Signal-to-noise analysis of Hopfield neural networks with a formulation of the dynamics in terms of transition probabilities

We study Hopfield neural networks with infinite connectivity using signal-to-noise analysis with a formulation of the dynamics in terms of transition probabilities. We focus our study on models where the strongest effects of the feedback correlations appear. We introduce an analysis of the path probability of one neuron in order to obtain the contribution of all feedback correlations to the dynamics of this neuron. In this way, we obtain a complete theory for dynamics with order parameter equations identical to those obtained with general functional analysis for finite temperature. In the first part of this work, we present our method in the fully connected Little-Hopfield neural network. We obtain, in a simple and direct way, the order parameter equations found by general functional analysis. In the second part, the exposed method is applied to the fully connected Ashkin-Teller neural network. It is shown that the retrieval quality is improved by introducing four-spin couplings. Simulation results support our theoretical predictions.