Hopf bifurcation stability in Hopfield neural networks

Abstract

In this paper we consider a simple discrete Hopfield neural network model and analyze local stability using the associated characteristic model. In order to study the dynamic behavior of the quasi-periodic orbit, the Hopf bifurcation must be determined. For the case of two neurons, we find one necessary condition that yields the Hopf bifurcation. In addition, we determine the stability and direction of the Hopf bifurcation by applying normal form theory and the center manifold theorem. An example is given and a numerical simulation is performed to illustrate the results. We analyze the influence of bias weights on the stability of the quasi-periodic orbit and study the phase-locking phenomena for certain experimental results with Arnold Tongues in a particular weight configuration.