts and how the brain responds to it. o Several neuromodulation techniques were performed in absence of a well developed theoretical model and in many cases their success could simply be randomly determined based on brain plasticity The neuroelectrodynamic model we now propose offers a new approach in this field. Since we have already demonstrated that information is carried by a specific changes in electrical activity, this novel model of computation with charges explains why these generated electric fields are important at micro-scale level within protein/ macromolecular structures where the first effects of neuromodulation occur. Two important local effects are commonly expected. First, neuromodulation changes the number Ν of charges involved in computation and second it provides a ‘modulation’ in the rhythms of electric flow determined by selected parameters of electric stimulation. Therefore, alterations of neuronal circuitry are expected to follow these changes at micro-scale level (Figure 21) which are modeled based on the previous equation that reflects the movement of electric charges: Vk +1 = ΑVk + Fk + K E ( Rk − X k ) + ΓVk

( 29 )

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where changes of values in Π matrix account for the effect of electric field on the distribution of electric charges within proteins and Λ determines the changes in rhythms of information processing seen as a neuromodulation effect. Additionally, the new model of computation with charges (Figure 21) illustrates the role of charge distributions within macromolecular structures. This mechanism is critical in providing several solutions to improve current DBS technique by a selection of electrical stimulation parameters. Recently, these techniques have been used to treat other neurological diseases expanding the domain of disorders treated by DBS by a stimulation of white matter tracts adjacent to the subgenual cingulate gyrus (Mayberg et al., 2005), which may be very useful in the treatment of resistant depression.

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5.2.6. Moving from Spike Timing to NeuroElectroDynamics We are aware that too much simplification can alter the nature of analyzed phenomena and first impressions may be misleading. As an example of when we know that the incoming light from the sun shapes every object in the room we are not able to perceive the intrinsic phenomenon of the incoming light being composed of particles (photons). However, the (quantum) theory of light has already addressed this issue and we know that photons can behave simultaneously like waves or like particles and importantly they can carry energy and momentum. By similitude we are aware that many neuroscientists have been lured by this spectacular occurrence of spikes with their millisecond time precision and their synchrony. The last sixty years of observations in electrophysiology were concentrated in the spike-timing domain as the keystone of understanding brain function. However, as shown in previous chapters, these opinions have swung back and forth in this field regarding the role of firing rate, interspike interval, spike timing patterns, synchrony and oscillations and we know nothing more about neural computation than we knew twenty years ago. The neuron doctrine combined with current spike timing framework does not seem to be able to explain several experimental results (Lehar, 2003) and one cannot help but feel that there are some unresolved deep rooted issues regarding the completeness of the current spike timing theory. The principles of spike timing alone fail to show how the brain works since spikes are not stereotyped events. This over simplification seems

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to be another important issue ignored by many scientists. In fact, spike timing theory pretends to be able to reconstruct the richness of input stimuli by knowing the points where the stereotypic wave occurs (see Gerstner and Kistler, 2002). Even though knowing the time of AP occurrence is useful, this information alone does not seem to be rich enough in its content and/or quality to capture all the important features of the analyzed system. Spike timing theory is expressed in a huge number of coding versions (firing rate, ISI, spike-timing-dependent plasticity etc). The observed accuracy of spike time patterns should have by itself earlier led to better predictions in terms of sensory response or behavior. Importantly, models that limit their observations to spike timing occurrence and assume the stereotypy of action potential (AP) seem to be inconsistent with imaging techniques developed at cell scale and cannot reliably describe the richness of brain computations. The physical mechanism of “reading the time” of spikes delivered by the presynaptic neurons to the postsynaptic neuron has never been explained. It is becoming clear that there is not enough experimental evidence to provide a coherent explanation even for ISI coding (Latham et al., 2006). Although grasping behavioral meaning from neuronal activity is important, current information processing theory of spike timing did not always relate these measures with phenomenological experience. Additionally, our understanding of brain activity is being shaped by imaging techniques (fMRI, PET, MRI) and less on spike timing framework. The new analysis of action potentials and spike directivity measure may create the impression that computations in the brain can be reduced to a sum of spikes vectors. However, as presented above we found semantics in simple spikes where electrical patterns are spatially distributed and directly related with input stimuli and changes in behavior. Importantly, for any brain model of computation, these observed neuronal activities have to be integrated in the larger universe of brain function and not to be seen in ‘small’ slices. Based on these new experimental observations of hidden electrical patterns in spikes, we have proposed the model of electrical charges combined with molecular and genetic mechanisms to account for observed neuronal activity and computations in brain. Current theory of neuronal activity assumes that the time of spiking is critical for computations in neuronal ensembles. However, extensive analyses at nerve cell scale using imaging techniques reveal patterns of electrical activities within recorded action potentials. The imaging of electrical patterns during several APs has been provided as experimental evidence for spike complexity (Aur and Jog 2006). We have postulated that the neuron membrane represents the physical barrier for charges to move. Charges can escape from this trap due to certain potential configurations. The result is that the movement and interaction of charges to specific spatial coordinates inherently performs intrinsic complex physical computations. Another novel aspect is that the presented experimental observations relate neuronal activity directly with the semantics of behavior. Since electrical patterns in spikes can be naturally associated to stimuli or behavioral events, as demonstrated in the previous chapter, the movement of charges across bounded spatial regions (e.g. neurons) generates meaningful representations. Importantly, the brain cannot be seen in terms of slices (Steriade, 2001) without losing its global interactive functions and this dynamics of charges can then be extended from neuronal spikes and synapses, to the entire brain as a whole. This new development of neuroelectrodynamics of charges as information carriers challenges current views of spike timing theories where reduced neural circuitry is able to perform autonomously disconnected tasks.

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While the time of AP occurrence seems to be a “poor” carrier of information the movement of charges in space and their interaction is informationally rich. Specifically, this outcome is supported at larger scales by current brain-imaging techniques where spatial occurrences of neuronal activities play an important role. At micro scale, the propagation of charges during action potential may fill the current gap between molecular computing techniques and time analyses of electrophysiological recordings. The charged particles in movement become physical carriers of information that perform complex computations within every spike (action potential, synaptic spike). Therefore, using above observations we propose here a new model for neural computation in the brain. o Visualizing electric charges and electric field as information carriers significantly alters fundamental characteristics of computational model. Spike timing model claims that information processing within synapses is accomplished by a ‘highly structured temporal code’. This is complemented with another biased hypothesis that synapses alone are able to provide long-term memory storage. Since “cells that fire together, wire together” firing is commonly seen as a method of information communication. This new description of computation with charges highlights a fundamental powerful way of information processing along with a strong capability of information communication. Our view differs radically from previous assertions and rightfully places computation inside neurons where charges, seen as carriers of information during their intrinsic movement and interaction with molecules, perform complex computations. o Computation is carried out by the movement and interaction of charges within neurons. The neuron is the place where several molecular mechanisms are involved in appropriate protein synthesis which is an important process required in long-term memory storage. Molecular structures that embed similar features are distributed in many different neurons. Once the stimulus which contains certain features is presented, some neurons are selectively activated creating the appearance of a heterogeneous sparse activity within the network (Yen et al., 2007; Tolhurst, et al., 2009). Reading of these memories is performed by electric charges during their interaction and movement and this process cannot be reduced solely to synapses. o Synapses merely allow the transfer of information between neurons while long term storage is provided by refining spatial distribution of charges within macromolecular structures (protein formations). This long term storage within molecular formations is not limited to synapses. The electrical patterns within spikes are the result of interaction between the dynamics of charges across neuronal soma, dendrites and axon and embedded charges within macromolecular formations (stored memories). The result of this interaction expressed in patterns of activation becomes correlated with emergent behavior (see Aur and Jog 2007a, Aur and Jog 2007b). o Semantics occurs naturally during computation with charges by an intrinsic interaction with previous stored memories embedded at molecular level. Therefore, we acknowledge the triple role played by charges in the brain computations since they are able to read information, transfer information by electric field and their movement and store information within a given spatial distribution of charges in macromolecular structures (e.g. proteins). Validated by experiments the model of interaction of charges can be extended from synaptic spikes to action potentials and further globally to the entire brain.

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The scalability property: The model of charges (the neuroelectrodynamic model) has the property to function and accommodate effectively at different scale/sizes where the model can be implemented and perform computations. The neuroelectrodynamic model can be seen as a compartmental model (Hines and Carnevale, 2001) that can be extended from a microscale level to larger scale neuronal areas in the striatum, hippocampus, cerebellum to model the dynamics and interaction of electric charges. An analysis of electric flux may allow the identification of dynamic parameters that on average can make the difference between normal states of computations and altered states of computations (known as severe neurological disorders) briefly analyzed above in this chapter. At microscale level, several processes are directly influenced by generated electric fields including new protein synthesis, a result of complex biomolecular computing mechanism (Nakanishi, 1997). Influenced by biological rhythms and thermodynamic properties (Wright and Dyson, 1999) it is likely that intrinsic quantum phenomena at nanoscale level determine DNA/RNA translation and gene selection for new protein synthesis (Khandogin et al., 2000; Khandogin and York, 2002; Khandogin and York, 2004) (see Figure 22).

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Interestingly, this basic scheme of biomolecular computing pictured in (Figure 22) is similar to the engineered model presented in Chapter 4 where the selection of ‘genes’ is used for changing the values of synaptic strengths and their spatial position (see Figure 18/ Chapter 4). Instead of spatially positioning synapses, the natural “genetic” program regulates the spatial distribution of atoms/electric charges at protein level which are continuously synthesized and replace the naturally degraded proteins. o Gene selection (regulation and expression) required for DNA-RNA transcription and new protein synthesis occurs as a result of various bimolecular mechanisms in response to a wide variety of extracellular signals including electric interactions and determines the synthesis of appropriate proteins. Automatically under the influence of various factors the genetic “program” is able to shape and ‘self-organize’ the distribution of atoms/charges in newly generated proteins that incorporate and store previous and new information. Conformational changes that occur in interconnected protein polymers can be triggered by several factors and coupled to charge redistribution phenomena. This view reinforces the description of molecular computation as previously described by others such as Frohlich, Hagan, Hameroff and Craddock (Frohlich, 1975; Hagan et al., 2002; Hameroff et al., 2005, Craddock et al., 2009). This entire activity is constructed as a bidirectional process since specific proteins can also control the expression of many different genes under the influence of various external factors (hormones, neurotransmitters, growth factors, see (Alberts et al., 2002). Summarizing the above general scheme, the coding phase that describes the dynamics of electric charges can be modeled by the following equation: Vk +1 = ΑVk + Fk + K E ( R new − X k ) + ΛΓVk ( 31 ) where Vk represents the matrix of charges velocity at step k , Α is an inertial/interaction matrix, Fk is the random R

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( R = [ Rk | ΠFkM ]) . This first stream resumes the dynamics of electrical charges during synaptic spikes, action potentials. Additionally, as shown above there is an important mechanism of coding that completes computation performed by charges, operates at gene level and builds new appropriate proteins with specific atomic/charge distribution. This mechanism can be considered the fundamental stream of coding/decoding that includes complex molecular computations of gene expression and regulation at the DNA/RNA level schematically presented in (Figure 22). Both the coding and de-coding streams of processing are highly inter-dependent as they are intricately regulated at the molecular scale. Charge migration in DNA, available tunneling channels and pure quantum transport through DNA demonstrate that the mechanisms of computing in brain are built starting from a quantum level (Chakraborty T., 2006). The mathematical formalism presented in Eq.( 33 ) that includes the effect of presence of charges from proteins shows that the dimension of the computing space increases with new synthesized proteins. It is likely that this dimension decreases with protein degeneration. Therefore, the process of protein generation/degeneration is another dynamic process that may highly influence the dimensionality of computational space within every neuron and networks of neurons. new

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The learning phase is a process of distribution/redistribution of electric charges that are increasingly organized within existent macromolecular formations, in new synthesized proteins that are transported to appropriate parts of the neuron. Such interactions of moving charges described in Eq.( 31 ) with electric charges distributed within proteins can be interpreted as reading/decoding and is modeled using a nonlinear function (see Chapter 4, Eq. 30): FkM = f M ( X kE ) ( 32 ) where X kE characterizes the position of all charges distributed in space and their local directivity ν k : ⎡X ⎤ X kE = ⎢ k ⎥ ( 33 ) ⎣ν k ⎦ which can be computed using the Eq.( 27 ). Therefore, the neuroelectrodynamic model schematically described in (Figure 22) includes several loops that describe electrical interactions and an important loop that regulates molecular computations. Additionally, the transportation of generated proteins to appropriate parts of the neuron has to be included within post-translational processing at molecular level. Importantly, this basic scheme shows that information cannot be limited to gene sequence or their combinations as described by Crick (Crick, 1958; Crick, 1970). The existence of this “feedback loop” mediated by electric interactions is critical and challenges the central dogma of molecular biology regarding the direction of information flow solely from DNA to proteins. Besides various molecular mechanisms, electric charges as active carriers of information may provide a vigorous feedback required for appropriate protein synthesis. It is likely that this feedback requires additional information transfer which may explain high values of information contained within a single spike (see Chapter 4, Eq. 102). For the first time to our knowledge this feedback mechanism that involves electric charges as carriers of information at molecular/genetic level is discussed as an important and possible component of computation in brain. o NeuroElectroDynamics can be extended to directly link ‘ neural’ computation to molecular processes and molecular computing. This new model reinforces the idea of neural Darwinism at this microscopic scale and relates computations performed by electric charges with genetic principles of neural plasticity. From this perspective the new model can be directly connected with functional molecular basis of neural development discussed earlier in the literature (Edelman et al., 1985; Edelman et al., 1987). Importantly, the conductance in specific dendritic, axonal branches changes with a new distribution of charges within new synthesized proteins and this process reinforces the theoretical concept of minimum path description principle as presented in Chapter 3. Furthermore, the selections of genes and gene regulation determine morphological changes within the neuron including changes in synaptic connectivity. These regulatory processes at gene level represented by coupled biochemical reactions can be described by applying quantum field theory in conceptual analysis of diffusion-limited reactions (Kim and Wang, 2007). Therefore this combination between electric properties and genetic mechanisms of protein synthesis at cell level is crucial in performing complex computations and could be related to research in molecular computing, modern study of theoretical models of

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DNA computation and their properties (Adleman, 1994; Braich et al., 2002; Kari et al., 2005). Also this scheme may explain why gene therapy is a very promising yet extremely difficult tool to use in the treatment of neurodegenerative disorders (Tuszynski et al., 1997; Tuszynski, 2007). It is likely that some form of gene therapy combined with induced electric modulations would be required in order to achieve true neuromodulation. Such neuromodulation could be expected to outperform other methods in the treatment of severe neurological disorders. o Based on this new model (NeuroElectroDynamics) a successful stable treatment of neurodegenerative diseases requires that the therapies target the mechanism of gene selection/regulation and protein synthesis as part of both loops; one of molecular computation and the other one governed by electric interactions. Without targeting changes at this fundamental level other therapies will improve the patient’s condition only for short periods of time since intricate adaptive mechanisms could not only restore previous condition but even worsen it (see L-Dopa, treatment). Presented in a form of challenges around traditional approaches of nervous system, the new framework emphasizes the critical importance of underlying physical laws by providing a unified description of brain computations. The analyses of physical movement of charges reveal that their interactions at molecular level are correlated with changes in thermodynamic entropy and information transfer since erasing information always requires a thermodynamic cost (Landauer, 1961; Aur et al., 2006, Aur and Jog, 2007c). The thermodynamic approach answers important questions regarding information transfer, deletion and storage of information during AP or synaptic spikes. As shown in Chapter 4, fractal and fuzzy mechanisms become native properties of computation with charges. The increase in the density of charges within neuronal structures through the neuron during every action potential can be seen to naturally perform as a sequence of fractal transformations. From micro-scale level to larger ensembles scales, granulation turns out to be an inherent property of computation performed by charges (Zadeh, 1997). Beside a top-down extension of ‘computational cube’ at neuron level from a cube to sub-cubes and sub-subcubes (Figure 23) similar models may develop a bottom-up hierarchy where supercubes may achieve functionalities to model defined brain regions as cerebellum, striatum, hippocampus. For the first time to our knowledge this new model (NeuroElectroDynamics) is able to directly relate molecular computing processes with electrophysiological data, neuronal models, mechanisms of computation and possible semantics. As presented above and in previous chapters this new vision should help to clarify some controversies regarding learning, cognition and behavior, allowing a new basis for all other sciences, for clinical treatment of brain disorders and artificial intelligence development. Viewing the entire brain as an integrated global dynamical system, where computation is the result of an intrinsic movement and interaction of charges, opens a new perspective in analyzing specific processes in time and space. As we have demonstrated in chapter 4 using extensive simulations, significant properties including spike patterns with millisecond time precision can be obtained with the model of computation with charges. The multidimensional space of charge dynamics seems to be the final frontier before beginning to describe the processes occurring within the brain

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VV α

E

αE

Figure 23: Successive parallel steps of coding and decoding at multiple scale levels. Within every computational cube the coding phase involves the movement of charges in space while the decoding phase ‘reads’ this spatial distribution

MST ⊂

NED ⊂

Brain Computations

MBC MBC

NeuroElectroDynamic Model

NED Spike Timing Model

MST Figure 24: Representation of hierarchy of models for brain computation may include spike timing models

M ST

(firing rate, interspike interval etc,) the neuroelectrodynamic model

NED and the real computational process, M BC . The dynamics and interaction of charges and molecular computing is able to add new properties to the ones demonstrated by the spike timing models (see fractality, granularity, fuzziness etc)

M BC (Figure 24) since as shown above the properties of spike timing model can be incorporated within models that describe computation with electric charges M ST ⊂ NED ⊂ M BC . However, some new characteristics such as fuzziness, fractality, granularity, overrepresentation seen as innate properties of computation with charges cannot be naturally found in any spike timing approach. This novel framework of dynamics and interaction of charges extends the view presented in spike timing theory and provides a general approach to help understand how the brain works starting from molecular level to cell scale and further to the whole brain viewed as a system. Unlike spike timing theory, the description of computations

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and information transfer using the dynamics and interaction of charges can be decoded within single spikes and related with behavioral semantics since semantics are gradually built during learning by complex regulatory processes at molecular genetic level (Aur et al., 2007a). o NeuroElectroDynamics represents a model of physical computation carried out by interactions of electric charges constrained in their dynamics and spatial distribution by neuronal architecture. The interactions of charges and molecular machinery defined by genetic regulatory mechanisms and protein synthesis included in this model are central to information processing, learning and memory formation. o NeuroElectroDynamics shows that computational properties of neurons are built and change at the quantum level. The higher the dimension of the system and the intrinsic interactions and dynamics, the higher the possible representations and emerging powers of computation. As a result, the whole brain can be regarded as a distributed computational machine where charges can perform the required computations by moving, interacting, performing hidden fine-grained manipulations. By using this new model, we predict that some issues not yet explained by the current view of spike time coding can be completely solved. As we developed our framework within this entirely different context than Hemmen and Sejnowski (Hemmen and Sejnowski, 2006) we discovered new answers for varying and differing views in this field using this unique description of charge neuroelectrodynamics. Question: What is the mechanism behind the semantics observed in spatial patterns of spikes? Answer: Reorganization/organization of electrical charges during learning within protein/macromolecular structures that are directly maintained, built or subtle changed by molecular genetic mechanisms. Question: What is the role of multiple ion fluxes in coding? Answer: Richness of information processing. Question: What are the fundamental limits imposed by this physics of computation? Answer: Thermodynamics, Landauer principle. Question: Why are electrical and chemical synapses all together in coding? Answer: Fast and slow learning; inhibitory and excitatory in chemical synapses. Question: What are the limits of information storage imposed by the properties of space and time in the brain? Answer: See memory formation and noise. However, explaining old paradigms is only a partial outcome of this new theoretical model. While previous challenges from spike timing theory can find quick fix answers, this broader view of the brain as an universe of charged particles raises new issues and challenges. Since information is carried by electric field, charges, and their interaction (e.g. ionic fluxes) this new development offers a unified approach whereas a neuron or few neurons can only partially describe/explain the intricacy of brain computations. Understanding the richness of neuronal code has also to be clarified in the context of the existent biological multi-compartmentalization and molecular genetic basis of computing at cell scale. Using charges as carriers of information reveals the importance of homeostatic processes where the glial cells have a vital role in assisting and maintaining the normal ionic environment around neurons.

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This new development regarding brain computation shows how the brain creates a whole information package out of all its parts whereas the electric charges represent the common denominator for computation. This new approach will no doubt raise other new questions and we anticipate that clarification of these new challenges is a unique experience in determining how this universe of interacting particles shaped by biological evolution computes and emerges in the mysterious brain intelligence. 5.2.7. Computation, Cognition and Artificial intelligence Understanding the brain language is likely to provide an important step to advance and guarantee some success in the field of artificial intelligence. Chalmers envisioned this possibility several years ago in a critique of sub symbolic computation (Chalmers, 1992). The current theory of singularity assumes that human-like artificial intelligence may arrive via emulation of the structural and functional properties of the human brain (Kurzweil, 2005). However, the exponential growth in computer science cannot generate by itself “superhuman intelligence” (McDermott, 2006). While biological intelligence is based on meaning extraction which is required for the survival of an organism, artificially built intelligence is based on techniques of information engineering (Jacob and Shapira, 2005). The mechanisms of intelligence lie deep within the neurons/cell processing (Ford, 2006). An amoeba searching for food has a rudimentary form of intelligence, adapting its motion without requiring a spike timing representation (Saigusa, 2008). The new model based on the dynamics and interaction of electrical charges with molecular computing intricacies performed at gene/DNA level will very elegantly explain even these processes. The progress in artificial intelligence is a reflection of understanding brain computation. At such larger scales, brain representations try to model the richly detailed sensual stimuli of the external world when everything is presented dynamically and simultaneously. The existence of such computation performed by charges raises several issues regarding a possible implementation in current non-biological computing systems. Is it possible to embody sensory information and to integrate perception, memory, reasoning and action in current computers? We believe that computation performed by electric charges as presented in this book can serve as a model to build “strong” intelligence often called universal intelligence (Hutter, 2005) or artificial general intelligence (AGI) (Wang et al., 2008). Constrained by biomolecular computing, gene selection, this model of computation (see Figure 21) is restricted to biointelligence a particular form of “general” intelligence. Therefore biologically inspired cognitive architectures (BICA) can be artificially designed starting from this basic model by a selective integration of “computational cubes” which can naturally develop their cognitive and learning functions by interacting and evolving in time reinforced by complex regulatory ‘genetic’ mechanisms. Thus having a strong theoretical and experimental framework that allows us to understand realistic brain models is a critical step in building artificial intelligent systems since information processing in brain represents the best known example of maturity in achieving natural intelligence. In the next few paragraphs, we will pose a diverse yet necessary set of questions and then provide answers. The objective here is to tie in such an assortment of queries that exist in the mindsets of computational and biological scientists, using the theoretical concepts that have been presented in this book. These concepts of

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computation with charges, thermodynamics, entropy, molecular and genetic framework will be our tools to answering the questions. The questions may appear to be somewhat disconnected but they represent the state of the filed and in a way are the reason why progress may have been slow in producing an approach to a more unified theoretical framework of brain computation. •

Is Computation Identical to Intelligence?

Why do our robots have physical and sensory capabilities of a two-and-a-half year old child? The first question that we have to ask is if computation and intelligence are reversible concepts. Does intelligence equal computation and vice versa? The current framework in computer science constrains algorithms to Turing models and it is our opinion that this barrier limits the development of artificial intelligent systems. o The failure of theoretical computer science to build intelligent systems lies in the fact that brain processes cannot be adequately modeled by “Turing” machines. Every neuron, the key to biological intelligence, is an open system with rich behavioral and computational capabilities beyond the Turing constraints. Following the proposed new model, dynamics and interaction between electric charges is an important process that provides a common denominator for processing and integration of diverse types of inputs (auditory, visual etc). It has already been shown in chapter 4 that simple features can be represented by a spatial distribution of charges/particles. In the brain the interaction between electric charges cannot be seen as being limited locally within certain boundaries and it is more likely that this process is a result of involvement of the entire brain. Therefore, it is expected that cognition emerges as a global dynamic process in the brain as a result of interaction and dynamics of electrical charges supported by a strong molecular genetic regulation of protein synthesis schematically presented in Figure 22. In chapter 4 we have shown a similar process, a simple model of pattern recognition and representation of letters using the computational cube framework. To be complete this interaction should include a memory formation like process that generates a selective accumulation of information. Since cognition is a dynamical phenomenon that follows the movement of charges it can be analyzed straightforwardly in terms of dynamical system theory where the past states seen as electric patterns commonly influences the future state interactions. •

The “incomputable” Issue

Another common perception in computer science is that if a mathematical function cannot be computed by a Turing machine, then the function is incomputable (Etesi and Nemeti, 2002) since computability is interpreted in terms of Turing machine capability. However, human minds may entertain some incomputable models since there are tasks where brain processing is able to surpass even the fastest supercomputer. Starting from Turing ideas, Copeland and Proudfoot have provided examples of models of computation more powerful than the ones of Turing machines (Copeland and Proudfoot, 1999). •

Uncertainty and the Inexact Reasoning Issue

The scientific aspect of how information is processed is directly related to physical implementation of computation in the brain. The system does not know the optimal solution; however, the mechanism to obtain the solution proves to be very efficient.

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Decision making under incomplete knowledge reinforces the fact that the brain ‘algorithm’ should explore more extensively than the usual search in the space of optimal solutions. Therefore, finding solutions for problems that are not completely defined cannot always be achieved using the precise deterministic formalism of symbol manipulation. Handling uncertainty and imprecision in a wide range of practical applications is well performed currently using a fuzzy approach. We have already shown that the model of computation with charges is intrinsically built under these fuzzy principles of spatial distribution of charges but utilizes acceptable, verified and well-grounded physical laws that all systems must obey. •

The Symbol Grounding Issue

Artificial intelligence was born on the issue of representation. However, perception, cognition and action are “deeply rooted in our interactions with the external world” (Brooks, 1991) and contrary to several frameworks presented in the literature perception and action schemes cannot be separated (Gallese et al. 1996; Gallese, et al. 1999). Additionally, in order to achieve human-like intelligence basic schemes that highlight the advantages of grounded systems (Harnad, 1999) have to allow several steps above the well known “reactive agents" endowed with only lower-level cognition. •

The Memory Issue

During the first years of our life we can see, hear and feel, however we are not able to remember these experiences later in life. One possible reason for this is that the system that can provide semantics for stimuli is not well developed at this stage. In order to build these associations the intrinsic structure needs time to assemble and refine the spatial arrangement of electrical charges embedded in proteins and amino acids macromolecular formations. Indeed, specific molecular structures and gene selection mechanisms need to be built within the system before there is any substantial growth in the semantics based associations. Refining and embedding this model of the external world shapes our memory. o For every incoming stimulus semantics are built during computation with charges by interactions with previously stored memories embedded at macromolecular level within neurons. Mathematical models of physical attractors as basins of attraction for electrical charges can be seen in experiments or in simulations. Certain electric interactions, electric fields, frequencies are required to preserve information in long-term memory and sleep phases are critical to maintain and build the system. Aging or sleep disorders may severely impair the laying down of memories that are required for later life. However, there is no need for an equivalent of memory formation and maintenance scheme such as sleep in any artificial built systems. Building synthetic intelligence has to start with a deep understanding of physical principles that govern memory formation since it is well demonstrated that the working memory shows a strong connection to fluid intelligence (Engle et al., 1999). However, the prevailing separation between physical memory allocation and data processing in current computers is a major issue for using past experience. •

Cognition and Computation

Understanding the mechanisms by which brain generates cognition is one of the grand problems facing neuroscience and philosophy. In fact this is the most difficult

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philosophical problem. The “undefined” entity like the mind which displays the faculty of cognition is essentially assumed to be separate from the physical brain which is considered merely to be an instrument of perception as well as action like the five sensory and action organs of the body. According to Vedantic philosophical thought even this cognitive mind is deemed to be nothing but a more subtle instrument of perception. Semantic properties of mental states differentiate computation performed in the brain and the ones embedded in digital computers. The algorithmic level can account for the formation of categorical decisions (Gold, and Shadlen, 2001) as an intermediary step to introduce semantics. It is likely that the new model of computation with charges deeply rooted in molecular processing will provide new insights into the basic computational operations that may acquire semantics within digital computations by incorporating the emergent collective computational abilities of interacting charges. Such methods of building new computational devices may also help in the development of a grounded theory of cognition for perception, action. However, modeling the dynamics of many charges requires a relatively large number of steps, huge memory and an increase in computational power. Once solved, this issue will influence the research on perception, memory, knowledge, language, thought and almost anything we can do will some day, be easily done by machines, cheaper and maybe faster. 5.3. Instead of Discussion The new model of computation substantively contradicts some well-established aspects of previous and current trend in research and has important cross-disciplinary ramifications. This new approach may raise several questions, therefore we anticipated some inquiries regarding the neuroelectrodynamic model and we will try to provide some answers in advance. • Why does NeuroElectroDynamics not comprise Maxwell’s equations, or any extension of them? First, not all of Maxwell equations are applicable to explain these computations. Selected representative Maxwell equations (Gauss's law, equation of continuity, the law of total currents, Lorentz force equation) are included in the book along with Coulomb law, Schrödinger’s equation scattered throughout chapter 3 to chapter 5. The goal of this book is to speak to a large audience therefore we preferred quoting simple examples to explain complex phenomena. In Chapter 4 we have chosen a parabolic potential field where charges have constrained dynamics and provide “solution” for a given problem. What is the action potential: The propagation of an electric wave, the afferent diffusion phenomena that include the dynamics and interaction of electric charges that carry and process information. All the required elements (from a spike) are present, including the electric field, the interaction and dynamics of charges. The dynamics of electric charges (electro-dynamics) follow physical principles and when the movement and interaction occurs within a neuron this phenomenon can be seen as neuro-electrodynamics. Energy is a clear physical requirement in this process. No matter how simple this method of explanation, this description of physical principles allows the reader to grasp advanced ideas of how computation is performed by electric charges. The temporal patterns occur from these interactions as a result of learning that organizes the distribution of electric charges at molecular level. This was demonstrated in Chapter 4. Therefore, the assumed “pre-eminence of neural spike timing” the core of

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spike timing paradigm is more like an epiphenomenon, a result of interaction between electric charges at molecular level where information is processed. •

What about all the intrinsic neuronal processes? Can you be so dogmatic about charges explaining it all? What about DNA computations? Can DNA-based molecular structures in the brain be responsible for processes such as perception, or movement? Can and do these fast processes carried by neurotransmitters make and break chemical bonds?

During daily activities in order to generate perceptions or movements, people need several hundred milliseconds for processing, to solve complex perception problems or movement actions. The movement of charges and their interaction indeed cannot explain all the above phenomena in the brain; however they are the essence of performing computation and information transfer related to processes as movement, image perception, object recognition. Electric fields propagate fast, charges interact and are influenced by subtle to large scale electric fields. This makes the flux of charges an ideal carrier of information and computation in a rapid and dynamic fashion that can process information within a few hundred milliseconds of viewing an image or a scene. Performed by physical movement of charges and their interractions brain computations are modulated by several neurotransmitters. There is no question that molecular computations/ protein-protein interactions/folding along with release of neurotransmitters are important factors concurrent in performing computation (e.g. memory formation/reading). Additionally, as shown above it is likely that gene regulatory mechanisms are involved in appropriate protein synthesis. However, some of these processes need longer periods of time, they may further modulate the dynamics of charges within synaptic spikes, action potentials. Indeed, this novel model builds on the fact that there is an intrinsic interaction between electric charges inside macromolecular structures within the neuron. The new model of dynamics of charges and interaction is able to bridge the current gap between molecular description and the macroscopic level of spike timing coding. •

What about spike timing theory, firing rate, interspike interval etc.

Finding the fundamental level where the model is built is an important step in describing the relationships between several parameters. Spiking is indeed determined by the flow and interaction of charges; however it is not a stereotyped event since one can, with adequate tools, record and observe electrical patterns of activity within every spike. This issue disqualifies the action potential as an atomic, “binary unit” that can be used to build a reliable computational system based on spike timing model alone. Additionally, averaging techniques always hide the dynamics of the phenomenon, which in this case is harmful. However, it is very likely that successive spikes from the same neuron, let alone from different ones, are definitely behaviorally and contextually dissimilar. Therefore, averaging spikes or computing the firing rate is like adding apples and oranges. Such a practice is scientifically flawed and incorrectly generalizes and simplifies the methods of computation that the brain is capable of. •

This model of computation with charges seems to be basically very simple, how come it was not proposed before?

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This new model of computation in the brain is a direct result of experimental observations by others, existing literature and knowledge in the theoretical aspects of information processing in various fields such as physics, mathematics, information theory and neuroscience, as well as our own observations of large quantity of data including information obtained during behavioral experiments in animals. In fact a rudimentary theory of charges and electrical field was proposed earlier in the 1930’s (Fulton, 1938). However, this initial conceptualization didn’t provide the required elements to explain information processing and computation. The theory was completely forgotten after the discovery of synaptic transmission and the role of neurotransmitters (Fulton, 1938). •

Mutual information analyses provide the amount of information shared by two variables not “transferred information” as authors claim in their analyses of ion fluxes.

Artificially we can generate signals that share large quantities of information; however, in the real world in order to obtain large values of mutual information, this information has to be transferred. Large correlations imply large MI either in classical or quantum systems (see Horodecki et al., 2005). Indeed, if X is the signal generated by an electromagnetic wave in point A in space and Y is measured in point B (A and B being two different points in space) then X and Y are not identical anymore. Once carried by the electromagnetic wave from A to B the signal has been distorted and there will be no X at point B. Therefore, in order for Y to have large values of information about X (or large values of mutual information between Y and X) information has to be physically carried, transferred. This is the gist of our framework developed in the third chapter and this is the essence of several experiments that measure MI and show information in “spike trains” (see Tiesinga et al., 2002). However, it is more likely that information can be transferred between electric fluxes within the same cell rather than between “wind” and “spike trains” somewhere in the brain as per several examples shown in the literature. Therefore, high MI values seen as shared information in brain requires a process where information is physically transferred. •

Many papers/theories present a multidimensional approach to neural computation, parallel neural processing. I do not see how your model is different?

Several authors describe multidimensionality but this pertains to the number of neurons in the network that compute in parallel as a measure of “dimension”. The computational units (artificial neurons) simultaneously process the input signal using specific functions (sigmoidal, RBF etc). This is the usual presentation of the concept of parallelism and multidimensionality in brain computation. However, neural computation and all these theories are built under the assumption of spike stereotypy. We have already discussed the limitations and inaccuracy of this bias and the implications towards these theories. The multidimensionality aspect of neuronal computation should then be seen at the fundamental level determined by the dynamics and interaction of charges during electric spikes that may include synaptic spikes or action potentials. Automatically, the

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parallelism and multidimensionality occur directly governed by the laws of physics as an intrinsic property of computation with charges. •

There is nothing new about your model since we already know that information is carried within trains of AP (see all references, Rieke et al., 1997)

It is an incorrect assumption that information is carried within trains of AP. As we have discussed in previous chapters, physical carriers of information are electric charges and electric field. In the nervous system, the fastest way that information transfer and processing occurs is through the flux of charges and electric interactions determined by the electric field and are not limited to AP trains. We have also shown in chapters 2 and 3 that the amount of information carried by charges is substantially higher than that carried by temporal occurrence of spikes. Such new conceptualization is a fundamental change in interpretation, which makes a huge difference in understanding computation in brain and relates it with molecular computing mechanisms. •

What about linking synaptic plasticity across different time scales with spike timing and protein synthesis as some scientists suggest lately?

As currently presented, spike timing theory is completely disconnected with any research in molecular biology and spike timing theory needs additional hypotheses to make these connections (see heterogeneity, sparse coding, oscillations, synchronicity in Vinje and Gallant, 2000; Maldonado et al., 2008; Poo and Isaacson, 2009; Jadhav et al.,. 2009). In science, this process of continuously adding new hypotheses was extensively analyzed by Thomas Kuhn and it is a process that marks “the transition from a paradigm in crisis” (Kuhn T. S., 1962). Time of spikes alone cannot be an adequately rich physical carrier for information and it is more likely that time patterns are a result of computations performed by electric charges and their interaction. This approach of neuroelectrodynamics of charges presented in this book is not an “articulation or extension” of existing spike timing paradigm and we feel that this new approach will be invaluable to other scientists interested in this field. Using simulations, we have been able to demonstrate that although the time of reading information using synapses is important, organization with learning occurs fundamentally in the spatial distribution of charges within macromolecular structures (e.g. proteins) and further this phenomenon determines observable/measurable outcome in firing rate and ISI. These efforts to incorporate current spike timing theory as a special circumstance of our broader theoretical construct will finally reveal that the new paradigm better describes these computations carried out at varying scales in brain. •

What about the theory of self-organization in spikes that is a well observed and studied phenomenon?

Patterns of activity within spikes in individual neurons and at the network level, observed by processing current electrophysiological recordings, emerge as a consequence of electric field propagation, charge interactions at microscopic levels with and within macromolecular formation (e.g. proteins). Self-organization of spike

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patterns is a direct consequence of changes in spatial distribution of charges that occur with learning and not vice versa. This is easily shown in simulations and in our data where during ‘spiking’ activity the distribution of charges becomes organized with learning and a spike directivity vector can be observed with every ‘action potential’. •

There is no evidence that proteins store information. Besides, I cannot imagine that charges will stay for a lifetime in specific spatial spots.

Several classes of proteins such as neurotransmitter receptors, cytoskeletal components and protein kinases have been related with memory formation (Johannsen, et al., 2008). Additionally, there is evidence that synaptic plasticity and long-term potentiation require rapid protein synthesis (Kennedy, 1998; Guzowski et al., 2000). Indeed, it is unlikely that these charges will stay during the life of the cell in the specific spots within some proteins. However, gradual degeneration of proteins in normal subjects is rapidly followed by protein synthesis that tends to preserve previous information embedded in spatial density of charges. Therefore, new macromolecular formations are likely to maintain significant chunks of prior information essentially copying themselves to retain the macromolecular structural consistency of the protein that is degrading. Further computations by the neuron can then subsequently change these structures, which will then be copied by the next set of newly synthesized proteins. In fact, in the extension of this model, rapid degeneration, abnormal protein folding etc can be related to cellular dysfunction seen in some disorders as memory loss that may severely impair performance. •

What about neurogenesis?

Seen often as a property of a “young” brain, neurogenesis was limited to only few brain areas such the hippocampus, epithelium and olfactory bulb. Several other regions such as the adult mammalian neocortex do not seem to have the ability for neurogenesis. The genesis of new neurons, in the adult human brain was demonstrated recently in the hippocampus (Eriksson, et al., 1998). Such hippocampal neurogenesis occurs throughout life and is present in animals and humans. Due to its importance, the olfactory system displays fast development and regeneration. Olfactory epithelium is another important system in the study of neurogenesis (Graziadei and Graziadei, 1979). It has been shown that in adult animals this ability to generate new neurons has a role in cognition and can improve memory. It is clear that the resulting neurons can migrate and expand neuronal electrophysiological properties by developing new active synapses. Additionally, injury related neurogenesis was also observed and is considered to be a form of recovery after brain lesions (Kornack and Rakic, 1999). The studies in adult rodents and monkeys that showed specific neurogenesis capable regions provide new insights regarding significant adaptability in the adult brain to function and elaborate cognitive abilities (Gould and Gross 2002; Kuhn et al., 1996). This may also reinforce the current model of charges where the placement of neurons in space, the position of synapses, is critical for performing computation and adapting to new input features. Neurogenesis seems to be missing in many descriptions of memory or cognitive studies and there are some other important issues that need to be answered. For example it is likely that selected old memories are transferred within newborn neurons.

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•

Many scientists have proposed exotic theories of how the brain works. What is your opinion about this fact?

There is scientific freedom to pursue any path in this field and indeed, there are several misunderstandings and ‘confabulations’ related to brain computations. The major problem today in brain science is that experiments do not validate these models. For a newcomer in this field it is hard to distinguish between a good theory and a product of imagination. Although these theories may seem attractive, there is very little validity to many of them. •

What is the role of neurotransmitters in spike computations? Where is the role played by the synapses?

Neurotransmitters can be seen as mediators of this transfer of information at ion channel level by selecting the number and kinetics of charges involved in the computational process (see amphetamine/schizophrenia experiments). Even though this is an indirect contribution, neurotransmitters have an important role in gating information flow by enabling the physical movement and interaction of charges. One can compare neurotransmitters with an example of the traffic lights at an intersection; they allow, disallow and control the traffic of cars but they cannot replace any car in the traffic. In this way neurotransmitters can be seen as facilitators of charge traffic. When they act, charges are preferentially enabled/disabled and redirected according to principles presented in chapter 4. Since there are several types of neurotransmitters with varying modes of action, a detailed analysis of neurotransmitter’s role in computation is beyond the immediate scope of this book. •

Your simulations are based on a classical model of charge movement, what about a quantum model?

Classical models are well known; however, lately quantum models have become better understood and familiar. Since we are talking about models, none of them can individually or together replace the real physical phenomenon. The quantum model was designed to overcome the limitations of the laws of classical mechanics which does not work at the atomic scale. Processes at neuron level utilize both scales. As we have presented in chapter 4, quantum phenomenon in spikes may be observed if a special “filtering” technique is used. However, this issue should not stop us from providing/using classical models for the charge movement since classical behavior can always provide a good model if k BT > hf . •

There are many models regarding neural computation. Why did you write this book?

Indeed, there are many theories of brain computations; however, many models persist in their effort to develop new spike timing concepts. Many neuroscientists and computationalists at academic institutions such as Berkeley, Caltech, CNRS, EPFL, MIT, Princeton, Stanford, UCLA and Yale are performing statistics, numbering spikes, counting intervals between spikes, claiming that they have indeed found a time pattern or an oscillation that solves the problem of coding in the hippocampus, cortex or

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elsewhere in brain. Others starting from a spike timing model have claimed that they will be able to soon duplicate how the brain works. After another year, other brave researchers may confirm or refute those experimental results. This development could go on forever since the amount of data about specific temporal patterns in the brain and their characteristics are infinite and observer dependent. Just by analyzing these patterns it is unlikely that a fundamental level of understanding let alone theoretical framework can ever be generated adequately. It is very interesting that after we launched our proposal to very few publishers, several attempts have been made in spike timing literature to adapt the timing of spike occurrence with macromolecular changes (see Clopath et al., 2008); however, at this moment spike timing development needs an additional fundamental hypothesis to relate molecular computation with “millisecond time patterns” and offers a very soft explanation compared to our direct approach. We therefore set out to put forward a new model on the basis established by the laws of physics. This makes our proposition and theoretical framework general and by default very amenable to expansion, with clear possibility for experimental verification. There were no additional assumptions that were made and no special requirements such as stereotypy of the spike or relationship to timing patterns. The well accepted laws of physics when applied in this field yielded a set of observations available within the literature and our own data. Further, the computational principles that we have proposed are independent of the anatomical structure, the cellular make-up, specific neurochemistry and express the essence of the phenomenon. Finally, the model has a natural scalable property which allows the development of experimental paradigms that can test the levels of theoretical constructs that we have proposed. •

Is there any difference between your model and the Restricted Coulomb Energy (RCE) model?

RCE is an algorithm similar to nearest neighbor classifier that may speculate some proprieties of electrical charges and that’s all. The difference in our approach from the nearest neighbor classifier is that the unknown input has to be close enough in distance to the neighbor in order to be classified. •

What about the Grossberg model of Adaptive Resonance Theory (ART)?

The main problem is that, as a model, ART does not provide a physical explanation for brain computation. By contrast in our model reading and writing information is carried out by an interaction of electric charges with a certain dynamics in every cell that is involved in the process. As we have shown in this book for reading and especially for writing in memory a number of repetitions are required to achieve a good signal to noise ratio. The “comparison” between input information and information in memory (e.g. distribution of charges in the protein matrix) is automatically performed by the charge/group of charges/ions that move. Organization/self-organization can be seen as a natural process of charge redistribution within the protein matrix in ion channels as an effect of information received from ions/charges that move and interact during action potentials/synaptic spikes. Information is indeed written/read within networks of neurons. However the main processing phases occur within every neuron where “self-organization” of spatial distribution of charges occurs within proteins, chain of proteins. Previously learned knowledge can be preserved if the proteins maintain a certain conformational state and is again written in new synthesized

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proteins. The turnover of proteins has its own molecular computing dynamics that allows newly manufactured proteins to rapidly embed the information that is continuously entering. Writing, transfer and reading of information is then maximized with organizations that take place at the molecular level and learning is said to be occurring. Acquiring new data should not change past memories since these new memories are usually written in new places/ new proteins, new synapses and sometimes new neurons. At the cell level charge/ion channels resonance type phenomena may naturally occur, however they manifest at a different scale than presumed resonance states described at network level presented in the ART model. •

Many labs develop their computational models based on brain slice analysis. What do you think about models of computation based on in vitro recordings?

Steriade has shown that slices cut at 500 µm have different electric patterns as opposed to 400 µm slices (Steriade, 2001). While brain slices can be used for several types of studies, such computational models based on recordings in brain slices are unlikely to preserve computational properties of the ‘intact’ brain. •

Do you have anything against statistical methods in neuroscience?

Statistical methods are very useful in science especially in neuroscience. In fact many results obtained and presented in this book as evidence are based on statistical analysis. These analyses however, cannot always explain the intrinsic phenomenon since by themselves statistical analyses are not able to reveal or build the laws that govern such processes. Every scientist needs to know these issues and use a multidisciplinary approach in order to understand these phenomena. •

The model you propose is not so critical regarding grandmother cells even though this theory has been rejected due its inconsistency long time ago. Why?

Jerzy Konorski showed indeed an intriguing phenomenon that was experimentally observed a long time ago. The existence of “gnostic” units has been recently confirmed by the experiments performed at Caltech. The recorded neurons responded to a set of images of Jennifer Aniston (see Quiroga et al. 2005). Indeed, neurons respond to specific set of input features, however it is unlikely to have dedicated neurons for Jennifer Aniston, Halle Berry or grandmother. It is likely that these neurons respond to other different images and problems may come from analysing firing rate data. Seen from this perspective the phenomenon reinforces the current model of the dynamics of charges, their interactions and the physical outcome in terms of particular characteristics stored in these neurons. In fact if Lettvin and colleagues had, instead of spending so much energy in disputing the observations of Konorski, considered the real cause of the unit response, they might have designed a similar model at least 30 years earlier. Additionally, beside this theory of gnostic units Konorski has contributed substantially in shaping the theories of associative learning. •

I can understand that this model of computation can be applied to large neural networks that compute in a cube. However, I do not see how your model can be applied at protein scale. Who plays the role of synapses in this case?

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If we use the dynamic model of charges then the embedded charges within the protein structure may play a similar role to what synapses provide at neural scale. Such charges spatially distributed within proteins are “sensitive” to dynamics of a charge or group of charges. Importantly, this is only a model that cannot replace the real phenomenon M BC (see M ST ⊂ NED ⊂ M BC ) since not all characteristics can be included in the model. •

Many models use inhibitory and excitatory synapses and I’m not sure that you understand any aspect regarding synapse processing?

Indeed most theories use a Hebbian model of learning as a simple step in building a neural network. The alterations of synapses provide plasticity; they can ‘strengthen’ their connection or weaken (depress it) depending on presynaptic or postsynaptic activity of neurons. This phenomenon is an effect of neurotransmitters and is also closely related to the spatial distribution and the dynamics of charges that in time become physical attractors and direct the electric flux. •

What about quantum theory of neural computation anticipated by Sir Roger Penrose and Stuart Hameroff?

Their theory was largely criticized by Tegmark (see Tegmark, 2000) due to some inconsistencies in presenting the quantum phenomenon at neural level. Classical mechanics can be a good approximation of quantum mechanics even when describing processes in proteins, acids if the analyzed motion (computations) is modeled at picoseconds time scale. However, as we have pointed out above, quantum theory is a model that should be definitely considered at this microscopic level within macromolecular structures (Hameroff and Penrose, 1996). We cannot imagine our world without these interactions. We think that the most important contribution of Dr. Hameroff is that he was able to provide a biological prototype that does not require spike timing theory as a model of computation. The example of paramecium locomotion that hunts and grazes the microbial world cannot be performed by delivering AP with “millisecond time precision”. Beside, the Penrose and Hameroff microtubulin construct is highly plausible as a model of interacting charges at microscopic level as we have proposed in this book and the paramecium locomotion is a good example in this case. •

In fact all theories that build neural networks models affirm that they are inspired from experimental data, I cannot understand how your new theory is different?

Many other theories started by using spikes or the dynamics of neurotransmitters as basic constituents for their model. We have shown above several deficiencies of considering spike timing or neurotransmitters as atomic units to build a coherent theory of brain computations. The single most significant deviation from previous frameworks is that we have demonstrated that charges are carriers of information and their dynamics is deeply rooted with information transfer/storage in neurons. This work is inspired and built upon the application of physical laws and it obeys physical laws. Additionally, we believe that the time for sophisticated models or products of

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imagination might be over since they will lose ground in favor of theoretical models that follow well accepted physical laws and experimental observations. •

I see that the book has few classical mechanics equations. Do you not think that your approach is too simple?

We wrote this book thinking initially that processes in brain are very complex. Indeed this very issue is the pitfall of simply measuring in ever more detailed way the “colors” of the system since these colors change constantly. However, our goal was to provide a path that would offer a set of rules and principles that were generalized, scalable and could be applied from the micro- to the macroscopic level. The extraction of these principles and their understanding should not be hindered by the complexity of brain processes. It was not necessary to know the mass and trajectory of every planet and star in the universe to be able to present and understand the theory of gravity. Similarly, the periodic table of elements in chemistry could be another excellent example of a similar situation wherein based on information of only a few elements the entire table was beautifully constructed. Initial observations followed by the application of fundamental principles of physics and mathematics can achieve what we have shown in our new proposed model. Indeed an understanding of what we are saying about these phenomena can be acquired starting with high school level mathematics since the brain story/language should be understandable with our common knowledge. •

Thermodynamics is used to study specific phenomena regarding heat. Why a thermodynamic interpretation for neuronal computation?

Thermodynamics is a general theory and should not be restricted to the analysis of heat processes. The second law of thermodynamics is a general law that can be used to explain several natural processes from the formation of the universe to simple chemical reactions. Additionally, as stated by Landauer, deleting information is a process that always generates heat and understanding memory formation/loss cannot be complete without a thermodynamic framework as this process also has a metabolic cost associated with it. •

Now if we have already understood how the brain works will we be able very soon to build human type intelligence?

Indeed, we are sure that several scientists are interested in building human–like intelligence since these projects may have diverse applications in all kind of fields. However, there are many speculations and a lot of confabulation in artificial intelligence (AI) as it is currently founded in computational and brain theories. Current artificial intelligence development is a direct reflection of an understanding of brain processes. Building AI is possible and feasible. However, it needs some amount of additional work fundamental theoretical work and merely application engineering of known theories will not be enough before the first functional bio-intelligent model can be assembled. •

Several computational models are built using fuzzy theory. What about a fuzzy theory as a model for brain computations. What about a Bayesian model?

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There is often a disconnect between a model and the corresponding natural phenomenon. A single natural phenomenon can shape several different models. Even though there are some issues regarding interpretation of fuzziness, Professor Zadeh has built a very successful theory. The principles of a fuzzy model are not far from the neuroelectrodynamic model since electrical charges by their spatial distribution provide natural intrinsic fuzzy properties. Additionally, we are sure that the development of fuzzy theory and granular computation can benefit from our new model of computation in the brain. Similar rationale can be applied for Markovian or Bayesian models used by theoretical neuroscientists to analyze the brain processes. •

Many scientists measure the frequency of electrical waves. Is computation the result of these electrical oscillations?

Generating waves or rhythms is a general property of elastic structures. The simple example in classical physics starts with the study of a harmonic oscillator involving a spring as a classical model. Therefore, the elastic interactions between several electric charges mediated by Coulomb forces are expected to generate oscillations. We would say indeed that these oscillations are emergent properties of existing computations with charges that mediate the search in the solution space. A good example in this area of research is the harmonic resonance theory that was recently proposed by Lehar (Lehar, 2005). The repertoire and diversity of these oscillation modes in many brain regions is far too complex to be fully modeled or understood (Buzsáki, 2006). •

You talk in your book about charges that move and other charges that somehow are hidden in macromolecules/proteins that may be influenced by this movement. How come this slight but significant movement of charges was never detected or talked about earlier?

This may be an important step in demonstrating the new model. A charge that moves can read and write information at the same time. It can read information due to intrinsic electric interactions with other charges and the existent electric field can change its trajectory in space. The question is how this charge can write information at the same time. Usually the charges that move are well known ions (Na+, K+, Ca2+,Cl-) and their movement is regulated by complex mechanisms (see ion channels etc). One can observe that few of these moving ions have a negative charge. Therefore, we expect negative charges to be distributed almost everywhere inside the neuronal cytoskeleton, in molecules, proteins. It is expected that once the positive ions move these negative charges will be attracted to these charges/electric fields. Since these electrons or ions cannot freely travel in space this process of interaction may generate polarizations, rotations or slight movements of molecules, protein polymorphism. Protein folding can be genuine proof of this phenomenon since the spatial distribution of electrical charges significantly changes. Current view of spike stereotypy does not incorporate such levels of detail. However, lately several techniques have shown that the movement of ions is not identical in case of two different spikes recorded from the same neuron; this is an indirect proof of changes of spatial distribution of charges at microscale level. •

Several papers claim that a theory of brain computation cannot be developed without considering some basic elements included from fractal theory.

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We have again not to forget that fractal theory, as well as the fuzzy theory, are both theoretical models. A model, no matter how elaborate or complex it is, cannot recreate the natural phenomenon or include all intrinsic properties (see Figure 24). Starting from the spatial arrangement of neurons to local irregularities and brain waves all can be seen as examples of fractal formations. In fact we have shown that electric field propagation during AP generation on dendrites and axonal branches does describe fractal transformations. •

It seems that your framework supports membrane computation principles.

Membrane computing is an innovative theory developed by Gheorghe Paun in the field of theoretical computation. However, this mathematical theory was not designed to represent a model of how computation is performed in the brain. This new model can benefit from the current presentation and also this development can benefit from a mathematical formalism developed in P-systems. •

You have mentioned only the interaction of charges as an important factor. What about Van der Walls forces?

Intermolecular forces are determined by a certain degree of covalent bonding. Even in a non-polar molecule the density of electrons is not uniformly distributed and there is a certain probability that they are asymmetrically distributed. This issue is well explained in a quantum mechanics framework since such polarization may exert forces that depend on the relative orientation of molecules. •

Is brain computation a reversible process?

Each action potential is a result of diffusive charge/ionic flow. Due to dissipative forces (e.g. friction, diffusion, thermal loss) this process cannot be in principle thermodynamically reversible. All logical operations that are carried out within suboptimal processes are in essence thermodynamically irreversible. If these phenomena were thermodynamically reversible then getting information would not necessarily have to increase the thermodynamic entropy. Therefore, the occurrence of ionic movement during an AP is expected to be a thermodynamically irreversible phenomenon. However, logical reversibility has to match a given input pattern to a unique output pattern and this behavior cannot be easily determined based on a thermodynamic analysis. Additionally, information processing can be seen as transformation of the incoming signals. This transformation of the input signal may preserve Ndimensionality of the input in case of a reversible framework. However, a different transformation from an N-dimensional space to a lower-dimensional space generates an irreversible transformation that reduces dimensionality in an irreversible way. •

Why do attention and learning usually increase spike synchrony?

Computers do not make such distinction; but our brain does make it since it tries not to spend energy on already solved problems since it has the ability to learn. The solution is provided faster by the brain if the number of charges simultaneously involved in this process increases and this requires an activation of ensemble of

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neurons and spike synchrony. The dynamics and interaction of charges would solve the problem and then, by storing the answer, a much faster solution is offered the next time. How is the answer stored? Our hypothesis is that the answer is written in a spatial distribution of charges within macromolecular structures/proteins that shape the memory. Many variables such as whether the answer already exists (problem has been solved before), if it is similar to other problems already solved, or if the situation and solutions are entirely new will determine what happens. •

What is wrong with spike timing theory?

The theory of spike timing can be shown to have limitations by spike directivity analysis on procedural T-maze learning experiment data. Behavioral meaning can be easily obtained if one analyzes spike directivity as presented in (Aur and Jog 2007). Spike directivity is a hidden feature that can be computed and graphically represented as a vector that points in space to the center of mass of charges which moves during an AP. After behavioral learning the changes in spike directivity are related to the selected arm of the T-maze. The selection of the T-maze arm based on tone is the most important characteristic that can link neural response to behavior. However this information cannot be obtained by analyzing the firing rate or interspike interval but can be easily revealed by analyzing the spatial modulations of spike directivity. This result is astounding and cannot be explained by the current theory of spike timing and is strong evidence against the theory of spike timing as a general theory for explaining neural code or brain computations. •

You are talking several times about motion principles. What are these motion principles?

Indeed we present computation as dependent on charge motion and we basically propose two models to describe this motion. At microscopic level this motion and interactions can be modeled using quantum principles and at macroscopic level classically by Newtonian equations. •

Your work almost ignores the well stated Hebbian theory? Why?

The brain has developed a mechanism to represent information from simple stimuli to abstract learned phenomena as a process of moving and interaction of electrical charges. Hebbian connectivity can be explained as a consequence of this process at molecular level of neurotransmitter release, new protein synthesis that is genetically regulated and occurs within synapses and significantly changes electrical properties and propagation of signals. We have already observed how learning organizes the flow and distribution of charges within spikes (Aur et al., 2006; Jog et al., 2007). Between spikes every neuron has a slightly negative polarity. The flux of ions synaptically / nonsynaptically received may change this polarity and determine the occurrence of action potentials. Therefore, it is likely that the neighboring neurons that have ‘synaptic connectivity’ to be activated together by the received flux of ions in the region and changes that occur can be interpreted as a wiring–rewiring mechanism within a certain volume/area (aka neural-network activity). From simple perceptions to complex cognitive processes that underlie learning and memory formation many of these processes could be shaped by this biophysical process

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of interaction, charge dynamics. As presented above, we believe that several controversies in neuroscience and neural computation can be resolved with the neuroelectrodynamic model. •

What do you think about mirror neurons? Several scientists claim this to be the most important discovery of the last decade.

As pointed out by many neuroscientists mirror neurons are considered to play a crucial role in social cognition (Gallese et al., 1996; Rizzolatti, and Craighero, 2004; Enticott, 2008). A mirror neuron always fires when the subject or another individual are performing the same action. In many cases the activity of mirror neurons is related to sensorial information which is distributed in the network (e.g. inferior parietal lobe or ventral premotor cortex). It is considered that these neurons are able to code/decode abstract properties or the meaning of actions (Kohler et al., 2002) and the absence of mirror neurons is related to severe brain disorders (Dapretto et al., 2006). Our new model of charges provides an explanation for the existence of mirror neurons since in order to perform certain action, information regarding some specific sub-tasks or characteristics is required. The neuroelectrodynamic model sustains the fact that information needed is embedded within the neural structures (macromolecules/proteins etc). The representation of the same or similar features is expected to be located in a bounded spatial location within a certain group of proteins within certain neurons. While performing a certain task the access to a certain characteristic is required and since information reading is performed by charges during their movement and interaction within a certain group of neurons, reading or writing such features will involve an increased density of charges in that region that activate some specific neurons. Therefore, these neurons that contain information regarding the meaning of the action will be necessarily activated (miror neurons in this case) independent of whether the action is performed by the subject or someone else. •

I do not see any difference between NeuroElectroDynamics and conductive models aka HH model. What is so special about this new model?

In case of conductive models (aka HH model) the inputs and outputs are electric signals (currents, voltages etc). The cable model basically describes the idea of a “memoryless” property since charge density in a cable is proportional to the distance of the charges from the cable axis and therefore this distribution is generally independent of the past history. The strength of this approach resides in various optimization techniques that highlight various quantitative aspects and provide models for currents, voltages (see HH model) or lately time patterns that approximate recorded data. Neuroelectrodynamics is fundamentally a completely different model. First, the inputs and outputs of the system are spatial distribution of charges. Second, the intrinsic processes describe the dynamics and interaction between charges. Third, such description can naturally include internal molecular processes, unique properties of DNA computing, complex genetic regulatory mechanisms and the effects of neurotransmitters. The subtle change that occurs at molecular level with learning and the resulting electric flow can be related with an organization in spike patterns. This qualitative information processing highlights semantic aspects which has no equivalent yet in current conductive models. At different scales, neuroelectrodynamics is engaged

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directly in information processing. However, it can also be adapted to provide quantitative models for currents, voltages, time patterns as in HH models and their extensions. •

As you know the cortex has a huge number of neurons. Therefore, I plan to emulate (simulate in real-time) a million spiking neurons. Do you think that this project will accelerate the progress in brain research?

Well, we think that such a project may take a long time and high cost to be implemented and the outcome is unsure. Other than being an interesting simulation where several students can learn how to do programming we have serious concerns if this approach based on spike timing paradigm alone could accelerate any progress in brain research. •

You put a heavy emphasis on Maxwell's demon. I am not sure that it is so important in physics.

Maxwell's demon may not be so important in physics but it makes the required connection between physical processes, second law of thermodynamics and information theory which is critical for understanding information processing in neurons. •

Often you refer to 'revolutions' in other subjects. This comes across as an attempt to justify your own work. However what happened in other areas has no weight as to the credibility of your work. In fact I find it rather offputting to keep raising these other 'great steps' since the natural implication is that your work is also one. This is a rhetorical trick of crackpots to justify their work without evidence. For example you talk about Copernicus - what has his work really got to do with yours? How does it relate to or justify what follows?

Indeed Copernicus had nothing to do with tetrodes, or brain recordings. He was living at a time when the universe was felt to revolve around the Earth. Based on astronomical observations he completely changed the view where the Earth was not at the center of the universe. Certainly, what happened in other areas has no weight or relevance regarding this work. However it shows that we can be prisoners of a dogma that flourishes, has no relevance, and is repeatedly presented. We should learn from this past since the history of science repeats itself in other terms in other fields. Such is the case with the current state of affairs with neural computation. Functional constructs of neural function revolve around spike timing models since previous observations accredited the idea of a temporal code. In this context, a major directional change from this spike timing view would in our view indeed be a revolution. Based on tetrode data recordings and analyses as supporting evidence, this view has to be changed in order to be able to better understand the complexity of computation in the brain and we feel that challenging this view could eventually result in a change that is substantial enough to become a ‘revolution’ in the field and connected areas.

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•

Moving on, your claim is that the shape of the spikes is caused by a propagating charge in the neuron. How fast does it move? A moving charge generates a magnetic field, does it have an influence in computation?

The shape of the spike is given by the propagation of the electric field and resulting distribution of charges. During an action potential the velocity of the electric field is highly fluctuant and it increases with axonal diameter up to 120 m/s in myelinated axons (Clamann and Henneman, 1976). Under the influence of this electric field, charges (e.g ions) move. However, their speed is 30-40 times lower than the speed of electric field (Li et al., 1998). This phenomenon is the basis of underlying assumption of charge movement model where every AP can be reduced to a distribution of charges in the three-dimensional space. The generated magnetic field is very small and has been assumed to be negligible. The implications of the presence of the magnetic field may have to be added as the theoretical construct proceeds further. However, since the charge velocity is not significant then the dynamics of charges are predominantly changed by electric interactions. Therefore, in these analyses in a reductionist approach we have ignored the influence of the magnetic field. •

Is the point of the first paragraph in the first Chapter suggests that a rational, axiomatic basis for neural understanding need not exist? By the way I have an old colleague who has a solid math background and then he went into neuroscience. He may help you to solve these things on an axiomatic basis.

Indeed, in physics, quantum mechanics is an example of a successful theory built from a set of postulates accepted as true. This model can be extended to other fields and NeuroElectroDynamics may turn out to be a framework that incorporates a quantum level formulation and is an attempt to move in this direction. However, if firing rate or ISI are uniquely seen as providers of “neural code” and this hypothesis is transformed into an axiom, then the consequences that follow from it may generate in the best scenario a new successful model in mathematics rather than a model of brain computations due to many significant assumptions and simplifications. The main problem here highlights the importance of vision in science. Referring back to the example of Copernicus, he may not have been the most gifted mathematician or physicist of his time but he had the vision that changed the approach to celestial motion. We hope that we have provided this beginning with the theory of NeuroElectroDynamics. •

Your approach seems to want to tear down many of the extant tenets of neuroscience. I rather prefer a gradual development of the field with some complementary modeling rather than a Kuhnian paradigm shift as you propose.

Current worldview on this subject does not allow connecting neural computation and molecular biology and it is difficult to understand how the brain processes information. Neural computation operates using several myths as presented in Chapter1 and ignores the existence of fundamental physical laws. With an unfulfilled promise to

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find the code in the temporal patterns of spikes, neural computation itself has failed to make major predictions about how the brain utilizes all of its machinery to perform computation for diverse functions such as information storage and retrieval and action performance linked to semantics of action. This is reflected currently in the currently disproportionate, diverse and independent development of molecular biology in rapport with neural computation. On the other hand molecular biology/molecular computing focuses on “local information” within a myriad of proteins, genes and the analyzed data must remain “local” as long there is no attempt to make any connection with neural computation principles. NeuroElectroDynamics builds a new model, bridges these two sub-fields and based on universal laws explains how information is carried and processed in the brain. To make this connection, the theoretical analyses of data and physical experiments have to move and focus in a different direction and this new view triggers indeed a “paradigm shift”. In this case the understanding cannot be “achieved by an articulation or extension of the old paradigm”. The development of the model in the last four chapters clearly shows that neuroelectrodynamics is not based on spike timing coding paradigm. •

Brain computation is still evolving i.e. each new discovery will continue to shape our understanding and the way we model after the brain. Would NeuroElectroDynamics become another myth by itself?

As long as NeuroElectroDynamics connects two existing subfields of neuroscience; namely neural computation and molecular biology, and it has strong experimental roots (patterns within spikes/spike directivity etc) it is unlikely to itself become a myth; it can either be incorporated by one of them if the progress completely eliminates the other subfield of research or, as we have presented in our book the new model incorporates and integrates knowledge from both areas in order to explain how the brain computes. •

Do you think that the exponential growth in computational power is going to produce human like artificial intelligence?

We believe that the current explosion in building high-performance computers can help emulate complex, natural phenomena. However to simulate brain processes one has to understand how the brain works, along with decoding the ‘brain language’. By itself, computer science cannot go too far in this endeavor and our book is a step further in understanding some principles that govern human intelligence embedded within a model of constrained dynamics and interaction of charges. •

Do you expect any future for the new model? I mean do you expect that Adaptive Resonance Theorists, spike timing theorists or confabulation theorists will adopt and teach this new model?

Independently, from our previous data that pictured an intracellular propagation of charges during AP occurrence (Aur and Jog, 2006, see also Chapter2), Milstein and Koch provided some new evidence from hippocampal cells within rat where “the action potential gives rise to a large quadrupole moment that contributes to the extracellular field up to distances of almost 1 cm (Milstein and Koch, 2008). Most importantly this

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new development shows that action potential cannot any longer be considered as ‘binary’ as was previously considered by Koch in 1999 and others (Koch, 1999). Clopath et al. try to explain memory consolidation as the “triggering of protein synthesis that needs the presence of neuromodulators such as dopamine” (Clopath et al., 2008). However, this presentation shows that memory consolidation has no explanation using basic concepts of spike timing theory and the development requires a different level of description, new hypotheses and a physical level as presented in our approach. A third paper (Johnson and Redish, 2008) reflects how the ‘decision point’ occurs at neuronal levels as a hippocampal spatial processing phenomenon and confirms our results obtained and published earlier by processing tetrode recordings from striatum that showed changes in spike directivity as a “spatial” outcome related to decision choice (see T-maze procedural tasks, learning, Aur and Jog 2007a). It is likely that similar results can be obtained from hippocampus using spike directivity analyses. We have a deep respect for the theoretical developments by many and their contribution to science in general and neural computation in particular since we have learned from them. In fact many of us started their journey to understand brain function by, for example, reading the “The Adaptive Brain” book and for a variety of reasons for some of us the enthusiasm of reading that book has never stopped. Of course, we do expect and want to have critical appraisal of our ideas. The same level of rigor and questions however that we would be expected to answer to should also be applied to all other theories that are currently popular. We also expect resistance from scientists that have spent their lifetime constructing their own theories and do not expect them to immediately embrace our new framework. In his scientific autobiography Max Planck remarkably envisioned this aspect: “a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it” (Planck, 1949). This view represents a central concept forwarded by Thomas Kuhn as a brilliant accurate analysis of how progress is commonly acquired in science (Kuhn, 1962). However, our sincere and honest hope is that scientists of the caliber who have already contributed so much developing computational neuroscience will begin incorporating these ideas into their own work. Summary of Principal Concepts • •

• •

The connectionist view (Hebb, 1949) assumes that synaptic plasticity is responsible for the memory phenomena and interprets this process in terms of wiring and re-wiring paradigm of Hebbian connectivity. Our new model demonstrates that each neuron can store information within its intrinsic structure within the soma, axon, dendrites and synapses where biological macromolecules, proteins can be viewed as holding spatial distributions of electric charges. The polarity of the system of electric charges at this microscopic level within molecules and protein structures amongst others is the machinery that builds our memories. Writing/reading the memory is performed by a dynamic movement and interaction of charges/ions which are the carriers of information during action potentials/ synaptic spikes etc;

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Rehearsals and brain rhythms play an important role and are the mechanisms which transfer information from short-term memory into stable long term memories. However, this happens within the neurons themselves at the same location or at a distance due to spatio-temporal information propagation with electric fields and charges’ interaction. At the most fundamental level this continuous flow of ions, their interaction at molecular level during AP, synaptic spikes, may induce conformational changes/polarization, determining a new distribution of electric charges, an alteration of old memories and generation of new ones. Building memory patterns is the result of protein polymorphism, proteinprotein associations, protein binding, and protein-protein docking mediated by electric and hydrophobic interactions. Within every neuron, the polymorphism of protein structures, protein conformational changes play a crucial role in building new memory patterns (e.g. electrical patterns within spikes, spike directivity changes ). The synthesis of appropriate proteins is a result of various bimolecular regulatory mechanisms in response to a wide variety of extracellular signals and electric interactions. Therefore, at the most significant level self-organization physically occurs at molecular level, is observable within new synthesized proteins, and their distribution within dendrites, axons etc, then is naturally reflected at macrolevel scale measurable within spike patterns (e.g. spike directivity). The distortion or memory loss occurs inevitably with the passage of time, protein degradation occurs when information is altered and new synthesized proteins were not effectively able to embed previous data. Information regarding an input feature event, pattern is carried by electric flux and written within groups/ensembles of neurons at the molecular level. Therefore, such information is highly distributed and is unlikely to be stored in a single neuron (e.g. see recordings/experiments). The universe is composed by an infinite number of particles; our brain is not substantially different, shaped by evolution the “brain universe” of many interacting charges emerges in a natural form as intelligence.

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