Some theoretical and numerical results for delayed neural field equations

Grégory Faye 1 Olivier Faugeras 2
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
2 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 : In this paper we study neural field models with delays which define a useful framework for modeling macroscopic parts of the cortex involving several populations of neurons. Nonlinear delayed integrodifferential equations describe the spatio-temporal behavior of these fields. Using methods from the theory of delay differential equations, we show the existence and uniqueness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymptotically stable. We also present a fairly detailed study of the numerical computation of these solutions. This is, to our knowledge, the first time that a serious analysis of the problem of the existence and uniqueness of a solution of these equations has been performed. Another original contribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our numerical schemes on a variety of examples that are relevant to modeling in neuroscience.
Type de document :
Article dans une revue
Physica D, Elsevier, 2010, 239 (9), pp.561--578. 〈http://www.sciencedirect.com/science/article/pii/S0167278910000229〉
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Soumis le : mardi 23 juillet 2013 - 15:39:54
Dernière modification le : jeudi 3 mai 2018 - 13:32:58

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  • HAL Id : hal-00847433, version 1

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Grégory Faye, Olivier Faugeras. Some theoretical and numerical results for delayed neural field equations. Physica D, Elsevier, 2010, 239 (9), pp.561--578. 〈http://www.sciencedirect.com/science/article/pii/S0167278910000229〉. 〈hal-00847433〉

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