The relativistic Hopfield model with correlated patterns

TL;DR

This paper introduces a relativistic Hopfield model with correlated patterns, analyzing its thermodynamic properties and phase diagram, revealing new correlated and symmetric phases influenced by pattern correlations.

Contribution

The study extends the Hopfield model by incorporating relativistic dynamics and pattern correlations, deriving self-consistent equations and analyzing phase behavior.

Findings

Existence of thermodynamic limit for the free-energy.

Identification of new correlated and symmetric phases.

Pattern correlations affect phase diagram regions.

Abstract

In this work we introduce and investigate the properties of the "relativistic" Hopfield model endowed with temporally correlated patterns. First, we review the "relativistic" Hopfield model and we briefly describe the experimental evidence underlying correlation among patterns. Then, we face the study of the resulting model exploiting statistical-mechanics tools in a low-load regime. More precisely, we prove the existence of the thermodynamic limit of the related free-energy and we derive the self-consistence equations for its order parameters. These equations are solved numerically to get a phase diagram describing the performance of the system as an associative memory as a function of its intrinsic parameters (i.e., the degree of noise and of correlation among patterns). We find that, beyond the standard retrieval and ergodic phases, the relativistic system exhibits correlated and symmetric regions -- that are genuine effects of temporal correlation -- whose width is, respectively, reduced and increased with respect to the classical case.