Robustness of connectome harmonics to local gray matter and long-range white matter connectivity changes

Abstract

Recently, it has been proposed that the harmonic patterns emerging from the brain's structural connectivity underlie the resting state networks of the human brain. These harmonic patterns, termed connectome harmonics, are estimated as the Laplace eigenfunctions of the combined gray and white matters connectivity matrices and yield a connectome-specific extension of the well-known Fourier basis. However, it remains unclear how topological properties of the combined connectomes constrain the precise shape of the connectome harmonics and their relationships to the resting state networks. Here, we systematically study how alterations of the local and long-range connectivity matrices affect the spatial patterns of connectome harmonics. Specifically, the proportion of local gray matter homogeneous connectivity versus long-range white-matter heterogeneous connectivity is varied by means of weight-based matrix thresholding, distance-based matrix trimming, and several types of matrix randomizations. We demonstrate that the proportion of local gray matter connections plays a crucial role for the emergence of wide-spread, functionally meaningful, and originally published connectome harmonic patterns. This finding is robust for several different cortical surface templates, mesh resolutions, or widths of the local diffusion kernel. Finally, using the connectome harmonic framework, we also provide a proof-of-concept for how targeted structural changes such as the atrophy of inter-hemispheric callosal fibers and gray matter alterations may predict functional deficits associated with neurodegenerative conditions.

Fig. 1

Overview of the workflow for the construction of the connectome harmonics. Local connectivity from cortical surface mesh (bottom left) and long-range connections from tractography (top left) are combined in a high-resolution structural connectome (middle), from which a graph Laplacian L is computed based on the adjacency (A) and the degree (D) matrices of the combined connectivities. Connectome harmonics (right) are the eigenvectors of the graph Laplacian.

Fig. 2

Weight-based thresholding of long-range white matter connections increases MI between connectome harmonics and the default mode network (DMN). (A) Proportion r of local gray matter connections in the adjacency matrix A of the graph Laplacian, for different threshold value z_C applied to the long-range white matter connectivity. (B) Distribution of connectome weights (number of white matter streamlines W between vertices) in log-log scale, described as probability given that the weight is positive (P(W|W > 0)). Vertical lines correspond to $z_C$ values with same color code as A). (C-D) Mutual information (MI) between the first 100 connectome harmonics and the DMN for a range of z_C resulting in a range of proportion r of local connections. Color code is consistent across panels. (See Suppl. Figure 3 for MI with other RSNs).

Fig. 3

Mutual Information (MI) between connectome harmonics and the DMN for several randomized versions of long-range white matter connectivity. Mean and standard deviation of connectome harmonics’ ${\psi }_{k\in K=\{7,8,9,10,11\}}$ MIs for different proportions of local to long-range connections (parameterized by the adjacency weight threshold $z_C$ as in Figure 2). Different types of connectome surrogates are shown: original (no randomization), inter (interhemispheric only), intra (intrahemispheric only), inter+intra (inter and intrahemispheric separately), and global (inter and intrahemisphere combined), see Methods). ⋆ indicates $p<0.05$ of Monte-Carlo statistical test of ${\psi }_{k\in K=\{7,8,9,10,11\}}$ MI values with the DMN for original harmonics vs. surrogate data MI distributions for $z_C=1$ of the different types of randomizations. MI with other RSNs for those randomizations are shown in Suppl. Figure 9.

Fig. 4

Effect of different white matter fiber lengths on the emergence of functional harmonics patterns. Mutual information (MI) between the DMN and connectome harmonics’ ${\psi }_{k\in K=\{7,8,9,10,11\}}$ for different percentages of trimming of the white matter connectivity. Trimming was performed by eliminating longest tracks first (A), shortest tracks first (B), and in random order (C). Baseline connectome used for $z_C=1$ corresponding to r ≃ 0.7 before trimming.

Fig. 5

Mutual Information between connectome harmonics and the DMN for gradual removal of inter-hemispheric white matter connectivity. Mutual information (MI) between DMN and harmonics ${\psi }_{k\in K=\{7,8,9,10,11\}}$ for gradual trimming of inter-hemispheric connections (termed callosectomy) by descending (A), ascending (B) orders of track length, and randomly (C). Adjacency weight threshold is set to $z_C=1$, corresponding to the proportion r ≃ 0.7 of local connections before alteration. ⋆ indicates $p<0.05$ of Monte-Carlo statistical test between distributions of MIs using 100 samples. Note that we used a maximum callosectomy of 99% in order to avoid totally disconnected hemispheres.

Fig. 6

Low frequency harmonics are robust to cortical surface changes. High-resolution connectome and connectome harmonics are recomputed for different smoothing levels of WMGM cortical surface (left), or pial surface (right) and compared to original WMGM surface mesh using Pearson correlation in atlas space for the first 100 harmonics. Adjacency weight threshold is set to $z_C=1$, corresponding to a proportion r ≃ 0.7 of local connections.

Fig. 7

Low frequency harmonics are robust to mesh resolution changes. Region-wise correlation of connectome harmonics ${\psi }_{k\in K=\{1,\dots ,100\}}$ using fsaverage4 (5,124 vertices) and fsaverage5 (20,484 vertices) cortical surface mesh templates from FreeSurfer. Proportion r of local connections is indicated by different degrees of connectome adjacency weight threshold $z_C$. Insets show a magnified version of the correlation matrix for connectome harmonics ${\psi }_{k\in K=\{1,\dots ,20\}}$.

Fig. 8

Low frequency harmonics are robust to local diffusion kernel width changes. Vertex-wise correlation of connectome harmonics ${\psi }_{k\in K=\{1,\dots ,100\}}$ using cvs_avg35 template decimated to 20,000 vertices, using ${\Lambda }_s=1$ (Λ₁) vs. ${\Lambda }_s=2$ (Λ₂) neighboring vertices as local connectivity kernel. Proportion r of local connections for each kernel sizes (separated by a dash ${\Lambda }_1-{\Lambda }_2$) is indicated by different degrees of connectome adjacency weight threshold $z_{{}_C}$. Insets show a magnified version of the correlation matrix for connectome harmonics ${\psi }_{k\in K=\{1,\dots ,20\}}$.

Fig. 9

Mutual Information between connectome harmonics and the DMN for gradual disruptions of local gray matter connectivity. Mutual information (MI) between DMN and harmonics ${\psi }_{k\in K=\{7,8,9,10,11\}}$ for gradual removal of local gray matter connections (termed anisotropy) by descending (A), ascending (B) order of edge length and in random order (C). Adjacency weight threshold is set to $z_C=1$ corresponding to a proportion r ≃ 0.7 of local connections before alteration. ⋆ indicates $p<0.05$ of Monte-Carlo statistical test between distributions of MIs using 100 samples.