Fig. 3.

Betti numbers detect the existence of geometric organizing principles in neural population activity from rat hippocampus. a Mean cross correlation of N=88 rat CA1 pyramidal cells activity during spatial navigation. b Betti numbers as a function of graph edge density (# edges / possible # edges) for the clique complex of the pairwise correlation network in (a). c Comparison of data Betti numbers (thick lines) to model random networks with (top) geometric weights given by decreasing distance between random points in Euclidean space and (bottom) with no intrinsic structure obtained by shuffling the entries of the correlation matrix. d Integrals of the curves from panel B show that the data (thick bars) lie in the geometric regime (g) and that the unstructured network model (s) is fundamentally different (p<0.001). Similar geometric organization was observed in non-spatial behaviors such as REM sleep. Figure reproduced with permission from (Giusti et al. 2015)