On the Convergence Properties of the Hopfield Model

The main contribution of the present work is showing that the known convergence properties of the Hopfield model can be reduced to a very simple case, for which an elementary proof is provided. The convergence properties of the Hopfield model are dependent on the structure of the interconnections matrix W and the method by which the nodes are updated. Three cases are known: (1) convergence to a stable state when operating in a serial mode with symmetric W, (2) convergence to a cycle of length 2, at most, when operating in a fully parallel mode with symmetric W, and (3) convergence to a cycle of length 4 when operating in a fully parallel mode with antisymmetric W. The three known results are reviewed and it is proven that the fully parallel mode of operation is a special case of the serial model of operation. There are three more cases than can be considered using this characterization: serial mode of operation, antisymmetric W; serial mode of operation, arbitrary W; and fully parallel mode of operation, arbitrary W. By exhibiting exponential lower bounds on the length of the cycles in other cases, it is proven that the three known cases are the only interesting ones.