What is Memory? A Homological Perspective

Abstract

We introduce the delta-homology model of memory, a unified framework in which recall, learning, and prediction emerge from cycle closure, the completion of topologically constrained trajectories within the brain's latent manifold. A Dirac-like memory trace corresponds to a nontrivial homology generator, representing a sparse, irreducible attractor that reactivates only when inference trajectories close upon themselves. In this view, memory is not a static attractor landscape but a topological process of recurrence, where structure arises through the stabilization of closed loops. Building on this principle, we represent spike-timing dynamics as spatiotemporal complexes, in which temporally consistent transitions among neurons form chain complexes supporting persistent activation cycles. These cycles are organized into cell posets, compact causal representations that encode overlapping and compositional memory traces. Within this construction, learning and recall correspond to cycle closure under contextual modulation: inference trajectories stabilize into nontrivial homology classes when both local synchrony (context) and global recurrence (content) are satisfied. We formalize this mechanism through the Context-Content Uncertainty Principle (CCUP), which states that cognition minimizes joint uncertainty between a high-entropy context variable and a low-entropy content variable. Synchronization acts as a context filter selecting coherent subnetworks, while recurrence acts as a content filter validating nontrivial cycles.