Biomathematics in 1980Papers presented at a workshop on Biomathematics: Current Status and Future Perspectives, Salerno, April 1980

313 Pages·1982·8.79 MB·English

by  Luigi Ricciardi and Alwyn Scott (Eds.)

BIOMATHEMATICS IN 1980
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NORTH-HOLLAND
MATHEMATICS STUDIES 58
Biomathematics in 1980
Papers presented at a workshop on
Biomathematics: Current Status and Future Perspectives,
Salerno, April 1980
Edited by
LUIGI RICCIARDI
University of Naples
Italy
and
ALWYN SCOTT
Los Alamos National Laboratory
N.M., U.S.A.
N·H
cp~C
I
1982
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM· NEW YORK· OXFORD
0) North-Holland Publishing Company, 1982
All riglus reserved. No pari of this publication may be reproduced, stored in a retrieval system,
or tran.l'milled, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner.
ISBN: 044486355 9
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM· NEW YORK· OXFORD
Sole distri~utorsfor the U.S.A. and Canada:
ELSEVIER NORTH-HOLLAND, INC.
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Library of Congress Cataloging in Publication Data
Main entry under title:
Biomathematics in 1980.
(North-Holland mathematics studies; 58)
Includes indexes.
1. Biomathematics. 2. Biology--Mathematical
models. I. Ricciardi, Luigi M., 1942 -
II. Scott, Alwyn, 1931 - . III. Series.
QH323.5.B48 591' .0724 81-22349
ISBN 0 ·444·86355 ·9 AACR2
PRINTED IN THE NETHERLANDS
PREFACE
Biomathematics, it is sometimes said, should fall between the cracks of
legitimate science. Papers get published i~area because biologists don't
know it isn't mathematics and mathematicians don't know it isn't biology.
Whatever the situation may have been in the past, we do not believe this
indictment to be true now. Good biomathematical research is appearing and the
range of this activity is impressive: from useful mathematical statements about
problems that are firmly rooted in the "wet" reality of biology to deep and
original theoretical speculations about outstanding puzzles.
The current state of affairs in biomathematics was surveyed at a workshop on
"Biomathematics: Current Status and Future Perspectives" which was held at the
University of Salerno during the middle of April, 1980. Flanked by the columns of
Paestum and the splendors of the Amalfi coast, we passed the week presenting and
listening to talks (of course) and considering the opening premise of this
preface. Of the many fine research summaries discussed at this workshop, we have
been constrained by space and time to select a subset for this book. The papers
presented here do not represent a complete spectrum of modern biomathematics, but
they do, we believe, clearly demonstrate the viability of the field. We hope this
book captures the intellectual excitement of the workshop; for an appreciation of
the soft beauty of the Salerno region, we suggest that the reader take advantage
of the next opportunity to travel there.
The first two chapters are the most "theoretical" of the collection. Rosen
uses some sophisticated concepts of modern communication theory as a context for
discussing a fundamental question: why do organisms age? The following chapter
by Haken takes advantage of recent progress in understanding the global behavior
of partial differential equations to consider another fundamental question of
biology: how do living systems organize themselves?
The next four chapters treat a subject where biomathematics has been most
successful: the dynamics of a single nerve cell. Holden gives us a most complete
survey of the current activities in this area, emphasizing the interest it has for
modern applied mathematics. Scott and Vota Pinardi describe some "wet" studies of
nerve pulse interaction and show how'useful the ideas presented by Holden can be.
Shingai demonstrates the value of a computer model for studying the dynamics of
single neurons and neuronic interactions in the retina. Finally, Angelini, Barbi,
Chi llemi and Petracchi show how a "leaky-integrator" model can be used to
understand the dynamics of phase locking between sensory neurons.
Eilbeck presents a chapter describing the numerical aspects of the path from
coupled nonlinear diffusion equations to pattern formation in chemical and
biological systems and de Mottoni discusses some corresponding theoretical
results. Parisi, Filosa and Monroy then tie these numerical and theoretical
considerations to a real biological problem: the development of form during
growth of a sea urchin embryo.
v
vi PREFACE
Next we turn to the question of complex neural systems, i.e., brains.
Braitenberg's chapter lays the groundwork with a careful survey of the relevant
facts from neurophysiology and anatomy. Dalenoort considers the long gap between
the facts of neurology and those of psychology, and employing the not ion of a
hierachical structure in self-organization, describes a system to do simple
arithmetic. The fundamental idea of Donald Hebb (i.e., cell assemblies) emerges
here to playa central role. Motivated by the performance of real neural systems,
Palm describes a design for an "associ at ive" memory, and Harth describes an
adaptive algorithm that models both the immune response and the evolution of
mimicry. Amari then presents a theory of self-organi zing nerve systems that
includes Hebbian, perceptron, correlation and orthogonal types of neural learning,
and Libos outlines some methodological and conceptual frames of neuronal analysis
and synthes is.
Turning to the question of real human visual perception, Radil-Weiss,
Radilova, Bozkov and Bohdanecky present a series of perceptual experiments
performed in a psychophysiology laboratory by nonmathematicians. Such experiments
are the reality toward which theoreticians must strive.
Nobile, Ricciardi and Sacerdote discuss dynamic properties of certain systems
of difference equations related to the classical logistic equation, and, in
particular, the Gompertz equation for population growth. Then Teramoto examines
the dispersive behavior of populations. His work is especially significant for at
least three reasons: i) The reappearance of "nonl inear diffusion" which was
previously discussed in Chapters III and IV in relation to nerve impulse dynamics,
in Chapters VII, VIII and IX as the basis for a theory of biological pattern
formation, in Chapter XIV in connection with the dynamics of nerve fields, and in
Chapter II from a general perspective. [Haken's claim that "most of the proposals
made with respect to self-organizing systems can be included by our approach" (see
Chapter II) does not appear to be an overstatement.] ii) The demonstration that
diffusive effects lead to new results, not previously obtained by Volterra, thus
demonstrating the continued fertility of this field, and iii) The close relation
of his work to biological reality. Finally De Klerk and Gatto discuss population
biology in the context of the original problem (fish harvesting) and provide a
theoretical basis for the strategy of "pulse" fishing .
.-
The last chapter, by Swiatek, in a sense closes the circle (that began with
a the?retical discussion of sensecence) by describing the practical problem of
modellng the pulmonary circuit of the cardiovascular system.
Looking back we see what a wide range of research topics has been considered:
electrophysiology, brain theories, embryonic development, self-organization,
sensescence, perception of form, memory structure, population dynamics and
circulatory system modeling. Yet many themes emerge that are central to several
fields: reaction diffusion effects, hierarchical structure, coupled nonlinear
diffusion equations, communication theory, self-organization, and the complex
reality of the biological world. One topic, only touched upon in the chapters
by Haken and by Nobile, Ricciardi and Sacerdote, is that of chaos. Dynamical
systems are now known to exhibit stochastic (noisy) behavior tha~a fundamental
property of the nonlinear equations. We expect this subject to playa much more
important role in future research.
Equally impressive is the wide geographical distrubution of this work;
chapters are written by scientists from: Japan, Italy, Czechoslovakia, West
Germany, the Netherlands, the United Kingdom, the U. S. A., Hungary, Canada, and
Poland. During the workshop we experienced very little difficulty in commun-
icating. Thus, in addition to demonstrating once again that people from many
diverse cultures can work together on problems of common interest, we are pleased
to have this book represent modern biomathematical research from an international
perspective rather than from that of a particular nation or school of thought.
PREFACE vii
This world-wide interest demonstrates that biomathematics is not "falling between
the cracks."
Finally we express our warmest thanks to those who helped us turn an idea
into a real book. Dr. Filomena ("Mena") De Santis played a central role
throughout the workshop: planning, organizing, worrying but always with that
brightness in the face of adversity that so characterizes the Neapolitans.
Without her contributions, the workshop itself would perhaps not have taken place,
not to mention this book. Lynne MacNeil put the manuscript together in two short
months which grew ever shorter as the deadl ine approached. To each of them we
say: "Brava."
Luigi Ricci ardi Alwyn Scott
Nap 1e s Los Alamos
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TABLE OF CONTENTS
PREFACE v
CHAPTER I: FEEDFORWARD CONTROL AND SENESCENCE
Robert Rosen
CHAPTER II: MATHEMATICAL METHODS OF SYNERGETICS FOR
APPLICATIONS TO SELF-ORGANIZING SYSTEMS 9
H. Haken
1. Introduction 9
2. The general approach 11
3. Some generalizations 12
4. Instability hierarchies, chaos, and how to escape it 13
5. Conclusion 13
CHAPTER III: THE MATHEMATICS OF EXCITATION 15
A. V. Holden
1. Introduction 15
2. The phenomenology of excitation 15
3. The geometry of excitation 20
4. Numerical solutions of the H-H equations 20
4a. Membrane equations 21
4b. Travelling wave equations 25
4c. Cable solutions 26
4d. Pharmacology 26
5. The FitzHugh equations 26
6. The FitzHugh-Nagumo equations 28
7. The Hodgkin-Huxley membrane equations 29
8. Generalized excitation-propagation equations 31
9. Alternatives to the Hodgkin-Huxley equations 33
10. Other excitation equations 34
lOa. Axons 34
lOb. Molluscan neuronal somata 35
lOco Cardiac muscle 36
11. General membrane excitation equations 38
12. Conclusions 39
ix