Generalized synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative

Abstract

In the present paper, the synchronization of the extended Hindmarsh–Rose neuronal model with fractional order derivative is presented in detail taking into consideration the effects of the very slow intracellular exchange of calcium ion (Ca2+) occurring between cytoplasm and its store. That is, we approach the synchronization by the Ge–Yao–Chen partial region stability theory. The method of resolution used is the powerful Adam–Bashforth–Moulton method which converges quickly to the exact solution. Numerical simulations of the error dynamics proved the effectiveness of the control method. The synchronization time is evaluated for different orders of the derivative and its analysis shows that, it is closely related to the order of the derivative and that fractional order induces quick synchronization compared to integer order.