Stability of discrete Hopfield neural networks with time-delay

It is well-known that discrete Hopfield neural networks (DHNNs) without delay converge to a stable state. Due to this property, DHNNs without delay have wide potential applications to many fields, such as associative memory devices and combinatorial optimization. A DHNN with delay, which can deal with temporal information, is a generalization of a DHNN without delay. This paper investigates the convergence theorems of DHNNs with delay, based on new updating modes. A new bivariate energy function is constructed which represents the relationships between application problems and DHNNs with delay. It is proved that DHNNs with delay converge to a stable state. These results extend the existing results corresponding to DHNNs without delay. We also relate the maximum of this energy function to a stable state of DHNNs with delay. Furthermore, we describe algorithms for DHNNs with delay in detail.