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Discrete Dynamics in Nature and Society
Volume 2014, Article ID 971520, 8 pages
Research Article

Equivariant Bifurcation in Coupled Two Neural Network Rings

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 6 November 2014; Accepted 1 December 2014; Published 30 December 2014

Academic Editor: Victor S. Kozyakin

Copyright © 2014 Baodong Zheng and Haidong Yin. 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.


We study a Hopfield-type network that consists of a pair of one-way rings each with three neurons and two-way coupling between the rings. The rings have symmetric group , which means the global symmetry and internal symmetry . We discuss the spatiotemporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatiotemporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling; that is, different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase, antiphase, , or periods out of phase depending on the delay. Some numerical simulations support our analysis results.