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

2026-03-04 · Source: cs.NE updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, long

Summary

This work develops a dynamical theory for sequential reasoning in Hopfield networks, addressing the limitation of existing models that largely rely on numerical evidence for sequential retrieval. The authors introduce an input-driven plasticity (IDP) Hopfield network, featuring a two-timescale architecture that couples fast associative retrieval with slow reasoning dynamics. They derive explicit mathematical conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes. Specifically, for a HardTanh activation function, they identify a critical gain threshold of $\kappa_{\text{critical}}=4$ for stable, recurrent transitions between memories. Below this threshold, activity either collapses or undergoes transient, weakening transitions. This framework provides a principled mathematical account of sequentiality in associative memory models, bridging classical Hopfield dynamics with modern reasoning architectures like Transformers.

Key takeaway

For AI Researchers and Research Scientists developing associative memory models, this work offers a robust theoretical foundation for implementing sequential reasoning. You should consider adopting the two-timescale IDP Hopfield framework to achieve predictable, self-sustained memory transitions. Understanding the derived gain thresholds and escape times will be crucial for designing systems that reliably perform multi-step reasoning without sacrificing analytical control, particularly when aiming for Transformer-like capabilities.

Key insights

A two-timescale IDP Hopfield network enables analytically tractable sequential memory retrieval with explicit transition conditions.

Principles

• Sequentiality emerges from slow reasoning dynamics.
• Gain thresholds determine self-sustained transitions.
• Timescale separation simplifies analytical tractability.

Method

The method involves coupling fast associative retrieval with slow reasoning dynamics in an IDP Hopfield network, using a circulant reasoning matrix to induce limit cycles over memories and deriving conditions for self-sustained transitions.

In practice

• Use IDP Hopfield for structured memory transitions.
• Set gain parameter $\kappa \geq 4$ for stable sequences.
• Employ two-timescale architecture for analytical control.

Topics

• Hopfield Networks
• Associative Memory
• Sequential Retrieval
• Dynamical Systems
• Input-Driven Plasticity

Best for: AI Researcher, AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.NE updates on arXiv.org.