Characteristic functional cores revealed by hyperbolic disc embedding and k-core percolation on resting-state fMRI

Abstract

Hyperbolic disc embedding and k-core percolation reveal the hierarchical structure of functional connectivity on resting-state fMRI (rsfMRI). Using 180 normal adults' rsfMRI data from the human connectome project database, we visualized inter-voxel relations by embedding voxels on the hyperbolic space using the [Formula: see text] model. We also conducted k-core percolation on 30 participants to investigate core voxels for each individual. It recursively peels the layer off, and this procedure leaves voxels embedded in the center of the hyperbolic disc. We used independent components to classify core voxels, and it revealed stereotypes of individuals such as visual network dominant, default mode network dominant, and distributed patterns. Characteristic core structures of resting-state brain connectivity of normal subjects disclosed the distributed or asymmetric contribution of voxels to the k_max-core, which suggests the hierarchical dominance of certain IC subnetworks characteristic of subgroups of individuals at rest.

Figure 1

Hyperbolic disc embedding and angular coherence of the voxels on the disc. (a) A brain network with 500 random nodes was displayed using 3-dimensional MNI coordinates projected on a 2-dimensional brain space. This visualization provides intricate edges and nodes that are not easily discernible. (b) Hyperbolic disc embedding provides easy-to-recognize visualization of the voxels on the hyperbolic disc. In this hyperbolic disc, 5937 voxels were used, which shows the intervoxel relationships between voxels in an unoverlapped way with 10 times more voxels than the one in (a). Specifically, the brain was resampled into 5937 6 × 6 × 6 mm³ voxels, which were assigned as voxels of subnetworks belonging to independent components (ICs). The hyperbolic distance between two voxels on this hyperbolic disc is equivalent to the correlation proximity between these voxels in Euclidean space. The radial coordinate responds to the degree of the voxel, i.e., the hub voxel is near the center of the disc, and the angular coordinate responds to the similarity of voxels. As an example, voxels from the independent component (IC) subnetworks are presented in different colors. The voxels from the salience network (SN) (red) are more widely distributed on angular coordinates than the default mode network (DMN) (blue) or visual network 4 (VN4) (green). (c) The angular coherence quantifies the degree of aggregation of a group of voxels based on coordinates. It ranges from 0 to 1, and a higher value indicates compact gathering with smaller differences in angles between voxels in the group. Widely spread voxels have a lower value of angular coherence. The angular coherences of voxels comprising (c) VN4 (ξ = 0.98), (c,d) DMN (ξ = 0.51) are shown.

Figure 2

Distribution of the angular coherences of 180 individuals’ voxels on hyperbolically embedded discs. Groups of voxels belonged to (a) functional labels derived from group independent component analysis (ICA) and (b) atlas-based anatomical labels. (a) Fifteen independent components were chosen from group ICA for the entire data: default mode network (DMN), anterior DMN (aDMN), precuneus network (PCN), salience network (SN), dorsal attention network (DAN), left central executive network (L CEN), right CEN (R CEN), sensorimotor network (SMN) 1/2, auditory network (AN), visual network (VN) 1/2/3/4, and visual attention network (VAN). The spatial maps of ICs were binarized (Z > 6), and voxels were classified to belong to each of the specific ICs. The coordinates of groups of voxels per specific IC were calculated (see “Methods” section). The values of angular coherence of SMN and VN3 and VN4 were the highest. (b) The whole brain was segmented into fifteen anatomical lobes based on the Brainnetome atlas to yield anatomical labels: bilateral frontal/temporal/parietal/limbic/occipital/subcortical and a cerebellum. The coordinates of groups of voxels per lobe were calculated (see “Methods” section). The median of each distribution is indicated with a circle, and the mean is indicated with a horizontal line.

Figure 3

Conceptual illustration of k-core percolation and plots describing k-cores and the k_max-core derived by k-core percolation. (a) k-core percolation renormalizes the brain network by peeling the layers with k-steps from k = 1 to k = k_max for the brain network. Intervoxel correlations were thresholded to yield an adjacency matrix after checking the scale freeness of the degree distribution of voxels and put into hyperbolic disc embedding and k-core percolation. The voxels with a degree equal to coreness k are eliminated, and recalculation of the voxels’ degree proceeds to the next step and continues until the remaining voxels forming the largest component at that step are disintegrated into many pieces at once. The voxels at this step k = max are called k_max-core voxels. (b) The k_max-core voxels included not only the voxels with the largest degree on the initial adjacency matrix but also the voxels with smaller degrees. This histogram shows the degree distribution of voxels from one subject (#100,206). The blue bins represent all the voxels, and the red bins represent k_max-core voxels. k_max was 240, and the degrees of k_max-core voxels ranged from 260 to 1088. The k-core percolation finds k_max-core voxels that have dense connectivity among themselves as well as hierarchically at the apex within their belonging independent components (ICs) and even the voxels with lower down to one-fourth of voxel with the highest degree. (c) A flag plot shows the changing k-cores of a subject that vary with the coreness k value during k-core percolation. Each voxel that belongs to a specific IC is shown on the y-axis, and the voxels comprising each k-core are colored. This subject has a k_max core with a 240 k-value and shows the first abrupt decrease during k-core percolation in DMN, DAN, CEN, and VN (k ≈ 156) and the second abrupt decrease in VN (k ≈ 172).

Figure 4

The k-cores and the k_max-core depicted by flag plots and hyperbolically embedded discs. Each individual has his/her own size of k_max-core and changes in the size of k-cores according to coreness k during k-core percolation. In individuals, a few abrupt decreases were observed over the gradual change of the largest component. (a) The coreness k and the size of the core S (k) of a subject were plotted, showing two abrupt changes. Specific k-cores that showed an abrupt decline (k = 7, 155, 156, 172, 173) were embedded on the hyperbolic discs to show the explosive decrease in core voxels. The voxels belonging to a k-core are denoted with red circles; otherwise, they are denoted with black circles on these hyperbolic discs. When the plot shows an abrupt decrease in S (k), voxels belonging to the k-core are reduced at once. (b) In an individual, k_max-core shows the various sizes and independent component (IC)-voxel compositions. The k_max-core (k = 240) of a subject is presented as an example. There were 694 voxels left on the k_max-core, and the voxels that belonged to the default mode network (DMN) were in blue, salience network (SN) in red, and visual network 3 (VN3) in green. Voxels other than k_max-core voxels are in pale circles. (c) The components of each k-core from one subject that vary with coreness k value are shown on the flag plot using functional IC labels (c) and anatomical labels (d) that annotate voxels to specific subnetworks. In the flag plot, every voxel is presented on the y-axis with labeling, and the horizontal bar of each voxel refers to the maximum k of k-cores to which the voxels belong. The voxels from each subnetwork on the y-axis were sorted in descending order of k. The bar plots show the affiliation of k_max-core voxels (e,f). This individual showed abrupt declines in k-core size in the DMN, dorsal attention network (DAN), central executive network (CEN), and VN by functional labels and in the frontal, temporal, parietal, and occipital lobes by anatomical labels. The k_max-core voxel was classified using larger functional labels (7 ICs) (e) and anatomical labels (8 lobes) (f). In this individual, VN or parietal/occipital lobe voxels belonged in the largest number to the k_max-core.

Figure 5

The plots show which subnetworks, in 30 individuals, the k_max-core voxels belong to. k-core percolation yielded k_max-core voxels for each individual and which independent components (ICs) or lobes those k_max-core voxels belonged. ICs were represented as 15 (a) or as the seven categorized (b). Five visual networks (VN1/2/3/4, VAN) into one visual network (VN), etc. Since some of the voxels belong to multiple ICs, categorized VN had slightly fewer voxels than the sum of the number of voxels of constituents (V1/2/3/4, VAN). (a) According to the fifteen functional labels, visual subnetworks, VN1, VN2, and VN3, and the visual attention network (VAN) were leading in the number of voxels among k_max-core voxels, followed by sensorimotor networks (SMN 1, 2) and the salience network (SN). (b) Once categorized, the propensity of VN among the seven was outstanding. (c) Anatomical labels for both lobes and cerebellum showed prominence of occipital and parietal lobes among the fifteen, and (d) once categorized to eight, parietal and occipital lobes were sustained.

Figure 6

The common core voxels were shared by 60% of individuals. k-core percolation disclosed to which IC subnetworks prevalently among individuals, the k_max-core voxels belonged. (a) A bar plot showing the affiliation of k_max-core voxels. Shared voxels of the precuneus network (PCN), visual network 1 (VN1), VN2, V3, and visual attention network (VAN) were easily found and rare voxels in salience (SN) and sensorimotor network 2 (SMN2). (b) The shared voxels on the template brain were visualized. Voxels of k_max-core are found in VN and PCN.

Figure 7

Three types of k_max-core voxel-independent component (IC) compositions at the end of k-core percolation. There were three types, named based on which ICs the k_max-core voxels belong: VN-dominant, DMN-dominant and distributed. Categorized functional labels, consisting of the default mode network (DMN), salience network (SN), dorsal attention network (DAN), central executive network (CEN), auditory network (AN), and visual network (VN), were used to classify k_max-core voxels according to their belonging to these categorized labels. In the top row, each individual’s k max core was embedded on the hyperbolic disc. The k_max-core was enlarged and shown in detail. In the middle, k_max-core was visualized on the 3-dimensional brain. At the bottom, the k_max-core voxels belonging to seven networks are shown separately. (a) Individuals with more than 40% of their k_max-core voxels in the DMN were classified as DMN-dominant. k_max-core voxels of one example (129,533) of the DMN-dominant type show blue regions indicating k_max-core voxels that belong to the DMN. More than 60% of k_max-core voxels were in DMN regions ranging over the precuneus, lateral parietal cortex, and medial prefrontal regions. (b) Individuals with more than 40% of k_max-core voxels being in VN were classified as VN-dominant. A VN-dominant individual (126,325) shows that more than 80% of k_max-core voxels belong to the VN, ranging over medial and lateral occipital and parietal regions. (c) Individuals were classified as having a distributed pattern when no dominant IC subnetworks were found. In an example case (110,411), every subnetwork voxel contributed to less than 20% of k_max-core voxels. We counted all duplicates when the k_max-core voxels belonged to multiple IC subnetworks.

Figure 8

The stacked histogram of the degree distribution of k_max-core voxels calculated from the adjacency matrix used as input. (a) After finding the k_max-core of a subject with k-core percolation, we read the degree of each k_max-core voxel on the adjacency matrix. We classified k_max-core voxels into seven categorical independent components (ICs) and produced a stacked histogram showing each k_max-core voxel’s affiliation and the voxel degree simultaneously. (b) We depicted k_max-core of 30 subjects. The voxel degrees of k_max-core voxels and their affiliation are in different colors. k_max-core voxels located on the right side of the histogram were deemed to have a greater degree initially in the adjacency matrix, indicating that they have connections with non-k_max-core voxels as well as within k_max-core voxels. In contrast, k_max-core voxels on the left side of the histogram denote a relatively smaller voxel degree, implying that it has fewer connections with non-k_max-core voxels and is thus almost confined to obtain connections with k_max-core voxels. The histograms of 30 subjects were sorted in ascending order with the mean degree of the k_max-core voxels in 3-dimensional space.