# Vidybida A. K.

### Delayed feedback makes neuronal firing statistics non-Markovian

Kravchuk K. G., Vidybida A. K.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1587-1609

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.

### Output stream of a binding neuron

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1619–1638

For a binding neuron with threshold 2 stimulated by a Poisson stream, we determine the intensity of the output stream and the probability density for the lengths of the output interpulse intervals. For threshold 3, we determine the intensity of the output stream.