# American Institute of Mathematical Sciences

2014, 11(2): 285-302. doi: 10.3934/mbe.2014.11.285

## On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model

 1 Dipartimento di Studi e Ricerche Aziendali (Management &Information Technology), Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy 2 Dipartimento di Matematica, Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy

Received  October 2012 Revised  January 2013 Published  October 2013

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.
Citation: Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285
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