A topological perspective on the dual nature of the neural state space and the correlation structure

When analysing neural activity, one often studies either the neural correlations or the state space of population vectors. However, the reason for choosing one over the other is seldom discussed. Here, with methods from the mathematical field of topology, we compare these approaches and propose using both for unsupervised inference of neural representations. If the recorded neurons have convex receptive fields on a single covariate space, there is a duality between the topological signatures derived from correlations on the one hand and population vectors on the other hand. However, in the presence of multiple neural modules with non convex receptive fields, this duality breaks down. We explain how to leverage complementary information derived from both approaches to sucessfully characterize the represented covariate spaces directly from the data also under these challenging circumstances. Furthermore, we prove appropriate reconstruction results and showcase applications to multiple neural datasets from various brain regions and diverse neural modules.