Delayed feedback makes neuronal firing statistics non-Markovian

Abstract

The instantaneous state of a neural network consists of both the degree of excitation of each neuron and the positions of impulses in communication lines between the neurons. In neurophysiological experiments, the times of neuronal firing are recorded but not the state of communication lines. However, future spiking moments substantially depend on the past positions of impulses in the lines. This suggests that the sequence of intervals between firing moments (interspike intervals, ISI) in the network can be non-Markovian. In the present paper, we analyze this problem for the simplest possible neural “network,” namely, for a single neuron with delayed feedback. The neuron receives excitatory input both from the input Poisson process and from its own output through the feedback line. We obtain exact expressions for the conditional probability density P(t n+1 | t n ,…,t 1, t 0)dt n+1 and prove that P(t n+1 | t n ,…,t 1, t 0) is not reduced to P(t n+1 | t n ,…,t 1) for any n ≥ 0: This means that the output ISI stream cannot be represented as a Markov chain of any finite order.