An Exactly Solvable Asymmetric Neural Network Model

We consider a diluted and nonsymmetric version of the Little-Hopfield model which can be solved exactly. We obtain the analytic expression of the evolution of one configuration having a finite overlap on one stored pattern. We show that even when the system remembers, two different configurations which remain close to the same pattern never become identical. Lastly, we show that when two stored patterns are correlated, there exists a regime for which the system remembers these patterns without being able to distinguish them. Spin glass models for associative memory have found increasing interest in the last few years. As first proposed by Little [l] and Hopfield [2], these models are based on an Ising Hamiltonian and hence can be treated by equilibrium statistical mechanics. A detailed discussion of the equilibrium properties of the Hopfield model is given in Amit et al. [3]. Two assumptions are crucial to allow for an exact solution of the equilibrium properties of the model: the synaptic connections are taken to be symmetric and each neuron is connected to an infinite number of other neurons. In biological networks the synapses are known t o be asymmetric and on the average a neuron is connected only to a fraction F = lov6 of all neurons. Hence it is important to study the effects of asymmetry and dilution. Nonsymmetric models have been investigated by several groups [4-81. One possible way to introduce asymmetry is to keep the .learning rules. of Hebb, but cut out some of the synaptic connections. As long as the fraction ,o remains finite, the memory states are not seriously degraded[4]. On the other hand the effect of extreme dilution, p=O(l/N) is expected to be much more drastic. Random systems with long-range interactions, but finite coordination number have been discussed in the context of diluted spin glasses [8], graph optimization [9] and random networks of automata [lo-121. In this paper, we give an exact solution for the dynamics of a dilute nonsymmetric version of the Little-Hopfield model [l, 21. The model consists of a system of N Ising spins cr1 = k 1, whose interactions Jij depend on p stored patterns. By definition of the model the Jij are given by