Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness

Kseniia Kravchuk; Alexander Vidybida

doi:10.3934/mbe.2014.11.81

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2014, Volume 11, Issue 1: 81-104. Doi: 10.3934/mbe.2014.11.81

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Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness

Kseniia Kravchuk^1, and
Alexander Vidybida^1,

1.
Bogolyubov Institute for Theoretical Physics, Metrologichna str., 14-B, 03680 Kyiv

Received: November 30, 2012

Revised: May 31, 2013

Published: August 2013

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Abstract

Spiking statistics of a self-inhibitory neuron is considered. The neuron receives excitatory input from a Poisson stream and inhibitory impulses through a feedback line with a delay. After triggering, the neuron is in the refractory state for a positive period of time.
Recently, [35,6], it was proven for a neuron with delayed feedback and without the refractory state, that the output stream of interspike intervals (ISI) cannot be represented as a Markov process. The refractory state presence, in a sense limits the memory range in the spiking process, which might restore Markov property to the ISI stream.
Here we check such a possibility. For this purpose, we calculate the conditional probability density $P(t_{n+1}\mid t_{n},\ldots,t_1,t_{0})$, and prove exactly that it does not reduce to $P(t_{n+1}\mid t_{n},\ldots,t_1)$ for any $n\ge0$. That means, that activity of the system with refractory state as well cannot be represented as a Markov process of any order.
We conclude that it is namely the delayed feedback presence which results in non-Markovian statistics of neuronal firing. As delayed feedback lines are common for any realistic neural network, the non-Markovian statistics of the network activity should be taken into account in processing of experimental data.

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Mathematics Subject Classification: Primary: 60G55, 92C20; Secondary: 90C15.

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