Qualitative Analysis Of Equilibria In A Ring Neural Network Model

The focus of this dissertation is to study issues related to existence, stabilityand bifurcation of equilibria in a ring of four identical neurons with self-feedbackand delays, which has an on-center o?-surround characteristic for some suitableconnection weights. Such a network has been found in a variety of neural struc-tures, such as neocortex, cerebellum, hippocampus, and even in chemistry andelectrical engineering, and can be studied to gain insight into the mechanismsunderlying the behavior of recurrent network. This dissertation is organized asfollows:Firstly, the background and the motivation for the study of artificial neuralnetworks are presented. Then, some known results of neural network models areintroduced. Occurrence of bifurcation and some traditional methods used to studybifurcation are introduced simply. Besides, the center manifold reduction andnormal form calculation in a finite-dimension space are introduced.Secondly, based on the isotropic subgroup of D4⊕Z2, we classify all thepossible equilibria of this neural network. Then, we obtain the su?cient conditionsensuring the existence of the equilibria of di?erent patterns.Thirdly, we calculate the characteristic equation of the equilibria via the lin-earized equation. By means of the characteristic equation, we discuss the stabilityof the equilibria.Fourthly, by regarding the two connection weights as bifurcation parameters,which is di?erent from traditional research using the time delay as bifurcationparameters, with the help of center manifold reduction and normal form theory,we investigate the equivariant Hopf bifurcation near the trivial equilibrium, andobtain some criteria about the bifurcation direction and stability of bifurcatingperiodic solutions.Finally, we discuss the bifurcation phenomena near asynchronous equilibriaas well.