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

2014 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 ith 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. J. Phys. Frarcce 51 ( 1990) 1797-1801 1er SEPTEMBRE 1990, Classification Physics Abstracts 87.30 75. 10H 64.60 In recent years, Hopfield neural networks [1] have been studied extensively (for recent reviews see e.g. [2-4]). The Hopfield network provides a standard for neural nets since its long time retrieval behavior could be treated analytically [5] by solving for the equilibrium statistical mechanics of the net. The solution could be achieved since all neurons in the net are interconnected. In this paper we report on numerical studies of linear Hopfield type nets of (usually N = 400) neurons where each neuron is connected only to a certain number of other neurons, and periodic boundary conditions are employed. Previous work in this direction was mainly devoted to nearest neighbor and next nearest neighbor connections, or to dilute (damaged) networks where a certain fraction of connections was randomly cut [6-10]. The network consists of lV neurons, each of them can be in two states ?: = ±1. The coupling strength Ji,j between pairs of connected neurons is determined by the M random patterns (g> ) (eï çfv), J-t = 1, 2,..., M, which are to be stored in the network: The state of neuron i at a given time t + 1 is obtained from the states of the z subsequent neurons at time t by