Simulating the electric activity of FitzHugh–Nagumo neuron by using Josephson junction model

The resistive-capacitive-inductance Josephson junction (RCLSJ) model can simulate the electric activities of neurons. In this paper, the RCLSJ system is controlled to reproduce the dynamical properties of the FitzHugh–Nagumo system neuron by using the improved adaptive synchronization scheme. Improved Lyapunov functions with two controllable gain coefficients (α,β) is constructed, and the controller is approached analytically to realize linear generalized synchronization defined as $x=k\hat{x}+C$. The summation of error function during the process of synchronization and the power consumption of controller are calculated in the dimensionless model to measure the effect of the two gain coefficients (α,β) by selecting different constants (k,C) to represent different kinds of generalized synchronization. The results are approached as follows: (1) the power consumption of the controller is independent of the selection of the two gain coefficients (α,β); (2) the synchronization region is marked in the phase space of the two gain coefficients; (3) the power consumption of controller is dependent on the selection of constants (k,C), smaller power consumption of the controller is required with larger k at fixed C; larger power consumption costs with larger C at fixed k. The specific case for C=0,k=1 is also discussed to understand the case for complete synchronization.