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Wald's identity and random walk models for neuron firing

Published online by Cambridge University Press:  01 July 2016

V. I. Kryukov
V. I. Kryukov*
Affiliation:
Computing Research Center, Pushchino, USSR Academy of Sciences, Moscow Region, USSR 142292
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Abstract

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

Keywords

WALD'S FUNDAMENTAL IDENTITYSEQUENTIAL ANALYSISFIRST-PASSAGE PROBLEMSOPTIMAL STOPPINGRANDOM WALKADDITIVE PROCESSORNSTEIN–UHLENBECK PROCESSNEURON FIRING
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 257 - 277
Copyright
Copyright © Applied Probability Trust 1976 

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