Workshop on Topology: Identifying Order in Complex Systems

Experimental neuroscience is achieving rapid progress in the ability to collect neural activity and connectivity data. Detecting meaningful structure in this data is challenging because the measured quantities are related to more "fundamental" variables by an unknown nonlinear transformation. We find that combinatorial topology can be used to obtain meaningful answers to questions about the structure of neural activity and introduce an approach that extracts features of the data invariant under arbitrary nonlinear monotone transformations. These features can be used to distinguish random and geometric structure, and depend only on the relative ordering of matrix entries. We apply our technique to neural activity in rat hippocampus, and find that the intrinsic pattern of correlations possesses a geometric organization in both spatial and non-spatial behaviors.