Global Exponential Stability of Discrete-Time Multidirectional Associative Memory Neural Network with Variable Delays

A discrete-time multidirectional associative memory neural networks model with varying time delays is formulated by employing the semidiscretization method. A sufficient condition for the existence of an equilibrium point is given. By calculating difference and using inequality technique, a sufficient condition for the global exponential stability of the equilibrium point is obtained. The results are helpful to design global exponentially stable multidirectional associative memory neural networks. An example is given to illustrate the effectiveness of the results. 1. Introduction The multidirectional associative memory (MAM) neural networks were first proposed by the Japanese scholar M. Hagiwara in 1990 [1]. The MAM neural networks have found wide applications in areas of speech recognition, image denoising, pattern recognition, and other more complex intelligent information processing. So they have attracted the attention of many researchers [2–7]. In [5], we proposed a mathematical model of multidirectional associative memory neural network with varying time delays as follows, which consists of the fields, and there are neurons in the field ( : where , , denote the membrane voltage of the th neuron in the field at time , denote the decay rate of the th neuron in the field , is a neuronal activation function of the th neuron in the field , is the connection weight from the th neuron in the field to the th neuron in the field , is the external input of the th neuron in the field , and is the time delay of the synapse from the neuron in the field to the th neuron in the field at time . We studied the existence of an equilibrium point by using Brouwer fixed-point theorem and obtained a sufficient condition for the global exponential stability of an equilibrium point by constructing a suitable Lyapunov function. However, discrete-time neural networks are more important than their continuous-time counterparts in applications of neural networks. One can refer to [8–10] in order to find out the research significance of discrete-time neural networks. To the best of our knowledge, few studies have considered the stability of discrete-time MAM neural networks. In this paper, we first formulate a discrete-time analogues of the continuous-time network (1.1), and in a next study the existence and the global exponential stability of an equilibrium point for the discrete-time MAM neural network. 2. Discrete-Time MAM Neural Network Model and Some Notations In this section we formulate a discrete-time MAM neural network model with time-varying delays by employing the