The Blume-Emery-Griffiths neural network: dynamics for arbitrary temperature

TL;DR

This paper analyzes the dynamics of the Blume-Emery-Griffiths neural network at any temperature using probabilistic and generating functional approaches, providing explicit formulas and confirming results with simulations.

Contribution

It introduces a recursive probabilistic scheme for the network's dynamics at arbitrary temperature and compares it with the generating functional method, highlighting their agreement and differences.

Findings

Explicit formulas for the first few time steps of dynamics.

Agreement between methods up to the third time step.

Differences beyond the third time step for certain parameters.

Abstract

The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied for arbitrary temperature. By employing a probabilistic signal-to-noise approach, a recursive scheme is found determining the time evolution of the distribution of the local fields and, hence, the evolution of the order parameters. A comparison of this approach is made with the generating functional method, allowing to calculate any physical relevant quantity as a function of time. Explicit analytic formula are given in both methods for the first few time steps of the dynamics. Up to the third time step the results are identical. Some arguments are presented why beyond the third time step the results differ for certain values of the model parameters. Furthermore, fixed-point equations are derived in the stationary limit. Numerical simulations confirm our theoretical findings.