Statistical dependency as a measure to evaluate Markov properties of stochastic point processes

Abstract

Neuronal spike trains are regarded as stochastic point processes. To estimate the order and value of Markov processes of the interspike interval sequences with small number of samples, we have proposed a new measure of simplified statistical dependencyd _m. This measure is derived from statistical dependencyd _m (T=τ) in the case of Gaussian process, and is obtained by the standard deviation and the matrices of the serial correlation coefficients. Sinced _m is a parametric measure, it is calculated from the interval sequence transformed into the aormal distribution. We designate this as normalized simplified statistical dependencyNd _m. The order and value of the maintained spike sequences recorded from the mesencephalic reticular formation, red nucleus, optic tract, and lateral geniculate nucleus neurons in cats have been estimated. It is indicated that there is a considerable correspondence between the value ofNd _m and that ofd _m (T=τ). This suggests thatNd _m is useful in practice to estimate the order and value of Markov process with small number of samples.