Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons

Aniello Buonocore; Luigia Caputo; Enrica Pirozzi; Maria Francesca Carfora

doi:10.3934/mbe.2014.11.189

Article Contents

2014, Volume 11, Issue 2: 189-201. Doi: 10.3934/mbe.2014.11.189 open access

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Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons

1.
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli
2.
Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli

Received: September 30, 2012

Revised: March 31, 2013

Published: September 2013

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Abstract

With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.

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Mathematics Subject Classification: Primary: 60J60, 60J70; Secondary: 92-08.

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