Dynamics of Two-Dimensional Discrete-Time Delayed Hopfield Neural Networks

This chapter is devoted to the analysis of the complex dynamics exhibited by twodimensional discrete-time delayed Hopfield-type neural networks. Since the pioneering work of (Hopfield, 1982; Tank & Hopfield, 1986), the dynamics of continuous-time Hopfield neural networks have been thoroughly analyzed. In implementing the continuous-time neural networks for practical problems such as image processing, pattern recognition and computer simulation, it is essential to formulate a discrete-time system which is a version of the continuous-time neural network. However, discrete-time counterparts of continuous-time neural networks have only been in the spotlight since 2000. One of the first problems that needed to be clarified, concerned the discretization technique which should be applied in order to obtain a discrete-time system which preserves certain dynamic characteristics of the continuous-time system. In (Mohamad & Gopalsamy, 2000) a semi-discretization technique has been presented for continuous-time Hopfield neural networks, which leads to discrete-time neural networks which faithfully preserve some characteristics of the continuous-time network, such as the steady states and their stability properties. In recent years, the theory of discrete-time dynamic systems has assumed a greater importance as a well deserved discipline. In spite of this tendency of independence, there is a striking similarity or even duality between the theories of continuous and discrete dynamic systems. Many results in the theory of difference equations have been obtained as natural discrete analogs of corresponding results from the theory of differential equations. Nevertheless, the theory of difference equations is a lot richer than the corresponding theory of differential equations. For example, a simple difference equation resulting from a first order differential equation may exhibit chaotic behavior which can only happen for higher order differential equations. This is the reason why, when studying discrete-time counterparts of continuous neural networks, important differences and more complicated behavior may also be revealed. The analysis of the dynamics of neural networks focuses on three directions: discovering equilibrium states and periodic or quasi-periodic solutions (of fundamental importance in biological and artificial systems, as they are associated with central pattern generators (Pasemann et al., 2003)), establishing stability properties and bifurcations (leading to the O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m