Adaptive Fuzzy Control for the Generalized Projective Synchronization of Fractional-Order Extended Hindmarsh-Rose Neurons

We investigate the generalized projective synchronization (GPS) control of fractional-order extended Hindmarsh-Rose (FOEHR) neuronal models with transcranial magneto-acoustical stimulation (TMAS) input. This improved neuronal model has advantages in describing the complex firing characteristics of neurons stimulated by alternating current. In this study, a master-slave neuron system consisting of two FOEHR neuronal models is assumed to be subject to uncertain model parameters and unknown external disturbances. To quantify the GPS error, we design a new error variable based on the properties of the fractional-order derivative and construct a related GPS error system. Fuzzy logic systems are introduced to approximate the unknown nonlinear dynamics of the error system. To ensure the synchronous firing rhythms of the master-slave neuron system, an adaptive fuzzy control algorithm is proposed under the Lyapunov approach, in which the adaptive parameters are robust to the estimation errors. By choosing the appropriate design parameters, the proposed control scheme enables the master-slave neuron system to achieve GPS in a finite amount of time and to be resilient to uncertain parameters and unknown disturbances. The simulation results demonstrate that after the designed control inputs are implemented, the states of the slave neuron synchronize with those of the master neuron in specified proportions, and the corresponding synchronization error converges towards an arbitrarily small neighborhood of zero.