Q-Ising neural network dynamics: a comparative review of various architectures

TL;DR

This review compares the dynamics of various Q-Ising neural network architectures using probabilistic analysis, deriving recursive schemes for their evolution and equilibrium states, highlighting differences due to architecture and feedback correlations.

Contribution

It provides a comprehensive probabilistic framework for analyzing the dynamics of multiple Q-Ising neural network architectures, including new recursive schemes and fixed-point equations.

Findings

Closed-form solutions for asymmetric and layered networks.

Feedback correlations complicate symmetric network analysis.

Equilibrium equations match replica-symmetric mean-field results.

Abstract

This contribution reviews the parallel dynamics of Q-Ising neural networks for various architectures: extremely diluted asymmetric, layered feedforward, extremely diluted symmetric, and fully connected. Using a probabilistic signal-to-noise ratio analysis, taking into account all feedback correlations, which are strongly dependent upon these architectures the evolution of the distribution of the local field is found. This leads to a recursive scheme determining the complete time evolution of the order parameters of the network. Arbitrary Q and mainly zero temperature are considered. For the asymmetrically diluted and the layered feedforward network a closed-form solution is obtained while for the symmetrically diluted and fully connected architecture the feedback correlations prevent such a closed-form solution. For these symmetric networks equilibrium fixed-point equations can be derived under certain conditions on the noise in the system. They are the same as those obtained in a thermodynamic replica-symmetric mean-field theory approach.