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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 (... - 2019), 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.
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https://hal.inria.fr/hal-00847433
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Submitted on : Tuesday, July 23, 2013 - 3:39:54 PM
Last modification on : Monday, October 12, 2020 - 2:28:06 PM

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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: Nonlinear Phenomena, Elsevier, 2010, 239 (9), pp.561--578. ⟨hal-00847433⟩

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