An increasingly popular approach to the analysis of neural data is to treat activity patterns as being constrained to and sampled from a manifold, which can be characterized by its topology. The persistent homology method identifies the type and number of holes in the manifold thereby yielding functional information about the coding and dynamic properties of the underlying neural network. In this work we give examples of highly non-linear manifolds in which the persistent homology algorithm fails when it uses the Euclidean distance which does not always yield a good approximation of the true distance distribution of a point cloud sampled from a manifold. To deal with this issue we propose a simple strategy for the estimation of the geodesic distance which is a better approximation of the true distance distribution and can be used to successfully identify highly non-linear features with persistent homology. To document the utility of our method we model a circular manifold, based on orthogonal sinusoidal basis functions and compare how the chosen metric determines the performance of the persistent homology algorithm. Furthermore we discuss the robustness of our method across different manifold properties and point out strategies for interpreting its results as well as some possible pitfalls of its application. Finally we apply this analysis to neural data coming from the Visual Coding - Neuropixels dataset recorded in mouse visual cortex after stimulation with drifting gratings at the Allen Institute. We find that different manifolds with a non-trivial topology can be seen across regions and stimulus properties. Finally, we discuss what these manifolds say about visual computation and how they depend on stimulus parameters.
Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex
