Back to the Continuous Attractor

TL;DR

This paper investigates the stability of continuous attractors in neural systems, revealing that bifurcations can produce structurally stable forms and that these attractors are functionally robust for analog memory despite fragility.

Contribution

It demonstrates that continuous attractors can be structurally stable through bifurcations and introduces a theoretical framework explaining their robustness in biological and neural network models.

Findings

Bifurcations from continuous attractors can be structurally stable.

Recurrent neural networks exhibit approximate continuous attractors with stable manifolds.

Continuous attractors remain useful for understanding analog memory despite fragility.

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.