Dynamic Analysis Of Several Classes Of Networks With A Ring Of Three Identical Neurons

The network with a ring of three identical neurons has significant biologicaland physical backgrounds and many models of neural networks have been proposedas this special case.This thesis deals with several classes systems of networks whichhas discontinuous message function. We obtain the convergence of the system undersome certain parameter conditions.First, we deal with a network with a ring of three identical neurons whichhas discontinuous message function. We obtain the convergence of the system toequilibriums with some certain parameter conditions.To go on with, we consider a class of delay di?erential systems which describingthe evolution of network with a ring of three identical neurons.By applying itssymmetry,the theory of di?erential di?erence equation and studying its limitingequation ,our analysis shows the convergence of the system when the threshold isnot less than 1/2.Finally,there is a class of network with a ring of three identical neurons whichhas a disproportion message function. We have some special study of the abovesystem with the limit that the threshold is larger than 1/2. We come to the conclusionthat it has abundant features:when we restrict initial values to the given phasespace, its solution may be convergent to some stable points, but others may beconvergent to limit cycle.From the analysis of the models above,we can come to the conclusion thattime delay and message function play vital roles in the dynamical features of theneural networks. There is no time delay in the first model , and we find that ithas very simple dynamic features. With time delay, the second model not only hasconvergence solution, but also has more stable points under some special thresholdvalues. Because of the time delay and disproportion message function,limit cyclesexist in the third model. With the same reason, it makes the third model the mostsuitable one to re?ect the real world.