Spin-glass models of neural networks

TLDR

The two dynamical models proposed by Hopfield and Little, which describe collective neural network behavior, are analyzed and their long‑time dynamics are governed by infinite‑range Ising spin‑glass Hamiltonians. The study focuses on the thermodynamic limit with a finite number of memorized spin configurations and investigates how more general memory distributions influence the system’s thermodynamic and dynamic properties. Memories are encoded as random spin configurations stored in quenched random couplings, and the analysis examines the resulting behavior in the thermodynamic limit. Both models display identical long‑time behavior below a transition temperature \(T_c\); they possess \(2p\) Mattis‑type ground states fully correlated with the stored patterns, and additional dynamically stable metastable states—specific mixings of patterns—emerge below approximately \(0.46\,T_c\).

Abstract

Two dynamical models, proposed by Hopfield and Little to account for the collective behavior of neural networks, are analyzed. The long-time behavior of these models is governed by the statistical mechanics of infinite-range Ising spin-glass Hamiltonians. Certain configurations of the spin system, chosen at random, which serve as memories, are stored in the quenched random couplings. The present analysis is restricted to the case of a finite number p of memorized spin configurations, in the thermodynamic limit. We show that the long-time behavior of the two models is identical, for all temperatures below a transition temperature ${T}_{c}$. The structure of the stable and metastable states is displayed. Below ${T}_{c}$, these systems have 2p ground states of the Mattis type: Each one of them is fully correlated with one of the stored patterns. Below T\ensuremath{\sim}0.46${T}_{c}$, additional dynamically stable states appear. These metastable states correspond to specific mixings of the embedded patterns. The thermodynamic and dynamic properties of the system in the cases of more general distributions of random memories are discussed.