Topological Data Analysis Reveals Robust Alterations in the Whole-Brain and Frontal Lobe Functional Connectomes in Attention-Deficit/Hyperactivity Disorder

The Betti-0 curve. A, Two-dimensional toy example: a set of 15 nodes, four filtration values ε, represented as the circle diameter and their corresponding Betti-0 (B₀). B, Betti-0 curve for a hypothetical brain network; each point in the curve represents the B₀ for each filtration value. In both cases, at ε = 0 the number of components is equal to the number of nodes, n. As the filtration value increases, the number of components reduces, and eventually will reach a single one containing all nodes. Brain views generated with brain-net (Xia et al., 2013), r stands for Pearson’s correlation.

KCC between brain parcellations for the explored properties of the B₀ curves: AUC, slope and kurtosis. KCC value is depicted in yellow-red, with p(KCC > 0.59) < 0.05, given two raters and 176 degrees of freedom.

B₀ curves for each group and brain parcellation. Group average with 95% confidence interval (CI) of the B₀ curves. Forest plot of OR with 95% CI for the logistic regression terms of AUC, kurtosis (K), and slope (S) are depicted for each brain atlas: AAL (top left), CC200 (top right), P264 (bottom left), and CC400 (bottom right). Gray dashed line stands for OR equal to one.

Edges with differences in the proportion of subjects between groups at ε = 0.35 (A) and ε = 0.5, 0.75, and 1 (B). Nodes from each lobe (AAL atlas) are represented with different colors in the chord diagrams. Only edges with a proportion difference at p < 0.01 (uncorrected) are depicted. For ε = 0.35 the edges are represented in the brain using brain-net (Xia et al., 2013). R stands for the right side of the brain.