Hyperbolic disc embedding of functional human brain connectomes using resting state fMRI

Abstract

The brain presents a real complex network of modular, small-world, and hierarchical nature, which are features of non-Euclidean geometry. Using resting-state functional magnetic resonance imaging (rs-fMRI), we constructed a scale-free binary graph for each subject, using internodal time-series correlation of regions-of-interest (ROIs) as a proximity measure. The resulted network could be embedded onto manifolds of various curvature and dimensions. While maintaining the fidelity of embedding (low distortion, high mean average precision), functional brain networks were found to be best represented in the hyperbolic disc. Using a popularity-similarity optimization model (PSOM) on the hyperbolic plane, we reduced the dimension of the network into 2-D hyperbolic space and were able to efficiently visualize the internodal connections of the brain, preserving proximity as distances and angles on the PSOM discs. Each individual PSOM disc revealed decentralized nature of information flow and anatomic relevance. Using the hyperbolic distance on the PSOM disc, we could detect the anomaly of network in autistic spectrum disorder (ASD) subjects. This procedure of embedding grants us a reliable new framework for studying functional brain networks and the possibility of detecting anomalies of the network in the hyperbolic disc on an individual scale.

Characteristic core voxels in normal individuals revealed by hyperbolic disc embedding and k-core percolation on resting state fMRI

“…To visualize the correlation structure of the voxels composition of the complex functional brain network, we adopted a method to transfer the high-dimensional connection (edge) information to the hyperbolic disc space. According to our previous investigation [34] having looked for an optimal non-Euclidean space for embedding the inter-voxel correlation structure, we simply chose 2-dimensional hyperbolic disc embedding. Hyperbolic disc representation reflected the original high-dimensional edge information similarly well to the high dimensional Euclidean embedding alternatives with regard to fidelity and reproducibility [34].…”

“…These networks were binarized after confirming the linearity on a log-log plot of the degree distribution of the output adjacency matrix, and the largest components of the network of having at least 80% of the entire 5,937 voxels were embedded on the hyperbolic discs using the previously described method [34]. Embedding was done on the hyperbolic disc using ॺ 1 / 2 model according to the methods previously reported [31,34]. BOLD time series of voxel pairs were used to calculate inter-voxel correlation and after confirming scale-freeness on the degree distribution, the correlation matrix was thresholded to yield the adjacency matrix.…”

“…The correlation coefficients between two voxels were calculated to define the edges of the network. These networks were binarized after confirming the linearity on a log-log plot of the degree distribution of the output adjacency matrix, and the largest components of the network of having at least 80% of the entire 5,937 voxels were embedded on the hyperbolic discs using the previously described method [34]. Embedding was done on the hyperbolic disc using ॺ 1 / ℍ 2 model according to the methods previously reported [31,34].…”

“…The resulting binary graph from each subject was mapped onto a hyperbolic disc, using [31,34]. Connectomes were assumed to exist in underlying geometric space, linked with the observed topologic properties through a law of connection probability that defined the likelihood if the two regions of the brain were linked.…”

“…We evaluated the size and degree distribution of this k max -core for ICs and lobes. independent components (ICs) [34]. The hyperbolic distance between two voxels on this hyperbolic disc is equivalent to the correlation proximity between these voxels on the Euclidean space.…”