Table Of ContentBioelectricity
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
ISBN 978-1-4757-9458-8 ISBN 978-1-4757-9456-4 (eBook)
DOI 10.1007/978-1-4757-9456-4
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
Softcover reprint of the hardcover 1st edition 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
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 es
tablished in physical science and engineering, but also applicable to biolog
ical 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 mem
brane 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 funda
mental 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 intro
duction of basic scientific principles, the subject under study is additionally
strengthened.
Robert Plonsey
Roger C. Barr
Durham, North Carolina
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. . . 1
Vectors and Scalars 1
Vector Algebra. . 2
Sum ..... 2
Vector Times Scalar 2
Unit Vector. . . 2
Dot Product. . . 2
Resolution of Vectors. 3
Cross Prod uct . . . 4
Gradient. . . . . . 5
Potential Change Written as Dot Product 6
Properties of G. . . . . . 7
Gradient V . . . . . . . 7
Comments about the Gradient 8
Divergence . . . . . . . . 8
Outflow through Surfaces 1 and 2 9
Outflow through All Six Surfaces 10
Divergence . . . . . . . . 10
Comments about the Divergence. 11
Laplacian. . . . . . . . . 12
Comments about the Laplacian 12
Vector Identities . . . . 13
Useful Vector Identities. . . 13
Verification of Eq. (1.38). . . 13
The Gradient of Source and Field Points 14
Gradient of (l/r) 15
Gradient of (l/r') 15
Gauss's Theorem. 16
Green's Theorem . 16
Green's First Identity. 16
Green's Second Identity. 17
xi
xii Contents
Comment on Green's Theorem 17
Summary of Operations 18
Exercises. . . . . . . . . 18
Chapter 2. Electrical Sources and Fields. 21
Fundamental Relationships . 21
Potentials, Fields, Currents. 21
Poisson's Equation 22
Duality 23
Monopole Field . . 24
Dipole Field. . . . 26
Expressing r in Terms of r 27
1
Evaluation of the I/r Derivative . 27
Taking the Gradient. . . . . 28
Units for Some Electrical Quantities 29
Exercises. . . . . . . . . . 30
Chapter 3. Introduction to Membrane Biophysics. 33
Introduction. . . . 33
Membrane Structure. . 33
Ionic Composition 35
Nernst-Planck Equation 36
Diffusion. . . . 36
Electric Field 37
Einstein's Equation 37
Total Flow . . . 38
Equivalent Conductance 38
Transference Numbers 40
Nernst Potential . . 41
Concentration Cell 41
Nernst Equilibrium 42
Biological Membrane. 43
Relative Charge Depletion . 43
Resting Potential. . 44
Donnan Equilibrium. . . . 44
Two Ion Species . . . . 44
More Than Two Ion Species 46
Distribution of Ions 46
Biological Systems. . 47
Goldman Equations. . 48
Analysis for One Ion. 49
Combined Flow of Several Ions 50
Goldman's Equation for the Membrane Voltage 51
Slope and Chord Conductance . 52
