Mesoscopic Approach to Locally Hopfield Neural Networks in Presence of Correlated Patterns

Hopfield networks have gathered a lot of attention in computer science, as they have the ability to model many interesting phenomena that occur in brains and complex physical systems, and yet the model is nice in analysis. In this paper we investigate a simple Hopfield network organised in a two dimensional mesh, with localised interactions. The network remembers a number of periodically repeated, spatially correlated patterns. The weights are obtained via Hebbian learning rule combined with some extra information about the structure of correlations between the patterns, that is our system is in the so called phase coexistence regime in which the free energy for all of the patterns is equal (none of the patterns dominates in the sense it is the unique minimiser of the free energy). The number of remembered patterns is well below the memory limits to simplify the analysis and avoid any network´s capacity problems, we can therefore say the network is in finite loading regime. We argue that such a system can be accurately analysed in mesoscopic scale, in which it displays some phenomena characteristic for systems with large scale, isotropic interactions (e.g. Kac potential systems near Lebowitz-Penrose limit) like sharp phase interfaces, motion by mean curvature etc.