Back to the Continuous Attractor

2024-07-31

arXiv.org

AI Summary

Continuous attractors in brain-inspired memory modeling face a fundamental tension between structural fragility—prone to bifurcations under parameter perturbations—and functional robustness required for reliable memory storage. Method: We introduce the “persistent manifold” theory, which characterizes continuous attractors as dynamically stable over finite timescales despite structural instability. Our approach integrates fast-slow dynamical decomposition, persistent manifold analysis, and recurrent neural network (RNN) training. Contribution/Results: We provide the first empirical evidence in RNNs of approximate continuous attractors exhibiting the predicted slow-manifold geometry. Critically, these manifolds retain functional stability—preserving memory encoding and retrieval—under substantial parameter perturbations. This validates continuous attractors as a general computational paradigm for memory representation, challenging the conventional emphasis on their structural fragility. We establish a new interpretive framework prioritizing *functional robustness* over structural rigidity, reconciling theoretical instability with biological plausibility.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals. Unfortunately, continuous attractors suffer from severe structural instability in general--they are destroyed by most infinitesimal changes of the dynamical law that defines them. This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations. We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms. Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar. We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors. Fast-slow decomposition analysis uncovers the persistent manifold that survives the seemingly destructive bifurcation. Moreover, recurrent neural networks trained on analog memory tasks display approximate continuous attractors with predicted slow manifold structures. Therefore, continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.

Problem

• Continuous Attractors
• Stability Issues
• Memory Tasks Simulation

Innovation

• Structural Stability
• Persistent Manifolds
• Continuous Attractors in Neural Networks