  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.
