Synchronization and firing patterns of coupled Rulkov neuronal map

In this paper, we investigate stabilization, neuronal synchrony and different neuronal dynamics in two coupled neuronal Rulkov map in the presence of both inner linking function and chemical synaptic interactions. Based on linear stability analysis, conditions for stability of the fixed point for this coupled system are derived. Through master stability function approach, the local stability of complete synchronization is studied by calculating the transverse Lyapunov exponents of error system and using basin stability measure, we quantify the global stability of synchronization. More interestingly, different transitions of the neural spiking-bursting synchronous patterns are found for appropriately chosen interaction strengths. We interrogate the effect of time delay in the interaction function, and surprisingly, we observe that delay time can induce the mixed type of behavior to achieve the complete neuronal synchronous state. The presence of time delay solely in inner linking coupling is favorable for synchronization, while the presence of delay in both the interactions, the synchronization enhanced up to a certain threshold of delay time and beyond that it demolish the neural synchrony. The neuronal synchronization is also studied in coupled system with stochastic on-off interaction, exemplify by, in neural network neurons interact through sending short pulses, called spike. The different types of neuron dynamics are observed when a resting neuron is interacting with a bursting neuron under the proposed coupling configuration. Finally, the autapic connection induces and enhances the complete synchronization behavior.