Kinetic Analysis Of The Discrete Neural Network Model

In this thesis, we describe some important properties of the dynamic behaviors of several class of time-discrete models of two neurons, which includes stability, asymptotic stability, the existence and attractability of periodic solutions, Flip-bifurcation and Neimark-Sacher-bifurcation. It is consists of five chapters. As the introduction, in Chapter 1, the background and history of neural networks are briefly addressed, and some notations and definitions are given in this chapter. In Chapter 2, a class of discrete-time systems with McCulloch-Pitts nonlinearity including the discrete versin of an artifical neural network of two neurons piecewise constant argument are investigated. We find that this system is convergent if the absolute values of threshold values are large. On the other hand, for the threshold values with small absolute values, using the analytical technique to construct return mappings, we not only discuss the existence and attractability of periodic solutions, but also we observe an interesting distribution of the various periodic solution. In Chapter 3, We consider the existence of periodic solution of certain discrete-time neural network systems. First, we show that all solution of such systems are bounded, and then we arrive at a very concise result: every solution is periodic eventually. In Chapter 4, the stablity and existence of periodic solutions are studied for two class of discrete-time models of two neurons(excitation-excitatiori connection type and excitation-inhibition connection type) with standard saturation function. Our results indicate that the dynamic behaviors of two models are very different. We also obtain some results about the coexistence of multiple attractive periodic solutions for the excitation-inhibition connection type model. In Chapter 5, stability and bifurcation analysis are studied for a discrete-time system modelling a network of two neurons with self-connections. Choosing an appropriate bifurcation parameter, we prove that the Flip-bifurcation and the Neimark-Sacker bifurcation occur when the bifurcation parameter exceeds critical values. The direction and stability of bifurcation are determined by the normal form theory and center manifold theorem. Results of some computer simulations are displayed graphically.