Bioelectricity A Quantitative Approach

Bioelectricity A Quantitative Approach

Robert Plonsey and

Roger C. Barr Duke University Durham, North Carolina

SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library of Congress Cataloging in Publication Data

Plonsey, Robert. Bioelectricity: a quantitative approach.

Bibliography: p. Includes index. 1. Electrophysiology- Mathematical models. 2. Electrophysiology- Method­

ology. 1. Barr, Roger C. Il. Title. QP341.P734 1988 599'.019127 88-22418

This limited facsimile edition has been issued for the purpose of keeping this title available to the scientific community.

1098765

© 1988 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1988

AII rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

Softcover reprint of the hardcover 1st edition 1988

ISBN 978-1-4757-9458-8 ISBN 978-1-4757-9456-4 (eBook)DOI 10.1007/978-1-4757-9456-4

To our unseen co-authors, our wives:

VIVIAN PLONSEY JEAN BARR

and our unnamed co-authors: The students in BME 101

Preface

This text is an introduction to electrophysiology, following a quantitative approach. The first chapter summarizes much of the mathematics required in the following chapters. The second chapter presents a very concise overview of the general principles of electrical fields and current flow, mostly es­tablished in physical science and engineering, but also applicable to biolog­ical environments. The following five chapters are the core material of this text. They include descriptions of how voltages come to exist across membranes and how these are described using the Nernst and Goldman equations (Chapter 3), an examination of the time course of changes in membrane voltages that produce action potentials (Chapter 4), propagation of action potentials down fibers (Chapter 5), the response of fibers to artificial stimuli such as those used in pacemakers (Chapter 6), and the voltages and currents produced by these active processes in the surrounding extracellular space (Chapter 7). The subsequent chapters present more detailed material about the application of these principles to the study of cardiac and neural electrophysiology, and include a chapter on recent developments in mem­brane biophysics.

The study of electrophysiology has progressed rapidly because of the precise, delicate, and ingenious experimental studies of many investigators. The field has also made great strides by unifying the numerous experimental observations through the development of increasingly accurate theoretical concepts and mathematical descriptions. The application of these funda­mental principles has in turn formed a basis for the solution of many different electrophysiological problems.

In past years, most introductory texts of electrophysiology have been primarily descriptive. In them, the more quantitative and theoretical aspects of the field have usually been left to footnotes, appendixes, and references. As a consequence, there has been little opportunity for a beginning student to approach this field quantitatively. The goal of this textbook is to introduce the field of electrophysiology from a frankly theoretical and

vii

viii Preface

quantitative perspective, to provide an approach to the subject from the perspective actually used in many advanced texts and research papers. We do not minimize the importance of descriptive material (and have included this as well) but feel that firmer understanding can be achieved through an examination of quantitative relationships. Since this requires the intro­duction of basic scientific principles, the subject under study is additionally strengthened.

Durham, North Carolina

Robert Plonsey Roger C. Barr

Acknowledgments

The authors acknowledge with appreciation the comments and suggestions of those students who have used earlier drafts of this text. They welcome further comments and suggestions.

The authors also greatly appreciate the hard work and wise counsel of Ms. Ellen Ray, who has typed and revised innumerable copies of inscrutable scribbling, corrected numerous mistakes, and done it all with good humor and great patience.

R.P. R.CB.

IX

Contents

Chapter 1. Vector Analysis Introduction. . .

Vectors and Scalars Vector Algebra. .

Sum ..... Vector Times Scalar Unit Vector. . . Dot Product. . . Resolution of Vectors. Cross Prod uct . . .

Gradient. . . . . . Potential Change Written as Dot Product Properties of G. . . . . . Gradient V . . . . . . . Comments about the Gradient

Divergence . . . . . . . . Outflow through Surfaces 1 and 2 Outflow through All Six Surfaces Divergence . . . . . . . . Comments about the Divergence.

Laplacian. . . . . . . . . Comments about the Laplacian

Vector Identities . . . . Useful Vector Identities. . . Verification of Eq. (1.38). . .

The Gradient of Source and Field Points Gradient of (l/r) Gradient of (l/r')

Gauss's Theorem. Green's Theorem .

Green's First Identity. Green's Second Identity.

1 1 2 2 2 2 2 3 4 5 6 7 7 8 8 9

10 10 11 12 12 13 13 13 14 15 15 16 16 16 17

xi

xii

Comment on Green's Theorem Summary of Operations Exercises. . . . . . . . .

Chapter 2. Electrical Sources and Fields. Fundamental Relationships .

Potentials, Fields, Currents. Poisson's Equation

Duality Monopole Field . . Dipole Field. . . .

Expressing r1 in Terms of r Evaluation of the I/r Derivative . Taking the Gradient. . . . .

Units for Some Electrical Quantities Exercises. . . . . . . . . .

Chapter 3. Introduction to Membrane Biophysics. Introduction. . . . Membrane Structure. . Ionic Composition Nernst-Planck Equation

Diffusion. . . . Electric Field Einstein's Equation Total Flow . . .

Equivalent Conductance Transference Numbers Nernst Potential . .

Concentration Cell Nernst Equilibrium Biological Membrane. Relative Charge Depletion .

Resting Potential. . Donnan Equilibrium. . . .

Two Ion Species . . . . More Than Two Ion Species Distribution of Ions Biological Systems. .

Goldman Equations. . Analysis for One Ion. Combined Flow of Several Ions Goldman's Equation for the Membrane Voltage Slope and Chord Conductance .

Contents

17 18 18

21 21 21 22 23 24 26 27 27 28 29 30

33 33 33 35 36 36 37 37 38 38 40 41 41 42 43 43 44 44 44 46 46 47 48 49 50 51 52

Contents

Role of Chloride Ion at Rest. . . Chloride Tracks Potassium. . . Experimental Study of the Resting Potential. Experimental EtTects of Chloride Ion

Exercises. . . . . . . . . . Design Project: AC Biogenerator Other Information. . . .

Chapter 4. Action Potentials Observed Action Potentials .

Earthworm Action Potentials. Earthworm Extracellular Potentials.

Nonlinear Membrane Behavior. . . Action Potentials in Crab Axon. . Stimulus and Response in Crab Axon . Nonlinear Membrane Measurements .

Origin of Action Potential, Resting and Peak Voltages. Changing Permeabilities. . . . . . . . Resting and Peak Voltages of Aplysia. . . Gross Explanation of Action Potential Origin Movements of Ionic Tracers . . . . . . Voltage Clamp. . . . . . . . . . .

A More Detailed Action Potential Explanation More Detailed Model . . . . . . . . Notation for Transmembrane Potential . . Notation for Intra- and Extracellular Potentials

Parallel-Conductance Model. Ionic Currents . . . . . . Capacitative Current. . . . Vm as Related to Total Current Example for Squid Axon

Voltage Clamp. . . . . . Origin of Voltage Clamp . Basic Voltage Clamp Design Voltage Clamp Records. Current-Voltage Curves .. Independence Principle. . Separation of Ionic Current into Components

Hodgkin-Huxley Equations Model for Potassium. . . . Model for Sodium. . . . . HH Method for Evaluating hoc

Simulation of Membrane Action Potential Analytical Evaluation. Numerical Procedure. Calculation Results .

xiii

52 52 52 54 56 60 61

65 65 65 67 68 69 70 71 71 71 72 73 73 74 74 74 75 75 75 75 77 77 77 78 78 79 80 82 83 84 85 86 88 91 91 92 93 94

xiv

Action Potential Characteristics. Refractory Periods. . . Anode Break Excitation.

Active Transport. . . . Pump's Characteristics . Formal Stoichiometric Approach Pump Included in Steady-State Model.

Exercises. . . . . . . . . . . .

Chapter 5. Propagation Introduction. . . . . Core-Conductor Model.

Resistance and Capacitance in a Cylindrical Fiber Electrical Model . . . . . . . Core-Conductor Model Assumptions. . . . .

Cable Equations . . . . . . . . . . . . . Relationship of Potential to Longitudinal Current Relationship of Longitudinal Intracellular Current

to Transmembrane Current. . . . . . . . Expression Relating Longitudinal Extracellular Current

to the Total Transmembrane Current (Including Applied Currents) . . . . . . . .

Spatial Derivatives of <1>. and <1>" . . Vrn Related to <1>, and <1>.. . • . . • Membrane Current Related to iJ2 Vm /OX2

Local Circuit Currents during Propagation. Mathematics of Propagating Action Potentials Numerical Solutions for Propagating Action Potentials Propagation Velocity Related to Radius. Propagation in Myelinated Nerve Fibers.

Myelin Sheath Propagation .

Exercises. . .

Chapter 6. Subthreshold Stimuli. Linear Subthreshold Conditions. . Space and Time Constants. . . . Stimulus Current at Origin (Steady-State Solution)

The Problem. . . . . Equations Governing Vrn

Region of the Stimulus . The Homogeneous Solution Imposing Boundary Conditions at Origin The Steady-State Solution. . . . . .

Contents

95 95 96 97 97 98 99

101

105 105 105 105 106 108 108 109

109

109 110 III lI2 112 114 lI5 116 118 118 119 120

125 125 127 128 128 128 129 129 130 131

Contents

Step Current at Origin. General Time-Varying Solution Laplace Transformation. Boundary Condition. . . . . Solution . . . . . . . . . Interpretation of Spatial and Temporal Response.

Cable Input Impedance. Cables of Finite Length.

Finding Zin in General Reflection Coefficient. Zin for a Terminated Cable. Cable of Finite Length .

Single Spherical Cell. . . Response to Current Step Rheobase. . . . . . Chronaxie. . . . . . Comparison to Experimental Findings.

Exercises. . . . . . . . .

Chapter 7. Extracellular Fields. Introduction. . . . Basic Formulation

Fiber Source Model Potentials from Source Elements. Potentials in Terms of Vm

Monopole Source Density . Dipole Source Density . . Modification for Thick Fiber

Fiber Source Models: Dipoles Depolarization and Repolarization Dipoles Quadrupolar Source. . . . Rectangular Action Potential .

Fiber Source Models: Monopoles Triangular Action Potentials Quadrupole Approximation .

Exercises. . . . . . . . . Extracellular Detection Design

References . . . . . . . .

Chapter 8. Membrane Biophysics Introduction. . . . . . Voltage Clamp. . . . .

Space-Clamp Uniformity Error in Sensing Vm . • .

Newer Voltage Clamp Methods. Sucrose Gap. . . . . . .

xv

131 132 132 134 135 135 138 138 139 140 140 141 141 143 143 144 145

149 149 149 150 150 151 152 153 154 154 155 156 156 157 159 159 160 163 163

165 165 166 166 167 169 169

xvi

Two- and Three-Microe1ectrode Voltage Clamp Spherical Cell . . . . . . . Cylindrical Cell. . . . . . . .

Single-Microelectrode Voltage Clamp. Patch Clamp . . . . . Single-Channel Morphology Single-Channel Currents . Single-Channel Kinetics. . F1uctuation-Dissipation Theorem Channel Statistics. . . . . . Membrane Current . . . . . Hodgkin-Huxley Potassium Channel-General Comments Hodgkin-Huxley Potassium Channel Fluctuation Noise Sources of Membrane Noise

Thermal Noise Shot Noise . . . . . II! Noise. . . . . .

Appendix: Random Variables, Autocorrelation Function, and Power Density Spectra

Random Variables. . Random Processes. . Correlation Functions Spectral Analysis . .

Exercises. . . . . .

Chapter 9. The Electrophysiology of the Heart C>verview. . . . . . . . . . . . . . . Electrical Nature of Intercellular Communication. Evidence for Functional Continuity in Cardiac Muscle. Free Wall Activation of the Heart Double-Layer Sources Heart Vector (Dipole) Lead Vector. . Standard Leads Lead Field . .

The Source-Field Description. Reciprocity . . Lead Field Multiple Dipoles . Lead System Design Application of Lead Field Theory to Standard

Electrocardiographic Lead I Recording . . . . . . . . .

Intracellular versus Extracellular. Extracellular Recordings Reference Electrodes. . . . .

Contents

171 171 172 175 176 180 181 182 184 185 187 191 192 195 195 196 196

196 197 197 198 199 202

205 205 207 208 210 213 216 217 218 222 222 223 224 225 225

225 226 227 227 228

Contents

Intramural Electrodes for Cardiac Activity Wave Thickness . . . . . . . . .

Human Cross-Sectional Anatomy . . . . Body Surface Potentials from Distributed Cardiac Potentials

Green's Theorem Applied to Body Volume Simplification of Integral . . . . . . . . . . . Introduction of Solid Angle . . . . . . . . . . Body Potential from Epicardial Potentials and Gradients Simplifications. . Transfer Coefficients .

Exercises. References

Chapter 10. The Neuromuscular Junction

xvii

229 230 234 234 235 236 237 237 238 238 239 243

245 Introduction. . . . . . . . . . . . 245 Neuromuscular Junction . . . . . . . 246 Evidence for Quantal Nature of Transmitter Release 248 Poisson Statistics for Transmitter Release-Single Trial 249 Expressions for Effect of Ca2 " and MgH on Transmitter Release. 251 Post junctional Response to Transmitter . 254 Exercises . 256 References . . . . . . . . . . . 257

Chapter 11. Skeletal Muscle Muscle Structure. . . . Muscle Contraction. . . Structure of the Myofibril . Sliding Filament Theory Excitation-Contraction. Exercises. References

Chapter 12. Functional Neuromuscular Stimulation Introduction. . . . . . Electrodes . . . . . . . . . Electrode-Tissue Behavior. . . . Electrode Operating Characteristics Electrode Materials . . . . . . Types of Electrode (for Specific Application) Nerve Excitation. . . . . . Secondary Pulse Considerations. Excitation of Myelinated Nerve. Cuff Electrodes. . . Recruitment. . . . Nerve Cuff Electrode.

259 259 260 262 265 269 270 270

271 271 271 272 274 278 278 280 283 284 286 291 292

xviii

Surface Electrode . Intramuscular Electrode . . . . . . . . . . Muscle Alterations Induced by Electrical Activation. Recruitment Regimen Clinical Applications. Exercises. References

Index

Contents

293 293 293 295 296 296 299

301

Bioelectricity A Quantitative Approach