New results for delayed neural field equations

Neural field models with delays define a useful framework for modeling macroscopic parts of the cortex involving sev-eral populations of neurons. Nonlinear delayed integro-differential equations describe the spatio-temporal behav-ior of these fields. Using methods from the theory of delay differential equations, we show the existence and unique-ness of a solution of these equations. A Lyapunov analysis gives us sufficient conditions for the solutions to be asymp-totically stable. We also present a study of the numerical computation of these solutions in a special case. 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 con-tribution of ours is the definition of a Lyapunov functional and the result of stability it implies. We illustrate our work on a variety of examples that are relevant to modeling in neuroscience