Theta phase precession reduces the duration of spurious loops in the simplicial complex.

<p>(<b>A</b>) The point clouds depict the core of the learning region, as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003651#pcbi-1003651-g002" target="_blank"><b>Figure 2B</b></a>, for the sake of legibility. In both the <i>θ</i>-off and <i>θ</i>-on cases, note the red circles at the lower boundary of L, indicating longer-persisting loops. (<b>B</b>) Histograms showing the distributions of spurious loop durations in the <i>θ</i>-off and <i>θ</i>-on cases. We performed the Kolmogorov-Smirnov (KS) test to statistically compare the five cases against one another: 0 = no theta, 1 = a single 8 Hz sinusoidal wave, 4 = a combination of four sinusoids, M = a subcortical EEG signal from a wild-type mouse, and R = a subcortical EEG signal recorded from a wild-type rat. Black squares indicate a significant difference (<i>p</i><0.05) in loop duration between <i>θ</i>-off and all other cases; the <i>p</i>-values for pairwise comparisons between different <i>θ</i>-on cases reveal no significant difference (gray squares, <i>p</i>>0.2). The statistical similarity of all the <i>θ</i>-on distributions enabled us to combine the <i>θ</i>-on loop duration data to obtain better statistics for the histogram. The structure of the histograms is fit by the gamma distribution (blue lines). We used the smooth histogram profile (dashed line) to show that the typical duration of the topological loops in the <i>θ</i>-on cases is about half that of the <i>θ</i>-off case. (<b>C</b>) The typical number of loops in the <i>θ</i>-on cases is ∼40% higher than in the absence of <i>θ</i>-precession. The <i>θ</i>-off histogram is fit by the GEV distribution (red line), while the <i>θ</i>-on histogram is better fit by gamma distribution (blue). The KS diagram on the right shows that while the simulated <i>θ</i>-oscillations produce similar results, the signals recorded from rat and mouse produce statistically different distributions from the simulated signals. (<b>D</b>) The maximal number of spurious loops observed in the <i>θ</i>-on cases, fit by the gamma distributions, is less than half that in the <i>θ</i>-off case.