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Abstract and Applied Analysis
Volume 2014, Article ID 368652, 18 pages
http://dx.doi.org/10.1155/2014/368652
Research Article

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

1Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
2Department of Mathematics, Tongji University, Shanghai 200092, China

Received 2 May 2013; Revised 15 March 2014; Accepted 17 March 2014; Published 29 April 2014

Academic Editor: Patricia J. Y. Wong

Copyright © 2014 Jiao Jiang and Yongli Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the dynamics of a delayed neural network model consisting of identical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory and -equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results.