Table Of ContentBiological Oscillators:
Their Mathematical Analysis
THEODOSIOS PAVLIDIS
Department of Electrical Engineering
Princeton University
Princeton^ New Jersey
ACADEMIC PRESS New York and London 1973
A Subsidiary of Hat court Brace Jovanovich, Publishers
COPYRIGHT © 1973, BY ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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Library of Congress Cataloging in Publication Data
Pavlidis, Theodosios.
Biological oscillators.
Bibliography: p.
1. Biology-Periodicity. 2. Biological control
systems—Mathematical models. 3. Oscillations.
I. Title.
QH527.P38 574.1 72-13615
ISBN 0-12-547350-8
PRINTED IN THE UNITED STATES OF AMERICA
Dedicated to the memory of my father
PAUL
Preface
The goal of this monograph is to present the main features of the
dynamic properties of biological oscillators and the mathematical
techniques necessary for their investigation. It is not a comprehensive
description of all known biological oscillators, since this would require
a much bigger volume as well as a different type of expertise. Certain
classes of biological oscillators are described and then only in as much
detail as is required for the study of their dynamics.
Biological oscillators present a host of problems which are peculiar to
them. Whereas in most technical applications the important variables
and equations of the system are well known, this is rarely, if ever, the
case in biological systems. Thus one must use indirect means for their
description. In many instances the usual problems are placed in a
converse form. In technological applications one usually starts with the
structural description of a system and then determines its behavior during,
say, nonlinear resonance (entrainment). In a biological system one
attempts to deduce its structure from experimentally obtained behavior.
In other cases the whole problem might be peculiar to living systems.
For example, there is strong evidence that many biological clocks
consist of a population of mutually coupled oscillators. There are very
few other applications where one deals with more than two or three
coupled oscillators. This fact together with the intrinsic mathematical
difficulty of the problem can explain the scarcity of the technical literature
about such systems.
The analysis of the various techniques emphasizes the use of mathe-
matics as tools and it does not include rigorous proofs. A number of
mathematical models of specific biological systems are presented together
with results of computer simulations. Hopefully this will make the
treatment accessible to anyone familiar with the basic concepts and the
formalism of elementary differential equations and linear algebra.
Most of the examples and many of the general questions discussed are
from the area of low frequency biological rhythms reflecting my own
personal interest in that subject.
xi
Acknowledgments
I owe much to Colin S. Pittendrigh, now at Stanford University, who
introduced me to the subject and helped me to start my research in this
area. He has continued to be a constant inspiration in my work on
biological rhythms. I also wish to thank his former students and associates
at Princeton and, in particular, William F. Zimmerman, Arthur
T. Winfree, and Victor G. Bruce for the many discussions I have had
with them while developing the ideas which eventually resulted in this
monograph. In this book I have used many examples from the literature
and have included descriptions of experimental results in order to
illustrate the various mathematical techniques discussed. I am grateful
to J. A. Bourret (California State University, Long Beach), Britton
Chance and E. Kendall Pye (University of Pennsylvania), Patricia
DeCoursey (University of South Carolina), Jerry F. Feldman, Ruth
Halaban, and Jon W. Jacklet (State University of New York, Albany),
Klaus Hoffmann (Max-Planck-Institut für Verhaltenphysiologie), S.
Jerebzoff (Universite Paul Sabatier, Toulouse, France), Colin
S. Pittendrigh (Stanford University), Shepherd K. Roberts (Temple
University), Richard H. Swade (San Fernando Valley State College,
Northridge, California), and Arthur T. Winfree (Purdue University) for
permitting me to reproduce figures and tables from their papers. I am
thankful to Mrs. Pearl Bazel for her excellent typing of the manuscript
and to Steven L. Horowitz for his help in some of the computer
simulations. Last but not least I want to express my gratitude to my
wife Marian and my children Paul, Karen, and Harry for their patience
and understanding while this book was completed.
xiii
C H A P T ER 1
Fundamentals of the Mathematical Theory
of Oscillators
1.1 Introduction
This chapter contains a review of fundamental mathematical concepts
and techniques which will be used in the remainder of the book. Most
of this material is covered in courses on feedback theory and/or nonlinear
systems of senior or first-year graduate level. Thus readers with strong
technical backgrounds could skip this chapter. Others may find it
necessary to read selected sections. The background necessary for
its comprehension is calculus plus elementary linear algebra. This
material is usually covered in the first two years of college courses.
Since an increasing number of biologists attend such courses we felt
that it was safe to assume it as reasonable background for the book.
The next two sections deal with the phase plane techniques, including
an example from the theory of population dynamics. The fourth section
points out the salient features of the study of w-dimensional dynamical
systems. Next the asymptotic techniques of Krylov, Bogoliubov, and
Mitopolski are presented. These are based in approximating nonlinear
systems by linear ones. The sixth section deals with another linearization
technique, that of the describing function. Finally we review briefly
certain aspects of conservative systems, a class which can be handled
analytically easier than others and for this reason is often discussed in
the literature.
A number of references are listed for further reading at the end of
each section. The following are some which we found particularly
useful for an overall treatment of nonlinear oscillations: And-I
(comprehensive), Bog-I, But-I, Dav-I (concise with many specific
examples), Haa-I, Haj-I, Hal-I (rigorous and concise), Hay-I (many
1
2 1 THE MATHEMATICAL THEORY OF OSCILLATORS
examples), Lef-I, Lef-II, Min-I (comprehensive), Min-II, Nem-I
(abstract treatment), Oga-I, Oga-II, Pli-I, Saa-I, San-I, Ura-I.
1 .2 Phase Plane Techniques
The dynamic behavior of many physical systems can often be described
by a second-order differential equation or a system of two first-order
differential equations. Such descriptions involve two variables, which
if known at any given time t allow the prediction of the behavior of the
0
system for any time t > t on the basis of the solution of the differential
0
equation(s). For example, the position p(t ) and velocity v(t) of a mass m
0 0
suspended from a spring with constant k determine its further motion
on the basis of the equation
md2p/dt2 + kp = 0 (1.2.1)
Indeed the solution to the above is
p(t) = A cosf^/m)1'2 · t + φ] (1.2.2)
for some constants A and φ. These can be determined uniquely from
the conditions
p(t) = A cos<£, v(t) = —(klm)1/2A ύηφ (1.2.3)
0 0
The two variables which characterize the behavior of the system are
usually called the state of the system. They can be represented as
coordinates in a plane and in this way the behavior of the systems can be
described in terms of plane curves. Thus the derivative of Eq. (1.2.2) is
v(t) = -(kjm)1'2 A sm[(kjmf'21 + φ] (1.2.4)
Eliminating the time-dependent terms between this equation and
Eq. (1.2.1) we obtain
p2 + (m/k)v2 = A2 (1.2.5)
which represents an ellipse in the plane with coordinates p and v.
The curves which result by eliminating the time from the solution of
the differential equation(s) are called the trajectories of the system and
the plane where they are plotted the phase plane.
This formalism is useful not so much in cases where the differential
equation(s) can be solved explicitly as where they cannot. There exist
graphoanalytical techniques which allow one to determine the trajec-
1.2 PHASE PLANE TECHNIQUES 3
tories or certain properties of them and in turn deduce the dynamic
behavior of the system. We will study some of these techniques next.
The general form of the mathematical equations describing a system
which can be studied on the phase plane with (x, y) coordinates is
idxldt=f(xy) (1.2.6)
9
\dyldt=g(xy) (1.2.7)
t
Such a system is often called one degree of freedom because its motion
is along a plane curve. This term, unfortunately, is often misunderstood
to imply that the dynamic behavior of the system can be described by a
single variable, which is certainly not true. Another common term is
second-order system, which does not leave room for misunderstanding
and which has been motivated by the fact that in general any system of
the form given by Eqs. (1.2.6) and (1.2.7) is equivalent to one described
by a single second-order differential equation.
Any such equation can be placed in the above form. Indeed consider
the following:
d2x\dt2 +/(*, dx\dt) = 0 (1.2.8)
This is obviously equivalent to
dxjdt=y (1.2.9)
dyjdt = -f{x,y) (1.2.10)
The converse is also usually true. If f(x>y) is differentiable with
respect to both of its arguments, then differentiating Eq. (1.2.6) with
respect to time we obtain
d2x _ df dx df dy
It2 ~ ~dx~dt + dy~dt
or
d2x dfdx df ,
x Λ / 1 η Λ ΐ
If Eq. (1.2.6) can be solved with respect to y, then an expression of the
form
y = F(x, dx/dt)
is obtained. Substituting this for y in Eq. (1.2.II) leaves an equation
in x, dx/dt, and d2x/dt2. Although these transformations (differentiation
and solution with respect to y) are not always possible, they are valid
4 1 THE MATHEMATICAL THEORY OF OSCILLATORS
in most cases of practical interest. A similar reasoning can be used to
obtain a second-order differential equation with respect to y.
The points (x, y) for which the functions f(x, y) and g(x> y) take
e e
the value zero are called points of equilibrium or singularities or critical
points of the system 5. If the initial values of (x, y) are equal to the
coordinates of such a point then they will remain the same since both
derivatives dx\dt and dyjdt are zero.
What will happen if the point (x, y) is close but not exactly equal to
a point of equilibrium ? Obviously there are only three possibilities:
(1) The distance between (x,y) and (x ,y) increases with time.
e e
Then the point (x , y) is called unstable.
e e
(2) The distance decreases with time; (x , y) is called asymptotically
e e
stable.
(3) The distance varies with time but it stays between an upper and
a lower bound; (x ,y) is called neutrally stable.
e e
In (1) and (2) the changes are not necessarily monotonie.
We can study the type of stability in the proximity of an equilibrium
point by using a linearization of Eqs. (1.2.6) and (1.2.7). Let
χ = xe + L y = y + η
e
Then substituting these quantities in the above equations and using
a Taylor series expansion for f(x y) and g(x, y) we obtain
9
The partial derivatives are all computed at the point (x , y). The
e e
first term of each expansion is by definition equal to zero. For sufficiently
small ξ and η the higher terms can be eliminated and we thus have
d i l d t = f - i + f , - (1.2.12)
e 1 v
dvldt=g-i+g- (1.2.13)
x yv
where f stands for df/dx, etc.
x
This is a linear system and it can easily be transformed into an
equation of the form
d*(ldt* - (fx + gv) dW + (fxgv ~ g*fv)i = 0 (1.2.14)
An identical equation can be obtained with respect to η.
1.2 PHASE PLANE TECHNIQUES 5
The solution of this differential equation is of the form
ξ = AeXlt + Bex* (1.2.15)
where A and Β depend on the initial conditions and λ , λ are the
1 2
two roots of the algebraic equation
λ2 - (fx + gy)* + (fxgy - gxfy) = 0 (1.2.16)
The solution for η is obtained by substituting Eq. (1.2.15) into
(1.2.12):
K - fx Ht K - fx x,t (1.2ΛΤ)
η = A e + B e
J y J y
If one or both of the X and λ have positive real parts, then ξ and η
1 2
will increase with time and the point (x , y) will be unstable.
e e
If both λ and λ have negative real parts, then ξ and η will tend to
1 2
zero and the point will be asymptotically stable.
If both λ and λ have zero real parts or one has zero and the other a
χ 2
negative real part, the point will be neutrally stable.
Note that the sum of the real parts of X and λ is f + g and thus an
x 2 x y
obvious necessary condition for stability is
fx+gy<0 (1.2.18)
The product A A equals f g — gf . For stability it is necessary
X 2 x y xy
that both roots have the same sign or that
fxgy- gxfy >0 (1.2.19)
It is easy to verify that (1.2.18) and (1.2.19) together are sufficient
conditions for the stability of x , y.
e e
We may further verify that the roots will be real if
(fx+gy)2-4(fxgy-g*fy) >0
or
(Λ-^)2 + 4^Λ>0 (1.2.20)
This analysis allows us to plot the trajectories of the system in the
neighborhood of an equilibrium point (see Fig. 1.2.1). Table 1.2.1 is
a list of the various types of possible configurations encountered together
with their usual nomenclature.
The behavior of the system away from an equilibrium point is more
difficult to study.
