Stability And Synchronization Control Of A Fractional-order Neuron Model

In the recent decades, with the leap forward development of computer science and technology, and the progressive maturation of theory of fractional order differential equations, the research of dynamic behavior of fractional order differential systems has become more and more popular. Furthermore, fractional order differential systems have showed the enormous application foundation and prospect in biology, physics, automation, electrical engineering, etc. Therefore, the research of the dynamic behaviors of fractional order differential systems has the theoretical value of and the application potentials. Note that the dynamic behaviors of fractional order Hindmarsh-Rose (HR) neuron models are important in the neural systems. In this paper, we mainly study the stability and synchronization of fractional order HR neuron models and some sufficient conditions ensuring the stability and synchronization of models are derived.The main contents include:1. The backgrounds of fractional order calculus and neurons are introduced. Some existing results about neural model are recalled. The main work is presented.2. The related concepts of fractional calculus and the preliminaries about stability of fractional order systems are mainly introduced. Three kind of definitions of fractional calculus and their relationships are presented. Then, the concepts of stability of general fractional order systems and the definition of the k-class function are showed.3. The feedback control and washout-filter control of the fractional order HR neuron model are mainly discussed on the basis of the classic integer order HR model. The influences of the order on the stability and other dynamics of systems under two kinds of different controllers are analyzed by making use of the stability theory of fractional order differential equations. Numerical simulations are given to show the feasibility of results.4. Synchronizations between the fractional order HR neuron models are discussed by designing the state observer and the linear input feedback controllers, respectively. Based on fractional order direct Lyapunov theorem and linear matrix inequalities, some sufficient conditions to realize synchronization under two kinds of controllers are obtained. Numerical simulations illustrate the feasibility of theoretical analysis.5. The work of this paper is summarized with the outlook for the future.