Synchronization of coupled memristive Hindmarsh–Rose maps under different coupling conditions

Abstract

The importance of the synchronization emergence in neuronal networks has motivated many researchers to study this phenomenon. However, dealing with the systems in the discrete-time domain is more straightforward and underemphasized. Therefore, neuronal maps have recently been widely used in investigating different collective behaviors of the interacting neurons, including synchronization. Besides, the study of the memristors in neuronal models or as a synaptic function in neuronal networks is another prominent subject of interest nowadays. In fact, the magnetic induction impact on the membrane potential can be considered by the memristors in neuronal models or networks. Hence, in this paper, we investigated the synchronization of the recently proposed memristive Hindmarsh–Rose neuron maps under different coupling conditions: electrical synapses, chemical synapses, inner linking functions, and hybrid synapses. In each case, we analyzed the stability of the synchronous solution using the master stability functions. Also, we calculated the time-averaged synchronization error as a numerical verification. We found that memristive Hindmarsh–Rose neurons synchronize when they are coupled through electrical and hybrid synapses, while through chemical synapses, they cannot reach a synchronous solution. Also, we showed that a slightly lower coupling value is needed to synchronize the neurons interacting through inner linking functions than the electrical synapses.