Hebb-Hopfield neural networks based on one-dimensional sets of neuron states

Neural Networks (NN), which interconnection matrix is the Hebb matrix of Hopfield (HH) [2,3] are considered. Quasi-continuos sets of neuron states are being used for network matrix production. It is shown, that in this case minima of Hopfield energy are at the bottom of deep ditches, corresponding to the basic set of network activity states for the HH NN. The corresponding states can be made to be stable states of the network. When neuron threshold fatigue is introduced, depending of its recent activity state, the network activity becomes cyclic, moving with a constant rate in one of the two possible directions in the ring, depending on the initial conditions. The phenomena described present novel robust types of NN behavior, which have a high probability to be encountered in living neural systems.