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 neuronal firing moments are registered, but not the state of communication lines. However, future spiking moments depend substantially 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 this paper, we address this question for a simplest possible neural "network", namely, 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 $P (t_{n+1} | t_n,... ,t_1, t_0)$ does not reduce to $P (t_{n+1} | t_n,... ,t_1)$ for any $n \geq 0$. This means that the output ISI stream cannot be represented as a Markov chain of any finite order.