On the Convergence Properties of the Hopfield Model

The main contribution is showing that the known convergence properties of the Hopfield model can be reduced to a very simple case, for which we have an elementary proof. 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 at most 2 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. We review the three known results and prove that the fully parallel mode of operation is a special case of the serial mode of operation, for which we present an elementary proof. The elementary proof (one which does not involve the concept of an energy function) follows from the relations between the model and cuts in the graph. We also prove that the three known cases are the only interesting ones by exhibiting exponential lower bounds on the length of the cycles in the other cases.