Topological inference on brain networks with application to lesion symptom mapping

Abstract

Persistent homology (PH) characterizes the shape of brain networks through persistence features. Group comparison of persistence features from brain networks can be challenging as they are inherently heterogeneous. A recent scale-space representation of persistence diagrams (PDs) through heat diffusion reparameterizes them using a finite number of Fourier coefficients with respect to the Laplace--Beltrami (LB) eigenfunction expansion of the domain, providing a powerful vectorized algebraic representation for group comparisons. In this study, we develop a transposition-based permutation test for comparing multiple groups of PDs using heat-diffusion estimates. We evaluate the empirical performance of the spectral transposition test in capturing within- and between-group similarity and dissimilarity under varying levels of topological noise and cycle location variability. In application, we propose a topological lesion symptom mapping (TLSM) method based on the proposed framework. The method is applied to resting-state functional brain networks of individuals with post-stroke aphasia to identify characteristic cycles associated with varying levels of speech-language impairment.

Figures (8)

• Figure 1: A 1-cycle emerges out of a Rips filtration constructed on a brain network and the corresponding persistence of the cycle is encoded in a persistence diagram (PD).
• Figure 2: Top half: Left Three: The evolving 1-skeleton of a 100-point point cloud sampled from a key shape with a cycle. Right: PD from the Rips filtration constructed on the 1-skeletons of the point cloud. The point in the PD that corresponds to the key cycle stands out with high persistence - much further away from the diagonal ($y = x$) line than the rest of the points. Bottom half: Heat kernel (HK) smoothing of the PD from the Rips filtration through Laplace-Beltrami (LB) eigenfunctions with respect to the bandwidths $\sigma = 0$ (original PD), 0.1, 1, 10. Top Row: Smoothed PDs. Bottom Row: Corresponding Fourier coefficients with respect to the LB eigenfunctions presented in matrix form.
• Figure 3: An example of a PD and PSS- and HK-estimated versions.
• Figure 4: Left: We randomly sampled 100 points from the image with an innate shape of a key. Top Right Row: Underlying key shape and possible locations of topological noise in the form of a small cycle. Bottom Right Row: Variants of the key shape in Group 2. They could appear in the 4 pre-specified forms or randomly out of the variants.
• Figure 5: An example of $n_1=n_2=n_3=5$ 100-point point clouds where the 100 points in each point cloud of the first two groups were generated randomly from the part of the rectangular image, whereas the 100 points in each point cloud of the third group were generated randomly with 95% of points from the shape of the key.