Existence and attraction of a phase-locked oscillation in a delayed network of two neurons

Abstract

In this article, we study the network of two neurons with delay. Using the discrete Lyapunov functional of Mallet-Paret and Sell and the techniques developed recently by Krisztin, Walther and Wu (for the scalar case), we obtain a two-dimensional closed disk bordered by a phase-locked periodic orbit and we have a complete description about the structure of various heteroclinic connections in the global forward extension of a three-dimensional $C^1$-submanifold contained in the unstable set of the trivial solution.