Firing statistics of inhibitory neuron with delayed feedback. II: Non-Markovian behavior

In this paper, we address this question for a simplest possible neural “net”, namely, a single inhibitory neuron with delayed feedback. The neuron receives excitatory input from the driving Poisson stream and inhibitory impulses from its own output through the feedback line. We obtain analytic expressions for conditional probability density P(tn+1|tn, …, t1, t0), which gives the probability to get an output ISI of duration tn+1 provided the previous (n + 1) output ISIs had durations tn, …, t1, t0. It is proven exactly, that P(tn+1|tn, …, t1, t0) does not reduce to P(tn+1|tn, …, t1) for any n ≥ 0. This means that the output ISIs stream cannot be represented as a Markov chain of any finite order.