Some theoretical and numerical results for delayed neural field equations

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• 作者：Gr&#xe9 ; gory Faye ; Olivier Faugeras
• 关键词：Neural fields ; Nonlinear integro-differential equations ; Delays ; Lyapunov functional ; Pattern formation ; Numerical schemes
• 刊名：Physica D
• 出版年：2010
• 出版时间：1 May 2010
• 年：2010
• 卷：239
• 期：9
• 页码：561-578
• 全文大小：9802 K

文摘

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 integro-differential 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.