Large-Scale Neural Networks With Asymmetrical Three-Ring Structure: Stability, Nonlinear Oscillations, and Hopf Bifurcation

A large number of experiments have proved that the ring structure is a common phenomenon in neural networks. Nevertheless, a few works have been devoted to studying the neurodynamics of networks with only one ring. Little is known about the dynamics of neural networks with multiple rings. Consequently, the study of neural networks with multiring structure is of more practical significance. In this article, a class of high-dimensional neural networks with three rings and multiple delays is proposed. Such network has an asymmetric structure, which entails that each ring has a different number of neurons. Simultaneously, three rings share a common node. Selecting the time delay as the bifurcation parameter, the stability switches are ascertained and the sufficient condition of Hopf bifurcation is derived. It is further revealed that both the number of neurons in the ring and the total number of neurons have obvious influences on the stability and bifurcation of the neural network. Ultimately, some numerical simulations are given to illustrate our qualitative results and to underpin the discussion.

I. Introduction

Nowadays, with the rise of artificial neural networks, the study of bionics and human brain mechanism has become a major scientific research task. Then, one area that is being seriously investigated is neural networks for dynamics, which have been successfully applied to optimization [1], [2], signal processing [3], image processing [4], pattern recognition [5], [6], associative memory [7], and many other subjects for several decades [8]. Since the application of neural networks depends heavily on network dynamics, a lot of in-depth research on its dynamic characteristics has been channeled out, and many valuable and meaningful results have been reported [9]–[11].