Existence and stability of a unique equilibrium in continuous-valued discrete-time asynchronous Hopfield neural networks

This paper investigates a continuous-valued discrete-time analog of the well-known continuous-valued continuous-time Hopfield neural network model, first proposed by Takeda and Goodman (1986). It is shown that the assumption of D-stability of the interconnection matrix, together with the standard assumptions on the activation functions, guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes. Conditions for local and global asymptotic stability are also derived, for both synchronous and asynchronous modes of operation. The results obtained are discussed both from the points of view of applications and robustness