Stability and synchronization of coupled Rulkov map-based neurons with chemical synapses

• As a function of the intrinsic control parameters, and the external chemical coupling strength, the conditions for stability of the fixed point of this system are derived.
• The stable regions of the master stability function and the strip-like chaotic structure in different parameter-spaces are obtained.
• Given some values of control parameter, we propose the interval ranges of the coupling strength in which the two chaotic Rulkov map-based neurons can be synchronized completely.
• Compared with the control parameter and the synaptic parameter, the chemical coupling strength plays an opposite role in the process of synchronization transition.
• These findings could be useful for further understanding the role of two identical Rulkov chaotic map neurons coupled via the chemical synapses in the field of cooperative behaviors of coupled neurons.

The stability and synchronization analysis of two chaotic Rulkov maps coupled by bidirectional and symmetric chemical synapses are taken into account. As a function of intrinsic control parameters α, σ, η, reversal potential v, synaptic parameters θ, k, and external chemical coupling strength gc, conditions for stability of a fixed point for this system are derived. Some typical domains are chosen for numerical simulations which include time evolution of transmembrane voltages and phase portraits, and both of them are presented for theoretical analysis. Based on the master stability functions approach and calculation of the maximum Lyapunov exponents of synchronization errors, synchronized regions of the coupled neurons and a strip-shaped chaotic structure in parameter-space are obtained. Specially, given some values of control parameter α, we propose interval ranges of coupling strength gc in which the two chaotic Rulkov map-based neurons can be synchronized completely. It is shown that there exist different transition mechanisms of the neuronal spiking and bursting synchronization. The synchronized regions will become smaller and smaller as control parameter α or synaptic parameter θ increases. Nevertheless, the coupled neurons can at first transit from desynchrony to in-phase synchronization, and then to complete synchronization as chemical coupling strength gc increases. Compared with control parameter α and synaptic parameter θ, chemical coupling strength gc plays an opposite role in the process of synchronization transition. These findings could be useful for further understanding the role of two chaotic Rulkov maps coupled by bidirectional and symmetric chemical synapses in the field of cooperative behaviors of coupled neurons.