Global Convergence and Asymptotic Stability of Asymmetric Hopfield Neural Networks

The global convergence and asymptotic stability of Hopfield neural networks are known to be bases of successful applications of networks in various computing and recognition tasks. However, all previous studies on the networks assumed that the interconnection matrix is symmetric or antisymmetric. In this paper the two fundamental properties of the networks are studied without a symmetry assumption. It is proved that the networks will be globally convergent to a stable state if the interconnection matrix is weakly diagonally dominant in a sense to be defined. Furthermore, under any one of conditions assuring global convergence of the network, the maximal attraction radius of any one of stable states is found to be half of the distribution distance of the state to the network. The obtained results not only generalize the existing results, but also provide a theoretical foundation of performance analysis and new applications of the Hopfield networks.