Memory flows: geometry and dynamics of sequential retrieval in input-driven Hopfield networks

Associative memory models classically describe retrieval as convergence to stored patterns, but the theory of sequential retrieval remains comparatively limited. We develop a two-timescale input-driven Hopfield network in which fast associative retrieval is modulated by a slow feedback variable evolving over a prescribed memory-transition graph. This yields autonomous, graph-constrained transitions while preserving analytical tractability. Using slow-fast and geometric singular perturbation theory, we characterize memory fixation via memory manifolds and identify the geometric regions that govern stability loss and escape from retrieved memories. We further derive reduced slow dynamics and explicit criteria for transition onset, self-sustained retrieval, and collapse, expressed through escape times, gain thresholds, and fixed points. Together, these results provide a tractable framework for analyzing feedback-driven sequential retrieval in Hopfield networks.