A markovian model for stochastic integrate-and-fire networks

Jonathan Touboul 1, 2
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : In this paper we introduce and study a mathematical framework in order to characterize and simulate networks of noisy integrate-and-fire neurons. This framework is based on a markovian modelization of the network, similar to the event-based modelization of deterministic networks. In these networks the value of interest at each neuron is not the membrane potential itself but the related \emph{countdown process}, which is defined loosely as the time remaining to the next spike if nothing occurs meanwhile in the network. The main issue of this modelization is to ensure that the dynamics of this countdown process, possibly supplemented with other variables, is an autonomous Markov process (i.e. that does not depend on the membrane's potential). We prove that a wide range of integrate-and-fire neuron models and different types of interactions fit into this general mathematical framework. This framework involves renewal processes and has already been studied in the field of random networks in a more restricted setting by Cottrell, Robert, Turova for instance \cite{cottrell:92, cottrell-turova:00, fricker-robert-etal:94, turova:96, turova:00}, and from a mathematical viewpoint, ergodicity matters have been discussed Fayolle, Menshikov, Malyshev and Borovkov \cite{fayolle-malyshev-etal:95,borovkov:98}. This modelization provides a very efficient algorithm to simulate large networks of noisy integrate-and-fire neuron models. We discuss different types of implementations, and developped together with Renaud Kervien and Alexandre Chariot a very efficient paralel simulator implement on GPU.
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[Research Report] RR-6661, INRIA. 2008
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  • HAL Id : inria-00323643, version 1

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Jonathan Touboul. A markovian model for stochastic integrate-and-fire networks. [Research Report] RR-6661, INRIA. 2008. 〈inria-00323643〉

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