Pattern recognition in Hopfield type networks with a finite range of connections

We study pattern recognition in linear Hopfield type networks of N neurons where each neuron is connected to the z subsequent neurons such that the state of the i th neuron at time t+1 is determined by the states of neurons i+1, ..., i+z at time t. We find that for small values of z/N the retrieval behavior differs considerably from the behavior of diluted Hopfield networks. The maximum number of random patterns that can be retrieved increases in a non linear way with z and the asymptotic mean overlap between input and output patterns decreases sharply as z is decreased and reaches zero at a finite value of z