Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T16:14:45.512Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bateman, H. and Erdélyi, A. (1953) Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York.Google Scholar
Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
Bondar, A. A. (1968) О вероятностном характере преобразования информации нейроном. В сб.: “Статистическая' электрофизиология”, часть I, Вильнюс,стр. 81–92.Google Scholar
Borovkov, A. A. (1965) О времени первого прохождения для одного класса процессов с независимыми приращениями. Теор. вероят. и её примен. X, 360–363.Google Scholar
Capocelli, R. M. and Ricciardi, L. M. (1971) Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8, 214–223.Google Scholar
Clay, J. R. and Goel, N. S. (1973) Diffusion models for firing of neuron with varying threshold. J. Theoret. Biol. 39, 633–644.Google Scholar
Cox, D.R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
Cowan, J. D. (1972) Stochastic models of neuroelectric activity. In Proc. I.U.P.A.P. Conference on Statistical Mechanics, Ed. Rice, S., Light, J. and Freed, K. University of Chicago Press.Google Scholar
Daniels, H. E. (1969) The minimum of stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Fienberg, S. E. (1970) A note on the diffusion approximation for the single neuron firing problem. Kybernetik 7, 227–229.CrossRefGoogle Scholar
Fienberg, S. E. (1974) Stochastic models for single neuron firing trains: a survey. Biometrics 30, 399–427.Google Scholar
Gerstein, G. L. and Mandelbrot, B. (1964) Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41–68.Google Scholar
Gikhman, I. I. and Skorokhod, A. V. (1973) Теория случайных процессов, т.Google Scholar
Gluss, B. (1967) A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density. Bull. Math. Biophys. 29, 233–243.Google Scholar
Johannesma, P. I. M. (1968) Diffusion models for the stochastic activity of neurons. In Neuronal Networks, ed. Caianiello, E. R. Springer, New York.Google Scholar
Kazanovitch, Ya. B. and Kryukov, V. I. (In press).Google Scholar
Keilson, J. (1963) The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34, 1003–1011.Google Scholar
Keilson, J. (1965) Green's Function Methods in Probability Theory. Griffin, London.Google Scholar
Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. Chicago University Press.Google Scholar
Kryukov, V. I. (1968) Теоретический анализ влияния рефрактерности на вариабельность нецрона. В сб.: “Статистическая электрофизиология”, часть I, Вильнюс, стр. 286–305.Google Scholar
Kryukov, V. I. (1973) Модели случайного блуждания для импульсной активности нейрб онов с рефрактерностью. Препринт, Пущино-на-Оке.Google Scholar
Miller, H. D. (1961) A generalization of Wald's identity with applications to random walks. Ann. Math. Statist. 32, 549–560.Google Scholar
Novikov, A. A. (1971) О моментах остановки винеровского процесса. Теор. вероят. и её примен. XVI, 458–465.Google Scholar
Roy, B. K. and Smith, D. R. (1969) Analysis of the exponential decay model of the neuron showing frequency threshold effects. Bull. Math. Biophys. 31, 341–357.Google Scholar
Shepp, L. A. (1967) A first passage problem for the Wiener processes. Ann. Math. Statist. 38, 1912–1914.CrossRefGoogle Scholar
Shepp, L. A. (1969) Explicit solutions to some problems of optimal stopping. Ann. Math. Statist. 40, 993–1010.Google Scholar
Siebert, W. M. (1969) On stochastic neural models of the diffusion type. Quart. Progr. Rep. 94, Electronics Research Lab. M.I.T., 281–287.Google Scholar
Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
Stein, R. B. (1967) Some models of neural variability. Biophys. J. 7, 37–68.Google Scholar
Sugiyama, H., Moore, G. P. and Perkel, D. H. (1970) Solutions for a stochastic model of neuronal spike production. Math. Biosci. 8, 323–341.Google Scholar
Zhitomirsky, I. S. (1963) О функции распределения момента первого выхода. Теор. вероят. и её примен. YIII, 156–165.Google Scholar