EXISTENCE AND UNIQUENESS OF A TRAVELING WAVE FRONT OF A MODEL EQUATION IN SYNAPTICALLY COUPLED NEURONAL NETWORKS

Consider the model equation in synaptically coupled neuronal networks@u@t+ m(u − n)= ( − au) Z 10(c) ZRK(x − y)H uy, t −1c|x − y| − dydc+ ( − bu) Z 10( ) ZRW(x − y)H(u(y, t − ) − )dyd.In this model equation, u = u(x, t) stands for the membrane potential of a neuron at position x andtime t. The kernel functions K 0 and W 0 represent synaptic couplings between neurons insynaptically coupled neuronal networks. The Heaviside step function H = H(u − ) represents thegain function and it is defined by H(u − ) = 0 for all u . The functions and represent probability density functions. The function f(u) m(u − n)represents the sodium current, where m > 0 is a positive constant and n is a real constant. Theconstants a 0, b 0, 0, 0 and > 0 represent biological mechanisms. This model equationis motivated by previous models in synaptically coupled neuronal networks.We will couple together intermediate value theorem, mean value theorem and many techniquesin dynamical systems to prove the existence and uniqueness of a traveling wave front of this modelequation. One of the most interesting and difficult parts is the proof of the existence and uniquenessof the wave speed. We will introduce several auxiliary functions and speed index functions to provethe existence and uniqueness of the front and the wave speed.