Dynamical Analysis Of Fractional-order Hopfield Neural Networks With Time Delays

Fractional calculus is a generalization of the ordinary differentiation and integra-tion to non-integer order. Fractional differentiation provides single neurons with a fun-damental and general computation that can contribute to efficient information process-ing, stimulus anticipation and frequency-independent phase shifts of oscillatory neu-ronal firing. In addition, fractional calculus can provide a concise model for the de-scription of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multi-scale processes. Frac-tional calculus is being applied to build new mathematical model of bioengineering re-search. Moreover, the neural network, as a large-scale complex system, exhibits the rich and colorful dynamical behaviors. In order to analyze and apply easily, the transmission delays are ignored in modeling for most of system. But it is demonstrated by theories and practices that time delay is unavoidable. So it is important to consider neural net-work with time delay. Compared with integer-order neural network, fractional-order neural network is just a new research direction, there are less theoretical results and applications. All in all, dynamical analysis of fractional-order neural model and neural networks is investigated just in the initial stage at present. There are not much pub-lished studies on this subject, and a lot of interesting and significant work to be studied. Therefore, the research of the fractional-order neuron models and neural networks is very promising.This dissertation focuses on dynamical analysis of the simplified Hodgkin-Huxley model and the fractional-order Hodgkin-Huxley (H-H). In addition, stability analysis of fractional-order Hopfield neural networks with time delays is also studied. The main achievements and originality contained in this dissertation are as follows:1. The dynamical behaviors of a two-dimensional simplified H-H model exposed to external electric fields are investigated through qualitative analysis and numerical simulation. A simplified H-H model with two parameters is given. The dynamical behaviors of the simplified model are consistent with the original H-H model. Then the dynamical behaviors of two-dimensional simplified H-H model exposed to exter-nal electric fields are studied. A necessary and sufficient condition is proposed for the existence of the Hopf bifurcation. The conditions of supercritical and subcritical Hopf bifurcation are also obtained. Furthermore, the stability of equilibrium points and limit cycles is also investigated. In addition, the canards and bifurcation are discussed in the simplified model and original model. Finally, the bifurcation curves with the coeffi-cients of different linear forms are shown. The numerical results demonstrate that some linear forms can retain the bifurcation characteristics of the original model, which are of great use to simplify the H-H model for the real-world applications.2. The dynamical analysis of the fractional-order H-H model is studied. Two crit-ical values of fractional-order H-H model between the stability and periodic solutions with fractional-order q are obtained. A fractional-order H-H neuron model is first intro-duced. Then the stable and unstable regions about control parameters are given. When the fractional-order q is getting larger, the stable region is reducing, while the unsta-ble region is increasing. Second, with fractional-order q changing, two critical values of fractional-order H-H model between the stability and periodic solutions are given. The critical value in which the stable state of fractional-order H-H model transforms to the periodic state decreases with fractional-order q increasing. Meanwhile, the other critical value in which the periodic oscillation of fractional-order H-H model decays to resting state increases with fractional-order q increasing. And the cycle interval about the control parameters is also increasing.3. This dissertation mainly studies the stability of fractional-order Hopfield neu-ral network with time delay. First, a stability theorem of fractional-order neural net-works with time delay is derived. The stability conditions of the fractional-order two-dimensional neural networks with time delay are obtained based on the stability the-orem of fractional-order system with time delay. Second, two fractional-order neural networks with different ring structures and time delays are developed. One ring struc-ture shows that every neuron in the network is only connected to its one closest neigh-bor. And the same ring structure time delay feedback is also considered in the network. The other ring structure shows that every neuron of the network is only connected to its two closest neighbors. By studying the developed neural networks, the correspond-ing sufficient conditions for stability of the systems are also derived. Furthermore, the fractional-order neural networks with hub structure and time delays are studied. Some sufficient conditions for stability of the systems are obtained. Last, when the initial conditions become complicated including random and periodic function, the fractional-order neural networks with hub structure and time delays are still stable under the given stability conditions by numerical simulation analysis.4. The global stability analysis of fractional-order neural networks with time delay is investigated. A comparison theorem for a class of fractional-order systems with time delay is shown. The existence and uniqueness of the equilibrium point for fractional- order neural networks with time delay are proved. Furthermore, the global asymptotic stability conditions of fractional-order neural networks with time delay are obtained. Moreover, the global stability analysis of fractional-order delayed neural networks with bounded perturbations is investigated. According to the comparison theorem and the stability theorem of linear system with time delay, the global uniformly stability con-ditions of fractional-order delayed neural networks with bounded perturbations are ob-tained.