Abstract
Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We comprehensively and rigorously assess its performance in simulated neural recordings of the brain's spatial representation system. Grid, head direction, and conjunctive cell populations each span low-dimensional topological structures embedded in high-dimensional neural activity space. We evaluate the ability for persistent cohomology to discover these structures for different dataset dimensions, variations in spatial tuning, and forms of noise. We quantify its ability to decode simulated animal trajectories contained within these topological structures. We also identify regimes under which mixtures of populations form product topologies that can be detected. Our results reveal how dataset parameters affect the success of topological discovery and suggest principles for applying persistent cohomology, as well as persistent homology, to experimental neural recordings.
Highlights
The enormous number of neurons that constitute brain circuits must coordinate their firing to operate effectively
We demonstrate that persistent cohomology can discover topological structure in simulated neural recordings with as few as tens of neurons from a periodic neural population (Figure 3)
It can discover more complex topological structures formed by combinations of periodic neural populations if each population is well-represented within the dataset (Figure 7)
Summary
The enormous number of neurons that constitute brain circuits must coordinate their firing to operate effectively This organization often constrains neural activity to low-dimensional manifolds, which are embedded in the high-dimensional phase space of all possible activity patterns (Gao and Ganguli, 2015; Gallego et al, 2017; Jazayeri and Afraz, 2017; Saxena and Cunningham, 2019). These low-dimensional manifolds exhibit non-trivial topological structure (Curto, 2016) This structure may be imposed externally by inputs that are periodic in nature, such as the orientation of a visual stimulus or the direction of an animal’s head. In either case, detecting and interpreting topological structure in neural data would provide insight into how the brain encodes information and performs computations
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