Synchronous Phenomena of Neural Network Models Using Hindmarsh–Rose Equation

Abstract

We study synchronous phenomena in neural network models with neurons described by Hindmarsh–Rose (HR) equation. Those neurons generate periodic spikes, quasiperiodic spikes and chaotic spikes in some range of bifurcation parameters. We propose two models: a model with synaptic connections described by a gap junction (model 1) and a model with synaptic connections described by a first-order kinetics (model 2). We calculate numerically Lyapunov exponents and quadratic deviations of membrane potentials among the neurons. By increasing the strength of excitatory synapses, we find that a chaotic synchronization occurs for model 1, but does not for model 2 when the values of the bifurcation parameters for each neuron are set to those for the chaotic spikes. On the other hand, by increasing the strength of inhibitory synapses, it turns out that the periodic spikes are generated in antiphase for model 1, and the chaotic spikes and the periodic spikes are alternately generated in antiphase for model 2 for those values of the bifurcation parameters.