Mathematical Modeling And Simulation In Enteric Neurobiology

MATHEMATICAL MODELING AND SIMULATION IN ENTERIC NEUROBIOLOGY

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Roustem Miftahof Pohang University of Science & Technology, Korea

Hong Gil Nam Pohang University of Science & Technology, Korea

David Lionel Wingate Queen Mary, University of London, UK

MATHEMATICAL MODELING AND SIMULATION IN ENTERIC NEUROBIOLOGY

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Miftahof, Roustem. Mathematical modeling and simulation in enteric neurobiology / Roustem Miftahof, Hong Gil Nam, David Lionel Wingate. p. ; cm. Includes bibliographical references and index. ISBN-13: 978-981-283-480-5 (hardcover : alk. paper) ISBN-10: 981-283-480-X (hardcover : alk. paper) 1. Gastrointestinal system--Innervation--Mathematical models. 2. Gastrointestinal system--Innervation--Computer simulation. I. Nam, Hong Gil. II. Wingate, David L. III. Title. [DNLM: 1. Gastrointestinal Motility--physiology. 2. Computer Simulation. 3. Gastrointestinal Tract--innervation. 4. Gastrointestinal Tract--physiology. 5. Models, Biological. WI 102 M633m 2009] QP145.M615 2009 616.3--dc22 2008052082

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Contents

Preface

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List of Symbols

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Acronyms

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Introduction 1

Overview of Enteric Neurobiology 1.1 1.2 1.3 1.4 1.5 1.6 1.7

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The Role of the Gut . . . . . . . . . . Regional Function in the Human Gut The Intrinsic Innervation of the Gut . Gastrointestinal Smooth Muscle . . . Extrinsic Innervation . . . . . . . . . The Effect of Food on the Gut . . . . Clinical Pharmacology . . . . . . . .

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Myoelectrical Activity of the Smooth Muscle 2.1 Myoelectrical and Mechanical Activity . . 2.1.1 Biological Background . . . . . . 2.1.2 Mathematical Model . . . . . . . 2.1.3 Numerical Algorithm . . . . . . . 2.1.4 Physiological Response . . . . . . L 2.1.5 Effect of Increase in g˜Ca . . . . . . T 2.1.6 Effect of Increase in g˜Ca−K and g˜Ca T 2.1.7 Effect of Cyclic Changes in g˜Ca . . vii

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viii Mathematical Modeling and Simulation in Enteric Neurobiology L T 2.1.8 Effect of Increase in g˜Ca and g˜Ca . . . . . . . . T 2.1.9 Effect of Increase in g˜Ca−K and Decrease in g˜Ca 2.1.10 Remarks . . . . . . . . . . . . . . . . . . . . . 2.2 Effects of Ion Channel Modulators . . . . . . . . . . . 2.2.1 Biological Background . . . . . . . . . . . . . . 2.2.2 Effect of Forskolin . . . . . . . . . . . . . . . . 2.2.3 Effect of Lemakalim . . . . . . . . . . . . . . . + . . 2.2.4 Effect of High Concentration of External Km 2.2.5 Effect of Phencyclidine . . . . . . . . . . . . . . 2.2.6 Effects of Selective K+ -Channel Agonists/ Antagonists . . . . . . . . . . . . . . . . . . . . 2.2.7 Remarks . . . . . . . . . . . . . . . . . . . . .

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3.1 Effects of Specific Inhibitors of the and the Ryanodine-Sensitive Ca2+ Channels of the Sarcoplasmic Reticulum . . . . . . . . . . . . . . 3.1.1 Effects of Cyclopiazonic Acid . . . . . . . . . . . 3.1.2 Effects of Cyclopiazonic Acid and Ryanodine . . . 3.1.3 Effects of Cyclopiazonic Acid and Non-Selective Ca2+ Channel Antagonists . . . . . . . . . . . . . 3.1.4 Effects of Cyclopiazonic Acid and Selective Ca2+ Channel Antagonists . . . . . . . . . . . . . 3.1.5 Effects of Thapsigargin, Non-Selective and Selective Ca2+ Channel Antagonists, and Ryanodine . . . . . . . . . . . . . . . . . . . 3.1.6 Effects of Thapsigargin and High [Ca2+ ]0 . . . . . 3.1.7 Effects of Thapsigargin, High [Ca2+ ]0 , [K+ ]0 and Selective Ca2+ Channel Antagonists . . . . . 3.2 Effects of 1,4-dihydropyridine Enantiomers (–)-(S)-Bay K 8644 and (+)-(R)-Bay K 8644 . . . . . . . . . . . . . 3.2.1 Effects of (–)-(S)-Bay K 8644 . . . . . . . . . . . 3.2.2 Effects of (–)-(S)-Bay K 8644 and High [K+ ]0 . . . . . . . . . . . . . . . . . . 3.2.3 Effects of (+)-(R)-Bay K 8644 . . . . . . . . . . . 3.3 Effects of Motilides . . . . . . . . . . . . . . . . . . . .

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3.3.1 Effects of Motilin and Erythromycin . . 3.3.2 Effects of Motilides and Ryanodine . . . 3.3.3 Effects of Motilides, a Non-Selective Ca2+ Channel Blocker and Thapsigargin . . . 3.4 Effects of Benzodiazepines . . . . . . . . . . . 3.4.1 Effects of Benzodiazepines Alone . . . . 3.4.2 Reverse of the Effects of Benzodiazepines 3.5 Remarks . . . . . . . . . . . . . . . . . . . . . 4

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Physicochemical Basis of Synaptic Transmission 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 Cholinergic Neurotransmission . . . . . . . . . . . . . 4.2.1 Biological Background . . . . . . . . . . . . . . . 4.2.2 Mathematical Model . . . . . . . . . . . . . . . . 4.2.3 Numerical Algorithm . . . . . . . . . . . . . . . 4.2.4 Physiological Neurotransmission . . . . . . . . . 4.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . 4.3 Inhibition of Cholinergic Neurotransmission . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.3.2 Biological Background . . . . . . . . . . . . . . . 4.3.3 Mathematical Model . . . . . . . . . . . . . . . . 4.3.4 Effect of Chloride Salts of Divalent Cations . . . . 4.3.5 Effect of β-Bungarotoxin . . . . . . . . . . . . . . 4.3.6 Effect of Botulinum Toxin . . . . . . . . . . . . . 4.3.7 Change in the Concentration of Extracellular Ca2+ 4.3.8 Effect of Cholinergic Antagonists . . . . . . . . . 4.4 Facilitation of Cholinergic Neurotransmission . . . . . . 4.4.1 Biological Background . . . . . . . . . . . . . . . 4.4.2 Mathematical Model . . . . . . . . . . . . . . . . 4.4.3 Effect of Cholinergic Agonists . . . . . . . . . . . 4.4.4 Effect of TTX . . . . . . . . . . . . . . . . . . . . 4.4.5 Effect of Repetitive Stimulation . . . . . . . . . . 4.4.6 Remarks . . . . . . . . . . . . . . . . . . . . . . 4.5 Adrenergic Neurotransmission . . . . . . . . . . . . . . 4.5.1 Biological Background . . . . . . . . . . . . . . . 4.5.2 Mathematical Model . . . . . . . . . . . . . . . .

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4.5.3 Physiological Adrenergic Transmission . . . 4.5.4 Remarks . . . . . . . . . . . . . . . . . . . 4.6 Altered Adrenergic Neurotransmission . . . . . . . 4.6.1 Effects of Extracellular Ca2+ Removal and Application of TTX . . . . . . . . . . . 4.6.2 Inhibition of Neuronal Uptake-1 Mechanism 4.6.3 Inhibition of Catechol-O-Methyltransferase 4.6.4 Mathematical Model . . . . . . . . . . . . . 4.6.5 Effect of α1 -Adrenoceptor Antagonists . . . 4.6.6 Mathematical Model . . . . . . . . . . . . . 4.6.7 Effect of the Repetitive Stimulation . . . . . 4.6.8 Remarks . . . . . . . . . . . . . . . . . . . 5

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Neuronal Assemblies 5.1 Planar Neuronal Network . . . . . . . . . . 5.1.1 Introduction . . . . . . . . . . . . . 5.1.2 Biological Background . . . . . . . . 5.2 Inhibitory Neural Circuit . . . . . . . . . . 5.2.1 Axo-Axonal Interaction . . . . . . . 5.2.2 Effect of COMT Inhibitors . . . . . . 5.2.3 Effect of α1 -Adrenoceptor Blockers . 5.3 A Model of the Sensory Pathway . . . . . . 5.3.1 Introduction . . . . . . . . . . . . 5.3.2 Biological Background . . . . . . . 5.3.3 Mathematical Model . . . . . . . . 5.3.4 Responses to Deformation . . . . . 5.3.5 Effect of Iberiotoxin and CHTX . . 5.3.6 Effect of ω-CgTX . . . . . . . . . . 5.3.7 Effect of TTX . . . . . . . . . . . . 5.3.8 Effect of Purinoceptor Agonists . . 5.3.9 Effect of Protein Kinase C Activator 5.3.10 Effect of DPDPE . . . . . . . . . . 5.3.11 Remarks . . . . . . . . . . . . . . . 5.4 Enteral Sympathetic Communication . . . 5.4.1 Introduction . . . . . . . . . . . . . 5.4.2 Mathematical Model . . . . . . . . .

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5.4.3 Effect of a Single Deformation . . . . . . . . . 5.4.4 Effect of a Periodic Deformation . . . . . . . . 5.4.5 Effect of Intermittent Deformation . . . . . . 5.4.6 Remarks . . . . . . . . . . . . . . . . . . . . 5.5 A Planar Neuronal Network . . . . . . . . . . . . . 5.5.1 Effect of Cholinergic and Adrenergic Agonists and Antagonists . . . . . . . . . . . . . . . . 5.5.2 Effect of Cholinesterase Inhibitors . . . . . . . 5.5.3 Effect of Cholinergic Antagonists . . . . . . . 5.5.4 Remarks . . . . . . . . . . . . . . . . . . . . 6

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Multiple Neurotransmission

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6.1 Co-transmission by Acetylcholine and Serotonin . . . . . . 6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Biological Background . . . . . . . . . . . . . . . . . 6.1.3 Mathematical Model . . . . . . . . . . . . . . . . . . 6.1.4 Electrical Activity of Mechanoreceptors . . . . . . . . 6.1.4.1 Physiological Response . . . . . . . . . . . . 6.1.4.2 Effect of 5-HT3 Receptors . . . . . . . . . . . 6.1.4.3 Effect of 5-HT3 -Receptor Antagonists . . . . 6.1.5 Electrical Activity of the Primary Neuron . . . . . . . 6.1.5.1 Stimulation of Mechanoreceptors . . . . . . 6.1.5.2 Effect of 5-HT3 Receptors . . . . . . . . . . . 6.1.5.3 Effect of 5-HT3 -Receptor Antagonists . . . . 6.1.5.4 Effect of 5-HT4 Receptors . . . . . . . . . . . 6.1.5.5 Effect of Co-activation of 5-HT3 and 5-HT4 Receptors . . . . . . . . . . . . . . . . . . . 6.1.5.6 Effect of Cisapride . . . . . . . . . . . . . . . 6.1.6 Electrical Activity of the Motor Neuron . . . . . . . . 6.1.6.1 Effect of nACh Receptors . . . . . . . . . . . 6.1.6.2 Effect of 5-HT3 and 5-HT4 Receptors . . . . 6.1.6.3 Effect of Co-activation of 5-HT3 and nACh Receptors . . . . . . . . . . . . . . . . . . . 6.1.6.4 Effect of Co-activation of 5-HT4 and nACh Receptors . . . . . . . . . . . . . . . . . . .

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6.1.6.5 Effect of 5-HT3 /5-HT4 -Receptor Agonists and Co-activation of nACh Receptors . . 6.1.6.6 Effects of 5-HT3 -Receptor Antagonist and Co-activation of nACh Receptors . . . . . 6.1.7 Electrical Activity of Smooth Muscle . . . . . . . 6.1.7.1 Effect of µACh Receptors . . . . . . . . . 6.1.7.2 Effect of 5-HT4 Receptors . . . . . . . . . 6.1.7.3 Effect of Co-activation of 5-HT4 and µACh Receptors . . . . . . . . . . . 6.1.7.4 Effect of 5-HT4 Receptor Antagonists and Co-activation of 5-HT3 and µACh Receptors . . . . . . . . . . . 6.1.8 Remarks . . . . . . . . . . . . . . . . . . . . . . 6.2 Co-transmission by ACh and Excitatory Amino Acids . . 6.2.1 Biological Background . . . . . . . . . . . . . . . 6.2.2 Electrical Activity of the Primary Neuron . . . . . 6.2.2.1 Effect of Mechanical Stimulation . . . . . 6.2.2.2 Effect of AMPA Receptors . . . . . . . . . 6.2.2.3 Effect of NMDA Receptors . . . . . . . . 6.2.2.4 Effect of Co-activation of NMDA and AMPA Receptors . . . . . . . . . . . 6.2.3 Electrical Activity of the Motor Neuron . . . . . . 6.2.3.1 Effect of nACh Receptors . . . . . . . . . 6.2.3.2 Effect of NMDA Receptors . . . . . . . . 6.2.3.3 Effect of Co-activation of nACh and NMDA Receptors . . . . . . . . . . . 6.2.3.4 Effect of Co-activation of nACh and AMPA Receptors . . . . . . . . . . . 6.2.3.5 Effect of Co-activation of AMPA, NMDA and nACh Receptors . . . . . . . . . . . 6.2.4 Smooth Muscle — Neuronal Chain Preparation . 6.2.4.1 Effect of nACh Receptors . . . . . . . . . 6.2.4.2 Effect of Co-activation of AMPA and nACh Receptors . . . . . . . . . . .

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6.2.4.3 Effect of Co-activation of nACh, AMPA and NMDA Receptors . . . . . . . . . . 6.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . 6.3 Co-transmission by ACh and Substance P . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . 6.3.2 Biological Background . . . . . . . . . . . . . . 6.3.3 Mathematical Model . . . . . . . . . . . . . . . 6.3.4 Effect of Randomly Applied High Frequency Stimuli . . . . . . . . . . . . . . . . 6.3.5 Effect of Low Frequency Stimulation . . . . . . 6.3.6 Remarks . . . . . . . . . . . . . . . . . . . . . 7

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Introduction . . . . . . . . . . . . . . Biological Background . . . . . . . . . Mathematical Model . . . . . . . . . . Numerical Algorithm . . . . . . . . . . Electromechanical Wave Phenomenon . Effect of Lidocaine N-ethyl Bromide Quartery Salt QX-314 . . . . . . . . . . 7.7 Effect of Changes in Extracellular Ca2+ 7.8 Effect of Cholinergic Antagonists . . . . 7.9 Remarks . . . . . . . . . . . . . . . . .

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Dynamics of Intestinal Propulsion 8.1 8.2 8.3 8.4 8.5 8.6

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Functional Unit 7.1 7.2 7.3 7.4 7.5 7.6

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Model Formulation . . . . . . . . . . . . . . . . Numerical Algorithm . . . . . . . . . . . . . . . Pendular Movements . . . . . . . . . . . . . . . Segmental Contractions . . . . . . . . . . . . . Peristaltic Reflex . . . . . . . . . . . . . . . . . . Effect of Multiple Neurotransmission and Drugs on Pellet Propulsion . . . . . . . . . . . . . . . . 8.6.1 Introduction . . . . . . . . . . . . . . . . 8.6.2 Effect of Co-Activation of 5-HT3 , nACh and µACh Receptors . . . . . . . . . . . .

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8.6.3 Effects of Alosetron . . . . . . . . . . . . . . . . 8.6.4 Effects of Selective 5-HT4 Receptor Agonists . . 8.6.5 Effects of Selective 5-HT4 Receptor Antagonists . 8.6.6 Effect of Cisapride . . . . . . . . . . . . . . . . 8.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

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References

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Index

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Preface

Integrative physiological phenomena are difficult to analyze because current experimental techniques do not provide comprehensive information for all hierarchical levels of biological systems. It is difficult to integrate the properties of individual cells to a higher level of organization using a reductionist approach. Interactions of components within a system are often so complicated that their combined activities can be understood only with the application of mathematical and computational tools that serve as universal quantitative approximators. Recent developments in forward dynamic modeling and simulation have attracted considerable attention as non-invasive methods to investigate complex biological phenomena. They have become an essential supplement to experimental techniques that explore inaccessible biological processes and thereby assist researchers to deduce intricate functional mechanisms. Currently, there are no texts for navigating the extensive field of mathematical and computational modeling in enteric neurobiology. The book is the first research monograph that provides insight into intricate biological mechanisms of intestinal motility that could not have been achieved through in vivo and in vitro experimental approaches and associated descriptive functional analyses. Although it is written for experts in the field, the book is also intended as a companion text for special courses in integrated system biology, applied mathematics, mathematical modeling, numerical computing, and engineering science for advanced undergraduate and graduate students. It cannot be used though as a traditional textbook to teach either “biology for computer scientists” or “mathematical modeling techniques for biologists.” Classical textbooks on human physiology, molecular biology and numerical xv

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mathematics should be used as a background reading. The emphasis is given to fundamental topics that lie at the interface between biomedical, chemical, physical, computer sciences and recent experimental developments in enteric neurobiology and engages the multidisciplinary approach. Readers learn and understand computer simulations best by following the process of model formulation, and the design and execution of numerical algorithms. We demonstrate applications of the results of computer simulations in several realms, including target identification to treat intestinal motor dysfunction, multiple drug effects on the system, and various medical applications. From a practical point of view, the book will help establish strategies for people with complementary backgrounds who are already involved, or who are planning to work, in multidisciplinary areas. Therefore, the approach used in the book has been to treat the subject matter with a degree of mathematical rigor while recognizing the complexity of biological structure. Special attention has been given to computer simulations for interpolation and extrapolation of electromechanical and chemoelectrical coupling phenomena, nonlinear self sustained electromechanical wave activity and pharmacological interactions concerned with the effects of co-localization and co-transmission by multiple neurotransmitters, receptor polymodality and heterogeneity, and drug interactions. A brief introduction to the physiology of the gastrointestinal tract is given in Chapter 1. A one-dimensional mathematical model for electromechanical phenomenon in smooth muscle fibers is developed in Chapter 2, based on actual morphological and electrophysiological data. Numerical algorithms and procedures are outlined. The results of pharmacological model validation are given in Chapter 3. The effects of drugs that alter the uptake and release of intracellular Ca2+ from stores in sarcoplasmic reticulum, specific L-type Ca2+ channel antagonists and non-specific weakly selective L-type Ca2+ channel agonist, motilides, and benzodiazepines are numerically examined. Mathematical models and numerical investigations into the processes of chemical neurotransmission are presented in Chapter 4. Special attention is given to electrochemical coupling phenomenon at the synaptic level, including cycles of biochemical conversions for acetylcholine and adrenaline and their derivatives. The effects of tetrodotoxin, β-bungarotoxin, salts of

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Preface xvii

divalent cations, inhibitors of catechol-O-methyltransferase on neuronal uptake mechanisms, and influence of external Ca2+ concentration on the dynamics of signal transduction are analyzed. Mathematical models of the enteric nervous system as a planar neuronal network are presented in Chapter 5. A traditional network model as a number of interconnected one-dimensional neurons is compared to a novel network model as a syncytium with spacially distributed weakly connected oscillators. The influence of synaptic plasticity on the spread of excitation/inhibition waves within the neuronal network is emphasized. Numerical procedures to solve the governing equations are outlined. Electrophysiological mechanisms of co-transmission by serotonin (5-HT), acetylcholine, α-amino-3-hydroxy-5-methyl-4-isoxalose propionic acid (AMPA), L-aspartate, N-methyl-D-aspartate (NMDA) and coexpression of different receptors types, including 5-HT type 3 and 4, nicotinic and muscarinic cholinergic, AMPA and NMDA are examined in Chapter 6. Interpretation of numerical results in conjunction with in vivo experimental recordings and their possible applications to the development of novel therapeutics to treat diseases associated with altered visceral nociception is provided. A novel, integrative, computational model that combines information across multiple levels of structural organization for a functional unit by substituting the entire gut with a continuum of overlapping dynamically stable functional elements is presented in Chapter 7. The unit is modeled as a soft biological shell with electromyogenic properties. A combined numerical algorithm is designed to solve the governing system of nonlinear partial and ordinary integro-differential equations. A new phenomenon which is discovered numerically and is related to self-sustained myoelectrical activity is presented and analyzed. Medical applications for the propulsion of a solid non-deformable bolus, and the neurochemical basis of visceral hypersensitivity are examined in Chapter 8. Results of numerical experiments are discussed in conjunction with the diseases affecting gastrointestinal motility. All simulations in the book are obtained with ABS Technologies
software. The complete version of software is available upon a request at: “[email protected].”

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xviii Mathematical Modeling and Simulation in Enteric Neurobiology

We are grateful to our colleagues who have assisted us in the writing of the book. We extend our special thanks to Ms. M. J. Hwang for her patient typing of the manuscript. We thank the I-BIO Program at Pohang University of Sciences and Technology, Republic of Korea, where most of the book was written. Mathematical Modeling and Simulation in Enteric Neurobiology was reviewed by an expert who checked the accuracy of what the reader will learn here, to help ensure that this book will provide the reader with all that he or she would need to know about this modern field of science. We greatly appreciate his or her contribution. Finally, our special thanks are extended to Ms. S. C. Lim, World Scientific Publishing, who supported the project from the very beginning and made its timely publication possible. Professor R. N. Miftahof Emeritus Professor D. L. Wingate Professor H. G. Nam

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List of Symbols

Cma Cp Cm(s) Cmd Ra Rv Rm(s) Rp ϕ ϕd ϕ1 ϕ2 ϕ3 ϕ4 ϕ∗ +(−) ϕp ϕrest ϕNa , ϕK , ϕCl ϕ¯ Na , ϕ¯ K , ϕ¯ θ T ϕ˜ Ca ∗ ϕ˜ Ca−K ϕ˜ K∗ ∗ ϕ˜ Cl L ϕ˜ Ca +(−) ϕR

capacitance of the nerve fiber subsynaptic membrane capacitance membrane capacitance membrane capacitance of dendritic membrane membrane resistance (axon) resistance of the extrasynaptic structures membrane resistance (muscle, synapse) resistance of the soma of the primary neuron membrane potential (smooth muscle) dendritic potential action potential (mechanoreceptors) action potential (primary neuron) action potential (secondary neuron) action potential (ICC) membrane potential (axon) excitatory ( + ) and inhibitory ( − ) postsynaptic potentials resting potential (neurons, smooth muscle, axon) reversal potentials for Na+ , K+ and Cl− currents reversal potentials for Na+ , K+ and Cl− currents reversal potential for fast Ca2+ current reversal potential for Ca2+ -K+ current reversal potential for slow K+ current reversal potential for slow Cl− current reversal potential for slow Ca2+ current resting potentials of the synapse xix

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ϕ0 ϕp∗ ϕl , ϕc gNa , gK , gCl g¯Na , g¯K , g¯θ T L g˜Ca , g˜Ca ∗ g˜K∗ , g˜Ca−K ∗ g˜Cl ∗ ∗ g0˜ s1 , g0˜s2 gi˜∗s1 , gi˜∗s2 m h n m ˜∞ h˜ n˜ αm,h,n βm,h,n α˜ m,h,n β˜ m,h,n Iext (i) T ICa L ICa ∗ ICa−K IK∗ ICl ICa ICa−K INa IK ˜INa I˜K ˜ICl Im1,m2

excitatory potential threshold potential for L-type Ca2+ channels electrical waves of depolarization (l — longitudinal, c — circular muscle layer) maximal conductances for Na+ , K+ , Cl− currents (axon) maximal conductances for Na+ , K+ , Cl− currents maximal conductance of fast and slow Ca2+ channels, respectively maximal conductance of K+ and Ca2+ activated K+ channels maximal conductance of leak Cl− channels maximal extracellular conductances maximal intracellular conductances activation gating variable of Na+ current inactivation gating variable of Na+ current inactivation gating variable of K+ current steady state of activation gating variable of fast Ca2+ current inactivation gating variable of fast Ca2+ current activation gating variable of fast K+ current activation time constants of Na+ and K+ channels deactivation time constants of Na+ and K+ channels activation time constants of fast Ca2+ and K+ channels deactivation time constants of fast Ca2+ and K+ channels external ion currents T-type Ca2+ current (smooth muscle) L-type Ca2+ current (smooth muscle) Ca2+ activated K+ current (smooth muscle) voltage dependent K+ current (smooth muscle) leak Cl− current (smooth muscle, ICC, neurons) calcium current (ICC, neurons) Ca2+ activated K+ current (ICC, neurons) sodium current (ICC, neurons) voltage dependent K+ current (ICC, neurons) sodium current (mechanoreceptors) voltage dependent K+ current (mechanoreceptors) leak Cl− current (mechanoreceptors) transmembrane currents

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List of Symbols xxi ∗ Iionic a L L0 l  ταCa αCa

α Mvs λl , λc l , c λ ρ Kc p(0) Tp Ta γ0 eij nr,s,z vr,s,z W θ t ts td t s˜1 , s˜2 c1−14 A X = (x1 , . . . , xn )T

total ion current cross-section diameter of the axon length of the axon length of the nerve terminal length of a functional unit empirical constant of synapse relaxation time constant of Ca2+ current relaxation parameter for activation of intracellular Ca2+ current numerical parameter volume-to-surface ratio stretch ratios in the longitudinal and circular smooth muscle layers strains ratios in the longitudinal and circular smooth muscle layers plant’s factor strength of the cellular Ca2+ buffering capacity intracellular Ca2+ current constant intraluminal pressure passive force active force linear density direction cosines of the outward normal n an outward normal components of the velocity vector strain energy density function temperature time time of excitation duration of excitation time delay t ∗ time of of depolarization in the presynaptic zone Lagrangian coordinates mechanical constants matrix vector column

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L1−4
c
c εp εN τ c(i) , b(i) , V0 , V1 AChc AChv AChf AChp R ACh − R E AChE NAc NAs NAp ARα2 , ARα1 COMT NAp − ARα2(α1) S, S A [. . .] [. . .] 2+ [Ca ]out [Ca 2+ ] ˜ 2+ ] [Ca [Ca 2+ ]n kc(A) kd(A) k+p (A) k+,−(i)

K ∗ , K ∗∗

weighted averages of A penalty operator spatial operator penalty parameter viscosity parameter time-step of integration parameters related to the activation of the release of endoplasmic Ca2+ fraction of acetylcholine in the cleft vesicular acetylcholine free fraction of acetylcholine postsynaptic fraction of acetylcholine choline receptors acetylcholine-receptor complex acetylcholinesterase enzyme acetylcholine-enzyme complex fraction of noradrenaline in the cleft vesicular noradrenaline postsynaptic fraction of noradrenaline adrenoceptors catechol-O-methyl transferase enzyme noradrenaline-adrenoceptor complexes products of chemical reactions equilibrium concentration of reacting components concentration of reacting components external calcium concentration (synapse) internal calcium concentration (synapse) internal calcium concentration (muscle) Free cytosolic calcium (ICC) k numerical constant affinity constants diffusion constants diffusion rate constant of acetylcholine on the sub-synaptic membrane rate constants of forward ( + ) and backward ( − ) chemical reactions Michaelis-Menten constants

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List of Symbols xxiii

SR v SR f SR c MAO SR c − MAO R5−HT 3 R5−HT 4 SR − R 5−HT 3 SR c − R5−HT3 G PhC PIP 3 IP 3 Dg RER − IP 3 Ca2+ ier Ca2+ ER PkC Pr PPhos SPv SPf SPc RNK Enz SPc − EMz SPc − RNK − C SRv − R5−HT − C

vesicular serotonin free fraction of serotonin serotonin in the cleft is monoamine oxidase serotonin — monoamine oxidase complex serotonin type 3 receptors serotonin type 4 receptors serotonin-5-HT type 3 receptor complex serotonin-5-HT type 4 receptor complex guanine-nucleotide G protein phospholipase C phosphatidyl inositol 1,4,5-triphosphate inositol 1,4,5-triphosphate diacylglycerol receptor on the endoplasmic reticulum calcium of the endoplasm total amount of calcium stored in the endoplasmic reticulum protein kinase C intracellular protein protein-phosphatase vesicular substance P free fraction SP substance P in the cleft neurokinin receptors enzyme SP-enzyme complex SP-receptor-guanine-nucleotide protein complex serotonin-receptor-guanine-nucleotide protein complex

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Acronyms

ACh AH AMPA
( − ) − (S) − BayK8644 ( + ) − (R) − BayK8644 CCPA CHTH CM CNQX CNS CPA CPP DNQX DPDPE D-AP5 EC ENS EPSP FGID FSK FU GABA

acetylcholine primary sensory neuron α-amino-3-hydroxy-5-methyl-4-isoxalose propionic acid 1,4-dihydropyridine (DHP) enantiomers 2-chloro-N6 -cyclopentyladenosine charybdotoxin circular smooth muscle 6-cyano-7-nitroquinoxaline-2,3-dione central nervous system cyclopiazonic acid 3-(2-carboxypiperazin-4-yl) propyl-1-phosphonic acid 6,7-dinitroquinoxaline-2,3-dione d-penicil-amine2 -D-penicillamine5 d-2-amino-5-phosphopentanoic acid enterochromaffin cell enteric nervous system excitatory postsynaptic potential functional gastrointestinal disorders forskolin functional unit γ-aminobutyric acid xxv

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Mathematical Modeling and Simulation in Enteric Neurobiology

GABAA,B Glu ICC ICC-MP IPSP LEM LM MMC NA NANC NECA NMDA PCP PDBu S SP SR SR (5-HT) TTX ω-CgTX

γ-aminobutyric acid receptors (types A and B) glutamate interstitial cell of Cajal interstitial cell of Cajal-myenteric plexus (pacemaker) inhibitory postsynaptic potential lemakalim longitudinal smooth muscle migrating myoelectrical complex noradrenaline non-cholinergic non-adrenergic 5  -N-ethylcarboxamidoadenosine N-methyl-D-aspartate phencyclidine phorbol 12,13-dibutyrate motor neuron substance P 5-hydroxytryptamine serotonin tetrodotoxin ω-conotoxin GVIA

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Introduction

Mathematical modeling has been used for decades to help scientists understand the mechanisms and dynamics behind experimental observations. Designing models requires experimental data, and knowledge or a hypothesis about how the components of the system are connected. Neurogastrobiology is largely an empirical science, much of it incomprehensible and unpredictable, because of a deficient theoretical foundation. Investigations into different aspects of morphology, electrophysiology, neurophysiology, neuropharmacology, and biomechanics have mostly utilized reductionist approaches to analyze the interactions of neuromuscular and regulatory mechanisms at molecular levels, leaving unstudied the operative integration of component processes involving electromechanical, chemoelectrical and electrochemical coupling. Our understanding of how the gastrointestinal tract regulates the flow of its contents in normal and diseased states will remain fragmentary and semi-quantitative, with correspondingly limited clinical applicability, until a theory that unifies the fine- and large-scale effects has been developed and demonstrated to adequately predict physiological behaviors. Existing models of the gastrointestinal system have proven helpful in predicting behavior under experimental conditions. However, most models suffer from biological naiveté: they can validate experimental hypotheses, but generally cannot provide any information about system function beyond that which the experimenters had already proposed to be true. New models are being developed that incorporate detailed dynamics for sets of interactions with varying degrees of complexity. Computational power can solve large sets of equations over multicellular networks, to effectively bridge the gap among subcellular, cellular, tissue, organ and system-level features. The range xxvii

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of problems that are being addressed is growing steadily, utilizing modeling methods that differ in their modularity and mathematical implementation. However, the current deficit in scientists equally trained and prepared to understand mathematics and biology/medicine hampers the development and application of computer simulations methods in biology. A range of mathematical models and numerical methods related to specific questions of neurogastrobiology with applications to the studies of gastrointestinal motility are presented in this book. The extent to which the models provide a clear understanding of how fine-scale mechanisms produce large-scale effects are analyzed using computer simulations. Attempts have been made to gain insight into the physics of complex wave phenomenon in the gut by offering new explanations for experimental data and re-interpreting old observations using novel conceptual views on hierarchical conjugation in low excitable biological media. With this approach we have tried to eliminate many of the mistakes and misinterpretations of physiological findings that have dominated the field for several decades. One regrettable consequence of this is the prominence of gastrointestinal motor disorders in medicine and the limited effectiveness of existing methods to treat it. The focus of this book is on neurobiological aspects of intestinal motility. A “bottom-up” approach is employed in the text. This means that to understand the function of a system, it is first necessary to understand the function and behavior of its components and their interactions. Such an approach is appealing because it suggests a modular organization in the gastrointestinal system, which is intuitively useful in understanding possible interactions among different components and their emergent properties. An ordered design process is used to understand biological processes in smooth muscle fibers, neurons and neuronal networks, and myoelectrical activity and signal transduction mechanisms. This strategy appears to be the best method to build a feasible model of the gut. The challenging question is how to manage the computational complexity of the model. The processes are described by nonlinear systems of partial and ordinary integro-differential equations that can only be solved numerically and simulated on a computer. It is not our intention to provide a detailed study of numerical implementations, but rather to introduce

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Introduction xxix

concepts of how numerical algorithms are designed and what are the requirements to be met in order to achieve the desired accuracy and efficiency in calculations. The resultant equations include parameters and constants, such as mechanical and cable electrical properties of the longitudinal and circular smooth muscle syncytia, rates of chemical reactions for the conversion of acetylcholine, noradrenaline and serotonin, and activation thresholds, which are, in most cases, unknown and must be identified using the available data. The procedure of parameter identification requires solving the model with a set of “guessed”input values. The results of simulations are then compared with experimental data. If the results do not match, a new guess is made and the process is repeated. This process can be computationally demanding, especially when using a large number of parameters and variables. However, it could be rationalized by using efficient search methods or by optimizationbased techniques. The inability to find a suitable parameter set might indicate that the equations are not a correct representation of the system. In such cases, the simulation results may question the original hypothesis from which the model was derived and predict the need for new changes to the model. Thus, numerical experimentation can lead to testable quantitative or qualitative predictions that distinguish between in vivo and in vitro experimental methods. Finally, after the model has been numerically validated, it can serve as a gnostic tool to explore a wide range of experimental conditions in a virtual environment; to understand the impact of physiological variables in ways that cannot be adequately represented in even the most complex animal models; to predict the effects of intrinsic/extrinsic interventions before expensive bench experiments are run; to reproduce and study “analytically” the effects of different classes of drugs; and to facilitate the investigation of issues of clinical safety and efficacy. Numerical studies play a crucial role in elucidating, explaining and understanding complex biological phenomena when seen together with experimental and existing knowledge. New successes emerge as the opportunity to use mathematical and computational models in biosciences continue to grow. Collaboration between modelers and experimentalists will result in accelerated integration of these methodologies into mainstream biology.

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CHAPTER 1

Overview of Enteric Neurobiology

1.1 THE ROLE OF THE GUT At the beginning of the 20th century, Ramón y Cajal revealed the intricacies of the structure of the central nervous system. The nerve plexuses in the wall of the gut were described by Leopold Meissner and Georg Auerbach in the mid-19th century, but their function remained obscure for another hundred years, while notable advances were made in the physiology of the central nervous system. The principal reason for this relative obscurity was that there were no obvious functions that could be ascribed to the enteric nerves. In contrast, the motor and sensory functions controlled by the central nervous system — perception and bodily movement — are part of universal human experience and readily susceptible to study. In health, apart from occasional minor discomforts, the only conscious events that can be related to the digestive system, are the ingestion of food and fluid, and the act of defecation; moreover, these activities appear to be modulated by social and cultural rather than biological constraints. The elucidation of gut function only became possible when advances in technology enabled dynamic imaging of the gut and sampling of its content. The gut is a specialized region of the body surface that, being an invaginated tube through the body, is protected from the external environment. Unlike the skin, the surface of the gut is permeable, allowing nutrients and water to be absorbed to provide energy substrates to the organism, and to maintain fluid balance. In terrestrial species, for whom food and water are only intermittently available, the gut tube is also a storage device that allows the absorption of solutes and fluid over a much longer period of time than 1

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the relatively brief periods available for the ingestion of food and water. In all species, the gut tube is normally closed at each end by muscular structures that only relax proximally for the ingestion of nutrients and distally for the expulsion of waste. But it is important to remember that the lumen of the digestive tube is external to the body. It is possible to insert a probe into one end of the gut and advance it to the other end without breaching the surface of the body. But the enclosure of the gut cavity within the body confers thermal equilibration of the gut contents with the “milieu interieur,” and protection from mechanical damage to a mucosal surface that is, in parts, highly permeable. There is a wide variation in the gross anatomy of the digestive tract between species, reflecting varying habitats and diets, but there are also some common properties. In vertebrate species, the abdominal cavity is separated from the mouth by the thorax, hence the first part of the gut is a transit segment. This empties into a digestive region, where mechanical and chemical breakdown of nutrients is assisted by the addition of fluid and enzyme secretions. The final section of the digestive tube stores the residue for the final extraction of water and electrolytes and the eventual expulsion of solid waste. The nomenclature of the different regions of the human digestive tract is shown in Fig. 1.1. The esophagus is the conduit traversing the thorax from the mouth to the stomach. The stomach is the digestive vat, and has two distinct regions. The corpus is the storage region, where food accumulates and hydrochloric acid and enzymes are secreted from the mucosal lining into the cavity. The antrum is the pump that delivers gastric contents into the small intestine, which is the main absorptive region. The proximal, middle, and distal portions of the small intestine are, respectively, the duodenum, jejunum, and ileum. There are no structures in the wall of the gut that mark the limits of the three divisions, only a gradual change in mucosal structure. Conventionally, the junction of the duodenum and jejunum is located at the ligament of Treitz, where the otherwise freely mobile intestine is anchored by fascia, but there is no stepwise change in the anatomy of the bowel at this point. There is no defined location for the junction between jejunum and ileum, only a gradual change in mucosal structure along the entire length of the small intestine. Finally, the large intestine is a capacious cavity for the storage of food residues; the colon is the site of bacterial degradation of

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Overview of Enteric Neurobiology 3

Organ

Esophagus

Corpus

Stomach

Antrum Duodenum Jejunum

Small Intestine

Ileum

Large Intestine (Colon)

Rectum

Fig. 1.1 Schematic representation of the anatomy of the human digestive tract.

the remaining solids, and of the final extraction of water and electrolytes to create solid feces. The final segment of the large intestine is the rectum; it is for the most part a potential cavity, except when the propulsion of feces from the distal colon provides the stimulus for defecation.

1.2 REGIONAL FUNCTION IN THE HUMAN GUT Unlike the locomotor system, where we have detailed knowledge of physical activity, both visually and through proprioception, the gastrointestinal system provides few clues to its owner about its operations. Swallowing and defecation, respectively the propulsion and expulsion of substances, are voluntary behaviors, but the progress and transformation of material between these two acts are, for the uninformed, hidden mysteries. In upright humans, the mouth is above the anus, and children often make the na¨ıve assumption

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that gravity is important in the transit of food through the body, but gravity is not a propulsive force in the digestive tract. Propulsion is determined by the two layers of smooth muscle that surround the entire digestive tract. The inner layer of circular muscle can contract to produce a constriction of the gut cavity, while contraction of the outer layer of longitudinal muscle can reduce the length of a gut segment. In general, all circular contractions move in a caudad direction; the distance traversed by a contraction varies between a few millimeters and many centimeters. The different functions of the segments of the digestive tube are summarized in Fig. 1.2. The duration of residence of ingested material in each segment differs greatly, and is related to the transformations that take place in each segment.

Boundaries

Regional Function

Upper and lower esophageal sphincters

Transit time

Propel

20 min

Store Mix

5 - 300 min

Pump

20 sec

Pylorus Mix

Absorb

90 - 180 min

Dessicate

Ileocaecal valve

Storage

Internal and external anal sphincters Propel

16 - 48 hrs

5 - 30 min

Fig. 1.2 Summary of the function of the digestive tract. The main function of the different regions (the central column); the location of the sphincters of the gut (the left-hand column), and the duration of transit of gut content through the different regions (the right-hand column).

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Overview of Enteric Neurobiology 5

Transit through the esophagus into the stomach is rapid — a matter of seconds — and is accomplished via peristalsis: a circular muscle contraction occluding the cavity and preceded by a phase of relaxation, that traverses the entire esophagus. The caudal end of the esophagus is normally closed by a muscular structure, the lower esophageal sphincter, but the intrinsic innervation of the muscle ensures that the sphincter relaxes to allow the swallowed bolus of solid or liquid to pass into the stomach. The stomach has two functional divisions: corpus and antrum. The distal end of the antrum leads to the duodenum through the pylorus which is also a smooth muscle sphincter. To understand the mechanical function of the stomach, it is important to appreciate that the corpus and antrum have different properties. The corpus is a mixing chamber that delivers gastric content to the antrum; the latter is essentially a bidirectional pump. As soon as material is delivered from the esophagus, regular contractions sweep down the stomach as far as the pylorus every 20 seconds. These contractions are not powerful enough to occlude the cavity of the corpus, but they serve to mix ingesta and secretions, and propel content into the antrum. Compared to the corpus, the antrum is a narrow tube, and is occluded by the contraction wave, while at the same time, the pyloric sphincter briefly relaxes. During the antral contraction, a portion of content is propelled as a bolus through the pylorus into the first part of the duodenum, while the remainder is forced back into the corpus and the pylorus closes. This cyclical activity continues until all the ingested content has been emptied from the stomach. Effectively, only a few milliliters of content are delivered with each gastric contraction. Gastric emptying starts almost as soon as a meal arrives, but complete emptying of a meal, depending upon its physicochemical composition, requires an average of about five hours. The colon is the reservoir for the accumulation of non-absorbed solids, and for the recovery of most of the remaining water and electrolytes. This is essentially a process of desiccation, in which fluid content is transformed into solid feces. This is a slow process; propulsion of feces through the colon may require 24–48 hours. The final internal destination of feces is the rectum, which is filled by infrequent high amplitude peristaltic contractions of the distal colon. Distension of the rectum by feces or by gas from bacterial digestion or swallowed air is a conscious perception, and results in the urge to release gas as flatus, or to expel feces in the act of defecation. Thus, while

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the transit of feces through the colon is slow, the residence of material in the rectum is brief, assuming that the “call to stool” is heeded. Constipation — fecal retention — is often the consequence of the individual failing to respond to the stimulus of rectal filling. It is important to bear in mind that the major internal sphincters of the digestive tube — lower esophageal, pyloric, and ileocecal — are not, as was once thought to be the case, structures that regulate forward flow. Their function is to prevent retrograde flow (reflux) of content from one segment to the preceding segment because of the damage that can be inflicted. Thus, the acid contents of the stomach damage the mucosal lining of the esophagus, and bile damages the gastric mucosa, while migration of material from the colon into the ileum leads to bacterial overgrowth of the small bowel. This account of regional gut function is no more than a brief summary of a complex series of processes; a detailed narrative of the variables and mechanisms that govern the system would fill an entire textbook. Nevertheless, it should serve to show that the digestive tract is a complex organ system, capable of responding to variable inputs of food and fluid with the efficient extraction of nutrients and the exclusion of waste. Moreover, the system is able to deliver ingested solids and liquids that range between pure water and complex solids to the permeable surface of the digestive tube at a rate that does not overwhelm the absorptive processes. Nothing is wasted, and that includes the large volume of body water and electrolytes that is secreted into the gut. Appropriate propulsive motor activity of the muscular wall of the digestive tube is clearly a key component of this adaptive ability, and as with other motor functions of the body, this requires neural control.

1.3 THE INTRINSIC INNERVATION OF THE GUT The intrinsic innervation of the gut consists of two networks of neurons that entirely surround the bowel (Fig. 1.3). The myenteric plexus (Auerbach’s plexus) is located in a plane of connective tissue between the outer longitudinal muscle and the inner circular muscle, and the submucosal plexus (Meissner’s plexus) lies between the circular muscle and the submucosal layer. Both have similar architecture; they consist of a network of ganglia interconnected by axon bundles. The ganglia contain the cell bodies of the neurons. These networks surround the digestive tube from the esophagus

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Overview of Enteric Neurobiology 7 Longitudinal muscle Connective tissue Circular muscle

Myenteric plexus Submucosal plexus

Fig. 1.3 The location of the myenteric and submucosal plexuses of the enteric nervous system in relation to the outer longitudinal and inner circular smooth muscle layers of the gut wall.

to the rectum. There are three morphologically distinct species of neuron, but histochemically a much greater variety when they are characterized by their neuropeptide content. The functional groups of neurons are afferent (sensory), efferent (motor), and interneurons that have synaptic connections with other plexus neurons. The myenteric plexus regulates the motor activity of the smooth muscle, and in particular, the circular smooth muscle. This property has been the subject of much study, and the outcomes of such studies form the substrate for the chapters that follow in this book. Action potentials in neuron cell bodies are transients, as are the action potentials of smooth muscle fibers. Also, it has been possible for electrophysiologists to record directly from neuronal cell bodies in isolated ex vivo preparations of gut because, in some species but particularly the guinea pig, the longitudinal muscle can be easily stripped away to reveal the myenteric plexus. The function of the submucosal plexus is more uncertain. It is likely that it is involved in the regulation of mucosal transport, mucosal blood flow, and also the immune system of the mucosa, but it is difficult to detect transient changes in these functions that mirror neuronal activity. Moreover, the submucosal plexus cannot be accessed for study in the same way as the myenteric plexus.

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The gut may be visualized as consisting of segments that are millimeters in length, each invested with the neuronal circuitry for regulating function and for sensing physicochemical and spatial changes in the gut lumen. Each of these functional units is connected, through interneurons, with the adjacent rostral and caudal functional units. The neuronal architecture of the plexuses mirrors the architecture of the cerebral cortex, while the range of neuropeptide species in enteric neurons is almost identical with those in the brain. Collectively, the intrinsic plexuses of the gut constitute the enteric nervous system (ENS). Nearly a century ago, Langley postulated that the plexi form the third division of the autonomic nervous system, alongside the sympathetic and parasympathetic divisions. This concept was refined by later workers, so that the enteric plexuses were then seen as the “final motor neurons” of the gut, categorized as cholinergic parasympathetic or adrenergic sympathetic neurons utilizing acetylcholine (ACh) and noradrenalin (NA), respectively, as neurotransmitters. Fifty years ago, a population of enteric neurones were identified as “non-cholinergic non-adrenergic” (NANC). The quest to find “the NANC neurotransmitter” segued into the explosion of information on regulatory peptides as neurotransmitters and neuromodulators, leading to the diverse neuronal types now recognized. In recent years, the enteric nervous system has been characterized as the “little brain in the gut,” whereas the sympathetic and parasympathetic divisions serve only as the pathways for communication between the ENS and the central nervous system. Unlike the autonomic nervous system, the ENS neural networks have the property of initiating and performing complex motor programs in response to changing sensory input. The difference between the CNS and the ENS is that the “big brain” — the cerebral cortex — is remote from the tissues that it controls and the sensory input that determines its operation, whereas the “little brain” is a cellular bilayer that is separated by only millimeters from sensory input and motor output. There is one other important difference between the CNS and ENS. The cerebral cortex is organized with different areas devoted to different functions. In contrast, there are no specialized areas in the ENS, and no need for such differentiation, since every part of the digestive tube contains the neural circuitry that enables the underlying tissues to carry out their functional role.

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Overview of Enteric Neurobiology 9

1.4 GASTROINTESTINAL SMOOTH MUSCLE Gastrointestinal smooth muscle is similar to smooth muscle elsewhere in the body. The myocytes are arranged in a syncitium, with individual cells being attached to neighboring cells. The contiguity is not only structural; it is also electrical, so that membrane depolarization can spread through the cell mass, enabling it to contract as a single muscle mass. There are no motor endplates linking nerve and muscle fiber; neural control is exercised by neurotransmitters released from varicosities along the axons that penetrate the muscle mass from the ganglia in the myenteric plexus. With the exception of the esophagus, it is also characterized by regular depolarization migrating through the muscle layer from an area that serves as a pacemaker (Fig. 1.4). This characteristic was described as “the basic electrical rhythm,” when it was first described by Alvarez in 1920, but the

Intrinsic electrical rhythmicity

ICC density

none

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3/min

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11 - 8/min

+++

variable and intermittent (3 - 20 min?)

+

none

+

pacemaker

pacemaker

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Fig. 1.4 The pacemaking properties of the digestive tract. The density of interstitial cells of Cajal (the right-hand column), showing a correlation between their density and pacemaking activity; pacemaking sites (the left-hand column), and the frequency of pacemaking activity (the central column).

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term “electrical slow wave” is now generally used. The smooth muscle layers of the stomach, small intestine, and colon are electrically isolated, and these organs have differing electrical slow wave frequencies. The “pacemakers” are not discrete structures; they are the regions of the viscus with the fastest intrinsic slow wave frequency. In the stomach and small intestine, the pacemaking sites are at the orad end of the organs, in the proximal gastric corpus and just distal to the pylorus respectively, so that the slow waves always propagate in a caudad direction. If the bowel is transacted and the cut ends then rejoined, a permanent stepwise change in frequency will occur at the site of the junction as the continuity of the smooth muscle layers is interrupted by fibrous scar tissue. In such a situation, there will be a new pacemaking site for the bowel distal to the site of the section, at a lower frequency than before. The situation in the colon is less certain; slow wave activity has been detected but the electrical activity of colonic smooth muscle is complex and remains to be fully characterized. Different layers of the colonic wall may have different slow wave frequencies, and pacemaking sites have not been identified. Understanding the motor physiology of the stomach and small bowel has been possible because of the close similarity in morphology, electrophysiology, and function between man and other mammalian species, but this not the case for the colon. The electrical activity of the stomach and small bowel can be clearly recorded by electromyography using electrodes attached to the serosal surface of the organs. Such recordings reveal the succession of slow waves, and the associated spike bursts. Slow waves, which reflect the synchronous depolarization of contiguous smooth muscle cells, mark the points in time when contractions of the myocytes are possible. The spike bursts are the summated action potentials of cells in the vicinity of the recording electrode. The correlation between spike bursts and contractile activity can be confirmed by simultaneous recording of pressure changes at the same location. The electrical slow wave activity of the stomach can also be recorded using electrodes on the surface of the abdomen because the relatively stable position of the stomach within the abdominal cavity provides a consistent electrical vector; this non-invasive technique is known as electrogastrography. The technique is of limited value because spike bursts associated with the gastric slow wave cannot be detected using this method.

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Overview of Enteric Neurobiology 11

The presence, rhythmicity, and propagation of the slow wave in the gut wall are linked to, and may be dependent upon, another tissue element of the gut wall. These are the interstitial cells of Cajal (ICC), first described by Ramón Y Cajal as simple interstitial cells. They are not neuronal elements, but they have the property of excitability. The density of the population of ICC is greatest where slow wave activity is dominant, particularly the small intestine. They are, however, absent, as is slow wave activity, in the esophagus. It will have occurred to the informed reader that there are close affinities between gastrointestinal smooth muscle and myocardium, albeit that the intrinsic electrical frequency of gut muscle is much slower than in the heart. There is, however, one major difference. Every propagated slow wave in the myocardium induces action potentials resulting in contraction. This is not the case in the gut; the passage of each slow wave provides the opportunity for a contractile event, but whether or not a contraction occurs is determined by the intrinsic innervation. This allows the creation of patterns of motor activity that reflect the need, at any locus, for gut content to be arrested, perturbed, or propelled, and confers great flexibility on the system. There is one further difference between the smooth muscle of the heart and the gut. The only possible variation in myocardial performance is a change in pacemaking frequency, so as to increase or decrease the heart rate and hence cardiac output. In the gut, pacing frequency does not change, and the motor activity at any point is varied by the presence or absence of action potentials during the passage of the wave of depolarization.

1.5 EXTRINSIC INNERVATION The co-ordinated motor response to a meal involves the entire gut. Local motor activity is determined by the intrinsic innervation of the gut; since this is continuous from one end of the digestive tract to the other, it might be thought that the transmission of information along the ENS is sufficient for co-ordination. Certainly it suffices for the propagation of peristalsis; distension of a gut segment by a solid bolus stimulates circular muscle contraction above the bolus, and relaxation below, and so the bolus is moved to the next segment. But there are also responses by segments to events that are occurring in a distant segment. One example is the “gastro-colonic response” in which distension of the stomach by a meal provokes the movement of stool

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BRAIN

CG

IMN SMG

Spinal Cord

IMN

IMG

Fig. 1.5 Anatomical connections of the enteric nervous system, prevertebral ganglia, and spinal cord. (CMG = celiac ganglion, SMG = superior mesenteric ganglion, IMG = inferior mesenteric ganglion, IMN = intermesenteric nerve).

from the distal colon into the rectum, provoking the urge to defecate. In contrast, painless distension of the rectum following the ingestion of a meal will diminish the vigor of propulsive activity in the proximal small bowel. Communication between stomach and distal colon along the ENS would involve many synapses. There is, however, a system for the rapid transmission of information between distant parts of the gut. There are three prevertebral ganglia — celiac, superior mesenteric, and inferior mesenteric — that are connected to the ENS (Fig. 1.5). These ganglia are connected by two intermesenteric nerves. A stimulus in one part of the bowel can travel by an afferent axon to one of the ganglia, along an intermesenteric nerve to the next ganglion, and thence back to the ENS in the axon of an efferent nerve; only three synaptic connections are required. These pathways are essential for the patterns of motor activity that involve the whole of the stomach and small intestine.

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Overview of Enteric Neurobiology 13

Finally, the neural connections between the gut and the brain deserve mention. It has long been known that the vagus nerve, the Xth cranial nerve, innervates the fore-gut as well as the heart. Pavlov’s discovery of the conditioned reflex at the end of the 19th century led to the belief that the CNS “controls” the gut. It is now known that the autonomic pathways to the gut are largely afferent, conveying information from sensory receptors in the gut to the brain. These pathways are the parasympathetic vagus and pelvic nerves, and the adrenergic spinal nerves, projecting from the prevertebral ganglia into the spinal cord. Gut-brain interaction has been the subject of serious study for the last three decades, but it would be inappropriate to deal at length with this subject in a book devoted to enteric neurobiology. Suffice it to say that the brain is kept fully informed of events in the digestive tract, even though little, if any, of this information is normally projected into consciousness.

1.6 THE EFFECT OF FOOD ON THE GUT The effect of food on gut function is not only a profound secretory response, but also a profound motor response. To understand this motor response, it is necessary to consider the activity of the gut in the absence of food — the fasting motor pattern. Recording of contractile events in the fasting gut reveals a stereotypic motor pattern throughout the stomach and small bowel that is repeated, on average, every 90 minutes. First, there is a phase (Phase 1) of motor quiescence, which dominates the greater part of the cycle (Fig. 1.6a). This is followed by a phase (Phase 2) of intermittent contractions. The final part of the cycle (Phase 3) consists of a relatively brief phase of powerful regular contractions, occurring at the frequency of the slow wave at that location: ν = 0.05 Hz in the human stomach, and ν = 0.15–0.2 Hz in the duodenum. This cycle is periodic motor activity, the period being the interval between two similar phases, for example between one Phase 3 and the next Phase 3. The effect of food is the abolition of this cyclical motor complex, with its replacement by intermittent contractile activity similar to Phase 2. The cardinal element of the fasting motor complex is its migratory nature and it is now known as the migrating motor complex (MMC) (Fig. 1.6b). Recording at multiple sites in the stomach and small bowel shows a slow but steady progression of each MMC along the bowel. MMC’s may start in

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(a)

Periodic motor activity

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Stomach Duodenum Jejunum Ileum

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Motor quiescence Irregular contractions Regular contractions

Postprandial (“fed”) pattern - resembles Phase 2

(b)

Fasting MMCs

0

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Fed

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Fig. 1.6 (a) Periodic motor activity of the stomach and small intestine (the upper panel) and data for the approximate duration of the phases of the motor complex (the lower panel). (b) Migrating motor complexes as they are recorded, and of the simultaneous arrest of MMCs on feeding at all levels of the small intestine. The icons for Phases 1–3 of the MMC are defined in Fig. 1.6a.

the stomach, but others are generated in the duodenum. Most MMC’s reach the terminal ileum, but some appear to fade at a more proximal locus. The migration velocity of the MMC is usually measured in centimeters/minute, whereas slow wave — and hence peristalsis — propagation is measured in

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Overview of Enteric Neurobiology 15

centimeters/second. Transit of an MMC from the stomach to the terminal ileum takes 1–2 hours. The MMC represents a biorhythm which is common to all mammalian species. In man, the periodicity of the MMC is similar to the periodicity of the nREM/REM sleep cycle, but this is not the case in other species, and there is no synchrony between the two biorhythms in man. In man and other carnivorous species, the effect of ingesting a meal is dramatic; the fasting pattern of MMC’s is abolished at all levels of the gut almost simultaneously (Fig. 1.6b). The periodic fasting pattern is replaced by a pattern of intermittent contractions, similar to the Phase 2 component of the MMC. This conversion is clearly not due to the local presence of nutrients, as MMC activity on feeding is abolished in the distal small bowel well before the arrival of any nutrient. MMC activity returns only when the last remnants of the meal pass out of the stomach into the duodenum. Clearly, the trigger for the conversion to the fed motor pattern is the detection of nutrients by mucosal chemoreceptors in the proximal duodenum. But these receptors are not connected to the ENS, but are vagal sensory receptors. The arrival of food stimulates vagal afferent input to the dorsal vagal nucleus (Fig. 1.7). In turn, the efferent vagus is activated, and it is the efferent vagal input into the ENS that effects the conversion. The conversion from fasting to fed activity can be abolished experimentally by section of the vagus nerves, and it is impaired in patients who have had truncal vagotomy. The significance of periodic activity and its abolition by food in terms of enteric neurobiology is that it is now clear that these are motor programs that reside within the enteric nervous system. There are parallels with the central nervous system in that the MMC program like the CNS programs for coordinated locomotor activity, is not expressed at birth. In human newborns, MMC’s start to appear 3–6 months after birth. The functional importance of these motor patterns in man is twofold. First, the MMC pattern is required for the propulsion of non-digestible solids that are not broken down to a small particle size in the stomach. During the gastric emptying of a meal, gastric contractions are not powerful, and only particles less than 5 millimeters in diameter are expelled through the pylorus. It is not until the more powerful contractions of the gastric MMC return, and all nutrients have been emptied, that larger non-digestible solids will be expelled from the stomach. Secondly, MMC’s play an important role in preventing bacterial overgrowth of the small

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C A

Fig. 1.7 The pathways for the switch between fasting and fed motor activity. The arrival of nutrient in the duodenum (A) is detected by vagal afferents, and transmitted to the dorsal vagal nucleus (B). This activates vagal efferent input to the enteric nervous system (C) that operates the change in motor program from periodic fasting activity to the postprandial pattern.

intestine. Although the ileocecal valve is a barrier to the reflux of bacterial flora from the colon, it is an imperfect barrier. The repetitive scouring action of MMC’s in sweeping all intestinal content into the colon is essential to the maintenance of a germ-free lumen in the small bowel. Professor C. Code described the MMC as “the intestinal housekeeper”; research since then has confirmed his prediction. Finally, some other salient points about periodic and postprandial motor activity: i) The genesis of an MMC in the duodenum is accompanied by the release of motilin, a peptide unique to the gut, from mucosal endocrine cells into the bloodstream. Whether this is the cause or the effect of the motor phenomenon remains unresolved. ii) Proof that the motor programs reside within the ENS is provided by evidence that the motor programs are damaged or absent in diseases in which the ENS is damaged. This fact allows diagnosticians to use manometric recordings of small bowel motor activity to test the integrity of the ENS in suspected motor disorders of the gut.

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Overview of Enteric Neurobiology 17

iii) As stated above, when the bowel is transacted and rejoined, there is a permanent change in slow wave frequency above and below the site of section. MMC migration is similarly interrupted, but after a few weeks, MMC’s cross the site of section and co-ordination is restored. This suggests that, while the continuity of the smooth muscle mass is permanently interrupted by scar tissue, neurons can grow across the gap to restore the integrity of the ENS. iv) Mechanoreceptors on the peritoneal surface of the gut respond to pressure — when, for example, the bowel is manhandled by a surgeon during a laparotomy — by inducing a reflex adrenergic blockade of all mechanical activity in the bowel wall. This is a state known as paralytic ileus, and it persists for hours after the stimulus has been removed.

1.7 CLINICAL PHARMACOLOGY The history of the use of medications intended to modify the motor activity of the gut falls into two distinct phases, namely before and after the identification of the neurotransmitters of the ENS, and the characterization of their many receptors. The first phase was the use of herbal extracts for the treatment of gastrointestinal ailments. The two commonly used herbal remedies were opium, extracted from the seeds of the poppy Papaverum Somniferum, and belladonna, extracted from the berries of Atropa belladonna. Opiates have long been used for the treatment of diarrhea, but also for the relief of pain and abdominal discomfort. Their significance in the present context is that opioids are neurotransmitters within the ENS, and the antidiarrheal actions of opiates depend, at least in part, on the diminution of the velocity of transit of bowel content. Belladonna extracts contain atropine alkaloids, which are strongly anticholinergic. Belladonna extracts diminish the propulsive force of the gut, and also the secretion of gastric acid, but their use, although widespread in antiquity, was limited by unwanted adverse effects on other body systems. The modern era of gut motor pharmacology began with the discovery of two drugs in the 1960s. One drug was Diphenoxylate (Lomotil), an opiate agonist that acts only on opiate receptors in the gut, and thus lacks the unwanted addictive effects of opiates on the central nervous system. It has

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proved to be an unrivalled preparation for the management of diarrhea, but its efficacy derives from its effect on absorption rather than propulsion, and it has been largely replaced by Loperamide (Imodium), which proved to be more effective. The other drug was Metoclopramide (Maxolon, Pimperan), a substituted benzamide. It was found to have potent anti-emetic properties, but also to accelerate gastric emptying, and from this stemmed the concept of “pro-kinetic” drugs. As receptors in the gut became identified, it became clear that Metoclopramide acts both on the dopamine D2 receptor, and on 5-hydroxytryptamine receptors, where it is a mixed (5-HT4 ) agonist and (5-HT3 ) antagonist. As pharmaceutical medicine moved from the age of accidental drug discovery to the new era of designer drugs, the search began for “pro-kinetic” drugs that act selectively on gut receptors. One of the drawbacks of the drug was its actions on the central nervous system; indeed, its anti-emetic potency derives from its action on the chemoreceptor trigger zone that initiates emesis. The aim was to find drugs with a “pro-kinetic” effect on the gut, preferably acting on a single receptor, and free of CNS side-effects because of an inability, in therapeutic dosage, to cross the blood-brain permeability barrier. The first apparent success was Cisapride (Prepulsid, Propulsid ), discovered at Janssen Pharmaceutica, Belgium. Cisapride is a parasympathomimetic that acts as a strong 5-HT3 - and weak 5-HT4 -receptor agonist. Stimulation of the serotonin receptors increases acetylcholine release in the ENS. Cisapride did indeed prove useful in the treatment of gastro-oesophageal reflux, by increasing the force of peristalsis and so increasing the clearance of acid refluxed from the stomach. But other claimed benefits were inconsistent, and the drug was withdrawn by the Food and Drug Administration (USA) in 2000 because of adverse cardiac side effects. Two other drugs acting on serotonin receptors in the gut are Tegaserod (Zelnorm), a 5-HT4 agonist, and Alosetron (Lotronex), a 5-HT3 agonist. Both have been promoted for different components of the irritable bowel syndrome, but their success has not been overwhelming. While the drugs mentioned simulated metoclopramide in their actions on the gut serotonin axis — but without any central effects — another drug, Domperidone, was intended as a dopamine D2 antagonist to have some of the properties of Metoclopramide. But its potency is limited, and it also stimulates the release of prolactin from the hypothalamus.

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Overview of Enteric Neurobiology 19

Two more pharmacological approaches deserve mention. Trimebutine is a gut-selective opioid agonist developed in France aimed at relieving various gastrointestinal discomforts. It was never submitted for licensing in the USA or the UK, but is widely used in Europe, Asia, and South America. However, such success as it has had owes more to successful marketing than proven efficacy. Secondly, there are the derivatives of erythromycin, a macrolide antibiotic. Erythromycin was known to be associated with gastrointestinal side-effects, and when the effect of the drug on gut motor activity was studied, it emerged that it mimics the effects of motilin. Manipulation of the molecular structure by substitution of the sugars in the macrolide produced molecules that preserved the motilin effect while making it devoid of any antibacterial action. An ideal candidate disease for such a drug is the gastroparesis (paralysis of gastric motor activity) that occurs in Type 1 diabetes mellitus, and this proved to be the case. Sadly, this relatively unusual condition did not seem to justify the cost of bringing the drug to the market. What all of these drugs share is dubious therapeutic efficacy. They have been designed to alleviate the spectrum of so-called “Functional Gastrointestinal Disorders” (FGID’s), which include such ill-defined entities as irritable bowel syndrome and non-ulcer dyspepsia. In these disorders, controlled randomized blinded trials show a high level of placebo response at about 30%, however, measured. Drugs such as Tegaserod and Alosetron show an improvement in 50% of subjects, but that is only an improvement of 20% over placebo. The reason for this is almost certainly that the indications for their use are vague, both in terms of diagnostic precision and pathophysiology. Pharmacological sophistication may have outstripped our understanding of the biology of these disorders. On the other hand, perhaps the best is yet to come. The chapters that follow describe methodologies that should lead to the development of other drugs, and at considerably lower cost than is the case today.

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CHAPTER 2

Myoelectrical Activity of the Smooth Muscle

2.1 MYOELECTRICAL AND MECHANICAL ACTIVITY Real muscle cells of the gastrointestinal tract have many conductances that participate in the generation of a variety of endogenous membrane oscillations described as the migrating myoelectric complex. Understanding the mechanisms of this biological phenomenon requires information about the dynamics and function of different “mixes” of ion conductances. This analysis becomes more complex due to the fact that the dynamics and activity of some of the ion channels are modulated by neurotransmitters and hormones. From this it is obvious that it is extremely difficult to discover the contribution of each conductance in controlling the excitability of muscle cells. Therefore, a quantitative mathematical model is helpful to do so. Many mathematical models have been developed to simulate patterns of electrical activity of excitable biological media: the Bonhoefer–van der Pol oscillator model; the FitzHugh–Nagumo model; the model of Plant and Kim; and the Chay and Keizer model. All these models are based on the reduced Hodgkin–Huxley equations and embody the behavior of the Na+ , K+ , Ca2+ and mixed Ca2+ –K+ channels. They have proved useful for gaining mathematical insights into the membrane nonlinearities underlying the electrical properties. However, the parameters used have no direct physical interpretation. Despite strenuous attempts to simulate the myoelectrical activity with relaxation oscillators, it becomes increasingly clear that this is not a useful model because it does not include the unique parameters that govern the behavior of specific ionic channels or the cable properties of smooth muscle cells. A class of Plant models differs from previous ones in which the 20

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Myoelectrical Activity of the Smooth Muscle 21

slow oscillation depends crucially on changes in transmembrane potential, and the mechanisms for spiking and slow oscillation are closely integrated. Numerical treatment of the models has shown that the typical slow waves are satisfactorily reproduced and the transition from slow wave to bursting represents the bifurcation from one periodic solution to another with the bifurcation parameter. While these models are too complicated for mathematical analysis and also computationally quite demanding, they are particularly attractive for the biologists because they are based on experimentally measurable biophysical entities.

2.1.1 Biological Background Considerable work in the past has been devoted to the isolation and investigation of ion channels in the membrane of smooth muscle cells. Although the precise detailed analysis of their electrical activity has not been possible, because of technical difficulties (isolated smooth muscle cells do not generate slow waves; in vivo small intestinal muscles are difficult to voltage clamp; there are no specific blockers of different channels involved that function properly in the physiological ionic gradients) the dynamics of slow waves and action potentials can be conceptualized as the resultant of the coordinated work of a mixture of channels. Pharmacological and electrophysiological studies based on the analysis of functional properties have uncovered three general types of Ca2+ channels: voltage-dependent, activated by the depolarization of the membrane; receptor-linked, controlled by the neurotransmitters, and leak Ca2+ channels. Recently, molecular cloning has revealed a greater diversity in the voltagedependent class: L, T, N and P types of voltage-dependent Ca2+ channels have been identified. L-type are high-voltage activated, “slow” channels, sensitive to 1,4-dihydropyridine compounds. They are found in virtually all excitable tissues and are the major pathway for voltage-gated calcium ions entry in smooth muscle cells. T-type are low-voltage activated,“fast” channels because they are opened by small depolarizations from relatively negative holding potentials and exhibit rapid, purely voltage-dependent inactivation. Their most prominent function is to support pacemaker activity and Ca2+ entry at negative membrane potentials. N-type show a greater tendency to inactivate with depolarized holding potentials, although their rate of inactivation

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is quite variable. It has been strongly argued that they are important in the mediation of neurotransmitter release. P-type are high-voltage activated Ca2+ channels and demonstrate slow inactivation. They are present in cerebellar Purkinje cells and are not found in the smooth muscle cells of the gastrointestinal tract. The effects of activating receptor-linked Ca2+ channels are quite complex and possibly modulate the properties of voltage-dependent Ca2+ channels. The experimental results of the activation of muscarinic receptors on the intestinal smooth muscle by iontophoretically applied acetylcholine showed a chronotropic effect on their electrical activity. An application of ACh increased the peak and slowed the rate of decay of depolarization wave induced by the inward calcium current. There was no apparent shift in the voltage-current relationship, suggesting that ACh does not affect the voltage dependence of Ca2+ channels. Another mechanism of modulation of voltage-gated Ca2+ channels is by interstitial cells of Cajal. The excitatory signal generated in the ICC usually has an amplitude of 60–70 mV, strong enough to coordinate the generation of slow waves and spikes in smooth muscle cells. Two populations of K+ channels are identified: Ca2+ -activated K+ channels and potential sensitive K+ channels. It is uncertain from experimental observations made on gastrointestinal smooth muscle cells to what extent these channels individually contribute to the potassium current dynamics of a single cell. However, it is quite clear that the biological significance of Ca2+ –K+ channels is to serve as a natural feedback mechanism to terminate periods of excitation by increasing K+ conductance. No obvious sign of Na+ channels or currents has been found in single visceral smooth muscle cells, yet the slow potential changes that are prominent in the whole tissue seem to involve an increase in sodium permeability. The existence of at least two types of Na+ channels, distinguished by their kinetics and sensitivity to tetrodotoxin, is strongly suggested in smooth muscle cells. Physiologically, sodium current may accelerate the generation of an action potential by triggering a rapid activation of calcium current, while not being essential for activation of action potentials. Since the Cl− permeability of smooth muscle cells is now believed to be low, Cl− channels and currents are likely to be, respectively, sparse and small.

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Myoelectrical Activity of the Smooth Muscle 23

The membrane potential difference in most smooth muscle of the gastrointestinal tract is not a steady value. Instead, it exhibits two types of spontaneous and repetitive depolarizations and repolarizations which are slow and of long duration, ∼ 2 s, oscillations, 0.01–0.5 Hz, of membrane potential of an amplitude of 15–30 mV known as slow waves, and brief short-term, 0.1–0.2 s, and high amplitude, 60–70 mV, action potentials, known as bursts. Extracellular recordings have demonstrated that the individual slow wave usually has a sinusoidal configuration that consists of a rapid depolarizing phase which is followed by a sustained positive plateau potential and a slow repolarization phase, although the shape and amplitude vary in different parts of the bowel. The precise ionic mechanism of the slow waves is still unknown, but pharmacological and ion replacement experiments have suggested that the rising phase is generated by a transient increase in membrane permeability to calcium ions via fast voltage-gated T-type Ca2+ channels, while the following plateau phase is the result of an increase in the intracellular concentration of Ca2+ which is due to ion influx through the voltage-gated L-, N-type Ca2+ channels. During the repolarization phase inactivation of Ca2+ -channels and opening of Ca2+ -activated K+ channels are observed. However, no clear solution to this problem exists at present, and a detailed quantitative description of the contributions of the various ionic channels at different phases of generation of slow waves is not available. Spike bursts consist of a regular sequence of action potentials lasting from 100 to 200 (ms) each. Bursts are not always present and occur only on the crest of a slow wave. Their occurrence depends on neurohumoral influences. Cholinergic agonists enhance spike potential activity whereas adrenergic amines inhibit it. Contraction of smooth muscle is the fundamental event in gastrointestinal motion. In gastrointestinal muscle, the cells are embedded in a connective tissue matrix and coupled electrically via gap junctions into a functional syncytium. The syncytium reacts to external and internal stimuli, both electrical, neural and chemical, with the generation of active forces. Although many biochemical details underlying excitation-contraction coupling are not yet defined, the key player in the coupling phenomenon is free cytoplasmic Ca2+ . Cytosolic free Ca2+ triggers the events leading to the interaction of the contractile proteins and so to force development. Contraction of smooth

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muscles is thought to be initiated by myosin light chain phosphorylation via the activation of Ca2+ calmodulin-dependent myosin light chain-kinase, an important prerequisite for myosin-actin interaction. Studies using aequorin and fura-2 suggest that the relationship between cytosolic Ca2+ , myosin light chain phosphorylation and force development changes as a function of free ˜ 2+ ]. Many details of this process have become well estabintracellular [Ca lished. However, a complete experimental correlation between the ultrastructural and mechanical measurements in smooth muscle is still lacking. In the resting state, the concentration of cytosolic calcium is not more than 0.1 µM. Upon stimulation, the cytosolic Ca2+ rises, mainly as a result of influx and partly from the mobilization of internal stores. There are at least four different pathways for Ca2+ influx. That Ca2+ which enters the cell through resting influx or leak mechanisms is taken up by storage sites, presumably the sarcoplasmic reticulum, and does not directly increase the ˜ 2+ ]. Calcium entering the cell through L-type Ca2+ cytoplasmic free [Ca channels is partly accumulated in the storage sites, while the rest reaches the cytoplasm to activate contractile filaments. Ca2+ entering through T-type Ca2+ channels does not fill the stores because the storage site cannot accumulate or retain calcium during spike bursts. Thus, all of the Ca2+ entering through voltage-dependent Ca2+ channels is available to activate contractile filaments. A fourth source for a rise in internal calcium is the Ca2+ -induced Ca2+ release from the sarcoplasmic reticulum which can be initiated by Ca2+ influx, by caffeine and/or by activation of the second messenger system, inositol triphosphate. However, many of the calcium handling mechanisms as well as the regulation of dephosphorylation in visceral smooth muscles have not been fully clarified. A spike discharge in intestinal smooth muscle induces a single phasic contraction (twitch). Bursts of multiple action potentials trigger fusion of separate twitches into a long-lasting tonic contraction. Rhythmic contractions can also be recorded in intact gastrointestinal muscle in the presence of slow waves only. Amplitudes and durations of contractions are related to the amplitudes and durations of the plateau phase of the slow wave. Slow waves can also regulate the rate and force of phasic contractions. In addition, smooth muscle can maintain constant tension in the absence of applied stimulation, called tonus.

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Myoelectrical Activity of the Smooth Muscle 25

2.1.2 Mathematical Model Consider a finite number of smooth muscle fibers which are arranged in a sequence. The myoelectrical activity of it is under the control of the pacemaker cell, which is morphologically identified as interstitial cell of Cajal. The principal assumptions embodied in the model are: (i) the smooth muscle fiber is a dynamical self-regulatory myogenic medium; (ii) its electrical activity, slow-wave or bursting, represents the integrated function of ionic channels, presumably: voltage-dependent Ca2+ channels of L-, T-type, Ca2+ -activated K+ -channels and potential sensitive K+ -channels, and leak Cl− -channels; (iii) ICC generates a high-amplitude spike that propagates along the fiber; it modulates the properties of L-type voltage-dependent Ca2+ channels; this effect is assumed to be mainly chronotropic, with an increase in the time of permeability for calcium ions; (iv) the muscle fiber possesses nonlinear viscoelastic properties; active forces of contraction are a result of activation of contractile protein system, which depends on the dynamic turnover of intracellular calcium, and passive relaxation forces are due to the elastic properties of the reinforcing collagenous and elastin fibers. The governing system of equations that describes the myoelectrical activity of the smooth muscle fiber consists of the following equations. The dynamics of generation and propagation of the excitation wave along the smooth muscle fiber is given by
 1 ∂ ∂ϕ ∂ϕ αCm = a(s) ∂t 2Rm ∂s ∂s T L ∗ + ICa + IK∗ + ICa−K + ICl ), − (ICa

s ∈ (0, L)

(2.1)

where T T ˜ − ϕ˜ Ca ) = g˜Ca m ˜ 3 h(ϕ ICa L L = g˜Ca x˜ Ca (ϕ − ϕ˜ Ca ) ICa

IK∗ = g˜K n˜ 4 (ϕ − ϕ˜ K )

(2.2)

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∗ ICa−K =

˜ 2+ ](ϕ − ϕ˜ K ) g˜Ca−K [Ca ˜ 2+ ] 0.5 + [Ca

ICl = g˜Cl (ϕ − ϕ˜ Cl ). L Here Cm is the membrane capacitance; ϕ is the membrane potential; ITCa , ICa ∗ are the fast (T-type) and slow (L-type) inward calcium currents; ICa−K , IK∗ are the outward Ca2+ -activated K+ and voltage-activated K+ currents; ICl is the leak Cl− current; ϕ˜ Ca , ϕ˜ K , ϕ˜ Cl are the equilibrium potentials for the T L respective currents; g˜Ca , g˜Ca , g˜K , g˜Ca−K , g˜Cl , are the maximal conductances ˜ 2+ ] is the intracellular concentration of Ca2+ of ionic channels involved; [Ca ions; and α is the numerical parameter; a(s) is the cross-section radius; Rm is the specific resistance of the membrane; s is the Lagrangian coordinate. The probabilities of activation and inactivation of ionic channels ˜ n, (m, ˜ h, ˜ x˜ Ca ) satisfy the following equations

m ˜ I = α˜ m /(α˜ m + β˜ m ) d h˜ ˜ − β˜ h h)/1.5 ˜ = (α˜ h (1 − h) dt d n˜ = (α˜ n (1 − n) ˜ − β˜ n n)/1.5 ˜ dt   d x˜Ca = 1 + [exp 0.15(−ϕ − 50)]−1 − x˜ Ca /60 dt

(2.3)

˜ 2+ ] d[Ca ˜ 2+ ]) = ρ(Kc x˜ Ca (ϕ˜ Ca − ϕ) − [Ca dt where α˜ m =

0.1(50 − ϕ) exp(5 − 0.1ϕ) ˜ −1

β˜ m = 4 exp(25 − ϕ)/18 ˜ ˜ α˜ h = 0.07 exp(25 − ϕ)/20 β˜ h = (1 + exp(5.5 − 0.1ϕ)) ˜ −1 α˜ n =

0.01(55 − ϕ) ˜ exp(5.5 − 0.1ϕ) ˜ −1

˜ β˜ n = 0.125 exp(45 − ϕ)/80.

(2.4)

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Myoelectrical Activity of the Smooth Muscle 27

Here α˜ (m,h,n) is the rate, at which the channels switch from a closed to an open state, and β˜ (m,h,n) is the rate for the reverse; ρ, Kc are parameters referred to the dynamics of Ca2+ -channel; and ϕ˜ = (127ϕ + 8265)/105. The dynamics of the action potentials at the ICC (ϕ4 ) is described by the function of voltage-dependent N-type Ca2+ , Ca2+ -activated K+ , Na+ , K+ and Cl− channels: Cs

dϕ(4) = −(ICa + ICa−K + INa + IK + ICl ) + Iext(4) dt

(2.5)

where Cs is the membrane capacitance; I... are ionic currents carried through different ionic channels and Iext(4) is the external membrane current. The equations for ionic currents are given by ICa = ICa−K = INa =

gCa(4) z (ϕ(4) − ϕCa ) 1 + ζ[Ca2+ ]n gCa−K ρ∞ [Ca2+ ]n (ϕ(4) − ϕCa−K ) 0.5 + [Ca2+ ]n 3 gNa(4) mNa hNa (ϕ(4)

(2.6)

− ϕNa )

IK = gK nK4 (ϕ(4) − ϕK ) ICl = gCl(i) (ϕ(4) − ϕCl ) where ϕCa , ϕCa−K , ϕNa , ϕK , ϕCl are the reversal potentials for the respective currents; gCa(4) , gCa−K , gNa(4) , gK , gCl(4) are the maximal conductances of Ca2+ (N-type), Ca2+ -activated K+ , Na+ , K+ and Cl− channels; ζ is the parameter of calcium inhibition of the Ca2+ -channels; [Ca2+ ]n is the intracellular concentration of free calcium; z, ρ∞ , mNa , hNa and nK are dynamic variables of the ion channels whose kinetics is described by dz dt dhNa dt dnK dt ρ∞

= (z∞ − z)/τz = λh (h∞ − hNa )/τh = λn (n∞ − nK )/τn = 1 + exp(0.15(ϕ(4) + 47))−1

(2.7)

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mNa = m∞ (ϕ(4) ) y∞ = αy∞ τy

(y = mNa , hNa , nK )

τy = 1/(αy∞ + βy∞ ) αm∞ =

0.12(ϕ(4) + 27) 1 − exp(−ϕ(4) − 27)/8

βm∞ = 4 exp(−ϕ(4) − 47)/15 αh∞ = 0.07 exp(−ϕ(4) − 47)/17 βh∞ =

1 1 + exp(−ϕ(4) + 22)/8

αn∞ =

0.012(ϕ(4) + 12) 1 − exp(−ϕ(4) − 12)/8

βn∞ = 0.125 exp(−ϕ(4) − 20)/67 z∞ =

1 . 1 + exp(−0.15(ϕ(4) + 42))

Changes in the free cytosolic calcium concentration yield d[Ca2+ ]n 0.2z(ϕCa − ϕ(4) ) − 0.3[Ca2+ ]n . = dt 1 + ζ[Ca2+ ]n

(2.8)

The membrane current Iext(4) = ϕ4 /Rs ; Rs is the specific resistance. L-type voltage-dependent Ca2+ -channels are under modulatory control of the ICC and are supposed to become active if the input, recorded as the excitatory potential, (ϕ+ ), exceeds a certain threshold (ϕ∗ ). Therefore, the evolution of channels is according to L L g˜Ca = δ(ϕ+ )g˜Ca

(2.9)

where δ(ϕ+ ) is the function defined as δ(ϕ+ ) = 1, ϕ+ ≥ ϕ∗ and δ(ϕ+ ) = 0 otherwise.

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Myoelectrical Activity of the Smooth Muscle 29

The dynamics of propagation of the electromechanical wave along the muscle fiber is given by
 ∂ ∂v ∂v p a γ0 = k +T +T (2.10) ∂t ∂s ∂s where v is the velocity of contraction-relaxation; γ0 is the linear density of muscle in an undeformed state; T p , T a are the passive and active components, respectively; k is the rheological parameter. The passive (T p ) component is calculated as  0, λ≤1 Tp = (2.11) c1 (exp(c2 (λ − 1) − 1)), λ > 1 and for the active force (T a ) components, we assume the empirical relationship  ˜ 2+ ] ≤ 0.1 µM 0, [Ca      ˜ 2+ ]4 + c5 [Ca ˜ 2+ ]3 c3 + c4 [Ca a T = (2.12) ˜ 2+ ]2 + c7 [Ca ˜ 2+ ], 0.1 < [Ca ˜ 2+ ] ≤ 1µM  +c6 [Ca     ˜ 2+ ] > 1 µM maxT a , [Ca where λ is the rate of elongation; c1−7 are mechanical constants. The initial conditions assume that the smooth muscle fiber and ICC are in the resting state t = 0: ϕ∗ (s, 0) = 0,

ϕ(+) (0) = 0,

ϕ(0) = ϕrest

ϕ0 = ϕ1 = 0;

ϕ2 = ϕ3 = −60 (mV);

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

(∼)

m (0) = α m (0)/( α m (0) + β m (0)) h (0) = α h (0)/( α h (0) + β h (0))

(2.13)

n (0) = α n (0)/( α n (0) + β n (0)) ˜ 2+ ](0, ϕrest ) = [Ca ˜ 2+ ]0 . x˜ Ca (0, ϕrest ) = (x˜Ca )0 , [Ca

The boundary conditions assume that the excitation of the system is due to the discharge of the ICC. The description provided constitutes the complete mathematical formulation of the model of the self-regulatory myoelectrical activity of the smooth

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muscle fiber. It reproduces accurately the following sequence of events: (i) the discharge of the ICC; (ii) the modulation of activity of the L-type Ca2+ -channels on the smooth muscle membrane; (iii) the increase in intracellular calcium ion concentration in the smooth muscle cell and activation of fast T-type Ca2+ -channels, and (iv) the development of the electromechanical reaction of the contractionrelaxation.

2.1.3 Numerical Algorithm The simulation of the propagation of electromechanical waves along the smooth muscle fiber requires the simultaneous integration of Eqs. (2.1), (2.3), (2.5), (2.7), (2.8), (2.10) with additional relations (2.9), (2.11), (2.12), and the initial condition (2.13). In the solution of non-stationary wave problems, when rapid transients are involved, finite difference schemes are more efficient, because the limitations of the time-step of integration imposed by the stability conditions are not cumbersome. Equations (2.1) and (2.10) can be written in vector form ut = T (u)s u(0, t ) = u(L, t ) = 0;

u(s, 0) = 0

φt = (k ∗ (u)φs )s − f (φ) φ(s, 0) = 0; where vectors

(2.14)

φ(0, t ) = φ0 , φ(L, t ) = 0




 ∂u ∂u T u= , ∂˜s ∂t

2
T ∂ u ∂u 1 a p k T (u) = + φT (u) + T (u) ∂t γ0 ∂t ∂˜s

k ∗ (u) =

a(s) 2Rm Cm (1 + ∂u/∂˜s )

T L + ICa + IK + ICa−K + ICl )/Cm . f (φ) = (ICa

(2.15)

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Myoelectrical Activity of the Smooth Muscle 31

The solution to (2.14) is approximated at mesh points (s, t n ) in the domain D = {s, t |s ∈ [0, L], t n ∈ [0, T ] and si = ih, t n = nτ, where i ∈ [0, M ], n ∈ (0, N ), hM = L, t n /τ = N }. Consider the Lax–Wendroff and hybrid schemes both O(τ + h 2 ). The Lax–Wendroff scheme is 1 τ n+1/2 n n ui+1/2 = (uin + ui+1 ) + (Ti+1 (u) − Tin (u)) 2 2h τ n+1/2 n+1/2 uin+1 = uin + (Ti+1/2 (u) − Ti−1/2 (u)), i = 2, M − 1. h

(2.16)

The higher-order approximation schemes, compared to first-order schemes, provide more accurate solutions on rough meshes, which is particularly important in multidimensional problems. However, they cause high-frequency oscillations in the numerical solution. To eliminate them, a corrected solution is obtained usually by adding a “dissipative” term of the form εN uss . The parameter εN is a “viscosity” coefficient. It can be assumed proportional to τ and is chosen according to a condition of maximum “smoothing” of the oscillations in the course of the numerical computation. Over long computations, however, the correcting term may severely distort the solution. To avoid this the hybrid finite-difference scheme becomes more useful τ n u˜ in+1 = uin + (Ti+1 (u) − Tin (u)) h 1 τ ˜ u˜˜ in+1 = (u˜ in+1 + uin ) + (Tin+1 (u)) 2 2h 1 n n n n (ui+1 − uin ) − γi−1/2 (uin − ui−1 )] + [γi+1/2 8

(2.17) i = 2, M − 1

where n n = max(γˆ i−1 , γˆ in ) γi−1/2 n n γi+1/2 = max(γˆ in , γˆ i+1 )

(2.18)

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and

γˆ in =

 | i+1/2 un | − | i−1/2 un |   , | i+1/2 un | + | i−1/2 un | > εn   | n n   i+1/2 u | + | i−1/2 u | (2.19)

| i+1/2 un | + | i−1/2 un | ≤ εn

0,      

i+1/2 u

n

n = ui+1 − uin ,

εn = 0.01 max| i+1/2 un |.

The above scheme is of second-order accuracy in the regions where solutions change smoothly, and of first-order accuracy with moderate “smoothing” in the regions where the gradients are high. To avoid a significant “smoothing” of the solution u˜˜ in+1 , the parameter n of hybridization γi+1/2 is corrected by means of a “compression factor” in the region of the leading front of the wave. Thus τ n n n n uin+1 = u˜˜ in+1 + (γi+1/2 Gi+1/2 (u) − γi−1/2 Gi−1/2 (u)) 2h

(2.20)

n where the compression factor Gi+1/2 n n n Gi+1/2 = gin − gi+1 − |gi+1 − gin | sgn(δki+1/2 )

(2.21)

and

gin

= min 0, min k

min(|δki+1/2 |, δki−1/2 sgn(δki+1/2 )) |δki+1/2 | + |δki−1/2 |



 n [u˜˜ i+1

−

u˜˜ in ]

. (2.22)

Here δki+1/2 = [ i+1/2 u˜˜ n+1 ]k , and k denotes the kth component of vector u. To solve equation for φt the Saulyev scheme of second-order approximation φin+1 =

n+1 n+1 n n φi−1 + (2/ri − 4)φin + 3φi+1 − φi−2 − 4φi−1 2τf (φin ) − , 3 + 2/ri 2 + 3/ri

i = 2, M − 1

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Myoelectrical Activity of the Smooth Muscle 33

φ0n+1 = φ1n+1 (1 + 1/r1 ) + φ1n (1 − 1/r1 ) − φ2 + τf (φ1n )/r1 φin+2 =

n+1 n+1 n+2 n+2 −φi+2 + 3φi−1 + 4φi+1 + (2/ri − 4)φin+1 + φi+1 2τf (φin+1 ) − , 3 + 2/ri 2 + 3/ri

i = M − 1, 2 n+2 n+1 φM +1 = −φM −2 +

n+1 ) (φM

2τf rM −1

, (2.23)

and the Evans–Abdullah scheme 1 n n [ri (1 + ri−1 )φi−1 + (1 − ri2 )φin + ri+1 (1 − ri+1 )φi+1 1 + 2ri 
M 2 n −1 i = 2m, m = 1, + ri+2 φi+2 ] − τf (φin ), 2 (2.24) 1 n+1 2 n n 2 n φi+1 = (r φ + ri (1 − ri )φi + (1 − ri+1 )φi+1 1 + 2ri+1 i−1 i−1 φin+1 =

n n + ri+2 (1 + ri+2 )φi+2 ) − τf (φi+1 ),

i = 2m + 1

are employed. The time-step of integration τ is chosen from  h2 hCN , , τ ≤ min max(∂T /∂λ) 2 max(k ∗ (u)) 

CN = 0.6

(2.25)

where the first term in brackets is the Courant–Friedrichs–Lewy condition, whereas the second term is the general criterium of computation stability for parabolic equations. The results of test calculations showed that the algorithm composed by the hybrid + Evans–Abdullah schemes was computationally efficient and accurate. It is 1.5 times faster than the Lax–Wendroff plus Evans–Abdullah algorithms, and more than twice faster than other combinations of schemes. This is because less strict limitations could be applied to τ in the Evans– Abdullah scheme.

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2.1.4 Physiological Response Under normal resting physiological conditions, the inward T-type Ca2+ current shows slow dynamics and it reaches the maximum amplitude 0.27 nA. This current is not sufficient to increase [Ca2+ ]i which continues to decline and the min[Ca2+ ]i = 0.36 mM is observed. At this moment, the L-type Ca2+ current of short duration and high rate is activated with the maximum amplitude, 0.21 nA, recorded. It ceases at the moment when the max[Ca2+ ]i = 0.46 mM is achieved. An outward, transient, voltagedependent K+ current and a more sustained Ca2+ -dependent K+ current follow the inward calcium currents. The IK∗ is of short duration and of ∗ small amplitude, max IK = 0.086 nA. The dynamics of ICa−K is coupled 2+ with [Ca ]i changes and with the depolarization of the membrane. It rises quickly from 0.3 to 0.59 (nA) and then declines to its initial value. ∗ ICa−K is only 0.48 nA when the max[Ca2+ ]i = 0.46 mM is reached. As a consequence of the balanced activity of the ionic currents, slow oscillations of the membrane potential are generated (Fig. 2.1). They have a frequency ν = 0.18 Hz and a constant amplitude of approximately 25 mV. The wave of depolarization achieves its maximum in 1.6 s, exhibits a short plateau of duration 0.4 s, and finally decreases slowly to the resting value, ϕ = −51 mV. The rate of influx/outflux of Ca2+ ions is 0.06 mM/s and a max[Ca2+ ]i = 0.49 mM is achieved. There is a delay, t = 0.9 s, in the oscillations in internal calcium concentration as compared to the wave of depolarization. The concentration of cytosolic Ca2+ reaches its maximum when the process of hyperpolarization has already started. Part of the free intracellular Ca2+ activates the contractile protein system of the smooth muscle to produce regular rhythmic contractions, concurrent in phase and time with the dynamics of the Ca2+ oscillations. That is, the rise in tension, T a , coincides with an increase in [Ca2+ ]i and the phase of relaxation corresponds to a decrease in the intracellular calcium concentration. The contractions have constant amplitude, 4.8 g, and the calculated velocity of twitch is 5.0 g/s. The maximum force of contraction generated by the smooth muscle syncytium equals 14 g.

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Myoelectrical Activity of the Smooth Muscle 35

Fig. 2.1 Electrical pattern, oscillation of cytosolic Ca2+ and force development in a locus following depolarization of the smooth muscle membrane.

2.1.5 Effect of Increase in g˜ LCa An increase in permeability of Ca2+ -activated K+ -channels causes changes in the electrical behavior of the locus (Fig. 2.2). The period of slow wave becomes longer with the appearance of a characteristic plateau at the top of duration  1.0 s. The intensity of depolarization also increases. The membrance potential shifts from its resting value to ϕ = −24 mV in 0.67 s and oscillates at a constant amplitude of 23.5 mV. The generation of spikes at a frequency ν = 3 Hz and maximum amplitude of 9 mV on the crests of slow waves is observed. An increase in intensity of depolarization increases the influx of Ca2+ . The Ca2+ -channels remain active for a longer period, 1.63 s, and a higher

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Fig. 2.2 An increase in the permeability of Ca2+ -activated K+ -channels result in a regular bursting regime of myoelectrical activity.

˜ 2+ — max[Ca ˜ 2+ ] = 0.47 µM — is regisconcentration of intracellular Ca tered. The calculated amplitude of cytosolic calcium oscillations is 0.18 µM. As a result, an increase in tension, max T a = 14.6 g, is observed while the intensity of contractions remains unchanged. During relaxation the force falls to 6.4 g.

2.1.6 Effect of Increase in g˜ Ca–K and g˜ TCa A simultaneous increase in the permeability of Ca2+ -activated K+ -channels and fast T-type Ca2+ -channels causes significant changes in electrical

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Myoelectrical Activity of the Smooth Muscle 37

Fig. 2.3 Effect of a simultaneous increase in the permeability of Ca2+ -activated K+ channels and T-type Ca2+ -channels on the electrical behavior and dynamics of tension development.

T behavior of the functional unit. Thus, a gradual increase in g˜Ca from 0.51 2 to 0.65 mSm/cm converts the regular slow wave pattern to bursting chaos (Fig. 2.3). Action potentials of varying frequency, ν = 3.5–5 Hz, and amplitude, ϕ = 40–60 mV, are generated on the crests of slow waves. The locus T fires spikes max ϕ = 60 mV, ν = 5 Hz when g˜Ca−K = 0.65 mSm/cm2 . With a decrease in the permeability of fast Ca2+ -channels to its initial T value, the amplitude of discharges decreases concomitantly and for g˜Ca = 2 0.65 mSm/cm only slow wave activity is recorded.

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Oscillations of calcium influx follow the dynamics of the excitatory electrical pattern. No significant increase in the concentration of free cytosolic ˜ 2+ ] = 0.48 µM is recorded Ca2+ is observed. The maximum value of [Ca even when the locus discharges bursts of the maximum amplitude. However, more Ca2+ ions are retained in the syncytium during the phase of repolar˜ 2+ ] of 0.34 µM is registered at t = 5.8 s, ization. Thus, the minimum [Ca while during the period 5.8 < t < 14.6 s, which coincides with the gen˜ 2− remains at a higher level, eration of intensive bursts, the level of free Ca 2+ ˜ [Ca ] = 0.48 µM. It falls to 0.45 µM at t = 19.1 s when the locus returns to the slow wave regime. The dynamics of active force generation is similar to the Ca2+ (t) curve. Readings of the maximum contractile force vary between 14–15 g. Again, during the period of intensive spike activity the amplitude of oscillations of the phasic contractions rises to 10 g. The minimal value of T a = 6.2 g ˜ 2+ . coincides with the drop in the concentration of Ca

2.1.7 Effect of Cyclic Changes in g˜ TCa Characteristic cyclic transformations of electrical activity are known as a migrating myoelectric complex. It comprises a slow wave regime (Phase I), irregular bursting (Phase II) and regular bursting (Phase III) with generation of spikes on the crests of slow waves, which converts back to irregular bursting and the slow wave mode. They were simulated in the model by changing the permeability of T-type Ca2+ -channels ˜ 2+ ] > 0.5 mM at 2 + bt + c, [Ca T (2.26) g˜Ca = ˜ 2+ ] ≤ 0.5 mM. 0.51, [Ca t is time, and a = −0.49 × 10−8 mSm/s2 · cm2 , b = 0.98 × 10−4 mSm/s · cm2 , c = 0.51 mSm/cm2 are empirical parameters. T As g˜Ca rises in time, the locus fires repetitive action potentials, ϕ = T 56–72 mV at a frequency ν  17 Hz (Fig. 2.4). With a decrease in g˜Ca ≤ 2 0.65 mSm/cm , the system transforms to an irregular bursting mode of short duration, 6.0 s. The amplitude and frequency of spikes gradually decrease, T and the amplitude of action potentials reduces to 9 mV. Finally, when g˜Ca = 2 0.51 mSm/cm , the locus reverts to the slow wave regime.

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Myoelectrical Activity of the Smooth Muscle 39

Fig. 2.4 Different phases of a migrating myoelectrical complex as a function of cyclic changes in the permeability of T-type Ca2+ -channels.

˜ 2+ influx is in During the regular bursting regime the dynamics of Ca phase with the wave of depolarization. Thus, the concentration of cytosolic Ca2+ rises concomitantly with the generation of spikes and achieves its ˜ 2+ ] = 0.47 µM just after the end of firing. The ampitude maximum [Ca of oscillations equals 0.17 µM and changes insignificantly during the whole cycle of transformations. As a result, phasic contractions of different amplitude are produced. The maximum active force generated is 15.1 g. Contractions are most intense

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Fig. 2.5 Beating as a result in maintenance of a high concentration of cytosolic Ca2+ at a constant level. The smooth muscle syncytium responds with a long-lasting tonic contraction.

when the amplitude of their oscillation is 8 g. The syncytium responds to a high-frequency excitation generating a single phasic contraction.

2.1.8 Effect of Increase in g˜ LCa and g˜ TCa Simultaneous increase in permeability of L- and T-type Ca2+ -channels L produced unexpected results. A slight increase in the value of g˜Ca = T 2 0.0042 mSm/cm with a gradual rise in g˜Ca from 0.51 to 0.55 mSm/cm2

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Myoelectrical Activity of the Smooth Muscle 41

caused a bursting of the locus (Fig. 2.5). Spikes are generated at a constant frequency, ν  4 Hz, and their amplitude increases concomitantly with the rise in permeability of T-type Ca2+ -channels. The maximum amplitude action potentials, max ϕ = 60 mV, are registered at 9.2 s after the excitation. It was obvious to expect a simultaneous or slightly delayed achievement of the maximum concentration of free intracellular Ca2+ with the electrical ˜ 2+ ] = 0.5 µM is recorded 3.6 s after the excitaresponse. However, max[Ca tion and falls to a constant value of 0.48 µM when the locus starts firing spikes of maximum amplitude. As a result, the locus reacts with the generation a long-term tonic contraction with T a = 15.6 g.

2.1.9 Effect of Increase in g˜ Ca–K and Decrease in g˜ TCa An increase in the permeability of Ca2+ -activated K+ -channels with a simultaneous gradual decrease in the permeability of fast T-type Ca2+ -channels T below the normal value, g˜Ca < 0.51 mSm/cm2 , abolishes slow wave electrical activity in the locus (Fig. 2.6). The content of free intracellular Ca2+ increases ˜ 2+ ] = 0.41 µM to 0.04 µM from its initial level and remains unchanged at [Ca throughout. As a result, the syncytium produces an active force of the maximum tension, T a = 13.4 g, which remains constant throughout.

2.1.10 Remarks The model accurately simulates the depolarization and transmembrane Ca2+ influx and further contractions of the smooth muscle syncytium. Results of simulations agree qualitatively and quantitatively with experimental observations. Thus, receptor-linked L-type Ca2+ -channels provide Ca2+ for sustained slow wave electrical activity. The Ca2+ influx through these channels fills storage sites and activates contractile filaments without production of action potentials. According to the results obtained, slow waves generate an ˜ 2+ reaches its maximum 0.45 µM active force of amplitude 4.8 g. Cytosolic Ca in 0.9 s after a maximum of depolarization has been reached. Ozaki et al. (1991), working on gastric smooth muscle, recorded a 0.8 ± (0.1) s delay. According to the same authors, force development occurred when cytosolic

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Fig. 2.6 A step increase in the permeability of Ca2+ -activated K+ -channels and a simultaneous decrease in the permeability of T-type Ca2+ -channels in time abolishes electrical activity.

Ca2+ reached  30% of the peak value. In our calculations the dynamics of tension development is concomitant in phase and time with fluctuations ˜ 2+ . An explanation for such a response lies in the model of the synin Ca ˜ 2+ . By adjusting the paramcytium and in the initial steady-state value for Ca eters of the model, it is possible to achieve experimentally observed phase ˜ 2+ and tension curves. shifts between the Ca Gradual changes in the permeability of Ca2+ -activated K+ -channels and fast Ca2+ -channels result in changes in the electrical pattern, i.e. conversion

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Myoelectrical Activity of the Smooth Muscle 43

from the slow wave mode to a bursting regime, and intensive transient contractions of the locus. The amplitude of force oscillation decreases when the amplitude and frequency of spikes increase. Apparently in this case the large amount of Ca2+ that enters the cell cannot be absorbed by an already filled buffer system and storage site. As a result, a larger proportion of the Ca2+ that enters the cell is available to activate contractile filaments, resulting in more intense transient contractions. A simultaneous increase in the permeability of L- and T-type Ca2+ -channels causes production of multiple action potentials of a high frequency which trigger fusion of separate twitches into a long-lasting tonic contraction. A gradual decrease in the amount of Ca2+ entering the cell via fast Ca2+ -channels and a simultaneous gradual increase in the activation of Ca2+ -activated K+ -channels converts the system into a silent non-osciliatory state during which the syncytium responds with tonic contraction. Such responses would have been hard to predict conceptually. The shapes of the calculated curves T a (t ) are very similar to those recorded experimentally, but a quantitative comparison is not possible. Although the results support the general concept that contraction of visceral smooth muscle is primarily regulated by cytosolic Ca2+ , experimental evidence suggests that it is not the only determinant of contractile force. Other regulatory mechanisms, such as the enteric nervous system, are involved in the generation of tension and relaxation. Thus, exposure of the guinea-pig duodenum to tetrodotoxin (TTX) abolished all active baseline tension. Ito et al. (1988) demonstrated that treatment of the guinea-pig ileum with atropine and TTX abolished its contractions, suggesting that the effect is due to the interruption of transmission among the enteric neurons. The above numerical results are in agreement with these observations. However, in some species TTX does not affect background neurotransmitter release and can induce myogenic phasic activity. It has been found that the percentage of slow waves producing spikes, and hence phasic contractile activity of feline circular muscle, increases when the enteric nervous system of intestinal segments is rendered generally inoperative by TTX. These findings indicate that the ENS can sometimes function to reduce myogenic activation of intestinal contraction, and that the amplitude and frequency of phasic contractions of circular muscle under certain conditions can be expected to increase in the absence of ENS activity. It should be remembered that the circular and the longitudinal visceral smooth muscle are controlled by different neural plexi

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and that the smooth muscle cells mobilize Ca2+ by different mechanisms depending on their location. Because of the assumptions of the model, we could not reproach a whole variety of effects of TTX.

2.2 EFFECTS OF ION CHANNEL MODULATORS 2.2.1 Biological Background In addition to investigation of the ionic transport system, understanding the nature of the myoelectrical phenomenon has been increased significantly by the application of pharmacological tools, i.e. selective ion channel modulators which affect excitability of smooth muscle cells by increasing or decreasing conductances of particular ionic channels. Thus minoxidil, a K+ -channel agonist, hyperpolarizes the membrane and abolishes mechanical activity of smooth muscle cells by elevating the conductance of K+ channels; iberiotoxin depolarizes the membrane, decreasing the permeability of K+ -channels, and does not have a significant effect on contractile activity; charybdotoxin, a Ca2+ -activated K+ -channel blocker, reduces the frequency of slow waves without changing the level of depolarization of the membrane; while forskolin, a Ca2+ -activated K+ -channel agonist, hyperpolarizes the membrane and reduces the frequency of spike discharges. In the study of the pharmacology of the myoelectrical phenomenon, the single smooth muscle cell technique offers several advantages over multicellular preparations. It permits an accurate assessment of observed responses which are properties of the muscle cell itself and provides exact data about the mechanism of drug-cell interaction. However, the potential applicability of results depends on the fulfillment of an implicit assumption, that the properties of single cells resemble the behavior of a syncytium and are unchanged by the isolation procedure. Also pharmacokinetic variables, tissue specificity and the experimental setup can all influence outcomes. Under these circumstances, it becomes attractive to have a numerical model which could reproduce, with sufficient accuracy, the general properties of a single smooth muscle cell as well as of an entire syncytium and could, therefore, serve as a tool to elucidate and predict the mechanisms of action of different pharmacological compounds.

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Myoelectrical Activity of the Smooth Muscle 45

2.2.2 Effect of Forskolin Forskolin (FSK) is a Ca2+ -activated K+ -channel agonist. To simulate its action during the numerical experiments we varied the parameter g˜Ca−K . A gradual increase in conductivity of Ca2+ -activated K+ -channels (g˜Ca−K = 0.08, 0.15 and 0.2 mSm/cm2 ) hyperpolarizes the membrane of smooth muscle cells. The level of hyperpolarization is dose-dependent: ϕ = −62.5 mV and −67.6 mV, respectively, for the minimal and maximal value of g˜Ca−K (Fig. 2.7). The hyperpolarization completely abolishes slow wave electrical

Fig. 2.7 Membrane potential, cytosolic calcium and mechanical activity of the smooth muscle syncytium in response to forskolin (FSK) alone and to excess extracellular K+ .

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activity. Activation of the T-type Ca2+ -channels of the syncytium pretreated with FSK does not initiate spike activity. The locus remains hyperpolarized throughout. Although the conductivity of Ca2+ -channels remains unchanged, the hyperpolarization is accompanied by a significant decrease in the con˜ 2+ ] = 0.048 µM centration of free intracellular Ca2+ . The maximum [Ca 2+ 2 ˜ for g˜Ca−K = 0.2 mSm/cm and the maximum [Ca ] = 0.09 µM for 2 g˜Ca−K = 0.08 mSm/cm are registered. As a result, the locus remains relaxed throughout: T a = 0.

2.2.3 Effect of Lemakalim Lemakalim (LEM) is a selective K+ -channel agonist. Its action in the model was simulated by an increase in the conductivity of K+ -channels. An increase in g˜K = 0.7; 0.9 and 1.5 mSm/cm2 depolarizes the smooth muscle syncytium. The resting value of membrane potential rises to ϕ = −46.3 mV immediately after the addition of LEM (Fig. 2.8). The amplitude and frequency of slow wave oscillations are reduced, ν = 0.5 Hz, and in 10 s no slow waves are recorded. Remarkably, a concomitant with LEM activation of the fast Ca2+ channels brings the locus to bursting activity with the generation of action potentials ϕ = 68–72 mV of high frequency. This effect is reversible, i.e. with an inactivation of the fast Ca2+ -channels the spikes disappear and the electrical pattern becomes similar to that observed in the presence of LEM alone. The dynamics of cytosolic Ca2+ changes and active force generation corresponds to the ϕ(t) curve. Under normal physiological conditions, phasic contractions vary from 9.3 g to 14.2 g. A tetanic contraction with the constant active force T a = 12.2 g and T a = 12.7 g is developed by the locus in 10 s after its treatment with LEM (g˜K = 0.7, 1.5 mSm/cm2 , respectively).

2.2.4 Effect of High Concentration of External K+ m Changes in the concentration of external potassium ions ([K+ ]0 ) directly affect the value of the equilibrium potential ϕ˜ K , which satisfies the Nernst relationship. In the numerical experiments the effect of a high concentration

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Fig. 2.8 Effect of lemakalim (LEM) alone and in a response to a high concentration of K+ .

of K+ was achieved by varying the ϕ˜ K value. An increase in ϕ˜ K to −55 mV causes depolarization of the locus by 30 mV. No slow wave electrical activity is observed. The membrane potential and the concentration of free cytosolic ˜ 2+ ] = 0.51 µM. calcium remain constant throughout: ϕ = −21.7 mV; [Ca The active force T a = 16.4 g is generated. Using the model we studied the combined effect of a high concentration of K+ ions with FSK and LEM on myoelectrical activity of the locus. An

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increase in the concentration of [K+ ]0 after the increase in conductance of Ca2+ -activated K+ -channels (g˜Ca−K = 0.08, 0.2 mSm/cm2 ) has a significant depolarizing effect (Fig. 2.7). The membrane potential rises from −69.8 mV to −21.5 mV. A concomitant increase in active force is recorded: T a = 16.4 g. Removal of the excess extracellular K+ hyperpolarizes the locus. The level of membrane potential observed corresponds to the value recorded with the presence of FSK alone. The locus returns to its relaxed state: T a = 0. According to the results of calculations, application of LEM to the locus exposed to high [K+ ]0 does not cause significant changes in the level of the membrane potential (Fig. 2.8). A slight hyperpolarization by 3.9–7.1 mV is observed for different values of g˜K = 0.7, 1.5 mSm/cm2 . The concentration ˜ 2+ ] = 0.512 and 0.524 (µM) are of cytosolic free calcium rises slightly: [Ca recorded, and the active force T a = 16.8 g and T a = 17.1 g is generated, respectively. As in the previous case, the washout of LEM brings the locus to its pretreated state.

2.2.5 Effect of Phencyclidine Phencyclidine (PCP) is a selective K+ -channel antagonist. We simulated its action assuming the conductivity of K+ -channels equals zero: g˜K = 0. Application of PCP abolishes slow wave electrical activity completely and depolarizes the smooth muscle syncytium (Fig. 2.9). The resting value of the membrane potential increases from −51 mV to −16.9 mV. The depolarization is accompanied by a significant increase in the concentration of free ˜ 2+ ] = 0.484 µM is recorded. As a result, the locus intracellular Ca2+ ; max[Ca undergoes a tonic contraction: T a = 15.4 g. Addition of FSK to the locus pretreated with PCP, which was achieved by a simultaneous increase in g˜Ca−K = 0.08, 0.2 mSm/cm2 while g˜K = 0, demonstrates a strong hyperpolarizing effect on the muscle cyncytium. The membrane potential drops to ϕ = −62.4 and −67.6 mV and indicates a “dose-dependent” response. Hyperpolarization is followed by a decrease ˜ 2+ ] = 0.09 and 0.047 µM; the in cytosolic calcium ion concentration: [Ca tetanus is changed into complete relaxation: T a = 0. Washout of FSK and PCP brings the locus to its initial physiological condition with normal myoelectrical activity.

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Fig. 2.9 Effect of phencyclidine (PCP) alone and in a response to addition of FSK on myoelectrical and mechanical activity of the smooth muscle syncytium.

2.2.6 Effects of Selective K+ -Channel Agonists/Antagonists Migrating myoelectrical complex patterns were reproduced by activating T T-type Ca2+ -channels. The parameter, g˜Ca , is assumed to be changing according to (2.26). The working assumption is that the syncytium displays three phases of myoelectrical activity (Fig. 2.10). In phase I the membrane potential oscillates

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Fig. 2.10 Effects of selective Ca2+ -activated K+ and voltage-activated K+ -channel antagonists and antagonists on the migrating myoelectrical complex. (A) Normal MMC. (B) Response to treatment with LEM alone and together with FSK. (C) Effect of PCP alone and together with FSK. (D) Response to an increase in [K+ ]0 and application of LEM. (E) Effect of LEM and charybdotoxin (CHTX). (F) Effect of CHTX and LEM on different phases of the MMC. CHTX is added during the third and LEM during the following second phase of the MMC.

T with an amplitude of 23 mV and frequency of 0.18 Hz. As g˜Ca increases, the locus starts firing irregular spikes of different amplitudes (phase II), which are followed by repetitive regular bursting. Action potentials of maximum amplitude, ϕ = 72 mV, and oscillating at maximum frequency, ν = 17 Hz,

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Fig. 2.10 (Continued )

are observed. With the inactivation of fast Ca2+ -channels in time, the locus converts back, via phase II, to phase I. Treatment of the locus with LEM changes the slow wave pattern of the MMC. The amplitude of oscillations of the membrane potential reduces, ϕ = 8–10 mV, while their frequency increases, ν = 0.2 Hz. The amplitude of action potentials does not change significantly, ϕ = 60–62 mV, while the frequency of firing decreases to 6–8 Hz in phase III and a value ν = 3.8 Hz is calculated in phase II. Addition of FSK hyperpolarizes the syncytium, ϕ = −64 mV and completely abolishes slow wave and bursting activity.

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Application of PCP in the third phase of MMC depolarizes the smooth muscle, ϕ = −7.4 mV, and inhibits the cyclic myoelectrical activity of the locus. Subsequent addition of FSK transforms the locus to a hyperpolarized state. The effect of LEM on the locus exposed to a high concentration of extracellular K+ ions was rather unusual. Thus, LEM added at t = 7 s (Fig. 2.10) causes the generation of high-amplitude action potentials. The amplitude of spikes is maximal immediately after the application of LEM: max ϕ = 28−30 mV. The spikes are produced at a frequency ν  60 Hz. Application of LEM to the locus pretreated with charybdotoxin (CHTX) initiates the generation of spikes at a frequency ν  6 Hz. Their amplitude remains small, ϕ = 6–8 mV, and constant throughout the presence of LEM. A comparative study of a subsequent action of CHTX and LEM demonstrated that lemakalim added in the second phase of the MMC mainly changes the frequency of bursting and only slightly increases the amplitude of action potentials.

2.2.7 Remarks Macroscopic K+ currents, essentially Ca2+ -activated K+ and voltageactivated K+ currents, play an important role in the myoelectrical activity of the gastrointestinal smooth muscle syncytium. These currents limit the amplitude of upstroke depolarization of slow waves and are assumed to determine the amount of Ca2+ entering the cell during the plateau phase of slow waves. It is noteworthy that the properties of the K+ currents vary over a large pharmacological and electrical range in different tissues and species. Thus, Carl (1995) suggests that K+ -channels comprise heterogeneous groups of channel with respect to the 4-amidopyridine binding site and its interaction with the voltage sensor of the K+ -channels molecule. Another type of voltage-activated K+ -channel identified in intestinal smooth muscle is the glibenclamide-sensitive K+ -channel, which is supposed to modulate activity of the voltage-dependent Ca2+ -channel. Lee et al. (1994) conclude, based on the results of their investigations of Ca2+ -activated K+ -channels of freshly dispersed gastric smooth muscle cells, that activity of the above channels is modulated by membrane-delimited protein kinase A and protein phosphatase. Most of the subtypes of K+ -channels involved in myoelectrical

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activity of smooth muscle cells have been identified pharmacologically but there are no data on their electrical properties. The results of experimental investigation of the effects of selective Ca2+ activated K+ and voltage-activated K+ -channel agonists and antagonists on the electrical activity of strips of gastric and colonic smooth muscle are consistent with our findings. The observations of Richardson et al. (1992) and Du et al. (1994) state that application of FSK hyperpolarizes muscle cells by 13.5 ± 1.4 mV. Their experimental measurements of the effect of an increase in [K+ ]0 show depolarization of the smooth muscle syncytium by 28.9 ± 1.5 mV. The level of hyperpolarization after addition of FSK varies between 12.5 and 17.6 mV and is dose-dependent. An increase in the concentration of external K+ depolarizes the syncytium by 30 mV. Recording of cumulative treatment of the specimen with FSK and high [K+ ]0 qualitatively and quantitatively resembles the data of membrane potential oscillations shown above. It was demonstrated that cromakalim (an analog of LEM) reduces the amplitude and frequency of slow waves with a concomitant inhibition of phasic mechanical activity. The authors suggest that inhibition of slow wave electrical activity by cromakalim and lemakalim might be through activation of glibenclamide-sensitive K+ -channels, while inhibition of phasic contractile activity could be partly due to an inhibitory action of activated cyclic adenosine monophosphate on the contractile machinery and/or modification of the process of refilling of intracellular Ca2+ stores. There is good qualitative agreement between experimental and theoretical data on the effects of PCP on smooth muscle cells, although a quantitative discrepancy is present. For example, treatment of a specimen with PCP causes its depolarization by 20 mV, while the computer model “depolarizes” it by 30 mV. The results of simulation of the cumulative addition of FSK to a specimen pretreated with PCP resemble multiple experimental recordings. In addition to the experimental data discussed above, which deal mainly with the registration of electrical currents and membrane potentials, it was possible to follow the dynamics of changes in the concentration of intracellar Ca2+ ions and to calculate the active force of contraction generated by a locus. FSK significantly reduces the free cytosolic Ca2+ ion concentration and causes relaxation of the locus. High concentration [K+ ]0 increases the level of intracellular Ca2+ and causes tetanus. Effects of application of

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LEM and PCP are similar to the effects of high [K+ ]0 , although the mechanisms of action of LEM and PCP are different from the increase in [K+ ]0 . These data indicate that the selective K+ agonists/antagonists studied have profound effects on excitability and motility of smooth muscle syncytium. Therefore, the results could be of potential interest in the prediction of the possible effects of synthesized drugs, such as fenamates, on motility of the small intestine. With the model as proposed it was possible to study effects of selective K+ -channel agonists/antagonists on the migrating myoelectric complex. Lemakalim added during the second phase of the MMC increases the frequency of slow waves and decreases the frequency of spikes. Subsequent addition of FSK totally abolishes the generation of the MMC. Although these results are in agreement with experimental findings, no detailed qualitative or quantitative comparison can be made. It is of interest to note that a summative effect of LEM and high [K+ ]0 induces a short-term beating chaos with the production of high-amplitude action potentials. The blocking action of CHTX on Ca2+ -activated K+ -channels and its potentiated effect on contractions have been demonstrated in many types of smooth muscles. Experimental results obtained on the circular smooth muscle strips of the guinea-pig stomach atrium show that the toxin depolarizes the smooth muscle membrane by 3–5 mV, and enhances spontaneous contractions. However, there is some controversy about the effects of CHTX on the slow wave and spike electrical activity. For example, Suzuki et al. (1993) observe an increase in the amplitude of slow wave and action potentials, while recordings obtained by Du et al. (1994) show no slow wave and burst activity. According to the results, application of CHTX at the beginning of the second phase of the MMC depolarizes the syncytium by 31.5 mV and eliminates the generation of phase III. Thus, the locus undergoes tetanic contraction with T a = 16 g. The addition of LEM when the T-type Ca2+ -channels are maximally activated, which corresponds to phase III of MMC, induces the generation of low-amplitude spikes which disappear after washout of LEM. One can only speculate that the effect observed could be ascribed to the block of large conductance Ca2+ -activated K+ -channels by CHTX. The blockade of these channels removes their counterbalancing effect on voltage-dependent L- and T-type Ca2+ -channels. This results in greater Ca2+ entry in the cell and a decrease in its excitability. With the activation of K+ -channels by LEM, an

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intensive outflux of K+ ions occurs. This increases excitability of the cell and, as a result, spikes are produced. To prove this, we simulated sequential treatment of the locus with CHTX and LEM. The toxin abolishes burst activity and phasic contractions in the smooth muscle syncytium. LEM, added during the second phase of the MMC, reduces the frequency and does not increase the amplitude of spikes significantly, which could be due to an increase in excitability of the syncytium. The above data support results of clinical studies of fenamates, e.g. flufenamic and mefenamic acids, which are K+ -channel openers, on gastrointestinal motility in irritable bowel syndrome. Fenamates alter the migrating MMC, reducing significantly or abolishing completely the slow waves and the occurrence of spike potentials.

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CHAPTER 3

Pharmacology of Myoelectrical Activity

The separate and combined effects of pharmacological compounds in four categories were studied: (i) agents that alter the uptake and release of intracellular Ca2+ from the stores of the sarcoplasmic reticulum (SR) such as cyclopiazonic acid, thapsigargin and ryanodine; (ii) specific L-type Ca2+ channel antagonists such as verapamil, methoxyverapamil, nifedipine, diazoxide, nimodipine, and (+)-(R)-Bay K 8644, and non-specific weakly selective L-type Ca2+ channel agonist (−)-(S)-Bay K 8644; (iii) motilides, such as erythromycin, which affect the dynamics of permeability of both T- and L-type Ca2+ channels; and (iv) benzodiazepines, which decrease the conductance properties of the L-type Ca2+ and Cl− channels.

3.1 EFFECTS OF SPECIFIC INHIBITORS OF THE Ca2+ -ATPase AND THE RYANODINE-SENSITIVE Ca2+ CHANNELS OF THE SARCOPLASMIC RETICULUM The storage of calcium in the sarcoplasmic reticulum of smooth muscle is important in regulating the free cytosolic calcium concentration ([Ca2+ ]i ) both at rest and in excited states. The SR constitutes central and peripheral compartments that form a continuous network. Extracellular calcium is the major source of Ca2+ for this intracellular store. The uptake of Ca2+ into the SR depends on Ca2+ -ATPase, while its release is achieved through two mechanisms that involve different types of Ca2+ channels in the SR: the inositol triphosphate receptor and the ryanodine-sensitive Ca2+ -induced Ca2+ release channels. Two agents have been described as selective inhibitors of the 56

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Ca2+ -ATPase: cyclopiazonic acid (CPA), a mycotoxin from Aspergillus and Penicillium, and thapsigargin, also a natural compound. CPA easily passes into the cytoplasm through the plasma membrane, reduces Ca2+ -ATPase activity, and can be partly washed away. The pharmacological effects of thapsigargin are more complicated. At concentrations <1 µM it prevents the filling of the Ca2+ stores, while at higher concentrations it interacts with voltage-dependent L-type Ca2+ channels in the plasma membrane. Ryanodine, a neutral alkaloid extracted from Ryania speciosa, on the other hand, has been demonstrated to open irreversibly Ca2+ channels in the SR and thus suppress phasic contractions in smooth muscle. Although there are appreciable pharmacological data about the effects of CPA, thapsigargin, and ryanodine on the dynamics of [Ca2+ ]i in smooth muscle cells, the effects of these agents on the integrated responses of a whole cell (that is, on the dynamics of membrane ion currents and on the electrical and mechanical activities) remain to be described.

3.1.1 Effects of Cyclopiazonic Acid The pharmacological effects of CPA, thapsigargin, and ryanodine were simulated by varying the model’s parameters, ρ (strength of the cellular Ca2+ buffering capacity), Kc (intracellular Ca2+ current constant) and α (numerical parameter), which act upon the dynamics of [Ca2+ ]i turnover. A normal reference recording is given at the beginning of the tracing. The overall effect of CPA, an increase in the frequency of oscillation of membrane ionic currents and the electromechanical activity of the smooth muscle syncytium, is time-dependent (Fig. 3.1). Thus, a change such that ρ = 2.5 × 10−4 and Kc = 3.4 × 10−3 nearly abolishes the inward I LCa . The L current gradually recovers and reaches an amplitude of 0.4 nA. Note that ICa demonstrates high frequency alternations of short duration that are not seen under normal conditions. The velocity of influx/outflux also increases for T both phases. Concurrent changes in ICa show a slight reduction of amplitude. A current of 0.24 nA is registered immediately after the “addition” of CPA, which then regains its strength to reach a maximum, 0.36 nA. The transient outward IK∗ ceases at first after the administration of CPA and then regains its strength. The dynamics of activation of the current is not affected. The influx/outflux phases are of the same intensity while the amplitude increases

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Fig. 3.1 Effects of cyclopiazonic acid (CPA), ryanodine, and nifedipine on the smooth muscle syncytium.

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Pharmacology of Myoelectrical Activity 59

Fig. 3.1 (Continued )

slightly from 0.19 nA to the maximum 0.2 nA. The effect of CPA on the dynamics of the Ca2+ -dependent K+ channel is also time-dependent. A tran∗ sient fall in the amplitude of ICa−K to 0.23 nA is observed at the beginning of the process, rising later to the maximum 0.6 nA. The velocity of the outward ∗ component of ICa−K remains unchanged from the norm, while the velocity of the inward phase increases. CPA first reduces the amplitude the slow waves to 11.6 mV and hypopolarizes the membrane so that the resting membrane potential is φrest = −57 mV. It causes the frequency of slow waves to increase continuously to 0.25 Hz and, in time, the amplitude of membrane potential oscillations reaches 34 mV. The smooth muscle syncytium shifts to a new stable excitable state in which small amplitude, 9.7 mV, and high frequency,

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10 Hz, action potentials are seen to be generated on the crests of slow waves. The estimated slopes of the depolarization and hyperpolarization phases are 67.8 and 26.9 mV/ms, respectively. The maximal concentration of free cytosolic Ca2+ increases slightly, [Ca2+ ]i = 0.51 mM, while the range of its fluctuation rises greatly to a maximum amplitude of 0.19 mM. As a result, phasic contractions of the muscle are enhanced with a maximal T a = 15.2 g recorded.

3.1.2 Effects of Cyclopiazonic Acid and Ryanodine The addition of ryanodine to the CPA-containing environment alters the dynamics of all the membrane ionic currents (Fig. 3.1). The effect was achieved by changing α = 0.18. The amplitudes of the ionic currents do not differ significantly from those values observed with simulated CPA application alone. There is a perceptible decrease in the frequency of slow waves in the smooth muscle syncytium to 0.13 Hz with the retention of the same pattern of excitability that was elicited by the simulated treatment with CPA alone. A slight increase in the concentration of free cytosolic calcium [Ca2+ ]i = 0.52 mM is registered in the presence of both agents. This occurs because the ryanodine-sensitive channels, now locked open, release stored calcium. Ryanodine does not affect the intensity of the phasic contractions which remain unchanged: max T a = 15.2 g.

3.1.3 Effects of Cyclopiazonic Acid and Non-Selective Ca2+ Channel Antagonists The response to CPA is sensitive to the presence of nifedipine, a non-selective blocker of T- and L-type Ca2+ channels. Its presence is modeled by a reduction in the values of the permeability of these channels: g LCa = 0.003 and L g TCa = 0.45 mSm/cm2 . The addition of nifedipine abolishes the ICa and IK∗ T ∗ currents and substantially diminishes the ICa and ICa currents. For examT ∗ ple, the amplitude of the ICa equals 0.19 nA and of the ICa−K , 0.11 nA. The changes in the ionic currents reflect a decrease in the variation of the membrane potential. The maximum depolarization recorded is −42 mV. Also, there is a significant decrease in the frequency of slow wave electrical activity, ν = 0.15 Hz. The simultaneous application of CPA and nifedipine

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Fig. 3.2 Response of the smooth muscle syncytium to treatment with CPA, verapamil, and (−)-(S)-BAY K 8644 in the presence of altered [Ca2+ ]0 and [K+ ]0 .

reduces the cytosolic calcium concentration to the min[Ca2+ ]i = 0.1 mM, and that leads to a substantial reduction in the active force of contraction, max T a = 6.7 g.

3.1.4 Effects of Cyclopiazonic Acid and Selective Ca2+ Channel Antagonists The analysis of the combined effects of CPA, verapamil (a selective L-type Ca2+ channel antagonist), (−)-(S)-Bay K 8644 (weakly selective L-type Ca2+

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channel agonist), and an increase/decrease in the extracellular concentration of calcium and potassium ions is presented in Fig. 3.2. The effect of verapamil is achieved in the model by the nearly complete blockade of the maximum permeability, g LCa = 4 × 10−4 mSm/cm2 . Verapamil in the presence of CPA abolishes the slow wave electrical activity, bringing the smooth muscle syncytium into a stable hyperpolarized state. A steady value of the membrane potential, ϕ = −65 mV, is recorded throughout. This change lowers the basal [Ca2+ ]i such that it declines quickly to its minimal level, 0.12 mM, and remains constant. The concentration of free cytosolic calcium is only 27% of normal, not enough to activate the contractile system. The muscle thus remains relaxed. The simulated addition of (−)-S-Bay K 8644 to the system (achieved in this case by making the value of g LCa to be 6×10−3 mSm/cm2 ) in the simulated presence of CPA converts the membrane to a stable depolarized state with a constant ϕ = −24 mV. The effect of the agent is time-dependent. [Ca2+ ]i rises at 1.29 mM/s from 0.12 mM to 0.55 mM and then more slowly reaches the maximum level, 0.6 mM. As a result, a long-lasting tonic contraction is generated with the max T a = 20 g. The simulation of an external Ca2+ -free environment is accomplished by assuming the equilibrium potential for Ca2+ ions ϕ˜ Ca = −10 mV. CPA in such a calcium-free solution abolishes slow waves and hypopolarizes the membrane. The membrane potential falls transiently to ϕ = −68 mV and then recovers to a steady level ϕ = −48 mV. CPA depletes intracellular Ca2+ stores in the SR. That is, there is an exponential decrease in [Ca2+ ]i to 0.1 mM. As a consequence, no active force of contraction is generated by the smooth muscle. The separate effects of increased/decreased concentrations of extracel+ lular Ca2+ 0 and K0 on the electromechanical activity of the syncytium are modeled by altering the values of the resting membrane potentials for calcium and potassium ions. An increase in [Ca2+ ]0 (ϕ˜ Ca = 130 mV) causes depolarization of the membrane, ϕ = −13 mV, with a substantial increase in [Ca2+ ]i to max[Ca2+ ]i = 0.58 mM. These changes are associated with an intense contraction, T a = 19.4 g. The subsequent further increase in [K+ ]0 (ϕ˜ K = −35 mV) hypopolarizes the membrane by 0.5 mV. [Ca2+ ]i declines to 0.53 mM with a corresponding fall in the active force of contraction, T a = 17.4 g.

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3.1.5 Effects of Thapsigargin, Non-Selective and Selective Ca2+ Channel Antagonists, and Ryanodine The results of the numerical simulation of the pharmacological effects of nimodipine (a selective L-type Ca2+ channel blocker), diazoxide (a nonselective L-type Ca2+ blocker and a K+ channel agonist), ryanodine, and thapsigargin are shown in Fig. 3.3. The application of nimodipine (g LCa = 4×10−6 mSm/cm2 ) abolishes slow wave activity and hypopolarizes the membrane, ϕ = −48 mV. The cessation of calcium influx through the L-type Ca2+ channels leads to the cessation of the Ca2+ -dependent Ca2+ release from the SR, which results in the rapid decline of [Ca2+ ]i to a low level, 0.11 mM. This concentration of free Ca2+ is insufficient to initiate contraction and the muscle remains relaxed. The washout of nimodipine followed by the addition of diazoxide (g LCa = 4 × 10−6 and g K = 0.45 mSm/cm2 ) has an effect similar to that described for nimodipine. The simulated addition of ryanodine to the system blocks the influx T of calcium ions via L-type of Ca2+ channels without affecting the ICa . T The amplitude of the ICa remains unchanged from normal. The current decreases quickly to reach a minimum of 0.25 nA, which coincides with the min[Ca2+ ]i = 0.35 mM. The rise in the influx of extracellular Ca2+ is concurrent with the increase in the concentration of cytosolic Ca2+ . The T max ICa = 0.38 nA corresponds with the max[Ca2+ ]i = 0.51 mM. The outward voltage-dependent K+ current becomes small, while the amplitude of ∗ ICa−K slightly decreases as compared to the norm. The above changes in membrane ionic currents cause a substantial decrease in the frequency of slow waves, ν = 0.25 Hz, and a slight reduction in their amplitude, 23 mV, developed immediately after the addition of the drug. The numerical results demonstrate that ryanodine slows the rate of turnover of intracellular Ca2+ . The calculated velocity of variation of Ca2+ equals 6.7 mM/s. The dynamics of [Ca2+ ]i affects the frequency but not the intensity of phasic contractions. The magnitude of the active force increases insignificantly. These effects are reversed after the simulated exposure to nimodipine, which hypopolarizes the membrane and inhibits both the inward calcium currents and the muscle contractions. The presence of thapsigargin in low concentration is modeled by making ρ = 2×10−4 and Kc = 4.6×10−3 . Thapsigargin causes a slight increase in the

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Fig. 3.3 Actions of L-type Ca2+ channel antagonists (nimodipine, diazoxide), ryanodine, and thapsigargin on myoelectrical activity of the smooth muscle.

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Fig. 3.3 (Continued )

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L T ICa and a greater rise in the ICa . The amplitude of the influx of Ca2+ through the L-type Ca2+ channel exceeds 0.64 nA and that through the T-type Ca2+ channel is 0.24 nA, both greater values than those observed under normal ∗ conditions. Also, there is amplification of the outward IK∗ and ICa−K currents. According to the numerical results, they have a fixed amplitude with the ∗ max IK∗ = 0.33 nA and max ICa−K = 0.42 nA. There is a marked rise in the frequency of activity of all the ionic channels. Thapsigargin has little effect on muscle resting membrane potential but it increases both the amplitude of slow waves, ϕ = 32.9 mV, and their frequency, ν = 0.57 Hz. It also increases the excitability of the muscle. Continuous action potentials, amplitude 19.6 mV and frequency 34 Hz, are generated on the crests of the slow waves. Thapsigargin does not affect the rising depolarization phase of the slow wave but it slows the phase of repolarization. There is a rise in the concentration of intracellular calcium, max[Ca2+ ]i = 0.48 mM, with greater amplitude in its variation, 0.19 mM, and a concomitant increase in the active force, max T a = 15.2 g, and in the frequency of phasic contractions. The subsequent addition of nimodipine abolishes electromechanical activities and shifts the system to a stable unexcitable state.

3.1.6 Effects of Thapsigargin and High [Ca2+ ]0 To see the dose-dependent effects of a high concentration of [Ca2+ ]0 on the smooth muscle, both in the presence and in the absence of thapsigargin, the system is excited with different [Ca2+ ]0 , gradually changing the ϕ˜ Ca from 130 to 150 mV (Fig. 3.4). The first increase in [Ca2+ ]0 transforms the system from L a steady state to a beating mode. The amplitudes of the ionic currents, ICa and ∗ IK , instantly rise to the maximum, 2.8 nA and 1.4 nA, respectively, and then T ∗ fall exponentially to a constant level of about 0.5 nA. Both the ICa and ICa−K T ∗ currents increase in magnitude to max ICa = 0.6 nA and max ICa−K = 0.9 nA and then alternate at fixed amplitudes, 0.05 nA and 0.4 nA, respectively. The membrane potential oscillates at a frequency ν = 40.5 Hz and an average amplitude ϕ = 25 mV. [Ca2+ ]i rises continually and reaches a maximum 0.62 mM. As a result, a long-lasting contraction develops with T a = 20.1 g. The subsequent addition of thapsigargin abolishes the entry of Ca2+ T through L-type Ca2+ channels and further stabilizes the ICa current. It fluctuates at a declining magnitude to reach its stable value, 0.55 nA. Similar

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Fig. 3.4 Changes in myoelectrical activity after addition of thapsigargin with high altered [Ca2+ ]0 .

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Fig. 3.4 (Continued )

∗ changes occur in the dynamics of the IK∗ and ICa−K currents: the IK∗ ceases, but the outflux of K+ ions is sustained through the Ca2+ -dependent K+ channel. Note that the addition of thapsigargin to the system with a high concentration of extracellular Ca2+ shifts the system to a steady non-oscillating state with ϕ = −40.5 mV. There is no significant decrease in [Ca2+ ]i . The concentration of free cytosolic Ca2+ varies within the steady value and no change in the active force is recorded. The further washout of thapsigargin and the increase in ϕ˜ Ca ≥140 mV converts the pattern of electrical activity of the muscle to a stable beating mode. The membrane ionic currents have constant amplitudes which L increase with the increment in [Ca2+ ]0 . Thus, the maximal ICa = 1.9 nA for

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ϕ˜ Ca = 140 mV and 2.7 nA for ϕ˜ Ca = 150 mV. The changes in the other currents are consistent with the observations described above. High frequency, ν = 38.5 Hz, and amplitude, ϕ = 28.7 mV, action potentials are generated. These changes are accompanied by increases in the intracellular concentration of free Ca2+ to 66.8 µM and in the active force to 22.3 g. Thapsigargin does not change the pattern of myoelectrical activity produced by an increased [Ca2+ ]0 . The syncytium remains in a high frequency beating mode. Thapsigargin reduces the amplitude of all ionic membrane currents by 10–20% as well as the level of maximum depolarization of the membrane: max ϕ = −15.1 and −4.1 mV are calculated for ϕ˜ Ca = 140 and 150 mV, respectively. However, the drug exerts a particular effect on [Ca2+ ]i , which rises by 0.5–0.7 mM with the addition of the drug, depending on the level of external Ca2+ , with max[Ca2+ ]i = 0.73 and 0.76 mM achieved at a steady state. As a result, there is a concurrent increase in the active force of contraction with max T a = 23.8 g (ϕ˜ Ca = 140 mV) and max T a = 24.5 g (ϕ˜ Ca = 150 mV).

3.1.7 Effects of Thapsigargin, High [Ca2+ ]0 , [K+ ]0 and Selective Ca2+ Channel Antagonists The results of the combined effect of thapsigargin, a high extracellular concentration of Ca2+ and K+ ions, and methoxyverapamil, a selective L-type Ca2+ channel antagonist, are presented in Fig. 3.5. The presence of methoxyverapamil is modeled by reducing the permeability of the L-type Ca2+ channel to g LCa = 4 × 10−6 mSm/cm2 . During the simulations, [Ca2+ ]0 is assumed to be elevated throughout (ϕ˜ Ca = 150 mV), while [K+ ]0 rises gradually. Thus, an increase in ϕ˜ K to −45 mV abolishes slow wave electrical activity and depolarizes the membrane, with ϕ = −17 mV. However, the influx of external Ca2+ remains unaltered. [Ca2+ ]i rises steadily to achieve a max[Ca2+ ]0 = 0.55 mM. No phasic contractile activity is seen. A rise in tension accompanies the dynamics of the change in concentration of free cytosolic Ca2+ to reach a max T a = 20.6 g. The addition of methoxyverapamil hypopolarizes the membrane to −43.6 mV and affects the inward current of Ca2+ through the L-type Ca2+ channel. As a result, [Ca2+ ]i decreases exponentially to 0.53 µM with a fall in the active force to 17.5 g. The incremental increase in [K+ ]0 , which

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Fig. 3.5 Effects of a graded increase in [K+ ]0 with the simultaneous application of methoxyverapamil and thapsigargin in the presence of high altered [Ca2+ ]0 . [K+ ]0 is raised in increments: ϕK = −45, −35, −5, 5, 15 and 35 (mV) (arrows 1–6, respectively).

is consistent with an increase in ϕ˜ K = −35 mV, reverses the effect of methoxyverapamil, again depolarizing the membrane and activating the transient rise of [Ca2+ ]i . Thapsigargin, on the other hand, does not alter electrical activity of the smooth muscle membrane, which remains unchanged. However, the concentration of intracellular calcium continues to rise to max[Ca2+ ]i = 0.61 mM. As anticipated, tension increases to T a = 20.3 g. Incremental increases in [K+ ]0 depolarize the membrane with a shift of ϕ to −2.8, 0.5, 1.47, and 5.4 mV. Note that there is no concurrent augmentation

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Pharmacology of Myoelectrical Activity 71

in the influx of extracellular calcium. Instead [Ca2+ ]i declines concomitant with the rise in [K+ ]0 . Thus, [Ca2+ ]i = 0.6 mM is recorded for ϕ˜ K = −5 mV; it continues to fall to 0.57 µM for ϕ˜ K = 5 mV, [Ca2+ ]i = 0.52 mM for ϕ˜ K = 15 mV to reach the min[Ca2+ ]i = 0.35 mM for ϕ˜ K = 55 mV. The decrease in [Ca2+ ]i is reflected by the fall in the active force to a min T a = 8.9 g.

3.2 EFFECTS OF 1,4-DIHYDROPYRIDINE ENANTIOMERS (–)-(S)-BAY K 8644 AND (+)-(R)-BAY K 8644 The α1 subunit of L-type Ca2+ channels on the smooth muscle membrane express receptors for 1,4-dihydropyridine (DHP). Pharmacological studies of the enantiomers of DHP, e.g. (−)-(S)-Bay K 8644 and (+)-(R)-Bay K 8644, show that the (−)-(S)-enantiomers are weakly selective agonists and that the (+)-(R)-enantiomers are full antagonists of L-type Ca2+ channels. (−)-(S)-Bay K 8644 uniquely possesses dual properties that are voltage- and dose-dependent: at hyperpolarized levels of the membrane potential low concentrations of the drug enhance the permeability of L-type Ca2+ channels, whereas at depolarized potentials high concentrations are inhibitory.

3.2.1 Effects of (–)-(S)-Bay K 8644 The results of numerical simulations of the effects of different concentrations of (−)-(S)-Bay K 8644 on myoelectrical activity of the smooth muscle syncytium are presented in Fig. 3.6. A slight increase in the g LCa = 4.8×10−3 mSm/cm2 at ϕ = −30 mV depolarizes the membrane, shifting it to a stable non-oscillatory state with ϕ = −29.1 mV. In addition to the induced enhanced activity of the L-type Ca2+ channel, there is an increase in the influx of Ca2+ through the T-type Ca2+ channel, so that max[Ca2+ ]i = 0.53 mM is achieved. These changes cause a rapid rise of [Ca2+ ]i to 0.48 mM, which is accompanied by an increase in the generated tension, T a = 17.1 g. A higher concentration of the drug (g LCa = 6.5 × 10−3 mSm/cm2 ), applied under the same condition, ϕ = −30 mV, again depolarizes the membrane to a slightly higher level ϕ = −23.9 mV. Although this change is followed by an increase in the content of the free intracellular calcium, the max[Ca2+ ]i achieved is less by 19 nM than that produced by simulation of

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Fig. 3.6 The response of the muscle syncytium to a low, intermediate and high concentrations of (−)-(S)-Bay K 8644 (arrows 1–3, respectively).

the application of the lower concentration of (−)-(S)-Bay K 8644. As a result, a relaxation of the muscle syncytium with a fall in the force of contraction to 16.3 g is observed. This tendency prevails with further augmentation of the concentration of the compound. Thus, for g LCa = 8.65 × 10−3 mSm/cm2 added at ϕ = −23.9 mV, a weak potentiation of depolarization is seen to produce ϕ = −21.4 mV with a concurrent slight decrease in [Ca2+ ]i by 20 µM and in the tension by 0.2 g. Note that the above changes are consistent with an increase

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Pharmacology of Myoelectrical Activity 73

in all membrane ionic currents under consideration to a different degree, i.e. the increased influx of Ca2+ ions is accompanied by intensified outward voltage- and Ca2+ -dependent K+ currents.

3.2.2 Effects of (–)-(S)-Bay K 8644 and High [K+ ]0 To study the dose-dependent effect of (−)-(S)-Bay K 8644, the muscle was depolarized to a constant level, ϕ = −18.6 mV, by simulating a high external [K+ ]0 , assuming ϕ˜ K = −45 mV (Fig. 3.7). Changing the permeability gLCa

Fig. 3.7 Effects of (−)-(S)-Bay K 8644 acting in the presence of a high [K+ ]0 . The concentration of drug increased incrementally (arrows 1–6).

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gradually from 3.4 × 10−4 to 1.25 × 10−5 (mSm/cm2 ) simulates the incremental increase in the concentration of the drug. The gradual decrease in the g LCa value to 1.2 × 10−5 mSm/cm2 causes a gradual hyperpolarization of the membrane. Interestingly, this is not followed by a concomitant drop in [Ca2+ ]i , which rises steadily to reach the maximal [Ca2+ ]i = 0.51 mM. The muscle dynamics corresponds with the changes in the concentration of free Ca2+ . The tension increases to achieve a max T a = 16.6 g. However, the further decrease in g LCa from 5.1 × 10−5 to 1.25 × 10−5 mSm/cm2 inverts the previous effect, causing a hyperpolarization of the membrane accompanied by a decline in the [Ca2+ ]i and in the active force. Thus, at g LCa = 5.1×10−4 mSm/cm2 , a min[Ca2+ ]i = 0.49 mM and min T a = 15.9 g are recorded and these continue to diminish to [Ca2+ ]i = 0.45 mM and T a = 13.9 g at g LCa = 1.25 × 10−4 mSm/cm2 .

3.2.3 Effects of (+)-(R)-Bay K 8644 (+)-(R)-Bay K 8644 causes a dose-dependent inhibition of the myoelectrical activity and relaxation of the contracted smooth muscle (Fig. 3.8). At a concentration equivalent to g LCa = 3 × 10−4 mSm/cm2 added at ϕ = −49.8 mV, the drug hypopolarizes the membrane, ϕ = −62.6 mV, and increases the frequency, ν = 0.5 Hz, of slow waves, but it does not affect slow wave amplitude. The changes in electrical activity affect [Ca2+ ]i , which diminishes about two-fold to vary about an average concentration of free cytosolic calcium of 0.26 mM. The smooth muscle continues to sustain phasic contractions after treatment with (+)-(R)-Bay K 8644 but at a lower intensity. The syncytium loses its initial tone and a min T a = 2.7 g is observed, while the amplitude of the contractions remains unchanged. The simulation of an increased concentration of (+)-(R)-Bay K 8644 (g LCa = 2.3 × 10−4 mSm/cm2 ) abolishes slow wave activity and invariably hyperpolarizes the membrane, ϕ = −56.5 mV. The inactivation of the L- and T-type Ca2+ channels results in the rapid depletion of intracellular Ca2+ to a minimal [Ca2+ ]i = 0.186 mM. A small tension is generated, T a = 2.1 g. Simulation of a further increase in the concentration of the drug (g LCa = 5.10−5 mSm/cm2 ), added at a hyperpolarized state, ϕ = −56.6 mV, has a weak potentiating effect on the membrane. There is a continuing fall of

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Fig. 3.8 Changes in myoelectrical activity of the smooth muscle syncytium after the addition of (+)-(R)-Bay K 8644. Arrows indicate the incremental increase in drug.

[Ca2+ ]i to reach a minimal [Ca2+ ]i = 0.11 mM with a concomitant nearly complete relaxation of the smooth muscle, T a = 0.16 g.

3.3 EFFECTS OF MOTILIDES Motilin, a 22-amino acid peptide produced by the mucosa of the small intestine, and its synthetic analogs, called motilides, e.g. erythromycin, GM611, and OHM 11526, all exhibit regulatory dose-dependent effects on phase III

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of the migrating myoelectric complex of the intestine and on the mechanical contractile activity of the gastrointestinal tract. Radioimmunochemical studies have revealed the existence of a specific receptor for motilin and its derivatives on the smooth muscle membrane. The pharmacological effect of the motilides is the result of the release of ACh from nerve terminals. It involves both Ca2+ influx from the extracellular space and, to a much lesser extent, Ca2+ release from the internal stores. Although the role of [Ca2+ ]0 has been proven experimentally, the types of Ca2+ channels responsible for this influx of calcium ions remain uncertain.

3.3.1 Effects of Motilin and Erythromycin Assuming that motilin and erythromycin alter T- and/or L-type Ca2+ channel permeability, their presence is simulated by the oscillation of the parameters of the maximal conductances, g LCa and g TCa . Thus, an activation of the T-type Ca2+ channel, g TCa = 0.58 mSm/cm2 , changes the pattern of dynamics of the ionic membrane currents and myoelectrical activity of the smooth muscle syncytium (Fig. 3.9). The amplitude of the transitory inward current rises T ICa = 0.25 nA with high frequency alternations observed at its maximum value. The oscillations are of short duration and occur at a frequency of 19.1 Hz. This change is accompanied by an increase in activity of the L-type Ca2+ channel. The current is of higher intensity and an average amplitude L ICa = 1 nA and a frequency of 20.8 Hz is registered. The outward repolar∗ izing IK∗ and ICa−K currents also increase in amplitude. IK∗ responds with a rapid rise in the amplitude, maximal IK∗ = 0.9 nA, which declines linearly and disappears. Its intensity depends on [Ca2+ ]i . The Ca2+ -dependent K+ current demonstrates high frequency oscillations when it reaches the maximum amplitude. The duration of low amplitude high frequency oscillations ∗ in ICa−K coincides with the dynamics of influx of extracellular Ca2+ . As a result of the addition of motilides, the syncytium is tranformed from a stable state of slow wave generation to a bursting mode of electrical activity. The resting membrane potential is hyperpolarized to −56.1 mV, while the amplitude of slow waves increases to ϕ = −26 mV. Action potentials of average amplitude, 14 mV, and frequency, ν = 19.5 Hz, are generated on the crests of slow waves. The oscillation in the concentration of free cytosolic calcium also increases. However, the maximal [Ca2+ ]i only slightly exceeds

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Fig. 3.9 Effects of motilides on the smooth muscle syncytium are achieved by simultaneous chronotropic changes in the premeability of T- and L-type Ca2+ -channels.

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Fig. 3.9 (Continued )

the value recorded under normal conditions by 7 nM. Therefore, there is an insignificant increase in the contraction force with the max T a = 13.6 g. Note, that the phasic mode of contractile activity continues. The subsequent simultaneous increase in g TCa to 0.65 mSm/cm2 and in g LCa to 4.2×10−3 mSm/cm2 , representing a higher concentration of motilides, L changes the Ca2+ current dynamics. While ICa rises in amplitude and freL quency, to a maximal ICa = 3 nA, its intensity declines exponentially to reach T the strength of 0.25 nA. In contrast, ICa reaches the maximum level of 0.48 nA with small amplitude, 0.025 nA, and high frequency alternations of the current. The outward potassium currents also increase in intensity. IK∗ has the

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Pharmacology of Myoelectrical Activity 79

maximum amplitude, 1.8 nA, immediately after activation of the L-type Ca2+ channel. However, it does not sustain this strength and quickly diminishes to a stable value of 0.17 nA with a lower frequency. There is significant increase ∗ in the amplitude of ICa−K current as compared to the previous results. The stable maximum of Ca2+ -activated K+ influx attained is 0.79 nA. According to the results of our simulations, motilides at higher concentrations convert the syncytium to a beating mode with depolarization of the membrane, ϕ = −32.3 mV, and abolition of slow waves. Action potentials of 12.4 mV and 11 Hz are recorded. [Ca2+ ]i rises to reach a steady maximum, 0.52 mM. Phasic contractions cease and a long-lasting tonic contraction is generated with a max T a = 17 g. To investigate further a quantitative reciprocal interaction of the T- and L-type Ca2+ channels as a possible mechanism of action of motilides, g LCa was increased to 4.5 × 10−3 mSm/cm2 while g TCa remained unchanged and elevated at 0.65 mSm/cm2 . As a result of these alterations, the syncytium continues to produce spikes of high frequency but of increased amplitude. This causes a slight increase in [Ca2+ ]i by 20 µM and a concurrent rise in tension T a by 0.1 g. The simultaneous increase in g TCa = 0.8 and g LCa = 4.2× 10−3 mSm/cm2 causes a further increase in amplitude of all ionic currents ∗ L and spikes. For example, a max ICa = 3 nA, max ICa−K = 1.05 nA and firing of action potentials of a max ϕ = 34.4 mV and ν = 9.8 Hz are recorded. Instead of the expected rise in free intracellular Ca2+ , its concentration falls slightly to reach [Ca2+ ]i = 0.51 mM. There is a concomitant small decline in active force of contraction, T a = 16.4 g. A further decrease in g TCa = 0.7 mSm/cm2 , maintaining g LCa = 4.2 × 10−3 mSm/cm2 , shifts the system from a beating to a regular bursting mode of myoelectrical activity (Fig. 3.9). There is an increase in intensity of Ca2+ influx via the T- and L-type Ca2+ channels. The membrane ionic currents follow a pattern of dynamics similar to that discussed above. However, they L show a longer duration and larger fluctuations in amplitude: ICa = 1.2−3 T and ICa = 0.4−0.58 nA. The amplitude of repolarizing outward voltageand Ca2+ -dependent K+ currents varies within the ranges 0–2.0 and 0.6– 1.05 nA, respectively. The slow wave activity remains preserved in the smooth muscle with action potentials being generated at a frequency, of ν  4 Hz. The average amplitude of action potentials is ϕ = 26.5 mV. Interestingly, the dynamics of [Ca2+ ]i variation is not coherent with spiking activity.

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Thus, the concentration of cytosolic Ca2+ increases slowly to achieve the max[Ca2+ ]i = 0.51 mM at the moment when the spike bursting ceases. Phasic contractions lengthen in time, diminish in amplitude, and become more sustained with a max T a = 16.4 g.

3.3.2 Effects of Motilides and Ryanodine It has been suggested that motilides evoke the release of free Ca2+ from intracellular stores, the SR in particular. The effect of the addition of ryanodine was simulated by making α = 0.2. The addition of ryanodine to the system containing a motilide decreases the frequency of firing without altering the magnitudes of ionic currents, membrane potentials, cytosolic calcium and active force (Fig. 3.10). The syncytium responds with a more prolonged contraction. The further activation of the L-type Ca2+ channel, g LCa = 5.7 × 10−3 mSm/cm2 , and a slight reciprocal decrease in the conductivity of the T-type Ca2+ channel, g TCa = 0.65 mSm/cm2 , again converts L the smooth muscle to a beating mode. The maximum of ICa diminishes to T 1.4 nA while a max ICa rises to 0.68 nA. The intensity of outward IK∗ and ∗ ICa−K currents remains unchanged with maximum values recorded 1.17 and 0.9 nA, respectively. The syncytium becomes depolarized, with a minimal ϕ = −30.6 mV, and generates spikes of amplitude 15.8 mV and frequency 33.4 Hz. [Ca2+ ]i rises to a constant level, 0.514 mM. The muscle reacts with a development of a tonic of contraction, max T a = 16.7 g.

3.3.3 Effects of Motilides, a Non-Selective Ca2+ Channel Blocker and Thapsigargin Treatment with nifedipine, g LCa = 0, abolishes myoelectrical and mechanical activity produced by a motilide (Fig. 3.11). The syncytium stays hyperpolarized and relaxed throughout. The combined effects of the application of a motilide, thapsigargin, and an increased [Ca2+ ]0 is achieved in the model by assuming g LCa = 4.5 × 10−3 , g TCa = 0.7 mSm/cm2 , ρ = 2 × 10−3 , Kc = 4.57 × 10−3 and ϕ˜ Ca = 130 mV. The results show an increase in intensity of all ionic currents and in the bursting rate in the smooth muscle membrane. Compared to the case of treatment with motilides alone, there

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Fig. 3.10 The partial reversal of the effects of motilides on myoelectrical activity of the syncytium by ryanodine.

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Fig. 3.10 (Continued )

are three distinct superimposed processes in the final record of membrane potential oscillations: low amplitude and high frequency; intermediate frequency and high amplitude; and low frequency and high amplitude. This pattern of myoelectrical activity remains throughout the action of the above substances. A steady augmentation of [Ca2+ ]i occurs to attain 0.65 mM, which is accompanied by a concomitant rise in the tone of contraction with a max T a = 21.6 g.

3.4 EFFECTS OF BENZODIAZEPINES Pharmacological and radiohistochemical studies have revealed two types of γ-aminobutyric acid (GABA) receptors, called GABAA and GABAB .

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Fig. 3.11 The effects of nifedipine and thapsigargin on the pharmacological action of motilides. Thapsigargin is added simultaneously with an increase in [Ca2+ ]0 (arrows 1, 2).

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Fig. 3.11 (Continued )

GABAA receptors are present on the somas of primary sensory neurons and smooth muscle cells, while GABAB receptors are located on nerve terminals. GABAA receptors are coupled to ligand gated L-type Ca2+ and Cl− channels on smooth muscle.

3.4.1 Effects of Benzodiazepines Alone The effects of benzodiazepines was simulated by altering values as follows: g LCa = 3 × 10−3 , g Cl = 4 × 10−4 , g Cl(3) = 0.13 mSm/cm2 . The exposure to benzodiazepines hyperpolarizes the membrane of the smooth muscle

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syncytium and alters slow wave electrical activity to produce oscillations in membrane potential of small amplitude, ϕ = 5.7 mV, and frequency, ν = 0.75 Hz (Fig. 3.12). The free intracellular concentration of Ca2+ remains steady at 0.3 mM, insufficient to sustain contractions. The tension of the smooth muscle syncytium remains at T a = 6 g, too low to excite the system.

Fig. 3.12 The pharmacological effect of benzodiazepines.

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Fig. 3.12 (Continued )

As a result, the syncytium remains in a steady, relaxed, and non-oscillatory state. We then investigated ways to alleviate the inhibitory effects of benzodiazepines on intestinal muscle function. As one possibility, we added

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Fig. 3.13 The partial reversal by ryanodine of the action of benzodiazepines on myoelectrical activity.

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Fig. 3.13 (Continued )

ryanodine to the smooth muscle depressed by the effect of benzodiazepines to raise the intracellular calcium concentration (Fig. 3.13). The exposure to ryanodine does not change the dynamics of the ionic membrane currents and action potentials on the primary sensory and secondary neurons. Significant

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changes are seen, however, in the dynamics of the membrane currents in L the smooth muscle syncytium. ICa alternates at a high frequency, with rapid T activation and inactivation, at amplitude 7.6 nA. The transient ICa current undulates at 6.2 Hz and an average amplitude 0.28 nA. It changes to high frequency alternating current when the L-type Ca2+ current is maximally active. ∗ The dynamics of the repolarizing outward ICa−K and IK∗ currents resembles the dynamics of those currents when the smooth muscle syncytium is in a stable oscillatory state with amplitudes increased by 0.5 nA. The chloride current follows the dynamics of the membrane electrical activity. Ryanodine restores the slow wave myoelectrical activity and the excitability of the smooth muscle in the presence of benzodiazepines. Resting membrane potential fluctuates at 23.2 mV with the generation of spikes on the crests of slow waves. The resultant brief action potentials are of a constant amplitude, 42.1 mV, and frequency, 6.25 Hz, and resemble a bursting mode, notably phase II of the migrating myoelectrical complex. Ryanodine does not raise significantly the concentration of free cytosolic calcium, [Ca2+ ]i = 0.3 mM. However, the syncytium responds with the generation of phasic contractions of a small strength, T a = 8 g, which is enough to evoke spikes, ϕp+ = 40 mV, at the mechanoreceptors and to sustain further excitability of the locus for t ≥ 5.68 s.

3.4.2 Reverse of the Effects of Benzodiazepines In pharmacological experiments benzodiazepines at very low concentrations potentiate GABAA receptor-mediated acetylcholine release to augment transient contractions. In smooth muscle preparations and at higher concentrations, this effect is concealed by the dominant inhibition produced by these agents in muscle. Consider the simultaneous addition of ACh, a motilide, and ryanodine to reverse the suppression caused by the exposure to benzodiazepines (Fig. 3.14). The effects are mainly exerted on the smooth muscle, with little or no change in the neuronal elements. The simulated application of ACh with the other agents increases the intensity of all ionic membrane currents, with a rise in both their amplitudes and frequencies L of fluctuation. Thus, the maximal ICa = 9–10 nA is reached simultaneously with the maximal ACh concentration; with the removal of ACh, it declines L T to ICa = 3 nA. ICa undulates at an amplitude of 0.26 nA and a frequency

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31.2 Hz and changes to a highly alternating current with an amplitude of superimposed oscillations of 0.16 nA. The dynamics of the hyperpolarizing ∗ ICa−K and IK∗ resembles the dynamics of the currents as described above, but they have considerably higher amplitudes, 0.7 and 5.8 nA, respectively.

Fig. 3.14 Changes in myoelectrical activity of the syncytium after the simultaneous application of benzodiazepine, ryanodine, motilide, and a release of acetylcholine from the nerve terminal of the effector neuron linked to the L-type Ca2+ channel.

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Fig. 3.14 (Continued ) ∗ The dynamics of the leak ICl corresponds to the dynamics of the membrane potential. Remarkable changes are seen in slow wave electrical activity. The amplitude and frequency of slow waves increase to the values, 24.5 mV and 0.17 Hz, respectively. High amplitude action potentials, ϕ = 61.4 mV, are

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generated on the crests of the slow waves, resembling those observed during phases II and III of the migrating myoelectrical complex. As a result, the free cytosolic Ca2+ rises to 0.43 mM with the development of contractions, max T a = 11.8 g.

3.5 REMARKS The exact effects of CPA on the electrical activity and/or membrane ionic currents are not known in any kind of cell, and experimental findings are controversial. Thus, physiological studies on the canine ileum circular muscle demonstrate that, at a concentration 10 µM, the drug increases the frequency of slow waves and spontaneous action potentials; at 10–30 µM, CPA slightly affects the amplitude of slow waves, while at higher concentration, >30 µM, it reduces the frequency, amplitude, and duration of slow waves and evokes a small increase in muscle tone. The results of our numerical simulations are consistent with the experimental observations that the application of CPA hyperpolarizes the membrane, reduces the amplitude and increases the frequency of slow waves. The smooth muscle syncytium shifts to a hyperexcitable state with the generation of spontaneous action potentials of small amplitude on the crests of slow waves. As a predicted result of the above changes, there is an increase in [Ca2+ ]i and in the active force of contraction. Available reports of biological experiments with CPA, because of technical limitations, do not provide a detailed quantitative description of ionic membrane currents, except for the Ca2+ -dependent K+ current. Patch clamp studies of single Ca2+ -dependent K+ channel activity in cells from the rabbit intestine show that CPA (1–3 µM) produces initial potentiation followed ∗ by a dose-dependent reduction of the peak amplitude of ICa−K . To explain that observation, a decrease in the activation of the voltage-dependent K+ channels due to a diminished Ca2+ uptake into the cell and a subsequent exhaustion of Ca2+ -induced Ca2+ release from the SR was postulated. It has ∗ also been suggested that since ICa−K is responsible for the action potential repolarization it is likely that Ca2+ -pump activity in the SR may indirectly affect the shape of action potentials and the frequency of bursting. Our observations disagree with these ideas in that, according to our numerical experimentation, there is a transient suppression of IK∗ followed by an increase in its intensity. Similar changes in the dynamics of other ionic currents involved

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occur as well. These discrepancies could reflect damage from experimental manipulations performed on the cells that alter the finely balanced interactions among ion channels in isolated cells. From biological experiments, it has been postulated that Ca2+ entry into the cell depends upon the filling state of the SR and so is strongly enhanced by CPA. The drug, in concentrations of 0.1–10 µM, does not affect the strength of ICa , causing only a slight decrease in its amplitude. It depletes intracellular Ca2+ stores and activates the influx of Ca2+ through L-type Ca2+ channels even when channel properties per se are not directly changed. The addition of ryanodine (0.1–10 µM) causes the release of Ca2+ from the SR via ryanodine sensitive channels, evokes a rise in [Ca2+ ]i and stimulates a Ca2+ influx that is sensitive to nifedipine and Bay K 8644. This sensitivity indicates that the influx occurs mainly through activated L-type Ca2+ channels. The results of our simulations of the application of CPA with ryanodine, and of CPA with nifedipine, are in good qualitative agreement with these experimental findings. We demonstrate that the effect of CPA depends on the concentration of Ca2+ in the external medium, i.e. electromechanical activity ceases when [Ca2+ ]0 = 0 and significantly increases when [Ca2+ ]0 rises. Nifedipine in the presence of CPA blocks the L-type Ca2+ current without affecting the dynamics of the transient Ca2+ current and without changing the frequency of slow waves. However, the model also responds with a reduction in the amplitude and an alteration in the shape of slow waves, changes inconsistent with data from the same report. Experiments on ileal smooth muscle reveal that CPA induces contractions and a rise in [Ca2+ ]i that are blocked by verapamil at 1–10 µM. The subsequent washout of verapamil and the addition of Bay K 8644 restore mechanical activity with the generation of tonic contraction. In our results with the model, the artificial syncytium responds with qualitative precision to the combined treatment with the above compounds. Thus, verapamil inhibits myoelectrical activity and Bay K 8644 brings the syncytium to a state of stable depolarization with tonic contraction. In vitro, an increase in [K+ ]0 in the presence of CPA causes a decrease in 2+ [Ca ]i . Tabo et al. (1996), in studies of intestinal muscle of the rat, showed a simultaneous decrease in the isometric tension and [Ca2+ ]i in muscle exposed to CPA with a high [K+ ]0 . We have been able to reproduce these observations numerically. Our results are in good qualitative agreement with

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experimentally recorded traces of changes in the active force and [Ca2+ ]i . They support the idea that this effect of an increase in the external concentration of potassium, in this situation, is not due to a concomitant depolarization of the membrane, which produces a rise in [Ca2+ ]i . The results of numerical simulations of the effects of nimodipine and diazoxide on the myoelectrical activity of the smooth muscle syncytium are consistent with in vitro observations of the action of these drugs. Thus, nimodipine and diazoxide, in concentrations of 1 µM and 500 µM, respectively, abolish spontaneous rhythmic slow wave electrical activity and decrease both the resting [Ca2+ ]i and muscle tension. The modeled incremental change in the maximal permeability of the L-type Ca2+ channel and, separately, the simultaneous change of the maximal permeabilities of the Ltype Ca2+ and K+ channels both significantly reduce the concentration of the free cytosolic Ca2+ and produce complete relaxation of the syncytium. The presence of ryanodine receptors on the SR of smooth muscle cells has been specifically shown. Dessy and Godfraind (1996), in studies of guineapig intestinal smooth muscle, demonstrated that ryanodine (10 µM) evokes a slow increase in the free intracellular Ca2+ and a consequent rise in the force of contraction. This response is highly sensitive to the blockade of voltage-dependent Ca2+ channels as it is reversed after exposure to nimodipine (1 µM). With the model we have been able to follow the changes in the dynamics of all the ionic currents after the application of the drug. Thus, L T ryanodine terminates ICa . This result is with little effect on the intensity of ICa consistent with the observation made by Gagov et al. (1993), but their experimental technique did not allow them to distinguish between the two Ca2+ currents. This explains the erroneous conclusion made by the authors that the drug fails to affect depolarization-induced Ca2+ currents. We showed that ryanodine reduces the frequency of slow waves and contractions and increases [Ca2+ ]i and contractile force. As expected, the effects are terminated by a subsequent addition of nimodipine. Thapsigargin at 1 µM has little or no effect on the resting membrane potential and the total Ca2+ current in voltage-clamped cells of guinea-pig ileum, but it slightly increases the basal [Ca2+ ]i , the excitability of the syncytium, the amplitude of action potential-triggered [Ca2+ ]i transients, and tension. Isolated rat ileal muscle responds to excitation with a high extracellular concentration of K+ in the presence of methoxyverapamil (10 mM) and

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Ca2+ (2.5 mM) with an increase in [Ca2+ ]i and contractile force. A higher concentration of thapsigargin (10 µM) with a gradual increment in [K+ ]0 produces a dose-dependent decrease in [Ca2+ ]i and in the force of contraction. These effects are explained by a decrease in the driving force for Ca2+ due to a decrease in the electrochemical gradient for Ca2+ across the plasma membrane. The results of our numerical simulations agree with the above biological data. Thapsigargin, under normal physiological conditions, increases the amplitude and frequency of slow waves, [Ca2+ ]i , and the amplitude of phasic contractions. A depletion of Ca2+ stores by thapsigargin potentiates the rise in [Ca2+ ]i in response to the application of external Ca2+ because of a strong increase in the influx of calcium ions through the L-type Ca2+ T L (and the channels. The decrease we observed in the intensity of ICa and ICa 2+ other ionic currents) with the drug is a result of the regulation of Ca influx by free cytosolic Ca2+ . The behavior of the model resembles the results of the experiments by Tabo et al. (1993). Our data are consistent with the explanation proposed by those authors. However, the results point to a much more complex integrating mechanism, a balanced ionic channel activity linked to intracellular control systems. The results of pharmacological experiments with dihydropyridine enantiomers, such as (−)-(S)-Bay K 8644 and (+)-(R)-Bay K 8644, have suggested multiple voltage-dependent allosteric interactions between the receptors for DHPs, for benzodiazepines, and for diphenylalkyl-amines on the L-type Ca2+ channel. (−)-(S)-Bay K 8644 is non-selective for the subtypes of DHP receptors and its effects are determined mainly by the voltageinduced state of the channel at the moment of application. For example, Ca2+ channel activation by (−)-(S)-Bay K 8644 is enhanced at negative potentials. The recordings of muscle contractions evoked by the (−)-(S)enantiomer show that the dynamics of force generation is dose-dependent. It rises steadily to a maximum as the concentration of the compound increases from 0.1 to 1 µM. Further increase in the concentration of the enantiomer, ≥ 1µM, causes muscle relaxation. These data support a high affinity binding of (−)-(S)-Bay K 8644 to the resting or open Ca2+ channel at negative potentials, resulting in prolonged channel opening. At depolarized potentials, the drug stabilizes the channel in an inactivated state and produces an inverse contraction-concentration relationship. Contractile responses to a high [K+ ]0 are potentiated by (−)-(S)-Bay K 8644 and are both membrane potential- and dose-dependent. The lowest concentration of the drug (1 nM)

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is most effective at hyperkalemic ([K+ ]0 = 30 mM) depolarized potentials. However, the potentiation progressively weakens with increasing concentration of (−)-(S)-Bay K 8644 (1 µM), having no effect at [K+ ]0 = 100 mM. The results of our simulations are in good agreement with the above experimental findings. The model reproduces with qualitative accuracy a reverse contractile response of the muscle syncytium to (−)-(S)-Bay K 8644 with high [K+ ]0 . (+)-(R)-Bay K 8644 causes the concentration-dependent relaxation of high [K+ ]0 -induced contractions, which is complete at 1 µM of the drug. The drug enhances L-type Ca2+ channel inhibition at depolarized membrane potentials, reflecting the 38- to 65-fold higher affinity of binding to activated channels than to non-activated channels. It has been postulated that low concentrations of the (+)-(R) enantiomers activate the agonist receptor, whereas high concentrations activate antagonist receptor. Our numerical results reproduce the experimental recordings and can be used as a proof of the proposed hypothesis. Because we adopt a pragmatic approach in the model to describe the structure of the L-type Ca2+ channel, it is impossible to distinguish different active sites and to study a simultaneous effect of dihydropyridine enantiomers and Ca2+ channel blockers. More elaborate models of the ionic channels are required to answer those questions. It has been demonstrated experimentally that motilin and motilides, e.g. erythromycin, GM61, affect phases II and III of the migrating myoelectrical complex. Costa et al. (1996) recorded on the pig small bowel preparation a marked increase in the frequency and number of action potentials on slow waves during phase II without significant change in the amplitude and the velocity of their propagation. The application of motilin has a similar effect on electrical activity of the smooth muscle syncytium. However, compared to erythromycin, its effect is chronotropic, i.e. the amplitude of spikes decreases in time. It has been suggested that this is a result of dose-dependent inhibition of inward Ca2+ currents via activation of the motilin receptor and through other [Ca2+ ]i -independent mechanisms. Nicardipine, nifedipine, verapamil, diltiazem, and the removal of extracellular Ca2+ all abolish motilin-induced tonic and phasic contractions, emphasizing the major contribution of Ca2+ influx via L- and T-type Ca2+ channels in sustaining mechanical activity. However, Matthijs et al. (1988) showed that motilin is able to induce contractions even in the absence of external Ca2+ . This observation supports the experimental data

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by Depoortere et al. (1995) with intracellular injection of heparin, which demonstrates that motilin/motilides can evoke the release of Ca2+ from the SR via activation of IP3 . Also, the application of erythromycin in the presence of thapsigargin and high [Ca2+ ]0 causes an increase in [Ca2+ ]i . Analysis of the mechanical activity of the intestinal smooth muscle revealed that motilin and motilides have a biphasic effect on phasic contractions of the rabbit duodenal smooth muscle. At low concentrations the drugs increase transitory contractions but inhibit them at as the concentration rises. The basal tension, on the other hand, increases independently of drug concentration. These effects are unlikely to be due to ACh release from the cholinergic nerve terminals because a pretreatment with TTX or atropine does not detectably affect the above results. Our modeling of the action of motilin and motilides on the smooth muscle syncytium reproduces (qualitatively) the reported experimental effects. The exact mechanism of action of the drugs and the precise location of motilin receptors are not known. From the results of the numerical simulations, we propose a complex chronotropic allosteric interaction between the T- and L-type Ca2+ channels together with the release of Ca2+ from intracellular stores. The short incremental increase in permeability of gTCa followed by a more prolonged and less intense rise in the gLCa cause a response similar to that from the addition of motilides, a rise in the frequency and decrease in the amplitude of spikes and a change in the pattern of phases II and III of the MMC. The fact that these effects are abolished by nifedipine supports the idea of the activation by motilin of L-type Ca2+ channels. With the model, we can reproduce the effects of erythromycin, thapsigargin, ryanodine, and high [Ca2+ ]0 simultaneously at a locus of the small bowel; the result is consistent with experimental observations and with the idea of Ca2+ i release from the SR. Using radioreceptor-binding assay and autoradiography techniques, the presence of GABA receptors in the postganglionic cholinergic neurons of the myenteric nervous plexus and the muscle layers of the intestine has been demonstrated. Electrophysiological and biochemical data suggest that GABAA -receptors are located on somas of primary sensory neurons, and GABAB -receptors are found on the nerve terminals of both sensory and motor type neurons. The results of pharmacological studies indicate that the α-subunit of the GABAA receptor is coupled to the chloride channel.

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Combination therapy with various psychotropic agents, including benzodiazepine s and barbiturates, can depress the motility of the gastrointestinal tract both in vivo and in vitro. The drugs act on GABA receptors through the two pharmacologically distinct benzodiazepine-binding sites, the central sites, located on GABAA -receptors, and the peripheral sites, on GABAB -receptors. These differ in their sensitivity to diazepam. The small intestine possesses mainly the peripheral type of receptors on the smooth muscle. The evidence regarding the effects of diazepam and pentobarbital on the electrical and mechanical activity of the small bowel is confusing. Thus, benzodiazepines potentiate a GABAA -mediated ACh release on the soma of sensory neurons in a dose-dependent manner. However, at low concentrations, they do not affect the contractile activity of smooth muscle. At high concentrations, benzodiazepines abolish GABAA -mediated ACh release and severely depress intestinal motility through the activation of GABAB receptors on motor neurons. It has also been suggested from experiments on the dog ileum that exogenously applied GABA causes transient relaxation by way of non-cholinergic and non-adrenergic neurons. GABA, at low concentrations, alters the migrating myoelectric complex profile, i.e. the cyclic pattern of intestinal motility is disrupted and replaced by a period of irregular activity. At increased concentrations, GABA abolishes spiking activity. Because of the multiple levels of action of benzodiazepines, we conducted our numerical experiments on a locus of the small bowel, which includes both the regulatory neuronal elements of the myenteric nervous plexus and the effector muscle compartment. The model displays a series of intrinsic processes that are obscure in the usual biological preparations: the ionic currents on the somas of the primary sensory and secondary neurons and in the muscle syncytium. The results of those simulations that can be compared to existing experimental data show good qualitative agreement. For example, the discharge of the primary sensory neuron is altered by the added benzodiazepines while the secondary neuron remains unaffected. Concurrence can be seen also in the significant decrease in the intensity of the myoelectrical activity, in the disruption of its periodicity, and in the decrease in [Ca2+ ]i with the cessation of contractions. However, only with the modeling approach can we observe in parallel the changes in the underlying processes that contribute to the net effects. This ability enabled us to assess pharmacological “interventions” in an effort to annul undesirable “side-effects” of

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these drugs on gastrointestinal motility. For example, the treatment of the locus with ryanodine restores the irregular pattern of phase II of the MMC with the generation of phasic contractions. The simultaneous application of erythromycin, ryanodine, and endogenous acetylcholine released from the terminals of the motor neuron returns the system to a behavior that resembles normal. The restoration of a stable periodic myoelectrical activity with the cyclic changes of phases II and III of the MMC and the return of intense muscle contractions all can be viewed as clinically desirable changes.

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CHAPTER 4

Physicochemical Basis of Synaptic Transmission

4.1 INTRODUCTION Coordinated motility of the intestine is managed by the autonomous enteric nervous system (ENS), which is a multilevel, highly organized, multicomponent network, structurally composed of interconnected blocks — ganglia with a large number of internal pathways. Rapid and precise communication among blocks is made possible by two mechanisms — axonal conduction and synaptic transmission. The principles of function of the neural control circuits include both afferent and efferent electrochemical signals interacting among spatially separated multipolar neurons. The functional significance of these nonlinear interactions, which can be excitatory or inhibitory, is to quantify the transmitter output so that a precise amount of information, coded as the electrochemical signal, can be passed or gated via the synapse, and to orientate optimal nerve-pulse propagation in one direction. The concept of electrochemical translation of presynaptic impulse patterns through the chemical synapses to the postsynaptic output signal is based on complicated cellular and subcellular mechanisms of synthesis, storage, release and utilization of transmitters, and their binding with receptors or enzymatic systems to convert the recognition into cellular function. Unfortunately, no direct experimental measurements of the reactions are available. The classically defined neurotransmitter that provides excitatory neuro-neuronal impulse transfer in autonomic ganglia of the enteric nervous 100

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system is acetylcholine. In recent years, however, more than 20 substances, many of them neuropeptides, have been found in the ENS or have been proposed to be neurotransmitters. While neurophysiologists are discovering new details of the neuroanatomical structure and identifying unknown enteric neurotransmitters, physiologists and artificial intelligence researchers have set up conceptual models of natural and artificial processing systems. Cybernetic approaches to feedback control are inherent in the construction of the heuristic models for such autoregulatory systems and for the quantitative analysis of their success. But it is relevant to note that these models suffer from “physiological insufficiency,” because they do not consider chemical mechanisms of signal transfer via the synapses. In these models the neurons are non-dimensional and connections among them are assumed a priori from the matrix of synaptic connection strengths. This severely limits the applications of these types of model in physiological and clinical investigations. In order to elucidate fully the mechanisms of ENS function, all available information on different levels of its structure, with the relative contributions of different factors to, and their roles in, the overall operation of the system, should be taken into account. There is a growing need for modified models of neuronal networks that incorporate specific anatomical properties of interconnections and for the metabolic, ionic, and chemical processes involved in the dynamics of electric signal transfer within them to be developed. The simultaneous treatment of electrophysiological and neurophysiological data in the mathematical models provides an integrative perspective about the ongoing processes and allows the prediction of interactions between different structures in biological systems.

4.2 CHOLINERGIC NEUROTRANSMISSION 4.2.1 Biological Background The enteric nervous system is a principal determinant in the control of small bowel motor activity. It operates with great flexibility and rapidly adapts its performance to even delicate external changes that refer its functional properties to the abstract neuronal network. Morphologically, three large

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expanded and vertically linked nerve plexuses are distinguished: the myenteric or Auerbach’s plexus, and the submucosal external and internal plexuses. The latter two are functionally considered as one, the submucosal or Meissner’s plexus. In pioneering work, Dogiel (1989) proposed that the shapes of nerve cells in the intestine are related to their function. According to his histological classification, three types of cells are distinguished. The first two are readily recognized in the ENS. The majority of neurons seem to be type I (47%) or type II (36%). Type I neurons have one long slender process, known as the axon, of relatively uniform diameter (0.2–1.2 µm), which travels several millimeters within the network and ends in a varicose presynaptic terminal, usually 1–1.5 µm in diameter. The nerve terminal contains localized collections of vesicles that are filled with neurotransmitter. The preand postsynaptic elements of neuro-neuronal synapses are separated by the synaptic cleft, 20–40 nm wide. The postsynatic membrane has morphologically specified receptive regions. Type II neurons have a spheroidal shape and are effectively multiaxonal with several, usually 3 to 10, short and long processes. According to their electrophysiological properties, the neurons of the ENS are classified as either S neurons or AH neurons. The vast majority of Dogiel type I cholinergic neurons show the electrophysiological properties of S-type neurons, i.e. their response to electrical stimulation usually consists of a burst of fast excitatory postsynaptic potential (EPSP’s) that develops rapidly and has a short duration, ≤50 ms. The average amplitude of depolarization exceeds 70–90 mV. All AH neurons have the shape of Dogiel type II neurons. They respond with a slow EPSP development, which persists for many seconds, and a prolonged hyperpolarization following their action potential. There is now compelling evidence that neurons store and release more than one transmitter. It has also been shown that some neurons in culture can produce different transmitters based upon alterations in culture conditions, which indicates highly adaptive properties of the neural system. The interstitial cells of Cajal are found within Auerbach’s plexus. The fundamental morpological characteristic of these cells is the absence of synaptic inputs, which suggests that they function as pacemaker cells.

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Physicochemical Basis of Synaptic Transmission 103

4.2.2 Mathematical Model Synaptic transmission in the cholinergic neuron includes the presynaptic and postsynaptic processes of acetylcholine transformations. The main modulator of the presynaptic processes is Ca2+ ions. The electrical signal, ϕ∗ , once produced, travels peripherally from the soma along the axon. Arriving at the terminal it activates the influx of Ca2+ ions through voltage-gated channels. The large and rapid intracellular Ca2+ increase causes acetylcholine exocytosis from the vesicular store via the activation of reactive Ca2+ centers on the vesicles. The released neurotransmitter, AChf , diffuses into the synaptic cleft, AChc , where part of it is utilized by acetylcholinesterase enzyme, E, for the development of acetylcholine enzyme complex, AChE; another part diffuses to the postsynaptic membrane, AChp , and binds with the choline receptors, R. The formation of (ACh–R) complex causes a reorientation of the molecular structure of the receptive zone and the generation of EPSP (ϕp+ ). After the released transmitter has exerted its effect it is degradated to choline (S). The temporal characteristic of the chemical synapse is a synaptic delay — the time that is required from the onset of the electrical signal in the presynaptic terminal to the onset of ϕp+ . The electrochemical events outlined above can be summarized by the following assumptions: i) a cholinergic neuron is a geometrically non-uniform cable core of a given length (Fig. 4.1); its diameter increases by a factor of two in the terminal end; the dynamics of the electric signal, ϕ∗ , propagation along the unmyelinated axon is described by the modified Hodgkin–Huxley

Fig. 4.1 A schematic illustration and geometrical characteristics of a generalized cholinergic neuron.

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model; the soma of the neuron located at the left boundary is stimulated by an external electric impulse of a known intensity and duration; the presynaptic terminal area remains in the unexcited state throughout; ii) the basic influx of Ca2+ ions into the cellular space is secured by their transport along voltage gated channels; iii) all the reactions of acetylcholine transformation are of the first order and satisfy Michaelis–Menten kinetics. The electrical signal propagation along the axon satisfies 1 ∂ ∂ϕ∗ Cm = ∂t 2Ra ∂s




∂ϕ∗ a (s) ∂s



2

− I ionic .

(4.1)

Here Cm is the membrane capacitance; Ra is the membrane resistance; a(s) is the diameter of the axon a(s) =

 a, 2a,

0 < s ≤ L − L0 s > L − L0

t > 0, s ∈ (0, L).

(4.2)

s is the Lagrangian coordinate; L, L0 are the length of the axon and nerve terminal, respectively; t is time. The total ionic current is calculated as Iionic = INa + IK + ICl .

(4.3)

The three currents are expressed in terms of the conductance multiplied by their respective driving forces INa = gNa m 3 h(ϕ∗ − ϕNa ) IK = gK n 4 h(ϕ∗ − ϕK ).

(4.4)

ICl = gCl (ϕ∗ − ϕCl ). Here gNa , gK , gCl are the maximal conductances of Na+ , K+ and Cl currents, respectively, and m, h, n are the probabilities of opening of these gates; ϕNa , ϕK , ϕCl are the reversal potentials of sodium, potassium and cloride currents.

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Physicochemical Basis of Synaptic Transmission 105

The dynamics of change in the probability variables is given by dy = αy (1 − y) − βy y dt

(y = m, h, n).

(4.5)

The activation, αy , and deactivation, βy , parameters are αm =

0.1θ(2.5 − ϕ∗ ) , exp(2.5 − 0.1ϕ∗ )

βm = 4θ exp(−ϕ∗ /18)

αh = 0.07θ exp(−0.05ϕ∗ ), αn =

βh =

ϕ∗ )

0.1θ(10 − , exp(1.0 − 0.1ϕ∗ ) − 1

θ 1 + exp(3 − 0.1ϕ∗ )

(4.6)

βn = 0.125θ exp(−0.125ϕ∗ ).

Here, θ denotes temperature (◦ C). The cycle of chemical reactions of ACh transformation in the synaptic zone is shown in Fig. 4.2. The system of kinetic equations of the reactions is d[Ca2+ ] dt d[AChv ] dt d[AChf ] dt d[AChc ] dt

= [Ca2+ ]out , ϕ∗ − k+5 [Ca2+ ] = −kc [Ca2+ ][AChv ] (4.7) = kc [Ca ][AChv ] − kd [AChf ] + k+6 [S] 2+

= kd [AChf ] − kp [AChc ]

Ca2+

k+p

kd

AChv −→ AChf −→ AChc −→ AChp k+1

k+4

AChp + R
ACh − R −→ S + R k−1 k+2

k+3

AChp + E
AChE −→ S + E k−2

Fig. 4.2 The scheme of acetylcholine conversion.

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d[AChp ] = k+p [AChc ] + K−1 [ACh–R] + k−2 [AChE] dt −k+1 [AChp ][R] − k+2 [AChp ][E] d[S] dt d[R] dt d[ACh–R] dt d[E] dt d[AChE] dt

= k+4 [ACh–R] − k+6 [S] + k+3 [AChE] = k−1 [ACh–R] + k+4 [ACh–R] − k+1 [AChp ][R] = k+1 [AChp ][R] − k−1 [ACh–R] − k+4 [ACh–R] = k+3 [AChE] + k−2 [AChE] − k+2 [AChp ][E] = k+2 [AChp ][E] − k+3 [AChE] − k−2 [AChE].

Here k+,−(i) is the rate constants of forward (+) and backward (−) chemical reactions (i = 1, 4); kc is the association constant of Ca2+ ions with the active centers on the vesicles (kc = 0, 1); k+5 is the rate of absorption of Ca2+ ions by the cell buffer systems; kd is the constant of the AChf fraction diffusion into the synaptic cleft; k+p is the diffusion rate constant of the AChc fraction on the subsynaptic membrane; k+6 is the rate of absorption of products of the chemical reactions; [Ca2+ ]out is the concentration of calcium ions in the external medium; [Ca2+ ] is the intracellular calcium content; […] are concentrations of reacting components. From the last four equations we have d[R] d[ACh–R] + =0 dt dt d[AChE] d[E] + =0 dt dt

(4.8)

or [R(t )] + [(ACh–R)(t )] = [R 0 ] = const. [(AChE)(t )] + [E(t )] = [E0 ] = const.,

(4.9)

where [R0 ] is the concentration of choline receptors on the postsynaptic membrane, and [E0 ] is the concentration of acetylcholinesterase enzyme.

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Physicochemical Basis of Synaptic Transmission 107

Substituting (4.9) into (4.7) one obtains d[Ca2+ ] dt d[AChv ] dt d[AChf ] dt d[AChc ] dt d[AChp ] dt

= [Ca2+ ]out , ϕ∗ − k+5 [Ca2+ ] = −kc [Ca2+ ][AChv ] = kc [Ca2+ ][AChv ] − kd [AChf ] + k+6 [S] = kd [AChf ] − k+p [AChc ] = k+p [AChc ] − [AChp ](k+1 [R 0 ] + k+2 [E0 ]) + [AChp ](k+1 [ACh–R] + k+2 [AChE]) + k−1 [ACh–R] + k−2 [AChE]

d[ACh–R] = k+1 [AChp ][R 0 ] − [ACh–R](k−1 + k+4 + k+1 [AChp ]) dt d[AChE] = k+2 [E0 ][AChp ] − [AChE](k−2 + k+3 + k+2 [AChp ]) dt d[S] = k+4 [ACh–R] − k+6 [S] + k+3 [AChE] dt [R] = [R 0 ] − [ACh–R] [E] = [E0 ] − [AChE]. (4.10) The dynamics of the fast excitatory postsynaptic potential generation on the postsynaptic membrane satisfies Cp

dϕp+ dt

+ ϕp+ (−[ACh–R] + Rv−1 ) =

ϕR+ . Rv

(4.11)

Here Cp is the capacitance of the subsynaptic membrane;  is the empirical constant of the synapse; Rv is the general resistance of the extrasynaptic structures; ϕR+ is the resting postsynaptic potential. The system of equations (4.1)–(4.6), (4.10) and (4.11) describe the processes of the action potential propagation from the soma of the neuron along its axon to the synaptic terminal, the chemical mechanisms of the nerve-pulse

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transmission onto the postsynaptic structure, and the generation of the fast excitatory postsynaptic potential on the adjacent neuron. Initial conditions assume that the system is at the resting state t = 0 : ϕ∗ (s, 0) = 0,

ϕp+ (0) = ϕR+

y(0) = αy (0)/(αy (0) + βy (0))

(4.12)

and the initial concentrations of reacting components are known [Ca2+ ] = 1 × 10−4 ,

[Ca2+ ]out = 1.0, [AChf ] = 0.1,

[R ◦ ] = 0.13,

[E◦ ] = 0.05 (mM),

[AChv ] = 80,

[AChc ] = 3.16 × 10−5 ,

(4.13)

[ACh–R] = [AChp ] = [AChE] = [S] = 0.

Boundary conditions assume that the soma of the cholinergic neuron is excited by the external electric impulse, and the synaptic end remains unexcited all the time  ϕ0 , 0 < t ≤ 0.002 s ∗ ϕ (0, t ) = (4.14) 0, t > 0.002 s, ϕ∗ (L, t ) = 0.

4.2.3 Numerical Algorithm The system (4.1)–(4.6), (4.10), (4.12), (4.13) written in vector form is   ϕt∗ = k ∗ (u)φs∗ s − f1 (ϕ∗ ) ϕ∗ (0, t ) = ϕ0∗ ,

ϕ∗ (L, t ) = 0

(4.15)

ϕ∗ (s, 0) = 0, X˙ = Ax + f2 (ϕ∗ ) X(0) = X0

(4.16)

where k ∗ (u) =

a(s) 2Rm Cm (1 + ∂u/∂s)

f1 (ϕ∗ ) = (INa + IK + ICl )/Cm x = (x1 , x2 , . . . , x8 )

T

f2 (ϕ∗ ) = ([Ca2+ ]out , ϕ∗ , 0, . . . , 0)T .

(4.17)

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Physicochemical Basis of Synaptic Transmission 109

Here x1 = [Ca2+ ],

x2 = [AChv ],

x3 = [AChf ],

x4 = [AChc ],

x5 = [AChp ],

x6 = [ACh–R],

x7 = [AChE],

x8 = [S],

(4.18)

with the meaning of parameters as described above. The matrix A (8 × 8) has the non-zero entries   0 0 0 0 0 0 0 −k+5   0 0 0 0 0 0 0  −kc x1    0 0 0 0 0 k+6  kc x1 −kd      0 0 kd −k+p 0 0 0 0    A= 0  a55 k−1 −k−2 0 0 k+p  0    ◦  0 0 0  0 0 0 k+2 [R ] a66     ◦] [E a 0 0 0 0 0 0 k   +2 76 0 0 0 0 0 k+4 k+3 −k+6 (4.19) a55 = k+1 (x6 − [R ◦ ]) + k+2 (x7 − [E◦ ]) a66 = k−1 + k+4 + k+1 x5 a76 = k−2 + k+3 + k+2 x5 . To solve (4.15), (4.16) the Evans–Abdullah algorithm (1.23) and the fourthorder Runge–Kutta method with adjustable time-step are used τ xn+1 = xn + [L1 + 2L2 + 2L3 + L 4 ] , n ≥ 0 (4.20) 6 where weighted averages of A (x, t) are L1 = A(xn , tn ) τ τ L2 = A xn + L1 , tn 2 2 τ τ L3 = A xn + L2 , tn 2 2 L 4 = A(xn + τL3 , tn + τ) τ is the time-step.

(4.21)

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The algorithm is computationally efficient. Results of calculations accurately reproduce the dynamics of nerve signal propagation along the unmyelinated axon and cholinergic signal transduction mechanisms at the synapse.

4.2.4 Physiological Neurotransmission The action potential elicited by the external stimulation has an amplitude of 69 mV and propagates along the uniform part of the unmyelinated axon at a velocity of 17.5 cm · s−1 . In the vicinity of the terminal ending the amplitude of ϕ∗ shortly decreases, but then increases again to ϕ∗ = 72 mV. The acceleration of the signal propagation to 21.6 cm · s−1 is also observed. The depolarization of the presynaptic membrane activates a short-term influx of calcium ions into the terminal through Ca2+ voltage-dependent channels (Fig. 4.3). In t ∗ = 0.75 ms (t ∗ is the time of the initiation of depolarization processes in the presynaptic zone) the concentration of cytosolic Ca2+ rises to 5.55 µM and at t ∗ = 1.4 ms reaches its max[Ca2+ ] = 19.4 µM. Some of the ions are immediately absorbed by the buffer system, others diffuse towards the vesicle. The cytosolic calcium falls to a resting level at two rates: 1.5 µM ms−1 for 0.6 ms and abruptly for the next 0.01 ms. Binding of Ca2+ ions with the active centers on the vesicles initiates the AChv release. This process is most rapid for 1.41 < t ∗ < 2.13 ms. The velocity of AChv release is not constant but depends on the concentration of cytosolic Ca2+ . Thus at the beginning of the process, the velocity of acetylcholine free fraction increase equals 0.5 mM·ms−1 . At t ∗ = 1.44 ms it reaches a maximum of 1.6 mM·ms−1 . Subsequently it decreases to zero at a steady rate. During the whole cycle about 10% of vesicular acetylcholine is released: [AChv ] = 78.2 mM. With the achievement of an AChf concentration of 10.6 mM, which is observed at t ∗ = 1.26 ms, acetylcholine release into the synaptic cleft starts. For the first 0.06 ms the acetylcholine cleft concentration rapidly reaches a value of 3.2 mM. After that, the velocity of AChf diffusion is stabilized and becomes 0.11 mM·ms−1 . According to the calculated data max[AChc ] = 5.38 mM, equivalent to 50% of AChf is diffused from the presynaptic terminal. The main part of AChc enters the postsynaptic membrane and reacts with the choline receptors. This process is most intense in the period 1.41 <

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Physicochemical Basis of Synaptic Transmission 111

Fig. 4.3 The dynamics of the cytosolic Ca2+ ion, vesicular and free acetylcholine fractions, concentrations and excitatory postsynaptic potential development in the case of norm (1), after application of divalent cations (2), and botulinum toxin (3).

t ∗ < 1.44 ms when the max[ACh–R] = 0.11 mM is formed. In the following 0.72 ms the content of acetycholine-receptor complex slightly increases to 0.123 mM. The total number of activated receptors equals 99.2% (6.99 µM). Part of the postsynaptic acetylcholine undergoes fission by the acetylcholine esterase enzyme. The maximum concentration of (AChE)-complex equals 0.47 mM. The complex quickly dissociates into enzyme and choline, S, which is reabsorbed by the nerve terminal and is drawn into a new cycle of acetylcholine synthesis. In the course of all transformations the max[S] = 0.12 mM is formed.

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The process of the fast EPSP generation starts with the beginning of [ACh–R] complex development. There is a slight time delay between the two processes, which equals 0.4 ms. The EPSP increases as a step function due to the rise of [ACh–R] and achieves its maximum 87.1 mV in 0.25 ms.

4.2.5 Remarks The basic hypotheses in our model of the cholinergic secondary neuron rely on data about axon cable characteristics, synapse geometry, and electrical and chemical mechanisms underlying electrical signal transfer. The electrochemical response of the neuron to external stimulation can be predicted accurately, assuming the accuracy of input data. In terms of the number of parameters involved, the neuron model is quite complex. In calculations, we used the values of parameters and constants obtained from direct measurements made in in vitro experiments on cholinergic neurons. Some of the parameters, e.g. k+5 , k+6 , k+p , kd , we have been unable to estimate from literature and they have been chosen during the numerical simulation to fit the experimental data well. According to the numerical results, an increase in the diameter by a factor of two in the vicinity of the presynaptic terminal causes an increase in the velocity and amplitude of the propagated signal. Its amplitude rises from 69 to 72 mV, and velocity increases from 17.5 cm·s−1 to 21.6 cm·s−1 . Intracellular recordings from S neurons of the ENS show that the amplitude of the action potential changes between 65 and 75 mV, and the propagation velocity 6–20 cm·s−1 and 20–50 cm·s−1 , respectively. The process of ACh release has a quantum character. According to the numerical results, a single impulse evokes release of 7.8 µM (10%) of the stored AChv . In the chemical synapse, the postsynaptic response is registered after a time delay (kd ), which varies in the range 0.3–100 ms. The factors determining its duration are: (a) neurotransmitter release from the vesicles; (b) its diffusion into the synaptic cleft and the postsynaptic membrane; and (c) binding of AChp with the receptors, and formation of the (ACh–R)complex. It is presumed that the phases (b) and (c) define the main part of the time delay. According to experimental findings, the postsynaptic depolarization has a short step-like active phase — the period of maximum excitatory

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Physicochemical Basis of Synaptic Transmission 113

postsynaptic potential attainment equals 0.1–0.3 ms. The level of depolarization and the velocity of chemical processes on the postsynaptic membrane are wholly determined by the intensity of the reactions of the associationdissociation of the acetylcholine-receptor complex and by the electrical properties of the membrane. In the case under modeling the development of [ACh–R] = 0.123 mM evokes the EPSP generation, which reaches its maximum 87.1 mV in 0.25 ms.

4.3 INHIBITION OF CHOLINERGIC NEUROTRANSMISSION 4.3.1 Introduction Pharmacological modulation of the small intestinal motor activity is widely used to improve the understanding of the regulatory mechanisms and to explain the kinetic effects of the compounds used to treat disorders related to motor dysfunction. However, most experimental studies do not allow the differentiation of the level and of drug action and only allow conclusions based on the recorded integrated physiological response of the organ. The majority of data available today on pharmacological investigations are obtained from in vivo and in vitro studies where a limited number of compounds have been tested. However, the development of drugs for clinical use requires standard test models in which a large number of substances can be analyzed simultaneously to determine whether, and how, drugs contribute to the overall pharamacophysiological response.

4.3.2 Biological Background The majority of pharmacological agents used in clinical practice can act in two different ways to alter the processes responsible for cholinergic neurotransmission, by inhibiting the release of acetylcholine, and/or interaction of ACh with postsynaptic receptors. Drugs that affect postsynaptic ACh receptors are classified as cholinergic antagonists. They exhibit higher affinity for choline receptors which allows faster binding and thus prevents activation of receptors by a native neurotransmitter. According to the chemical activity and the stability of the antagonist-receptor complex that is formed they are subdivided into competitive and non-competitive antagonists. Each of

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these groups is subdivided into reversible and irreversible antagonists. Competitive interaction assumes finite rates for the reactions of association and dissociation between the compound and receptor with the development of an unstable bond. In the case of irreversible interaction, the velocity of dissociation of the antagonist-receptor complex virtually equals zero, i.e. the antagonist forms a strong stable bond with the receptors. Drugs that inhibit the release of ACh can either suppress the release of ACh by occupying the active centers for Ca2+ ions on the vesicles, or reduce the ACh-free fraction diffusion from the terminal into the synaptic cleft by changing the permeability of the membrane to the neurotransmitter. The observable response to the action of both groups of drugs is doseand putative receptor-dependent. In the ENS, neural receptors for ACh are of the nicotinic and muscarinic type. However, this classification is strictly pharmacological, based on experimental recordings for the effects of nicotine and muscarine drugs, and does not yield quantitative information about their chemical activity. We shall study numerically the effect of cholinergic antagonists and inhibitors of the release of ACh, irrespective of putative receptor classification, on the intensity of the cholinergic synaptic transmission. Regulation of the free Ca2+ concentration in the presynaptic terminal is the basis of a variety of mechanisms that confer the properties of plasticity on chemical synapses. They are controlled by processes within the neurons, such as changes in the resting membrane potential, duration of the excitation of the nerve-terminal, extracytosolic calcium-ion concentration, and extrinsic perturbations, such as the synaptic input from other neurons. Ca2+ enters the cell during its depolarization via N-type voltage-dependent Ca2+ channels. Unlike the voltage-gated sodium channels, the Ca2+ channels in the terminals do not deactivate quickly, but stay open as long as presynaptic depolarization lasts. It is supposed that free cytosolic calcium ions bind with calmodulin protein. The developed complex affects the movement of vesicles towards the active sites by dissolving some actin filaments.

4.3.3 Mathematical Model The basic assumptions used in the model have been given earlier. Consider the scheme of ACh conversion in the synaptic zone in the case of action of competitive cholinergic antagonists (C) (Fig. 4.4).

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Physicochemical Basis of Synaptic Transmission 115 Ca2+

k+p

kd

AChv −→ AChf −→ AChc −→ AChp k+1

k+4

AChp + R
ACh–R −→ S + R k−1 k+2

k+3

AChp + E
AChE −→ S + E k−2 k+7

C + R
CR k−7

Fig. 4.4 A chemical cycle of acetylcholine transformation in the case of cholinergic antagonist (C) action.

First-order chemical kinetics equations describe the pharmacodynamics of the proposed cycle d[Ca2+ ] dt d[AChv ] dt d[AChf ] dt d[AChc ] dt d[AChp ] dt

= [Ca2+ ]out , ϕ∗ − k+5 [Ca2+ ] = −kc [Ca2+ ][AChv ] = kc [Ca2+ ][AChv ] − kd [AChf ] + k+6 [S] = kd [AChf ] − k+p [AChc ] = k+p [AChc ] + k−1 [ACh–R] + k−2 [AChE] − k+1 [ACh−R p ][R] − k+2 [AChp ][E]

d[R] = k−1 [ACh–R] + k+4 [ACh–R] − k+1 [AChp ][R] dt − k+7 [C◦ ][R] + [CR](k+7 [R] + k−7 ) d[ACh–R] = k+1 [AChp ][R] − k−1 [ACh–R] − k+4 [ACh–R] dt d[AChE] = −[AChE](k+3 + k−2 + k+2 [AChp ]) + k+2 [AChp ][E◦ ] dt d[S] = k+4 [ACh–R] − k+6 [S] + k+3 [AChE] dt

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d[CR] = k+7 [C◦ ][R] − [CR](k+7 [R] + k−7 ) dt [E] = [E◦ ] − [AChE] [C] = [C◦ ] − [CR].

(4.22)

After prolonged treatment of the neuron with the antagonist the rates of forward and reverse reactions between competitive antagonist (C) and choline receptors (R) become equal. This condition corresponds to the partial chemical equilibrium and thus k+7 [C][R] − k−7 [CR] = 0

(4.23)

[R ◦ ] = [R] + [CR].

After simple transformations a steady concentration of blocked receptors and be found [CR] =

K ∗ [R ◦ ] , K ∗ + [C]−1

K∗ =

k+7 . k−7

(4.24)

K ∗ is the Michaelis–Menten constant. Hereafter, an overbar signifies an equilibrium concentration. The concentration of reacting receptors is [R] = [R ◦ ] − [ACh–R] − [CR].

(4.25)

Substituting (4.24) in (4.10) we obtain for [AChp ] and [ACh–R] d[AChp ] = k+p [AChc ] − [AChp ](k+1 [R ◦ ] + k+2 [E◦ ]) dt + k−1 [ACh–R] + [AChE](k−2 + k+2 [AChp ]) + k+1 [AChp ][ACh–R] + k+1 [AChp ][CR]

(4.26)

d[ACh–R] = −[ACh–R](k−1 + k+4 + k+1 [AChp ]) dt + k+1 [AChp ][R ◦ ] − k+1 [AChp ][CR]. Other equations in (4.10) remain unchanged. The model responses were simulated with the initial and boundary conditions and the sets of kinetic, electrical and geometrical parameters and constants as described above. The

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Physicochemical Basis of Synaptic Transmission 117

variables were the rate constants of binding and dissociation of cholinergic antagonist with the receptors on the postsynaptic membrane: k−7 = 2 ms−1 , k+7 = 5 × 105 mM·ms−1 , the concentrations of competitive antagonists [C◦ ] = 0.13, 0.26 mM; the concentration of inactivated choline receptors (CR) at the initial moment of time equalled zero.

4.3.4 Effect of Chloride Salts of Divalent Cations The mechanisms of action of the chloride salts of Mg2+ , Co2+ , Cd2+ and Ni2+ are based on the displacement of the free cytosoic Ca2+ ions in the synaptic terminal without changes in the intensity of spontaneous vesicular ACh release from the store. During the numerical simulation we varied the constant, k+5 , of absorption of calcium ions by the cell buffer systems. According to the results obtained, an increase in k+5 to 10 ms−1 caused a rapid decrease in the intracellular Ca2+ concentration: [Ca2+ ] = 5.4 µM was registered in 1.41 ms from the beginning of depolarization of the presynaptic terminal. At the value k+5 = 20 ms−1 only 2.8 µM of calcium ions enter the cell. As a result the reduction in the vesicular ACh release max[AChv ] = 79.5 mM was registered, and the accumulation of its free fraction and the suppression of AChf diffusion into the synaptic cleft were observed (Fig. 4.3). This caused the suppression of the generation of ϕp+ = 0, and a blockade of the nerve-pulse transmission via the synapse.

4.3.5 Effect of β -Bungarotoxin β-BuTX and its derivatives suppress the diffusion of the free fraction of ACh into the synaptic cleft. The thousand-fold decrease in the diffusion constant, kd = 0.04 ms−1 , caused a decrease in the diffusion velocity of AChf from the terminal and its content in the cleft to [AChf ] = 3.9 mM. At kd = 0, the accumulation effect of the free acetylcholine fraction was clearly seen: the maximum [AChf ] = 11.7 mM in the terminal was registered. In the case that kd = 0.04 ms−1 , the slow rise of EPSP to max ϕp+ = 41.09 mV is seen, while at kd = 0, ϕp+ = 0 throughout. It is important to note that the intensities of voltage-dependent Ca2+ ion influx and ACh release from the vesicles remain unchanged and coincide with the data observed for the case of the physiological norm.

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4.3.6 Effect of Botulinum Toxin The mechanism of action of botulinum toxin is based on a blockage of the active centers of Ca2+ ions binding on the vesicles. The effect of action was modeled assuming kc = 0. This did not influence the influx of Ca2+ ions into the nerve terminal, but completely suppressed the release of AChv : the concentration of vesicular acetylcholine remained at a constant level, [AChv ] = 80 mM. As a result the blockade of the nerve-pulse transmission was observed: ϕp+ = 0.

4.3.7 Change in the Concentration of Extracellular Ca2+ A decrease in extracellular calcium reduces the rate of Ca2+ ion flow into the nerve-terminal. In the case that [Ca2+ ]out = 0.5 mM (normally [Ca2+ ]out = 1.0 mM) the increase in the free cytosolic calcium content equalled 10 mM·ms−1 ; for [Ca2+ ]out = 0.1 mM the measured velocity of influx was 1.5 mM·ms−1 . Consequently, the amount of free calcium available for binding with the active sites on the vesicles was diminished. The maximum level of the internal calcium, [Ca2+ ] = 9.2 µM, was obtained when [Ca2+ ]out = 0.5 mM. max[Ca2+ ] = 2 µM was obtained when the concentration of [Ca2+ ]out was only 10% of the norm. In the first case, where [Ca+2 ]out = 0.5 mM, the amount of discharged vesicular ACh was 0.9 mM, which is only 1% of the whole stored neurotransmitter: [AChv ] = 79.1 mM is retained in a vesicular form. When the free fraction of ACh in the presynaptic terminal reached the maximum [AChf ] = 10.6 mM, it started to diffuse into the cleft. The maximum concentration of the neurotrasmitter that arrived at the receptive zone was AChp = 4.4 mM. The change in extracellular calcium content did not influence the velocity of the binding of ACh with receptors and its reaction with enzyme. The maximum concentration of the highly reactive acetylcholine-receptor complex of (ACh–R) of 0.121 mM was recorded and the generation of fast excitatory postsynaptic potential with an amplitude of ϕp+ = 85.8 mV was observed. In the case where [Ca2+ ]out = 0.1 mM the process of release of the vesicular ACh was completely suppressed, and in this respect a total blockage of synaptic transmission was observed: ϕp+ = 0.

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Physicochemical Basis of Synaptic Transmission 119

4.3.8 Effect of Cholinergic Antagonists The addition of ACh-antagonists causes a deactivation of chemical processes on the postsynaptic membrane. These effects are dose-dependent. According to the numerical results, the addition of compounds in concentrations of [C◦ ] = 0.13–0.26 mM caused a complete blockade in excitation transmission, ϕp+ = 0; while at lower concentrations [C◦ ] = 0.026 mM, a decrease in the number of choline receptors which were available for interaction was observed, [R] = 17 µM. Subsequently, the concentration of (ACh–R)-complex reduced to 0.016 mM, which, however, is enough to produce a depolarization of the postsynaptic membrane with the generation of EPSP, ϕp+ = 69.6 mV. In the case of a partial chemical equilibrium the addition of compounds in a concentration of [C◦ ] = 0.26 mM, which is two times higher than the whole number of choline receptors on the postsynaptic membrane, did not cause the blockade of the synaptic transmission. The maximum of [ACh–R] = 1.87 µM was formed, which caused the generation of EPSP with an amplitude of 53.2 mV (Fig. 4.5). The increase in the rate constants, k−7 = 20 ms−1 , k+7 = 2.5 × 106 (mM·ms)−1 , reduced the amplitude of the recorded excitatory postsynaptic potential to 20.9 mV.

Fig. 4.5 Changes in the concentration of receptor availability on the postsynaptic membrane and excitatory postsynaptic potential development in the case of norm (1), after addition of a cholinergic antagonist [C0 ] = 0.026 mM (2), and in the case of chemical equilibrium (3).

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4.4 FACILITATION OF CHOLINERGIC NEUROTRANSMISSION 4.4.1 Biological Background With drugs that completely or partially block synaptic transmission, there is a large group of chemical agents that facilitate cholinergic transmission. Their primary mechanism of action is to inhibit the activity of acetylcholinesterase enzyme in the synaptic cleft, and to increase the effectiveness of ACh release from the nerve-terminals. Anticholinesterase drugs with antiacetylcholinesterase activity used in medicine can be divided into two main classes according to their chemical structure and binding with acetylcholinesterase enzyme. The first class — reversible anticholinesterases — have chemical affinity with natural ACh and form an unstable enzyme-inhibitor complex. The second — organophosphorus or irreversible anticholinesterases — are chemically unrelated to ACh and form a long-lasting stable compound with the enzyme. Both classes of drugs react with the enzyme in a similar manner to natural ACh and the terms“reversible” and “irreversible” only reflect a difference in duration of their interaction. Most of the anticholinesterase agents used in medical practice are of the reversible type. The presence and localization of cholinesterase enzyme are determined histochemically. It is found in cholinergic terminals, and is also located postjunctionally in ganglionic synapses. Cholinesterases are classified into: true or specific cholinesterases, which exhibit the highest activity against the liberated ACh than against other choline esters; pseudo- or non-specific cholinesterases, which are responsible for controlling the effect of ACh acting as a local hormone to maintain tone. Two binding sites at the active center of the enzyme have been demonstrated: the anionic site, which reacts with the cationic head of the ACh molecule; and the esteric site, which combines with its ester group. Kinetic studies have shown that the anticholinesterase drugs inhibit the enzyme in a competitive manner. Both types of enzyme catalyze the hydrolysis of the neurotransmitter but only true cholinesterase acts on acetyl-β-methylcholine. Strictly reversible anticholinesterase agents, such as galantamine, ambenonium, endorphonium and aceclidine, are not hydrolyzed by cholinesterase and relatively quickly dissociate intact from the enzyme. Others, such as neostigmine and its derivatives, remain bound to the esteric site of enzyme for a long time and undergo the reaction of

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Physicochemical Basis of Synaptic Transmission 121 Ca2+

k+p

kd

AChv −→ AChf −→ AChc −→ AChp k+1

k+4

AChp + R
ACh–R −→ S + R k−1 k+2

k+3

AChp + E
AChE −→ S + E k−2 k+8

E + B
EB k−8

Fig. 4.6 A cycle of acetylcholine transformation in the case of a competitive acetylcholinesterase enzyme inhibitor (B) action.

hydrolysis, just as ACh does, but at a very slow rate. The fact that a particular cholinesterase is most active against a certain substrate does not necessarily mean that its normal function is limited to that compound. The anticholinesterase drugs used in clinical practice inhibit both types of enzyme, but they are, in general, slightly more potent against pseudo- than against true cholinesterases. The organophosphorus cholinesterase inhibitors react only with the esteric site of the enzyme in a competitive and irreversible manner. However, because of their high toxicity they are not used in medical practice.

4.4.2 Mathematical Model The cycle of ACh conversion in the cholinergic synapse, in the case of the addition of reversible anticholinesterase drug (B), is shown in Fig. 4.6. The equivalent system of kinetic equations is given by d[Ca2+ ] dt d[AChv ] dt d[AChf ] dt d[AChc ] dt

= [Ca2+ ]out , ϕ∗ − k+5 [Ca2+ ] = −kc [Ca2+ ][AChv ] (4.27) = kc [Ca ][AChv ] − kd [AChf ] + k+6 [S] 2+

= kd [AChf ] − k+p [AChc ]

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d[AChp ] = k+p [AChc ] + k−1 [ACh–R] + k−2 [AChE] dt − k+1 [AChp ][R] − k+2 [AChp ][E] d[ACh–R] = k+1 [AChp ][R ◦ ] − [ACh–R](k−1 + k+4 + k+1 [AChp ]) dt d[AChE] = −[AChE](k+3 + k−2 ) + k+2 [AChp ][E] dt d[E] = [AChE](k+3 + k−2 ) − [E](k+2 [AChp ] + k+8 [B◦ ]) dt + [EB](k−8 + k+8 [E]) d[EB] = k+8 [E][B] − k−8 [EB] dt d[S] = k+4 [ACh–R] − k+6 [S] + k+3 [AChE] dt [R] = [R ◦ ] − [ACh–R] [B] = [B◦ ] − [EB]. Here the same abbreviations, meaning of parameters and constants as in (4.10) are used. When the anticholinesterase drug is administered for a long period, the rates of forward and reverse reactions between competitive agonist (B) and the enzyme (E) become equal. This condition corresponds to a partial chemical equilibrium. After rearrangements similar to those done in (4.22)–(4.25), the fractions of (AChp ) and (AChE) are calculated from d[AChp ] = k+p [AChc ] − [AChp ](k+1 [R ◦ ] + k+1 [AChp ][ACh–R] dt + k+2 [E◦ ]) + k+2 [AChp ]([AChE] + [EB]) + k−1 [ACh–R] + k−2 [AChE] d[AChE] = −[AChE](k+3 + k−2 + k+2 [AChp ]) dt + k+2 [AChp ][E◦ ] − k+2 [AChp ][EB]. Other equations in (4.10) remain unchanged.

(4.28)

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Physicochemical Basis of Synaptic Transmission 123

During numerical simulations, the following parameters were varied: the rate constants of association and dissociation of cholinergic agonists, k+8 = 1 × 107 mM·ms−1 , k−8 = 40 ms−1 , respectively; the concentrations of cholinesterase inhibitors were [B◦ ] = 0.1 mM, [B◦ ] = 0.05 mM; the concentration of inactivated enzyme [EB] (0) = 0. In the case of the modeling of repetitive stimulation, trains of 3, 10 and 20 excitatory impulses applied at the soma of the neuron of amplitude, ϕ0 = 100 mV, duration, ts = 2 ms, were considered. The effect of the action of tetrodotoxin was modeled by a change in the maximal conductance of the axonal membrane for sodium ions, gNa = 12 mS·cm−2 .

4.4.3 Effect of Cholinergic Agonists The treatment of the nerve-terminal by cholinesterase inhibitors in concentration [B◦ ] = 0.1 mM caused a significant decrease in the active acetylcholinesterase content in the synaptic cleft [E] = 3.5 µM. The presence of cholinesterase inhibitor leads to an expected accumulation of free fraction of ACh in perfusate: max[AChc ] = 5.42 mM. The amount of inactivated neurotransmitter in the cleft equalled [AChE] = 0.22 mM, which was only 47% of the norm (Fig. 4.7).

Fig. 4.7 Change in the concentrations of acetylcholinesterase enzyme and inactivated acetylcholine in the synaptic cleft, in the case of norm (1), and the action of anticholinesterase drug in concentrations [B0 ] = 0.1 mM (2), [B0 ] = 0.15 mM (3).

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Consequently, 0.124 mM of ACh-receptor complex was developed on the postsynaptic membrane, which induced the generation of excitatory postsynaptic potential of amplitude ϕp+ = 89.08 mV. In the case of the addition of [B◦ ] = 0.05 mM, ϕp+ = 89 mV was registered. In the case of a partial chemical equilibrium between the inhibitor and the enzyme, the addition of [B] = 0.15 mM, which is three times higher than the concentration of acetylcholinesterase, did not completely abolish the enzyme activity. The concentration of free enzyme diminished to 1.3 µM. A slight increase in the neurotransmitter content in the cleft, max[AChc ] = 5.44 mM, and a significant decrease in inactivated acetylcholine [AChE] = 1.22 µM were observed. The amplitude of EPSP reached its maximum value ϕp+ = 89.1 mV.

4.4.4 Effect of TTX Artificial application of TTX in the vicinity of the presynaptic terminal significantly diminished the inward sodium flux and, as a result, completely blocked the action potential generation and propagation in and through the affected zone. The nerve terminal remained in an unexcited state, ϕ∗ = 0, and Ca2+ ion flow was abolished. Intracellular content of free calcium remained at a constant level, [Ca2+ ] = 0.1 mM, throughout. The concentration of ACh in the cleft equalled 0.0316 µM, only enough to induce a small-amplitude postsynaptic potential: ϕp+ = 0.5 mV, which lasted less than 0.5 ms.

4.4.5 Effect of Repetitive Stimulation High-frequency stimulation of the cholinergic neuron by a train of 3, 10 and 20 impulses showed that the increase in frequency of excitation caused an increase in the amount of released neurotransmitter from the vesicles. In the case of the action of 20 impulses, 2.2 mM of ACh was released from the store. The velocity of exocytosis was not constant. The maximum was 0.5 mM · ms−1 , which occurred at the beginning of the process, when the main part of the vesicular fraction of ACh was released, and sharply decreased to 0.001 mM · ms−1 after that. Concentration of AChf quickly reached its maximum value of 10.6 mM and diffused at a velocity of 25.6 mM·ms−1 from the presynaptic terminal

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Physicochemical Basis of Synaptic Transmission 125

into the cleft. At time t ∗ > 1.44 ms the velocity of diffusion started to decrease in an exponential, manner, which coincided with the process of saturation of the neurotransmitter in the cleft. For t∗ ≥ 2.4 ms the velocity of diffusion of AChf equalled zero. After stimulation by a series of three impulses the maximum concentration of ACh on the postsynaptic membrane equalled 5.7 mM, while with 20 impulses, max[AChp ] = 6.3 mM was observed. The analysis of the dynamic of the (ACh–R)-complex development showed that the binding of AChp with the choline receptors had a rate of 2.5 mM·ms−1 . The time of achievement of max[ACh–R] = 1.24 mM was less than 0.05 ms. During the high-frequency stimulation the postsynaptic potential increased in size. Fast excitatory postsynaptic potential of amplitude ϕp+ = 89.1 mV was recorded after the action of 10 impulses. In the case of stimulation by a train of 20 impulses the amplitude of excitatory postsynaptic potential increased slightly to ϕp+ = 89.4 mV.

4.4.6 Remarks It would be ideal to know the precise molecular mechanism by which the drug produces its biological effect, and it is tempting to ascribe a site of action anatomically close to the response observed. However, often even the site of action may have not been established with precision because of the ability of neural systems to operate across large distances. Traditional pharamacological experiments directly activate final output cells thereby masking the actions of neurons. Therefore, the point of application of a drug may be quite far off from the real physiological response. Many drugs that influence intestinal motor activity do so by interactions with specific receptors. In the enteric nervous system, neural receptors for ACh are of the nicotinic and muscarinic types. Their coexistence suggests a feedback regulatory interaction within the neuronal network and even within a single neuron, e.g. muscarinic cholinergic receptors located on nerve terminals may inhibit the release of ACh. In the present model we have not identified the types of ACh-receptors, which, however, does not limit the model’s validity, when run with their general chemical characteristics — the rate constants of association and dissociation of the drug-receptor complex.

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The model can be adjusted for the analysis of cholinergic neuron function in the case of muscarinic or nicotinic synapses. The properties of unidirectional sensitivity and informational capacitance of the cholinergic synapse are related to changes in Ca2+ concentration. The main source of calcium is external entry into the nerve-terminal via voltage-operated channels. In normal physiological conditions the concentration of the free cytosolic calcium rises to 19.4 µM. As our numerical results showed, a decrease in [Ca2+ ]out to 0.5 mM diminished the amplitude of EPSP to 85.8 mV, and in the case of [Ca2+ ]out = 0.1 mM a blockade of the transsynaptic transmission was produced. One interesting point that emerged during the modeling was that a decrease in external Ca2+ concentration caused an increase in the time delay, td = 0.82 ms (normally td = 0.4 ms). This is explained by the slow rates of Ca2+ ion influx and accumulation of the free fraction of ACh in the presynaptic terminal. The modeling of the action of chloride salts of divalent cations showed a decrease in the cytosolic content of Ca2+ to 5.4 µM and 2.8 µM. This would release 0.5 mM of AChv , which is insufficient for synaptic transmission. However, the increase in intracellular calcium did not alone stimulate neurotransmitter release. Calcium combines with the active centers on the vesicles and activates AChv exocytosis. One of the possible mechanisms of the action of botulinum toxin is inhibition of the active centers for Ca2+ ions on the vesicles. In numerical simulation the effect of botulinum toxin action prevented ACh release. The release of ACh from the terminal is derived from two principal sources: those susceptible to blockade by TTX, and those that are TTXresistant. Beani et al. showed that 90% of spontaneous output can be blocked by TTX. The remaining available fraction of ACh diffuses passively into the synaptic cleft and induces the depolarization of the postsynaptic membrane. The generated potential is of low amplitude, 0.5 mV, and is called a secretory potential. In simulation, the effect of TTX application on the axon of a cholinergic neuron was to block action potential propagation through the affected area, as a result of the lack of Ca2+ influx in the nerve terminal. However, the presence of the free fraction of acetylcholine ([AChf ] = 3.16 × 10−2 µM) in the cleft caused a postsynaptic depolarization of amplitude 0.65 mV that lasted 0.01 ms.

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Physicochemical Basis of Synaptic Transmission 127

The neural release of ACh can also be inhibited by application of βbungarotoxin, causing a blockade through an increase in the aggregation and fusion of the synaptic vesicles without an influence on the mechanisms of neurotransmitter accumulation in the terminal area. According to the results obtained, a fractional decrease in the diffusion constant caused a decrease in the amplitude of EPSP. In the case of kd = 0, a total blockade of excitation transmission occurs while the concentration of AChf in the terminal remains at a high constant level, [AChf ] = 11.7 mM. However, β-bungarotoxin is extremely hazardous and is not used in clinical practice in its pure form, but only as a tool in pharmacological experiments. The drugs generally designated as anticholinesterases are those whose primary action is to inhibit cholinesterase enzyme activity and thus preserve and augment the action of liberated ACh. Because of their resemblance in molecular architecture to native ACh, their pharmacological effect is based on binding with the active centers of the enzyme. However, they differ in their affinity to the enzyme and remain bound to it for a longer period. During the simulation these features were modeled by assuming the constants of association of the reversible anticholinesterase compound with acetylcholinesterase enzyme greater than the rate of (ACh–R)-complex formation by the factor of 400 and the ratio of dissociation constants k−8 /k−1 equalled 15. This caused a ten-fold decrease in the concentration of unbounded enzyme and a two-fold decrease of inactivated ACh in the cleft. As a result an increase in the level of EPSP was achieved. It should be stressed that we only considered competitive inhibition of acetylcholinesterase enzyme. However, this approach makes it possible to model noncompetitive and uncompetitive types of interactions. In the case of non-competitive inhibition, the reaction of binding of the drug (B) with the acetylcholine-enzyme complex should be taken into consideration, while in the case of uncompetitive inhibition only the reaction of the drug (B) with the AChE-complex should be taken into account. Radioligand binding studies have shown that drug-receptor interactions follow simple mass-action relationships, from where it can be assumed that the magnitude of the response is directly proportional to the fraction of total receptor sites occupied by the drug molecules. After the application of 0.13–0.26 mM of the irreversible antagonist, a complete blockade of transmission is observed. However, as experimental findings show, the occupation

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of receptors by competitive irreversible antagonists can be reduced by the simultaneous presence of a high concentration of an agonist or a competitive antagonist; that is, the receptors can be protected from blockade by a competitive irreversible antagonist. The addition of irreversible antagonist in a lower concentration, [C◦ ] = 0.026 mM, causes a reduction of 86.7% in the total number of available ACh-receptors, which instantaneously reflects on the amplitude of EPSP. It decreases to 69.6 mV, which is 79% of the magnitude registered in normal physiological conditions. Even when the amount of free receptors available for interaction with ACh is only 1.5% away from the norm ([R] = 1.93 µM, in the case of partial chemical equilibrium), they are able to provide generation of EPSP with an amplitude of 53.2 mV. The (CR)-complex concentration and the velocity of association or dissociation of a drug with receptors have a significant influence on the processes of depolarization of the postsynaptic membrane. Thus, a simultaneous ten-fold increase in k−7 and ten-fold decrease in k+7 , which correspond to a competitive type of interaction, diminish the postsynaptic response to 20.9 mV. Under normal physiological conditions neurons of the enteric nervous system are activated by series of excitatory impulses which follow at a high frequency, 5 × 102 –103 changes per second. The main feature of the synaptic structures, which provides the transformation and transmission of incoming information, is their capability for temporary accumulation and analysis of signals. In these cases a larger postsynaptic response of the neuron can be recorded than after a single stimulation. As the results of the numerical simulation show, the increase in frequency of excitation causes an increase in the amplitude of EPSP to 89.1 mV.

4.5 ADRENERGIC NEUROTRANSMISSION 4.5.1 Biological Background The sympathetic pathway to the intestine involves preganglionic cholinergic neurons with cell bodies located in the thoracolumbar spinal cord and postganglionic adrenergic neurons located in the coelica, superior and inferior mesenteric, and pelvic ganglia. The ramifications of the postganglionic nerve fibers within the intestinal wall are extensive and run to all structural layers. The majority of adrenergic terminals within the ENS synapse to the nerve cell and only a small number supply the smooth muscle of the intestine

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directly. The presence of noradrenergic nerve terminals and the ability of noradrenaline to block neural control of small bowel motility suggest that the sympathetic nerves act as a brake on the ENS. There are four recognized types of adrenergic receptors, α1 , α2 , β1 , β2 , which are differentiated pharmacologically. According to the anatomical position of the receptors on neurons and smooth muscle, they are classified as pre- and postsynaptic adrenoceptors. Postsynaptic adrenoceptors, located in the end organs, are the targets of neurotransmitters and drugs, and their stimulation or blockade will be translated into a variety of physiological and pharmacological effects. Presynaptic receptors are located at the membranes of the presynaptic vesicles, and their role is to modulate the release of noradrenaline from its sesicular storage sites. Intracellular recordings of electrical activity of neurons in the myenteric plexus have suggested that concentration-dependent hyperpolarization with a decrease in the amplitude of the nicotinic fast excitatory postsynaptic potentials is due to stimulation of α1 -adrenoceptors located on cholinergic neurons. Inhibitory postsynaptic potentials (IPSP) of short duration have been recorded in 5% of S-type and 15% of AH-type myenteric neurons and in a large proportion of submucous neurons. α2 -adrenergic receptors found presynaptically serve to inhibit the transmitter release from the neuron. β-adrenergic receptors are generally thought to occur post-synaptically on neurons and smooth muscle.

4.5.2 Mathematical Model Noradrenaline (NA) at the synaptic terminal is distributed into three subcompartments: free noradrenaline in cytoplasm; “releasable” noradrenaline in vesicles (Nas ), which releases a fraction of NA into the synaptic cleft (Nac ); and bound noradrenaline in vesicles. The release of transmitter is preceded by the process of depolarization-exocytosis coupling. The nature of the coupling mechanism has not yet been fully elucidated, but as with acetylcholine, it is generally assumed that release is Ca2+ -dependent. The entry of Ca2+ ions into the axonal varicosities through the voltage-gated Ca2+ channels during the excitation of the membrane raises the intracellular calcium concentration. An increase in intracellular calcium ion content is the trigger for the extrusion of the fraction of noradrenaline (Nas ), from the vesicles into the synaptic cleft (Nac ), where part of it diffuses passively to the post-synaptic

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membrane (Nap ), and binds with the α1 -adrenoceptors (ARα1 ); another part binds with the α2 -adrenoceptors (ARα2 ), located on the presynaptic membrane. Excess NA is removed by two mechanisms. Uptake-1 mechanisms involve re-uptake of NA into the axon terminal where it is further deaminated by mitochondrial monoamine oxidase (MAO). The second mechanism uptake-2, re-uptakes the neurotransmitter intracellularly where it is degradated by catechol-O-methyltransferase enzyme (COMT) to metabolic products (SA ). After inactivation the deaminated metabolites diffuse into the blood flow. There are two other mechanisms that lower the concentration of Nac : diffusion away from the site and binding to acidic extracellular polymers, but these are less important than uptake-1 and -2 mechanisms. The formation of the Nap –ARα1 complex causes a reorientation of the molecular structure of the receptive zone on the post-synaptic membrane and the generation of the inhibitory postsynaptic potential. The reaction scheme is shown in Fig. 4.8. The dynamics of electrochemical coupling at the adrenergic neuron and the generation of inhibitory postsynaptic potential assumes that i) an adrenergic S-type neuron is a geometrically non-uniform cable core of a given length; its diameter increases by a factor of two in the terminal area; the dynamics of electric signal propagation along the axon is described by the modified Hodgkin–Huxley equations; the soma of the Ca2+

kdA

NAs −→ NAc −→ NAp A k+1

A k+2

NAc + AR α2
NAc − AR α2 −→ AR α2 + SA A k−1 A k+3

A k+4

NAp + AR α1
NAp − AR α1 −→ AR α1 + SA A k−3

A k+6

NAc

 uptake-2  A k+5

A k+3

uptake-1 A k+8

NAp + COMT
NAp − COMT −→ COMT + SA A k−3

Fig. 4.8 A cycle of noradrenaline transformation.

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Fig. 4.9 A schematic illustration and geometrical characteristics of a generalized adrenergic neuron.

ii)

iii) iv)

v) vi)

neuron located at the left boundary is stimulated by an external electric impulse of a known intensity and duration; the presynaptic terminal area remains in the unexcited state throughout (Fig. 4.9); the synaptic zone is a null-dimensional, three-compartment open pharmacokinetic model, i.e. presynaptic terminal, synaptic cleft and postsynaptic membrane, in which all processes are assumed to be of order one; all the chemical reactions of noradrenaline transformation in each compartment are described by first order Michaelis–Menten kinetics; the basic influx of Ca2+ ions into the nerve terminal is secured by their transport through voltage-gated Ca2+ channels; decay of the free calcium level in the terminal is due to the buffering capacity of the cytoplasm; the dynamics of the IPSP generation depends on the concentration of activated α1 -adrenoceptors on the postsynaptic membrane; autoinhibition of the adrenergic synapse is due to activation of presynaptically located α2 adrenoceptors.

The above considerations lead to the following system of equations for the conduction of the action potential, ϕ∗ , along the unmyelinated axon (4.1)– (4.6), and the neurotransmitter conversion d[Ca2+ ] A = [Ca2+ ]out , ϕ∗ − k+9 [Ca2+ ] dt d[NAs ] = −kc [Ca2+ ][NAs ] dt d[NAc ] A A = kc [Ca2+ ][NAc ] − (kdA + k+5 + k+6 )[NAc ] dt A A [NAc ][AR α2 ] + k−1 [NAc − AR α2 ] − k+1

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d[NAp ] A A = kdA [NAc ] + k−3 [NAp − AR α1 ] − k+3 [NAp ][AR α1 ] dt A A [NAp −COMT] − k+7 [NAp ][COMT] + k−7

d[SA ] A A [NAp −AR α1 ] = k+2 [NAc −AR α2 ] + k+4 dt A [NAp −COMT] + k+8

d[AR α2 ] A A A + k+2 )[NAc −AR α2 ] = −k+1 [NAc ][AR α2 ] + (k−1 dt (4.29) A A A + k+2 )[NAc −AR α2 ] d[NAc −AR α2 ] = k+1 [NAc ][AR α2 ] − (k−1 d[AR α1 ] dt d[NAp −AR α1 ] dt d[COMT] dt d[NAp − COMT] dt

A A A + k+4 )[NAp −AR α1 ] = −k+3 [NAp ][AR α1 ] + (k−3 A A A [NAp ][AR α1 ] − (k−3 + k+4 )[NAp −AR α1 ] = k+3 A A A = −k+7 [NAp ][COMT] + (k−7 + k+8 )[NAp −COMT] A A A [NAp ][COMT] − (k−7 + k+8 )[NAp −COMT], = k+7

where kA+,−(i) are the rate constants of forward (+) and backward (−) chemical reactions (i = 1–4, 7, 8); kc is the rate association constant of Ca2+ ions with the vesicles (kc = 0, 1); kA+9 is the rate constant of absorption of Ca2+ ions by the cell buffer systems; kAd is the constant of the NAc fraction diffusion on the subsynaptic membrane; kA+(j) is the rate constant of re-uptake of the fraction of NAc from the synaptic cleft by uptake-1 and -2 mechanisms (j = 5, 6) and the meaning of other parameters as described in (4.10). The summation of the last six equations in (4.29) gives d[AR α2 ] d[NAc −AR α2 ] + =0 dt dt d[AR α1 ] d[NAp −AR α1 ] + =0 dt dt d[COMT] d[NAp −COMT] =0 + dt dt

(4.30)

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or [AR α2 (t )] + [NAc −AR α2 (t )] = [AR 0α2 ] = const. [AR α1 (t )] + [NAp −AR α1 (t )] = [AR 0α1 ] = const.

(4.31)

[COMT(t )] + [NAp −COMT(t )] = [COMT ] = const. 0

Assume that the total concentration of α1 , α2 -adrenoceptors [AR0α1 ], [AR0α2 ] and catechol-O-methyltransferase enzyme [COMT0 ] remains constant throughout. Substituting (4.31) into (4.29) we obtain d[Ca2+ ] A = [Ca2+ ]out , ϕ∗ − k+9 [Ca2+ ] dt d[NAs ] = kc [Ca2+ ][NAs ] dt d[NAc ] = kc [Ca2+ ][NAs ] dt A A [AR 0α2 ])[NAc ] − (kdA + k+5 + k+6 A A )[NAc −AR α2 ] + (k+1 [NAc ] + k−1

d[NAp ] A A = kdA [NAc ] + (k+3 [NAp ] + k−3 )[NAp −AR α1 ] dt A a [NAp ] + k−7 )[NAp −COMT] + (k+7 A A [AR 0α1 ] + k+7 [COMT0 ])[NAp ] − (k−3

d[NAc −AR α2 ] A A A = k−1 [NAc ][AR 0α2 ] − (k−1 + k+2 )[NAc −AR α2 ] dt A − k+1 [NAc ][NAc −AR α2 ]

d[NAp −AR α1 ] A A A = k+3 [NAp ][AR 0α1 ] − (k−3 + k+4 )[NAp −AR α1 ] dt A − k+3 [NAp ][NAp − AR α1 ]

d[NAp −COMT] A A A = k+7 [NAp ][COMT0 ] − (k−7 + k+8 )[NAp −COMT] dt A [NAp ][NAp − COMT] − k+7

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d[SA ] A A = k+2 [NAc −AR α2 ] + k+4 [NAp −AR α1 ] dt A + k+8 [NAp −COMT]

[AR α2 ] = [AR 0α2 ] − [NAc −AR α2 ] [AR α1 ] = [AR 0α1 ] − [NAp −AR α1 ] [COMT] = [COMT0 ] − [NAp −COMT].

(4.32)

The dynamics of the inhibitory postsynaptic potential, ϕp− , development satisfies Cp

dϕp− dt

+ ϕp− (−[NAp −AR α1 ] + Rv−1 ) =

ϕR− , Rv

(4.33)

where Cp is the capacitance of the subsynaptic membrane;  is the empirical constant of the synapse; Rv is the general resistance of the extrasynaptic structures; and ϕR− is the resting postsynaptic potential. Initial conditions suggest that the whole system is at the resting state and the initial concentrations of reacting components are given by [Ca2+ ]out = 1.0,

[Ca2+ ] = 1 × 10−4 ,

[NAc ] = 3.16 × 10−5 ,

[AR 0α1 ] = 0.1,

[NAs ] = 80, [AR 0α2 ] = 5 × 10−2 ,

[COMT0 ] = 0.13 (mM),

(4.34)

[NAc −AR α1 ] = [NAp −AR α2 ] = [NAp ] = [NAp −COMT] = [SA ] = 0. Boundary conditions assume that the soma of the neuron is excited by the external electric impulse. The system of equations (4.1)–(4.6), (4.32), (4.33), and initial and boundary conditions, (4.12)–(4.14), (4.33), provide a complete description of the simplified electrochemical coupling and the dynamics of the generation of the inhibitory postsynaptic potential in the adrenergic neuron. A hybrid second-order accuracy in both time and space finite-difference scheme and the second-order Runge–Kutta scheme have been used to solve the problem. The results of calculations are obtained for the values of electrical and morphological parameters of the neuron consistent with published estimates.

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4.5.3 Physiological Adrenergic Transmission The dynamics of the propagation of the wave of depolarization along the unmyelinated axon and the depolarization of the presynaptic terminal is similar to that described above (Sec. 4.2.4). An increase in cytosolic concentration of Ca2+ ions initiated the process of noradrenaline release from the free “releasable” store. The velocity of NA release was not constant but depended on the concentration of cytosolic Ca2+ . Thus at the beginning of the process, the velocity of the neurotransmitter free fraction increase equalled 0.66 µM·ms−1 and it reached a maximum of 2.87 µM·ms−1 . Subsequently, it decreased to zero at a steady rate. The free fraction of noradrenaline content in the terminal at this moment of time equalled 97 µM. During the whole cycle about 2.1% of noradrenaline was released, [NAs ] = 78.29 µM. The released noradrenaline diffused into the synaptic cleft where the maximum concentration, max[NAc ] = 86.3 µM, was registered (Fig. 4.10). It was quickly distributed among α1 , α2 -adrenoceptors, and another part was utilized by neuronal and extraneuronal re-uptake mechanisms. The concentration of NAp equalled 6.47 µM·ms−1 at the beginning and it reached a

Fig. 4.10 Changes in the concentration of noradrenaline in the cleft and noradrenaline — α1 receptor complex in the case of physiological neurotransmission (1) and after addition of inhibitors (2).

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maximum of 48.6 µM·ms−1 . The maximum concentration of the neurotransmitter available for binding with the α1 -adrenoceptors was [NAp ] = 35.2 µM. During electrochemical transmission, 8.8% ([ARα1 ] = 91.2 µM) of α1 -adrenoceptors were activated and the maximum of the (NAp –ARα1 ) complex equal to 8.53 µM was developed. The complex quickly dissociated, releasing a receptive zone. Further, the concentration of the postsynaptic fraction of noradrenaline quickly diminished at a velocity of 28.1 µM·ms−1 and [NAp ] = 26.3 µM was recorded. The amplitude of ϕp− quickly increased to −78.77 mV and then transformed into a part of the plateau with a small amplitude rise: ϕp = −76.2 mV. This reflected the time which was necessary to inactivate the (NAp –ARα1 ) complex on the postsynaptic structures. The dynamics of the (NAc –ARα2 ) complex development showed that the concentration reached the maximum [NAc –ARα2 ] = 12.03 µM and the share of the activated ARα2 -receptors equalled 37.97 µM, which was 21% of the whole number of available adrenoceptors on the presynaptic membrane (Fig. 4.11). The uptake-2 mechanism of inactivation is effective for a large range of neurotransmitter concentrations and has a lower affinity to the neurotransmitter than the uptake-1 mechanism. The velocity of association and dissociation of NAp with catechol-O-methyltransferase was constant at

Fig. 4.11 The dynamics of inactivation of noradrenaline on the postsynaptic membrane by catechol-O-methyltransferase enzyme under normal conditions (1) and after treatment with inhibitors of neuronal uptake (2).

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40 µM·ms−1 . The maximum concentration [NAp –COMT] = 32.9 µM was registered and only 33 µM of the enzyme was utilized. During the whole cycle of noradrenaline transformation, 355.3 µM of the metabolic product was being developed.

4.5.4 Remarks The model quantitatively reproduces the dynamics of electrical signal transfer along the axon and noradrenaline conversion: release from the free “releasable” store, diffusion into the synaptic cleft, binding with α1 -adrenoceptors located postsynaptically and α2 -adrenoceptors located on the presynaptic membrane, utilization of NAc by uptake-1 and -2 mechanisms, and the generation of inhibitory postsynaptic potential on the adjacent neuron. A single impulse evokes the release of 2.1% of the wholly stored NA. The concentration of NA on the postsynaptic membrane depends on the geometrical parameters of the synapse and varies in different tissues from 6 µM to 100 µM–1 mM. According to our numerical results, max[NAp ] = 35.2 µM was observed. The level of hyperpolarization of the postsynaptic membrane is wholly determined by the intensity of the reactions of association-dissociation of the noradrenaline-adrenoceptor complex and by the electrical properties of the postsynaptic structures. In the case being modeled the development of [NAp –ARα1 ] = 8.53 µM evokes IPSP generation, which reached 76.2 mV in 0.36 ms. These values agree with those previously reported. An important system of the mediator utilization is the uptake-2 mechanism, which is carried out by the entire surface of an adrenergic neuron and is capable of removing noradrenaline from the extracellular space in a short period of time. This type of absorption is most intensive during the intervals between the nerve-pulse actions. According to the numerical data only 41% of NAp was involved in the reaction of binding. The major part of the mediator was reabsorbed and utilized. The mobilized part of NA underwent a metabolic inactivation in the cytoplasm of adjacent neurons under the influence of catechol-Omethyltransferase enzyme. However, in the model we did not consider the

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processes of resynthesis of the neurotransmitter, and hence they were not analyzed here.

4.6 ALTERED ADRENERGIC NEUROTRANSMISSION 4.6.1 Effects of Extracellular Ca2+ Removal and Application of TTX The process of noradrenaline release from the store is controlled by Ca2+ dependent cytosolic mechanisms. A decrease in extracellular calcium reduces the concentration of Ca2+ ions in the nerve terminal. During simulations, the initial concentration of Ca2+ ions in the external medium was assumed to be: [Ca2+ ]out = 0.5 mM, and [Ca2+ ]out = 0.1 mM. In the case of [Ca2+ ]out = 0.5 mM, the maximum level of the internal calcium [Ca2+ ] = 9.2 µM was obtained; and max[Ca2+ ] = 2 µM was registered when the concentration of [Ca2+ ]out was 0.1 mM. In both cases NA overflow was completely abolished and no response in the postsynaptic membrane was observed: ϕp− = 0. The effect of the action of TTX was modeled by the change in maximal conductance of the axonal membrane for sodium ions: gNa = 12 mS·cm−2 . According to the results obtained numerically, the applications of TTX in the vicinity of the presynaptic terminal significantly diminished the Na+ -influx and, as a result, blocked the nerve-pulse propagation in/through the affected zone. The nerve terminal remained in an unexcited state throughout: ϕ∗ = 0.

4.6.2 Inhibition of Neuronal Uptake-1 Mechanism The effects of released NA are attenuated and terminated rapidly by re-uptake of a large proportion into the axon (uptake-1 mechanism) where it is further taken up and stored in the dense-cored synaptic vesicles, while its free fraction in the cytoplasm runs the gamut of mitochondrial monoamine oxidase. The rate of intraneuronal uptake is high at low concentrations of the neurotransmitter and is saturated by about 1 µM·l−1 noradrenaline. Neuronal uptake is inhibited by cocaine, imipramine, and sympathomimetic amines, such as metaraminol, dexamphetamine, phenylethylamine, isoprenaline, and their derivatives, methylphenidate, amantadine, chlorpromazine, guanethidine, which are structurally analogous to the native noradrenaline and have a similar action to cocaine. However, when considering the pharmacological

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effects of these compounds, it is important to remember that they each possess a number of side effects, and may interfere with adrenergic mechanisms in more than one way. The mechanism of the action of cocaine is to prevent the uptake of amines into the neuron. During the numerical simulation of the inhibition of neuronal uptake we assumed kA+5 = 0 in (4.32). According to the results obtained, it caused an increase in the amount of diffused NA into the synaptic cleft: max[NAc ] = 115.3 µM (86.3 µM; hereafter the values given in parentheses refer to the “norm”) but did not influence the time-characteristics of the process (Fig. 4.10). The maximum concentration attained coincided for both cases. 54.35 µM of noradrenaline (versus 35.2 µM) capable of binding with the α1 -adrenoceptors was registered on the postsynaptic membrane. 13% of the total number of adrenoceptors located on the postsynaptic membrane appeared to be activated and the max[NAp –ARα1 ] = 12.2 µM (8.52 µM) was developed (Fig. 4.11). As a result the inhibitory postsynaptic potential of amplitude ϕp− = −82.1 mV (−78.7 mV) was generated. Further, with the decrease in concentration of the (NAp –ARα1 )-complex to 10.5 µM, the amplitude of IPSP slightly increased at 81.0 mV and then remained constant throughtout. The concentration of bound α1 -adrenoceptors was [ARα1 ] = 89.46 µM (93.2 µM). There is a dynamic equilibrium between neuronal and extraneuronal uptake mechanisms: when neuronal uptake is inhibited, more noradrenaline is utilized by extraneuronal uptake and vice versa. Thus, extraneuronal uptake provides a kind of buffer, attenuating the peak of the response by lowering the concentration of the neurotransmitter in the vicinity of adrenoceptors. Inside the cells NA is exposed to the action of intracellular catechol-Omethyltransferase. In the case under analysis an increase in the intensity of this process was reproduced. The maximum amount of inactivated noradrenaline equalled [NAp –COMT] = 21.8 µM. Its concentration rapidly doubled in the following 0.6 ms and reached the maximum value −44.7 µM (32.9 µM). About 34.4% (25.3%) of the total enzyme content was consumed: [COMT] = 92.3 µM. The dynamics of binding of the free fraction of noradrenaline in the synaptic cleft with α2 -adrenoceptors located on the presynaptic membrane showed that the maximum conductration of the formed

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(NAc –ARα2 )-complex equalled 15.4 µM, which was slightly higher than the value obtained for the “norm”; 12 µM and 30.8% of the total number of ARα2 were bound. The amount of free adrenoceptors equalled [ARα2 ] = 34.6 µM and it increased to 38.7 µM.

4.6.3 Inhibition of Catechol-O-Methyltransferase Catechol-O-methyltransferase is widely distributed in the smooth muscle, in certain adrenergic neurons and other tissues, being located intracellularly as a cytoplasmic enzyme. There are considerable differences in the amounts of enzyme activity at sites of adrenergic transmission, but in some neuronal junctions it appears to play a significant role in terminating NA action and in modulating the action of exogenous catecholamines. There are a number of inhibitors of COMT: catechol itself, pyrogallol, quercin, rutin, substances described as flavenoids of the vitamin P group, tropolones and a group of benzoic acid derivatives. The most effective inhibitors of COMT are the tropolones, the most potent of which is methyltropolone. This is effective in vivo as well as in vitro, producing almost complete inhibition of short duration.

4.6.4 Mathematical Model The cycle of adrenaline conversion in the synapse, when a COMT inhibitor, (IK), is introduced is shown in Fig. 4.12. The kinetic equations for interaction of IK and COMT are d[IK] A A = −k+10 [IK][COMT] + k−10 [IKK] dt d[IKK] A A = −k+10 [IK][COMT] − k−10 [IKK] dt (4.35) d[COMT] A A A = −k+7 [NAp ][COMT] + k−10 [IKK] − k+10 [IK][COMT] dt A A + k+8 )[NAp − COMT]. + (k−7

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kdA

NAs −→ NAc −→ NAp A k+1

A k+2

NAc + AR α2
NAc − AR α2 −→ AR α2 + SA A k−1 A k+3

A k+4

NAp + AR α1
NAp − AR α1 −→ AR α1 + SA A k−3

A k+6

NAc

 uptake-2  A k+5

uptake-1

A +3

k

A k+8

NAp + COMT
NAp − COMT −→ COMT + SA A k−3

A k+10

IK + COMT
IKK A k−10

Fig. 4.12 The scheme of noradrenaline conversion in the synapse after treatment with inhibitors (IK) of catechol-O-methyltransferase.

After simple rearrangements in (4.35) we obtain d[IK(t )] d[IKK(t )] + =0 dt dt

(4.36)

[IK(t )] + [IKK(t )] = [IK0 ] = const.

(4.37)

or

The substitution of (4.34), (4.35) in (4.32) provides the kinetic equations for the cycle of NA conversion d[Ca2+ ] A = [Ca2+ ]out , ϕ∗ − k+9 [Ca2+ ] dt d[NAs ] = kcA [Ca2+ ][NAs ] dt d[NAc ] A A [NAc ] + k−1 )[NAc −AR α2 ] = kcA [Ca2+ ][NAs ] + (k+1 dt A A A − (kdA + k+5 + k+6 + k+1 [AR 0α2 ])[NAc ]

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d[NAp ] A A = kdA [NAc ] + (k+3 [NAp ] + k−3 )[NAp −AR α1 ] dt A A + k−7 [NAp −COMT] − (k+3 )[AR 0α1 ] A [COMT])[NAp ] + k+7

d[NAc −AR α2 ] A A A = k+1 [NA+1 ][AR 0α2 ] − (k−1 + k+2 )[NAc −AR α2 ] dt A − k+1 [NAc ][NAc −AR α2 ]

d[NAp −AR α1 ] A A A [NAp ][AR 0α2 ] − (k−3 + k+4 )[NAp −AR α1 ] = k+3 dt A − k+3 [NAp ][NAp −AR α1 ]

d[NAp −COMT] A A A = k+7 [NAp ][COMT] − (k−7 + k+8 )[NAp −COMT] dt d[COMT] A A A = −k+7 [NAp ][COMT] + (k−7 + k+8 )[NAp −COMT] dt A A A + [IKK](k−10 + k+10 [COMT]) − k+10 [IK0 ][COMT]

d[IKK] A A A = k+10 [IK0 ][COMT] − (k−10 + k+10 [COMT])[IKK] dt d[SA ] A A = k+2 [NAc −AR α2 ] + k+4 [NAp −AR α1 ] dt A [NAp −COMT] + k+8 [AR α2 ] = [AR 0α2 ] − [NAc −AR α2 ] [AR α1 ] = [AR 0α1 ] − [NAp −AR α1 ] [IK] = [IK0 ] − [IKK].

(4.38)

After a prolonged treatment of the neuron with the inhibitor, a partial chemical equilibrium is achieved. The rates of forward and backward reactions between inhibitor and catechol-O-methyltransferase enzyme become equal. Hence A A k+10 [COMT][IK] − k+10 [IKK] = 0

[COMT] = [COMT] + [IKK].

(4.39)

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From (4.39) we obtain for a steady concentration of blocked enzyme [IKK] =

K ∗ [COMT0 ] , K ∗ + [IK]−1

K∗ =

A k+10 A k−10

.

(4.40)

K ∗ is the Michaelis–Menten constant. The concentration of enzyme is [COMT] = [COMT0 ] − [NAp − COMT] − [IKK].

(4.41)

After the substitution of (4.39) in (4.38) and simple algebra, the expressions for the concentration of NA on the postsynaptic membrane and the inactivated complex take the form d[NAp ] A A = kdA [NAc ] + k−3 [NAp −AR α1 ] + k−7 [NAp −COMT] dt A + k+3 [NAp ][NAp −AR α1 ] A [NAp ][NAp −COMT] + k+7  A  A − k+3 [AR 0α1 ] + k+7 [COMT0 ] [NAp ] A [IKK][NAp ] − k+7

  A d[NAp −COMT] A A [NAp −COMT] = − k−7 + k+8 + k+7 dt  A  + k+7 [NAp ][COMT0 ] − [IKK] . (4.42) Other equations in (4.38) remain unchanged. The values of chemical, electrical and geometrical parameters and constants of the model correspond to those used in Sec. 3.5.3. The rates of association and dissociation constants of inhibitor IK with COMT were: kA+10 = 1 × 107 mM·ms−1 , kA−10 = 5 × 102 ms−1 ; the concentrations of enzyme inhibitors were: [IK0 ] = 0.15 mM, and [IK0 ] = 0.2 mM; the concentration of inactivated COMT enzyme, IKK, at the initial moment of time equalled zero. The addition of the inhibitor caused a significant increase in the concentration of free noradrenaline fraction on the postsynaptic membrane. The response was nonlinear and depended on the concentration of IK compound. Thus in the case of treatment of the adrenergic neuron with 0.15 mM of inhibitor, [NAp ] = 58.2 µM was registered, and an addition of 0.2 mM of inhibitor, max[NAp ] = 80.5 µM was observed (Fig. 4.13).

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Fig. 4.13 Effects of inhibitors of catechol-O-methyltransferase (IK) on the dynamics of noradrenaline and noradrenaline — α1 receptor complex on the postsynaptic membrane. The curves (1) and (2) refer to the case of partial chemical equilibrium: [IK] = 0.2 mM and 0.13 mM, respectively; the curves (3) and (4) correspond to treatment of the synapse with [IK] = 0.15 mM and 0.2 mM, respectively.

This reflected the amount of (NAp –ARα1 )-complex development. In the former case, [NAp –ARα1 ] = 15.5 µM (8.52 µM) was formed and 18.8 µM of the α1 -adrenoceptors were occupied; in the latter case, the concentration of active complex achieved 19.8 µM and 13.8 µM of ARα1 were bound. The amount of utilized noradrenaline by COMT also decreases. According to the numerical results in the case of treatment with [IK] = 0.15 mM of inhibitor, the concentration of bound (NAp –COMT)-complex was 25.2 µM, while after treatment with a higher dose [IK] = 0.2 mM, the concentration level decreased to 20.04 µM. These changes influence the level of hyperpolarization of the postsynaptic membrane, which was dose-dependent. After the addition of 0.15 mM of IK the IPSP of amplitude −83.3 mV (−78.7 mV) was recorded, and the increase in the dose of inhibitor to 0.2 mM caused a further decrease in ϕp− = −84.6 mV. In the case of a partial chemical equilibrium between the inhibitor and the enzyme, the addition of IK in concentrations of 0.13 mM and 0.2 mM caused an increase in NAp content to 119.3 µM and 135.5 µM, respectively. Consequently, the concentration of bound noradrenaline with adrenoceptors on the postsynaptic membrane increased: max[NAp –ARα1 ] = 24.5 µM

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and max[NAp –ARα1 ] = 26.98 µM was obtained. In the case under consideration 24.5–27% of the total amount of ARα1 was occupied. The blockage of 81–87% of COMT caused a decrease in the intensity of utilization of NAp . Only 9.59 µM and 13.0 µM of the (NAp –COMT) complex was formed. As a result the amplitude of IPSP increased: ϕp− = −85.6 mV (in the case of [IK] = 0.13 µM), and ϕp− = −86.1 mV (in the case of [IK] = 0.2 mM).

4.6.5 Effect of α 1 -Adrenoceptor Antagonists Drugs that block the α1 -receptors for sympathomimetic amines are classified into groups on the basis of their chemical structure: haloalkylamines (phenozybenzamine), imidazolines (phentolamine, tolazoline), phenoxyalkylamines (thymoxamine). The mechanism of their action is rather complex and not completely understood. It is supposed that the antagonism is competitive and slowly reversible due to the high strength of the covalent bond formed with the receptor. As well as acting on α1 -receptors they can affect adrenergic transmission preventing the uptake of released NA into neuronal and extraneuronal sites, and inhibiting α2 -receptors on the presynaptic membrane. The second mechanism is more important since compounds that inhibit noradrenaline uptake cause only a slight increase in noradrenaline release. Here we shall only consider the action of α1 -receptor antagonists on the dynamics of adrenergic transmission.

4.6.6 Mathematical Model Define the α1 -adrenoceptors antagonists as IRα1 . In a case of the addition of IRα1 to the system the last reaction (Fig. 4.12) should be changed to A k−11

IR α1 + AR α1
I AR α1 A k−11

while other reactions remain unchanged. The kinetic equations for IRα1 and α1 -adrenoceptors interaction are d[IR α1 ] A A [IR α1 ][AR α1 ] + k−11 [IAR α1 ] = −k+11 dt (4.43) d[IAR α1 ] A A = k+11 [IR α1 ][AR α1 ] − k−11 [IAR α1 ] dt

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 A  d[AR α1 ] A A [NAp ][AR α1 ] + k−3 + k+4 [NAp −AR α1 ] = −k−3 dt A A + k−11 [IAR α1 ] − k+11 [AR α1 ][IR α1 ].

After rearrangements similar to those done in the previous section, and substitution of the result into (4.32) we obtain d[Ca2+ ] A [Ca2+ ] = [Ca2+ ]out , ϕ∗ − k+9 dt d[NAs ] = −kc [Ca2+ ][NAs ] dt   A d[NAc ] A [NAc −AR α2 ] [NAc ] + k−1 = kc [Ca2+ ][NAc ] + k+1 dt   A A A − kdA + k+5 + k−6 + k+1 [AR 0α2 ] [NAc ] d[NAp ] A A [COMT][NAp ] [NAp −AR α1 ] − k+7 = kdA [NAc ] + k−3 dt A A + k−7 [NAp −COMT] − k+3 [AR α1 ][NAp ]

 A  d[NAc − AR α2 ] A A [NAc ][AR 0α2 ] − k−1 + k+2 [NAc −AR α2 ] = k+1 dt (4.44)  A  − k+1 [NAc ][NAc −AR α2 ]   A d[NAp − AR α1 ] A A [NAp −AR α1 ] = k+3 [NAp ][AR α1 ] − k−3 + k+4 dt   A d[NAp − COMT] A A [NAp −COMT] = k+7 [NAp ][COMT0 ] − k−7 + k+8 dt A − k+7 [NAp ][NAp −COMT]

  A d[AR α1 ] A A [NAp −AR α1 ] = −k+3 [NAp ][AR α1 ] + k−3 + k+4 dt A A + k−11 [IAR α1 ] + k+11 [AR α1 ]([IAR α1 ] − [IR 0α1 ])

 A  d[IAR α1 ] A A = k+11 [AR α1 ][IR 0α1 ] − k−11 + k+11 [AR α1 ] [IAR α1 ] dt

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Other equations in the model and initial and boundary conditions remain unchanged. In the case of chemical equilibrium between the compound IRα1 and the α1 -adrenoceptors A A k+11 [IAR α1 ] = 0 [AR α1 ][IR α1 ] − k−11

(4.45)

[AR 0α1 ] = [AR α1 ] + [IAR α1 ], and a steady concentration of blocked α1 -adrenoceptors is [IAR α1 ] =

K ∗∗ [AR 0α1 ] , K ∗∗ + [IR α1 ]−1

K ∗∗ =

A k+11 A k−11

.

(4.46)

K ∗∗ is the Michaelis–Menten constant. The concentration of adrenoceptors is [AR α1 ] = [AR 0α1 ] − [NAp −COMT] − [IAR α1 ].

(4.47)

After the substitution of (4.48) in (4.44) and rearrangements, the expressions for the concentration of noradrenaline on the postsynaptic membrane and the (NAp –ARα1 )-complex become d[NAp ] A A = kdA [NAp ] + k−3 [NAp −AR α1 ] + k−7 [NAp −COMT] dt A [NAp ][AR 0α1 ] − [NAp −AR α1 ] − k+3 A [NAp ]([NAp −COMT] − [COMT0 ]) − [IAR α1 ] + k+7

    A d[NAp −AR α1 ] A A + k+4 [NAp ] AR 0α1 [NAp − AR α1 ] + k+3 = − k−3 dt  (4.48) − [NAp −AR α1 ] − [IAR α ] , while other equations in (4.44) remain unchanged. The rates of association and dissociation constants of antagonist with the α1 -adrenoceptors were assumed to be kA+11 = 7 × 105 mM·ms−1 , kA−11 = 70 ms−1 , respectively; the concentrations of enzyme inhibitors were: [IR0α1 ] = 0.15 mM, and [IR0α1 ] = 0.25 mM; the concentration of (IARα1 ) at the initial moment of time equalled zero.

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The addition of IRα1 caused a block of 55.4% and 67.5% of the receptors on the postsynaptic membrane, respectively. Accordingly, the concentration of available α1 -adrenoceptors diminished to 44.6 µM and 33.51 µM. The velocity and the maximum concentration of the (NAp –ARα )-complex development also decreased. Thus after the treatment with [IR0α1 ] = 0.15 mM, max[NAp –ARα1 ] = 4.81 µM (8.42 µM) was recorded (Fig. 4.14). An increase in concentration of inhibitor to 0.25 mM caused a further decrease in the amount of active complex: max[NAp –ARα1 ] = 3.49 µM was obtained. The level and intensity of postsynaptic response development was also dosedependent: the amplitudes and velocities of IPSP generation decreased to ϕp− = −71.42 mV (−78.7 mV) and −66.24 mV, respectively. In the case of partial chemical equilibrium, the pharmacodynamic effects of the addition of antagonist were also dose-dependent. When 0.2 mM of antagonist was introduced, about 69.2% of the α1 -receptors appeared to be blocked; an increased IRα1 concentration to 0.3 mM caused further increase — 78.3% in the amount of blocked adrenoceptors. This influenced the concentration of the (NAp –ARα1 )-complex development: max[NAp – ARα1 ] = 3.12 µM (8.52 µM) (in the case of [IR α1 ] = 0.2 mM) and the max[NAp –ARα1 ] = 2.36 µM (in the case of [IR α1 ] = 0.3 mM). As a

Fig. 4.14 Effect of α1 receptor antagonist (IRα1 ) on the development of noradrenaline — α1 receptor complex dynamics. The curves (1) and (2) refer to addition of [IRα1 ] at concentrations 0.15 mM and 0.25 mM, respectively; the curves (3) and (4) correspond to the case of partial chemical equilibrium: [IRα1 ] = 0.2 mM and 0.3 mM, respectively.

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Physicochemical Basis of Synaptic Transmission 149

result, a higher level of hyperpolarization was attained: the amplitudes of the IPSP’s developed on the postsynaptic membrane were ϕp− = −64.3 mV and ϕp− = −58.86 mV, respectively.

4.6.7 Effect of the Repetitive Stimulation High-frequency stimulation of the adrenergic neuron by a train of 20 impulses caused an increase in the amount of released noradrenaline. About 9.04% (versus 2.1%) of noradrenaline was released from the store for a period of 5.37 ms after the beginning of excitation: [NAs ] = 72.77 µM (78.29 µM). During the whole cycle, the velocity of neurotransmitter released remained constant −1.67 µM·ms−1 . Free noradrenaline quickly diffused into the synaptic cleft where its concentration increased to max[NAc ] = 99.1 µM (86.3 µM). Visible oscillations of the NAc (t)-curve are due to the release of new portions of the neurotransmitter as a result of the following stimulation. According to the numerically obtained results, each portion contained 2.53 µM of noradrenaline. The maximum concentration of active complex on the postsynaptic membrane, [NAp –ARα1 ] = 10.3 µM (8.53 µM), was developed and 10.4 µM (8.8 µM) of α1 -adrenoceptors were occupied. The first changes in the amplitude of IPSP are registered 0.27 ms after the stimulation starred: ϕp− = −1.3 mV. The level of hyperpolarization rapidly attained the max ϕp− = −80.8 mV (−78.77 mV), which then slowly increased to −81.2 mV. The part of the free noradrenaline in the synaptic cleft bound with the α2 -adrenoceptors located on the presynaptic membrane. A concentration of the (NAc –ARα2 )-complex equalling 13.5 µM (12.03 µM) was recorded and 13.4 µM of α2 -adrenoceptors were occupied. They remained at this level during the whole period of stimulation and only a small amount of them, 3 µM, remained unbound. The nonreacting part of the NAp was utilized by uptake-1 and -2 mechanisms. The intensity of degradation did not depend on the frequency of stimulation. The maximum concentration inactivated by catecholO-methyltransferase neurotransmitter was: [NAp –COMT] = 39.3 µM (32.9 µM). During the whole cycle of chemical transformations, 33.5 µM (33 µM) of COMT was used.

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4.6.8 Remarks The details of neuroanatomy and interneuronal function within the enteric nervous system are both myriad and incompletely known. There is substantial evidence that postganglionic sympathetic nerves that innervate the gastrointestinal tract over extrinsic pathways serve primarily to reduce the excitability of cholinergic neurons. Owing possibly to the previous lack of neurophysiological information, coupled electrochemical analysis of the adrenergic transmission in the ENS under the influence of adrenergic agonists/antagonists has not been quantitatively explored. Here we sought to validate the general neuropharmacological principles of adrenergic transmission. The similarity in the behavior of the theoretical and in vivo/in vitro recorded responses shows that the model can adequately explain the dynamics of generation of the IPSP and thus the underlying mechanisms of electrochemical coupling in the adrenergic synapse. The treatment with uptake-1 and COMT inhibitors and repetitive stimulation led to an increase in the amount of the released NAs from the vesicular store and in the amplitude of the postsynaptic potential, e.g. after the repetitive stimulation 9.04% of the NAs was released and the amplitude of the inhibitory postsynaptic potential increased to −82.1 mV ÷ −83.3 mV versus −78.77 mV recorded under the “normal” physiological condition. The addition of α1 -antagonists caused a decrease in the concentration of the (NAp –ARα1 )-complex and as a result a decrease in the amplitude of the IPSP to −58.86 mV÷−66.24 mV, and all of these effects were dose-dependent. The advantage of the proposed pharmacokinetic study is that the above methodology could be easily extended to the investigation of ensembles of neurons within the enteric plexus, taking into account their morphology, neuroanatomy, pharmacology and electrophysiology.

REFERENCE Beani L, Bianchi C, Siniscalci A, et al. (1969). The effect of catecholamines and sympathetic simulation on the release of acetylcholine from the guinea-pig colon. Br J Pharm 36: 1–17.

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CHAPTER 5

Neuronal Assemblies

5.1 PLANAR NEURONAL NETWORK 5.1.1 Introduction Over the last few decades, a new framework has emerged for investigating informational and computational capabilities of neural networks. Research in the area of neural modeling has brought forward a number of different types of models. Based on the same general scheme — one-layer feedback or multi-layer feedforward performance, with neurons linked by “artificial” synapses — they differ considerably with respect to their capability for solving technical problems and to their biological applicability. Most of the proposed models of neural networks are “morpholess,” — they neglect the dendritic and axonal geometry of nerve cells, and express the electrical output of a neuron as a single number that represents the rate of firing. These models are essentially mathematical realizations of Hebb’s theory of cell assemblies and are widely used to explore auto- and hetero-associative capabilities of large neural networks. However, they are a drastic simplification with respect to the real biological systems and therefore their medical functionality is highly limited. In our approach we put special emphasis on biological plausibility of the model to be developed, including consideration of morphological and neurochemical principles of coding and transmitting of electrical nerve impulses.

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5.1.2 Biological Background The enteric nervous system is a principal determinant of control of the small bowel motor activity. It operates with great flexibility and rapidly adapts performance to delicate internal/external changes which is comparable to functional properties to abstract neuronal networks. These are supported by the structural analysis of polysynaptic interconnections among neurons, which are organized in positively coupled feedback elementary units — ganglia. Connected by bundles of neuron processes, they form a rectangular multi-layer network with the distance between nodes of about 2–5 mm. Morphologically, three large, expanded and vertically linked nerve plexuses are distinguished: the myenteric or Auerbach’s plexus, and the submucosal or Meissner’s plexus. Elements of Auerbach’s plexus run parallel to the muscle fibers throughout the thickness of the wall and dominate the coordination of muscular activity. Sensory information arriving from the effector units and the central nervous system is summarized, analyzed, and transformed as a regulatory modulated signal to the effector elements, smooth muscle cells. Meissner’s plexus is located in the submucosal layer and its major target is the intestinal mucosa. The response properties of the ENS are determined by the type, density and distribution of ganglia. This idea has recently received extensive experimental confirmation with the use of double-labeling immunohistochemical methods, which provide a unique opportunity to identify functionally distinct populations of cholinergic and non-cholinergic neurons and to reveal topographic maps of their projections within the intestinal plexuses. The existence of orally and anally directed excitatory cholinergic neurons with short local (< 2 mm) and long axonal (up to 10 mm in length) projections has been demonstrated. The projections of inhibitory neurons of a short length and anally long directed projections, up to 30 mm, were traced. The collection of inhibitory and excitatory neurons in a single row of ganglia can be considered as the simplest functional unit: it is the repeating neuronal units that control intestinal motility. The relative temporal and spatial contribution of excitatory and inhibitory inputs at any individual neuron depends on the morphological characteristics of the synapse and the properties of neuronal integration. Depending on the location of contact on the neuron the following synapses can be distinguished: axo-axonic, axosomatic and axodendritic.

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Axosomatic inputs often provide the inhibitory effect; axodendritic are mostly excitatory; while axo-axonic synapses play a modulatory role by indirectly depressing or enhancing the amount of transmitter released by the postsynaptic neuron. The nerve cells interact in a variety of ways. However, input signals mediated by synaptic connections dominate and the results of synaptic interactions eventually appear as an output impulse, either excitatory or inhibitory, in adjacent neurons.

5.2 INHIBITORY NEURAL CIRCUIT 5.2.1 Axo-Axonal Interaction The main function of the adrenergic nerves in the ENS is to reduce the excitability of the cholinergic neurons. Thus, suppressing the intrinsic reflex pathways, leading to an inhibition of all types of intestinal movements, showed that cholinergic neurons of the enteric nervous system are under double adrenergic control: a stimulating effect of catecholamines mediated by excitatory β-adrenoceptors is counteracted by an inhibitory effect of α1 -adrenoceptor activation. Intracellular recordings of the electrical activity of neurons in the myenteric plexus suggest that a concentration-dependent hyperpolarization with a decrease in the amplitude of the nicotinic fast excitatory postsynaptic potentials is due to stimulation of α1 -adrenoceptors located on cholinergic neurons. Inhibitory postsynaptic potentials have been recorded in 5% of S-type and 15% of AH-type neurons of the myenteric plexus and in a large proportion of submucosal neurons. α2 -adrenergic receptors found presynaptically serve to inhibit the transmitter release from the neurons. β-adrenergic receptors are generally thought to occur postsynaptically on neurons and on smooth muscle. Particular combinations of such substances may correlate with specific functional classes of nerves. The conclusion that these transmitters must be released simultaneously with the axonal action potential is inescapable. These transmitters may interact at many levels in their synthesis, release, degradation, or postsynaptic actions. The possible interactions vary, of course, with the particular substances that are co-localized. Consider the neural circuit composed of excitatory (cholinergic) and inhibitory (adrenergic) neurons that are arranged as shown in Fig. 5.1.

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Mathematical Modeling and Simulation in Enteric Neurobiology Cholinergic neuron

Axo-axonal synapse

Adrenergic neuron

Fig. 5.1 An inhibitory neuronal circuit.

In addition to the general assumptions related to the cholinergic and adrenergic transmission, we slso assume that the excitatory potential propagating along the cholinergic axon and the fast inhibitory potential at the axo-axonal synapse sum linearly: ϕsum = ϕ|s=s ∗ + ϕp−

(5.1)

where s = s ∗ is the Lagrange coordinate of the synapse. The above considerations lead to the simultaneous solution of the following equations: the action potential propagation along the axons of cholinergic and adrenergic neurons, (4.1)–(4.6); chemical kinetics for the adrenergic synapse function, (4.32); the inhibitory postsynaptic potential development, (4.33); synaptic interaction at the axo-axonal synapse, (5.1); with initial and boundary conditions defined by (4.12), (4.14) and (4.34). Changes in the resulting potential at the synaptic zone under normal physiological conditions are shown in Fig. 5.2. The critical value of the time lag, t, at which there was no influence of the IPSP on the dynamics of the excitatory potential propagation, equaled 1.68 ms. The amplitude and velocity of the propagating spike ϕ remained unchanged; it reached the cholinergic synapse where it initiated further mechanisms of nerve-pulse transmission. As the t interval was shortened to 0.6 ms the inhibitory influence began to take effect. The intensity of depolarization phase decreased with a decrease in the maximum amplitude of ϕsum to 44.2 mV. At t = 0.39 ms the amplitude and the duration of ϕ reduced significantly. A spike of duration of 0.85 ms and a maximum amplitude of 6.5 mV was generated. This level of depolarization was insufficient to initiate further propagation of the excitation along

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φ sum (mV)

1 40 3 0.0 2 4 -40 5 -80 0.0

2.0

4.0

Time (s) Fig. 5.2 Changes in the resulting potential at the axo-axonal synapse of the inhibitory neuronal circuit for t = 1.68; 0.6; 0.39; 0 and −1.5 (ms) (lines 1 through 5, respectively).

the axon of cholinergic neurons and, as a result, the blockade of propagation developed. In the case of the simultaneous or delayed arrival of the wave ϕ at the axo-axonal synapse, an alteration in the dynamics of IPSP generation or, a complete suppression of the propagation of electrical signal was observed.

5.2.2 Effect of COMT Inhibitors The addition of the inhibitor of COMT caused a significant increase in the concentration of free NA fraction on the postsynaptic membrane. The response was nonlinear and depended on the concentration of the added compound. Thus, in the case of treatment of the adrenergic neuron with 0.15 mM of inhibitor, the IPSP of amplitude −83.3 mV was generated. After the addition of 0.2 mM of inhibitor ϕp− = −84.6 mV was recorded. These changes had no influence on the time characteristics of the interaction process at the axo-axonal synapse but significantly reduced the amplitude of the resulting potential to 8% from the “norm.”

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5.2.3 Effect of α 1 -Adrenoceptor Blockers The addition of α1 -adrenoceptor blockers in concentrations of 0.15 mM and 0.2 mM caused a blocking of 55.4% and 67.5%, respectively, of the total amount of receptors on the postsynaptic membrane. The rate and the maximum concentration of active (NAp −AR α1 ) complex development also decreased. The maximum content of the noradrenaline-adrenoceptor complex equaled 4.81 µM and 3.49 µM. This influenced the level and intensity of inhibitory potential generation. The amplitude decreased to ϕp− = −71.42 mV and 66.24 mV, respectively. These changes again had no effect on the dynamical characteristics of excitation-inhibition interaction but caused an increase in the amplitude of the resulting potential to 10–12% from the “norm.”

5.3 A MODEL OF THE SENSORY PATHWAY 5.3.1 Introduction Movements can be elicited in the bowel in vivo and in vitro that do not have extrinsic neural connections with the central nervous system. Thus, mechanical stimulation of the intestinal wall of an isolated segment by infusion of liquid, increase in intraluminal pressure, distention, and so on evokes stereotyped motility patterns consisting of contraction proximal to the point of excitation and relaxation distal to the point of indentation. These observations indicate that motility of the organ is entirely mediated by reflexes originated in the enteric nervous system, which contains intrinsic primary afferent neurons and interneurons, as well as sensory receptors and motor neurons. Arranged in neural circuits, they provide forward-feedback regulatory inputs to the contractile elements of the organ. Though the efferent excitatory pathways are well established, the afferent pathways are still poorly understood. Numerous models of the Hodgkin–Huxley type have been developted to study different hypotheses for the ionic mechanisms underlying electrical behavior in neuronal cicuits. Some unexpected properties of transitions between various modes of electrical behavior, association learning and adaptation have been elucidated. Only a few studies though have been concerned

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with modeling of electromechanical coupling in mechanoreceptors. The coupling mechanism in the above models is based on a hypothesis that the induced strain from deformation provides energy to activate ionic channels of the membrane of dendrites, which results in initiation of a receptor potential. However, all these models deal with single excitable cells such as pancreatic β-cells, molluscan Aplysia R-15 cells, pyramidal cells of the cerebral cortex and Pacinian corpuscles; few of them consider coupled bursters. Here we shall simulate an afferent pathway of the ENS to reveal electrophysiological activity of the interconnected primary and secondary sensory neurons, and we shall study the effects of charybdotoxin (CHTX), iberiotoxin, tetrodotoxin (TTX), ω-conotoxin GVIA (ω-CgTX), the A1 purinoceptor agonist 2-chloro-N 6 -cyclopentyl-adenosine (CCPA), the A2 purinoceptor agonist 2-[p-(carboxyethyl)phenylethylamino]-5 -Nethylcarboxamidoadenosine (CGS 21680), the mixed A1 -A2 agonist 5 N-ethylcarboxamidoadenosine (NECA), the protein kinase C activator phorbol 12,13-dibutyrate (PDBu), and the δ-opioid receptor agonist D-penicil-amine2 -D-penicillamine5 (DPDPE) on their electrical behavior.

5.3.2 Biological Background Numerous electrophysiological analyses combined with dye filling of neurons of the ENS have revealed that the pathways of intrinsic reflexes include primary sensory neurons, a number of secondary neurons, and final motor neurons. The primary afferent neurons have not been definitely identified and there is still considerable controversy concerning their location and projections. It is conceivable that they belong to a class of electrophysiologically defined AH neurons with a distinct Dogiel type II morphology. The primary sensory neurons have smooth cell bodies, are principally adendritic, pseudouniaxonal or multiaxonal (with two or more long processes) with a tendency to primary and secondary branching of the neurites close to the soma, and show a few or no synaptic inputs. Their receptive fields, i.e. mechanoreceptors, are free nerve endings and are located in the mucosa and the submucous layer. We could not find in the available literature any data about the morphology, electrophysiology or pharmacology of the receptive field of mechanoreceptors of the gut. The situation can be explained by the current inaccessibility of this part of the ENS to existing methods of investigation.

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According to extracellular recordings the primary sensory neurons are classified as slowly adapting mechanosensitive units. They discharge continuously without signs of adaptation during mechanical distortion and the frequency of discharge increases as a direct function of the intensity of stimulation. The resting potential of these neurons ranges from −55 to −75 mV and their mean input resistance varies from 20 to 190 M. Action potentials have amplitude 60–80 mV and duration 2.5–3 ms, and are followed by characteristic long-lasting after-hyperpolarizations of 1–2 s. The waveform of the action potentials is usually complex multiphastic with a prominent “shoulder”on the repolarizing slope. A pharmaclogical characteristic of soma action potentials of AH neurons is that they are resistant to TTX, and the slow afterhyperpolarization phase is blocked by CHTX and iberiotoxin. Simultaneous recordings made from a cluster of neurons demonstrated that many AH neurons provide identifiable synaptic inputs to neighboring cells, which would be predicted for sensory neurons. The identity of the secondary sensory neurons is also uncertain. There are different opinions about their morphological and electrophysiological characteristics. Thus, these were descsribed by some researchers as Dogiel type I cells with S-type electrical activity, while others refer to them, morphologically, as Dogiel type III neurons and, electrically, as tonic-type mechanosensitive units. The neurons are uniaxonal and multidendritic. Dendrites are of intermediate length, relatively little branched and project in the connectives orally and aborally to adjacent cell clusters. The distinguishing characteristic of their electrical behavior is that they discharge long trains of spikes of ∼21 s in a set pattern even after withdrawal of the mechanical stimulus. These patterns resemble all-or-nothing events and are independent of the initial stimulus. Action potentials of amplitude 70–80 mV are generated at a high frequency of 10–40 Hz. The frequency is relatively constant during the first 3–5 s of bursts and then declines linearly. Waveforms of the spikes are biphasic or triphasic and the duration of each spike varies from 2.5 to 4.5 ms. In some cases tonic-type units fire in beating mode, which can last for approximately 40 min. No distinct pharmacological properties of action potentials of the secondary sensory neurons have been described as yet. Multiunit extracellular recordings from the myenteric plexus of the small bowel revealed that the discharge of the slowly adapting neuron (primary sensory neuron) always proceeded with a constant time interval the discharge of

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the tonic-type unit (secondary sensory neuron). This suggests that the secondary neurons may be triggered by synaptic input from a primary sensory neuron.

5.3.3 Mathematical Model Consider a neuronal pathway which consists of the primary and secondary sensory neurons connected in sequence by an axodendritic synapse as shown in Fig. 5.3. Deformation of the mucosa and submucous layer initiates an impulse at free nerve endings of the mechanoreceptors. The excitatory signal then passes along the axon to the soma of the primary sensory neuron and further, along another axon and the axodendritic synapse, to the secondary neuron. From the physiological remarks it is clear that neuronal elements of the afferent neural circuits within the ENS transfer sensory information from the receptive field in the form of a temporal sequence of pulses. Furthermore, they exhibit single spike and rhythmic oscillatory patterns, i.e. slow alterations between a silent phase and an active phase of rapid bursts. Our objective is to analyze the electrical behavior of the afferent pathway at three particular levels, i.e. the mechanoreceptors, and somas of the primary and secondary sensory neurons. Therefore, the dynamics of the propagation of nerve signals along unmyelinated axons and processes of electrochemical coupling at the axodendritic synapse are not included in the simulation. There is experimental evidence that mechanoreceptors convert mechanical stimuli (stretch) to the receptor potential via activation of ion channels. These are called stretch-activated channels and they are selective to sodium

Fig. 5.3 Schematic diagram of the afferent pathway that consists of the primary and motor neurons linked in a sequence. The receptive field is stimulated by external applied deformation.

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ions. This idea has been successfully implemented in the simulation of electrical behavior of mechanoreceptors. The dynamics of the dendritic receptor potential is given by dϕd = −(I˜Na + I˜K + I˜Cl ) + (ϕ1 − ϕd )/Rm dt dϕ1 = −ϕ1 + k(ϕd − ϕ1 )/Rm γ dt

Cm

(5.2)

where ϕ1 , ϕd are the nerve ending and dendritic receptor potentials, respectively; Cm is the dendritic membrane capacitance; Rm is the membrane resistance; γ is the membrane time constant; k is a numerical constant; t is time. I˜Na , I˜K , I˜Cl are the sodium, potassium and chloride ionic currents, respectively, ˜ d − ϕ˜ Na ) I˜Na = g˜Na m ˜ 3 h(ϕ I˜K = g˜K n˜ 4 (ϕd − ϕ˜ K ) I˜Cl = g˜Cl (ϕd − ϕ˜ Cl ).

(5.3)

Here g˜Na , g˜K , g˜Cl are the maximal conductances of Na+ , K+ and Cl− channels; m, ˜ h˜ and n˜ are probabilities of opening of these channels; ϕ˜ Na , ϕ˜ K , ϕ˜ Cl are the reversal potentials of sodium, potssium and chloride currents. The activation and deactivation of the ionic channels is described by dy ∗ = ψ(α˜ y (1 − y ∗ ) − β˜ y y ∗ ) dt

˜ n) (y ∗ = m, ˜ h, ˜

(5.4)

where ψ is the temperature scale factor; α˜ is the rate at which the channels switch from a closed to an open state, and β˜ is the rate for the reverse. They satisfy the modified Hodgkin–Huxley formulation and are obtained from the approximation of experimental data α˜ m = 0.221 exp(ε(t ) + 0.01ϕd ) β˜ m = 4.5 exp(−ϕd /18) α˜ h = 0.048 exp(−ϕd /36) β˜ h = 0.12/(1 + exp(3.4 − 0.2ϕd )) α˜ n = 0.33 exp(1.1 − 0.1ϕd ) β˜ n = 0.185 exp(−ϕd /80), ε(t ) is the applied strain.

(5.5)

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The dynamics of soma action potentials at the primary, ϕ2 , and secondary, ϕ3 , sensory neurons is described by the integrated function of five ion channels: voltage-dependent N-type Ca2+ channels, Ca2+ -dependent K+ channels, and the Hodgkin–Huxley-type Na+ , K+ and Cl− channels Cs

dϕ(i) = −(ICa + ICa−K + INa + IK + ICl ) + Iext(i) dt

(i = 2, 3).

(5.6)

Cs is the membrane capacitance, I(··· ) are ionic currents carried through different ion channels and Iext(i) is the external membrane current. The equations for ion currents are ICa = ICa−K =

gCa(i) z (ϕ(i) − ϕCa ) 1 + ζ[Ca2+ ] gCa−K ρ∝ [Ca2+ ] (ϕ(i) − ϕCa−K ) 0.5 + [Ca2+ ]

INa = gNa(i) m 3 h(ϕ(i) − ϕNa )

(5.7)

IK = gK n 4 (ϕ(i) − ϕK ) ICl = gCl(i) (ϕ(i) − ϕCl ) (i = 2, 3) where ϕCa , ϕCa−K , ϕNa , ϕCl are the reversal potentials for the respective currents; gCa(i) , gCa−K , gNa(i) , gK , gCl(i) are the maximal conductances of voltagedependent Ca2+ channels (N-type), Ca2+ -dependent K+ channels, Na+ , K+ and leak Cl− channels; ζ is the specific parameter of calcium inhibition of the Ca2+ channels; [Ca2+ ] is the intracellular concentration of free calcium; z, ρ∝ , m, h and n are dynamic variables of the ion channels are given by dz = (z∝ − z)/τz dt dh = λh (h∝ − h)/τh dt dn = (n∝ − n)/τn dt
−1 ρ∝ = 1 + exp 0.45(ϕ(i) + 57) m = m∝ (ϕ(i) ) (i = 2, 3).

(5.8)

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Here y∝ = αy∝ τy

(y = m, h, n)

τy = 1/(αy∝ + βy∝ ) αm∝ =

0.12(ϕ(i) + 27) 1 − exp(−(ϕ(i) + 27)/8)

βm∝ = 4.4 exp (−ϕ(i) + 47)/25 αh∝ = 0.09 exp −(ϕ(i) + 47)/27 βh∝ =

1 1 + exp(−(ϕ(i) + 33)/8)

αn∝ =

0.012(ϕ(i) + 12) 1 − exp(−(ϕ(i) + 12)/8)

(5.9)

βn∝ = 0.12 exp (−ϕ(i) + 26)/87 z∝ =

1 . 1 + exp(−0.15(ϕ(i) + 42))

Changes in the free cytosolic calcium concentration yield d[Ca2+ ] = 0.234 × 10−4 z(ϕCa − ϕ(i) )/(1 + ζ[Ca2+ ]) − 0.003[Ca2+ ]. dt (5.10) The membrane current Iext(i) (i = 2, 3) is a result of the receptor potential, ϕ1 , and the soma action potential, ϕ2 , respectively. It is calculated from Iext(i) = ϕ(i−1) /R(j)

(j = p, s; i = 2, 3)

(5.11)

where Rp is the input resistances of the primary sensory neuron, and Rs is the resistance of the extrasynaptic structures in the afferent pathway. The dynamics of electrical signal propagation along the unmyelinated axons of the primary and secondary sensory neurons is described by the modified Hodgkin–Huxley equations. The electrochemical coupling at the axosomatic cholinergic synapse and the generation of the fast EPSP on the soma

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of the secondary neuron satisfy the general principles of cholinergic neurotransmission (see Chap. 4 for details). Initial conditions assume that the whole system is in the resting state. It is excited by an externally applied mechanical stimulus of deformation of a known intensity. The system of equations was solved numerically using the methods as described above in Chap. 4.

5.3.4 Responses to Deformation A single stretch of the wall of the small bowel evokes a single receptor potential at the dendrites of the mechanoreceptors (Fig. 5.4). The amplitudes and durations of the responses depend on the amplitudes and durations of the stimuli. Thus, the level of deformation ε = 0.1 and of duration td = 1.0 s is not enough to alter the receptor potential. The dendrites of the mechanoreceptors become hyperpolarized. A gradual increase in the intensity of stimulus leads to an increase in the amplitude of the electrical response. For ε = 0.5 the dendritic potential, ϕd , of maximum amplitude 23.6 mV is registered. Its dynamics is very slow, i.e. phases of depolarization and repolarization last approximately 0.5 s. However, when the stimulus ε = 0.75 is applied, the mechanoreceptor fires with the generation of high-amplitude spikes, ϕd = 62.5 mV, of short duration 2.3 ms. A characteristic feature of the electrical pattern is the long-lasting “step” of approximate amplitude 45 mV, which is omnipresent for ε ≥ 1.0. As is assumed in the model, the dendritic potential causes the membrane current Iext(2) to the soma of the primary sensory neuron. It fires with a delay which ranges from 0.1 to 0.4 s, with the generation of the soma action potential. The neuron remains hyperpolarized for ε = 0.1, bursting with the production of spikes of maximal amplitude ϕ2 = 20 mV when the level of deformation exceeds 0.75. The amplitude of bursts increases with stretch over the entire ε range and the maximum ϕ2 = 42.5 mV is recorded for ε = 0.5. It is worth pointing out that this level of ϕ2 remains constant throughout: ε ≥ 0.5. The pattern of discharges resembles a bursting chaos type of electrical activity with the frequency ν = 8–10 Hz. The duration of spikes being produced varies from 2.5 to 2.8 ms. A characteristic feature of

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Fig. 5.4 The dynamics of development of the dendritic and the soma action potentials at the primary sensory neuron as a result of mechanical stretch of different intensities.

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the pattern is the presence of a prominent “shoulder,” which is followed by a long-lasting slow after-hyperpolarization of 1.6 s. The response of the whole system to a single stimulus of deformation ε = 0.75 and duration td = 1.0 s shows that the dendritic and soma action potentials of maximum amplitude ϕd = 81.3 mV and ϕ2 = 42.5 mV are generated at the mechanoreceptor and primary sensory neuron, respectively. Discharge of the primary sensory neuron precedes electrical activity of the secondary sensory neuron. The time delay between the two coupled events is approximately 0.5 s. The neuron responds by firing a train of pulses of amplitude ϕ3 = 52.1 mV at a constant frequency ν = 10 Hz. Its duration is 1.4 s. The observed pattern resembles a regular bursting mode of electrical activity. Periodic stimulation with four stretch impulses, ε = 0.75, td = 0.5 s, following at a frequency ν = 0.36 Hz, causes multiple electical responses in the elements of the afferent pathway (Fig. 5.5). Thus, the dendritic and soma action potentials of amplitude ϕd = 76.8 mV and ϕ2 = 42.5 mV are being generated at a frequency which coincides with the frequency of excitation. The secondary sensory neuron fires a regular train of spikes in response to the first discharge of the primary sensory neuron. However, the second and third trains are delayed compared with discharges of the primary neuron. They start firing when the primary sensory neuron is already exhibiting the phase of slow after-hyperpolarization: ϕ2 ≤ −60 mV. The end of the third train of pulses overlaps with the beginning of the fourth impulse in the primary neuron. The secondary neuron does not respond to the last action potential and remains silent. Increase in the frequency of periodic stimulation ν = 0.75 Hz has little effect on the electrical pattern of dendritic potentials (Fig. 5.6). The primary sensory neuron responds to the first two excitations. The time delay between discharges of the second dendritic potential and the second soma action potential of the primary sensory neuron is reduced to 0.01 s. They fire practically simultaneously. The neuron fails to generate the third potential while the fourth discharge appears as a normal event. A somewhat surprising result is the behavior of the secondary sensory neuron. It produces only two identical trains of spikes, ϕ3 = 52.1 mV, ν = 10 Hz, duration 1.4 s, separated by a long period ∼5.6 s of hyperpolarization.

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Fig. 5.5 The effect of periodic stimulation on the electrical behavior of neurons.

5.3.5 Effect of Iberiotoxin and CHTX Iberiotoxin and CHTX are specific inhibitors of high conductance Ca2+ activated K+ channels. Their effects on electrical behavior of the afferent

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Fig. 5.6 Increase in the frequency of periodic stimulus disrupts regular bursting of the primary sensory neuron and significantly suppresses the firing rate of the motor neuron.

pathway were simulated assuming gCa−K = 0. (Hereafter, the mechanoreceptor field was stimulated by a single stretch ε = 0.75, td = 2.0 s, unless otherwise stated.) The results of calculation show that treatment with CHTX and iberiotoxin changes dramatically the pattern of electrical activity of

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Fig. 5.7 The effects of specific inhibitors of high conductance Ca2+ -activated K+ channels, iberiotoxin and charybdotoxin.

both sensory neurons (Fig. 5.7). The amplitude and frequency of soma action potentials of the primary sensory neuron increase: max ϕ2 = 50 mV, ν = 25 Hz while the slow after-hyperpolarization phase is blocked. The level of the reversal potential is raised to −11.2 mV. The secondary sensory neuron converts into regular beating mode and fires spikes of constant amplitude max ϕ3 = 52.5 mV at a frequency of ν = 14 Hz.

5.3.6 Effect of ω-CgTX ω-CgTX is a selective, long-lasting blocker of N-type Ca2+ channels. With the model we studied its action by assuming gCa(i) = 0 (i = 2, 3). Application of

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ω-CgTX completely blocks electrical activity at the level of the soma of the primary sensory neuron. It remains hyperpolarized throughout and, as a result, no action potentials are generated by the second sensory neuron.

5.3.7 Effect of TTX The mechanism of action of TTX is a 10-fold decrease in the permeability of Na+ and K+ channels. It is also suggested that in neuronal cells treated with TTX, sodium ions can enter the cell via voltage-dependent Ca2+ channels. During the numerical experiments we assumed that: g∗Na = 144 m−1 , gNa(2) = 10, gNa(3) = 0.8 (mSm/cm−2 ), g∗K = 14.4 m−1 , gK = 0.3, gCa(2) = 0.24, gCa(3) = 0.4 (mSm/cm−2 ). In this case it was of particular interest to analyze the influence of the intensity of mechanical stimuli on the electrical response in the sensory pathway. Therefore, the mechanoreceptors were excited by single stretches: ε = 0.75, td = 2.0 s and ε = 0.6, td = 1.0 s. According to the results obtained, application of TTX does not abolish electrical activity at the levels of mechanoreceptors and primary sensory neuron. Dendritic potentials of amplitude ϕd = 84 mV are generated in both cases. The amplitude and pattern of the soma action potential ϕ2 are significantly altered. Thus, max ϕ2 = 20 mV, max ϕ2 = 2.5 mV are resistered and the phase of slow after-hyperpolarization is completely blocked. The secondary sensory neuron appears to be very sensitive to TTX because it remains unexcited throughout.

5.3.8 Effect of Purinoceptor Agonists There is strong pharmacological evidence that two distinct inhibitory adenosine subtype receptors, namely A1 and A2 , are present on the somas of AH-type myenteric neurons. We studied the effects of A1 receptor agonist, CCPA, the A2 receptor agonist, CGS 21680, and the mixed A1 –A2 agonist, NECA. One of the possible mechanisms of their action is a simultaneous decrease in input resistance by 40–50% of the normal value, and an increase in conductance of Ca2+ -activated K+ channels. Hence, their action was simulated by varying the following parameters: input resistance for the primary sensory neuron was Rp = 42.5 M, and the parameter of permeability gCa−K ranged from 0.66 to 0.685 (mSm/cm2 ).

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Fig. 5.8 Treatment of the afferent pathway with A1 –A2 purinoceptor agonists.

The results of calculations demonstrate that application of the above compouds has no effect on electrical activity of the mechanoreceptors. However, they do alter excitability of the primary sensory neuron. The time of discharge is reduced by 50% of the norm and the soma action potentials have amplitudes ϕ2 = 10.1 mV when gCa−K = 0.685 mSm/cm2 , and ϕ2 = 23.7 mV for gCa−K = 0.66 mSm/cm2 (Fig. 5.8). Purinoceptor agonists do not affect the slow after-hyperpolarization, which remains unchanged at 1.6 s. In both cases under consideration, the secondary sensory neuron remains hyperpolarized.

5.3.9 Effect of Protein Kinase C Activator A proposed effect of PDBu on neurons is a simultaneous increase in the permeability of voltage-dependent Ca2+ channels and Ca2+ -activated K+ channels. During numerical simulation the following values were chosen: gCa−K = 0.5, gCa(2) = 0.15, gCa(3) = 0.35 (mSm/cm2 ). The result of treatment with PDBu was that the primary sensory neuron fires characteristic electrical patterns at a fixed frequency, ν = 15 Hz, and an amplitude of 48.3 mV (Fig. 5.9). However, their duration is not constant and varies with time. The secondary sensory neuron responds by regular bursting of action potentials of amplitude ϕ3 = 52.1 mV. The trains of pulses have a frequency

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Fig. 5.9 Effects of application of protein kinase C activator, phorbol-12, 13-dibutyrate. The compound changes the electrical portrait of the motor neuron which displays regular bursting and beating modes.

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of ν = 17 Hz and last 1.8–2.0 s. The neuron fails to produce the action potentials to each second and fifth excitatory stimuli of the primary neuron. This type of activity is repeated within an 11 s period. Further increase in conductance of gCa(2) = 0.2 mSm/cm2 causes repetitive regular bursting of the primary sensory neuron. The patterns of discharges have a constant duration and appear at a constant frequency. The amplitude of spikes is slightly enhanced: max ϕ2 = 49.8 mV is recorded. In response to an increase in gCa(3) = 0.5 mSm/cm2 the secondary sensory neuron displays a beating regime of electrical activity at the maximal frequency ν = 20 Hz. The amplitude of generated action potentials remains constant at 52.1 mV throughout the calculations.

5.3.10 Effect of DPDPE δ-opioid receptors are present at the guinea-pig ileum submucous plexus. Intracellular recordings from the submucous neurons have shown that δ-opioid receptors are coupled to the potassium conductance. Stimulation of δ-receptors with opioid agonists increases gK . We studied the effect of the δ-opioid receptor agonist DPDPE on the electrical behavior of the afferent pathway. The surprising result of simulation is that even a 10-fold increase in gK from the normal value does not cause any significant change in the electrical behavior of the primary and sensory neurons. Deformation initiates the dendritic potential at the mechanoreceptors, which is followed by generation of the soma action potentials ϕ2 and ϕ3 . However, when the type of stimulus is changed to electrical excitation of the free nerve endings, no dendritic potential is generated and the whole afferent pathway remains unexcited.

5.3.11 Remarks The numerical results presented here are consistent with results obtained experimentally from neurons of the submucous and myenteric nerve plexi, i.e. patterns of discharge, shape, amplitude, frequency and duration of spikes generated by the AH-type and secondary neuron. With the model proposed, we are able to calculate electrical responses at the level of mechanoreceptors which cannot be recorded experimentally. This puts mathematical modeling in an advantageous position to unveil “hidden” properties.

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Although the description of activity of each neuron employs general concepts of excitable cell models, the behavior of the coupled neurons is sometimes unexpected. The neurons respond adequately to single external stimuli, i.e. an increase in the amplitude and duration of applied distortion results in greater excitation of the mechanoreceptors and the somas of sensory neurons. The sequence of electrical discharges within the pathway is also consistent with experimental obsevations: the dendritic potential precedes soma action potential generation at the primary sensory neuron and the later one triggers discharge of the secondary sensory neuron. However, the response of the system to periodic stimulation is unusual. The electrical behavior of the primary and secondary sensory neurons depends greatly on the frequency of applied stimuli. The neurons fail to discharge with each applied stimulus, especially when the frequency of indentation is ν ≥ 0.75 Hz. A possible explanation for the observed phenomenon lies in the electrical properties of the sensory neurons. Thus, the prolonged after-hyperpolarization in AH neurons limits their ability to maintain high-frequency bursting with repeated stimuli, which reflects the electrical activity of secondary sensory neurons. However, it may not be the only reason. More work remains to be done to elucidate the mechanism of post-stimulus depression. Although the soma action potentials of the primary sensory neuron display the exact electrical behavior of AH-type neurons observed experimentally, to verify the model further we simulated the pharmacological response of the neuron to CHTX, iberiotoxin, TTX and ω-CgTX. Computer simulation of the application of CHTX and iberiotoxin demonstrates blocking of the slow after-hyperpolarization, a result which is consistent with experimental observations. Such a response is a distinguishing feature between AH- and S-type neurons. An interesting output was recorded from the secondary sensory neuron, which fired a continuous train of spikes. Such a change is due to the blocking of Ca2+ -dependent K+ channels, which results in a decrease in the hyperpolarizing potassium current and enhancement of excitability. We cannot provide an experimental justification for the numerical results, because such data are not available. In light of this, it might be interesting to carry out experiments to record electrical activity directly from the somas of the primary and secondary sensory neurons after treatment with the above toxins.

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The calculations demonstrate that the primary sensory neuron is “insensitive” to TTX, which is another pharmacological characteristic of the AH-type neuron, while electrical activity of the secondary sensory neuron is completely abolished. The effect of ω-CgTX on the afferent pathway is not unexpected. The block of inward calcium current results in the suppression of excitability of both neurons. Inhibitory adenosine receptors, which are found on myenteric neurons, produce cellular actions through a number of different effector systems. In neurons, one effect of increasing K+ conductance is to decrease the influx of Ca2+ ions via voltage-dependent Ca2+ channels. This could be achieved either by hyperpolarizing the membrane potential or by direct activation of G proteins, which are inhibitory to Ca2+ channels. It was suggested that A1,2 receptors may be linked to an adenosine 3 ,5 -cyclicmonophosphate (AMP)dependent pathway, leading to an increase in Ca2+ -dependent K+ conductance and, as a result, enhancement of the after-hyperpolarizing potential. The analysis of the two proposed mechanisms of action of adenosine receptor agonists shows that a simultaneous increase in gK and decrease in gCa conductances altered only the amplitude of the soma action potential, ϕ2 = 26.47 mV. Further changes in the above parameters had no significant effect on excitability of the sensory neurons. However, when we followed the second hypothesis, i.e. an increase in gCa−K(i) and a decrease in the input resistance of the primary sensory neuron, the results obtained agree with experimental observations: the excitability of both neurons is suppressed and hyperpolarization effects are more significant. From these findings one can postulate that the most likely mechanism of action of purinoceptor agonists is the increase in Ca2+ -dependent K+ conductance and the decrease in input resistance of the somas of neurons. There is a general belief that second messenger systems in neurons serve to regulate their electrical behavior. This study demonstrates that a protein kinase C activator powerfully increases the excitability of sensory neurons via the increase in permeability of voltage-dependent Ca2+ channels and Ca2+ dependent K+ channels. These results are consistent with some experimental observations from sympathetic nerve terminals of guinea pigs. Three major types of opioid receptors are now recognized: µ, κ and δ. Neurons in the submucous plexus contain mainly δ opioid receptors. Based on experimental obsevations of modulation of peristalsis by exogenous and

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endogenous opioids acting at µ, κ, and δ opioid receptors, it was suggested that δ agonists act primarily on sensory neurons or interneurons. Measurements of ionic currents under patch-clamp from neurons of the guinea pig submucous plexus revealed that application of the δ-opioid receptor agonist DPDPE increased the conductance of an inwardly rectifying potassium current and strongly hyperpolarized the membrane of the cell, a result which agrees with numerical simulations. Interestingly, the effect of DPDPE depends on the type of stimulus, i.e. mechanical deformation or electrical excitation. Thus, DPDPE had no effect on the electrical activity of the afferent pathway when the mucosa of the gut was stretched. An explanation for this lies in the assumptions of the model for simulation of the receptive field. Parameter ε(t) appears as the main indentation factor in the activation constant of Na+ channels at free nerve endings. Although precise the mechanisms of the mechanically induced transitions of ionic channels from the closed configuration into the open state have not yet been established, kinetic analyses of single-channel events have shown that stretch increases the relative probability of opening of stretch-activated channels, which is largely accounted for by: decreases in the duration of a long closed time, and/or activation of “a cytoprotein which controls a second messenger, thereby up-regulating the channel.” Measurements of permeability of stretch-activated channels have demonstrated that activation of less that 1% of the channels could support physiologically significant influxes of ions and could explain the phenomenon of stretch-induced contraction, and activation of voltage-dependent Na+ and Ca2+ channels, etc. However, when the Na+ channel remained under-formed and the mechanoreceptors were excited by an electrical stimulus, application of DPDPE had a strong hyperpolarizing effect at all levels of the afferent pathway. No dendritic and soma action potentials were recorded.

5.4 ENTERAL SYMPATHETIC COMMUNICATION 5.4.1 Introduction Here we are concerned with the mathematical formulation of a model of a smooth muscle fiber which is under the control of a neuronal ganglion. We shall analyze motility patterns generated by the model in response to

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mechanical deformation. Although the reader may find some repetition from previous chapters, we think that it is necessary for a better understanding of the concepts and the processes linking different constructive elements, e.g. primary and motor neurons, smooth muscle syncytium, into a single structure.

5.4.2 Mathematical Model Consider a finite number of smooth muscle fibers arranged in a sequence. Assume that the fibers are interconnected by gating mechanisms. The principle hypotheses embodied in the model are: i) The muscle fiber is a dynamic self-regulatory myogenic medium; its myoelectrical activity is under the control of a ganglion, which maintains the stability of electrical patterns (the amplitude, duration and frequency of slow waves and spike activity). The ganglion consists of a series of neural circuits morphologically identified as the effector pathway composed of interstitial cells of Cajal, cholinergic and adrenergic neurons interconnected by way of an axo-axonal synapse, and the afferent pathway made of the primary and motor neurons connected in sequence through an axo-somatic synapse. ii) Propagation of the electrical impulse along the unmyelinated axons of cholinergic and adrenergic neurons is described by the modified Hodgkin–Huxley equations. iii) Synapses are modeled as a null-dimensional three-compartment open pharmaco-kinetic model (the presynaptic terminal, synaptic cleft and postsynaptic membrane). All chemical reactions of neurotransmitter transformation are adequately described by first order Michaelis– Menten kinetics. iv) The dynamics of nerve-pulse transmission in a cholinergic synapse includes the following electrochemical reactions: activation of Ca2+ influx into the presynaptic terminal through voltage-gated channels; acetylcholine release from the vesicles; diffusion of the ACh free fraction into the synaptic cleft; ACh binding with receptors located on the postsynaptic membrane; utilization of unreacted neurotransmitter by the acetylcholinesterase enzyme; and excitatory postsynaptic potential generation.

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v) The electrochemical processes at the inhibitory axo-axonal adrenergic synapse include: activation of Ca2+ influx into the presynaptic terminal; extrusion of the fraction of noradrenaline from free releasable stores into the synaptic cleft; NA binding with the α1 -adrenoceptors on the postsynaptic membrane; NA binding with the α2 -adrenoceptors on the presynaptic membrane; removal of NA by uptake-1 and uptake-2 mechanisms where it is degraded by catechol-O-methyltransferase enzyme; generation of the inhibitory postsynaptic potential. vi) The electrical activity of the myogenic syncytium of each fiber, either slow wave or bursting, represents the integrated function of ion channels, presumably: voltage-dependent Ca2+ channels of L- and T-types, Ca2+ -activated K+ channels and potential sensitive K+ channels, and leak Cl+ channels. The properties of L-type voltage-dependent Ca2+ channels are modulated by the cholinergic neuron. This effect is assumed to be mainly chronotropic with an increase in the time of permeability for calcium ions. vii) Excitation-contraction coupling assumes that the “internal” Ca2+ , which enters the cell during the period of excitation, activates the contractile protein system. The force-intracellular Ca2+ activity relationship is an approximation of the experimental data. The governing system of equations that describes the self-regulatory myoelectrical activity of a smooth muscle fiber includes: (2.1)–(2.13), (4.1)–(4.6), (4.10), (4.12), (4.13), (4.32)–(4.34), (5.2)–(5.11). We simulated independently both the mechanical responses due to excitation of the afferent pathway and those due to excitation of the efferent pathway. Excitation by the deformation of mechanoreceptors in the afferent pathway involves the following sequence of events: i) deformation of the free nerve endings of the mechanoreceptors; ii) action potential propagation along the unmyelinated fiber of the primary sensory neuron; iii) action potential generation at the soma of the primary sensory neuron and propagation of the wave of depolarization along the nerve axon towards the motor neuron; iv) electrochemical coupling (acetylcholine release, diffusion and binding with choline receptors) at the axo-dendritic synapse and generation of the EPSP;

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v) generation of the action potential at the soma of the motor neuron and its propagation along the nerve axon towards the cholinergic synapse located on the smooth muscle membrane; vi) initiation of chemical mechanisms of acetylcholine release and the generation of an excitatory postsynaptic potential; vii) activation of slow L-type Ca2+ channels and depolarization of the smooth muscle membrane; viii) active force generation as a result of an increase in intracellular [Ca2+ ]; ix) calculation of the deformation in terms of the force being generated, which leads to excitation of the free nerve endings of the mechanoreceptors. Excitation of the locus by way of discharge of the cholinergic neuron in the efferent pathway involves the following events: i) discharge of the soma of the cholinergic neuron and action potential propagation along the unmyelinated axon; ii) electrochemical coupling (initiation of chemical mechanisms of acetylcholine release and the generation of an excitatory postsynaptic potential) at the L-type Ca2+ channels; iii) depolarization of the smooth muscle membrane; iv) active force generation as a result of an increase in intracellular [Ca2+ ]; v) steps (i)–(ix) are repeated from the series in the preceding paragraph. The boundary conditions assume that the excitation of the system is due to the deformation of the afferent pathway by an externally applied mechanical stimulus, or the excitation of the soma of the cholinergic neuron of the efferent pathway by a discharge of the electric impulse. The system was integrated numerically using a combination of the Evans–Abdullah and fourth-order Runge–Kutta methods.

5.4.3 Effect of a Single Deformation A single stretch of the fiber of the intensity ε = 0.75 and td = 1.5 s evokes a single receptor potential at the dendrites of the mechanoreceptors of maximum amplitude 82.5 mV and duration 2.3 ms (Fig. 5.10). A characteristic feature of the electrical pattern is the long-lasting “step” with an amplitude

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Fig. 5.10 Response of the whole system to a single mechanical stimulus.

of approximately 45 mV. The dendritic potential causes the development of an action potential with an amplitude of 69 mV. It propagates along the unmyelinated axon of the primary sensory neuron towards its soma where it initiates an excitatory membrane current, Iext(2) , with the generation of a

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Fig. 5.10 (Continued )

single action potential of average amplitude ϕ2 = 81.3 mV. The pattern of discharges resembles a bursting chaos type of electrical activity with the frequency ν = 8–10 Hz. The duration of spikes being produced is ∼2.6 ms. A characteristic feature of the pattern is the presence of a prominent “shoulder” which is followed by a long-lasting slow after-hyperpolarization of 1.6 s. Discharge of the primary sensory neuron precedes electrical activity of the motor neuron. The action potential elicited at the soma of the primary sensory neuron propagates along the unmyelinated axon and reaches the axo-somatic synapse. The depolarization of the presynaptic membrane activates a short-term influx of calcium into the terminal. The concentration of cytosolic Ca2+ quickly reaches a maximum of 19.4 mM. Some of the ions are

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immediately absorbed by the buffer system while others diffuse towards the vesicle. They bind with the active centers and initiate AChv release. During the whole cycle about 10% of AChv is released. With the achievement of an AChf concentration of 10.6 mM, acetylcholine release into the synaptic cleft begins. According to the calculated data, max[AChc ] = 5.38 mM diffuses from the presynaptic terminal. The main part of AChc reaches the postsynaptic membrane and reacts with the choline receptors on the soma of the secondary sensory neuron. The development of (ACh–R)-complex results in the generation of fast EPSP, which increases as a step function due to the rise of [ACh–R] and achieves its maximum, 87.1 mV, in 0.25 ms. This level of EPSP is sufficient to excite the motor neuron. It discharges a single train of pulses of amplitude ϕ3 = 102.1 mV at a constant frequency ν = 10 Hz and duration 1.4 s. The observed pattern resembles a regular bursting mode of electrical activity. The time delay between the two coupled events (the end of the firing of the primary and the beginning of the firing of the motor neuron) is ∼0.6 s. Both the dynamics of propagation of the wave of excitation along the unmyelinated axon of the motor neuron and the dynamics of electrochemical coupling at the L-type Ca2+ channels on the smooth muscle membrane are essentially the same (quantitatively and qualitatively) as the dynamics of the electrical processes described above. The generated EPSP of amplitude 87.1 mV exceeds the threshold value for the activation of the permeability of L-type Ca2+ channels. The above changes cause cyclic transitory changes in the observed myoelectrical pattern: (i) slow wave mode transforms to bursting chaos, (ii) bursting chaos transforms to regular bursting with the generation of spikes on the crests of slow waves, and (iii) regular bursting converts back to bursting chaos and the slow wave mode. Thus, for T g˜Ca = 0.51 mSm/cm2 , the system displays periodic activity with the generation of waves of depolarization of amplitude 22.5 mV and frequency 0.44 Hz. The wave of depolarization achieves its maximum in 1.6 s, followed by a short plateau of 0.4 s, and finally slowly decreases to the resting value, ϕ = −49 mV. T T As g˜Ca increases in time, 0.51 ≤g˜Ca ≤ 0.59 mSm/cm2 , the regular slow wave pattern changes to bursting chaos discharges. The frequency and the amplitude of slow waves remain unchanged. Fast action potentials are generated at the top of the plateau at a frequency of ν = 19.5 Hz. Their firing level is high at the beginning, ϕ = 23 mV, and decreases towards the end of the

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burst, ϕ = 8 mV. For 6.5 < t ≤ 9.5 s, the pattern displays repetitive regular bursting. The burst amplitude, burst duration and number of spikes per burst all increase. The action potentials have a maximum amplitude of 72 mV and oscillate at a maximum frequency of 21 Hz. With the decrease in T g˜Ca ≤ 0.65 mSm/cm2 , the system transforms to a short-term chaotic burst mode of duration 2 s. A gradual increase in the concentration of intracellular calcium ions, mainly due to the influx of ions through the activated slow Ca2+ channels, leads to an increase in the permeability of Ca2+ channels. Oscillations of internal calcium are delayed compared to the dynamics of the depolarization of the muscle membrane. The rate of influx/outflux of Ca2+ ions is ˜ 2+ ] = 0.46 µM, 0.06 µM/s. Maximum internal calcium concentration, [Ca is achieved just before the slow wave reaches its resting value and each minimum is observed when the level of depolarization is near its maximum. This pattern of behavior changes with the appearance of the first irregular spikes followed by bursts. During this period, the concentration of internal calcium smoothly increases concomitantly with bursting and reaches its maximum just after the end of bursting. It continues to decrease to its minimal value until a new burst pattern begins and the process repeats. With the return of the burst chaotic oscillatory mode, the pattern of oscillations of [Ca2+ ] again becomes out of phase with the electrical pattern. The amplitude of oscillations, 0.049 µM, changes insignificantly during the cycle of electromechanical transformations. Part of the free intracellular Ca2+ reaches the contractile protein system and, as a result, rhythmic contractions of the fiber occur. The contractions are concomitant in phase and time with the dynamics of calcium oscillations. The rise and relaxation of tension coincide with the increase and decrease in ˜ 2+ ]. The contractions have an average amplitude 4.8 g, and the calculated [Ca velocity of twitch is 5.0 g·s. The maximum force generated by the smooth muscle syncytium is 14.4 g. As a response to the above changes in force, deformations of different amplitude are developed in the smooth muscle fiber under consideration. Thus, at t = 5.75 s a threshold wave of deformation, ε = 0.3, sufficient to excite the mechanoreceptors, is generated. The following waves of amplitude ε = 0.2 and 0.3 are recorded at t = 14.1 s and 9.5 s, respectively. Their occurrence coincides in time with the firing of the fast adapting mechanoreceptors

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of the afferent pathway. The spikes of the amplitudes ϕ0d = 40.3 mV, 16.1 mV and 40.3 mV are generated at the mechanoreceptors. This level of depolarization is sufficient to excite the soma of the primary sensory neuron, which fires three bursts at t = 7 s, 15.4 s and 20 s. The dynamics of the electrical patterns quantitatively and qualitatively resemble the pattern of the first spike observed at the beginning of the process. The generation of the EPSP at the soma of the motor neuron causes the generation of long-lasting ∼2 s tonic-type discharges of a high frequency ν = 10 Hz and amplitude ϕ3 = 102.1 mV. Within the recording time interval, 0 ≤ t ≤ 24 s, only three EPSP’s are generated on the L-type Ca2+ channels of the f smooth muscle membrane. Periodic alterations in g˜Ca cause regular periodic changes in the myoelectrical pattern of the muscle fiber. They produce bursting chaos discharges, and repetitive regular bursting of different frequency and amplitude. An increase in the intracellular concentration of Ca2+ causes subsequent changes in the tension. The amplitude and duration of contractions vary with time and agree with the dynamics of the myoelectrical behavior of the muscle.

5.4.4 Effect of a Periodic Deformation Periodic stimulation of the muscle with five stretch impulses, ε = 0.8, td = 0.5 s within the time interval 0 ≤ t ≤ 8.5 s at a frequency ν = 0.75 Hz causes multiple electrical responses in the afferent pathway. The characteristic dendritic action potentials of amplitude ϕd = 82.5 mV are generated at a frequency that coincides with the frequency of mechanoreceptor excitation. However, the primary sensory neuron responds only to the first three dendritic action potentials with three bursts of spikes of maximum amplitude ϕ2 = 81.3 mV (Fig. 5.11). It fails to respond to the fourth excitatory stimulus while the fourth discharge appears as a normal event and coincides with the fifth discharge at the mechanoreceptors. (The dynamics of the processes described below is similar to the dynamics of the electrical processes described in the previous section. Hereafter we shall point out only qualitative differences of the process. Quantitative information will be provided only where it differs from that in the analysis above.) Three EPSP’s of constant amplitude 87.1 mV and duration 0.5 s are recorded from

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Fig. 5.11 The effect of high frequency periodic stimulation on the electrical behavior of neurons.

the soma of the secondary sensory neuron. As a result the motor neuron produces three bursts of high amplitude spikes, ϕ3 = 102.1 mV, at a frequency of 10 Hz. The three resultant excitatory postsynaptic potentials activate the L-type Ca2+ channels to generate a long-lasting period

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Fig. 5.11 (Continued )

of myoelectrical activity in the smooth muscle fiber. During this period repetitive bursts of maximum frequency ν = 21 Hz are observed. The following short interval corresponds to irregular bursting discharges, which are generated at a fluctuating frequency. The spikes produced maintain excitatory activity of the system, which again converts into a bursting mode of electrical activity and fires high amplitude spikes at ν = 21 Hz. The above changes increase the intracellular calcium ion influx and, as a result of the activation of the contractile protein system, more prolonged periodic contractions, with an active force of amplitude 5.2 g, are produced.

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Beginning from the time t = 8.5 s the system demonstrates selfexcitatory behavior. Thus, sustained regular phasic contractions cause the generation of three distinct waves of deformation. The maximum ε = 0.3, td  0.7 s is sufficient to initiate a discharge of the free nerve endings with the generation of an action potential ϕ0d = 30 mV. This excitation is enough to sustain electrical activity at the afferent pathway neurons. As a result, the smooth muscle responds with a burst of high amplitude membrane potentials, ϕ = 72 mV, which oscillate at a frequency ν = 20 Hz. This level of myoelectrical activity is enough to keep the whole system excited.

5.4.5 Effect of Intermittent Deformation A 9-s regime of stimulation of a smooth muscle fiber with external impulses of deformation of variable intensity 0.3 ≤ ε ≤ 0.8, 0.25 ≤ td ≤ 0.75 s and frequency 0.5 < ν < 1 Hz demonstrates the intrinsic adaptive properties of the system (Fig. 5.12). The free nerve endings respond to low frequency excitation with the generation of action potentials of maximum amplitude, ϕd = 82.5 mV, and characteristic configuration. A simultaneous increase in frequency, ν = 1 Hz, and decrease in amplitude of stretches, ε = 0.5, distorts the shape of the action potential produced and reduces the amplitude of the spikes to 50 mV. The discharges of the mechanoreceptors coincide with the frequency of the indentations. A further decrease in the rate of the stimulus and increase in its duration change the pattern of electrical response to a prolonged depolarization at a constant level, ϕd = 32 mV. As a response to the six action potentials generated at the level of the free nerve endings, the soma of the primary sensory neuron discharges four high-amplitude characteristic spikes of max ϕ2 = 81.3 mV (Fig. 5.12). The number of EPSP’s recorded at the soma of the motor neuron is reduced to two and, as a result, only two long-lasting discharges are seen. Their quantitative characteristics do not differ from those discussed in the previous paragraphs. The myoelectrical activity demonstrates cyclic changes in the pattern of discharges from slow wave electrical activity to bursting chaos followed by regular bursts. The periodic shift between regular bursting activity and irregular bursts is characteristic of the self-excitatory property of the smooth muscle fiber. The frequency of recorded oscillations varies from 10 to 21 Hz.

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Fig. 5.12 The result of aperiodic stimulation on the response of the afferent pathway neurons.

Calcium ion influx through the activated Ca2+ channels in the smooth muscle membrane results in a phasic tension response. The muscle generates active force, max Ta = 9 g, during the regular bursting mode of myoelectrical activity. During the irregular bursts the amplitude of periodic contractions is lower and varies within 8.5 < T a < 13.2 (g).

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Fig. 5.12 (Continued )

The intermittent regime of excitation does not provide sustained selfexcitation of the muscle fiber. The maximum force of contraction is sufficient to cause a single wave of deformation, ε = 0.32, which is strong enough to stimulate the sensory elements of the afferent pathway with excitation of the fast calcium channels in the smooth muscle membrane. However, the muscle fiber fails to maintain a self-excitatory regime and it returns to an unexcited silent phase with the generation only of characteristic slow waves.

5.4.6 Remarks The model proposed is based on electrophysiological data of smooth muscle cells and extends the formalism proposed by Plant (1978) and Rinzel and

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Lee (1987) for excitable neurons. The model assumption is that the activity of voltage-gated L-type Ca2+ -channels is modulated by cholinergic neurons, which is consistent with physiological evidence. Their main effect is known to be chronotropic. Therefore we included the electrochemical processes of the nerve-pulse transmission via a cholinergic synapse. The results of numerical simulations predict the generation of oscillations of the membrane potential of amplitude 22.5 mV and frequency 0.44 Hz, which are in good quantitative agreement with experimental observations in which amplitude varied from 20 to 27 mV and the frequency ranged between 0.3–1.1 Hz. Before we proceed further with the analysis of the numerical results, we must point out the following. Although we have simulated real conditions, i.e. the electromechanical response of a muscle fiber of the small bowel to mechanical deformation of different intensities, our numerical results cannot be fully compared to published experimental data because it is technically impossible to record simultaneously in tissue the electrochemical and electromechanical processes from the neuronal elements of the enteric nervous system and the smooth muscle syncytium. Only the integrative model approach adopted here allows us to display concurrently the complex sequence of events and to study their dynamic interactions. Therefore, we can only compare the behavior of separate structural elements of the system to existing experimental results. The model is able to simulate accurately the depolarization, the transmembrane Ca2+ influx and the consequent contractions of the smooth muscle syncytium. The amount of Ca2+ influx through the receptor-linked L-type Ca2+ -channels fills storage sites and activates contractile filaments without production of action potentials. According to the results obtained ˜ 2+ slow waves generate an active force of amplitude 4.8 g. Cytosolic Ca reaches its maximum (0.45 µM) in 0.9 s after a maximum depolarization has been reached. Ozaki et al. (1991) working on the gastric smooth muscle recorded 0.8 ± 0.1 (s) delay. According to the same authors, force development occurred when cytosolic calcium reached ∼30% of the peak. In our calculations, the dynamics of tension development is concomitant in ˜ 2+ . An explanation for such a response phase and time with fluctuations of Ca ˜ 2+ . is in our model of the syncytium and in the initial stable-state value for Ca By adjusting the parameters of the model, it is possible to achieve experimen˜ 2+ and tension curves. Although tally observed phase shifts between the Ca

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the shape of the calculated curves T a (t) are very similar to those recorded experimentally, a quantitative comparison is not possible. An important step towards elucidating the cellular mechanisms of myoelectrical activity was achieved with the employment of intracellular recording techniques. It has been shown that excitatory synaptic transmission mediated by acetylcholine receptors is required for slow wave generation. Thus, application of ACh at a concentration of 0.1 mM causes a rise in frequency with a simultaneous increase in the rate of depolarization and amplitude of slow waves. According to the numerically obtained results a highly active acetylcholine-receptor complex in a concentration of 0.11 mM was formed, causing the generation of an EPSP of amplitude 87.1 mV. As shown above, the model predicts various electrical patterns of a T smooth muscle fiber. The simulations reveal that if g˜Ca is used as a variable parameter, the muscle fiber changes its patterns of firing in a cycle very similar to the three phases of the MMC: a quiescent phase when only slow waves are recorded; a period of bursting chaos or irregular activity, regular bursting, with the generation of spikes of a high amplitude; and back again to irregular activity followed by the quiescent phase again. Therefore, we conclude, based on the agreement of theoretical and experimental data, that the mechanism of the transitory changes of the myoelectrical activity of the small bowel is a change in the permeability of the voltage-gated L-type Ca2+ channel in response to the release of ACh from the effector neuron. No other supporting experimental evidence can be given, because simultaneous recording of muscle contraction and the L-type Ca2+ channel activity have not been done. One could argue that a variety of perturbations, corresponding to different pharmacological interventions in the system, can reproduce the MMC patterns. The numerical results are consistent with results obtained experimentally from neurons of the submucous and myenteric nerve plexi. A comparison of the theoretical results obtained with experimental data published in physiological media, i.e. patterns of discharges, shape, amplitude, frequency and duration of spikes generated by the AH-type and secondary neuron, demonstrates qualitative and quantitative agreement. The recordings of electrophysiological activity from the somas of the primary sensory and secondary myenteric neuron demonstrate that the amplitude of discharges varies between 75 and 85–90 mV. Our numerical results of the firing

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activity of the primary and secondary neurons, respectively, are 81.3 mV and 102.1 mV. The average experimentally observed frequency of firing of the secondary neuron is 10–15 Hz, which is consistent with the results of our simulations, 10 Hz. In the polymodal afferent fibers, the mechanisms of mechanosensitivity and chemosensitivity and the neurotransmitters involved remain unknown. The role of bradykinin, tachykinins, substance P, serotonin, prostaglandins and adrenaline have been studied extensively, but it is not clear how the free nerve endings distinguish and encode complex stimuli. Therefore, in the present approach we consider only mechanical deformation as the excitatory stimulus to the muscle fiber. Electrophysiological studies demonstrate that afferent fibers adapt rapidly to a mechanical stimulus, giving an on and off response, respectively, when a mechanical stimulus is applied and removed. It has also been proposed that mechanoreceptors encode faithfully the increase and decrease in active and passive tension during peristalsis by increasing the permeability of the sodium channels. Although the description of activity of each neuron employs general concepts of excitable cell models, the behavior of the coupled neurons is sometimes unexpected. The neurons respond adequately to single external stimuli. The sequence of electrical discharges within the pathway is also consistent with experimental observations. The dendritic potential precedes action potential generation in the soma of the primary sensory neuron and the discharge of the primary neuron triggers discharge of the motor neuron. However, the response of the system to periodic stimulation is unusual. The electrical behavior of the primary and secondary sensory neurons depends greatly on the frequency of applied stimuli. The neurons fail to discharge with each applied stimulus, especially when the frequency of deformation is ν  0.75 Hz. It is worth noting that the results correspond with intracellular recordings of electrical activity from the guinea-pig ileum. In the above experiments, sensory reflexes were elicited by distending the gut, a condition that we have modeled. A possible explanation for the observed phenomenon lies in the electrical properties of the sensory neurons. Thus, the prolonged after-hyperpolarization in AH neurons limits their ability to maintain high-frequency bursting with repeated stimuli, which reflects the electrical activity of motor neurons.

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5.5 A PLANAR NEURONAL NETWORK Consider a neuronal plexus that consists of neurons arranged in an overlapping series of functional modules connected by a horizontal multibranched system of the dendrites and interneurons (Fig. 5.13). The following connections among neurons within the module exist: i) direct — connects n and the adjacent (n + 1) neuron; ii) bypass — passes signal from n to (n + 2) neuron; iii) divergent — sends signals from (n + i) to (n + i) (i = 0, 1, 2, . . .) interneurons; iv) convergent — (n + i) neuron receives signals from (n + i) interneurons; and v) backward inhibitory — provide inhibition in the neuronal network. Such histoarchitectonics allows a pacemaker cell to modulate the rhythmical work of a great number of dispersed units. The discharge of the pacemaker

Fig. 5.13 A model of the enteric plexus as a planar neuronal network.

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causes the development of a wave of excitation that propagates along the network. In addition to the general assumptions related to electropharmacology of the chohinergic and adrenergic neurons, we further assume that the EPSP’s from different inputs are summed linearly on the soma of the neuron. The summatory excitatory signal is weighted: if the resulting value exceeds a threshold value, a new spike is generated, otherwise the neuron remains unexcited. The value of “weight” (ω) was adjusted during the numerical calculations to fit this model to experimentally observed results. In simulations we used ω = 0.5. In response to a discharge of the pacemaker cell the action potential, ϕ∗ , of amplitude 69.6 mV was generated. It propagated along the pathways at a constant velocity and at 69.5 ms reached the nerve Terminal that is in contact with the neighboring interneuron. Here the depolarization wave initiated the chemical mechanisms of transsynaptic transmission and a fast EPSP was generated on the soma of the interneuron. However, this level of depolarization was insufficient to generate a new impulse and to maintain the process of the propagation of excitation in the network. The interneuron was changed from the silent into the trigger state. A new spike developed only after receiving additional excitatory inputs from the neighboring neurons along the convergent pathways at t = 174.9 ms. Analogous changes were observed in other spatially distributed ganglia. The impulse from the pacemaker cell reached the neighboring neuron along the bypass pathway at t = 137.1 ms, while the signal from the proximally located neuron arrived at the axosomatic synapse along the direct pathway with a slight delay at t = 139 ms. At t = 139.65 ms a summatory fast EPSP with an amplitude of 87.09 mV occurred, which led to the development of a new spike (Fig. 5.14). Discharge of the inhibitory neuron caused the release of NA at the axosomatic synapse and the generation of a fast IPSP with an amplitude of −78.8 mV. This blocked the backward propagation of the wave of depolarization within the network. For all stages of the dynamic process, the intensity of the propagation of excitation was higher in the axial direction and was due to the structural anisotropy included in the model: the length of convergent and divergent connections was less than the length of direct and bypass pathways.

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Fig. 5.14 The dynamics of the propagation of excitation in the planar neuronal network.

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5.5.1 Effect of Cholinergic and Adrenergic Agonists and Antagonists With this model it was possible to analyze quantitatively and qualitatively the influence of cholinergic and adrenergic agonists/antagonists on the dynamics of nerve-pulse transmission. The results of numerical simulation showed that a decrease in the concentration of extracellular Ca2+ ions to 50% of the norm slightly reduced the amplitude of the EPSP but significantly increased the synaptic transmission time delay; in the case of 90% decrease of external Ca2+ , a blockade of the depolarization wave propagation within the networks was observed. The effect of the action of tetrodotoxin was the blockade of the propagation of excitation in the network. The addition of acetylcholinesterase inhibitor augmented the action of ACh and increased the amplitude of the EPSP without a visible increase in the intensity of the excitation wave propagation. The addition of drugs that inhibit catecholO-methyltransferase caused the augmentation of the action of NA and an increase in the amplitude of the IPSP, with, as a consequence, the blockade of the propagation of excitation.

5.5.2 Effect of Cholinesterase Inhibitors The treatment of the ENS by cholinesterase inhibitors at a concentration of 0.1 mM caused a significant decrease in the active acetylcholinesterase content in the synaptic cleft, which led to an increase in the free fraction of ACh in perfusate: max[AChc ] = 5.42 mM was registered. Consequently 0.124 mM of acetylcholine-receptor complex was developed on a postsynaptic membrane, which induced the generation of an excitatory postsynaptic potential with an amplitude ϕp+ = 89.08 mV. In the case of the addition of cholinergic agonist at a concentration of 0.05 mM, ϕp+ = 89.0 mV was registered. That level of depolarization of somas of the adjacent neurons was enough to initiate a new signal that further propagated within the ENS.

5.5.3 Effect of Cholinergic Antagonists The addition of cholinergic antagonists caused an inactivation of chemical processes on the postsynaptic membrane. According to the numerical results, the addition of cholinergic antagonists at concentrations of 0.13– 0.26 mM caused complete blockade of excitation transmission, ϕp+ = 0. At

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lower concentrations of the drug, 0.026 mM, the decrease in the number of choline receptors available for interaction was observed. Subsequently, the concentration of the ACh–R complex reduced to 0.016 mM. This was enough though to generate a fast EPSP of 69.6 mV on the soma of the second neuron. However, this excitation was not enough to sustain the propagating wave of depolarization along its axon. As a result, the transmission of the wave of excitation was blocked.

5.5.4 Remarks At present, theories describing biological network function are inadequate, much as the knowledge of their working principles remains frustratingly sketchy. The study of “hypothetical” neuronal networks may serve as a basis for understanding qualitatively and quantitatively the behavior of the more efficient real networks whose analysis is impossible. A good model is one that reduces the complexity of the system under consideration significantly while preserving its essential features. The degree of reduction needed clearly depends on the question posed. In our case the primary goal was to construct an explicit model for the dynamics of the transmission of the nerve pulse within the myenteric plexus. The modeling of the ENS as a planar neural network required an innovative approach, namely, to include into consideration the morphological and neurochemical aspects of the function of its basic units, the neurons. The ENS is arranged in a strict hierarchical manner. Even in ganglia considerable divergence and convergence of synaptic pathways exist. The pattern of information flow in the myenteric plexus during the performance of regulatory tasks cannot be easily determined from circuit diagrams constructed from anatomical studies. Neural correlates have to be established that are responsible for sophisticated function. However, even if all connections are identified in a network, the functionally important connections can be difficult to discern. One advantage offered by mathematical modeling is the possibility of examining the intrinsic properties of subunits, both as single neurons and groups, and to estimate their contribution to emergent properties of the real network. It was established that the longitudinal connective pathways play the major role in the process of excitation transmission in the myenteric plexus.

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Delays in synaptic transmission and in the propagation of impulses along axons produce a delay between the input and output signals. Previously, such delays have been modeled by a single parameter, the stochastic mean processing time, while with our model we were able to estimate a time delay from the analysis of the electrochemical mechanisms of synaptic interaction. Pre-modulatory associative mechanisms, which can be either excitatory or inhibitory, are supposed to be the main learning elements of neural networks. Time emerged as an important variable. With the model of the inhibitory circuit we have demonstrated the effect of temporary interaction between the fast IPSP and the excitatory signal. The minimum time lag in the generation of the IPSP at the axo-axonal adrenergic synapse, which had no effect on the dynamics of propagating action potential along the axon of the cholinergic neuron, was ≥ 1.68 ms. Experimental recordings from the neuromuscular synapse obtained by a method of intracellular registration gave 1.55 ms. The decrease in the time lag diminished the amplitude of the spike and significantly shortened it. The results of numerical simulations are similar to the experimental results of interaction between excitatory and inhibitory postsynaptic potentials in the soma of motor neurons. The planar network model proposed describes the spatio-temporal characteristics of the neuronal interactions within the myenteric plexus. Extracellular recordings of the excitation wave propagation in the myenteric plexus revealed the properties of electrical anisotropy, i.e. the nerve pulse propagated without damping in the longitudinal direction over 10 cm, while in the transverse direction only to 4 cm (longitudinal versus transverse ratio of 5:2). These experimental results are in good agreement with our results of numerical simulation from where it can be found that the longitudinal to transverse ratio equals 5:1. A certain quantitative discrepancy between theoretical and experimental findings can be explained by the inconsistency of the modeling of the transverse connections among ganglia. In our model we have considered only one interneuron, as a transverse link between two ganglia, but Yokayama and Ozaki (1978) have commented on the existence of a multiple chain of polysynaptic contacts. No single presynaptic neuron in the ENS is capable of exciting a postsynaptic cell sufficiently to reach the threshold for an action potential. Also the number of neurons that must be activated before transmission emerges is not known, but the data are consistent with discrete rather than global

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or diffusive activation. With the model we suppose that synaptic potentials evoked in the cells are linearly additive both spatially and temporally. This is based on experimental findings that some neurons can show linear summation of presynaptic potential over a wide voltate range while more commonly the relation is highly nonlinear, and the nonlinearity of interaction provides the means for high density information processing for the ENS. However, even with the assumption of linear interaction among neurons and numerical adjustment of some missing parameters, i.e. the weight constant and threshold potential, it was possible to reproduce accurately the effects of synaptic plasticity.

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CHAPTER 6

Multiple Neurotransmission

6.1 CO-TRANSMISSION BY ACETYLCHOLINE AND SEROTONIN 6.1.1 Introduction Until recently, the analysis of drug effects has involved the study of signal and receptor transduction mechanisms in a linear steady-state fashion. Although many biological and clinical effects have been predicted, with some rate of success through linear information transfer, such approaches do not include the complexity of nonlinear dynamic interactions that exist among different signaling pathways. Awareness of the nonlinear correlation of molecular interactions and net biological results is also increasing with a burgeoning understanding of cellular mechanisms of drug actions: receptor coexpression, multiple neurotransmission, non-covalent modifications of the active center, allosteric modulation, and cooperative ligand-protein binding. Recent advances in molecular biology and the increasing ease with which matter can be studied and manipulated on the nanometer scale have provided opportunities for the development of novel, highly effective therapies. Successful identification of drug targets requires an understanding of the intricacies of the biological regulatory network interactions between the key components of cells, organs and systems, and how these interactions change in normal and diseased states. This information does not reside in the genome or even in the individual proteins that genes code for, but rather in the integrative data of how proteins behave in context as they interact with the rest of the relevant cellular machinery to generate function at a macro level.

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There is considerable interest in the electro-pharmacological role of acetylcholine and serotonin (5-HT) as neurotransmitters, the effects of their co-transmission, and multiple receptor expression, in the pathogenesis of various diseases. However, understanding of the cellular mechanisms involved in the intricate pathways of processes responsible for the biological manifestations remains unsatisfactory. First, the combined processes of cotransmission cannot be studied effectively by any of the existing experimental techniques currently available to researchers for determining intrinsic mechanisms. Secondly, traditional in vivo and in vitro experimental approaches fail to provide desired quantitative information about synaptic neurotransmission. Thus, a new technique is needed that amalgamates interdisciplinary data and provides the basis for an integrated, rather than reductionist, analysis of complex biological phenomena.

6.1.2 Biological Background Effects of acetylcholine are mediated by nicotinic and muscarinic receptors. The nACh receptors are non-selective ionotropic receptors, whereas µACh receptors are part of the metabotropic G-protein-coupled receptor family. Activation of nACh receptors results in the generation of an inward calcium current with the production of fast or slow excitatory postsynaptic potentials, respectively. In contrast to the diverse effects of serotonin in the gastrointestinal tract, ACh always has an excitatory effect and enhances myoelectrical activity. The ubiquitous biogenic amine, 5-hydroxytryptamine (serotonin), is present in the neurons of the enteric nervous system, largely in the enterochromaffin (EC) cells. Serotonin acts as a neurotransmitter and a paracrine messenger in the gastrointestinal tract to mediate a wide range of physiological functions. These effects are achieved through activation of 5-HT1 –5-HT7 receptors. A distinct neural receptor, 5-HT3 , belongs to the family of ligand-gated ion channels. Serotonin applied by ionophoresis to a neuron with 5-HT3 receptors causes a short latency, < 100 ms, and a duration, < 2 s, depolarization by invoking a fast inward excitation current. The latter is due to an increase in permeability of calcium, potassium and sodium channels.

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The 5-HT4 receptors belong to a Gαs -protein-coupled family. They are positively linked to adenylyl cyclase in the second messenger signal transduction mechanism. The biological effect of their pharmacological activation is correlated with an increase in permeability of calcium-activated potassium and sodium channels and a decrease in permeability of potassium channels. The 5-HT3 and 5-HT4 receptors, and ACh-receptors are expressed on longitudinal and circular smooth muscle cells as well as the somas and presynaptic nerve terminals of the primary sensory (AH), motor (S), and possibly interneurons (S- and AH-types) of the myenteric nervous plexus. Electrochemical coupling at the neuro-neuronal and neuro-muscular synapses entails the initiation of cascade reactions of the second messenger system and activation of ligand-operated channels. Differences in dynamics of the implicated biochemical pathways, complex allosteric neurotransmitter-receptor interactions, species and regional organ dependence, result in highly variable biological responses. Thus, indefinite responses, i.e. an increase or failure to affect gastrointestinal propulsion in the ileum of mice with an invariable inhibition of the lower intestinal motor activity, are observed after the selective blockade of 5-HT3 receptors by Ondansetron and Alosetron. The selective stimulation of 5-HT4 receptors by mosapride consistently enhances the propulsion of the intraluminal content in the distal colon of rats and mice while prucalopride, a 5-HT4 -receptor agonist, induces motility of the entire colon of the dog and mediates relaxation and inhibition of spontaneous contractions of the colonic circular muscle with concurrent enhancement of longitudinal muscle contractility in men. Administration of a non-selective, strong 5-HT3 - and weak 5-HT4 -receptor agonist, Cisapride, increases motor activity, while the blockade of 5-HT3 and 5-HT4 receptors decreases the frequency of migrating motor complexes in the isolated murine terminal ileum and colon. Application of a mixed 5-HT4 -receptor agonist and weak 5-HT3 receptor antagonist, TS-951, causes a concentration-dependent increase in the amplitude of electrically induced sub-maximal twitch contractions of the longitudinal muscle of the ileum and distal colon of the guinea pig.

6.1.3 Mathematical Model The following experimental set-up has been designed: a one-dimensional model, composed of interconnected smooth muscle cells, and the primary

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Mathematical Modeling and Simulation in Enteric Neurobiology Smooth muscle syncytium

Neuro-muscular synapse Primary sensory neuron Motor neuron Neuro-neuronal synapse 5-HT3 – receptors

Mechanoreceptors

5-HT4 – receptors Mucosa nACh – receptors Enterochromaffin cells

µACh - receptors

Fig. 6.1 Scheme of the one-dimensional model used in numerical experiments. The distribution of serotonergic and cholinergic receptors is as indicated.

sensory and motor neurons (Fig. 6.1). The model of the bursting AH- and S-type neurons is based on the Hodgkin–Huxley formalism and includes activity of the voltage-dependent Na+ , K+ , Ca2+ , Ca2+ –K+ and leak Cl− channels. The dynamics of the propagation of electrical signals along the unmyelinated axons satisfies the classical Hodgkin–Huxley model. The synapse and the drug-receptor interaction models are equivalent to the pharmacodynamic model as described above (see Chap. 4 for details). Description of the electromechanical activity of smooth muscle cells and syncytia includes the intracellular processes of calcium conversion and function of the ligand gated L-type Ca2+ , and voltage-dependent T-type Ca2+ , K+ , Ca2+ -activated K+ and Cl− channels. The general topology of the intercellular arrangements accurately replicates anatomical, histomorphological and physiological data. Thus, a onedimensional model mimics the following sequence of events: i) deformation of free nerve endings of mechanoreceptors by mechanical stimuli of a known intensity and duration;

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Multiple Neurotransmission 203

ii) generation of action potentials (AP) and their propagation along the unmyelinated fiber toward the soma of the AH-neuron; iii) the action potential generation at the soma of the AH-neuron and the propagation of the wave of depolarization along the nerve axon toward the soma of the S-neuron; iv) electrochemical coupling at the axo-dendritic synapse on the soma of the S-neuron and the generation of fEPSP; v) discharge of the soma of the S-neuron and the propagation of AP along the nerve axon toward the neuro-muscular synapse; vi) electrochemical coupling at the synapse and the generation of fEPSP; vii) activation of L-type Ca2+ channels of the smooth muscle membrane; viii) active force generation and deformation of smooth muscle. The biomechanical principles of the model are described and studied numerically in Chap. 1. In addition to kinetic equations for the cycle of chemical transformations of acetylcholine at the cholinergic synapses, the system of the dynamics of serotonin (SR) conversion and the activation of serotonin type 4 receptors is given by d[SR v ] = −1/2kd [Ca2+ ]out ε(t )[SR v ] dt d[SR f ] ∗ = kd [Ca2+ ]ε(t )[SR v ] − k+2 [SR f ] dt d[SR c ] ∗ ∗ ∗ [SR f ] + [SR c −MAO](k−3 + k−3 [SR c ]) = k+2 dt ∗ ∗ [E0 ] + k+5 [R5-HT4 ] − [SR c ](k+3 ∗ [SR c − R5-HT4 ] + k−5

dt [SR c −MAO] ∗ = k+3 [SR c ][MAO0 ] dt ∗ ∗ ∗ + k+4 + k+3 [SR c ]) − [SR c −MAO](k−3 dt [SR c − R5-HT4 ] ∗ = k+5 [SR c ][R5-HT4 ] − [SR c −R5-HT4 ] dt ∗ ∗ ∗ + k+6 + k+7 [G0 ][SR c −R5-HT4 −G]) × (k−5

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d[R5-HT4 ] ∗ ∗ ∗ + k+6 )[SR c −R5-HT4 ] − k+5 [SR c ][R5-HT4 ] = (k−5 dt d[SR c −R5-HT4 −G] = k+7 [SR c −R5-HT4 ]([G0 ] − [SR c −R5-HT4 −[G]) dt ∗ [SR c −R5-HT4 −G] − k−7 d[PhC] ∗ [SR c −R5-HT4 −G] = k+10 dt ∗ [PhC] × ([PhC]inact − [PhC]) − k+15 d[PIP3 ] ∗ ∗ + k+12 )[PIP3 ][PIP] = −(k+11 dt d[Dg] ∗ [PhC][PIP3 ] = k+12 dt ∗ [Dg][Ca2+ ]4ier ([PhC]inact − [PhC]) − k+18 ∗ ∗ [PIP3 ][PhC] − k+17 [IP3 ][RER ] d[IP3 ]dt = k+11 ∗ ∗ − k−19 )[IP3 −RBR ] + (k−17

d[IP3 −RBR ] = b5 [IP3 −RBR −Ca2+ ] dt − (a2 + a5 )[Ca2+ ]ier [IP3 −RBR ]

(6.1)

d[Ca2+ ]ier = τ −1 (V0 − c1 V1 [IP3 −RBR −Ca2+ ]3 [Ca2+ ]ier dt V3 [Ca2+ ]2ier − [Ca2+ ]BR ) − ∗ ([Ca2+ ]2ier − k+13 ) d[PkC] ∗ ∗ [Dg][Ca2+ ]4ier ([PhCinact ] − [PhC]) − k−8 [PkC] = k+8 dt d[Pr]p ∗ ∗ = k+9 [PkC]([Pr] − [Pr]p ) − k−9 [PPhos][Pr]p dt c0 = [Ca2+ ]ier + c1 [Ca2+ ]BR [MAO] = [MAO0 ] − [SR c −MAO] [G] = [G0 ] − [SR c −R5-HT4 −G] [RBR ] = c2 [IP3 ]. ∗ Here k±i are rate constants of backward (−) and forward (+) chemical reactions; c(i) , b(i) , V0 , V1 are parameters related to the activation of the release of endoplasmic Ca2+ ; SR v is vesicular serotonin; SR f is the free fraction of

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serotonin; SR c is the SR content in the cleft; MAO is the monoamine oxidase; SR c −MAO is the serotonin-monoamine oxidase complex; R5-HT4 are serotonin type 4 receptors; SR c − R5-HT4 is the serotonin-receptor complex; G is the guanine-nucleotide G protein; PhC is phospholipase C; PIP3 is phosphatidyl inositol 1,4,5-triphosphate; IP3 is inositol 1,4,5-triphosphate; Dg is diacylglycerol; RER is IP3 receptor on the endoplasmic reticulum; Ca2+ ier is calcium of the endoplasm; Ca2+ is the total amount of calcium stored in the ER endoplasmic reticulum; PkC is protein kinase C; Pr is intracellular protein; PPhos is protein-phosphatase. The system of kinetic equations of the cycle of chemical transformations of serotonin and the activation of serotonin type 3 receptors is d[SR v ] dt d[SR f ] dt d[SR c ] dt d[SR p ] dt

= −1/2kc∗∗ [Ca2+ ]out ε(t )[SR v ] ∗∗ = kc∗∗ [Ca2+ ][SR v ] − kd∗∗ [SR f ] + k+6 [S] ∗∗ = kd∗∗ [SR f ] − k+p [SR c ] ∗∗ ∗∗ ∗∗ [SR 05-HT3 ] + k+2 [MAO0 ]) [SR c ] − [SR p ](k+1 = k+p ∗∗ ∗∗ × [SR p ](k+1 [SR] − R5-HT3 + k+2 [SR c −MAO]) ∗∗ ∗∗ [SR c −R5-HT3 ] + k−2 [SR c −MAO] + k−1

d[SR c −R5-HT3 ] ∗∗ = k+1 [SR p ][R50-HT3 ] dt ∗∗ ∗∗ ∗∗ + k+4 + k+1 [SR p ]) − [SR p −R5-HT3 ](k−1 d[SR c −MAO] ∗∗ = k+2 [MAO0 ][SR p ] dt ∗∗ ∗∗ ∗∗ + k+3 + k+2 [SR p ]) − [SR−MAO](k−2 d[S] ∗∗ ∗∗ ∗∗ = k+4 [SR c −R5-HT3 ] − k+6 [S] + k+3 [SR c −MAO] dt [R5-HT3 ] = [R50-HT3 ] − [SR−R5-HT3 ] [MAO] = [MAO0 ] − [SR c −MAO]

(6.2)

∗∗ are constants of backward (−) and forward (+) chemical reacwhere k−,+ tions; kc is the affinity constant; kd is the constant of diffusion; k+p is the

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velocity of diffusion of SR on the subsynaptic membrane; R5-HT3 are serotonin type 3 receptors; SR–R5-HT3 is the serotonin-receptor complex; other parameters as above. The generation of excitatory postsynaptic potential is calculated from Cp

dϕp+

ϕ+ + ϕp+ ( − [· · · ] + Rv−1 ) = R dt Rv   [ACh−R] [ · · · ] = [SR−R5-HT3 ]  [SR−R5-HT4 ]

(6.3)

where Cp is the capacitance of the subsynaptic membrane,  is the empirical constant, Rv is the general resistance of the extra synaptic structures, and ϕR+ is the resting potential. The system is at the resting state and the initial concentrations of reacting components are known [initial concentrations for ACh are given in (4.13)] [SR v ] = 60,

[SR f ] = 0.5,

[PhC]inact = 0.25, [R50-HT4 ] = 0.04,

[Ca2+ ]ier = 0.08, [G0 ] = 0.05,

[PIP3 ] = 0.1 × 10−3 , [PPhos] = 0.38,

[SR c ] = 2 × 10−4 ,

[SR p ] = 0.08

[R50-HT3 ] = 0.04

[MAO0 ] = 0.12,

[RER ] = 0.075,

[IP3 ] = 0.03

[PkC] = 4.0,

[Dg] = 4.5

(6.4)

[Ca2+ ]BR = 0.5 (mM)

[SR−MAO] = [SR c −R5-HT3 ] = [SR c −R5-HT4 ] = [S] = 0 [SR c −R5-HT4 −G] = [IP3 −RBR ] = [PhC] = [Pr]p = 0. Excitation to the system is provided through deformation of the mechanoreceptors. The system (4.1)–(4.6), (4.10), (4.11), (5.1)–(5.3), initial and boundary conditions (4.12), (4.13), (5.4) allowed us to study the effects of 5-HT and ACh alone and as a result of their interactions through the process of cotransmission and co-activation of different receptor types. The effect of drugs was modeled by varying conductances for affected channels or parameters involved in the second messenger system pathway. The numerical algorithms as described in Chaps. 1 and 3 were used throughout simulations.

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6.1.4 Electrical Activity of Mechanoreceptors 6.1.4.1 Physiological response Under normal physiological conditions, deformation of the wall of the gut with multiple stretches, ε = 0.17, n = 10, of duration, td = 0.9 s, followed at intervals, 1.5 s, initiates the voltage-dependent inward sodium, INa , outward potassium, IK , currents and chloride, ICl , current at the free nerve endings of the mechanoreceptors. The sodium influx has an average rate 1 nA/s and reaches the maximum amplitude 1.14 nA (Fig. 6.2). It has an almost instant recovery phase, during which the strength of the current reduces to 0.23 nA, followed by a prolonged period, 0.8 s, of slow decline to the resting value: min INa = 0.042 nA. An outward K+ current shows longer duration. The dynamics of activation, IK , is linear with the exponential inactivation. The current rises to its maximum of 0.04 nA at a constant rate 0.04 nA/s, and

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Fig. 6.2 The dynamics of the K+ and Na+ ion currents at the free nerve endings — mechanoreceptors: (A) physiological condition; (B) after activation of the 5-HT3 receptors, (C) following application of the selective 5-HT3 receptor antagonists.

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declines rapidly to a level 0.075 nA. The balanced activity of the ion currents, results in the production of the dendritic action potentials at the free nerve endings of the mechanoreceptors. They have maximum amplitude 15.7 mV and duration ∼2.3 ms.

6.1.4.2 Effect of 5-HT3 receptors Mechanical deformation causes the release of endogenous 5-HT from the EC cells. Free serotonin binds to the 5-HT3 receptors on the free nerve endings of the mechanoreceptors with a subsequent increase in permeability of ligand-gated ion channels. The INa current demonstrates nonlinear exponential behavior with an initial rapid rise, at a rate 2.04 nA/s, followed by a slow dynamic, 0.2 nA/s (Fig. 6.2). It reaches max INa = 1.77 nA. The recovery phase of the channel remains unchanged from that seen above. The dynamics of potassium channel activity have an exponential phase of activation with a slow start at the beginning, 0.05 nA/s, and a quick climb, 0.3 nA/s, to the maximum value, 0.188 nA. In the presence of endogenous serotonin the process of inactivation of IK goes on faster, 0.6 nA/s. The above change in the ionic currents reflects an increase in the amplitude of AP’s to 26 mV. The presence of endogenous 5-HT reduces the threshold for mechanical activation of the mechanoreceptors by 17.6%. Multiple stretch impulses of ε = 0.14 applied to the wall of the gut are sufficient to initiate the cascade of electrical events as described above.

6.1.4.3 Effect of 5-HT3 -receptor antagonists Treatment of the mucosa of the gut with selective 5-HT3 receptor antagonists, Ondansetron or Granisetron, abolishes the potassium current, IK  0 nA. There is a significant reduction in the sodium influx. The INa current shows an activation rate of 0.16 nA/s and reaches the maximum amplitude of 0.38 nA. As a result, dendritic action potentials of amplitude 3 mV are generated. They are not strong enough to set off electrical signals at the free nerve endings and to sustain their propagation along the unmyelinated fibers. However, an increase in the intensity of the mechanical stimulus by 35% (ε = 0.23) results in an unexpected response from the mechanoreceptors. There is an increase

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Multiple Neurotransmission 209

in both the INa and IK currents: max INa = 1.04 nA and max IK = 0.2 nA are recorded (Fig. 6.2). The mechanoreceptors produce action potentials, ϕd = 22.9 mV.

6.1.5 Electrical Activity of the Primary Neuron 6.1.5.1 Stimulation of mechanoreceptors The dendritic action potentials elicited at the free nerve endings propagate along the unmyelinated axon and reach the soma of the neuron. There they activate the voltage-dependent Na+ , K+ , Ca2+ , Ca2+ –K+ and leak Cl− chanp p nels. The dynamics of the inward INa and outward IK currents demonstrate high frequency oscillatory behavior with fast alternations of activation and p p inactivation. The maximal currents, INa = 81.5 nA and IK = 29.8 nA are registered immediately after excitation of the membrane. In 0.2 s both currents p p reduce in strength: INa = 40.8 nA and IK = 15 nA are recorded. The sodium and potassium channels remain active for 0.8 s. The dynamics of activation of Ca2+ channels has a slow initial phase, 1.9 nA/s (Fig. 6.3). This is followed by a fast phase, 14.8 nA/s, when the maximum value, 11.65 nA, is reached. There are high frequency oscillations, 0.8–3.8 nA, in the current behavior that occur when the channel starts p approaching its peak of activation. The recovery of ICa shows the linear dynamics at a rate ∼23 nA/s. A similar pattern is seen with the outward p ICa−K . It quickly attains the maximum level and sustains it for 0.6 s. The p max ICa−K = 10 nA. There are high fluctuations of the current, which are only present during activation of the channel. In the absence of free endogenous serotonin the AH-neuron responds only to odd excitatory signals arriving from the mechanoreceptors with the bursts of action potentials of maximum amplitude 90 mV. The pattern of discharges resembles a bursting chaos type of electrical activity at a variable frequency, ν = 8–10 Hz.

6.1.5.2 Effect of 5-HT3 receptors A selective stimulation of the 5-HT3 receptors by serotonin has a profound effect on the activity of all ion channels. The duration of the active state of

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-0.56

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Fig. 6.3 The dynamics of the Ca2+ ion currents and on the soma of the primary sensory neuron after: (A) selective stimulation of 5-HT3 receptors; (B) stimulation of 5-HT4 receptors; (C) concomitant stimulation of the 5-HT3 and 5-HT4 receptors, and (D) treatment with Cisapride.

the Na+ and K+ channels increases to 1.54 s. There is a significant rise in p p intensity of the currents: max INa = 154.3 nA and max IK = 52.3 nA. The brisk increase in the currents is followed by a lengthy period of recovery when fluctuations of the currents of average amplitudes, 61.7 nA and 35 nA, respectively, are observed. p The amplitude of calcium activated potassium current is: max ICa−K = 14 nA. It demonstrates a slight increased activation rate, 25 nA/s. The inactip vation phase of ICa−K remains intact from the one considered above (Fig. 6.4). p There are no changes in ICa . However, the channel sustains its active state for a longer period.

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C) -64.3

D) -62.5

0.0

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Time (s)

2.0

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Fig. 6.4 The dynamics of the Ca2+ -activated K+ current on the soma of the primary sensory neuron after: (A) selective stimulation of 5-HT3 receptors; (B) stimulation of 5-HT4 receptors; (C) concomitant stimulation of the 5-HT3 and 5-HT4 receptors, and (D) treatment with Cisapride.

As a result of the above alterations the primary sensory neuron generates max ϕ = 111.3 mV. It is important to note that the neuron requires a lower level of deformation, min ε = 0.14, to respond to the production of action potentials to each impulse of excitation arriving from the free nerve endings. Its electrical pattern of discharges resembles a beating mode with a frequency of 7 Hz.

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6.1.5.3 Effect of 5-HT3 -receptor antagonists Ondansetron and Granisetron applied separately completely abolish any electrical activity of the neuron. The neuron remains silent after external periodic stimulations of intensities 0.14–0.17 at the free nerve endings of the mechanoreceptors. However, larger deformation, ε = 0.24, causes an unexpected production of action potentials of short duration, 0.3 s, and max ϕ1 = 91 mV. A gradual increase in the concentration of the selective 5-HT3 antagonists has a dose-dependent effect on the reduction of excitability of the neuron.

6.1.5.4 Effect of 5-HT4 receptors Excitation of the 5-HT4 receptors on the soma of the primary sensory neuron p by the free serotonin results in the production of high amplitude, INa = p 166.6 nA and IK = 46.1 nA and short duration, 0.2 s, sodium and potassium currents. The Na+ channel returns quickly to its unexcited state, while K+ channels require ∼1 s to recover. The calcium current shows fast activation, p at a rate of 36 nA/s, and reaches the maximal amplitude, max ICa = 21.7 nA. The dynamics of its recovery has two distinct phases: the initial phase with p a rate 5.6 nA/s, and the faster phase of 8.64 nA/s. ICa−K shows a relatively p smooth rise to the maximum value, 12.6 nA. ICa−K remains active for 2.5 s (Fig. 6.4). There is an increase in depolarization of the soma with a shift of the resting membrane potential to 50 mV. The neuron fires a few high amplitude AP’s, max ϕ1 = 96.2 mV, followed by a slow process of decay of depolarization. There is also a reduction in duration, 0.6 s, of the period of afterhyperpolarization.

6.1.5.5 Effect of co-activation of 5-HT3 and 5-HT4 receptors A concomitant stimulation of the co-localized 5-HT3 and 5-HT4 receptors on the soma of the primary neuron results in an increase in its excitability. At the beginning of the process, when activity of the ligand-gated 5-HT3 p p receptors dominates, the currents of amplitude INa = 25 nA and IK = 36 nA are recorded. As a result of a full effect of the 5-HT4 receptors the dynamics

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of the sodium and potassium channels change to a high frequency oscilp p latory mode. Ion currents of intensity INa = 45 nA and IK = 12 nA are generated. The Ca2+ channel reiterates the type of activity observed under a separate stimulation of the 5-HT3 receptors. With concurrent excitation of p 5-HT4 receptors the pattern of ICa changes to a beating mode with constant amplitude of fluctuations, 4 nA (Fig. 6.3). The neuron produces spikes of high amplitude, max ϕ1 = 81 mV, followed by a long quiescent period of sustained depolarization, −13 < ϕ1 < 25 mV. This response is characteristic of a separate activity of the fast ligand-gated 5-HT3 receptors. A transitory period of irregular chaotic type discharges of AP’s corresponds to the gradual activation of the 5-HT4 receptors. The amplitude of ϕ1 increases from 5 to 85 mV and the frequency of firing varies between 18 < ν < 37 Hz. As the 5-HT4 receptors become fully active the pattern of electrical activity of the soma begins to resemble a beating mode. Action potentials of average amplitude ϕ1 = 50 mV and frequency ν = 28 Hz are generated. The selective 5-HT3 receptor antagonist, Ondansetron, abolishes the production of spikes on the soma. Continuous activation of the 5-HT4 receptors results in a translation of the electrical pattern of the neuron from the beating to a self-excitatory mode. Regular rhythmic undulations of the membrane potential of amplitude 30 mV with single spikes, max ϕ1 = 40 mV, on crests of the waves are produced. This effect persists while the 5-HT4 receptors remain active.

6.1.5.6 Effect of Cisapride A non-selective strong 5-HT3 - and weak 5-HT4 -receptor agonist Cisapride, increases the influx of calcium ions into the cell. The maximal amplitude of p p ICa equals 17.2 nA. There are changes in the patterns of activity of the INa , p p ICa−K and IK currents with a period of irregular high frequency oscillatory p activity and an interim quiescent period. Thus, the maximal values of INa p and IK of 228.5 nA and 57.6 nA, respectively, are achieved immediately after the application of the drug. The second peak of activity of duration 0.5 s p p p shows max INa = 38 nA and IK = 18.8 nA. The ICa−K current demonstrates p a steady rise in amplitude with max ICa−K = 13 nA at the beginning of the p process followed by the second peak of ICa−K = 19.2 nA (Figs. 6.3–6.5).

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8.68

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B)

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Fig. 6.5 The dynamics of the action potentials on the soma of the primary sensory neuron after: (A) selective stimulation of 5-HT3 receptors; (B) stimulation of 5-HT4 receptors; (C) concomitant stimulation of the 5-HT3 and 5-HT4 receptors, and (D) treatment with Cisapride.

Cisapride depolarizes the soma of the primary neuron and elevates the resting membrane potential, ϕ1(rest) = −62.5 mV. The neuron generates spikes of high amplitude, max ϕ1 = 111 mV, and frequency, 7 ÷ 18 Hz. There is a period of 0.65 s when no action potentials are produced. It is followed by a short period of bursting activity with max ϕ1 = 62 mV and duration 0.5 s.

6.1.6 Electrical Activity of the Motor Neuron 6.1.6.1 Effect of nACh receptors Depolarization of the presynaptic membrane at the neuro-neuronal synapse activates release of vesicular acetylcholine and its diffusion into the

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Multiple Neurotransmission 215

synaptic cleft. The main part of ACh reaches the postsynaptic membrane and reacts with the receptors on the soma of the motor neuron. The generation of fast excitatory potential starts with the beginning of (ACh–R)-complex development. It increases as a step function and achieves its maximum 87.1 mV in 0.25 ms. The level of fEPSP is sufficient to excite the neuron. It discharges a single train of pulses of amplitude 102.1 mV at a frequency 8.4 Hz and duration 1.9 s. The observed pattern of electrical activity resembles a regular bursting mode.

6.1.6.2 Effect of 5-HT3 and 5-HT4 receptors A selective stimulation of 5-HT3 receptors by the free endogenous 5-HT in the absence of ACh in the system does not have an excitatory effect on the soma of the motor neuron. In contrast, activation of the 5-HT4 receptors changes the pattern of neuronal electrical activity into a long-lasting excitatory mode. Action potentials, ϕ2 , of constant amplitude 114 mV and frequency 7 Hz are produced. The ion m m currents INa of magnitude 22.5 × 102 nA, IKm  148 nA, ICa−K  16.2 nA and m ICa  6 nA are recorded (Fig. 6.6).

6.1.6.3 Effect of co-activation of 5-HT3 and nACh receptors Simultaneous excitation of the 5-HT3 and nACh receptors on the soma of the motor neuron results in its hyperpolarization, ϕ2(rest) = 66.8 mV. The neuron generates action potentials of the maximal amplitude at a frequency 7.8 Hz. The observed electrical activity is due to an increase in the dynamics m m of ion currents: INa = 18.5×102 nA, IKm = 18 nA and ICa−K = 14.8 nA. They demonstrate a regular high frequency mode of activation-inactivation. The calcium current shows an intensive oscillatory phase of activation followed m by a slow recovery period: max ICa = 11.9 nA is recorded (Fig. 6.7).

6.1.6.4 Effect of co-activation of 5-HT4 and nACh receptors Concurrent stimulation of co-localized 5-HT4 and nACh receptors initiates a m strong large magnitude sodium ion current, max INa = 27 × 102 nA. There is m a concomitant slight decrease in amplitude of the IKm , ICa−K and Im Ca currents,

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Fig. 6.6 The dynamics of different ion currents and the action potential on the soma of the motor neuron after selective stimulation of the 5-HT4 receptors.

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Fig. 6.7 The dynamics of ion currents and the action potential on the soma of the motor neuron after co-joint stimulation of 5-HT3 and nACh receptors.

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compared to a separate excitation of 5-HT4 and/or nACh receptors. These changes lead to the generation of spikes of amplitude ϕ2 = 116.5 mV.

6.1.6.5 Effect of 5-HT3 /5-HT4 -receptor agonists and co-activation of nACh receptors Cisapride and Renzapride added into the system have a pronounced effect on m the dynamics of the calcium current. Thus, max ICa = 7.04 nA is observed. 2+ The Ca channel shows oscillatory behavior with fluctuations of the current of amplitude 4.2 nA and constant frequency ν = 6.4 Hz. Ion currents of m m intensity IKm = 210 nA, ICa−K = 16.7 nA and INa = 28.5 × 102 nA are recorded. The soma of the neuron is hyperpolarized to −70 mV. The neuron remains in a highly excitable state for the duration of 3.44 s and generates AP’s of max ϕ2 = 130 mV and frequency ν = 6.4 Hz. No significant changes are observed in the pattern of behavior of the IKm , m m ICa−K and INa ion currents after simultaneous application of Cisapride and the m release of ACh into the system. However, the dynamics of ICa demonstrates two distinct peaks of activation (Fig. 6.8). Immediately after activation of m the 5-HT3 , 5-HT4 and nACh receptors max ICa = 6.8 nA is observed. The current slightly decreases in intensity to 6.45 nA during the following 1.2 s. m The second maximum of the influx of calcium ions with max ICa = 9.5 nA is recorded 1 s later. The current quickly subsides to its inactive state. -0.52

61.4 B)

-3.53

17.7

φ (mV)

I (Ca) (nA)

A)

-6.55

-25.9

-9.56

-69.5 0.0

2.0

Time (s)

4.0

0.0

2.0

4.0

Time (s)

Fig. 6.8 Changes in the dynamics of the Ca2+ ion current and the action potential on the soma of the motor neuron after application of Cisapride and co-stimulation.

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Interestingly, the duration of electrical excitability of the neuron reduces to 2.8 s. There is a decrease in the maximal amplitude of AP’s, max ϕ2 = 120 mV and an increase in frequency of spikes, ν = 7.8 Hz. The level of hyperpolarization of the soma remains unchanged.

6.1.6.6 Effects of 5-HT3 -receptor antagonist and co-activation of nACh receptors Ondansetron and Granisetron block the serotonergic pathway between the primary sensory and motor neurons. The normal electrical signal transduction is maintained through the co-existing cholinergic mechanisms. The latter sustain the neuro-neuronal synaptic connectivity. They are responsible for the generation of short duration, <100 ms, 85–90 mV amplitude excitatory potentials on the soma of the adjacent motor neuron.

6.1.7 Electrical Activity of Smooth Muscle 6.1.7.1 Effect of µACh receptors Fast excitatory postsynaptic potentials at the neuro-muscular synapse activate the L-type Ca2+ channels located on the smooth muscle membrane. Alterations of the permeability of the channels cause cyclic transitory changes in the myoelectrical pattern. The slow wave mode transforms to bursting chaos with the generation of fast AP’s at a frequency of ν = 17 Hz. Their amplitude is 23 mV at the beginning and decreases towards the end of the burst, ϕm = 8 mV. The bursting chaos transforms to regular bursting with the generation of spikes on the crests of slow waves. The burst amplitude, burst duration and number of spikes per burst all increase. The action potentials have a maximum amplitude of 72 mV and oscillate at a maximum frequency of 19 Hz. The regular bursting converts back to bursting chaos and slow wave mode. There is a gradual increase in the concentration of intracellular calcium ions, mainly due to the influx of ions through the activated slow Ca2+ channels. Maximum internal calcium concentration, [Ca2+ ]i = 0.46 µM, is achieved. As a result of activation of the contractile protein system, rhythmic contractions of the longitudinal smooth muscle, max T a = 14.4 mN/cm, are produced. The duration of electromechanical activity is 2.6 s.

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6.1.7.2 Effect of 5-HT4 receptors Selective 5-HT4 receptor agonists, TS-591, Prucalopride and ML10302, applied onto the longitudinal muscle of the gut do not have any effects on its electromechanical activity.

6.1.7.3 Effect of co-activation of 5-HT4 and µACh receptors Excitation of the 5-HT4 receptors with TS-591, Prucalopride and ML10302 in conjunction with µACh receptors evokes an increase in intensity of the membrane ionic currents and electromechanical activity of the smooth L muscle syncytium. ICa shows high frequency alternations, 18.5 Hz, of short duration. The average amplitude of the L-type Ca2+ current is 7 nA and L T max ICa = 10.6 nA. The dynamics of the transitory calcium current, ICa , remains unchanged. It oscillates at a frequency 21 Hz and has average amplitude 0.30 nA. The outward voltage-dependent K+ and Ca2+ –K+ currents s are reduced in strength. Maximum ICa−K = 1.37 nA and IKs = 4.6 nA are recorded. The above changes in membrane ionic currents cause an increase in the frequency of the production of AP’s, ν = 20 Hz. The amplitude of spikes is slightly decreased, ϕm = 68 mV, compared to that observed when only µACh receptors are active. The concentration of free cytosolic calcium is 0.46 µM. As a result, the longitudinal smooth muscle responds with the production of phasic contractions. The maximum active force generated equals 15.6 g. The contractions are of duration 3.45 s, concurrent in phase and time with the dynamics of the intracellular Ca2+ oscillations and have constant amplitude, 6 g. The calculated velocity of twitch is 5.8 g/cm·s (Fig. 6.9).

6.1.7.4 Effect of 5-HT4 receptor antagonists and co-activation of 5-HT3 and µACh receptors Addition of the selective 5-HT4 antagonist, GR113808A, strongly inhibits contractions evoked by the free endogenous serotonin. However, electromechanical activity can be sustained through the function of the neuronal 5-HT3 and nACh receptors and µACh receptors located on the smooth muscle

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Multiple Neurotransmission 221 0.50

25.3

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a

T

Ca (mM)

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3.55 0.0

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Fig. 6.9 Changes in the action potential, intracellular calcium and active force dynamics in the longitudinal smooth muscle after co-joint activation of the 5-HT4 and µACh receptors.

membrane. Thus, application of Cisapride leads to twitch contractions of smooth muscle of amplitude T a = 13.8 mN/cm.

6.1.8 Remarks Comparison of numerical results to experimental data shows satisfactory qualitative and quantitative agreement. Using the isolated guinea-pig ileum, Tuladhar et al. (1997, 2000) showed that treatment of the preparation with selective 5-HT3 receptor antagonists, Ondansetron and Granisetron, resulted in an average 44% increase in the level of mechanical stimulus needed. Our results indicate that a possible mechanism is associated with an increase in the conductivity of the Na+ and K+ channels and changes in the dynamics of activation of the INa and inactivation of the IK currents. Interestingly, the inhibitory effect of Ondansetron and Granisetron with a complete suppression of the potassium current, IK  0 nA, and a significant reduction in strength of the INa current can be overcome by a higher level of mechanical deformation. This fact suggests that the 5-HT3 receptors may be directly linked to the mechanosensitive Na+ channels or are located in close proximity to them. However, cautious interpretation of the proposed mechanism is necessary. Currently, there is no affirmative experimental evidence on the dynamics of the Na+ and K+ currents at the mechanoreceptor level to support or disapprove this view. Experimental intracellular recordings from the soma of the primary neurons of the guinea-pig ileum myenteric nervous plexus demonstrate that after application of 5-HT the neuron produces long-lasting, 2–4 s trains of

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action potentials of amplitude 75–80 mV and frequency ranging from 0.1 to 10 Hz. Numerical analysis of the dynamics of the ion currents helps suggest the mechanisms responsible for the observed response. Thus, major p p changes are registered in the INa and IK currents with high-frequency fluctuations recorded during the recovery phase. It is possible to assume that the increased excitability of the mechanoreceptors and instability of the Na+ and K+ currents decrease the threshold of activation of the AH neuron. Selective activation of the 5-HT3 receptors on the soma of the S-neuron of the guinea-pig ileum induces action potentials of 5 mV and of low frequency. These experimental findings are of particular interest because the physiological saline in the experimental preparation contained nicardipine and scopolamine — the drugs that affect cholinergic transmission. The numerical simulation of co-transmission by ACh and 5-HT shows that the motor neuron generates high amplitude and frequency, 7.8 Hz, action potentials. This response can be attributed to a heteromeric co-assembly of the 5-HT3 receptors with nicotinic ACh (α4 subunit) receptors. Interestingly, application of Cisapride results in a two or three times higher concentration of intracellular Ca2+ than after treatment with the natural endogenous serotonin. While these experimental results concur with the results of our numerical simulations, the putative role of the 5-HT3 /nACh ligand-channel co-assembly require further pharmacological investigation. The effects of co-expression of the 5-HT3 , 5-HT4 and nACh receptors on the soma of the motor neuron on the dynamics of the electrical signal transduction were analyzed in this study, demonstrating a minor role for 5HT type 3 receptors, in contrast to the 5-HT type 4 receptors. The activation of 5-HT4 receptors evoked a long-lasting train of fast action potentials. Costimulation of 5-HT3 –nACh and 5-HT4 –nACh receptors has a considerable excitatory effect on neurons, resulting in a significant increase in the amplitude and frequency of firing rate. These findings are in line with a possible co-assembly of serotonergic and cholinergic receptors. The results indicate that Prucalopride alone fails to augment contractility of the longitudinal smooth muscle in the absence of ACh. Excitation of 5-HT4 receptors only in conjunction with µACh receptors is able to initiate longlasting phasic contractions. These findings are experimentally supported by the selective activation of 5-HT4 receptors on the longitudinal smooth muscle of the large intestine of different species.

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6.2 CO-TRANSMISSION BY ACh AND EXCITATORY AMINO ACIDS 6.2.1 Biological Background Experiments with mRNA encoding and immunohistochemical staining for excitatory amino acids, L-aspartate (L-Asp), glutamate (Glu), α-amino3-hydroxy-5-methyl-4-isoxalose propionic acid (AMPA) and N-methylD-aspartate (NMDA) have provided direct evidence for the existence of glutamatergic neurons in the ENS, the dorsal root ganglion and the spinal dorsal horn. Excitatory amino acids, Glu along with serotonin and substance P (SP) have been implicated in the pathogenesis of visceral hyperalgesia and have been proposed as pharmacological targets for drug design to treat conditions like irritable bowel syndrome. Immunostaining experiments revealed the presence of Glu, L-Asp, NMDA in peripheral cholinergic terminals of a subset of enteric neurons. Although it was not possible to distinguish clearly the contents of small synaptic vesicles, it was presumed that they co-stored ACh and Glu. The smaller population of large, dense-cored vesicles in the same terminals colocalized Glu and substance P. A smaller amount of Glu was also detected in enteric glia. Glutamine and AMPA play a pivotal role in excitatory neurotransmission and neuro-modulation of myoelectrical activity in the gut. They act at NMDA and AMPA receptors, respectively. AMPA and NMDA receptors are homo- or hetero-tetramers of subunits with three trans-membrane domains. Antibodies to the C-terminal portions of NMDA and AMPA receptor subunits NR1, NR2 and GluR1, GluR2/3 and GluR4 have been identified immunohistochemically on AH- and S-type neurons. Furthermore, the immunoreactivity of the glutamate transporter 1 (EEAC1) was expressed on the somas of the enteric neurons and in the ganglionic neuropil. NMDA receptors assemble as heteromers of NR1 and NR2 subunits that form a functional ion channel. The channels are permeable to Ca2+ . Alternative splicing can generate eight isoforms of receptors with different pharmacological properties. In a silent cell, receptors remain inactive. This is due to a voltage-dependent block of the channel pore by Mg2+ . Activation of the receptor is a multistep process and requires a fast depolarization of the membrane, which is usually accomplished by activation of AMPA receptors, and an allosteric binding of two molecules of Glu and glycine to topographically

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distinct domains, S1 and S2, located on the homologous regions of the NR2 and NR1 subunits. AMPA receptors are homo- or heteromers that consist of four subunits, GluR1–4. The receptors exist as two variants generated by alternative splicing and differ in their desensitization in the presence of cyclothiazide. Native AMPA receptors are permeable to Na+ , K+ and Ca2+ . The biological effects produced by endogenous and externally applied excitatory amino acids on neurons and intrinsic afferent/efferent neuronal pathways of the ENS are very diverse. Monophasic fast, slow, or biphasic-depolarizing responses were observed in AH- and S-type neurons after activation of NMDA receptors. The early fast phase was mimicked by AMPA and reduced in amplitude by local application of 6-cyano-7nitroquinoxaline-2,3-dione (CNQX) and 6,7-dinitroquinoxaline-2,3-dione (DNQX)-selective AMPA antagonists. The immediate slower depolarization phase was due to the pharmacodynamic effect of NMDA and was abolished after treatment with MK-801, a non-competitive NMDA antagonist, D-2-amino-5-phosphopentanoic acid (D-AP5), or 3-(2-carboxypiperazin4-yl) propyl-1-phosphonic acid (CPP). It has been proposed that AMPA, NMDA and nACh receptors are co-localized. The computational model arrangement includes the primary sensory and motor neurons linked in sequence to form the afferent pathway, terminating at the smooth muscle syncytium (Fig. 6.10). It reproduces the following sequence of events: (i) deformation of the free nerve endings of the mechanoreceptors by mechanical stimuli; (ii) generation of action potentials and their propagation along the unmyelinated fiber towards the soma of the AH neuron; (iii) action potential generation at the soma of the AH neuron, and propagation of the wave of depolarization towards the soma of the S neuron; (iv) electrochemical coupling at the axo-dendritic synapse on the soma of the secondary neuron, and generation of the EPSP; (v) discharge of the soma of the S neuron, and propagation of the wave of depolarization along the nerve axon towards the neuromuscular synapse; (vi) electrochemical coupling at the synapse and EPSP generation;

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Multiple Neurotransmission 225 Smooth muscle syncytium

Neuro-muscular synapse Primary sensory neuron (AH) Motor neuron (S)

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Fig. 6.10 Scheme of the one-dimensional model used in numerical experiments. The distribution of NMDA, AMPA, glutamate and cholinergic receptors is as indicated.

(vii) activation of L-type Ca2+ channels of the smooth muscle membrane; and (viii) active force generation and deformation of smooth muscle. We studied the effects of NMDA, Glu, L-Asp, and ACh, both alone and as a result of their interactions through cotransmission and coactivation of NMDA, AMPA, and nACh receptors. In all experiments, the system was excited with multiple stretches ε = 0.8, n = 10, of duration 1.0 s, and followed at intervals of 1.0 s. Mechanical stimuli were applied at the level of mechanoreceptors.

6.2.2 Electrical Activity of the Primary Neuron 6.2.2.1 Effect of mechanical stimulation Multiple stretches elicited the dendritic action potentials at the free nerve endings that propagated along the unmyelinated axon activated the voltagedependent Na+ , K+ , Ca2+ , Ca2+ –K+ and leak Cl− channels (Fig. 6.11).

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The dynamics of inward INa and outward IK currents demonstrated high p frequency oscillatory behavior. The maximal currents of amplitudes INa = p −81.5 nA and IK = 29.8 nA were recorded immediately after excitation of the p membrane. They subsided in strength in 0.2 s and sustained INa = 40.8 nA p and IK = 15 nA values. The sodium and potassium channels remained active for 0.8 s. The dynamics of activation of Ca2+ channels had a slow initial phase, p 1.9 nA/s, followed by a fast phase, 14.8 nA/s, when max ICa = 11.65 nA was recorded. There were high frequency oscillations of amplitudes 0.8–3.8 nA in the current behavior that occurred when the channel approached its peak p activity. The recovery of the ICa current showed the linear dynamics at a

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rate ∼23 nA/s. A similar pattern was seen in the dynamics of the outward p p ICa−K . It quickly attained the maximum, ICa−K = 8 nA, and sustained it for 0.6 s. The neuron generated action potentials of maximum amplitude ϕ = 105 mV at a frequency of 10 Hz.

6.2.2.2 Effect of AMPA receptors A selective stimulation of AMPA receptors on the soma of the AH neuron increased the permeability of Na+ , K+ and Ca2+ channels. It did not significantly change the dynamic portraits of their activity. There was an increase p p in intensity of the currents: max INa = −158 nA, max IK = 47.2 nA, and p max ICa−K = 10 nA were recorded. All channels sustained a longer period of active state. This resulted in a prolonged depolarization of the cell. The resting membrane potential shifted up by 5 mV and the neuron fired spikes of amplitude ϕ = 105 mV at a frequency of 14 Hz (Fig. 6.12).

6.2.2.3 Effect of NMDA receptors Glutamate alone acting at NMDA receptors increased the permeability of Ca2+ channels. Although the dynamics of activation and inactivation of the calcium current was not altered, there was an increase in 45

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strength: max ICa = −18 nA was observed. The currents remained active for 2.7 s. There was a prominent plateau of duration ∼2.0 s during which a stap ble current of average intensity ICa = −13.5 nA was recorded. Concomitant changes were seen in the dynamics of the calcium-activated potassium current. The neuron responded with the generation of high amplitude action potentials of max ϕ = 105 mV and frequency 7 Hz that had a short duration. It was followed by a slow long-lasting component when a relatively constant level of depolarization, ϕ = −25 mV, was observed (Fig. 6.13). Treatment of the neuron with glycine transporter inhibitor −1 ALX 5407 resulted in a prolonged >4 s depolarization.

6.2.2.4 Effect of co-activation of NMDA and AMPA receptors Co-stimulation of AMPA and NMDA receptors with AMPA and Glu resulted in a biphasic electrical response of the neuron (Fig. 6.14). The first phase was associated with activation of AMPA receptors and fast depolarization of the neuron. The maximum amplitude of action potentials and their frequency matched with those produced by activation of AMPA receptors alone. The second phase occurred at the end of the plateau of the slow component and was due to activation of NMDA receptors. The cell generated a few spikes of maximum amplitude 39 mV at a frequency 9.5 Hz. The analysis of the dynamics of Ca2+ and Ca2+ –K+ currents showed p an increase in the intensity of ion flow. max ICa = −16.5 nA and

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max ICa−K = 9.15 nA were recorded with a pattern of activation and inactivation similar to that described above. Application of MK-801, D-AP5 and CPP, NMDA receptor antagonists, did not affect the fast phase of depolarization. The neuron continued to fire high amplitude action potentials of unaltered amplitude and frequency. However, it reduced the excitability of the neuron, which failed to respond to every stimulus applied at the mechanoreceptor site (Fig. 6.15). Pretreatment of the neuron with CNQX and DNQX resulted in total inhibition of electrical activity.

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6.2.3 Electrical Activity of the Motor Neuron 6.2.3.1 Effect of nACh receptors Depolarization of the presynaptic membrane at the neuro-neuronal (AH-S) synapse initiated the release of vesicular acetylcholine and processes of electrochemical coupling. Fast EPSPs were produced on the postsynaptic membrane that depolarized the soma of the neuron. Currents of average p p amplitudes INa = −1250 nA and IK = 105 nA were recorded immediately after excitation of the membrane (Fig. 6.16). The sodium and potassium channels remained active for 1.9 s. A similar pattern was observed in the p dynamics of ICa−K . The current quickly attained the maximum strength of p 13 nA and sustained it for 0.8 s. The ICa current demonstrated high frequency p oscillatory behavior with max ICa = −12 nA. A biphasic behavior was characteristic of the dynamics of inactivation of Ca2+ channels. It had a rapid initial phase, 80 nA/s, followed by a slow recovery phase, 4 nA/s. The neuron discharged a train of action potentials of amplitude 123 mV for a period of

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1.6 s at a frequency of 8 Hz. The pattern of its electrical activity resembled regular bursting.

6.2.3.2 Effect of NMDA receptors Glutamate acting alone at NMDA receptors failed to induce electrical responses in the motor neuron.

6.2.3.3 Effect of co-activation of nACh and NMDA receptors Activation of nACh receptors with endogenously released ACh followed by an immediate stimulation of co-localized NMDA receptors with Glu led to a prolonged neuron depolarization. The neuron generated action potentials of average amplitude ϕ = 123 mV for a duration of 3 s. The dynamic portrait of ion currents did not differ from that observed under the effect of nACh receptors alone although the channels remained active for a longer time. Treatment of the S neuron with nicotinic antagonist hexamethonium did not affect electrical responses induced by Glu acting at NMDA receptors. Application of ALX 5407 and ORG 24598 increased the time of depolarization of the soma of the neuron to 10 s. However, it did not change the amplitude and frequency of action potentials. The effect was dose-dependent. MK801, D-AP5 or CPP inhibited the responses induced by Glu. The neuron continued to generate action potentials as a result of activation of intact nACh receptors. Tetrodotoxin applied onto the soma of the neuron, when both nACh and NMDA receptors were active, abolished action potentials and steadily depolarized the cell, ϕ = −7 mV.

6.2.3.4 Effect of co-activation of nACh and AMPA receptors A selective activation of AMPA receptors caused a slight hyperpolarization of the neuronal cell, ϕrest = −72 mV, and induced a single train of high frequency 6 Hz, action potentials of max ϕ = 128 mV and duration 2.0 s. The neuron failed, however, to sustain continuous firing activity and quickly shifted to a silent state. Co-stimulation of nACh and AMPA receptors converted the neuron to a highly excitable state. The cell generated repeated sets of action potentials. The effect was reversed with hexamethonium.

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Mathematical Modeling and Simulation in Enteric Neurobiology 65

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Fig. 6.17 Pattern of electrical activity recorded from the soma of the motor neuron as a result of co-joint stimulation of nACh, AMPA and NMDA receptors.

6.2.3.5 Effect of co-activation of AMPA, NMDA and nACh receptors Additional cojoint activation of NMDA receptors prolonged further depolarization of the neuron. It generated spikes of maximum amplitude 128 mV for 4.2 s (Fig. 6.17). No significant changes were seen in the dynamics of ion channels apart from them being active for a longer period of time. TTX, hexamethonium, CNQX and DNQX applied separately did not have any effect on the dynamics of the generation of action potentials. Cojoint application of hexamethonium and CNQX abolished electrical activity of the neuron. The result of treatment with D-AP5, a selective NMDA receptor antagonist, was a train of high amplitude spikes of max ϕ = 120 mV, and short duration ∼1.4 s.

6.2.4 Smooth Muscle — Neuronal Chain Preparation 6.2.4.1 Effect of nACh receptors Excitation of mechanoreceptors initiated the wave of depolarization that propagated along the axon to the soma of the AH neuron and further to the dendro-somatic synapse on the S neuron. The release of ACh and activation of nACh receptors caused the depolarization of the neuron and the generation of action potentials that propagated towards the neuro-muscular synapse. There they activated the L-type Ca2+ channels and caused the

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generation of action potentials max ϕ = 72 mV and the contractile protein system. The longitudinal smooth muscle produced rhythmic contractions of max T a = 14.2 mN/cm. TTX blocked the propagation of action potentials along the axons and dendrites of the primary sensory and motor neurons. Hexamethonium had a selective inhibitory effect on electrical activity of the S neuron.

6.2.4.2 Effect of co-activation of AMPA and nACh receptors AMPA acting alone at AMPA receptors located at the somas of AH/S neurons caused phasic contractions in the longitudinal smooth muscle. A maximum active force of T a = 13 mN/cm was recorded. Co-activation of nACh receptors on the soma of the motor neuron resulted in a slight increase in the strength of contractions. A maximum active force of T a = 14.9 mN/cm was registered. These effects were reversed with CNQX and DNQX.

6.2.4.3 Effect of co-activation of nACh, AMPA and NMDA receptors Co-excitation of nACh, AMPA and NMDA receptors increased the duration of electromechanical activity of the longitudinal smooth muscle. It produced prolonged phasic contractions of max T a = 16 mN/cm. The duration of each contraction was 4.8 s and was concurrent in phase and time with the dynamics of intracellular Ca2+ oscillations. The calculated velocity of twitch was 4.18 g/cm · s and remained unchanged from that observed when only nACh receptors were active. Hexamethonium lowered the strength, max T a = 12.4 mN/cm, but did not affect the duration of contractions. Hexamethonium and CNQX applied jointly abolished the mechanical activity of the longitudinal muscle.

6.2.5 Remarks A comparison to equivalent experiment data of the numerical results obtained under the proposed modeling setting cannot be provided. Furthermore, available results of the actions of L-Asp, NMDA and glutamate on electrical activity of enteric neurons are highly controversial.

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Liu et al. (1997) recorded slow, fast and biphasic depolarization from 70% of AH neurons while Ren et al. (2000) reported no depolarizing effects in any AH neurons after application of glutamate. Slowly activating and fast, concentration-dependent depolarization were observed in 75–84% of motor neurons by the same authors. An apparent excitatory effect of AMPA and NMDA receptors acting alone on the soma of the AH neuron was shown in the above numerical experiments. A possible mechanism was associated with an increase in the conductivity of the Na+ , K+ and Ca2+ channels and p p p p changes in intensity of the INa , IK , ICa−K , and ICa currents. Comparison of the dynamic profile of electrical responses to experimentally recorded signals indicates satisfactory qualitative and quantitative agreement with those data published by Liu et al. (1997). Fast electrical responses induced by activation of AMPA receptors were abolished by CNQX and DNQX. The inhibitory effect of CNQX and DNQX corresponded to a significant reduction in strength of the INa and IK currents. This fact concurs with the notion that AMPA receptors are directly linked to ionotropic channels. NMDA receptors caused the long-term, ∼2.7 s, depolarization of the AH neuron. The mechanism responsible for the observed response was in p prolonged activity of the inward ICa current. The effect was potentiated by a glycine transporter inhibitor −1, ALX 5407. Interestingly, MK-801, DAP5 and CPP, applied when both AMPA and NMDA receptors were active, reduced excitability of the neuron but did not affect its spiking activity. The basis of the effect may lie in the allosteric interaction between AMPA and NMDA receptors. This fact could have enormous implications for the pharmacological control of visceral nociception. However, affirmative experimental evidence is needed to support or disprove the simulated results. Glutamate acting alone at NMDA receptors did not have any depolarizing effect on the motor neuron. AMPA alone caused the generation of high amplitude action potentials of a short duration but failed to sustain periodic excitatory activity. The presence of ACh released from the nerve terminal of the primary neuron at the neuro-neuronal synapse and the activation of nACh was required for normal function of AMPA and/or NMDA receptors. As an explanation for the numerical results we can only propose: (i) a nACh/AMPA receptor co-assembly, and (ii) a dominant regulatory role of acetylcholine in the function of the multimeric protein receptor unit. Thus, binding of ACh molecules triggers conformational changes in the structure of

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the L-Asp-AMPA receptor complex and opens the AMPA ligand ion channel. A subsequent depolarization of the membrane causes the release of the Mg2+ block at the NMDA ligand ion channel which becomes permeable to Ca2+ . An increase in influx of free calcium ions results in prolonged depolarization of the cell. Although conventional experimental methods cannot easily deal with such complexities, it would be of great interest to confirm or refute the suggested mechanism. Our results indicate that AMPA alone augmented contractility of the longitudinal smooth muscle in a longitudinal smooth muscle-neuronal chain preparation. Co-joint activation of nACh, AMPA and NMDA receptors prolonged contractions and increased their strength. The effect was achieved through a sustained firing activity of the neuronal chain and was reversed with hexamethonium and CNQX. Application of selective NMDA antagonists, though, subdued the tonic component of contractions only. These findings are supported by in vitro studies of the effects of AMPA and NMDA receptors on electrically induced contractions of the longitudinal muscle on longitudinal muscle-myenteric plexus preparations in the guinea-pig ileum. In vivo studies of Ifenprodil, a selective NR2B blocker, induced a transient hypertonus of the dog gut which was abolished by atropine, a selective µACh receptor antagonist.

6.3 CO-TRANSMISSION BY ACh AND SUBSTANCE P 6.3.1 Introduction The tachykinins are a family of structurally related neuropeptides, which include three major mammalian endogenous ligands, namely substance P (SP), neurokinin A and neurokinin B, all widely distributed within subpopulations of nerve cells. Neurohistochemical staining and the radio-ligand binding technique demonstrate that 59–66% of cells in the enteric nervous system of the guinea-pig ileum show presynaptic co-localization of SP and ACh. Substance P exhibits a wide range of biological effects by its selective binding to two pharmacologically distinct ionotropic and metabotropic receptors — NK1 and NK3 . Molecular cloning has revealed the tachykinin receptors to be members of the G-protein coupled receptor superfamily, which employs ionositol 1,4,5-triphosphate IP3 signaling cascades. SP at least

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partially mediates slow excitatory synaptic potentials via NK1 receptors and produces neurogenic contractions/relaxations of intestinal smooth muscle. Acetylcholine, on the other hand, initiates fast EPSP through the activation of ligand-gated Ca2+ channels.

6.3.2 Biological Background Both presynaptic and postsynaptic chemical processes are involved in the cotransmission by ACh and SP. The main trigger in the initiation of neurotransmitter release is the entry of Ca2+ ions, which enter the presynaptic nerve terminal through voltage-gated calcium channels during the short period of its depolarization. The large and rapid intracellular increase in Ca2+ causes ACh and SP exocytosis from the vesicular stores via the activation of reactive Ca2+ centers on the vesicles. It is important to note that: (i) cholinergic transmission precedes SP transmission, and (ii) ACh and SP interact in a synergistic rather than an additive manner. The released neurotransmitters, AChf /SPf diffuse into the synaptic cleft, AChc /SPc , where part of the AChc is utilized by acetylcholinesterase enzyme (E) and a part of SPc undergoes enzymatic inactivation by neutral endopeptidase, aminopeptidase, and angiotensin converting enzymes (Enz) through the development of substance-enzyme complexes. Fractions of both neurotransmitters, AChp /SPp , diffuse to the postsynaptic membrane where they bind to acetylcholine (R), and neurokinin type-I, (RNK ) receptors. Subsequent events follow two different pathways. The formation of (ACh–R)-complex reorients the molecular structure of the receptive zone with a consequent increase in the permeability of membrane ion channels and, as a result, the generation of the EPSP. After the released ACh has exerted its effect, it is enzymatically degraded to choline. A general tachykinin conformer of the RNK receptor on the postsynaptic membrane binds to substance P and forms a SP–RNK complex. The latter activates guanine-nucleotide G protein that mediates the effects of SP–RNK complex via the phospholipase C (PhC) signaling pathway. Activated PhC cleaves phosphatidyl inositol 1,4,5-triphosphate, PIP3 , to release two fragments, inositol 1,4,5-triphosphate, IP3 , and diacylglycerol, Dg. These molecules have synergistic actions in the second messenger cascade. The former binds to the IP3 receptor, RER , on the endoplasmic reticulum causing

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the rapid release of Ca2+ into the cytoplasm. The resulting increase in the concentration of calcium permits the formation of a Ca2+ -calmodulin complex that serves as a co-factor in the activation of protein kinase C, PkC. This also requires the presence of a diacylglycerol molecule. The final step in the cascade is the phosphorylation/dephosphorylation of intracellular proteins, Pr, by protein-phosphatase, PPhos, with the consequent alteration of permeabilities of K+ , Ca2+ –K+ , and certain Ca2+ channels on the membrane.

6.3.3 Mathematical Model The above biological mechanisms are adequately described by the following system of equations d[Ca2+ ] dt d[SPv ] dt d[AChv ] dt d[AChf ] dt d[SPf ] dt d[AChc ] dt d[SPc ] dt

= [Ca2+ ]out ϕ∗ − k+5 [Ca2+ ] = −1/2kv [Ca2+ ][SPv ] = −1/2kc [Ca2+ ][AChv ] = kc [Ca2+ ][AChv ] − kd [AChf ] + k+6 [S] ∗ = kv [Ca2+ ][SPv ] − k+2 [SPf ]

= kd [AChf ] − k+p [AChc ] ∗ ∗ + k−3 [SPc ]) = k2+ [SPf ] + [SPc −Enz](k+3 ∗ ∗ ∗ [E0 ] + k+5 [RNK ] + k−5 )[SPc −RNK ] − [SPc ](k+3

d[AChp ] = k+p [AChc ] − [AChp ](k+1 [R 0 ] + k+2 [E0 ]) dt + [AChp ](k+1 [ACh−R] + k+2 [AChE]) + k−1 [ACh−R] + k−2 [AChE] d[ACh−R] = k+1 [AChp ][R 0 ] dt − [ACh−R](k−1 + k+4 + k+1 [AChp ])

(6.5)

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d[SPc −RNK ] ∗ [SPc ][RNK ] − [SPc −RNK ] = k+5 dt ∗ ∗ ∗ + k+6 + k+7 ([G0 ] − [SPc −RNK −G])) ×(k−5 d[AChE] = k+2 [E0 ][AChp ] − [AChE](k−2 + k+3 + k2 [AChp ]) dt d[S] = k+4 [ACh−R] − k+6 [S] + k+3 [AChE] dt d[SPc −Enz] ∗ = k+3 [SPc ][Enz0 ] dt ∗ ∗ ∗ + k+4 + k+3 [SPc ]) −[SPc −Enz](k−3 d[RNK ] ∗ ∗ ∗ + k+6 )[SPc −RNK ] − k+5 [SPc ][RNK ] = (k−5 dt d[SPc −RNK −G] ∗ = k+7 [SPc −RNK ]([G0 ] − [SPc −RNK −G]) dt ∗ [SPc −RNK −G] −k+7 d[PhC] ∗ = k+10 [SPc −RNK −G]([PhC]inact − [PhC]) dt ∗ −k+15 [PhC] d[PIP3 ] ∗ ∗ = −(k+11 + k+12 )[PIP3 ][PhC] dt d[Dg] ∗ = k+12 [PhC][PIP3 ] dt ∗ [Dg][Ca2+ ]4ier ([PhC]inact − [PhC]) −k+18 d[IP3 ] ∗ ∗ [PIP3 ][PhC] − k+17 [IP3 ][RER ] = k+11 dt ∗ ∗ − k−19 )[IP3 −RER ] +(k−17 d[IP3 −RER ] = b5 [IP3 −RER −Ca2+ ] dt −(a2 + a5 )[Ca2+ ]ier [IP3 − RER ] d[Ca2+ ]ier = τ 1 (V0 − c1 V1 [IP3 −RER −Ca2+ ]3 dt ×([Ca2+ ]ier − [Ca2+ ]ER ) ∗2 )) −V3 [Ca2+ ]2ier /([Ca2+ ]2ier − k+13

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d[PkC] ∗ ∗ [Dg][Ca2+ ]4ier ([PhC]inact − [PhC]) − k−8 [PkC] = k+8 dt d[Pr]p ∗ ∗ [PPhos][Pr]p = k+9 [PkC]([Pr] − [Pr]p )k−9 dt [R] = [R 0 ] − [ACh−R]; [E] = [E0 ] − [AChE] [Enz] = [Enz0 ] − [SPc −Enz]; c0 = [Ca2+ ]ier + c1 [Ca2+ ]ER ;

[G] = [G0 ] − [SPc −RNK −G] [RER ] = c2 [IP3 ].

The meaning of parameters is as described above. The initial conditions assume that the neurons are in the resting state. The boundary conditions assume that the excitation of the system is due to the application of an excitatory stimulus of amplitude 100 mV and duration 0.2 s to the soma of the primary neuron. The description provided above constitutes the mathematical formulation of the model of the co-localization and co-transmission of ACh and SP in a pre-synaptic nerve terminal and the co-existence of NK1 and muscarinic receptors on the post-synaptic membrane of a neuron. The numerical algorithm used in simulations is similar to that described in Chap. 4. The results of the numerical calculations are presented below. The dynamics of ACh conversion under normal physiological conditions in a synapse was studied in Chap. 4. Here we shall concentrate on the analysis of the effects of low and high frequency stimulation on substance P transformations. Concentrations of co-transmitted fractions of ACh are given in brackets for comparison.

6.3.4 Effect of Randomly Applied High Frequency Stimuli After the random application of stimuli at frequency 1 Hz to the soma of a neuron, an action potential of amplitude 72 mV was elicited in the vicinity of the terminal ending. The depolarization of the presynaptic membrane activates the short-term influx of calcium ions into the terminal through voltage-dependent Ca2+ channels. The concentration of cytosoilc Ca2+ rises to 1.9–2.41 µM. Some of the ions are immediately absorbed by the buffer system while others diffuse towards the vesicular stores of neurotransmitters. The binding of Ca2+ ions to the active centers on the vesicles initiates SP release. The velocity of SP release is not constant but depends on the concentration of cytosolic Ca2+ . As a response to a single excitation, about 0.2 µM

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0.04

79

0.03

SPv - R ( µM) NK

S P (µM) v

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0.02

0.01

77 0

16

Time (s)

32

0

16

32

Time (s)

Fig. 6.18 Response of a postsynaptic membrane to a coherent release of substance P and acetylcholine.

of vesicular SP is released ([AChv ] = 78.2 mM — 1% of vesicular ACh). Concentrations of cotransmitted fractions of ACh are given for comparison. The maximum concentration of 2.9 µM of SP is released in response to a train of twelve action potentials in 24 s ([SPv ] = 77.1 µM) (Fig. 6.18). The free fraction of neurotransmitter diffuses into the cleft. The concentration of SPc in the cleft varies in time. Changes depend on the dynamics of binding to the receptors and the activity of enzymes. Thus, the max[SPc ] = 0.056 µM (max[AChc ] = 5.38 mM) and min[SPc ] = 0.019 µM are registered at different times in the process. The main part of the SPc reacts with NK1 receptors on the postsynaptic membrane where max[SPc –RNK ] = 0.042 µM and min[SPc –RNK ] = 0.012 µM are formed. The maximum number of activated receptors equals 0.027 µM (6.99 µM). Another part of the SPc undergoes fission by enzymes with the formation of SPc –Enz complex. The complex quickly dissociates into free enzyme and reaction products that are reabsorbed by the nerve terminal to be drawn into a new cycle of substance P synthesis. With the formation of (SPc –RNK –G), the activation of the second messenger system in the underlying neuron with the transformation of the inactive form of phospholipase C to its active form and the production of inositol 1,4,5-triphosphate and diacylglycerol are observed. The concentration of Dg constantly rises, reaching the maximum steady state value, 50 nM. These changes are concomitant with a decrease in the [IP3 ]. Dg serves as a co-factor in the phosphorylation of the intracellular proteins. IP3 is important in

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Fig. 6.19 The dynamics of conversion of substance P and other fractions at the neuronal junction as a response to a single electrical stimulation.

the initiation of the release of calcium ions from the endoplasmic reticulum. It binds to the receptors on the endoplasmic reticulum and forms the (IP–3–RER )-complex, max[IP3 –RER ] = 0.3 µM (Fig. 6.19). The subsequent occupation of Ca2+ -activating sites by the complex induces a flux of Ca2+ ier , which reaches the maximum of 0.745. An increase in free cytosolic calcium and the binding of four molecules of Ca2+ to calmodulin, together with Dg, triggers the activation of protein kinase C. As a result, the maximum 166 nM of phosphorylated protein, PrP, is formed. This declines exponentially to reach picomolar concentrations at t = 324 s. The duration of the presence of phosphorylated protein is 244 s. The latter changes the permeability of K+ , Ca2+ –K+ , and Ca2+ ion channels on the membrane, producing slow potentials of amplitude 20–25 mV and duration 244 s, and thus altering the firing pattern of the adjacent neuron. The final effects of ACh, in contrast, is

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the generation of fast excitatory postsynaptic potentials of very short span, 50–100 ms, and high amplitude, 90 mV. Acting coherently, they superimpose on the long-term potentiation curve of the slow EPSP.

6.3.5 Effect of Low Frequency Stimulation Low frequency stimulation of a neuron with a train of 20 impulses of duration 10 s at frequency 0.025 Hz revealed a quantitative difference in the dynamics of the production of protein phosphatase. The qualitative dynamics of transformations of other fractions of SP involved in the process of release, binding to receptors, and activation of the second messenger system is similar to that described previously. However, the neuron responds with a periodic pattern of IP3 with the subsequent production of (IP3 –RER ) and the activation and release of free calcium ions from the intracellular stores. Thus, the maximum 0.7 µM of inositol 1,4,5-triphosphate is produced and it slowly fluctuates at a frequency 0.01 Hz and an amplitude of 0.35 µM. The max[IP3 – RER ] = 0.33 µM is formed, which results in a periodic release of Ca2+ ier . Its concentration quickly reaches the maximum value 0.745 µM. Free calcium is quickly removed from the cytoplasm and a min[Ca2+ ier ] = 0.01 µM is achieved. As a result, in response to excitation by the first nine stimuli a max[PrP] = 78 nM is being produced. The concentration of PrP declines exponentially to its resting level and to the followng four impulses only a small amount (0.12 µM) of phosporylated protein is being created. There is a slight increase in the concentration of PrP as a response to the succeeding stimuli when a maximum 0.27 µM and 0.18 µM of PrP are produced. As anticipated, slow excitatory signals of variable amplitude 5–25 mV and duration 200–244 s are generated on the postsynaptic membrane of the soma of the adjacent neuron. The co-release of ACh and the activation of muscarinic receptors causes the production of fast excitatory potentials of a constant amplitude 90 mV and duration 50 ms. The result of the cooperative effect of SP and ACh is an irregular summatory long-term EPSP of variable amplitude.

6.3.6 Remarks Despite the long recognition of the co-localization of SP and ACh in autonomic nerves and the familiarity of the concept of cotransmission,

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the limitations of biological experimentation have restricted the further development of this idea. Our approach reveals a wealth of detail that cannot be obtained otherwise. Full quantitative validation from the biological laboratory awaits the development of adequate methods. However, partial comparison to existing sparse experimental data is possible. Thus, Konishi et al. (1992) measured the SP in sympathetic neurons of isolated rate celiac-superior mesenteric ganglia using pressure liquid chromatography and radioimmunoassay. According to their results the neural content of SP is approximately 2 mM as compared to our value of 1.61 mM SPf . The authors also recorded depolarizing responses elicited by SP, which lasted 120–340 s. If we assume that the phosphorylated protein accomplishes the depolarization, then our numerical data indicating the presence of Prp in the postsynaptic structure for 260 s is within the observed range. Acetylcholine and substance P represent transmitters of contrasting character. ACh, released in milliseconds and in millimolar quantities, acts promptly through a membrane receptor linked directly to an ion channel. On the other hand, SP, released over seconds in micromolar quantities, acts on a second messenger system that includes a cascade of cytosolic reactions and protein phosphorylation before the target channels are activated. As a result, ACh elicits a rapid depolarization with the generation of a fast EPSP, while SP produces a slow long-lasting, 4–5 min depolarization. This fact may be directly related to what has been proposed as a synaptic plasticity — a rapid response to an external stimulus with subsequent adaptive changes and processing of the received information. Slight changes in the dynamics of the release of ACh and SP could greatly alter transmission at the synapse. If a regulatory system exists within the neuron that can alter these dynamics, the means by which it does so constitutes a matter of great importance.

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CHAPTER 7

Functional Unit

7.1 INTRODUCTION The processing of information, which is presented in the form of chemical, mechanical, or hormonal and paracrine signals, to the small and large bowel is intrinsically dynamic. The effectiveness and diversity of stereotypical responses of the system, e.g. pendular movements, segmental contractions, peristaltic and gradual reflexes, depend on the inherent activity of its structural components: neurons, smooth muscle cells and their topographical organization, the migrating myoelectric complex. Interstitial cells of Cajal distributed in the gut wall generate pacemaker potentials, smooth muscle syncytia possess electrical properties of the generation of slow waves and spikes, and myenteric neurons provide a spatio-temporal integration and synchronization through a large array of electrochemical synapses across the myenteric nervous plexus. Despite considerable experimental progress in the past decade in the understanding of the intricate biological mechanisms involved in the regulation of gastrointestinal transit, our conceptual knowledge of the basic processes responsible for intestinal movements is still lacking. This is mainly because of the complexity of the analyses that require considering biological phenomena simultaneously at the macroscopic and microscopic levels. Until now, the analysis and interpretation of the myoelectric phenomenon have been represented by in vivo and in vitro experimental studies conducted in a linear fashion. The approach adopts linear information transfer within the multi-hierarchical system and has been demonstrated to be adequate to explain many important biological phenomena. For example, the study of the 244

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effects of different drugs on the biomechanics of an isolated segment of the gut has helped to elucidate intrinsic mechanisms of the coordinated motility of the organ and to assist in the designing of new therapeutics, although with limited clinical success, to treat diseases associated with motor dysfunction. However, biological responses of the gut are shaped by multi-variant nonlinear interactions within the myenteric nervous plexus-smooth muscle syncytia network. Thus, the gut’s ability to respond to external/internal perturbations with a variety of motility patterns depends on “dynamic” rather than “static” interplays which occur simultaneously at different levels, including genes, proteins, cells, tissues and the whole gut. To attain deeper insight into the quantitative behavior of the organ, there is a need to develop a conceptual framework that can reproduce the multiple integrated physiological phenomena and to operate on it in ways that experimental approaches do not allow.

7.2 BIOLOGICAL BACKGROUND The intestine is a long cylindrical tube that extends from the stomach to the cecum. The absolute length of the small bowel generally makes up 80–90% of the entire gut length. It is accommodated in the abdominal cavity; as most of the intestine is loosely suspended by the mesentery, it is looped upon itself. Its diameter is not constant and gradually decreases from the proximal to distal parts. Thus, the diameter of the duodenum varies from 25 to 35 mm: in the jejunum it is 30 mm, and in the ileum 20–25 mm. The intestinal wall is a biological multicomponent composite formed of four histologically identified layers: mucosa, submucosa, muscle layer, and serosa. The mucosa is the innermost layer and is concerned primarily with the processes of digestion and absorption. The submucosa consists mainly of connective tissue and serves a purely mechanical function. The thin collagen fibrils are gathered into large fibers of 4–6 mm in diameter. Septa of connective tissue carrying nerves, blood and lymphatic vessels penetrate into the muscle layer and form a fibrillar sustained stroma for separate muscle bundles. Such structural organization of the “skeletal” three-dimensional network

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maintains a stable organization of the wall and allows the intestine to undergo reversible changes in length and in diameter, offering remarkable properties of stiffness and elasticity. The muscle coat is made of two main layers of musculature: a thicker inner layer, in which the long axis of the smooth muscle cells is orientated circumferentially, and a thinner outer layer in which the long axis of the structural muscular elements is orientated in the longitudinal axis of the bowel. The two layers are distinct and separate, although there exist intermediate bundles that pass from one layer to the other. The smooth muscle cells all lie on planes and run orthogonal to one another in the outer and inner layers. They are not just embedded in the complex fibrillar network of the stroma, but form cell-stroma junctions with it that are mainly of mechanical significance, i.e. an equivocal stress strain distribution during the reaction of contraction-relaxation. The thickness of the muscle layers, h0 , can vary greatly between animal species and the anatomical part of the organ. For example in the non-contracted ileum in humans, h0 ≈ 0.5 mm, in sheep h0 ≈ 150–200 µm, and in guinea pigs and mice h0 < 50 mm. The thickness of the small intestinal muscle coat in the guinea pig is relatively constant along its length. The outermost layer, the serosa, is composed of a thin sheet of epithelial cells and connective tissue. The mechanical activity of muscle layers is subject to regulation by reflexes originating in the enteric nervous system. A basic neural-effector circuit of the small bowel, reproducing a variety of motor patterns, is morphologically uniform and can be viewed as a self-regulatory functional unit (FU) with multiple stable states. Multistability is not a generic property of the gut but requires the precise tuning of parameters consistent with the processing of motility signals. Experiments done to measure the length of contractions in the small bowel revealed that the minimum length where a local contraction can be visualized and recorded myoelectrically varies from 1 to 2 cm. Structurally combined together and arranged by gating mechanisms to respond as a united entity they form a dynamical stable biological system — the intestine. Gating mechanisms are provided by the enteric nervous system and postulated as the controlling factors that determine the distance, velocity and intensity of propagation of the motor pattern.

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7.3 MATHEMATICAL MODEL The above considerations can be specified by the following basic assumptions defining the motor activity of the functional unit and were used in the construction of a model (Fig. 7.1): i) The small intestine is a cylindrical tube formed of similar overlapping myogenic functional units (loci). ii) The wall of the small intestine is composed of two muscle layers, embedded in the connective tissue network. Muscle fibers in the outer layer are orientated in a longitudinal direction of the organ and muscle fibers of the inner layer are arranged in an orthogonal, circular direction. Mechanical properties are different for the two layers but are assumed to be uniform along the wall. iii) The undeformed tubular intestinal locus of length l, and radius r0 , is modeled as a thin soft orthotropic cylindrical shell, h0 /r0 ≈ 0.05. iv) Both muscle layers are assumed to be electrogenic two-dimensional bisyncytia with electrical cable properties. The outer longitudinal layer exhibits anisotropic and the inner circular layer has isotropic electrical properties. Electromechanical coupling in muscle layers with the generation of deformations and forces are considered as a consequence of the evolution of the excitatory wave. v) Smooth muscle cells are spatially distributed autonomous oscillators connected homogeneously and weakly to form a myogenic syncytium; Pacemaker Circular muscle

S2 S1 Longitudinal muscle

Fig. 7.1 An intestinal segment modeled as a thin soft cylindrical shell. The Lagrangian coordinates s˜1 , s˜2 coincide with orientation of the longitudinal and circular muscle fibers, respectively.

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vi)

vii)

viii)

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all oscillators can be divided into pools according to their natural frequencies (
i ) (i = 1, L). If two oscillators have nearly equal frequency (
i ∼
j ) (i, j = 1, L), then they weakly communicate; if (
i =
j ) (i, j = 1, L), then they are strongly connected; if (
i  =
j ) (i, j = 1, N), then they are disconnected. Strong connections among pools are frequency modulated. Each oscillator is in a stable (silent) state. Transition to an excitable (firing) state occurs as a result of an external input. The electrical activity of the myogenic syncytium, either slow wave or bursting, represents the integrated function of ion channels, presumably: voltage-dependent Ca2+ channels of L- and T-types, Ca2+ activated K+ channels and potential sensitive K+ channels, and leak Cl− channels. Slow waves have a natural frequency; the discharge of pacemaker cells (ICC-MP) is responsible for their generation. The pacemaker cell, ICC-MP, modulates the properties of L-type voltage-dependent Ca2+ channels via cholinergic synapses. This effect is mainly chronotropic with an increase in the time of permeability for calcium ions. The synapses are modeled as a null-dimensional threecompartment open pharmacokinetic model (the presynaptic terminal, synaptic cleft and postsynaptic membrane). The dynamics of eletrochemical coupling in a synapse includes: reactions of activation of Ca2+ influx into the presynaptic terminal through voltage-gated channels, neurotransmitter release from the stores, diffusion of the free fraction into the synaptic cleft, binding with receptors located on the postsynaptic membrane, utilization of unreacted neurotransmitter by the enzyme(s), and excitatory postsynatic potential generation. Forces of contraction are a result of the activation of contractile proteins of the smooth muscle cells, which depends on the dynamics of intracellular calcium. Passive relaxation forces are predominantly due to the elastic properties of the connective tissue elements, the collagenous and elastin fibers. They are arranged in a planar network and provide structural support to the muscle cells. Until now no description has been given for the intraluminal bolus as a rheological medium. Therefore, we based our assumptions on the physiological observation that in a fasting state, the bowel contains gas (under pressure p), which is a product of metabolic reactions of intestinal microflora.

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The system of equations of the proposed model includes the functional unit dynamics  

 vs2 ∂ ∂vr ∂(λc − 1) √ a p γ0 − = + T (λc ) + T (λc , λl ) e1r g22 κ ∂t r ∂˜s1 ∂t   
∂(λl − 1) ∂ √ a p + T (λl ) + T (λc , λl ) e2r g11 κ + ∂˜s2 ∂t √ + p g nr  

 vs vr ∂ ∂vz ∂(λc − 1) √ − = + T a (λc ) + T p (λc , λl ) e1s g22 κ γ0 ∂t r ∂t ∂˜s1  
 ∂(λl − 1) ∂ √ a p + T (λl ) + T (λc , λl ) e2s g11 κ + ∂˜s2 ∂t √ + p g ns 
  ∂ ∂vz ∂(λc − 1) √ = + T a (λc ) + T p (λc , λl ) e1z g22 κ γ0 ∂t ∂˜s1 ∂t 
  ∂(λl − 1) ∂ √ a p κ + T (λl ) + T (λc , λl ) e2z g11 + ∂˜s2 ∂t √ + p g nz (0 ≤ s˜1 ≤ 2πr, 0 ≤ s˜2 ≤ l) (7.1) where the radial (vr ), circumferential (vs ) and longitudinal components (vz ) of the wall velocity vector are dr , dt

vr =

vs =

ds , dt

vz =

dz . dt

(7.2)

The rate of elongation [hereafter, the subscripts (l ) and (c) are related to the longitudinal and circular muscle layers, respectively] is calculated as λc =

ds ◦

λl =

,

d s˜1

ds ◦

.

(7.3)

d s˜2

The components and determinant of the metric tensor gij =

∂r ∂r ∂s ∂s ∂z ∂z + + ∂˜si ∂˜sj ∂˜si ∂˜sj ∂˜si ∂˜sj

g = g11 g22 − g12

2

(7.4)

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and the direction cosines of the outward normal n to the surface with respect to the cylindrical j-axis (i = 1, 2; j = r, s, z) are 1 ∂z 1 ∂r 1 ∂s , eir = √ , eis = √ , eiz = √ gii ∂˜si gii ∂˜si gii ∂˜si √ √ (e1s e2s − e1z e2z ) g11 g22 nr = √ g (7.5) √ √ (e1z e2r − e1r e2s ) g11 g22 ns = √ g √ √ (e1r e2s − e1s e2r ) g11 g22 . nr = √ g Here, the following notation is used: λ0 is the linear density of a biomaterial in an undeformed state; T p , T a are the passive and active components, respectively, of the total force (Tc,l ) of the wall; κ is rheological parameter; p is intraluminal pressure; ϕl , ϕc are the electrical waves of depolarization of the longitudinal and circular muscle layers; s˜1 , s˜2 are Lagrangian coordinates of the bioshell. p The passive (Tc,l ) components are calculated from p

Tc,l =

∂γ0W . ∂(λc,l − 1)

(7.6)

W is the strain energy density function of the connective tissue and passive muscular components given by 1 γ0W = [c1 (λl − 1)2 + 2c3 (λl − 1)(λc − 1) + c2 (λc − 1)2 2 + c14 exp(c4 (λl − 1)2 + c5 (λc − 1)2 + 2c6 (λl − 1)(λc − 1))].

(7.7)

a ) components, the empirical relationship is assumed For the active force (Tc,l  0, [Ca2+ ] ≤ 0.1 µM    2+ 4 2+ 3 c7 + c8 [Ca ] + c9 [Ca ] a (7.8) = Tc,l  + c10 [Ca2+ ]2 + c11 [Ca2+ ], 0.1 < [Ca2+ ] ≤ 1 µM   max T a , [Ca2+ ] > 1 µM a where Tc,l is the active force generated by the smooth muscle syncytium; c1−11 are mechanical constants.

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The characteristic feature of soft shells is the possibility of the simultaneous coexistence of unstressed, uniaxial and biaxial stress-strained zones. The creaseless shape occurs at λl,c > 1.0; in the case of the development of creases, when λl ≤ 1.0 or λc ≤ 1.0, the wrinkled area is modeled by a “smooth zone” made up with a system of unbound filaments. This is determined by requirements which relate geometrically to the conservation of smoothness of the surface of the bioshell, and force conditions relating to the continuity of membrane forces. Thus, if creases appear on the surface of the bioshell, p then for Tc,l at the uniaxial stress-strained state, it is assumed 0, λ(c,l) ≤ 1 p T(c,l) = (7.9) c12 (exp(c13 (λ(c,l) − 1) − 1) , λ(c,l) > 1. The dynamics of propagation of the electrical wave, ϕl , along the anisotropic longitudinal muscle layer is defined as ∂ϕl ∗ = Im1 (˜s1 , s˜2 ) + Im2 (˜s1 − s˜1 , s˜2 − s˜2 ) + Iionic (7.10) ∂t where Im1 , Im2 are the transmembrane currents per unit volume

  ∗ g0,˜ −2(µs˜1 − µs˜2 ) d˜s1 Gs˜2 s2 Im1 (˜s1 , s˜2 ) = Mvs + arctan (1 + µs˜1 )(1 + µs˜1 ) d˜s2 Gs˜1 Gs˜1
∗ 
∗ 
∂ g0,˜s2 ∂ϕl ∂ g0,˜s1 ∂ϕl + × ∂˜s2 λl ∂˜s2 ∂˜s1 λc ∂˜s1  µs˜1 − µs˜2 (7.11) Im2 (˜s1 , s˜2 ) = Mvs s 2π(1 + µs˜1 )(1 + µs˜1 ) Cm

(˜s2 − s˜2 )/Gs˜2 − (˜s1 − s˜1 )/Gs˜1 2 (˜s1 − s˜1 )/Gs˜1 − (˜s2 − s˜2 )/Gs˜2
∗ 
∗ 
∂ϕl ∂ϕl ∂ g0,s ∂ g0,s 1 2 + d˜s1 d˜s2 . × ∂˜s1 λc ϕ˜s1 ∂˜s2 λl ϕ˜s2

×

Here µs˜1 = Gs˜1 =

∗ ∗ g0,˜ s1 + gi,˜s1

λc

,

∗ g0,˜ s1 ∗ gi,˜ s1

,

Gs˜2 =

µs˜2 =

∗ g0,˜ s2 ∗ gi,˜ s2

∗ ∗ g0,˜ s2 + gi,˜s2

λl

,

 G = Gs˜1 Gs˜2 ,

(7.12)

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and the following notation is used: Cm is the capacitance of smooth muscle; g∗i,˜s1 , g∗i,˜s2 , g∗0,˜s1 , g∗0,˜s2 are the maximal intracellular [subscript (i)] and interstitial space [subscript (0)] conductivity of the longitudinal and circular muscle layers in the longitudinal and circumferential directions, respectively; Mvs is the membrane volume-to-surface ratio, I ∗ionic is the total ionic current given by ∗ Iionic = g¯Na m ∗3 h ∗ (ϕ(c,l) − ϕ¯ Na )

+ g¯Kn ∗4 (ϕ(c,l) − ϕ¯ K ) + g¯θ (ϕ(c,l) − ϕ¯ θ ).

(7.13)

are the maximal conductances; m∗ , h∗ , and n∗

g¯Na , g¯K , g¯θ are the probabilities of opening of the ion gates, and ϕ¯ Na , ϕ¯ K , ϕ¯ θ are the reversal potentials of Na+ , K+ , and Cl− currents, respectively, in smooth muscles. The dynamics of change in the probability variables is calculated as dy = α∗y (1 − y) − βy∗ y (7.14) dt where y refers to m∗ , h∗ and n∗ , respectively. The activation, α∗y , and deactivation, βy∗ , parameters for smooth muscle satisfy α∗m =

0.005(ϕ(c,l) − ϕ¯ m ) exp(0.1(ϕ(c,l) − ϕ¯ m )) − 1

∗ βm = 0.2 exp(ϕ(c,l) + ϕ¯ m )/38

α∗h = 0.014 exp − (ϕ¯ h + ϕ(c,l) )/20 βh∗ =

0.2 1 + exp 0.2(ϕ¯ h − ϕ(c,l) )

α∗n =

0.006(ϕ(c,l) − ϕ¯ n ) exp 0.1(ϕ(c,l) − ϕ¯ n ) − 1

(7.15)

βn∗ = 0.75 exp(ϕ¯ n − ϕ(c,l) ). The dynamics of propagation of the electrical wave, ϕc , along the isotropic circular muscle layer is
∗ 
∗  Mvs ∂ϕc ∂ g0,˜s1 ∂ϕc ∂ g0,˜s1 ∂ϕc ∗ Cm = + − Iionic (7.16) ∂t 1 + µs˜1 ∂˜s1 λc ∂˜s1 ∂˜s2 λl ∂˜s2 where the above-mentioned abbreviations and relationships for I ∗ionic are used.

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The intraluminal pressure, p, changes according to the adiabatic law p = p0 V 1.41 .

(7.17)

p0 is the value of p at t = 0; V is the ratio of a current intraluminal (V ) volume of the bioshell to its initial value (V0 ): V = V0 /V . The myoelectrical activity in a locus of the small bowel is governed by the dynamics of voltage-dependent Ca2+ channels (L- and T-type), mixed Ca2+ – K+ and K+ channels and chloride channels. Thus, the system of equations is a linear sum of the five ionic currents αCm

dϕ T L ∗ ∗ = −(ICa + ICa + ICa−K + IK∗ + ICl ). dt

(7.18)

Here Cm is the membrane capacitance, ϕ is the membrane potential and I(i) are ionic currents carried by ITCa , ILCa , the fast and slow inward calcium currents via T- and L-type Ca2+ channels, respectively T T ˜ − ϕ˜ ∗ ) ICa = g˜Ca m ˜ I3 h(ϕ Ca L L ∗ = g˜Ca x˜Ca (ϕ − ϕ˜ Ca ). ICa

(7.19)

I ∗Ca−K , I ∗Cl are the outward calcium-activated potassium and voltageactivated potassium currents IK∗ = g˜K∗ n˜ 4 (ϕ − ϕ˜ K∗ ) ∗ = ICa−K

∗ g˜Ca−K [Ca2+ ](ϕ − ϕ˜ K∗ )

(7.20)

0.5 + [Ca2+ ]

and ICl∗ is the chloride current ∗ ∗ ∗ = g˜Cl (ϕ − ϕ˜ Cl ). ICl

(7.21)

∗ ∗ , ϕ˜ K∗ , ϕ˜ Cl are the reversal potentials for the respective currents; [Ca2+ ] Here ϕ˜ Ca T is the intracellular concentration of calcium ions in the functional unit; g˜Ca , 2+ 2+ L ∗ ∗ ∗ g˜Ca , g˜K , g˜Ca−K , g˜Cl are, respectively, the maximal conductances of Ca , Ca ˜ n˜ and x˜Ca are dynamic ˜ h, activated K+ channels and a Cl− conductance; m,

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variables of the channels whose kinetics is m ˜ I = α˜ m /(α˜ m + β˜ m ) d h˜ dt d n˜ dt d x˜Ca dt dCa2+ dt

˜ ˜ − β˜ h h)/αλ = (α˜ h (1 − h) ˜ − β˜ n n)/αλ ˜ = (α˜ n (1 − n)

(7.22)

= (1 + exp(−0.15(ϕ + 50))−1 − x˜Ca )/ατx˜Ca =

ρ (Kc x˜Ca (ϕ¯ Ca − ϕ) − [Ca2+ ]) α

where α˜ m =

0.1(50 − ϕ) ˜ exp(5 − 0.1ϕ) ˜ −1

˜ β˜ m = 4 exp (25 − ϕ)/18 ˜ α˜ h = 0.07 exp (25 − 0.1ϕ)/20 ˜ −1 β˜ h = (1 + exp (5.5 − 0.1ϕ)) α˜ n =

(7.23)

0.01(55 − ϕ) ˜ exp(55 − ϕ)/10 ˜ −1

˜ β˜ n = 0.125 exp (45 − ϕ)/80. Here ϕ˜ = (127ϕ + 8265)/105; τxCa is the time constant, and ρ, Kc are parameters referred to the dynamics of Ca2+ channels; α is a constant; parameters α˜ and β˜ have the same meaning as above. The evolution of L-type voltage-dependent Ca2+ channels is given by L L g˜Ca = δ(ϕp+ )g˜Ca

(7.24)

where δ(ϕp+ ) = 1 when δ(ϕp+ ) ≥ ϕp∗ , the threshold value, and δ(ϕp+ ) = 0 otherwise. The dynamics of the action potentials at the ICC (ϕ4 ) is described by the function of voltage-dependent N-type Ca2+ , Ca2+ -activated K+ , and Na+ ,

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K+ and Cl− channels: Cs

dϕ(4) = −(ICa + ICa−K + INa + IK + ICl ) + Iext(4) . dt

(7.25)

Cs is the membrane capacitance, I · · · are ionic currents carried through different ion channels and Iext(4) is the external membrane current. The equations for ion currents are gCa(i) z(ϕ(4) − ϕCa ) 1 + ζ[Ca2+ ]n gCa−K ρ∞ [Ca2+ ]n = (ϕ(4) − ϕCa−K ) 0.5 + [Ca2+ ]n 3 = gNa(4) mNa hNa (ϕ(i) − ϕNa )

ICa = ICa−K INa

(7.26)

IK = gKnK4 (ϕ(4) − ϕK ) ICl = gCl(i) (ϕ(4) − ϕCl ). Here ϕCa , ϕCa−K , ϕNa , ϕK , ϕCl are the reversal potentials for the respective currents; gCa(4) , gCa−K , gNa(4) , gK , gCl(4) are the maximal conductances of Ca2+ (N-type), Ca2+ -activated K+ , Na+ , K+ and Cl− channels; ζ is the parameter of calcium inhibition of the Ca2+ channels; [Ca2+ ]n is the intracellular concentration of free calcium; z, ρ∞ , mNa , hNa and nK are dynamic variables of the ion channels whose kinetics is dz = (z∞ − z)/τz dt dhNa = λh (h∞ − hNa )/τh dt dnK = λn (n∞ − nK )/τn dt 1 ρ∞ = 1 + exp 0.15(ϕ(4) + 47) mNa = m∞ (ϕ(4) ) y∞ = αy∞ τy

(y = mNa , hNa , nK )

τy = 1/(αy∞ + βy∞ ) αm∞ =

0.12(ϕ(4) + 27) 1 − exp(−ϕ(4) − 27)/8

(7.27)

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βm∞ = 4 exp(−ϕ(4) − 47)/15 αh∞ = 0.07 exp(−ϕ(4) − 47)/17 βh∞ =

1 1 + exp(−ϕ(4) − 22)/8

αn∞ =

0.012(ϕ(4) + 12) 1 − exp(−ϕ(4) − 12)/8

βn∞ = 0.125 exp(−ϕ(4) − 20)/67 z∞ =

1 . 1 − exp 0.15(ϕ(4) + 42)

Changes in the free cytosolic calcium concentration yield 0.2z(ϕCa − ϕ(4) ) d[Ca2+ ]n − 0.3[Ca2+ ]n . = dt 1 + ζ[Ca2+ ]n

(7.28)

The membrane current Iext(4) = ϕ4 /Rs ; Rs is the specific resistance. The dynamics of the propagation of the wave of depolarization, ϕ∗ , along the unmyelinated fiber satisfies
 ∗ 1 ∂ ∂ϕ∗ a ∂ϕ 2 Cm = a (s) − Iionic (0 ≤ s ≤ L) (7.29) ∂t 2Ra ∂s ∂s where C am is the specific capacitance of the nerve fiber, Ra is the membrane resistance and a(s) is the diameter of the fiber, s is the Lagrange coordinate; L, L0 are the length of the fiber and the synaptic terminal, respectively. The total ion current is ∗ ∗ ∗ ∗ Iionic = gNa m 3 h(ϕ∗ − ϕNa ) + gK∗ n 4 (ϕ∗ − ϕK∗ ) + gCl (ϕ∗ − ϕCl )

(7.30)

where g∗Na , g∗K , g∗Cl are the maximal conductances of Na+ , K+ and Cl− channels, respectively, and m, n, h are the probabilities of opening of these ∗ ∗ channels; ϕNa , ϕK∗ , ϕCl are the reversal potentials of sodium, potassium and chloride currents. The activation and deactivation of the ion channels are given by dy = αy (1 − y) − βy y dt

(y = m, n, h).

(7.31)

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α is the rate at which the channels switch from a closed to an open state and β is the rate for the reverse. They depend only on the membrane potential αm =

0.11θ(ϕ∗ − 25) 1 − exp(−ϕ∗ − 255)/10

βm = 4.1θ exp(−ϕ∗ /18) αn =

0.005θ(ϕ∗ − 10) 1 − exp(10 − ϕ∗ )10

(7.32)

∗

βn = 0.075θ exp(−ϕ /80) αh = 0.02θ exp(−ϕ∗ /20) βh =

0.5θ 1 + exp(−ϕ∗ − 30)/10

where θ is the temperature (◦ C). The system of kinetic equations for the cycle of chemical transformations of acetylcholine at the synapse is given by d[Ca2+ ] = [Ca2+ ]out ϕ∗ − k+5 [Ca2+ ] dt d[AChv ] = −kc [Ca2+ ][AChv ] dt d[AChf ] = kc [Ca2+ ][AChv ] − kd [AChf ] + k+6 [S] dt d[AChc ] = kd [AChf ] − k+p [AChc ] dt d[AChp ] = k+p [AChc ] − [AChp ](k+1 [R 0 ] + k+2 [E0 ]) dt + [AChp ](k+1 [ACh−R] + k+2 [AChE]) + k−1 [ACh−R] + k−2 [AChE] d[ACh−R] = k+1 [AChp ][R 0 ] − [ACh−R] dt × (k−1 + k+4 + k+1 [AChp ])

(7.33)

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d[AChE] = k+2 [E0 ][AChp ] dt − [AChE](k−2 + k+3 + k+2 [AChp ]) d[S] = k+4 [ACh−R] − k+6 [S] + k+3 [AChE] dt [R] = [R 0 ] − [ACh−R] [E] = [E0 ] − [AChE] where k−,+ are the constants of backward and forward chemical reactions, kc is the affinity constant, kd is the constant of diffusion, k+p is the diffusion velocity of ACh on the subsynaptic membrane, AChv is vesicular acetylcholine and its free (AChf ) and postsynaptic (AChp ) fractions, R are receptors, E is acetylcholin-esterase enzyme (superscript 0 indicates the initial concentration), ACh−R is acetylcholine-receptor complex, AChE is acetylcholine-enzyme complex; S is the product of chemical reactions; and (Ca2+ )out is the external calcium ion content. The generation of the fEPSP is calculated from Cp

dϕp+ dt

+ ϕp+ (−
[ACh−R] + Rv−1 ) =

ϕR+ . Rv

(7.34)

Cp is the capacitance of the subsynaptic membrane;
is the empirical constant; Rv is the general resistance of the extra synaptic structures; ϕR+ the resting potential. The initial conditions assume that the system is in the resting state t = 0: vr = vs = vz = 0 ϕl = ϕc = ϕd = ϕ1 = ϕ∗ = ϕp+ = 0.

(7.35)

The oral (left) and aboral (right) ends of the bioshell are rigidly fixed t > 0: r(˜s1 ) = r(˜s2 ) = r0 s(˜s1 ) = s(˜s2 ) = 0,

z(˜s1 ) = 0,

z(˜s2 ) = l.

(7.36)

The system (7.1)–(7.36) provides a description of the following processes: i) discharge of a pacemaker cell and/or activation of mechanoreceptors of primary sensory neurons; ii) action potential generation and propagation along unmyelinated axons of the sensory neurons;

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iii) discharge of the soma of the primary sensory neurons and propagation of excitation waves along axons to the synaptic terminal on the motor neurons; iv) release of ACh at the synapse and generation of the EPSP on the soma of motor neurons; v) propagation of the action potential along the axon to the synaptic connection on the longitudinal and circular smooth muscle syncytia; vi) generation of depolarization waves and connection of spatially distributed weak oscillators on the surface of smooth muscle syncytia; vii) development of electromechanical reactions with contractionrelaxation of longitudinal and circular smooth muscle layers; and viii) activation of mechanoreceptors (step (i)).

7.4 NUMERICAL ALGORITHM Let ω¯ h = {(s1i , s2j ) = (ih 1 , jh 2 ), i = 0, N1 , j = 0, N2 , N1 h1 = 2πR0 , N2 h2 = 1} be a mesh on the cylinder S with the boundary γ = ω¯ h ∩ ∂S. Assume ωτ = {jτ, j = 0, 1, . . . ; τ > 0} is the temporal mesh in R+ 1 . The unknown functions are defined at the points: ωh × ωτ . The myoelectrical pools are located at the subregions: Kh = Kh,i,j = {(s1,j , s1,i+1 ) × (s2,j , s2,j+1 )}. The additive finite-difference scheme of alternating directions was adopted to solve (7.10) j+1/2

Cm

ϕh

j

− ϕh

τ j+1

Cm

ϕh

    j+1/2 j j+1/2 2 2 ∗ − 0.5I¯m1,h ∇h,1 ϕh + ∇h,2 ϕh + Ih,ionic ϕh =0 j+1/2

− ϕh τ

    j+1/2 j+1 j+1/2 2 2 ϕh + ∇h,2 ϕh = Im2,h ϕh − 0.5 ∇h,1 (7.37)

where
β ϕh

ϕs¯α sα

= ϕh |t =βτ ,

2 ∇h,α ϕ

=

g0∗sα



ϕs¯α sα , α = 1, 2; λh,α {ϕ(s + hα eα0 ) − 2ϕ(s) + ϕ(s − hα eα0 )} = , α = 1, 2 hα2

(7.38)

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are finite-difference spatial derivatives in the directions sα ; e0α is unit vector along the sα -axis; and λhα are mesh-point approximations of the rates of j+1/2 elongations λc or λl , respectively. I˜m1,h and I˜ym1,h (ϕh ) approximate functions Im1 , Im2 (ϕ) on the temporal level t + τ/2. They are calculated using the j+1/2 2 bilinear interpolation of ∇h,α ϕh on a finite element Kh . The Chebyshev’s cubic formulas with 3×3-axis are employed to calculate the integrals on Kh . ∗ Ih,ionic is calculated using the Euler implicit scheme of first-order accuracy: The following iterative method is used to solve (7.1)  Cm

k+1j+1/2 ϕ  h

j

− ϕh

τ

 j+1/2

2 k+1  − 0.5Im1,h (∇h,1 ϕh

j

2 − ∇h,2 ϕh )

(7.39) k j+1/2

∗ + Ih,ionic (ϕ h

) = 0,

0 j+1/2

k = 0, 1, . . . , ϕh

.

Vectors v = (vr , vs , vz ) and u = (r, s, z) are calculated from ˜ t + (vh , uh )(t ) +  c (uh )(t ) = 0 γ0 vh,t (t ) + R λ(u) uh,t = vˆ h (t ), uh |γ = 0,

t ∈ ωτ

uh |t =0 = u0 ,

(7.40)

vh |t =0 = 0.

Here λ˜ = (λh,c , λh,l ) is the approximation of the vector (λc , λl ); ηt (t) = (η(t + τ) − η(t))/τ is the finite difference partial derivative in the direction t. The operators R and  are approximations of corresponding parts of the spatial operator. The finite element method with bilinear finite elements is used to build R, ;  c is the penalty operator for c : 



 c (uh ), ηh =


 s

max{0, 1 − σ} √ g (nr ηrh + ns ηsh + nz ηzh )dS εp

∀ηh = (ηrh , ηsh , ηzh ) :

η h |γ = 0



(7.41)

 where σ = R/ rh2 + (zh − Zc )2 , εp > 0 is the penalty parameter. Integrals of each rectangle Kh in the region S are calculated using the formulas of central rectangles.

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Vectors uh (t + τ), vh (t + τ) are calculated from the following two-step iterative process: k+1 h

i) compute v

k+1

B

v

k

h

− vh k = −γ0−1 r ν (7.42)

k k k r = (v h − vh )(t ) + R(λ (uh ) − λ (uh )) k

k

k

+ τ(v h , uh + τ v h )(t ) + τ c (uh + τ v h )(t ), k+1 h

ii) compute u

k+1

u

k+1

h

= uh + τ v h .

(7.43)

Here B = diag(B1 , B2 , B3 ) and Bβ are operators of the form Bβ = (I + ωωk Bs1 )(I + ωωk Bs2 ),

β = 1, 2, 3.

(7.44)

The operators Bsα η = −ηs¯α sα

(α = 1, 2)

(7.45)

act in the specific spaces of the mesh functions. They are defined by the boundary conditions (Dirichlet for Bs2 and periodic for Bs1 ), ωk = max{kc , kl }, ω > 0, ν > 0, parameters of iteration. For k = 0, 1, . . . , and sufficiently small εp > 0 the steps (i) and (ii) are repeated until the condition k+1

max{ v

k

h

k+1

− v h ,  u

k

h

k+1

k+1

− uh } ≤ ε max{ v h ,  u h , }

(7.46)

is satisfied. The finite difference schemes have second-order accuracy with respect to spatial and time variables.

7.5 ELECTROMECHANICAL WAVE PHENOMENON Under resting physiological conditions, the functional unit displays stable periodic electrical activity, which is a result between the dynamic interplay between the ion currents through the voltage-dependent L- and T-type Ca2+ , Ca2+ -activated K+ , potential sensitive K+ , and leak Cl− channels. The inward

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T-type Ca2+ current shows slow dynamics and a maximum amplitude of 0.27 nA. The L-type Ca2+ current of short duration and high rate has a maximum amplitude of 0.21 nA. A transient, voltage-dependent K+ current has a small amplitude, max I ∗K = 0.086 nA. A Ca2+ -dependent K+ current demonstrates a dynamics similar to the dynamics of the calcium current. The maximum I ∗Ca−K = 0.035 nA is recorded. As a result of the balanced activity of the ionic currents, slow oscillations of the membrane potential ϕ are observed. The oscillations have a constant amplitude of approximately 25 mV and a frequency of 0.18 Hz. In the resting state, the concentration of intracellular calcium, [Ca2+ ]i , remains at a low level, min[Ca2+ ]i = 0.36 mM. A small part of the free Ca2+ i reacts with a contractile protein system of the smooth muscle to produce small regular rhythmic contractions. They are concurrent in phase and time with the dynamics of Ca2+ oscillations. There are no high amplitude action potentials and no changes in the shape of the functional unit at this stage. Discharges from the interstitial cell of Cajal at a constant frequency 0.04 Hz precedes electrochemical coupling at the ICC-MP-smooth muscle syncytium synapse. The wave of depolarization activates a short-term influx of calcium ions into the presynaptic terminal. The concentration of cytosolic Ca2+ quickly rises to its maximum of 19.4 µM. Some of the ions are immediately absorbed by the buffer system while others actively diffuse towards vesicles containing acetylcholine, AChv . They bind with the active centers and initiate the release of the neurotransmitter. The velocity of AChv release is not constant but depends on the concentration of cytosolic Ca2+ . Thus, at the beginning of the process, the rate of the acetylcholine free fraction increase equals 0.5 µM/ms and reaches a maximum of 1.6 mM/ms. With the achievement of an AChf concentration of 10.6 mM, acetylcholine release into the synaptic cleft begins. According to the calculated data, max[AChc ] = 5.38, equivalent to 50% of AChf , diffuses from the presynaptic terminal. The main part of AChc reaches the postsynaptic membrane and reacts with the choline receptors on the soma of the secondary sensory neuron. Fast EPSP generation starts with the beginning of (ACh–R) active complex development. It increases as a step function with the rise in concentration of the active complex and achieves its maximum at ϕp+ = 87.1 mV. Part of the postsynaptic acetylcholine undergoes fission by acetylcholine esterase enzyme. The maximum concentration of (AChE)-complex is 0.047 mM. The complex then quickly dissociates into enzyme and choline.

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The generated ϕp+ exceeds the threshold value for the activation of smooth muscle syncytia, longitudinal and circular muscle layers. The waves of depolarization, ϕl and ϕc , have an average amplitude of 68–70 mV. They propagate along the surface of the longitudinal and circular smooth muscle layers. The velocity of the wave of excitation is not constant but associated with the mechanical deformation of the wall. Thus, at the beginning of the process, while the deformations of the wall are small, the wave ϕl has a velocity 0.45 cm/s and ϕc is 0.5 cm/s. With the development of active forces of contraction and deformation of the gut, the velocities of propagation decrease to 0.37 cm/s for ϕl and 0.38 cm/s for ϕc . At this time the functional unit of the gut accommodates three waves of depolarization (Fig. 7.2) (because of the similarities in appearance of the results only the dynamics of the wave ϕl is shown). The anisotropic electrical cable properties of the longitudinal muscle layer make the front of the wave ϕl non-uniform, while in the electrically isotropic circular layer, ϕc remains uniform throughout. The waves ϕl and ϕc establish strong connections among spatially distributed oscillators on the syncytia. As a result of activation of L-type Ca2+ channels, cyclic transitory changes in their myoelectrical patterns occur: a stable slow wave mode, which is characteristic of the resting state, transforms to bursting chaos with the occurrence of irregular spikes of the crests of slow waves, followed by regular bursting with the generation of high amplitude action potentials on the crests, and finally converts back to a slow wave mode. The action potentials generated have a frequency of 19.5 Hz and a maximum amplitude of 72 mV. During this period, there is a massive influx of extracellular calcium ions through the L- and T-type Ca2+ channels. The concentration of the free internal calcium [Ca2+ ]i rises significantly to 0.59 mM. The free Ca2+ ions activate the contractile protein system with the development of the propagating mechanical wave of contraction-relaxation. The contractions are concomitant in phase and time with the dynamics of calcium oscillations. The active force of contraction has an amplitude of T al = 17.2 g/cm in the longitudinal layer, and T ac = 26.9 g/cm in the circular layer. The waves of contraction propagate in the aboral direction at a velocity 0.08 cm/s. The dynamics of the total forces, Tl and Tc , are shown in Fig. 7.3. The waves have a length 0.3–0.35 cm and amplitude 15.6 g/cm for Tl and 20.6 g/cm for Tc , respectively. Because of the electrical anisotropy of the longitudinal layer, the anterior front of the

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t =3.8 s

t = 6.8 s

t = 13.3 s

t = 20.8 s

t = 30.0 s

t =33.0 s

Fig. 7.2 Development of the wave of depolarization in the longitudinal smooth muscle syncytium at the times as indicated. The point (˜s1 = 0, s˜2 = 0) corresponds the location of the pacemaker.

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t = 3.8 s

t = 6.8 s Fig. 7.3 Dynamics of the development of the total forces in the longitudinal (left) and circular (right column) smooth muscle syncytia.

wave Tl has the form of an ellipse while the front of the Tc has the shape of a circle. Changes in the stress state lead to deformation of the segment of the gut. Thus, at t = 13.3 s the central part of the organ becomes overextended: λl = 1.35, λl = 1.42. There is a reduction of the maximum amplitude of the waves ϕl and ϕc to 55.7 mV. As a result, there is a loss of connectivity among oscillators and a disruption of coordinated electromechanical activity in that part. At t = 30 s the wave of depolarization, ϕl , splits into two separate waves in the middle part of the functional unit (Fig. 7.4). It is important to note that at this stage of the dynamic process the deformation of elongation is highest in this region. The velocity of the propagation of ϕl is not equal but

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t =20.8 s

t = 33.0 s Fig. 7.3 (Continued )

is slower along the stretched regions of the bioshell. As the waves of average amplitude 55.7 mV reach the right boundary, they vanish while their tails collide with a significant rise in amplitude to 70.8 mV. After reflection from the boundary a spike of amplitude 65 mV is generated that propagates backwards. It reaches the middle of the segment and produces a circular wave. At t = 42.2 s the wave ϕl covers the entire surface of the gut. Again, two tails are seen in the proximity of the right boundary. They again collide and another set of spiral waves is generated. The process is repeated as above. The amplitude of the spiral waves is above 65 mV and they provide strong connections among the spatially-distributed oscillators. Therefore, the pattern of electrical activity causes retrograde propagation of the mechanical wave of contraction-relaxation in the longitudinal syncytium. The above changes

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t = 40.6 s

t = 42.2 s

t = 43.6 s

t = 44.6 s Fig. 7.4 Generation of self-oscillatory myoelectrical activity in the longitudinal smooth muscle layer of the functional unit. Results are displayed on the cylindrical envelope (left column) and the surface of the bioshell (right column) for comparison.

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with the formation of the self-sustained spiral waves are not produced in the electrically isotropic circular smooth muscle layer. There is no split of the wave of depolarization, ϕc , at any time in the dynamic process. The waves ϕc propagate without any disruption and, after they reach the right unexcitable end, they disappear.

7.6 EFFECT OF LIDOCAINE N-ETHYL BROMIDE QUARTERY SALT QX-314 In an attempt to convert the system back to normal, i.e. to the aboral spread of the electromechanical wave, we simulated the effect of a sodium channel blocker, Lidocaine. Quaternary Lidocaine derivative blocks both fast, Na+ -dependent action potentials and voltage-dependent, non-inactivating Na+ conductance. Its effect has been achieved in the system by decreasing the permeability of sodium channels: from gNa = 80 mSm/cm2 to 70 and 50 mSm/cm2 , respectively. A slight decrease in gNa does not affect the dynamics of the electrochemical processes. The self-excitatory spiral wave activity persists. Further decrease in the permeability of sodium channels, gNa = 50 mSm/cm2 , abolishes the generation of the excitation wave ϕl , and the spiral wave phenomenon. The segment of the intestine then returns to a resting state.

7.7 EFFECT OF CHANGES IN EXTRACELLULAR Ca2+ A decrease in extracellular calcium affects multiple levels of the system: processes of electrochemical coupling at the ICC-MP-smooth muscle junction, the dynamics of the waves ϕl and ϕc , and contractile activity of smooth muscle syncytia. These effects are dose dependent. In the case where [Ca2+ ]ext = 0.5 mM (1 mM is the normal value) there is a significant reduction in the amount of ACh released from vesicular stores. The concentration of the free fraction of ACh is 0.9 mM and is only 1% of the whole stored neurotransmitter. However, it is sufficient to sustain the electrochemical coupling at the synapse with the generation of an excitatory postsynaptic potential with an amplitude of 85.8 mV. Slow waves retain their periodic oscillatory pattern of activity with unaffected amplitude, 25 mV. There is a slight increase in frequency and decrease

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in wavelength. There is a reduction in maximum intracellular calcium ions to 0.35 mM with a concurrent decrease in the active force of contraction by 48% (9.9 g/cm) and 65% (11.1 g/cm) from the norm in the longitudinal and circular smooth muscle layers, respectively. The level of generated ϕp+ can sustain the electrical waves of excitation, ϕl and ϕc , which propagate along the smooth muscle membrane and establish strong connections among oscillators in the syncytia. However, no spiral wave phenomenon can be seen under this condition. In the case where [Ca2+ ]ext = 0.1 mM, the process of acetylcholine release is completely suppressed. The pacemaker cells are chemically uncoupled with smooth muscle syncytia. No slow wave activity and no mechanical activity are recorded in the smooth muscle. The functional unit of the intestine is in a resting state.

7.8 EFFECT OF CHOLINERGIC ANTAGONISTS The pharmacological effects of cholinergic antagonists are acquired through a deactivation of the neurotransmitter-receptor binding process. With the model proposed we have been able to study the response of the system to application of antagonists in a wide range of concentrations. According to the numerical results, the concentration of compounds, 0.13–0.26 mM, causes a total blockade of synaptic transmission. The ICC-MP and both smooth muscle syncytia become pharmacologically disconnected. However, their physiological, electromechanical properties remain intact. Treatment of the gut with a lower concentration, 0.026 mM, produces a marked decrease in the number of choline receptors accessible to ACh. Subsequently, there is a reduction in the level of depolarization of the postsynaptic membrane. The ϕp+ produced has an amplitude of 69.6 mV and generates the normal response of electromechanical wave activity in the functional unit.

7.9 REMARKS Since the “rediscovery” of the migrating myoelectrical complex phenomenon in 1969, [for a critical historical review, see Wingate (1981)], this subject has been the focus of enormous scientific research and debate. Much in vivo and

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in vitro work has been carried out to investigate into the physiology of the phenomenon and its role in the pathogenesis of gastrointestinal dysfunction. Thus, it has been postulated that the migrating myoelectrical complex, which is a merely in vivo phenomenon, is a result of interrelated electrical (slow wave and spiking) and mechanical (contractions of smooth muscle) processes. Combined at the level of smooth muscle syncytia they form bands of regular contractions, which periodically migrate along the gastrointestinal tract. Based on a pure visual perception and on a priori assumptions of the propagation rather than on a scientific analysis of the event, the terminology and quantitative measures from the physics of waves have been adopted to describe the dynamics of migration. However, whether slow waves and the migrating complex are indeed propagating phenomena has never been questioned. If one assumes that slow waves do really propagate, then a simple analysis of the experimental data indicates that the wavelength of the slow wave varies within 2–110 cm! This result is based on the fact that the frequency of slow waves varies between 0.01–0.3 Hz and velocity ranges from 0.6–1.1 cm/s. Even if we disregard wide-range variability, which is not unusual in biological observations, the results imply that smooth muscle syncytia could be viewed as an infinite number of dispersed areas of myoelectrical activity, which are either connected over a long distance (110 cm) of the gut or represent spatially distributed units 2 cm in length. The convincing support for that approach comes from experiments on isolated preparations of smooth muscle syncytia by Lammers et al. Using a brush of (24 × 10) electrodes arranged in a rectangular array, the authors studied patterns of electrical activity over a large area, and the effect of single and multiple dispersed pacemakers on the spatial conduction of the wave of excitation. However, within the framework of old concepts of understanding of gastrointestinal motility, the authors failed to give any reasonable explanation for their remarkable observations. Thus, a superposition of slow-wave traces obtained simultaneously from 224 points shows that at the resting state, when no pacemaker activity is present, there are no significant phase and amplitude differences among all slow waves recorded. This simple analysis leads to suggestions that: (i) the smooth muscle syncytium may represent a continuum of spatially distributed autonomous oscillators, and (ii) slow waves do not propagate in the silent state!

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It is universally accepted that a discharge of a pacemaker, ICC-MP, precedes the spread of high amplitude action potentials on crests of slow waves. Remarkably, the analysis of results presented in Lammers et al. (1993) shows that in excited media slow waves recorded at different points of the syncytium have equal frequencies. This condition persists with every excitatory input provided by the pacemaker. This experimental fact confirms our idea that the external excitatory input can play the role of a connector among spatially distributed oscillators. It has also been suggested that a transient influx of calcium ions through the T-type Ca2+ channels is responsible for the shift of the membrane potential and the activation of the intracellular contractile protein system. The propagating calcium waves have been offered as a possible mechanism that sustains the spread of the electromechanical wave. However, the electrical wave propagates at higher velocities of 2.3–10.8 cm/s, depending on the species and tissue, compared to the always delayed mechanical wave of contraction-relaxation that moves at a much lower speed, 0.2–0.41 cm/s, and the intracellular calcium waves cannot provide the extensive, tens-of-inches conduction within morphologically inhomogeneous smooth muscle syncytia. Also, action potentials in smooth muscle propagate over short distances, 1.3–12.8 cm. Therefore, another mechanism must be involved in the efficient spread of the excitation along smooth muscle syncytia. There is compelling evidence for the crucial role of a planar neural network of the interstitial cells of Cajal, which topographically lies between the elements of the enteric nervous system and smooth muscle. The concepts and assumptions employed in the above model approach comprise real anatomical, morphological, physiological and pharmacological data about the function of the gut. We consider smooth muscle syncytia as homogeneous low excitable media. The results of numerical simulations have allowed us to get an insight into the processes of electromechanical wave formation and to discover numerically a phenomenon of self-sustained spiral wave in electrical activity in the longitudinal smooth muscle. This result has been anticipated from previous investigation of the effects of mechanical deformation of the longitudinal smooth muscle fiber on the propagation of the electrical wave of excitation. Analysis of the process helped reveal a condition that blocked the spread of excitation in the overextended segment of the fiber. Extrapolating these results onto a two-dimensional syncytium

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explains the occurrence of a split in the wave of depolarization, ϕl . Nonuniform deformation of the segment“provides”pathways for the propagation of excitation within it and thus maintains its mechanical activity. Remarkably, the coordinated wave of contraction is lost in the longitudinal layer with the development of electrical spiral waves. This could serve as an explanation for a pathological condition known as gastrointestinal dysrhythmia, which is characterized by the failure of propulsive activity in the gut. In view of the proposed new conceptual framework of electromechanical coupling there are other experiments that support the results. Thus, what the authors refer to a spatial distribution of multiple spike patches, could actually represent self-sustained electrical activity demonstrated in our numerical exercises. A mixed pattern of motor activity, including “broad peristaltic, narrower propagated (ripples) and longitudinal muscle rhythmic contractions” observed and described by D’Antona et al. (2001) is comparable to the total force, Tl , dynamics reproduced in the longitudinal smooth muscle syncytium. According to the results of the above study the velocity of propagation of the wave of contraction is 0.41 cm/s, while in our numerical simulations it varies from 0.3 to 0.4 cm/s. With our model we were able to study the effects of drugs on the dynamics of electromechanical conjugation. For example, we demonstrated that Lidocaine suppresses self-sustained spiral wave formation in the longitudinal syncytium. The numerical experiments also show that the effect of cholinergic antagonists results in the disconnection of the ICC-MP’s from the smooth muscle syncytia. At the moment there is no direct, confirmable experimental data either to reject or to support our theoretical findings; the basis of the theoretical results presented above needs to be established and verified experimentally.

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CHAPTER 8

Dynamics of Intestinal Propulsion

An enormous variety of biological functions exhibited by the gastrointestinal tract, including the coordinated propulsive activity of contents, is the result of the dynamic interplay among multiple structural elements comprising the organ. The methods of colonic scintigraphy, magnetic resonance imaging, radio-labeled markers (pellets) and video mapping allow physiologists and clinicians to appreciate gross variations in motility patterns, relative dwell times of content, and regional differences in motility indexes. However, these technologies do not have sufficient resolution to identify the relationship between spatio-temporal patterning of combined motor events and do not offer sufficient depth of accessibility to combined analysis for intricate mechanisms of function of the biological system. As a result, most of the conclusions are drawn from “technically deficient” experiments, since they only offer an implicit partial insight into how the system works under real physiological conditions and in a diseased state. The lack of understanding of physiology of the processes involved affects our approach to treating various gastrointestinal motor disorders, e.g. irritable bowel syndrome, idiopathic esophageal motor dysfunction — that remains unsatisfactory. During the last decade, considerable effort has been directed towards investigating peristaltic propulsion of, mainly, Newtonian and nonNewtonian fluids. There are large numbers of original publications and excellent reviews on the subject. In contrast, the research into propulsion of solids is very limited. Kydoniefs (1969) studied various configurations of an elastic, homogenous, isotropic cylindrical tube that contained a solid ellipsoidal body. His approach was adopted by Bertuzzi et al. (1983) in their 273

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pioneering work to simulate the dynamics of bolus propulsion in an elastic tube. The authors formulated a mathematical model of a segment of the small intestine as a circular cylinder with an enclosed non-deformable sphere (bolus). The wall of the cylinder was reinforced by a family of separate longitudinal (LM) and circular (CM) smooth muscle fibers of an orthogonal type of weaving, resembling the wall of the small intestine. The constitutive relationships were nonlinear and deduced from uniaxial tension-length and tension-velocity experiments. Excitation to the system was provided by an electrical signal in the form of a harmonic traveling wave of a given wavelength and amplitude. The bolus in motion was subjected to dry and viscous friction and the inertia forces acting both on the wall and the sphere were neglected. Even with imposed severe mechanical limitations of the axial symmetric deformations and biological naivete, it was possible to reproduce the dynamics of the propagation of a “ring”-like electromechanical wave, and to simulate propulsion of the bolus. The model presented satisfactory quantitative information about changes in configuration of the organ, the distribution of contact forces over the sphere, and predicted the average velocity of transit. However, the results were only applicable to the third phase of the migrating myoelectric complex. They are described as short-lasting segmental contractions that “squeeze” the intestinal content aborally.

8.1 MODEL FORMULATION Different classifications of movements observed in the gut reflect a large repertoire of functions performed by the organ. The movements are broadly divided into local-segmental contractions and pendular movements, and peristaltic waves. The most common are segmental contractions. They are produced by the CM layer, do not propagate, occur simultaneously or sequentially in different parts of the organ, and promote expulsion of the intraluminal content. In contrast, pendular movements are associated with contraction-relaxation of the LM layer, propagate over short distances aborally, and are assumed to mix and grind the intraluminal content. The peristaltic reflex is referred to a coordinated migrating myoelectrical activity of the LM and CM layers. Regulatory signals, provided by the

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Ganglion

Pellet

EC cells

Mucosa

Circular muscle layer

Longitudinal muscle Myenteric nervous plexus

5-HT3 - receptors 5-HT4 - receptors nACh - receptors ACh - receptors

Fig. 8.1 Schematic diagram of the spatio-temporal model used in numerical experiments.

myenteric plexus and interstitial cells of Cajal, operate with a great level of flexibility to adapt performance to even delicate informational changes. The peristaltic reflex begins with a preliminary, gradual reflex, represented by longitudinal contractions, followed by a phase of broadly spread segmentations. Peristaltic waves are recorded in two forms: slowly advancing contractions, and the peristaltic rush. The latter is manifested as rapid contractions that propagate over a long distance, usually seen in pathological states. Consider the functional unit that contains a solid, non-deformable bolus (Fig. 8.1). The biomechanical principles and the mathematical model are as described above. The model integrates biomechanical properties, electrophysiological, neuropharmacological and regulatory processes and reproduces accurately the following combined events: i) signal transduction among different neurons and effector cells; ii) electrochemical coupling, i.e. neurotransmission, at the synaptic level; special attention is given to the simulation of co-localization and cotransmission by multiple neurotransmitters, including acetylcholine, substance P, and serotonin, and receptor polymodality: µ1, µ2, NK1, NK2, 5HT3 and 5HT4 type receptors;

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iii) propagation of the wave of depolarization along the longitudinal and circular smooth muscle layers of the gut wall with electromechanical conjugation and the development of forces of contraction-relaxation; iv) shape changes; and v) propulsion of the pellet. Plastic or metallic radio-opaque biological markers used in acute or chronic experiments to study the gut transit time have the shape of a sphere or ellipsoid. It is assumed that during all stages of propulsion the wall is in contact with the pellet. The motion of the pellet is due to the propagation of the electromechanical wave of deformation along the functional unit and is subjected to dry and viscous friction. The contact forces between the wall of the bioshell and the pellet are orthogonal to the surface of the pellet. The kinematic equation of the motion of the bolus along the functional unit is added to the system:
z2
r dZ c η + Fd = Fc dzdζ (8.1) dt z1 r0 where Fc , Fd are the contact force and the force of dry friction, respectively; η is the coefficient of viscous friction. During all stages of dynamic reaction, the points of the wall lie over the surface of the sphere:   Kc = (Zc − uz )2 + (r0 + ur )2 + (r0 + us )2 − R 2 < 0 z ∈ [z1 , z2 ]

(8.2)

where u = (uz , ur , us ) is the displacement vector; Zc is the position of the center of the sphere at time t; R is the maximal radius of the sphere; z1 , z2 are the boundary points of contact of the sphere and the wall. The initial conditions assume that the system is in the resting state and the position of the center of the bolus is given. The oral and aboral ends of the bioshell are rigidly fixed.

8.2 NUMERICAL ALGORITHM The numerical procedure is similar to that described above (see Chap. 7 for details). An additional step is added to the iterative process to compute the

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bolus center position Zc,h (t + τ): η where

k+1 Z c,h

k − Zc,h + Fd = F c,h τ


 k √ √  Tc e1z g22 + TL e2z g11 sign(Vz )ηh dS F c,h , ηh =

(8.3)

(8.4)

sk

and Sk is the contact surface of the shell with the Kc on the kth step of integration. In all experiments the functional unit was excited by a series of electrical discharges of ICC located at the left boundary. They had a constant amplitude of 100 mV, duration 1.5 s and frequency 0.33 Hz. Depending on the type of movements under consideration, either longitudinal (pendular movements) or circular (segmental contractions) smooth muscle layers were excited. In the model of the peristaltic reflex, a reciprocal excitation was established in a coordinated reaction of contraction-relaxation between the LM and CM syncytia. The first contractions started in the outer layer followed by the activation of the inner circular layer in 0.5 s. All results are presented for the developed view on the cylinder, where s1 and s2 are the arc and the axial coordinates of the numerical mesh, respectively.

8.3 PENDULAR MOVEMENTS The generation and propagation of active Tla and total Tl forces along the longitudinal smooth muscle syncytium of FU are shown in Fig. 8.2. The circular smooth muscle layer remained inactive throughout. At t = 4.8 s after the excitation, the wave of contraction of average intensity Tla = 8.2 mN/cm was observed. It had a wavelength of 0.6 cm and propagated aborally at a velocity of 0.35 cm/s. The active force increased in strength and Tla = 10.3 mN/cm was generated at t = 11 s. At t = 25.6 s, the maximum force, max Tla = 16.9 mN/cm, was registered in the zone of contact of the wall with the pellet at t = 25.6 s. On the excitation of the entire segment of the gut an even distribution of the active forces of contraction, Tla = 16 mN/cm, developed. The dynamics of the spread of total force Tl , which is the sum of the active and passive components, along the surface of the FU corresponded

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Fig. 8.2 Development of active (left column) and total forces (right) in the longitudinal smooth muscle syncytium in the functional unit during pendular movements.

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to the dynamics of Tla . At the beginning, 0 ≤ t ≤ 4.8 s, the magnitude of Tl was determined mainly by the active forces. On the involvement of the connective tissue stroma in the dynamic process, the FU underwent a uniform stress distribution with Tl = 16.5 mN/cm for t ≥ 11 s. The maximum of the total force, max Tl = 19.5 mN/cm, was recorded in the contact zone at t = 25.6 s and was consistent with the most intensive propulsion of the pellet along the segment. For t > 25.6 s the total force of Tl = 16.8 mN/cm was recorded along the entire FU. The pellet was subjected to continuous back-and-forth movements (Fig. 8.3). In the time interval 0 ≤ t ≤ 4.8 s, it was displaced orally from its initial position by 0.06 cm. A period of incessant propulsive activity was recorded for 4.8 ≤ t ≤ 25.6 s during which the bolus was moved aborally by 0.25 cm at an average velocity of 0.01 cm/s. At t ≥ 25.6 s the reflected a 0 . 60 0 0 . 55 0

Z c (c m)

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0 . 69 3 0 . 33 6 -0 . 0 20 -0 . 3 76 -0 . 7 32 -1 . 0 88 0 . 00 0

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Tim e (s)

Fig. 8.3 Dynamics of the propulsion and velocity profiles of the pellet during pendular movements.

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wave Tl = 19.6 mN/cm pushed the bolus backward and at t = 38 s its center was located at 0.4 cm. For t > 38 s the pellet showed relatively small displacements ∼0.03 cm about the point 0.41 cm. The velocity of transit was not constant but varied from 1.05 to −1.1 cm/s at different moments of the dynamics process. The results of numerical simulations yield valuable insights about gaps in biological understanding of intricate mechanisms of the propulsive activity of the gut. Activation of the LM layer of the functional unit alone is consistent with the physiologically described gradual reflex. During this preliminary phase an aboral transit and mixing of the intraluminal content take place. Interestingly, the gradual reflex itself holds to a strict sequence of events: an aboral propulsion 4.8 < t < 25.6 s is followed by intensive mixing of the pellet, 25.6 < t < 35 s. As a result of pendular movements the bolus is displaced by a short distance, 0.06 cm, from its initial position. Because of different experimental conditions a quantitative comparison of our result to in vivo and in vitro measurements may not be accurate. An average velocity of the transit of a spherical metal pellet in a segment of the colon of a guinea pig of 0.75–0.8 cm/s was reported while in our experiments it was 1.05 cm/s.

8.4 SEGMENTAL CONTRACTIONS The dynamics of segmental contractions, i.e. the propagation of electromechanical waves of contraction-relaxation in electrically and mechanically isotropic circular smooth muscle syncytia, is shown in Fig. 8.4. The longitudinal smooth muscle layer remained inactive. There was a uniform distribution of active forces, Tca , along the surface of the FU throughout the process. The initial wave of contraction of intensity 10 mN/cm (t = 4.8 s) and length 0.5 cm had the shape of a ring. In less than 2 s it enveloped the entire segment of the gut with the development of a contraction averaging 12 mN/cm. With time the strength of the active force increased and max Tca = 16.9 mN/cm was recorded for t > 20 s. After the excitation of the CM syncytium was complete, t > 5 s, the total force, Tc = 35 mN/cm was generated in the wall. The wave Tc did not propagate. Only a slight increase in the magnitude, max Tc = 41 mN/cm, in the zone of contact with the bolus was observed for 5 ≤ t < 38.5 s. As

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Fig. 8.4 Development of active (left column) and total forces (right) in the circular smooth muscle syncytium of a segment of the gut during segmental contractions.

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Z c (c m )

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0 . 00 0

8 . 33 6

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2 5 .0 0 8

3 3 .3 4 3

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50

Tim e (s)

Fig. 8.5 Transit of the pellet during segmental contractions.

a result the pellet remained at standstill for 0 ≤ t < 11 s and only small librations were seen about the point 0.35 cm for t > 11 s (Fig. 8.5). Excitation of the CM layer alone revealed the production of nonpropagating contractions that envelope the entire segment. The total force distribution along the surface of the functional unit is symmetric with higher values of Tc in the contact zone with the pellet. There is no gradient of Tc along the length of the segment; therefore the pellet remains unmoved with only small librations observed for t > 11 s. One can only speculate that if the intraluminal bolus under consideration were deformable, the strong occluding contractions, max Tc = 41 mN/cm, could break it into two parts with the subsequent displacement of the fragments orally and aborally. This process is well documented experimentally. However, from the point of view of the mechanics of solids, this problem poses a great mathematical challenge.

8.5 PERISTALTIC REFLEX Both longitudinal and circular smooth muscle syncytia were reciprocally activated. At t = 4.8 s after the discharge of ICC the wave Tl of amplitude 15.8 mN/cm is generated in the longitudinal syncytium (Fig. 8.6). It propagated at a constant velocity of 0.35 cm/s in the aboral direction. Maximal total forces, max Tl = 18.2 mN/cm and max Tc = 23.5 mN/cm were recorded in the wall of the gut in the zone of its contact with the pellet. As the electrical excitation enfolded the whole FU a uniform distribution of the total force in

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Fig. 8.6 Dynamics of total forces in the longitudinal (left column) and circular (right) smooth muscle syncytia in the functional unit during the peristaltic reflex.

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the LM syncytium was observed. For t > 21.2 s the total forces in the CM syncytia prevailed. They continued to increase in intensity, Tc = 27.4 mN/cm, t = 21.2 s, and Tc = 28.3 mN/cm, t = 34.8 s, and shift over a short distance, 0.3 cm, along the surface of the FU. For t > 40 s the gut was evenly stressed with averages of Tl = 18.2 mN/cm and Tc = 25 mN/cm. The peristaltic propulsion, as in the case of pendular movements, began with the aboral displacement of the pellet accompanied by non-intensive “mixing” (Fig. 8.7). It moved at an average velocity of 0.01 cm/s and at t = 21.2 s it reached the point 0.57 cm. A brisk strictly aboral movement of the bolus over a distance of 0.21 cm was observed for 28.5 < t < 34 s. It was followed by librations when small back-and-forth movements over 0.03 cm about the point 0.68 cm were recorded: 34 < t < 45 s. Finally, the pellet was pushed orally and at t = 50 s its center was placed at 0.51 cm. The velocity profile indicates that the pellet moved at a maximum velocity of 1.25 cm/s in the aboral direction, and at a velocity of 0.84 cm/s in the oral direction (Fig. 8.8). A characteristic stereotypical phenomenon of the gut is the peristaltic reflex. It represents a higher level of organization and regulation of myoelectrical activity and includes the compulsory role of the myenteric nervous plexus in its space-temporal arrangement. In the model as described above the latter is represented by a pacemaker and is analogous to interstitial cells of Cajal. Even with this simplified assumption it was possible to simulate with great accuracy different motility patterns in a functional unit of the gut. The effective propulsion and mixing of the intraluminal content is accomplished by a coordinated mechanical reaction of contractionrelaxation of the LM and CM layers. Their anatomically distinct physiological significance still remains a subject of scientific debate. For example, simultaneous intracellular recordings from LM and CM on a guinea-pig isolated segment of the gut have revealed sequential spontaneous rhythmical depolarizations of the two layers and synchronous neuromuscular inputs during ascending excitation. To test this hypothesis we assumed the reciprocal relationship between the LM and CM syncytia in the generation of the peristaltic reflex. The results of numerical simulations reproduced with great accuracy a complex sequence of movements of the pellet: pendular-like movements 4.8 < t < 28 s, followed by a brisk squeeze, 28 < t < 34 s, and finally intensive mixing t > 34.5 s. In contrast, pendular contractions

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Fig. 8.7 Dynamics of the propagation of the pellet during pendular (left column) and peristaltic (right column) movements.

only provide back-and-forth movements of the pellet and not mixing, while segmental contractions alone fail to sustain the transit. At t = 50 s the center of the bolus was recorded at 0.51 cm in the case of the peristaltic reflex, at 0.41 cm during pendular and at 0.35 cm during segmental contractions,

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a 0 . 71 3

Z c (c m)

0 . 64 5 0 . 57 6 0 . 50 8 0 . 43 9 0 . 37 1 0 . 30 3 0 . 00 0

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Fig. 8.8 Dynamics of the propulsion and velocity profiles of the pellet during the peristaltic reflex.

respectively. In the case of synchronous activation of the two smooth muscle layers the transit of the pellet resembled movements recorded during the gradual reflex only. The dynamics of propulsion concurs with the total force distribution in the syncytia. Thus, equal in magnitude and similar in dynamics, waves of Tl are generated during the peristaltic and the gradual reflex. On the other hand, the values of Tc registered at all times during segmental contractions exceed the values observed during the peristaltic reflex by ∼45%. The overstressed state does not affect the propulsion of the pellet. This analysis suggests that the coordinated reciprocal contractile activity of the two muscle layers is necessary to maintain normal physiological transit of the intraluminal content. However, no detailed quantitative comparison of the theoretical and experimental results is possible at this stage.

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8.6 EFFECT OF MULTIPLE NEUROTRANSMISSION AND DRUGS ON PELLET PROPULSION 8.6.1 Introduction There is considerable interest in the electro-pharmacological role of acetylcholine and serotonin in the pathogenesis of irritable bowel syndrome. However, understanding of the complex mechanisms involved in intricate processes responsible for the biological manifestations of the disease, i.e. visceral hypersensitivity, constipation/diarrhea, bloating, and abdominal pain, remains unsatisfactory. Significant current issues are: (i) the processes of cotransmission by multiple neurotransmitters and co-activation of different receptor types cannot be studied effectively by any of the existing experimental techniques, and (ii) traditional in vivo and in vitro approaches have failed, so far, to provide required quantitative information about the role of synaptic neurotransmission in the pathogenesis of the syndrome. To gain insight into phenomenological mechanisms of co-transmission by ACh and 5-HT and to study their effects on the motor and propulsive activity of the gut, we analyzed numerically the concurrent effects of co-activation of muscarinic (µACh), nicotinic (nACh) and 5-HT types 3 and 4 receptors on the dynamics of signal transduction and electromechanical responses in the gut, and the role of pharmacological agents with selective, nonselective and mixed 5-HT3 - and 5-HT4 -receptor agonists/antagonists mechanisms of action in its propulsive activity.

8.6.2 Effect of Co-activation of 5-HT3 , nACh and µACh Receptors Co-activation of the 5-HT3 and nACh receptors affects mainly the dynamics of the spread of excitation within the myenteric nervous plexus. A chaotic pattern of electrical activity with the generation of high amplitude action potentials, ϕ = 89 mV, is dominant. The spikes occur randomly at different sites of the functional unit (Fig. 8.9). Disrupted strong waves of contractionrelaxation are produced by the longitudinal and circular muscle syncytia, Tl = 145 mN/cm and Tc = 1980 mN/cm (Figs. 8.10 and 8.11), which leads to uncoordinated movements of the pellet. The bolus demonstrates a “mixing” type of motion, with aboral propulsion at an average velocity of 0.87 cm/s.

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Fig. 8.9 Effects of co-activation of the 5-HT3 and nACh/µACh receptors on propagation of the wave of depolarization along the myenteric nervous plexus.

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Fig. 8.10 Effect of co-activation of the 5-HTs and nACh/µACH receptors on development of total forces in the longitudinal smooth muscle syncytia.

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Fig. 8.11 Effect of co-activation of the 5-HT3 and nACh/ACH receptors on development of total forces in the circular smooth muscle syncytia.

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8.6.3 Effects of Alosetron The pharmacological mechanism of action of the drug is a selective 5-HT type 3 receptor antagonism. Under the assumption that there is no ACh in the system, application of Alosetron causes a disruption of the leading front of the depolarization wave and a slow down in the velocity of the spread of excitation within the plexus to 0.04 cm/s. Alosetron has a prominent inhibitory affect on the firing rate and the amplitude of action potentials that are recorded on the crests of slow waves (Fig. 8.12). However, the frequency of slow waves is increased, 27 < t < 54 s. Smooth muscle syncytia become hyperpolarized with the resting membrane potential: ϕ(rest) = −68.2 mV. The above changes in electrical activity cause a significant reduction in the intensity of the active force of the contraction in the circular, Tca = 1.3 × 10−2 mN/cm, and longitudinal, Tla = 70 mN/cm, smooth muscle layers. As a result there is an impaired propulsive activity of the pellet which moves at a speed of 0.4 cm/s. Co-release of ACh at the neuro-neuronal and neuro-muscular junctions at 54 < t < 126 s with a subsequent activation of the nACh and µACh

Fig. 8.12 Effects of Lotronex on gut motility. The upper trace shows myoelectrical activity and the lower trace is the concomitant force of contraction in the circular smooth muscle layer.

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receptors restores normal dynamics of the generation of high amplitude action potentials of average amplitude max ϕ = 67.4 mV. Long-lasting, 72 s, tonic-type contractions of max Tca = 4.5 × 10−2 mN/cm are recorded in the circular layer. The dynamics of the total force distribution in the longitudinal and circular smooth muscle layers along the surface of the functional unit is shown in Fig. 8.13. Intense non-propagating contractions of max Tla = 15 × 10−1 mN/cm and max Tca = 3.4 × 10−2 mN/cm are observed in the vicinity of the right boundary of the gut. They significantly impair propulsion of the pellet, which comes to a complete standstill. Only after the removal of ACh from the system does the functional unit regain its propulsive activity in the presence of Alosetron.

8.6.4 Effects of Selective 5-HT4 Receptor Agonists Treatment of a segment of the gut with 5HT4 receptor agonists, Tegaserod (Novartis) or TS-951, sustain the reciprocal relationship in contractionrelaxation between the longitudinal and circular smooth muscle layers. No significant changes are observed in the dynamics of the propagation of the electrical wave of excitation within the myenteric plexus after application of the drugs. The results show an increase in tone of the longitudinal muscle layer, Tl = 205 mN/cm, with no changes in the total force in the inner circular layer. Regular contractions are evenly distributed along the surface of the functional unit and on the contact surface with the pellet. The bolus sustains pure expulsive aboral movement at velocity 1.2 cm/s with no mixing being observed.

8.6.5 Effects of Selective 5-HT4 Receptor Antagonists Neither GR 113808A nor SDZ205557 selective 5HT4 receptor antagonists has any effect on the propagation of the wave of excitation along the myenteric nervous plexus. However, the application of these drugs causes disruption of the migrating myoelectrical complex. Multiple, irregular, high amplitude (max ϕ = 54 mV) action potentials are recorded from different sites of the syncytia. Although active forces produced by the longitudinal and circular smooth muscle layers are strong enough to initiate and maintain propulsion

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Fig. 8.13 Dynamics of the development of total forces in the longitudinal (left column) and circular (right column) smooth muscle syncytia after co-release of ACh and application of Lotronex.

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of the pellet, there is an overall slowdown in movement. The maximal velocity of the pellet is only 0.5 cm/s.

8.6.6 Effect of Cisapride The application of Cisapride results in an increase in excitability of the primary, motor and interneurons of the myenteric nervous plexus. The neurons generate high amplitude and frequency action potentials. The drug does not affect the intensity of the propagation of electrical signals within the myenteric nervous plexus. However, an increase in the frequency of slow waves, ν = 30 Hz, in the longitudinal and circular smooth muscle syncytia is observed. Long-lasting tonic contractions are induced by both smooth muscle layers with max Tl = 200 mN/cm, and max Tc = 2100 mN/cm. As a result, squeezing of the pellet towards the aboral end, rather than a mixing type of propulsion, is recorded.

8.7 REMARKS With our simulations we were able to study numerically the effects of cotransmission by ACh and 5-HT on the propulsion of a solid non-deformable bolus in the intestine. The results indicate that excitation of 5-HT4 receptors only in conjunction with µACh receptors is able to initiate long-lasting phasic contractions. These findings are supported experimentally by the selective activation of 5-HT4 receptors on longitudinal smooth muscle of the large intestine of different species. The in vivo and in vitro studies of the effects of Alosetron on contractility and compliance of the colonic muscle demonstrated: (i) the decrease in contractions in the small and large intestines, (ii) the increase in wall compliance of the colon, and (iii) the delay of gut transit time. These results support our numerical simulations of pellet propulsion in a segment of the gut. Alosetron causes a decrease in the velocity of propulsion by 50%. Disruption in the wave of depolarization and a delay of the spread of excitation within the myenteric nervous plexus is suggested as a mechanism for the slowdown. Interestingly, co-activation of nACh and µACh receptors results in the development of long-lasting tonic-type contractions that bring the

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pellet to a standstill. The loss of the reciprocal relationship in contractionrelaxation between the two muscle layers is a possible mechanism. Selective 5-HT4 receptor agonists, Tegaserod and/or TS-591, alone do not appear to affect reciprocal relationships in mechanical activity between two smooth muscle layers. Strong contractions in the longitudinal and circular muscle layers cause a rapid expulsion of the pellet with a loss of physiological “mixing.” That may explain low clinical effectiveness of the drug (only 10–12% over the effect of the placebo) in the treatment of constipation. These results concur with previous observations of the effects of TS-591, which demonstrate a significant prokinetic effect on motility of the gut. The computer-based integrative approach enables one to gain insight into the intricate pharmacodynamic and physiological effects of various drugs, which cannot be achieved in either in vivo or in vitro preparations. This study confirmed numerically that a non-selective strong 5-HT3 and weak 5-HT4 receptor agonist, Cisapride, attains its prokinetic effect through the increase in influx of Ca2+ ions and excitability of the primary and motor neurons that lead to twitch contractions of the smooth muscle syncytium. However, these changes at a cellular level are not correlated linearly with the effects of the drug on the myenteric plexus. Thus, Cisapride does not alter the dynamics of the propagation of electrical signals within the planar neural network formed of AH and S neurons. At the functional unit level, Cisapride causes an increase in frequency of slow waves and fEPSP’s which results in the development of tonic-type contractions in the longitudinal and circular smooth muscle layers. The pattern of mechanical propulsive activity changes from “mixing” to the “expulsive” type.

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INDEX

cholinergic, 8, 23, 97, 101–103, 112–115, 117, 119–121, 152–154, 176–178, 195–197, 219, 222, 236, 269, 272 competitive, 113, 114, 116, 117, 120–122, 127, 128, 145 allosteric binding, 223 Alosetron, 18, 19, 201, 291, 292, 294 aminopeptidase, 236 AMPA antagonist, 224 6-cyano-7-nitroquinoxaline-2,3dione, 224 6,7-dinitroquinoxaline-2,3-dione, 224 AMPA receptor, 223–225, 227–229, 231–235 GluR1 subunit, 223, 224 GluR2/3 subunit, 223, 224 GluR4 subunit, 223, 224 angiotensin enzyme, 236 anisotropy, 195, 197, 263 electrical, 9–11, 20–23, 57, 102–104, 153, 163, 190–192, 207–209, 211–215, 228–234, 250–252 mechanical, 2, 5, 17, 20, 24, 50, 54, 93, 96–98, 156, 175–179, 244–247, 250 antagonist adrenergic cholinergic, 8, 23, 97, 110, 112–115, 117, 152–154, 162, 202 competitive, 113, 114, 116, 117, 120–122, 127, 128, 145 Auerbach, 1 autoinhibition, 131 autonomous oscillators, 247, 270

A acetyl-β-methylcholine, 120 acetylcholine, 8, 18, 89, 99, 101, 103–105, 110, 150, 177, 199, 200, 234, 244, 275 cleft, 102, 103 free, 102 postsynaptic, 102 vesicular, 103 acetylcholinesterase, 103, 106, 177, 195, 236 inhibitor, 56, 114, 120, 130, 134–136, 140–144, 152, 155, 166, 195, 221, 228 irreversible anticholinesterases, 113, 114, 120, 121, 127, 128 reversible anticholinesterases, 47, 113, 120, 121, 127, 145, 246 true, 120, 121 action potential, 7, 10, 11, 153, 177, 197, 198, 224–234, 287, 294 active center, 106, 110, 114, 118, 120, 126, 127, 181, 199, 240 adrenoceptor, 129–131 α1 , 71, 129–131, 133 α2 , 129–131 β1 , 129 β2 , 129 afferent pathway, 156, 157, 159, 186–188, 224 agonist, 17–19, 45–47, 54–56, 123, 157, 172, 174, 175, 195, 218, 220, 287, 292 adrenergic, 8, 13, 128–131, 134, 150, 153–155, 177, 193 313

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axon, 6, 9, 12, 102–104, 130, 131, 154, 159, 176, 178 B beating, 41, 55, 66, 68, 79, 158, 168, 171, 211 benzodiazepines, 56, 82, 84–90, 95, 98 GABAA receptor, 82, 84, 89, 97, 98 GABAB receptor, 82, 84, 97, 98 biorhythms, 15 bioshell, 250, 251 bisyncytia, 247 bolus, 5, 11, 248, 274–277, 279 kinematic equation, 276 shape, 251, 262, 265, 276, 280 botulinum toxin, 111, 118, 126 brush of electrodes, 270 buffer, 44, 106, 110, 117, 132, 139, 181, 240, 262 β-bungarotoxin, 117, 127 bursting, 21, 25 bursting chaos, 38, 180–183, 190 C 2-[p-(carboxyethyl)phenylethylamino]-5
N-ethylcarboxamidoadenosine, 157 charybdotoxin, 45, 51, 53, 157, 168 chemical equilibrium, 116, 119, 122, 124, 128, 142, 144, 147 partial, 81, 87, 116, 119, 122, 124, 128, 142, 144, 148 chemical reaction, 105, 106, 131, 132, 176, 204, 205, 258 chloride salt, 117, 126 2-chloro-N 6 -cyclopentyladenosine, 157 choline, 103, 106, 110, 111, 119, 178 cholinergic receptor, 117, 125 muscarinic, 22, 114, 125, 126, 200, 240, 243, 287 nicotinic, 114, 125, 126, 129, 153, 200, 222, 231, 287 chronotropic, 22, 25, 77, 96, 97, 177, 189, 248 circuit, 100, 153–156, 159, 176, 197, 246 neural, 6, 8, 9, 13, 23, 44, 100, 102, 114, 151, 153, 156, 176, 196, 197, 200, 244, 246, 271, 295 neural-effector, 246

Cisapride, 18, 201, 210, 211, 213, 214, 218, 220, 222, 294, 295 cocaine, 138, 139 colonic scintigraphy, 273 compression factor, 33 COMT (catechol-O-methyltransferase), 130, 132–134, 136, 137, 139–147, 149, 150, 155, 177, 195 COMT inhibitor, 140, 150, 155 methyltropolone, 140 tropolone, 140 connection, 7, 12, 13, 101, 153, 156, 192, 195, 197, 198, 248, 259, 263, 266, 269 backward inhibitory, 193 bypass, 192, 193, 195 convergent, 193, 195 direct, 20, 100, 112, 158, 174, 192, 193, 195, 223, 272 strong, 248, 266, 269 weak, 18, 72, 74, 201, 213, 259, 295 constitutive relationship, 274 co-transmission, 199, 200, 206, 222 coupling, 23, 129, 150, 157, 162, 177, 201, 247 electrochemical, 95, 100, 130, 134, 150, 176–178, 201, 203, 262 electromechanical, 30, 57, 157, 202, 259, 261, 265, 268, 269, 271, 272, 274, 276, 280, 287 Courant–Friedrichs–Lewy, 34 cromakalim, 54 cyclopiazonic acid, 56–58, 60, 61 D dendrite, 157, 158, 163, 179, 192, 233 dendritic potential, 163, 165, 169, 172, 173, 180, 191 depolarization, 9–11, 21–23, 154, 200, 244, 272 diacylglycerol, 205, 236, 237, 241 diazoxide, 56, 63, 64, 94 Diphenoxylate, 17 Dirichlet, 261 Dogiel, 102, 157, 158 Domperidone, 18 D-penicil-amine2-D-penicillamine5, 157

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Index E elasticity, 246 electrogastrography, 10 electromechanical wave, 30, 31, 261, 268, 269, 271, 274, 276, 280 enteric nervous system, 7, 8, 12, 15, 16, 44, 100, 150, 200, 235, 246, 271 enterochromaffin cell, 200 erythromycin, 19, 56, 75, 76, 96, 97, 99 esophagus, 2, 5, 6, 9, 11 excitatory amino acids, 223, 224 F fenamate, 55, 56 fiber, 25, 128, 152, 176, 177, 224, 245, 247, 256 collagen, 245 elastin, 25, 248 finite element, 260 force, 4, 5, 23–25, 250, 251, 274, 276–284, 286, 289–293 active, 23, 25, 29, 250, 277–281 contact, 274, 276, 277, 279, 280, 282 dry friction, 274, 276 passive, 25, 30, 191, 248, 250, 277 forskolin, 45, 46 functional unit, 8, 38, 152, 244 G galantamine, 120 ganglion, 12, 176, 223 dorsal root, 223 inferior mesenteric, 12, 128 pelvic, 13, 128 prevertebral, 12, 13 superior mesenteric, 12, 128, 244 gap junction, 23 glutamate, 223, 225, 227, 231, 233, 234 glutamate transporter 1, 223 Granisetron, 208, 212, 219, 221 guanine-nucleotide G protein, 205, 236 H Hebb’ rule, 151 hexamethonium, 231–233, 235 Hodgkin–Huxley formalism, 202

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hyperpolarization, 35, 46, 47, 49, 60, 74, 102, 129, 137, 144, 149, 153, 158, 165, 168–170, 173, 174, 180, 192, 212, 215, 219, 231 I iberiotoxin, 45, 157, 158, 166–168, 173 imipramine, 138 immunohistochemical, 152, 223 inositol 1,4,5-triphosphate, 205, 236, 241, 243 interstitial cell of Cajal, 9, 11, 22, 25, 102, 176, 244, 262, 271 ion channel, 20, 21, 28, 45, 177 activation, 25, 27, 174–178, 203–205, 252, 271 Ca2+ (L, T, N and P types), 21 Ca2+ -activated K+ , 22, 23, 25, 27, 28, 36–38, 42–46, 49, 51, 53–55, 79, 166, 168–170, 177, 202, 211, 226, 228–230, 248, 253–255, 261 Cl− , 22, 25, 27, 28, 56, 84, 160, 161, 202, 209, 225, 248, 252, 253, 255, 256, 261 glibenclamide-sensitive K+ , 53, 54 inactivation, 21–23, 27, 47, 52, 74, 89, 130, 136, 137, 196, 207–210, 215, 221, 227, 229, 230, 236 K+ , 20, 22, 23, 25, 27, 28, 35–38, 42–51, 53–56, 59, 63, 68, 69, 73, 76, 79, 92, 94, 95, 104, 160, 161, 166, 168–170, 173, 174, 177, 202, 207, 209–212, 220–222, 224–230, 234, 237, 242, 248, 252–256, 261, 262 modulators, 45 Na+ , 20, 22, 28, 104, 138, 160, 161, 169, 175, 202, 207, 209, 210, 212, 221, 222, 224, 225, 227, 230, 234, 252, 254, 255, 268 permeability, 18, 22, 23, 25, 175, 177, 200, 248 IP3 signaling, 236 irritable bowel syndrome, 18, 19, 56, 223, 273, 287 isotropy, 247, 252, 263, 268, 273, 280 electrical, 9–11, 20–23, 25, 151, 207–209, 242, 244, 247, 248

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Mathematical Modeling and Simulation in Enteric Neurobiology mechanical, 24, 30, 156, 208, 221, 224, 233, 244–247, 263, 266, 269–272, 274, 284

L Lagrangian coordinate, 27, 104, 154, 247, 250, 256 L-aspartate, 223 lemakalim, 47, 48, 53–55 Lidocaine, 268, 272 linear density, 30, 250 linear summation, 198, 253 locus, 11, 14, 36, 38, 39, 42, 44, 47–49, 51–56, 89, 97–99, 178, 247, 253 Lotronex, 18, 291, 293 M magnetic resonance imaging, 273 mechanoreceptor, 17, 89, 159, 160, 163, 202, 206–209, 212, 221, 222, 224, 225, 229, 232, 258, 259 Meissner, 1, 6, 102, 152 methoxyverapamil, 56, 69, 70, 95 metoclopramide, 18 Michaelis–Menten, 104, 116, 131, 143, 147, 176 migrating myoelectric complex, 20, 39, 40, 50, 51, 55, 56, 76, 89, 92, 96, 98, 244, 269, 270, 274, 292 phase I, 13, 14, 39, 50, 52 phase II, 13–15, 39, 51, 52, 89, 92, 96, 97, 99 phase III, 13, 14, 39, 52, 55, 75, 92, 96, 97, 99 minoxidil, 45 mixing, 5, 280, 284, 285, 287, 292, 294, 295 model, 20, 21, 156 Bonhoefer–van der Pol, 20 Chay–Keizer, 20 dynamic, 20–22 FitzHugh–Nagumo, 20 null-dimensional, 131, 176, 248 pharmacokinetic, 45, 131, 150, 248 Plant–Kim, 20 monoamine oxidase, 130, 138, 205 mosapride, 201

motilide, 56, 75–81, 83, 89, 90, 96, 97 motility pattern, 156, 176, 245, 273, 284 back-and-forth, 279, 284 pendular, 244, 274, 277–280, 284, 285 segmental, 244, 274, 277, 280–282, 285, 286 muscle layer, 7, 9, 10 circular, 4–7, 11 longitudinal, 4, 6, 7 myoelectrical activity, 20, 25, 176, 200, 223, 253, 274, 284 myosin light chain-kinase, 24 N Nernst relationship, 47 5
-N-ethylcarboxamidoadenosine, 157 network, 6, 8, 56, 100–102, 125, 151, 152, 192–197, 199, 245–248, 271 connective tissue, 6, 23, 245–248, 250, 279 multi-layer, 151, 152 multi-layer feedforward, 151 one-layer feedback, 151 planar, 151, 192, 194, 196, 197, 248, 271, 295 three-dimensional, 245 neurokinin A, 235 neurokinin B, 235 neurokinin receptor NK1 , 235, 236, 240, 241, 275 NK3 , 235 neuron, 6–8, 17, 44, 149–159, 221–234, 240–244 adrenergic, 8, 13, 17, 98, 128–131, 134, 135, 137–140, 143, 145, 149, 150, 153–155, 176, 177, 193, 195, 197 AH type, 129, 153, 172–174, 190, 201–203, 209, 222–224, 232–234 cholinergic, 8, 23, 97, 101–103, 108, 150, 152–154, 195–197, 202, 203, 219, 222 interneuron, 7, 8, 156, 175, 192, 193, 198, 201, 294 motor, 1, 6–17, 19, 171 postganglionic, 97, 128, 150

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Index primary sensory, 157–159, 162–165, 172–175, 178, 180, 191, 201, 210–212, 224, 226–229, 258 S-type, 102, 129, 130, 153, 158, 173, 202, 223, 224 type I, 102, 158 type II, 102, 157 neurotransmitter acetycholine, 111 adrenaline, 140, 191 neurokinin A, 235 neurokinin B, 235 noradrenaline, 8, 129–131, 135–139, 141, 143–145, 147–149, 156, 177 serotonin, 18, 203–206, 275, 287 substance P, 235, 236, 240–242 neutral endopeptidase, 236 Newtonian fluid, 273 nifedipine, 56, 58, 60, 80, 83, 93, 96, 97 nimodipine, 56, 63, 64, 66, 94 NMDA antagonist, 224, 229, 232, 235 1-phosphonic acid, 224 3-(2-carboxypiperazin-4-yl)propyl-1phosphonic acid, 224 D-2-amino-5-phosphopentanoic acid, 224 NMDA receptor, 223, 224, 227–229, 231–235 NR1 subunit, 223, 224 NR2 subunit, 223, 224 N-methyl-D-aspartate, 223 non-Newtonian fluid, 273 non-oscillatory state, 71, 86 noradrenaline, 8, 129–131, 135–139, 141, 143–145, 147–149, 156, 177 cleft, 129–132 free, 129 postsynaptic, 128–131 vesicular, 124 numerical scheme Euler, 260 Evans–Abdullah, 34, 109, 178 explicit, 196 finite-difference, 31, 32, 134, 259–261 hybrid, 32, 34, 134 implicit, 45, 260, 273

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Lax–Wendroff, 32, 34 Runge–Kutta, 109, 134, 178 Saulyev, 33 O ω-conotoxin, 157 Ondansetron, 201, 208, 212, 213, 219, 221 opioid receptor, 157, 172, 175 δ, 157, 172, 175 κ, 175 µ, 175 P pacemaker, 9, 10, 21, 25, 102, 193, 244, 248, 258, 264, 269–271, 284 pellet, 273, 276, 277, 279, 280, 282, 284–287, 291, 292, 294, 295 penalty operator, 260 peristaltic rush, 275 pharmacokinetic, 45, 131, 150, 248 phasic contraction, 24, 39–41, 44, 47, 56, 57, 60, 63, 66, 74, 79, 80, 89, 95–97, 99, 186, 220, 222, 233, 294 phencyclidine, 49, 50 phorbol 12,13-dibutyrate, 157, 171 phosphatidyl inositol 1,4,5-triphosphate, 205, 236 phospholipase C, 205, 236, 241 plexus, 6, 7, 9, 150, 152, 158, 244, 245, 275, 284, 288, 292, 294 Auerbach’s, 1, 6, 102, 152 Meissner’s, 1, 6, 102, 152 postsynaptic potential, 102, 107, 134, 153, 177, 200, 248 excitatory, 22, 100, 102, 107, 108, 152–154, 243 inhibitory, 129, 134, 150, 152–156, 234, 291 protein-phosphatase, 53, 205, 237, 243 prucalopride, 201, 220, 222 R radio-labeled markers, 273 radioreceptor-binding assay, 97 Ramón y Cajal, 1, 11

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rate constant, 106, 117, 119, 123, 125, 132, 204 (+)-(R)-Bay K 8644, 56, 71, 74, 75, 95, 96 receptor, 15, 18, 22 adrenergic, 128–131, 134, 150 AMPA, 223–225, 227–229, 231–235 cholinergic, 101–103, 152–154 GABAA , 82, 84, 89, 97, 98 GABAB , 82, 84, 97, 98 G-protein coupled, 235 neurokinin, 235, 236 NMDA, 223–225, 227–229, 231–235 opioid, 17, 19, 157, 172, 175 serotonergic, 202, 222 reciprocal relationship, 284, 292, 294 reflex, 13, 17, 153, 274, 275, 277, 280, 282–286 gradual, 2, 38, 41–44, 46, 74, 95, 163, 182, 212, 213, 219, 244, 275, 280, 286 peristaltic, 5, 244, 272–275, 277, 282–286 Renzapride, 218 rhythmic contraction, 24, 35, 182, 219, 233, 262, 272 ryanodine, 56–58, 60, 63, 64, 80, 81, 87–90, 93, 94, 97, 99 S sarcoplasmic reticulum, 24, 56 (−)-(S)-Bay K 8644, 56, 61, 62, 71–73, 95, 96 second messenger, 24, 174, 175, 201, 206, 236, 241, 243, 244 self-regulatory, 25, 30, 176, 177, 246 self-regulatory myogenic medium, 25, 176 sensory pathway, 156, 169 serotonergic receptor, 202, 222 5-HT3 type, 18, 200, 201, 203, 205–215, 217–222, 275, 287, 288, 290, 295 5-HT4 type, 18, 201, 203, 205, 206, 210–216, 218, 220–222, 275, 287, 292, 294, 295 cleft, 205 free, 202, 204

postsynaptic, 200, 206 vesicular, 204 slow wave, 10, 11, 14, 17, 21–25, 36, 38, 39, 42, 44–49, 52–56, 59, 60, 62, 63, 66, 69, 74, 76, 79, 84, 89, 176, 177, 181, 182, 186, 188–190, 219, 244, 248, 263, 268–271, 291 amplitude, 22–25 crest, 23 frequency, 9–11, 35 period, 36 plateau, 23, 24, 35, 36 smoothing, 32, 33 soft shell, 251 cylindrical, 245, 247, 250, 267, 273 orthotropic, 247 sphincter, 4–6 ileocaecal, 6, 16 lower esophageal, 5, 6 pylorus, 5, 10, 15 spike, 22–25 squeezing, 294 stomach, 2, 5, 6, 10–15, 18, 55, 245 antrum, 2, 5 corpus, 2, 5, 10 strain energy, 250 stress-strain, 246, 251 biaxial, 251 uniaxial, 251, 274 substance P, 191, 223, 235, 236, 240–242, 244, 275 sympathomimetic amine, 138, 145 amantadine, 138 chlorpromazine, 138 dexamphetamine, 138 guanethidine, 138 haloalkylamines, 145 imidazolines, 145 isoprenaline, 138 metaraminol, 138 methylphenidate, 138 phenoxyalkylamines, 145 phenylethylamine, 138 synapse, 12, 150–155, 176–178 artificial, 93, 101, 124, 151 axo-axonic, 152, 153 axodendritic, 152, 159

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Index axosomatic, 152, 193, 194 neuromuscular, 197, 224, 284 neuro-neuronal, 100, 102, 201, 214, 219, 230, 234, 291 postsynatic membrane, 102, 107, 110, 131, 144, 147–149, 155, 156, 176, 177 presynaptic terminal, 102–104, 114, 135, 176, 177, 248 strengths, 101, 221 transmission, 44, 100, 103, 108, 114, 117–120, 135, 136, 140, 145, 150, 153, 176, 189, 190, 193, 222, 236, 244 zone, 103, 105, 110, 114, 154, 236, 251 syncytium, 23, 176, 182, 247, 248, 250, 262, 264, 266, 270–272, 277, 278, 280–282, 284 electrogenic, 247 myogenic, 25, 44, 176, 177, 247, 248 smooth muscle, 4, 5, 7, 9–11, 17, 20–25, 244–248, 250, 252, 259, 262–265, 267–272, 274, 276–278, 280–283, 286, 289–295 T Tegaserod, 18, 19, 292, 294 tetrodotoxin, 22, 44, 123, 157, 195, 231 thapsigargin, 56, 57, 63, 64, 66–70, 80, 83, 94, 95, 97 tonus, 25

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U uptake-1 mechanism, 130, 132, 136–138, 149, 150, 177 uptake-2 mechanism, 130, 132, 136, 137, 149, 177 V vagus, 13, 15 verapamil, 56, 61, 62, 93, 96 video mapping, 273 viscoelasticity, 25 W Wave contraction-relaxation, 30, 246, 266, 271, 280, 284, 294 depolarization, 9–11, 21–23, 200, 244 electrical, 9–11, 20–23, 25, 242, 244 mechanical, 2, 5, 17, 20, 24, 156, 158, 202, 244–247, 250 non-stationary, 31 self-excitatory, 186–188, 213, 268 spiral, 266, 268, 269, 271, 272 wrinkled area, 251 Z Zelnorm, 18