Stability and periodicity of discrete Hopfield neural networks with column arbitrary-magnitude-dominant weight matrix

Stability and periodicity of neural networks is important behavior in biological and cognitive activities. In order to better simulate a biological genuine model, a special kind of discrete Hopfield neural networks (SDHNNs) in which every neuron has only one input is considered. By applying permutation theory and mathematical induction, we prove that the SDHNN always converges to a stable state or a limit cycle. The SDHNN is extended to the discrete Hopfield neural networks with column arbitrary-magnitude-dominant weight matrix (DHNNCAMDWM) in which there only exits a magnitude-dominant element in every column. Some important results, especially the periodic stability of the DHNNCAMDWM, are obtained. And the XOR problem is successfully solved by the results.