Some theoretical results for a class of neural mass equations

Grégory Faye 1 Pascal Chossat 2 Olivier Faugeras 1
1 NEUROMATHCOMP
CRISAM - Inria Sophia Antipolis - Méditerranée , INRIA Rocquencourt, ENS Paris - École normale supérieure - Paris, UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR8548
Abstract : We study the neural field equations introduced by Chossat and Faugeras in their article to model the representation and the processing of image edges and textures in the hypercolumns of the cortical area V1. The key entity, the structure tensor, intrinsically lives in a non-Euclidean, in effect hyperbolic, space. Its spatio-temporal behaviour is governed by nonlinear integro-differential equations defined on the Poincaré disc model of the two-dimensional hyperbolic space. Using methods from the theory of functional analysis we show the existence and uniqueness of a solution of these equations. In the case of stationary, i.e. time independent, solutions we perform a stability analysis which yields important results on their behavior. We also present an original study, based on non-Euclidean, hyperbolic, analysis, of a spatially localised bump solution in a limiting case. We illustrate our theoretical results with numerical simulations.
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https://hal.inria.fr/hal-00846150
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Submitted on : Thursday, July 18, 2013 - 4:05:37 PM
Last modification on : Wednesday, January 30, 2019 - 2:28:03 PM

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  • HAL Id : hal-00846150, version 1
  • ARXIV : 1005.0510

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Grégory Faye, Pascal Chossat, Olivier Faugeras. Some theoretical results for a class of neural mass equations. 2010. ⟨hal-00846150⟩

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