Dynamic Analysis Of Delay Networks With A Ring Of Three Identical Neurons

In this thesis, some important dynamics of two class of delay network modelswith a ring of three neurons are discussed. For the case of discontinuous mes-sage function, we obtain a su?cient condition ensuring the existence of attractiveperiodic solution. On the other hand, we obtain the su?cient condition for theexistence of periodic solution for the case of model with small parameter.The paper is composed of three chapters.In the first chapter, the background and history of neural networks are brie?yreviewed. Furthermore, we raise some problems which will be investigated. In theend, we simply introduce the main work in this paper.In the second chapter, we consider a class of delay network with a ring of threeidentical neurons which has discontinuous message function, where each neuronreceives two inputs: one from the next neuron by inhibitory interaction and anotherfrom the previous neuron by excitative interaction. Using its limiting equationand its symmetry, we obtain a su?cient condition for the existence of attractiveperiodic solution by restricting initial values to a given phase space. Also, we studythe convergence of the solutions.The third chapter is concern with the three neurons model with small param-eter. Employing Walther's method, we get an estimate for the Lipschitz constantof the returning map if the delay is su?ciently small. Based on the estimate of theLipschitz constant, we obtain the su?cient condition ensuring the existence of thestable slowly oscillatory periodic solution.