The Spike Noise Based on the Renewal Point Process and Its Possible Applications

Abstract

We consider a non-Markovian random process in the form of spikes train, where the time intervals between neighboring delta-pulses are mutually independent and identically distributed, i.e. represent the renewal process (1967). This noise can be interpreted as the derivative of well-known continuous time random walk (CTRW) model process with fixed value of jumps. The closed set of equations for the characteristic functional of the noise, useful to split the correlations between stochastic functionals (2008), is obtained. In the particular case of Poisson statistics these equations can be exactly solved and the expression for the characteristic functional coincides with the result for shot noise (2005). Further we analyze the stability of some first-order system with the multiplicative spike noise. We find the momentum stability condition for arbitrary probability distribution of intervals between pulses. The general condition of stability is analyzed for the special probability distribution of intervals between pulses corresponding to so-called dead-time-distorted Poisson process. It means that within some time interval after each delta pulse the occurrence of new one is forbidden (like in neurons). The possible applications of the model to some problems of neural dynamics, epidemiology, ecology, and population dynamics are discussed.