Introduction to Membrane Noise

Introduction to Membrane Noise

Louis 1 Defelice Emory University School of Medicine

Atlanta, Georgia

PLENUM PRESS . NEW YORK AND LONDON

Library of Congress Cataloging in Publication Data

DeFelice, Louis J Introduction to membrane noise.

Includes index. 1. Membranes (Biology)-Electric properties. 2. Electrophysiology. I. Ti­

tle. II. Title: Membrane noise. [DNLM: 1. Membranes. 2. Electrophysiol­ogy. QH 601 D313i) QH601.D4 574.19'127 80-16163

ISBN-13: 978-1-4613-3137-7 DOl: 10.1007/978-1-4613-3135-3

e-ISBN-13: 978-1-4613-3135-3

© 1981 Plenum Press, New York Softcover reprint ofthe hardcover lst edition 1981

A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,

recording, or otherwise, without written permission from the Publisher

Preface

I started working on membrane noise in 1967 with David Firth in the Department of Physiology at McGill University. I began writing this book in the summer of 1975 at Emory University under a grant from the National Library of Medicine. Part of the writing was also done at the Marine Biological Laboratory Library in Woods Hole and in the Library of the Stazione Zoologica in Naples.

I wrote this book because in the intervening years membrane noise became a definable subdivision of membrane biophysics and seemed to deserve a uniform treatment in one volume. Not surprisingly, this turned out to be much more difficult than I had imagined and some areas of the subject that ought to be included have been left out, either for reasons of space or because of my own inability to keep up with all aspects of the field.

This book is written for biologists interested in noise and for physicists and electrical engineers interested in biology. The first three chapters attempt to bring both groups to a common point of understanding of electronics and electrophysiology necessary to the study of noise and impedance in membranes. These chapters arose out of a course given over a period of six years to electrical engineers from the Georgia Institute of Technology and biologists from Emory University School of Medicine.

K. S. Cole (1979) has pointed out that to some extent "the computer succeeds calculus as a way of intellectual life," and I have to agree with him. Nevertheless I have stressed the analytic solution whenever possible. In doing so, I have tried to make the book mathematically self-contained. The principles behind integral and differential calculus are assumed, but otherwise the mathematics are developed as needed. For those readers

v

vi Preface

who require it, I recommend the excellent self-study review of elementary calculus found in the first few chapters of Franklin's Advanced Calculus.

The centerpiece of the mathematics used in this book is the Fourier transformation. In developing this subject I have followed the outstanding book The Fourier Transform and Its Applications by Bracewell. Fourier transformation is a special class of integral transformation; other types of transforms arise throughout this text and as an aid to the reader I have collected the most important equations from four major works in an appendix.* For those unacquainted with integral equations, some time will have to be spent on the purely mathematical aspects of this book to become familiar with the use of tables like those in the Appendix. The introductory chapter in Campbell and Foster's Fourier Integrals for Practical Applications is highly recommended as is Bracewell's practical dictionary of transforms at the end of his book. For a more advanced treatment, I recommend the elegant development of Integral Equations by Hochstadt. I have included many problems and exercises in this book that require mathematical solutions. The problems can be worked out using only the material presented in this volume and in some cases parallel examples given in the text. The exercises will require outside reading and may assume a mathematical background beyond that presented in this book.

The first use of noise analysis to study biological membranes is certainly not obvious. The idea appears to have been around for some time and one often finds oblique reference to the notion in unexpected places. Consider the figure below taken from Bacq (1976). The figure represents Bacq's view of chemical transmission taken from an earlier work of his pub­lished in 1934. Although I am probably reading more into Bacq's figure than he intended, I can almost hear the noise from the postganglionic cell in his figure, and I think any of the earlier workers on noise in electronic tubes would probably agree with me. One finds more explicit references to membrane noise analysis from sources not normally associated with the field but obviously very much in sympathy with the idea. From Frankenhaeuser (1960) we have the following comparison of the membrane of the node of Ranvier and the squid giant axon: "The toad fiber is clearly

* For convenience, text references to the equations from the four major works are given in forms such as CF # 1-201, BMP # 1.1-2; the initials stand for the work from which the equation is taken [CF for Campbell and Foster (1948), BMP for Bateman Manuscript Project (1954), OR for Oradshteyn and Ryzhik (1965), and BH for Bierens de Haan (1939)1, and the numbers are those used in the reference work (and in the Appendix for equations listed there). For example, BH # 151-9 is equation (9) of Table 151 in Bierens de Haan (1939).

Preface

::P:O:S:t~g:8:ng:':lo:n:IC=fl:b:re=====~~t:J;\f:::/ of the 8utomlc nerve system ~~~if'·::.

I " Chemical transmitter

Smooth muscle or glandular cell

vii

Schematic presentation of chemical transmission proposed by Bacq in 1934. (From Bacq, 1976.)

more permeable than the squid fiber. There is at present no way of deciding whether the difference depends on differences in the number of sites or in the efficiency of single sites." I'm not sure that Frankenhaeuser would agree, but I think we now know that the difference is basically in the number of more or less similar sites. The strongest evidence for this comes from noise analysis.

Some subjects only touched upon in this book obviously deserve more attention, for example, the use of inherent membrane noise to measure intercellular communication. The measurements are difficult and the theory is not completely worked out, so it may be some years before this technique is used regularly.

Another topic that I have not developed as far as I would have liked is the relationship between membrane noise and membrane impedance. The two are derived from the kinetics of channel conductance, and the formula that relates membrane noise to impedance, Sv = 1 Z 12 S[ , implies linearity. Yet even for small perturbations, excitable membranes may be significantly nonlinear, as shown in Figure 109.2. The above equation is a reasonable approximation in some cases, but it is unclear whether it holds for all voltage and frequency ranges or for all types of excitable membranes, some of which may be more nonlinear than others.

The most significant recent advance in membrane noise analysis is the measurement of currents through single channels in membranes. Bean et al. (1969) were the first to do this in model membranes. The technique was immediately exploited by Ehrenstein et at. (1970), who described the mechanism of the voltage-dependent conductance in this model membrane system at the level of single channels. Not long afterward, single-channel currents were measured in biological membranes (Neher and Sakmann,

viii Preface

1976a). The influence of this work on membrane noise analysis and membrane biophysics in general has been enormous.

The study of single-channel currents is just beginning. Preliminary reports of currents from single channels in excitable membranes have already been given at several recent conferences and the publications will be out before this book appears. David Clapham and I have been working on the measurement of several channels working together in small patches of cardiac membrane. The theoretical and experimental question of an action potential or an action current constructed from only a few channels is an unexplored area of noise analysis relevant to small cells or portions of cells in which the density of channels is low.

The channel models we develop from noise measurements or single­channel data must also describe macroscopic phenomena. The question is always the same: Is the macroscopic property being considered also a property of the single channel, or does the property result from an ensemble of channels, not to be found in anyone channel taken individually? An outstanding example of this type of problem is the explanation of the time variant conductance. I have taken a particular viewpoint in Figure 74.1, but the question is by no means resolved. One deep-seated prejudice that such models share is that all channels are alike and that the fluctuations we observe are due to statistical variations of identical subunits. This is implicit in the statistics we use and is essentially the same assumption one makes about electrons when analyzing Johnson noise.

Such problems will have to wait. In the meantime, I hope that the present work fulfills the expectations of those who supported it. I would like to thank certain people who, though they are not responsible for the shortcomings of this book, have influenced me and to whom I would like to express my appreciation: Hans Plendl; Cyril Challice; David Firth; Bert Verveen; Hans Michalides; Chuck Stevens; Enzo Wanke; Franco Conti; my mentor and friend, Alex Mauro; Bob DeHaan for his fruitful collaboration and for the Tender Heart Club in which many of the ideas presented in this book were first tested; Bill Adelman and Alberto Monroy for their laboratories and for the libraries of their institutions, the Marine Biological Laboratory in Woods Hole and the Stazione Zoologica in Naples, in which much of this book was written; Susan Clapham for typing, editing, and proofreading the manuscript and for helping not only with the organiza­tion of the book but also with the organization of the Erice Conference on Noise in Biological Membranes, held in Sicily in 1977; and David Clapham, Barry Sokol, Dick Ypey and John Clay, who helped me with many aspects of this work. I would also like to thank Vanni Taglietti for

Preface ix

orgamzmg a course on Noise Analysis in Biological Membranes in Pavia in 1978 that in some ways served as a model for the final form of this book, and Jerry Sutin for providing the environment in which this work could be done. This work was supported in part by NIH Grant LM02505 from the National Library of Medicine. Lastly, I wish to thank my family, Louie and Evelyn, Rachel, Emile and Anna Catherine, and Jean and Louis, for years of support and understanding.

It seemed appropriate to end the book more or less where it began, with the use of equivalent circuits to describe bioelectric phenomena. The papers in the last chapter were selected to give the reader a sense of the development of the subject as well as a summary of data. The list is incomplete but I hope adequate to the task.

Louis J. DeFelice

Contents

1. Animal Electricity 1. Newton's Opticks and Ganot's Physics 2 2. Pre-Galvani Experiments and the Leyden Jar. 4 3. Benjamin Franklin and the Magic Square 7 4. Volta's Electrophorus . 9 5. Galvani's First Experiment 12 6. Galvani's Second Experiment 13 7. Animal Electricity Described in Ganot's Physics 16 8. The Voltaic Pile and the Electric Fish 18 9. Examples 21 10. Models and Analogies Used in Electrophysiology 23

2. Basic Electrophysiology. 27

11. Salt Water Conducts Electricity 27 12. Resistance of Salt Water 31 13. How Ions Move: The Flux Equation 32 14. First Application of the Flux Equation: The Nernst Relation 36 15. Second Application of the Flux Equation: The Diffusion Potential 38 16. An Example of the Nernst Relation: The AgCl Electrode 40 17. An Example of the Diffusion Potential: The Agar Bridge 41 18. Steady Current in Ionic Solutions 42 19. The Integral Resistance 46 20. The Flux Equation and Potential Profile 50 21. An Example: The Glass Microelectrode 52 22. The Effect of Pressure on Integral Resistance 54 23. Membrane Rectification and Reactance . 58 24. Inductance and Capacitance: Time Domain . 60 25. Fourier Transformation: The Delta Function 63 26. Inductance and Capacitance: Frequency Domain 65 27. Equivalent Circuits of Membranes 71

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xii

28. Equivalent Circuits of Cells . . . . . . . 29. Cable Equation: Passive Properties. . .. 30. Equivalent Circuits and Active Membranes

Contents

73 76 80

3. Basic Circuit Theory ...... . 85

85 90 94 96 99

31. Voltage and Current Sources 32. Frequency Composition of Signals 33. The Mean and the Variance. . . 34. Spectral Density and Rayleigh's Theorem 35. Spectral Density and Source Impedance 36. Examples . . . . . . 37. Power Spectral Density . . . . . . . .

4. Noise Analysis .....

105 113

115

38. Filtering . . . . . . . . . . . . 115 39. Measurement of Spectral Density 124 40. Effect of Filter Bandwidth. 126 41. The Convolution Theorem. . . . 130 42. The Correlation Theorem . . . . 133 43. Measurement of Correlation Functions 137 44. Correlation Functions: Examples. . . 141 45. Integral Spectra . . . . . . . . . . 146 46. Relationship between the Integral Spectrum and the Correlation

Function. . . . . . . . . . . . 152 47. Examples of Integral Spectra 155 48. Inversion of the Integral Spectrum 158 49. The Integral Spectral Density . . 161 50. Multiple Lorentzians . . . . . . 164 51. Examples of the Inversion Formula 171 52. Correlation Functions of Filtered Noise 175 53. The Exponential Integral . . . . . . . 184 54. The Effect of Finite Time Measurements 196 55. Examples of Error Calculations . . . . 198 56. Comparison of Errors in Correlation Functions, Integral Spectra,

and Spectral Densities ........... . 218 57. Correcting Correlation Functions for ac Coupling . . . 224

5. Noise Sources 231

A. White Noise. 232

58. Johnson Noise 233 59. Derivation of the Nyquist Formula 236 60. Nyquist Formula for an Arbitrary Impedance . 243 61. Quantum Theory Formulation of Nyquist's Equation 245 62. Johnson Noise and the Nernst Equation 250 63. Measurement of Johnson Noise ......... . 252

Contents

B. l/f Noise

64. Excess Noise in Carbon Resistors 65. Excess Noise Depends on Current 66. The Effect of Resistor Size on Excess Noise 67. Resistor Noise: A Selected Chronological Bibliography. 68. Excess Noise in Ionic Conductors . . 69. Excess Noise in Lipid Bilayers. . . . 70. l/f Noise and Concentration Gradients 71. Theories of Excess Noise

C. Lorentzian Noise. . . . .

72. The Two-State Channel . 73. The Relationship between Channel Noise and Current Noise 74. Two-State Channels in Series . . . . . . . . . . . . . . 75. Bernoulli's Distribution for Two Independent Two-State Subunits 76. Correlation Functions for Two-State Channels in Series . . . . 77. The Relationship between Channel Models and Kinetic Schemes

D. Campbell's Theorem ...

78. The Mean and the Variance 79. Noise Spectra from Campbell's Theorem

6. Afer.nbrane Ir.npedance . .................. .

80. Equivalent Circuits of Kinetic Equations . . . . . . . . . . 81. The Small-Signal Impedance of a Population of Ionic Channels 82. The Small-Signal Impedance of Channels in a Membrane . 83. Transient Response to the RrLC Circuit . . . . . . . . 84. Voltage Noise from Channels Embedded in a Membrane 85. The Equivalent Noise Source for Channel Noise 86. Current Noise Parameters Derived from Voltage Noise and

Impedance ................... . 87. Small-Signal Impedance of the HH-Axon Membrane. 88. The Heaviside Line and the RrLC Cable. . . . . .

7. Experir.nental Results . . . . . .

89. Miniature End-Plate Potentials . 90. Acetylcholine Noise. . . . . . 91. Other Types of Chemically Induced Noise 92. ACh Noise under Voltage Clamp 93. Other Types of Chemically Induced Noise under Voltage Clamp 94. Effect of Procaine on ACh Noise . . . . . . . 95. Effect of Dithiothreitol on ACh Noise . . . . . 96. Current Noise from Denervated Skeletal Muscle. 97. Single-Channel Currents. . . . . . . . . . . . 98. Ion Flow through the ACh Channel. A Noise Analysis 99. Glutamate Noise I . . . . . . . . . . . . . . . .

xiii

257

258 260 262 272 274 282 287 289

291

296 309 313 316 319 320

323

324 329

333

333 337 340 343 349 352

355 356 361

371

371 376 384 386 393 395 398 398 402 405 408

xiv Contents

100. Glutamate Noise II. . . . . . . . 411 101. Electrical Noise from Motoneurons 413 102. Excitability Noise in Neurons 415 103. Nerve Membrane Noise. . . 417 104. Voltage Noise from the Node of Ranvier. 420 105. The Squid Giant Axon . . . . . . . . . 424 106. Current Noise from the Squid Giant Axon 426 107. Current Noise from the Node of Ranvier I . 432 108. Current Noise from the Node of Ranvier II 438 109. Small-Signal Impedance of Nerve and Heart Cell-Membranes. 443 110. Photoreceptor Noise. An Early Study . . . . . . . . . .. 448

ApPENDIX . . 453

REFERENCES 473

INDEX 491

Introduction to Membrane Noise