A New Framework for Analysis on Stability and Bifurcation in a Class of Neural Networks With Discrete and Distributed Delays.

Abstract

This paper studies the stability and Hopf bifurcation in a class of high-dimension neural network involving the discrete and distributed delays under a new framework. By introducing some virtual neurons to the original system, the impact of distributed delay can be described in a simplified way via an equivalent new model. This paper extends the existing works on neural networks to high-dimension cases, which is much closer to complex and real neural networks. Here, we first analyze the Hopf bifurcation in this special class of high dimensional model with weak delay kernel from two aspects: one is induced by the time delay, the other is induced by a rate parameter, to reveal the roles of discrete and distributed delays on stability and bifurcation. Sufficient conditions for keeping the original system to be stable, and undergoing the Hopf bifurcation are obtained. Besides, this new framework can also apply to deal with the case of the strong delay kernel and corresponding analysis for different dynamical behaviors is provided. Finally, the simulation results are presented to justify the validity of our theoretical analysis.