Multistability in a class of stochastic Hopfield neural networks

Abstract

In this paper, the problem of multistability analysis for a class of stochastic Hopfield neural networks is considered. By utilizing the properties of activation functions and applying Schauder's fixed-point theorem, a sufficient condition for the existence of multiple equilibria is derived. Then applying stochastic analysis technique and Lyapunov approach, a criterion is established for ensuring these equilibria to be locally exponentially stable in mean square. Estimation of positively invariant sets with probability 1 and basins of attraction for these equilibria are also obtained. Finally, an example is given to show the effectiveness of the derived results. Since the seminal work for Hopfield neural networks in (1), artificial neural networks have drawn much attention due to their successful applications in many kinds of research fields such as classification, associative memories, pattern recognition, optimization, image processing, cryptography and decision making and so on. These successful applica- tions heavily rely on dynamical properties of neural network- s. Therefore, the theoretical study on dynamics of neural networks has been an important research topic. Stability, as one of the most important issues related to dy- namics of neural networks, has attracted the interest of many researchers and a great number of papers have been report- ed in the literature. Most of them focus on monostability of neural networks, see, for instance (2-6) and the references therein. Monostability means that the neural network has a unique equilibrium which is globally attractive. However, in many applications of neural networks, the designed neu- ral networks are required to possess multiple locally stable equilibriums. For example, when a neural network is ap- plied to image processing, associative memory, and pattern recognition, the existence of multiple equilibriums is a nec- essary feature. So it is of importance to investigate the mul- tistability problem of neural networks. In the last few years, the multistability of neural networks has been investigated in (7-12) and the references therein. To mention a few rep- resentative works, the multistability of Hopfield-type neural networks with delay and without delay was investigated in (7), where basins of attraction for these equilibriums were estimated. In (8), the multistability issue for Bidirectional Associative Memory (BAM) neural networks was studied, and it was proved that the 2n-dimensional BAM neural net- works can have n 3 equilibria and n 2 equilibria of them are locally stable. In (9), by decomposition of state space, some sufficient conditions were derived for ensuring multistabili- ty of delayed Cohen-Grossberg neural networks. New mul- tistability criteria for delayed Cohen-Grossberg neural net- works were presented in (10). The multistability issue for delayed competitive neural networks and delayed Hopfield