Convergence and Chaos of a Class of Discrete-Time Background Neural Networks with Uniform Firing Rate

The dynamical properties of a class of discrete-time background network with uniform firing rate are investigated. The conditions for stability are derived. To guaranteed the boundness of all trajectories of the discrete-time background network, several invariant sets are obtained. It´s then proved that any trajectories of the network starting from each of the invariant sets will converge. In addition to the stability and convergence analysis, bifurcation and chaos are also discussed. It´s shown that the network can engender bifurcation and chaos with the increase of background input. The Lyapunov exponents are finally computed to confirm the existence of chaos. Since the background networks originate from the study of the activities of brain and chaotic activities are ubiquitous in the human brain, the chaos analysis of the background networks is significant.