Existence, Uniqueness Of Traveling Wave Solution Of Some Itegral Differntial Equations Arising Form Neuronal Networks

This paper is concerned with existence and uniqueness of traveling wave solution of the integral differential equation:arising from neuronal networks. The theorem of existence and uniqueness was studied by the article[1],and the conclusions of the theorem were proved by analysis of vector field and functional analysis methods, but the proof is not strict.This paper was based on the conclusions of the article [1], transformed the integral differential equation into the autonomous system of differential equations,obtained fixed points of the autonomous system, analyzed vector field near the fixed points and its surrounding. In accordance with the vector field existence and uniqueness of the traveling wave solution were proved.Firstly,for the parameter w = 0,by analysing the vector field around of four fixed points, the tranjectory from the saddle point (-1,0) to the stable node (1,β) was determined,which was traveling wave solution (Ï„(z),U(z)). Further more, the existence and uniqueness of the velocity v₀ satisfying the conditions for traveling wave solutionsU(v₀,0)=θ.Secondly,for parameter w∈(w_-,w₊),analyzing vector fields of two fixed points,four fixed points and six fixed points. Determined the existence and uniqueness of the traveling wave solution.At last,the existence and uniqueness of the traveling wave solution depended on synaptse constantα, large synapse constant was proved.