Interdisciplinary Applied Mathematics Volume 25 Editors S.S. Antman J.E. Marsden L. Sirovich S. Wiggns Geophysics and Planetary Sciences Mathematical Biology L. Glass, J.D. Murray Mechanics and Materials R.V.Kohn Systems and Control S.S. Sastry, P.S. Krishnaprasad Problems n engineerng, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines has led to the establishment of the series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouragng the ways that that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and, secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods and suggest innovative developments within mathematics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology. Interdisciplinary Applied Mathematics Volumes published are listed at the end of the book. Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo Anne Beuter Leon Glass Michael C. Mackey Michele S. Titcombe Editors Nonlinear Dynamics in Physiology and Medicine With 162 Illustrations 'Springer Anne Beuter Leon Glass Institut de Biologic Department of Physiology Uiversite de Montpellier I McGill University 4, Boulevard Henri IV Montreal,Quebec H3G I Y6 Montpellier Cex I, 34060 Canada France [email protected] .ca anne. beuter@wanadoo. fr Michael C. Mackey Michele S. Titcombe Department of Physiology Dpartment of Physiology McGill University McGill University Montreal, Quebec H3G I Y6 Mo ntreal, Queb HJG I Y6 Canada Canada ma [email protected] [email protected] Editors S.S. Antman J.E. Marsden Department of Mathematics Con trolan d Dynamical Systems and Mail Code 107-81 Institute for Physical Science and Technology California Institute of Techn ology University of Maryland Pasadena, CA 91125 College Park, MD 20742 USA USA [email protected] [email protected] L. Sirvich S. Wiggins Division of Applied Mathmatics School of Mathemaics Brown University University of Bristol Providence, RI 02912 Bristol BS8 ITW USA UK [email protected] [email protected] Mathematics Subject Classification (2000): 92Cxx, 37-01 Library of Congress Calaloging-in-Publication Oala Nonlinear dynamics in physiologyand medicine I editors, nne Beuter ... (et al.). p. cm.-{Jnterdisciplinary appliedmathematics ; v.25) Includes bibliographical references and inde><. ISBN 0-387-004-1 (hbk. : alk. paper} I. Physiology-Mathematical models. 2. Nonlinear systems. 3. Dynamics. I. Beute r, nne. 11. Series. QP33.6.M3 6N663 2003 6t2'.001'51J�c21 200304436 ISBN 0-387-0044-1 Printed on acid-free paper. @ 2003 Springer-Verlag New York., Inc. ll rights reserved. This work may not be translated or copied in whole or in part without the written pennission of the publisher (Springer-Verlag New York, Inc., 175 Fifth venue, New York., NY 10010, US), e><cept for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any fonn of infonnation storage and retrieval, electronic adaplation, computer sol\ware, or by similar or dissimilar methodology now known or hereal\er developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar tenns, even if they are not identi­ fied assuch, is not to belaken as an expression of opinion as towhether or not they are subject to proprielary rights. Prined in the United Slates of America. 98765432 1 SPIN 10t 172 Typesetting: Pages created by the editors using a Springer LaTeX macro package. www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member ofBerlelsmannSpringer Science+ Business Media GmbH Contributors Jacques Beair Caroline Haurie Departement de Mathematiques Department of Physioogy et de Statistique McGil University Universite de Montrea Anne Beuter Andre Longtin Laboratoire de Physiologie Department of Physics Institut de Biologie University of Ottawa Universite de Montpelier 1 Marc Courtemanche Michael C. Mackey Centre de Recherche Department of Physioogy Institut de Cardioogie de Montrea McGi University Roderick Edwards John Miton Department of Mathematics Department of Neuroogy and Statistics The University of University of Victoria Chicago Hospitas Leon Glass Michee S. Titcombe Department ofPhysiology Department of Physioogy McGi University McGi University Michae R. Guevara Aain Vinet Department of Physioogy Departement de Physioogie McGi University Universite de Montrea v We dedicate this book to the memory of Arthur Winfree, a man before his time in life and death. Preface Je tiens impossible de connaˆıtre les parties sans connaˆıtre le tout, non plus que de connaˆıtre le tout sans connaˆıtre particuli`erement les parties –Pascal The eternal mystery of the world is its comprehensibility –Einstein This book deals with the application of mathematical tools to the study of physiological systems. It is directed toward an audience of physiologists, physicians, physicists, kinesiologists, psychologists, engineers, mathemati- cians, and others interested in finding out more about the complexities and subtleties of rhythmic physiological processes from a theoretical per- spective. We have attempted to give a broad view of the underlying notions behind the dynamics of physiological rhythms, sometimes from a theoretical perspective and sometimes from the perspective of the experimentalist. This book can be used in a variety of ways, ranging from a more tra- ditional approach such as a textbook in a biomathematics course (at either the advanced undergraduate or graduate level) to a research re- source in which someone interested in a particular problem might look at the corresponding discussion here to guide their own thinking. We hope that researchers at all levels will find inspiration from the way we have dealt with particular research problems to tackle completely new areas of investigation, or even approach these in totally new ways. Mathematically, we expect that readers will have a solid background in differential and integral calculus as well as a first course in ordinary differ- ential equations. From the physiological side, they should have a minimal training in general physiology at the organismal level. We have endeav- ored in every area to make the book as self-contained as possible given this mathematical and physiological preparation. Furthermore, many of the later chapters stand on their own and can be read independently of the others. When necessary we have added references to other chapters or to appendices. This book is divided roughly into two sections. Following an introduc- tory Chapter 1, the first section (sort of theory) mainly introduces concepts from nonlinear dynamics using an almost exclusively biological setting for viii Preface motivation. This is contained in Chapters 2 through 5. Chapter 2 intro- duces concepts from nonlinear dynamics using the properties of difference and differential equations as a basis. Chapter 3 extends and draws on this material in considering bifurcations from fixed points and limit cycles, illus- trating a number of bifurcation patterns with examples drawn from data. This is continued in Chapter 4, which focuses on excitable cell electrophys- iology. The first section concludes with a consideration in Chapter 5 of the properties of biological oscillators when perturbed by single stimuli or periodic inputs. The second part (applications, more or less) consists of five in-depth examples of how the authors have used the concepts of the first part of this book in their research investigations of biological and physiological systems. Thus, Chapter 6 examines the influence of noise in nonlinear dynamical systems using examples drawn from neurobiology. Chapter 7 looks at the properties of spatially extended nonlinear systems as illustrated by the properties of excitable cardiac tissue. In Chapter 8 attention is directed to the dynamics of cellular replication, illustrating how periodic diseases can illuminate underlying dynamics. Chapter 9 returns to the neurobiology arena in considering the properties of a simple neural feedback system, the pupil light reflex. The book continues with this neurobiological emphasis to conclude in Chapter 10 with an examination of the dynamics of tremor. We have not considered data analysis techniques as a separate subject, but rather have included an Appendix C, explaining those techniques we have found valuable (e.g., Fourier analysis, Lomb periodogram analysis). No doubt, other investigators will have their favorites that are not mentioned. Since a combination of both analytical and computational techniques are essential to maximize understanding of complex physiological dynamics, we have included both analytic and computer exercises throughout the book using either the freeware XPP or the generally available commercial package Matlab. Introductions to both XPP and Matlab are to be found in Appendices A and B, respectively. The source code and data files (and in some cases, help files) for the