New results for delayed neural field equations

Grégory Faye (1) , Olivier Faugeras (1)

1 NEUROMATHCOMP - Neurosciences Mathématique et Computationnelle

Neural field models with delays 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 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 contribution 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.

Mots clés: System dynamics

Domaines: Neurosciences [q-bio.NC]; Neurosciences

Grégory Faye, Olivier Faugeras. New results for delayed neural field equations. Cinquième conférence plénière française de Neurosciences Computationnelles, "Neurocomp'10", Aug 2010, Lyon, France. ⟨hal-00553401⟩