Approximate continuous attractor theory

Continuous attractors offer a unique class of dynamical systems solutions for storing continuous-valued variables in recurrent neural 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 may limit their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations. We indeed 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, empirically their finite-time behaviors are similar. We theoretically show that continuous attractors are functionally robust and remain useful as a universal idealized pattern for understanding analog memory. Specifically, we show that a persistent slow manifold survives the unavoidable and seemingly destructive bifurcations. We use fast-slow decomposition to relate the flow within the manifold to the size of the perturbation. Conversely, an approximate continuous attractor with a slow manifold is necessarily near an ideal continuous attractor in the space of vector fields. Moreover, we show that the memory performance is bounded by the perturbative distance to the ideal continuous attractor. We observe that recurrent neural networks trained on analog memory tasks support these continuous attractor approximations as the universal solution class and link their generalization capabilities to their topologies. Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal idealized pattern for understanding analog memory.