Local and long-range connectivities can be altered by changing parameters at several stages of the framework and are summarized in Table 1. At early stages, smoothing of the cortical surface mesh (fi) and the number of streamlines to retain from tractography are important parameters to consider in building connectivity matrices (we used by default all available streamlines). At later stages, different parameter selections are possible for creating adjacency matrices from weighted connectivity matrices by thresholding (zC), for trimming (η, κ) of the long-range white matter connections, and for the diffusion kernel width (Λs), mesh resolution (nv), and anisotropy (ρ) of the local gray matter connections. These parameters influence the outcome of the framework by modifying the graph structure, the proportion of local or long-range connections in the combined connectome used for computing harmonics, or selectively removing specific connections or connectivity patterns.

Summary table of parameters.

Default parameter values were chosen based on a preliminary parameter space exploration so that harmonics could be generated most reliably while varying a subset of other parameters. Mapping of the Default Mode Network (DMN) to the Desikan-Killiany atlas is provided in Suppl. Table 1.

Cortical surface mesh smoothing. Cortical surface meshes are often averaged or smoothed in studies looking at averaged brain properties (Fischl et al., 1999). Here, smoothing was performed to soften the strong curvature of the surface prior to computing the connectome by the surface-tracts intersection routine (see Suppl. Figure 2). Because linearly extended tracts can terminate obliquely or not cross the mesh, this routine can result in a reassignment of the track bound to a different cortical surface node, or the track not being assigned at all, for computing the brain wide connectivity matrix. Here when indicated, cortical surface meshes from Freesurfer’s templates were smoothed using the smoothPatch function from an open source MATLAB toolbox (https://www.mathworks.com/matlabcentral/fileexchange/26710-smooth-triangulated-mesh) using the Laplacian smoothing with inverse vertice-distance based weights and by varying the smoothing coefficient fi. The numbers of smoothing iterations were taken from the Fibonacci sequence, up to 89 iterations, and visually inspected.

Local diffusion kernel width and mesh resolution.We applied two widths Λs of diffusion kernels over the cortical surface to compute the local gray matter connectivity matrix (Atasoy et al., 2016). Widths Λs comprised either one or two nearest neighbors, each resulting in a different proportion of local to long-range connection when the number of long-range white matter connections are fixed. If instead the proportion of local to long-range connections was held constant, a larger local diffusion kernel width allows to increase the number of long-range connections incorporated to the combined connectome. For example, a local:long-range proportion of 1: 1 with a local kernel of only immediate neighbors resulted in a combined connectome comprising around 100,000 long-range connections. The same 1: 1 proportion with a local kernel spanning immediate neighbors and their neighbors resulted in a combined connectome comprising around  400,000 long-range connections. The default local kernel width chosen throughout the study was two, but the results do not change with only direct neighbor connections (Fig. 8). To vary the mesh resolution, we used a mesh with 20,484 vertices (fsaverage5) and a coarser version of 5,124 vertices (fsaverage4). Changes in mesh resolution affect the position of vertices and can result in a different vertex attribution of track bounds or in discarded tracks.

Low frequency harmonics are robust to local diffusion kernel width changes. Vertex-wise correlation of connectome harmonics ψkK={1,,100} using cvs_avg35 template decimated to 20,000 vertices, using Λs=11) vs. Λs=22) neighboring vertices as local connectivity kernel. Proportion r of local connections for each kernel sizes (separated by a dash Λ1Λ2) is indicated by different degrees of connectome adjacency weight threshold zC. Insets show a magnified version of the correlation matrix for connectome harmonics ψkK={1,,20}.

Anisotropy.We introduce the removal of cortical surface mesh edges in a process termed anisotropy, whereby mesh edges are removed either randomly (with probability ρ), by ascending order, or by descending order of edge lengths. A visualization of the resulting graphs structure for gradual changes to ρ is provided in Suppl. Fig. 10 for illustrative purpose.

We performed two distinct operations to alter the long-range connectivity: thresholding, which is based on the number of tracks between vertices (weight-based), and trimming, which is based on the average track length between vertices (distance-based).

Long-range connectivity thresholding.For thresholding, the z-scored weighted long range connectivity matrix Cz (Eq. 4) was binarized according to an adjacency weight threshold value zC. The ratio r of local versus long-range connections is defined as r=tr(A)/(tr(A)+tr(Ac)), which is the number of local connections divided by the summed numbers of local and long-range connections. Because the local gray matter connectivity matrix is determined by the cortical surface mesh and the diffusion kernel, we kept it constant (unless otherwise mentioned) and varied the white matter connectivity threshold zC, which thereby controls the proportion r of local connections. As weights represent the number of streamlines connecting distant nodes, when zC increases, only the most prominent tracks of the white matter remain.

Long-range connectivity trimming.Trimming was simulated by removing some percentage η of the long range connectivity entries based on their average track length. Different scenarios were implemented: 1) removing the longest tracks first; 2) removing the shortest tracks first; and 3) removing tracks in random order. Note that we applied trimming to the long-range white matter connections after applying the threshold zC=1, thereby setting the proportion of local connections to r ≃ 0.7. Thus, the reported percentage of trimming affected only the remaining long-range white matter connections after thresholding.

Callosectomy.Finally, we introduce the removal of inter-hemispheric connections in a process termed callosectomy, whereby inter-hemispheric connections are removed either randomly (with probability κ) or by descending order of track lengths. A visualization of the resulting graphs structure for gradual values of κ is provided in Suppl. Figure 10 for illustrative purpose.

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