Finite-Time and Fixed-Time Synchronization of Coupled Chaotic HR Neurons With Chemical and Electrical Autapses: Sensitivity Analysis and Control Design

Synchronization of biological neurons plays a significant role in the processing and transmission of biological information, especially for therapeutic purposes. Therefore, designing accurate synchronization control methods is of great importance. On the other hand, the behavior of neurons is strongly influenced by the autapse terms, which are usually not considered in the design of synchronization control. In this paper, using sensitivity analysis, we demonstrate the important role of autapse effects on the behaviors of an isolated Hindmarsh-Rose (HR) neuron. Also, the effects of the chemical and electrical autapses on the behavioral properties of the neuron, such as spike frequency, bursting phenomena, etc., are investigated in several simulations. Furthermore, using a Lyapunov-like method, an asymptotic synchronization controller is designed for coupled HR neurons that are affected by the autapses. Using numerical simulations, the average norm of the synchronization error in 100 seconds without considering the autapse term is 2.1, and with considering its effect, it is 0.02. To ensure synchronization in finite time as well as convergence independent of initial conditions, a finite-time controller and a fixed-time controller are also designed, respectively. Simulations show faster convergence of the finite-time and fixed-time controls, 1.8 seconds and 1 second, respectively, compared to the asymptotic control scheme. Also, the average norm of the synchronization error using these methods is about 0.001 at 100 seconds, while it is 0.02 by using the asymptotic control method. Practically, these findings could directly assist therapists in applying appropriate control signals to achieve desired responses from neuronal networks for therapeutic purposes.