Mathematical Biosciences and Engineering, 2014, 11(1): 81-104. doi: 10.3934/mbe.2014.11.81

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

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

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

Spiking statistics of a self-inhibitory neuron is considered.The neuron receives excitatory input from a Poisson streamand inhibitory impulses through a feedback linewith a delay. After triggering, the neuron is in the refractorystate for a positive period of time.
    Recently, [35,6], it was proven for a neuron withdelayed 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 thespiking process, which might restore Markov property to the ISI stream.
    Here we check such a possibility. For this purpose, we calculatethe 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 stateas well cannot be represented as a Markov process of any order.
    We conclude that it is namely the delayed feedback presencewhich results in non-Markovian statistics of neuronal firing.As delayed feedback lines are common forany realistic neural network, the non-Markovian statistics of the networkactivity should be taken into account in processing of experimental data.
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