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

2026-03-03

AI Summary

This work addresses the lack of a dynamic theoretical framework in existing associative memory models for sequential retrieval and multi-memory integration, which hinders their ability to capture sequential reasoning mechanisms in modern machine learning. The paper proposes an Input-Driven Plasticity (IDP) Hopfield network featuring a dual-timescale architecture that couples fast associative retrieval with slow inferential dynamics, offering the first rigorous mathematical characterization of sequentiality in associative memory. Through nonlinear dynamical and stability analyses, the authors derive explicit conditions for self-sustained memory transitions—including gain thresholds, escape times, and collapse regions—thereby establishing a theoretical bridge between classical Hopfield dynamics and contemporary inference architectures, and providing a verifiable foundation for sequential reasoning.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• sequential retrieval
• associative memory
• Hopfield networks
• reasoning dynamics
• memory transitions

Innovation

• sequential retrieval
• Hopfield networks
• input-driven plasticity
• two-timescale dynamics
• associative memory