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Selective interaction between two independent stationary recurrent point processes

Published online by Cambridge University Press:  14 July 2016

S. K. Srinivasan
Affiliation:
Indian Institute of Technology, Madras
G. Rajamannar
Affiliation:
Indian Institute of Technology, Madras

Extract

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).

Type
Short Communications
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
[2] Ramakrishnan, A. (1950) Stochastic processes relating to particles distributed in a continuous infinity of states. Proc. Camb. Phil. Soc. 46, 595602.Google Scholar
[3] Srinivasan, S. K. (1969) Stochastic Theory and Cascade Processes. Elsevier, New York.Google Scholar
[4] Srinivasan, S. K. (1966) Stochastic integration and differential equation–physical approach. Symp. Mathematics and Theoretical Physics 9. Plenum Press, New York.Google Scholar
[5] Ten Hoopen, M. and Reuver, H. A. (1965) Selective interaction of two independent processes. J. Appl. Prob. 2, 286292.CrossRefGoogle Scholar
[6] Ten Hoopen, M. and Reuver, H. A. (1967) Interaction between two independent recurrent time series. Information and Control 10, 149158.CrossRefGoogle Scholar
[7] Ten Hoopen, M. and Reuver, H. A. (1967) On a first passage problem in stochastic storage systems with total releases. J. Appl. Prob. 4, 409412.CrossRefGoogle Scholar