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Journal Articles Physica D: Nonlinear Phenomena Year : 2010

Some theoretical and numerical results for delayed neural field equations

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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Dates and versions

hal-00847433 , version 1 (23-07-2013)

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  • HAL Id : hal-00847433 , version 1

Cite

Grégory Faye, Olivier Faugeras. Some theoretical and numerical results for delayed neural field equations. Physica D: Nonlinear Phenomena, 2010, 239 (9), pp.561--578. ⟨hal-00847433⟩
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