Spaces and sequences in the hippocampus: a homological perspective

Topological techniques have become a popular tool for studying information flows in neural networks. In particular, simplicial homology theory is used to analyze how cognitive representations of space emerge from large conglomerates of independent neuronal contributions. Meanwhile, a growing number of studies suggest that many cognitive functions are sustained by serial patterns of activity. Here, we investigate stashes of such patterns using path homology theory-an impartial, universal approach that does not require a priori assumptions about the sequences' nature, functionality, underlying mechanisms, or other contexts. We focus on the hippocampus-a key enabler of learning and memory in mammalian brains-and quantify the ordinal arrangement of its activity similarly to how its topology has previously been studied in terms of simplicial homologies. The results reveal that the vast majority of sequences produced during spatial navigation are structurally equivalent to one another. Only a few classes of distinct sequences form an ordinal schema of serial activity that remains stable as the pool of sequences consolidates. Importantly, the structure of both maps is upheld by combinations of short sequences, suggesting that brief activity motifs dominate physiological computations. This ordinal organization emerges and stabilizes on timescales characteristic of spatial learning, displaying similar dynamics. Yet, the ordinal maps generally do not reflect topological affinities-spatial and sequential analyses address qualitatively different aspects of spike flows, representing two complementary formats of information processing.

This study employs path homology theory to examine serial patterns of neuronal activity in the hippocampus, a critical region for learning and memory. While the traditional, simplicial homology approaches used to model cognitive maps, path homology provides a universal framework for analyzing the ordinal arrangement of neuronal sequences without presupposing their nature or function. The findings reveal that a limited number of distinct sequence classes, supported by combinations of short activity motifs, form a stable ordinal schema over timescales corresponding to periods of spatial learning. Notably, the ordinal maps derived from these sequences do not capture topological affinities, highlighting that spatial and sequential analyses address distinct but complementary dimensions of neural information processing.