Fig 1.
Eigenmodes of the network diffusion model: Brain network architecture determined by averaged connectomes from ten healthy adults (two scans each) using three different atlases: Desikan-Killiany, Destrieux, and functional (FXCN).
We then computed the network Laplacian which models diffusion through the brain as shown in Eq 1. Colored dots represent the value of the eigenmode at any given node of the graph, both positive and negative, and are overlayed on top of a green, glass brain shown in the coronal, sagittal, and axial plane. These eigenmodes represent the four slowest diffusion processes in the brain network where diffusion occurs quickly between similarly colored nodes, and slowly from bright red to dark blue (or vice versa as eigenmodes are identical under sign reversal).
Fig 2.
A white matter voxel’s importance for a given eigenmode is defined as the decrease in the corresponding eigenvalue when a 12mm-diameter lesion is placed at that voxel, damaging encompassed connections proportionally to the edge volume removed.
These importance maps were averaged from the MNI152 1mm-registered maps of ten healthy adults, two scans each. The brighter the voxel, the larger the corresponding decrease in eigenvalue. The X and Z coordinates of each taken axial and sagittal plane are shown for reference: increasing X denotes the sagittal plane moving from the right to left side of the brain, while increasing Z denote inferior to superior axial planes.
Fig 3.
The procedure used to generate the importance maps shown in Fig 2 was repeated while testing the effect of partial lesions.
Instead of simply weakening edges that pass through a lesion based on the volume of the edge removed, we modulated this edge loss by a “stoppage parameter” varying from (0,1) since it is possible for a region of damaged white matter to not fully stop streamlines from passing through it. Here, brighter colors represent that placing a lesion in that location caused a comparatively greater decrease in the second eigenvalue, weakening the rate of diffusion along the second eigenmode. Each column represents a different stoppage parameter, each row a different axial plane with the Z coordinate given in MNI152 1mm space. Here, increasing Z denote inferior to superior axial planes. For brevity, only the second eigenmode’s importance map is shown as the same geographical areas are highlighted regardless of this stoppage parameter, indicating that our analysis is robust and reliable under different interpretations of white matter damage.
Fig 4.
A-C) The average angle variance of the eigenvectors from normal volunteers were calculated along with 95% confidence bounds and are shown in blue, with higher variance indicating less reliability in the eigenvector of interest. For comparison, the average angle variance of eigenvectors for randomly generated networks are shown in red. Note that for the second and third eigenvector, the average angle variance is significantly lower than the average angle between eigenvectors from random networks, whereas the same does not hold true for the fourth eigenvector in the FXCN atlas. D) The distribution of the eigenvalues across all healthy subjects is shown in blue, with the distribution of eigenvalues in random networks shown in red. The eigenvalues of ten healthy adults were calculated individually using brain graphs averaged from two MRI scans from each subject and were then normalized by their mean. The resulting eigen-spectra were then averaged across all ten subjects and its 95% confidence interval was plotted as error bars. Eigenmodes are numbered in increasing order of eigenvalues. Insets for the lowest and highest eigenmodes were added for clarity. Statistical significance marked as * was determined via two-sample t-tests with an α of 0.05. E-F) The second eigenmode for two randomly generated brains are shown for comparison.
Fig 5.
Reliability of mean importance for the second, third, and fourth eigenmodes.
The heights of the bars show the mean importance in the white matter tract of interest and the error-bars show the standard deviation. The red points indicate ICC. The importance of a white matter voxel with respect to an eigenmode is defined as the decrease in the corresponding eigenvalue when a lesion of diameter 12 millimeters, centered on the voxel of interest, is cut, damaging all connections that pass through the lesion proportionally to the volume of the edge removed by the lesion. Here the labels on the x-axis are the abbreviations for the white matter tracts defined in the Johns Hopkins University (JHU) DTI-based white matter atlas [27].
Fig 6.
A) The Pearson’s correlation coefficient was calculated between the eigenmode importance maps and edge density maps from [15, 16]. Each edge density map (rich club, feeder, and local connections) represent the density of white matter connections throughout the brain and were calculated on the same group of healthy adult subjects used to generate the averaged importance maps shown in Fig 2. Each edge density map was then averaged across all subject and then the Pearson correlation of white matter voxels was taken between each density map and the averaged eigenmode importance maps in Fig 2. The error bars reflect the 95% confidence intervals of the Pearson’s correlation coefficient. B) The Pearson’s correlation coefficient was calculated between each averaged eigenmode importance map and shown as a heatmap where brighter squares indicate a higher correlation between the pair of eigenmodes shown on the x and y axes.
Fig 7.
Effect of corpus callosum on connectome eigenmodes.
Top) brain network architecture determined by averaged connectomes from eleven healthy adults for the control, seven cases of complete agenesis of corpus callosum (AgCC), and the same eleven healthy adults with virtual callosotomies. Note how the second eigenmode in all three cases represents diffusion between both hemispheres, albeit more polarized in cases of AgCC and virtual callosotomies, due to the lack of an intact corpus callosum. For higher eigenmodes, we see that each eigenmode in the control group is split into two separate modes in the AgCC or virtual callosotomy cases, one for each hemisphere. Intuitively, the corpus callosum synchronizes diffusion across both hemispheres, and its absence allows transmission to occur within one hemisphere with no effect on the other which is reflected in the changed eigenmodes. Bottom) the eigenvalue corresponding to the speed of diffusion in the second eigenmode (left-right communication) is shown along with 95% confidence bounds. All differences in eigenvalues are statistically significant.