Diffusion Processes and Related Topics in Biology

206 Pages·1977·3.525 MB·English

by  Prof. Luigi M. Ricciardi (auth.)

Lectu re Notes in
Biomathematics
Managing Editor: S. Levin
14
Luigi M. Ricciardi
Diffusion Processes
and Related Topics in Biology
Notes Taken by Charles E. Smith
Spri nger-Verlag
Berlin· Heidelberg· New York 1977
Editorial Board
W. Bossert· H. J. Bremermann . J. D. Cowan' W. Hirsch
S. Karlin' J. B. Keller' M. Kimura' S. Levin (Managing Editor)
R. C. Lewontin . R. May· G. F. Oster' L. A. Segel
Author
Prof. Luigi M. Ricciardi
Universita di Torino
Istituto di Scienze Dell'lnformazione
Corso Massimo D'Azeglio 42
10125 T orinolltaly
Notes taken by
Mr. Charles E. Smith
The University of Chicago
Dept. of Biophysics and
Theoretical Biology
Chicago, Illinois 60637/USA
Libr.uy of Congress Cataloging in Publication Data
Ricciardi, Luigi M 1942-
Diffusion processes and related topics in biology.
(Lecture notes in biomathematics; 14)
Bibliography: p.
Includes indexes.
1. Biomathematics. 2. Diffusion processes.
1. Title. II. Series.
QJ!323.5.R5 574' .01'82 77-7464
AMS Subject Classifications (1970): 60-01, 60J 60, 60 J 70, 92-02, 92 A 05,
92A15
ISBN-13: 978-3-540-08146-3 e-ISBN-13: 978-3-642-93059-1
001: 10.1007/978-3-642-93059-1
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation, re
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machine or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to be
determined by agreement with the publisher.
© by Springer-Verlag Berlin· Heidelberg 1977
Softcover reprint of the hardcover 1s t edition 1977
2145/3140-543210
CONTENTS
I. PRELIMINARIES. • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1
1. Terminology and Examples ••••••••••••••••••••••••••••••••••• 1
2. Markov Processes and the Smolukowski Equation •••••••••••••• 19
3. Auxiliary Definitions •••••••••••••••••••••••••••••••••••••• 25
II. DIFFUSION PROCESSES ••••••••••••••••••••••••••••••••••••••••••..• 31
1. Kinetic Equations ••••••••••••••••••••••••••••••••••••••••.• 31
2. Diffusion Equations 38
3. The Stationary Case 43
4. The Wiener Process ••••••••.•••.•••••••••••••••••••••••••••• 51
III. THE FIRST PASSAGE TIME PROBLEM •••••••••••••.•••••••••••••••••••• 61
1. Schr8dinger Equation ••••••••••••••••••••••••••••••••••••••• 61
2. Siegert's Method ••••••••••••••••••••••••••••••••••••••••••• 66
3. Siegert's Moment Relation •••••••••••••••••••••••••••••••••• 70
IV. DISCRETE MODELS ••••••••••••••••••••••••••••••••••••••••••••••••• 73
1. Random Walks ••••••••••••••••••••••••••••••••••••••••••••••• 73
2. Other Limits of Random Walks ................................ 81
3. Extinction and Absorption •••••••••••••••••••••••••••••••••• 85
4. Growth and Extinction in a Random Environment •••••••••••••• 98
5. Feller's Diffusion Equation •••••••••••••••••••••••••••••••• 113
6. Diffusion Models for Neuronal Activity ••••••••••••••••••••• 118
V. CONTINUOUS I-IODELS ..•....•...•....••..........•.•..•.•••.•..•.•.•.• 148
1. Stochastic Differential Equations .••..••..•.•.•...•...•.••... 148
2. The White Noise •......•.......................••...•..••....• 151
3. Special Cases and Examples ..•.•.........•...•...•............ 163
4. Transformations to the Wiener Process
5. Transformations to the Feller Process 180
Bibliography 189
Author Index 195
Subject Index .......•....................•......•••...•..........•.... 197
Preface
These notes are based on a one-quarter course given at the Department
of Biophysics and Theoretical Biology of the University of Chicago in 1916.
The course was directed to graduate students in the Division of Biological
Sciences with interests in population biology and neurobiology. Only a slight
acquaintance with probability and differential equations is required of the
reader. Exercises are interwoven with the text to encourage the reader to play
a more active role and thus facilitate his digestion of the material.
One aim of these notes is to provide a heuristic approach, using as
little mathematics as possible, to certain aspects of the theory of stochastic
processes that are being increasingly employed in some of the population biol
ogy and neurobiology literature. While the subject may be classical, the nov
elty here lies in the approach and point of view, particularly in the applica
tions such as the approach to the neuronal firing problem and its related dif
fusion approximations.
It is a pleasure to thank Professors Richard C. Lewontin and Arnold
J.F. Siegert for their interest and support, and Mrs. Angell Pasley for her
excellent and careful typing.
I . PRELIMINARIES
1. Terminology and Examples
Consider an experiment specified by:
a) the experiment's outcomes, ~, forming the space S;
b) certain subsets of S (called events) and by the probabilities of
these events.
Let us now associate with every outcome ~, according to a certain rule, a real
function, X(t,~), of the time variable. Thus doing we have constructed a family of
functions, one for each~. This family of functions is called a stochastic pro-
~, or a random function.
A stochastic process can thus be viewed as a function of two variables, t
and~. The domain of ~ is the set S of all possible outcomes or states of the
experiment, often called the state space of the process. This space may consist of
a discrete set of points, in which case our process is denoted as a discrete state
process; or of a continuum of states, in which case we have a continuous state pro-
cess. The domain of t is a set of real numbers, which likewise may be discrete or
continuous. Unless specified otherwise, we shall assume that the domain of t is
the entire time axis, i.e., the real line.
In summary, the stochastic process X(t,~) has the following properties:
1. For a specific outcome ~. ,X(t,~.) is a single deterministic function
J. J.
of time. This time function is called a realization or sample path
of the stochastic process.
2. For a specific time t. ,X(t. ,~) is a quantity depending on ~, i.e., it
J. J.
is a random variable.
3. X(t. ,~.) is a number.
J. J
In the following, a stochastic process will simply be denoted by X(t), i.e.,
we shall omit the explicit specification of the dependence on~. However, this
dependence is to be always implicitly assumed.
From the above considerations we see (c.f. Fig. 1.1) that X(t) may denote
four different things:
2
1. A family of fUnctions of time (t and ; variables)
2. A single fUnction of time (t variable and; fixed)
3. A random variable (t fixed and ; variable)
4. A single number (t fixed and ; fixed).
Figure 1.1 Illustrating the definition of stochastic process. The time fUnc
tions X(t,;r),X(t,;j), and X(t,;k) are sample paths. X(ti,;) where; takes the
values ;r' ;j' and ;k' is a random variable labelled by the chosen instant ti'
Up to now we have considered one-dimensional stochastic processes. However,
the above definition can be easily extended to the general case of an n-dimensional
stochastic process. Instead of X(t,;), we would consider the vector fUnction
X(t,;), or simply X(t), where each component of this vector is a one-dimensional
stochastic process.
Example 1. Let X(t) be the displacement (from an arbitrarily fixed origin) of a
colloidal particle subject to the impacts of molecules in the surrounding medium.
A specific outcome of the underlying experiment is the selection of a colloidal
particle and X(t) determines the trajectory of the particle. This trajectory looks
very irregular and cannot be described by a formula. Moreover, if one knows X(t)
up to a certain time tl , one cannot determine the process at times t > tl ( cf.
3
Fig. 1.2). We shall return to this particular process later on. For now, it suf-
fices to remark that this is a three-dimensional stochastic process in continuous
time with a continuum of states.
x
Figure 1.2. One sample path of the three-dimensional process of example 1,
depicting the trajectory of a colloidal particle.
Example 2. A simple, biologically motivated, example is offered by the time course
of a model neuron's threshold. After a spike (action potential) is generated, the
neuron's threshold may be described by a deterministic function of time; for
instance, by:
= So exp[a/(t-T)] a > O,t ~ T
S(t,T) {
, if the neuron has never fired up to time t.
~
where T is the instant at which the most recent neuronal firing occurred. Here T
is a random variable, whose value depends on the outcome of the underlying experi-
ment, i.e., the choice of a given neuron from a population of neurons. Thus setting
= =
S(t,T) S(t) X(t), we recognize X(t) as a one-dimensional stochastic process.
For a fixed T, X(t) = S(t) is a (deterministic) function of time expressed by the
mathematical relationship above. In figure 1.3 a realization of the process is
shown.
4
S(t)
So ~ ____- -l- ____ _
t
o
Figure 1.3. Time course of the threshold function of example 2.
It is easy to come up with other examples of stochastic processes, as all
of the works in References well illustrate. Probably the most commonly encountered
ones deal with fluctuations due to noise in physical devices and biological struc-
tures such as thermal noise, shot noise, membrane noise, etc. Depending upon the
physical experiment and our model of it, we may have one of four types of one-
dimensional stochastic processes: discrete or continuous time variable, and dis-
crete or continuous state variable. For clarity, we illustrate each case with an
example below. These examples are chosen for their simplicity as well as their
role in considerations to come.
A process discrete in state and time. A simple, ubiquitous, but important example
of this type of stochastic process comes from considering the experiment of repeat-
edly tossing a fair (unbiased) coin. We start at time 0 and toss our coin, say,
5
every second. A particle or. if you like. a pointsize frog. on the x-axis moves up
a unit step for each head. moves down a unit step for each tail. and starts at the
origin at time O. So outcomes of our experiment occur at integer values of time.
t=n. (n=0.1.2 •••• ) and X(t)=Xn• where Xn is the position of our frog at time n
and X = O. Two realizations of our process are shown in figure 1.4.
o
X(t)
4 n
n
1
0
r"'~ ~ o~
2 o
U I
~
· . I
·x . ... : .
· .
o · .!r;""i •; ..'.-..
t
.
. . .
x····· JiC ••••• X· ••••
. .
.
.
x···· . x····· x···
-2
Figure 1.4. Two sample paths of a one-dimensional random walk.
At n=l. there is a jump of one step. upward or downward. each with probability
1/2. For n=2. and each subsequent integer. we also have an equally likely upward
or downward unit excursion of our frog. and each jump is independent of all pre-
ceding jumps. Thus we can write a "stochastic difference equation" describing the
frog's position:
Xn = Xn_l + Wn_l (n=1.2.3 •••• ) (1.1)
where W is a random variable specifying the jump of the nth step by the distribu
n
tion function:
1/2 • (n=1.2.3 ••• .) (1.2)