Spatial representability of neuronal activity

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.

Significance statement

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.

Advantages of Persistent Cohomology in Estimating Animal Location From Grid Cell Population Activity

Many cognitive functions are represented as cell assemblies. In the case of spatial navigation, the population activity of place cells in the hippocampus and grid cells in the entorhinal cortex represents self-location in the environment. The brain cannot directly observe self-location information in the environment. Instead, it relies on sensory information and memory to estimate self-location. Therefore, estimating low-dimensional dynamics, such as the movement trajectory of an animal exploring its environment, from only the high-dimensional neural activity is important in deciphering the information represented in the brain. Most previous studies have estimated the low-dimensional dynamics (i.e., latent variables) behind neural activity by unsupervised learning with Bayesian population decoding using artificial neural networks or gaussian processes. Recently, persistent cohomology has been used to estimate latent variables from the phase information (i.e., circular coordinates) of manifolds created by neural activity. However, the advantages of persistent cohomology over Bayesian population decoding are not well understood. We compared persistent cohomology and Bayesian population decoding in estimating the animal location from simulated and actual grid cell population activity. We found that persistent cohomology can estimate the animal location with fewer neurons than Bayesian population decoding and robustly estimate the animal location from actual noisy data.