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A birth and death model of neuron firing

Published online by Cambridge University Press:  14 July 2016

B. F. Logan  and
L. A. Shepp
B. F. Logan
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
L. A. Shepp
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
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Abstract

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

Keywords

BIRTH AND DEATHNEURON FIRINGSGEOMETRIC WAITING TIMESSUCCESSIVE DEATHSWAITING TIMESPOPULATION MODELREVERSED TIMEMARKOV CHAINSQUEUING THEORY
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 2 , June 1974 , pp. 369 - 373
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. 1, 2nd ed. Wiley, New York.Google Scholar
[2] Schroeder, M. R. and Hall, J. L. A model for mechanical to neural transduction in the auditory receptor. To appear.Google Scholar
[3] Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
[4] Burke, P. J. (1956) The output of queuing systems. Operat. Res. 4, 699–704.Google Scholar