Synchronization of two identical and non-identical Rulkov models

• Two electrical coupled Rulkov neurons are likely to achieve complete synchronization.
• A synchronization scheme makes the complete synchronization attainable.
• The complete synchronization of two discrete-time Rulkov neurons is different from that of continuous-time models.

In this paper, the synchronization of two chaotic Rulkov map-based neurons is taken into account. Firstly, based on the master stability function (MSF) analysis, the complete synchronization of two electrical coupled chaotic Rulkov neurons is investigated in detail. The two-dimensional parameter-space plot that displays directly the values of the MSF in different colors is numerically obtained. The numerical values of the MSF show that the two electrical coupled Rulkov neurons are likely to achieve the complete synchronization when each single neuron is in a silent state or a period-1 bursting state, while are unable to reach the complete synchronous state when each single neuron is in a chaotic bursting state or a spiking state. Secondly, Pearson’s correlation coefficient is employed to measure the synchronization degree, which demonstrates the nonexistence of the complete synchronization for non-identical electrical coupled Rulkov neurons. Importantly, the complete synchronization can not be reached with the increase of the electrical coupling strength, which is different from the continuous-time neuronal models. Finally, based on the active control method, a synchronization scheme is presented to study the complete synchronization for two Rulkov neurons no matter whether they are identical or not. The scheme is also applied to investigate the anticipated synchronization and the lag synchronization for any two Rulkov neurons. Numerical simulations verify the correctness of our analytical results and the effectiveness of our methods.