Abstract
A diffusion equation for the transition p.d.f. describing the time evolution of the membrane potential for a model neuron, subjected to a Poisson input, is obtained, without breaking up the continuity of the underlying random function. The transition p.d.f. is calculated in a closed form and the average firing interval is determined by using the steady-state limiting expression of the transition p.d.f. The Laplace transform of the first passage time p.d.f. is then obtained in terms of Parabolic Cylinder Functions as solution of a Weber equation, satisfying suitable boundary conditions. A continuous input model is finally investigated.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Blanc-Lapierre, A., Fortet, R.: Theòrie des fonctions alèatoires. Paris: Masson & Cie. 1953.
Darling, D. A., Siegert, A. J. F.: The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639 (1953).
Gerstein, G. L., Mandelbrot, B.: Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41–67 (1964).
Hagiwara, S.: Analysis of interval fluctuation of the sensory nerve impulse. Jap. J. Physiol. 4, 234–240 (1954).
Helstrom, C. W.: Markov processes and applications. In: A. V. Balakrishnan (ed.), Communication theory. New York: McGraw Hill 1968.
Johannesma, P. I. M.: Diffusion models for the stochastic activity of neurons. In: E. R. Caianiello (ed.), Neural networks. Berlin-Heidelberg-New York: Springer 1968.
Middleton, D.: An introduction to statistical communication theory. New York: McGraw Hill 1960.
Ricciardi, L. M., Esposito, F.: On some distribution functions for non-linear switching elements. Kybernetik 3, 148–152 (1966).
— Ventriglia, F.: Probabilistic models for determining the input-output relationship in formalized neurons. I. A theoretical approach. Kybernetik 7, 175–183 (1970).
Roy, B. K., Smith, D. R.: Analysis of the exponential decay model of the neuron showing frequency threshold effects. Bull. Math. Biophys. 31, 341–357 (1969).
Siegert, A. J. F.: On the first passage time probability problem. Phys. Eev. 81, 617–623 (1951).
Tricomi, F. G.: Funzioni Ipergeometriche Confluenti. Roma: Cremonese 1954.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Capocelli, R.M., Ricciardi, L.M. Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8, 214–223 (1971). https://doi.org/10.1007/BF00288750
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00288750