Nonlinear feedback coupling in Hindmarsh–Rose neurons

Abstract

We analyse the model of neurons described by Hindmarsh–Rose (H–R) system with various coupling schemes. We have examined the scope of synchronization, anti-phase synchronization and amplitude death for linear indirect synaptic coupling of the H–R neurons. The work is extended to coupling of the form nonlinear cubic feedback. The coupling between two neurons using memristor is also examined. A memristor is now identified as the fourth fundamental circuit element which can be considered as an electrical synapse. Mutual coupling of H–R systems using cubic flux-controlled memristor exhibits the properties of bursting and amplitude death. With unidirectional coupling one neuron exhibits tonic spiking or bursting while the other neuron shows amplitude death. Mutual coupling with quadratic flux-controlled memristor model shows the possibilities of synchronization, oscillation death and other interesting dynamics like near-death rare spikes. Exponential flux-controlled memristor coupling in H–R neuron presents synchronization and oscillation death. We have examined the stability of different coupled systems and the Lyapunov exponent plots. It is shown that among different memristor couplings in H–R neurons, cubic flux-controlled memristor has got highest Lyapunov exponent. The present work summarizes the simulation results of different coupling schemes in H–R neurons which show many interesting dynamical characteristics of coupled neuron cells.