On Delta-Homology Analogy: Memory as Structured Trajectories

• URL: http://arxiv.org/abs/2303.04203v2
• Date: Fri, 15 Aug 2025 22:10:07 GMT
• Title: On Delta-Homology Analogy: Memory as Structured Trajectories
• Authors: Xin Li,
• Abstract summary: We introduce the emphdelta-homology analogy, which formalizes memory as a set of sparse, topologically irreducible attractors.<n>Based on the analogy, we propose a topological framework for memory and inference grounded in the structure of spike-timing dynamics and persistent homology.
• Score: 5.234742752529437
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We introduce the \emph{delta-homology analogy}, which formalizes memory as a set of sparse, topologically irreducible attractors. A \emph{Dirac delta-like memory trace} \( \delta_\gamma \) is identified with a nontrivial homology generator \( [\gamma] \in H_1(\mathcal{Z}) \) on a latent manifold of cognitive states. Such traces are sharply localized along reproducible topological cycles and are only activated when inference trajectories complete a full cycle. They encode minimal, path-dependent memory units that cannot be synthesized from local features alone. Based on the analogy, we propose a topological framework for memory and inference grounded in the structure of spike-timing dynamics and persistent homology. Starting from the observation that polychronous neural groups (PNGs) encode reproducible, time-locked spike sequences shaped by axonal delays and synaptic plasticity, we construct \emph{spatiotemporal complexes} whose temporally consistent transitions define chain complexes over which robust activation cycles emerge. These activation loops are abstracted into \emph{cell posets}, enabling a compact and causally ordered representation of neural activity with overlapping and compositional memory traces.

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