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Abstract:  Brains use a variety of coordinate systems to encode information. Sometimes these coordinate systems are linear and can be recovered from population activity using standard techniques. Often, however, they are not: many coordinate systems exhibit nonlinear global topology for which such tools can be less effective. Notably, grid cells in the entorhinal cortex comprise two linearly independent circular coordinate systems that, together, exhibit toroidal topology. Recent recordings using high-density probes confirm this toroidal topology persists during spatial and non-spatial behavior, and can be quantified and decoded with persistent (co)homology.

We ask a next natural question: is the propagation of circular coordinate systems through neural circuits a generic feature of biological neural networks, or must this be learned? If learning is necessary, how does it occur? We apply methods from topological data analysis developed to quantitatively measure propagation of such nonlinear manifolds across populations to address these problems. We identify a collection of connectivity and parameter regimes for feed-forward networks in which learning is required, and demonstrate that simple Hebbian spike-timing dependent plasticity reorganizes such networks to correctly propagate circular coordinate systems.