Cognitive Map as Reference Path Measure: The Hippocampus as a Schr\"odinger Bridge Sampler

A cognitive map enables animals to recall past journeys, plan future routes, and imagine unexperienced paths. How a single neural circuit generates such diverse sequential experiences from learned knowledge remains a fundamental open question. We propose that a cognitive map defines a reference path measure, a probability distribution over sequence spaces, and that the hippocampus constructs goal-directed episodes by solving a Schr\"odinger bridge problem under endpoint constraints. In this framework, present cues and terminal evidence modulate sequence generation through a continuation field that evaluates path reachability to target outcomes. We demonstrate that this computation is naturally realized by hippocampal circuit dynamics: recurrent relaxation computes the continuation field, dendritic gain modulates assembly transitions, and inhibitory competition governs event completion. Furthermore, offline replay supplies internal endpoint expectations, enabling contrastive calibration of boundary representations against waking experience. Across network simulations and large-scale hippocampal electrophysiological recordings (HC-3, HC-11, and DANDI 000978), our model recovers exact path laws, reveals reusable temporal structure, accurately predicts complete-event trajectories and durations under boundary conditioning, and accounts for experience-dependent shifts in sleep replay. This work unifies attractor dynamics with conditional generative modeling, providing a shared computational foundation for memory retrieval, prospective planning, and offline consolidation.