Stochastic Models of Spike Trains

Abstract

This chapter reviews the theory of stochastic point processes. For simple renewal processes, the relation between the stochastic intensity, the inter-spike interval (ISI) distribution, and the survival probability are derived. The moment and cumulant generating functions and the relation between the ISI distribution and the autocorrelation is investigated. We show how the Fano factor of the spike count distribution depends on the coefficient of variation of the ISI distribution. Next we investigate models of renewal processes with variable rates and CV2, which is often used to assess the variability of the spike train in this case and compare the latter to the CV. The second half of the chapter deals with stochastic point processes with correlations between the intervals. Several examples of such processes are shown, and the basic analytical techniques to deal with these processes are expounded. The effect of correlations in the ISIs on the Fano factor of the spike count and the CV2 are also explored.