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A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

Olivier Faugeras 1 Jonathan Touboul 1, * Bruno Cessac 1, 2 
* Corresponding author
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique - ENS Paris, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS-PSL - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide, by a fixed point method, a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit (jansen and Rit;1995): their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.
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Submitted on : Monday, August 4, 2008 - 5:17:10 PM
Last modification on : Wednesday, October 26, 2022 - 8:16:45 AM
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Olivier Faugeras, Jonathan Touboul, Bruno Cessac. A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs. [Research Report] RR-6454, INRIA. 2008, pp.58. ⟨inria-00258345v3⟩



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