Diffusion Processes and Related Topics in BiologyThese 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. |
Contents
PRELIMINARIES | 1 |
DIFFUSION PROCESSES | 31 |
THE FIRST PASSAGE TIME PROBLEM | 61 |
Copyright | |
3 other sections not shown
Other editions - View all
Diffusion Processes and Related Topics in Biology Luigi M Ricciardi,Charles E Smith No preview available - 1977 |
Common terms and phrases
A₁ A₁(x A₂(x assume asymptotic boundaries calculate consider defined denote density derived determine the transition deterministic differential equation diffusion equations diffusion interval diffusion process discrete drift and infinitesimal dx dt dx R(x example Exercise extinction Feller finite firing p.d.f. Fokker-Planck equation function Gaussian given increment independent infinitesimal moments infinitesimal variance initial condition instance integral Kolmogorov equation Laplace transform Markov process membrane potential n-tuple neuron obtain Ornstein-Uhlenbeck process particle passage time p.d.f. Poisson distributed Poisson process population probability mass process with drift process X(t Prove pulses r₁ random variable random walk result Ricciardi sample paths Siegert Smolukowski equation solution solve specified stationary steady state distribution stochastic process t₁ Taylor series threshold tion transition p.d.f. Wiener process zero drift ητ ητ μθ ахо дх