Interdisciplinary Applied Mathematics

EditorsS.S. Antman J.E. MarsdenL. Sirovich

Geophysics and Planetary Sciences

Mathematical BiologyL. Glass, J.D. Murray

Mechanics and MaterialsR.V. Kohn

Systems and ControlS.S. Sastry, P.S. Krishnaprasad

Problems in engineering, computational science, and the physical and biologicalsciences are using increasingly sophisticated mathematical techniques. Thus, thebridge between the mathematical sciences and other disciplines is heavily traveled.The correspondingly increased dialog between the disciplines has led to the estab-lishment of the series: Interdisciplinary Applied Mathematics.

The purpose of this series is to meet the current and future needs for the interactionbetween various science and technology areas on the one hand and mathematics onthe other. This is done, firstly, by encouraging the ways that mathematics may beapplied in traditional areas, as well as point towards new and innovative areas ofapplications; and, secondly, by encouraging other scientific disciplines to engage in adialog with mathematicians outlining their problems to both access new methodsand suggest innovative developments within mathematics itself.

The series will consist of monographs and high-level texts from researchers workingon the interplay between mathematics and other fields of science and technology.

Volume 8/I

Interdisciplinary Applied Mathematics

1. Gutzwiller: Chaos in Classical and Quantum Mechanics2. Wiggins: Chaotic Transport in Dynamical Systems3. Joseph/Renardy: Fundamentals of Two-Fluid Dynamics:

Part I: Mathematical Theory and Applications4. Joseph/Renardy: Fundamentals of Two-Fluid Dynamics:

Part II: Lubricated Transport, Drops and Miscible Liquids5.

6. Hornung: Homogenization and Porous Media7.8.

9. Han/Reddy: Plasticity: Mathematical Theory and Numerical Analysis10. Sastry: Nonlinear Systems: Analysis, Stability, and Control11. McCarthy: Geometric Design of Linkages12. Winfree: The Geometry of Biological Time (Second Edition)13. Bleistein/Cohen/Stockwell: Mathematics of Multidimensional Seismic

Imaging, Migration, and Inversion14. Okubo/Levin: Diffusion and Ecological Problems: Modern Perspectives15. Logan: Transport Models in Hydrogeochemical Systems16. Torquato: Random Heterogeneous Materials: Microstructure

and Macroscopic Properties17. Murray: Mathematical Biology: An Introduction18. Murray: Mathematical Biology: Spatial Models and Biomedical

Applications19. Kimmel/Axelrod: Branching Processes in Biology20. Fall/Marland/Wagner/Tyson: Computational Cell Biology21. Schlick: Molecular Modeling and Simulation: An Interdisciplinary Guide22. Sahimi: Heterogenous Materials: Linear Transport and Optical Properties

(Volume I)23. Sahimi: Heterogenous Materials: Non-linear and Breakdown Properties

and Atomistic Modeling (Volume II)24. Bloch: Nonhoionomic Mechanics and Control25. Beuter/Glass/Mackey/Titcombe: Nonlinear Dynamics in Physiology

and Medicine26. Ma/Soatto/Kosecka/Sastry: An invitation to 3-D Vision27. Ewens: Mathematical Population Genetics (Second Edition)28. Wyatt: Quantum Dynamics with Trajectories29. Karniadakis: Microflows and Nanoflows30. Macheras: Modeling in Biopharmaceutics, Pharmacokinetics

and Pharmacodynamics31. Samelson/Wiggins: Lagrangian Transport in Geophysical Jets and Waves32. Wodarz: Killer Cell Dynamics33. Pettini: Geometry and Topology in Hamiltonian Dynamics and Statistical

Mechanics34. Desolneux/Moisan/Morel: From Gestalt Theory to Image Analysis

Keener/Sneyd: Mathematical Physiology, Second Edition:

From Equilibrium to ChaosSeydel: Practical Bifurcation and Stability Analysis:

II: Systems Physiology

Simo/Hughes: Computational Inelasticity

I: Cellular Physiology

James Keener James Sneyd

Mathematical PhysiologyI: Cellular Physiology

Second Edition

123

Series EditorsS.S. Antman J.E. MarsdenDepartment of Mathematics and Control and Dynamical Systems

Institute for Physical Science and Mail Code 107-81Technology

University of Maryland Pasadena, CA 91125College Park, MD 20742 USAUSA marsden@cds.caltech.edussa@math.umd.edu

L. SirovichLaboratory of Applied MathematicsDepartment of BiomathematicsMt. Sinai School of MedicineBox 1012NYC 10029USA

All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarlyanalysis. Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now known orhereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, evenif they are not identified as such, is not to be taken as an expression of opinion as to whetheror not they are subject to proprietary rights.

springer.com

James KeenerDepartment of Mathematics

Salt Lake City, 84112USAkeener@math.utah.edu

University of Utah

James Sneyd

ISBN 978-0-387-75846-6 e-ISBN 978-0-387-75847-3

Library of Congress Control Number: 2008931057

Printed on acid-free paper.

DOI 10.1007/978-0-387-75847-3

sneyd@math.auckland.ac.nzAuckland, New Zealand

California Institute of Technology

Department of Mathematics U niversity of Auckland Private Bag 92019

Lawrence.Sirovich@mssm.edu

© 2009 Springer Science+Business Media, LLC

To Monique,

and

To Kristine, patience personified.

Preface to the Second Edition

If, in 1998, it was presumptuous to attempt to summarize the field of mathematicalphysiology in a single book, it is even more so now. In the last ten years, the numberof applications of mathematics to physiology has grown enormously, so that the field,large then, is now completely beyond the reach of two people, no matter how manyvolumes they might write.

Nevertheless, although the bulk of the field can be addressed only briefly, thereare certain fundamental models on which stands a great deal of subsequent work. Webelieve strongly that a prerequisite for understanding modern work in mathematicalphysiology is an understanding of these basic models, and thus books such as this oneserve a useful purpose.

With this second edition we had two major goals. The first was to expand ourdiscussion of many of the fundamental models and principles. For example, the con-nection between Gibbs free energy, the equilibrium constant, and kinetic rate theoryis now discussed briefly, Markov models of ion exchangers and ATPase pumps are dis-cussed at greater length, and agonist-controlled ion channels make an appearance. Wealso now include some of the older models of fluid transport, respiration/perfusion,blood diseases, molecular motors, smooth muscle, neuroendocrine cells, the barore-ceptor loop, tubuloglomerular oscillations, blood clotting, and the retina. In addition,we have expanded our discussion of stochastic processes to include an introduction toMarkov models, the Fokker–Planck equation, the Langevin equation, and applicationsto such things as diffusion, and single-channel data.

Our second goal was to provide a pointer to recent work in as many areas as we can.Some chapters, such as those on calcium dynamics or the heart, close to our own fieldsof expertise, provide more extensive references to recent work, while in other chapters,dealing with areas in which we are less expert, the pointers are neither complete nor

viii Preface to the Second Edition

extensive. Nevertheless, we hope that in each chapter, enough information is given toenable the interested reader to pursue the topic further.

Of course, our survey has unavoidable omissions, some intentional, others not. Wecan only apologize, yet again, for these, and beg the reader’s indulgence. As with thefirst edition, ignorance and exhaustion are the cause, although not the excuse.

Since the publication of the first edition, we have received many comments (someeven polite) about mistakes and omissions, and a number of people have devoted con-siderable amounts of time to help us improve the book. Our particular thanks are dueto Richard Bertram, Robin Callard, Erol Cerasi, Martin Falcke, Russ Hamer, HaroldLayton, Ian Parker, Les Satin, Jim Selgrade and John Tyson, all of whom assisted aboveand beyond the call of duty. We also thank Peter Bates, Dan Beard, Andrea Ciliberto,Silvina Ponce Dawson, Charles Doering, Elan Gin, Erin Higgins, Peter Jung, Yue XianLi, Mike Mackey, Robert Miura, Kim Montgomery, Bela Novak, Sasha Panfilov, EdPate, Antonio Politi, Tilak Ratnanather, Timothy Secomb, Eduardo Sontag, Mike Steel,and Wilbert van Meerwijk for their help and comments.

Finally, we thank the University of Auckland and the University of Utah for contin-uing to pay our salaries while we devoted large fractions of our time to writing, andwe thank the Royal Society of New Zealand for the James Cook Fellowship to JamesSneyd that has made it possible to complete this book in a reasonable time.

University of Utah James KeenerUniversity of Auckland James Sneyd2008

Preface to the First Edition

It can be argued that of all the biological sciences, physiology is the one in whichmathematics has played the greatest role. From the work of Helmholtz and Frank inthe last century through to that of Hodgkin, Huxley, and many others in this century,physiologists have repeatedly used mathematical methods and models to help theirunderstanding of physiological processes. It might thus be expected that a close con-nection between applied mathematics and physiology would have developed naturally,but unfortunately, until recently, such has not been the case.

There are always barriers to communication between disciplines. Despite thequantitative nature of their subject, many physiologists seek only verbal descriptions,naming and learning the functions of an incredibly complicated array of components;often the complexity of the problem appears to preclude a mathematical description.Others want to become physicians, and so have little time for mathematics other thanto learn about drug dosages, office accounting practices, and malpractice liability. Stillothers choose to study physiology precisely because thereby they hope not to studymore mathematics, and that in itself is a significant benefit. On the other hand, manyapplied mathematicians are concerned with theoretical results, proving theorems andsuch, and prefer not to pay attention to real data or the applications of their results.Others hesitate to jump into a new discipline, with all its required background readingand its own history of modeling that must be learned.

But times are changing, and it is rapidly becoming apparent that applied mathe-matics and physiology have a great deal to offer one another. It is our view that teachingphysiology without a mathematical description of the underlying dynamical processesis like teaching planetary motion to physicists without mentioning or using Kepler’slaws; you can observe that there is a full moon every 28 days, but without Kepler’slaws you cannot determine when the next total lunar or solar eclipse will be nor when

x Preface to the First Edition

Halley’s comet will return. Your head will be full of interesting and important facts, butit is difficult to organize those facts unless they are given a quantitative description.Similarly, if applied mathematicians were to ignore physiology, they would be losingthe opportunity to study an extremely rich and interesting field of science.

To explain the goals of this book, it is most convenient to begin by emphasizingwhat this book is not; it is not a physiology book, and neither is it a mathematicsbook. Any reader who is seriously interested in learning physiology would be welladvised to consult an introductory physiology book such as Guyton and Hall (1996) orBerne and Levy (1993), as, indeed, we ourselves have done many times. We give only abrief background for each physiological problem we discuss, certainly not enough tosatisfy a real physiologist. Neither is this a book for learning mathematics. Of course,a great deal of mathematics is used throughout, but any reader who is not alreadyfamiliar with the basic techniques would again be well advised to learn the materialelsewhere.

Instead, this book describes work that lies on the border between mathematicsand physiology; it describes ways in which mathematics may be used to give insightinto physiological questions, and how physiological questions can, in turn, lead to newmathematical problems. In this sense, it is truly an interdisciplinary text, which, wehope, will be appreciated by physiologists interested in theoretical approaches to theirsubject as well as by mathematicians interested in learning new areas of application.

It is also an introductory survey of what a host of other people have done in em-ploying mathematical models to describe physiological processes. It is necessarily brief,incomplete, and outdated (even before it was written), but we hope it will serve as anintroduction to, and overview of, some of the most important contributions to thefield. Perhaps some of the references will provide a starting point for more in-depthinvestigations.

Unfortunately, because of the nature of the respective disciplines, applied mathe-maticians who know little physiology will have an easier time with this material thanwill physiologists with little mathematical training. A complete understanding of allof the mathematics in this book will require a solid undergraduate training in mathe-matics, a fact for which we make no apology. We have made no attempt whatever towater down the models so that a lower level of mathematics could be used, but haveinstead used whatever mathematics the physiology demands. It would be misleadingto imply that physiological modeling uses only trivial mathematics, or vice versa; theessential richness of the field results from the incorporation of complexities from bothdisciplines.

At the least, one needs a solid understanding of differential equations, includingphase plane analysis and stability theory. To follow everything will also require an un-derstanding of basic bifurcation theory, linear transform theory (Fourier and Laplacetransforms), linear systems theory, complex variable techniques (the residue theorem),and some understanding of partial differential equations and their numerical simu-lation. However, for those whose mathematical background does not include all ofthese topics, we have included references that should help to fill the gap. We also make

Preface to the First Edition xi

extensive use of asymptotic methods and perturbation theory, but include explanatorymaterial to help the novice understand the calculations.

This book can be used in several ways. It could be used to teach a full-year course inmathematical physiology, and we have used this material in that way. The book includesenough exercises to keep even the most diligent student busy. It could also be used asa supplement to other applied mathematics, bioengineering, or physiology courses.The models and exercises given here can add considerable interest and challenge to anotherwise traditional course.

The book is divided into two parts, the first dealing with the fundamental principlesof cell physiology, and the second with the physiology of systems. After an introduc-tion to basic biochemistry and enzyme reactions, we move on to a discussion of variousaspects of cell physiology, including the problem of volume control, the membrane po-tential, ionic flow through channels, and excitability. Chapter 5 is devoted to calciumdynamics, emphasizing the two important ways that calcium is released from stores,while cells that exhibit electrical bursting are the subject of Chapter 6. This book isnot intentionally organized around mathematical techniques, but it is a happy coinci-dence that there is no use of partial differential equations throughout these beginningchapters.

Spatial aspects, such as synaptic transmission, gap junctions, the linear cable equa-tion, nonlinear wave propagation in neurons, and calcium waves, are the subject of thenext few chapters, and it is here that the reader first meets partial differential equations.The most mathematical sections of the book arise in the discussion of signaling in two-and three-dimensional media—readers who are less mathematically inclined may wishto skip over these sections. This section on basic physiological mechanisms ends witha discussion of the biochemistry of RNA and DNA and the biochemical regulation ofcell function.

The second part of the book gives an overview of organ physiology, mostly fromthe human body, beginning with an introduction to electrocardiology, followed by thephysiology of the circulatory system, blood, muscle, hormones, and the kidneys. Finally,we examine the digestive system, the visual system, ending with the inner ear.

While this may seem to be an enormous amount of material (and it is!), there aremany physiological topics that are not discussed here. For example, there is almostno discussion of the immune system and the immune response, and so the work ofPerelson, Goldstein, Wofsy, Kirschner, and others of their persuasion is absent. An-other glaring omission is the wonderful work of Michael Reed and his collaboratorson axonal transport; this work is discussed in detail by Edelstein-Keshet (1988). Thestudy of the central nervous system, including fascinating topics like nervous control,learning, cognition, and memory, is touched upon only very lightly, and the field ofpharmacokinetics and compartmental modeling, including the work of John Jacquez,Elliot Landaw, and others, appears not at all. Neither does the wound-healing work ofMaini, Sherratt, Murray, and others, or the tumor modeling of Chaplain and his col-leagues. The list could continue indefinitely. Please accept our apologies if your favoritetopic (or life’s work) was omitted; the reason is exhaustion, not lack of interest.

xii Preface to the First Edition

As well as noticing the omission of a number of important areas of mathematicalphysiology, the reader may also notice that our view of what “mathematical” meansappears to be somewhat narrow as well. For example, we include very little discussionof statistical methods, stochastic models, or discrete equations, but concentrate almostwholly on continuous, deterministic approaches. We emphasize that this is not fromany inherent belief in the superiority of continuous differential equations. It resultsrather from the unpleasant fact that choices had to be made, and when push came toshove, we chose to include work with which we were most familiar. Again, apologiesare offered.

Finally, with a project of this size there is credit to be given and blame to be cast;credit to the many people, like the pioneers in the field whose work we freely bor-rowed, and many reviewers and coworkers (Andrew LeBeau, Matthew Wilkins, RichardBertram, Lee Segel, Bruce Knight, John Tyson, Eric Cytrunbaum, Eric Marland, TimLewis, J.G.T. Sneyd, Craig Marshall) who have given invaluable advice. Particularthanks are also due to the University of Canterbury, New Zealand, where a signifi-cant portion of this book was written. Of course, as authors we accept all the blamefor not getting it right, or not doing it better.

University of Utah James KeenerUniversity of Michigan James Sneyd1998

Acknowledgments

With a project of this size it is impossible to give adequate acknowledgment to everyonewho contributed: My family, whose patience with me is herculean; my students, whohad to tolerate my rantings, ravings, and frequent mistakes; my colleagues, from whomI learned so much and often failed to give adequate attribution. Certainly the mostprofound contribution to this project was from the Creator who made it all possible inthe first place. I don’t know how He did it, but it was a truly astounding achievement.To all involved, thanks.

University of Utah James Keener

Between the three of them, Jim Murray, Charlie Peskin and Dan Tranchina have taughtme almost everything I know about mathematical physiology. This book could not havebeen written without them, and I thank them particularly for their, albeit unaware,contributions. Neither could this book have been written without many years of supportfrom my parents and my wife, to whom I owe the greatest of debts.

University of Auckland James Sneyd

Table of Contents

CONTENTS, I: Cellular Physiology

Preface to the Second Edition vii

Preface to the First Edition ix

Acknowledgments xiii

1 Biochemical Reactions 11.1 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thermodynamics and Rate Constants . . . . . . . . . . . . . . . . . . . . 31.3 Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Enzyme Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 The Equilibrium Approximation . . . . . . . . . . . . . . . . . . 81.4.2 The Quasi-Steady-State Approximation . . . . . . . . . . . . . 91.4.3 Enzyme Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.4 Cooperativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.5 Reversible Enzyme Reactions . . . . . . . . . . . . . . . . . . . 201.4.6 The Goldbeter–Koshland Function . . . . . . . . . . . . . . . . 21

1.5 Glycolysis and Glycolytic Oscillations . . . . . . . . . . . . . . . . . . . . 23

xvi Table of Contents

1.6 Appendix: Math Background . . . . . . . . . . . . . . . . . . . . . . . . . 331.6.1 Basic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.6.2 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 371.6.3 Enzyme Kinetics and Singular Perturbation Theory . . . . . . 39

1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 Cellular Homeostasis 492.1 The Cell Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2.1 Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.2 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 532.2.3 Diffusion Through a Membrane: Ohm’s Law . . . . . . . . . . 542.2.4 Diffusion into a Capillary . . . . . . . . . . . . . . . . . . . . . . 552.2.5 Buffered Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 Facilitated Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3.1 Facilitated Diffusion in Muscle Respiration . . . . . . . . . . . 61

2.4 Carrier-Mediated Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 632.4.1 Glucose Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 642.4.2 Symports and Antiports . . . . . . . . . . . . . . . . . . . . . . . 672.4.3 Sodium–Calcium Exchange . . . . . . . . . . . . . . . . . . . . . 69

2.5 Active Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.5.1 A Simple ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.5.2 Active Transport of Charged Ions . . . . . . . . . . . . . . . . . 762.5.3 A Model of the Na+– K+ ATPase . . . . . . . . . . . . . . . . . . 772.5.4 Nuclear Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.6 The Membrane Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.6.1 The Nernst Equilibrium Potential . . . . . . . . . . . . . . . . . 802.6.2 Gibbs–Donnan Equilibrium . . . . . . . . . . . . . . . . . . . . 822.6.3 Electrodiffusion: The Goldman–Hodgkin–Katz Equations . . 832.6.4 Electrical Circuit Model of the Cell Membrane . . . . . . . . . 86

2.7 Osmosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.8 Control of Cell Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.8.1 A Pump–Leak Model . . . . . . . . . . . . . . . . . . . . . . . . . 912.8.2 Volume Regulation and Ionic Transport . . . . . . . . . . . . . 98

2.9 Appendix: Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 1032.9.1 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.9.2 Discrete-State Markov Processes . . . . . . . . . . . . . . . . . . 1052.9.3 Numerical Simulation of Discrete Markov Processes . . . . . 1072.9.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.9.5 Sample Paths; the Langevin Equation . . . . . . . . . . . . . . 1102.9.6 The Fokker–Planck Equation and the Mean First Exit Time . 1112.9.7 Diffusion and Fick’s Law . . . . . . . . . . . . . . . . . . . . . . 114

2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Table of Contents xvii

3 Membrane Ion Channels 1213.1 Current–Voltage Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

3.1.1 Steady-State and Instantaneous Current–Voltage Relations . 1233.2 Independence, Saturation, and the Ussing Flux Ratio . . . . . . . . . . 1253.3 Electrodiffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.3.1 Multi-Ion Flux: The Poisson–Nernst–Planck Equations . . . . 1293.4 Barrier Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.4.1 Nonsaturating Barrier Models . . . . . . . . . . . . . . . . . . . 1363.4.2 Saturating Barrier Models: One-Ion Pores . . . . . . . . . . . . 1393.4.3 Saturating Barrier Models: Multi-Ion Pores . . . . . . . . . . . 1433.4.4 Electrogenic Pumps and Exchangers . . . . . . . . . . . . . . . 145

3.5 Channel Gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.5.1 A Two-State K+ Channel . . . . . . . . . . . . . . . . . . . . . . . 1483.5.2 Multiple Subunits . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.5.3 The Sodium Channel . . . . . . . . . . . . . . . . . . . . . . . . . 1503.5.4 Agonist-Controlled Ion Channels . . . . . . . . . . . . . . . . . 1523.5.5 Drugs and Toxins . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

3.6 Single-Channel Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.6.1 Single-Channel Analysis of a Sodium Channel . . . . . . . . . 1553.6.2 Single-Channel Analysis of an Agonist-Controlled Ion

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.6.3 Comparing to Experimental Data . . . . . . . . . . . . . . . . . 160

3.7 Appendix: Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623.7.1 The Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . 1633.7.2 A Fokker–Planck Equation Approach . . . . . . . . . . . . . . . 1653.7.3 Reaction Rates and Kramers’ Result . . . . . . . . . . . . . . . 166

3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4 Passive Electrical Flow in Neurons 1754.1 The Cable Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.2 Dendritic Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

4.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 1814.2.2 Input Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1824.2.3 Branching Structures . . . . . . . . . . . . . . . . . . . . . . . . 1824.2.4 A Dendrite with Synaptic Input . . . . . . . . . . . . . . . . . . 185

4.3 The Rall Model of a Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.3.1 A Semi-Infinite Neuron with a Soma . . . . . . . . . . . . . . . 1874.3.2 A Finite Neuron and Soma . . . . . . . . . . . . . . . . . . . . . 1894.3.3 Other Compartmental Models . . . . . . . . . . . . . . . . . . . 192

4.4 Appendix: Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . 1924.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

xviii Table of Contents

5 Excitability 1955.1 The Hodgkin–Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . 196

5.1.1 History of the Hodgkin–Huxley Equations . . . . . . . . . . . 1985.1.2 Voltage and Time Dependence of Conductances . . . . . . . . 2005.1.3 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 210

5.2 The FitzHugh–Nagumo Equations . . . . . . . . . . . . . . . . . . . . . . 2165.2.1 The Generalized FitzHugh-Nagumo Equations . . . . . . . . . 2195.2.2 Phase-Plane Behavior . . . . . . . . . . . . . . . . . . . . . . . . 220

5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6 Wave Propagation in Excitable Systems 2296.1 Brief Overview of Wave Propagation . . . . . . . . . . . . . . . . . . . . 2296.2 Traveling Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

6.2.1 The Bistable Equation . . . . . . . . . . . . . . . . . . . . . . . . 2316.2.2 Myelination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2366.2.3 The Discrete Bistable Equation . . . . . . . . . . . . . . . . . . 238

6.3 Traveling Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2426.3.1 The FitzHugh–Nagumo Equations . . . . . . . . . . . . . . . . 2426.3.2 The Hodgkin–Huxley Equations . . . . . . . . . . . . . . . . . . 250

6.4 Periodic Wave Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2526.4.1 Piecewise-Linear FitzHugh–Nagumo Equations . . . . . . . . 2536.4.2 Singular Perturbation Theory . . . . . . . . . . . . . . . . . . . 2546.4.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

6.5 Wave Propagation in Higher Dimensions . . . . . . . . . . . . . . . . . . 2576.5.1 Propagating Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . 2586.5.2 Spatial Patterns and Spiral Waves . . . . . . . . . . . . . . . . . 262

6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

7 Calcium Dynamics 2737.1 Calcium Oscillations and Waves . . . . . . . . . . . . . . . . . . . . . . . 2767.2 Well-Mixed Cell Models: Calcium Oscillations . . . . . . . . . . . . . . . 281

7.2.1 Influx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827.2.2 Mitochondria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827.2.3 Calcium Buffers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2827.2.4 Calcium Pumps and Exchangers . . . . . . . . . . . . . . . . . . 2837.2.5 IP3 Receptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857.2.6 Simple Models of Calcium Dynamics . . . . . . . . . . . . . . . 2937.2.7 Open- and Closed-Cell Models . . . . . . . . . . . . . . . . . . . 2967.2.8 IP3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2987.2.9 Ryanodine Receptors . . . . . . . . . . . . . . . . . . . . . . . . . 301

7.3 Calcium Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.3.1 Simulation of Spiral Waves in Xenopus . . . . . . . . . . . . . . 3067.3.2 Traveling Wave Equations and Bifurcation Analysis . . . . . . 307

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7.4 Calcium Buffering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.4.1 Fast Buffers or Excess Buffers . . . . . . . . . . . . . . . . . . . 3107.4.2 The Existence of Buffered Waves . . . . . . . . . . . . . . . . . 313

7.5 Discrete Calcium Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 3157.5.1 The Fire–Diffuse–Fire Model . . . . . . . . . . . . . . . . . . . . 318

7.6 Calcium Puffs and Stochastic Modeling . . . . . . . . . . . . . . . . . . 3217.6.1 Stochastic IPR Models . . . . . . . . . . . . . . . . . . . . . . . . 3237.6.2 Stochastic Models of Calcium Waves . . . . . . . . . . . . . . . 324

7.7 Intercellular Calcium Waves . . . . . . . . . . . . . . . . . . . . . . . . . 3267.7.1 Mechanically Stimulated Intercellular Ca2+ Waves . . . . . . . 3277.7.2 Partial Regeneration . . . . . . . . . . . . . . . . . . . . . . . . . 3307.7.3 Coordinated Oscillations in Hepatocytes . . . . . . . . . . . . . 331

7.8 Appendix: Mean Field Equations . . . . . . . . . . . . . . . . . . . . . . . 3327.8.1 Microdomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3327.8.2 Homogenization; Effective Diffusion Coefficients . . . . . . . 3367.8.3 Bidomain Equations . . . . . . . . . . . . . . . . . . . . . . . . . 341

7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

8 Intercellular Communication 3478.1 Chemical Synapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

8.1.1 Quantal Nature of Synaptic Transmission . . . . . . . . . . . . 3498.1.2 Presynaptic Voltage-Gated Calcium Channels . . . . . . . . . . 3528.1.3 Presynaptic Calcium Dynamics and Facilitation . . . . . . . . 3588.1.4 Neurotransmitter Kinetics . . . . . . . . . . . . . . . . . . . . . 3648.1.5 The Postsynaptic Membrane Potential . . . . . . . . . . . . . . 3708.1.6 Agonist-Controlled Ion Channels . . . . . . . . . . . . . . . . . 3718.1.7 Drugs and Toxins . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

8.2 Gap Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3738.2.1 Effective Diffusion Coefficients . . . . . . . . . . . . . . . . . . . 3748.2.2 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3768.2.3 Measurement of Permeabilities . . . . . . . . . . . . . . . . . . 3778.2.4 The Role of Gap-Junction Distribution . . . . . . . . . . . . . . 377

8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

9 Neuroendocrine Cells 3859.1 Pancreatic β Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

9.1.1 Bursting in the Pancreatic β Cell . . . . . . . . . . . . . . . . . . 3869.1.2 ER Calcium as a Slow Controlling Variable . . . . . . . . . . . 3929.1.3 Slow Bursting and Glycolysis . . . . . . . . . . . . . . . . . . . . 3999.1.4 Bursting in Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 4039.1.5 A Qualitative Bursting Model . . . . . . . . . . . . . . . . . . . . 4109.1.6 Bursting Oscillations in Other Cell Types . . . . . . . . . . . . . 412

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9.2 Hypothalamic and Pituitary Cells . . . . . . . . . . . . . . . . . . . . . . 4199.2.1 The Gonadotroph . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

10 Regulation of Cell Function 42710.1 Regulation of Gene Expression . . . . . . . . . . . . . . . . . . . . . . . . 428

10.1.1 The trp Repressor . . . . . . . . . . . . . . . . . . . . . . . . . . . 42910.1.2 The lac Operon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

10.2 Circadian Clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43810.3 The Cell Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

10.3.1 A Simple Generic Model . . . . . . . . . . . . . . . . . . . . . . . 44510.3.2 Fission Yeast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45210.3.3 A Limit Cycle Oscillator in the Xenopus Oocyte . . . . . . . . . 46110.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Appendix: Units and Physical Constants A-1

References R-1

Index I-1

CONTENTS, II: Systems Physiology

Preface to the Second Edition vii

Preface to the First Edition ix

Acknowledgments xiii

11 The Circulatory System 47111.1 Blood Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47311.2 Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47611.3 The Microcirculation and Filtration . . . . . . . . . . . . . . . . . . . . . 47911.4 Cardiac Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48211.5 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

11.5.1 A Simple Circulatory System . . . . . . . . . . . . . . . . . . . . 48411.5.2 A Linear Circulatory System . . . . . . . . . . . . . . . . . . . . 48611.5.3 A Multicompartment Circulatory System . . . . . . . . . . . . 488

11.6 Cardiovascular Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . 49511.6.1 Autoregulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49711.6.2 The Baroreceptor Loop . . . . . . . . . . . . . . . . . . . . . . . 500

11.7 Fetal Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50711.7.1 Pathophysiology of the Circulatory System . . . . . . . . . . . 511

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11.8 The Arterial Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51311.8.1 The Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . 51311.8.2 The Windkessel Model . . . . . . . . . . . . . . . . . . . . . . . . 51411.8.3 A Small-Amplitude Pressure Wave . . . . . . . . . . . . . . . . . 51611.8.4 Shock Waves in the Aorta . . . . . . . . . . . . . . . . . . . . . . 516

11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

12 The Heart 52312.1 The Electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

12.1.1 The Scalar ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52512.1.2 The Vector ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

12.2 Cardiac Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53412.2.1 Purkinje Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53512.2.2 Sinoatrial Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54112.2.3 Ventricular Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 54312.2.4 Cardiac Excitation–Contraction Coupling . . . . . . . . . . . . 54612.2.5 Common-Pool and Local-Control Models . . . . . . . . . . . . 54812.2.6 The L-type Ca2+ Channel . . . . . . . . . . . . . . . . . . . . . . 55012.2.7 The Ryanodine Receptor . . . . . . . . . . . . . . . . . . . . . . 55112.2.8 The Na+–Ca2+ Exchanger . . . . . . . . . . . . . . . . . . . . . . 552

12.3 Cellular Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55312.3.1 One-Dimensional Fibers . . . . . . . . . . . . . . . . . . . . . . . 55412.3.2 Propagation Failure . . . . . . . . . . . . . . . . . . . . . . . . . 56112.3.3 Myocardial Tissue: The Bidomain Model . . . . . . . . . . . . . 56612.3.4 Pacemakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

12.4 Cardiac Arrhythmias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58312.4.1 Cellular Arrhythmias . . . . . . . . . . . . . . . . . . . . . . . . . 58412.4.2 Atrioventricular Node—Wenckebach Rhythms . . . . . . . . . 58612.4.3 Reentrant Arrhythmias . . . . . . . . . . . . . . . . . . . . . . . 593

12.5 Defibrillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60412.5.1 The Direct Stimulus Threshold . . . . . . . . . . . . . . . . . . . 60812.5.2 The Defibrillation Threshold . . . . . . . . . . . . . . . . . . . . 610

12.6 Appendix: The Sawtooth Potential . . . . . . . . . . . . . . . . . . . . . . 61312.7 Appendix: The Phase Equations . . . . . . . . . . . . . . . . . . . . . . . 61412.8 Appendix: The Cardiac Bidomain Equations . . . . . . . . . . . . . . . 61812.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

13 Blood 62713.1 Blood Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62813.2 Blood Cell Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630

13.2.1 Periodic Hematological Diseases . . . . . . . . . . . . . . . . . . 63213.2.2 A Simple Model of Blood Cell Growth . . . . . . . . . . . . . . 63313.2.3 Peripheral or Local Control? . . . . . . . . . . . . . . . . . . . . 639

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13.3 Erythrocytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64313.3.1 Myoglobin and Hemoglobin . . . . . . . . . . . . . . . . . . . . 64313.3.2 Hemoglobin Saturation Shifts . . . . . . . . . . . . . . . . . . . 64813.3.3 Carbon Dioxide Transport . . . . . . . . . . . . . . . . . . . . . . 649

13.4 Leukocytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65213.4.1 Leukocyte Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . 65313.4.2 The Inflammatory Response . . . . . . . . . . . . . . . . . . . . 655

13.5 Control of Lymphocyte Differentiation . . . . . . . . . . . . . . . . . . . 66513.6 Clotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

13.6.1 The Clotting Cascade . . . . . . . . . . . . . . . . . . . . . . . . . 66913.6.2 Clotting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67113.6.3 In Vitro Clotting and the Spread of Inhibition . . . . . . . . . . 67113.6.4 Platelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

13.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678

14 Respiration 68314.1 Capillary–Alveoli Gas Exchange . . . . . . . . . . . . . . . . . . . . . . . 684

14.1.1 Diffusion Across an Interface . . . . . . . . . . . . . . . . . . . . 68414.1.2 Capillary–Alveolar Transport . . . . . . . . . . . . . . . . . . . . 68514.1.3 Carbon Dioxide Removal . . . . . . . . . . . . . . . . . . . . . . 68814.1.4 Oxygen Uptake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68914.1.5 Carbon Monoxide Poisoning . . . . . . . . . . . . . . . . . . . . 692

14.2 Ventilation and Perfusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 69414.2.1 The Oxygen–Carbon Dioxide Diagram . . . . . . . . . . . . . . 69814.2.2 Respiratory Exchange Ratio . . . . . . . . . . . . . . . . . . . . 698

14.3 Regulation of Ventilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 70114.3.1 A More Detailed Model of Respiratory Regulation . . . . . . . 706

14.4 The Respiratory Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70814.4.1 A Simple Mutual Inhibition Model . . . . . . . . . . . . . . . . 710

14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

15 Muscle 71715.1 Crossbridge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71915.2 The Force–Velocity Relationship: The Hill Model . . . . . . . . . . . . . 724

15.2.1 Fitting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72615.2.2 Some Solutions of the Hill Model . . . . . . . . . . . . . . . . . 727

15.3 A Simple Crossbridge Model: The Huxley Model . . . . . . . . . . . . . 73015.3.1 Isotonic Responses . . . . . . . . . . . . . . . . . . . . . . . . . . 73715.3.2 Other Choices for Rate Functions . . . . . . . . . . . . . . . . . 738

15.4 Determination of the Rate Functions . . . . . . . . . . . . . . . . . . . . 73915.4.1 A Continuous Binding Site Model . . . . . . . . . . . . . . . . . 73915.4.2 A General Binding Site Model . . . . . . . . . . . . . . . . . . . 74115.4.3 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . 742

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15.5 The Discrete Distribution of Binding Sites . . . . . . . . . . . . . . . . . 74715.6 High Time-Resolution Data . . . . . . . . . . . . . . . . . . . . . . . . . . 748

15.6.1 High Time-Resolution Experiments . . . . . . . . . . . . . . . . 74815.6.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . . 749

15.7 In Vitro Assays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75515.8 Smooth Muscle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756

15.8.1 The Hai–Murphy Model . . . . . . . . . . . . . . . . . . . . . . . 75615.9 Large-Scale Muscle Models . . . . . . . . . . . . . . . . . . . . . . . . . . 75915.10 Molecular Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759

15.10.1 Brownian Ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . 76015.10.2 The Tilted Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 76515.10.3 Flashing Ratchets . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

15.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770

16 The Endocrine System 77316.1 The Hypothalamus and Pituitary Gland . . . . . . . . . . . . . . . . . . 775

16.1.1 Pulsatile Secretion of Luteinizing Hormone . . . . . . . . . . . 77716.1.2 Neural Pulse Generator Models . . . . . . . . . . . . . . . . . . 779

16.2 Ovulation in Mammals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78416.2.1 A Model of the Menstrual Cycle . . . . . . . . . . . . . . . . . . 78416.2.2 The Control of Ovulation Number . . . . . . . . . . . . . . . . . 78816.2.3 Other Models of Ovulation . . . . . . . . . . . . . . . . . . . . . 802

16.3 Insulin and Glucose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80316.3.1 Insulin Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 80416.3.2 Pulsatile Insulin Secretion . . . . . . . . . . . . . . . . . . . . . 806

16.4 Adaptation of Hormone Receptors . . . . . . . . . . . . . . . . . . . . . 81316.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

17 Renal Physiology 82117.1 The Glomerulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

17.1.1 Autoregulation and Tubuloglomerular Oscillations . . . . . . . 82517.2 Urinary Concentration: The Loop of Henle . . . . . . . . . . . . . . . . 831

17.2.1 The Countercurrent Mechanism . . . . . . . . . . . . . . . . . . 83617.2.2 The Countercurrent Mechanism in Nephrons . . . . . . . . . . 837

17.3 Models of Tubular Transport . . . . . . . . . . . . . . . . . . . . . . . . . 84817.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

18 The Gastrointestinal System 85118.1 Fluid Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

18.1.1 A Simple Model of Fluid Absorption . . . . . . . . . . . . . . . 85318.1.2 Standing-Gradient Osmotic Flow . . . . . . . . . . . . . . . . . 85718.1.3 Uphill Water Transport . . . . . . . . . . . . . . . . . . . . . . . 864

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18.2 Gastric Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86618.2.1 A Steady-State Model . . . . . . . . . . . . . . . . . . . . . . . . 86718.2.2 Gastric Acid Secretion and Neutralization . . . . . . . . . . . . 873

18.3 Coupled Oscillators in the Small Intestine . . . . . . . . . . . . . . . . . 87418.3.1 Temporal Control of Contractions . . . . . . . . . . . . . . . . . 87418.3.2 Waves of Electrical Activity . . . . . . . . . . . . . . . . . . . . . 87518.3.3 Models of Coupled Oscillators . . . . . . . . . . . . . . . . . . . 87818.3.4 Interstitial Cells of Cajal . . . . . . . . . . . . . . . . . . . . . . . 88718.3.5 Biophysical and Anatomical Models . . . . . . . . . . . . . . . 888

18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 890

19 The Retina and Vision 89319.1 Retinal Light Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

19.1.1 Weber’s Law and Contrast Detection . . . . . . . . . . . . . . . 89719.1.2 Intensity–Response Curves and the Naka–Rushton Equation . 898

19.2 Photoreceptor Physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . 90219.2.1 The Initial Cascade . . . . . . . . . . . . . . . . . . . . . . . . . . 90519.2.2 Light Adaptation in Cones . . . . . . . . . . . . . . . . . . . . . 907

19.3 A Model of Adaptation in Amphibian Rods . . . . . . . . . . . . . . . . 91219.3.1 Single-Photon Responses . . . . . . . . . . . . . . . . . . . . . . 915

19.4 Lateral Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91719.4.1 A Simple Model of Lateral Inhibition . . . . . . . . . . . . . . . 91919.4.2 Photoreceptor and Horizontal Cell Interactions . . . . . . . . . 921

19.5 Detection of Motion and Directional Selectivity . . . . . . . . . . . . . . 92619.6 Receptive Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92919.7 The Pupil Light Reflex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933

19.7.1 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 93519.8 Appendix: Linear Systems Theory . . . . . . . . . . . . . . . . . . . . . . 93619.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939

20 The Inner Ear 94320.1 Frequency Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946

20.1.1 Cochlear Macromechanics . . . . . . . . . . . . . . . . . . . . . 94720.2 Models of the Cochlea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949

20.2.1 Equations of Motion for an Incompressible Fluid . . . . . . . 94920.2.2 The Basilar Membrane as a Harmonic Oscillator . . . . . . . . 95020.2.3 An Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . 95220.2.4 Long-Wave and Short-Wave Models . . . . . . . . . . . . . . . . 95320.2.5 More Complex Models . . . . . . . . . . . . . . . . . . . . . . . . 962

20.3 Electrical Resonance in Hair Cells . . . . . . . . . . . . . . . . . . . . . . 96220.3.1 An Electrical Circuit Analogue . . . . . . . . . . . . . . . . . . . 96420.3.2 A Mechanistic Model of Frequency Tuning . . . . . . . . . . . 966

Table of Contents xxv

20.4 The Nonlinear Cochlear Amplifier . . . . . . . . . . . . . . . . . . . . . . 96920.4.1 Negative Stiffness, Adaptation, and Oscillations . . . . . . . . 96920.4.2 Nonlinear Compression and Hopf Bifurcations . . . . . . . . . 971

20.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

Appendix: Units and Physical Constants A-1

References R-1

Index I-1