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

• URL: http://arxiv.org/abs/2603.03201v2
• Date: Thu, 05 Mar 2026 11:03:58 GMT
• Title: A Dynamical Theory of Sequential Retrieval in Input-Driven Hopfield Networks
• Authors: Simone Betteti, Giacomo Baggio, Sandro Zampieri,
• Abstract summary: This work develops a theory of sequential reasoning in Hopfield networks.<n>We derive explicit conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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