A Dynamical Theory of Sequential Retrieval in Input-Driven Hopfield Networks

TL;DR

This work considers the recently proposed input-driven plasticity Hopfield network and analyzes a two-timescale architecture coupling fast associative retrieval with slow reasoning dynamics, providing a principled mathematical account of sequentiality in associative memory models.

Abstract

Reasoning is the ability to integrate internal states and external inputs in a meaningful and semantically consistent flow. Contemporary machine learning (ML) systems increasingly rely on such sequential reasoning, from language understanding to multi-modal generation, often operating over dictionaries of prototypical patterns reminiscent of associative memory models. Understanding retrieval and sequentiality in associative memory models provides a powerful bridge to gain insight into ML reasoning. While the static retrieval properties of associative memory models are well understood, the theoretical foundations of sequential retrieval and multi-memory integration remain limited, with existing studies largely relying on numerical evidence. This work develops a dynamical theory of sequential reasoning in Hopfield networks. We consider the recently proposed input-driven plasticity (IDP) Hopfield network and analyze a two-timescale architecture coupling fast associative retrieval with slow reasoning dynamics. We derive explicit conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes. Together, these results provide a principled mathematical account of sequentiality in associative memory models, bridging classical Hopfield dynamics and modern reasoning architectures.

Figures (3)

• Figure 1: Schematic illustration of the proposed two-timescale mechanism for sequential memory transitions. At the initial time, the fast state variable is at a stable equilibrium corresponding to memory $\upxi_1$. The slow reasoning variable $z$ progressively destabilizes this equilibrium while simultaneously stabilizing the equilibrium associated with memory $\upxi_2$. At the escape time $T_{\mathrm{escape}}$, the equilibrium corresponding to $\upxi_1$ loses stability and the system transitions toward $\upxi_2$. This transition occurs only when the gain parameter $\kappa$ of the slow dynamics exceeds a critical threshold $\kappa_{\mathrm{critical}}$. We analytically characterize both $k_{\mathrm{critical}}$ and the resulting escape time $T_{\mathrm{escape}}$.
• Figure 2: Cyclic memory transitions in one- and two-timescale Hopfield models. (A) Limit-cycle structure induced by the transition matrices $Q$ in the one-timescale dynamics and $A$ in the two-timescale reasoning dynamics (\ref{['eq: ttreas']}), with both systems initialized near memory $\upxi^{1}$. (B-C) One-timescale model: sequential retrieval proceeds through mixed memory states, with partial overlaps and unpredictable escape times across memories. (D-E) Two-timescale reasoning model: transitions occur between individual memories without mixing, with uniform, quantifiable escape times and overlaps reaching their maximal value, indicating exact memory alignment.
• Figure 3: Dependence of the slow reasoning dynamics on the gain $\kappa$ and the initial condition $Z_0$. (A) Solutions of the discrete iteration $\mathcal{P}(Z)=Z/\kappa$ as a function of the gain $\kappa$. Real solutions $Z_\pm>1$ exist only for $\kappa\geq \kappa_{\text{critical}}=4$ and sequential transitions are self-sustaining only if the initial saliency weight satisfies $Z_{0}>Z_{-}$. (B) HardTanh activation function, linear in $(-1,1)$ and saturated outside this interval. (C-D) Component-wise slow reasoning dynamics for different gain regimes. (C) Subcritical gain $\kappa=3$: oscillations decay and the dynamics collapse to the origin, leading to loss of retrieval. (D) Supercritical gain $\kappa=5$: oscillations persist and entrain to the stable fixed point $Z_{+}$, enabling reliable sequential memory retrieval.

Theorems & Definitions (1)

• Remark 1