The multiscale self-similarity of the weighted human brain connectome

Abstract

Anatomical connectivity between different brain regions can be mapped to a network representation, the connectome, where the intensities of the links, the weights, influence resilience and functional processes. Yet, many features associated with these weights are not fully understood, particularly their multiscale organization. In this paper, we elucidate the architecture of weights, including weak ties, in multiscale human brain connectomes reconstructed from empirical data. Our findings reveal multiscale self-similarity, including the ordering of weak ties, in every individual connectome and group representative. This phenomenon is captured by a renormalization technique based on a geometric network model that replicates the observed structure of connectomes across all length scales, using the same connectivity law and weighting function for both weak and strong ties. The observed symmetry represents a signature of criticality in the weighted connectivity of the human brain and raises important questions for future research, such as the existence of symmetry breaking at some scale or whether it is preserved in cases of neurodegeneration or psychiatric disorder.

Fig 1. Self-similarity of weighted human brain connectomes across scales. A) Scheme of the anatomically hierarchical reconstruction of the multiscale connectomes. The sketch of the brain has been adapted from “Wisdom Memory Anatomical Intelligence" from ArtDraw.org, under license CC0 1.0 Universal. B, F) Complementary cumulative weight distribution across layers for UL Human 14 and the group-representative, respectively. C, G) Complementary cumulative strength distribution. D, H) Strength-degree relationship. We have computed the average strength for nodes in each degree class. The exponent of the relationship between the two magnitudes in layer l = 0 is displayed in the bottom right corner. E, I) Disparity of weights of connections around nodes. The straight line corresponds to the completely heterogeneous scenario and the orange area to the expectation for a random null model that distributes the strength of nodes locally uniformly at random.

Fig 2. Weak-ties spectra. A) Scheme of the filtering procedure and definition of the weak ties statistic. The filters select a percentage of connections with the lowest weight or confidence, respectively. B, E) Normalized density of intermodular connections versus percentage of edges remaining in the subgraphs filtered by thresholding weights. X-axis in log scale. Results for UL Human 14 and the group-representative, respectively. C, F) Same as B and E, where subgraphs are obtained by thresholding confidence 1 − α. D, G) Average weight of intramodular links compared with average weight of intermodular links.

Fig 3. The WS1 model for l = 0 of UL Human 14. Red dots correspond to the empirical data using 30 bins (20 bins in the inset). The blue lines correspond to the average value obtained from 100 synthetic networks generated with the model, and the pink regions show the standard deviation around the expected value. A) Complementary cumulative weight distribution. B) Complementary cumulative strength distribution, and average weight by degree class in the inset. C) Strength-degree relationship. D) Normalized density of intermodular connections. E) Disparity in weights around nodes. F) Topological properties: complementary cumulative degree distribution, and degree-dependent clustering coefficient in the inset. G) Louvain modules in hyperbolic space. H) Louvain modules in a dorsal view of the brain from above (in Euclidean space).

Fig 4. Geometric renormalization of weighted connectomes. A) Figure adapted from Ref. [18]. Each scaled layer is obtained after a GRW step with resolution r starting from the embedding of the original connectome at l = 0. Each ROI i in blue has an angular position on the similarity space and its size is proportional to the logarithm of its hidden degree. Straight solid lines represent the links in each layer with weights denoted by their thickness. Coarse-graining blocks correspond to the orange shadowed areas, and dashed blue lines connect ROIs to their supernodes in layer l + 1. Two supernodes in layer l + 1 are connected if some ROI of one supernode in layer l is connected to some ROI of the other, with the supremum of the weights of the links between the constituent ROIs serving as the weight of the new connection. Note that the transformation with r = 4 goes from l = 0 to l = 2 in a single step due to the semigroup property. B–G) Self-similarity of UL connectome 14 along the GRW flow. Comparison of values obtained from the empirical data (points) and the model (solid lines). B) Complementary cumulative weight distributions. C) Complementary cumulative strength distributions. Inset: strength-degree relationship, where the average strength is shown for each degree class. D) Disparity in weights around nodes. E) Normalized density of intermodular connections thresholding by weight, X-axis in log scale. F) Normalized density of intermodular connections thresholding by confidence, X-axis in log scale. G) Average weight of intramodular links compared to average weight of intermodular links. In B, C and G weights, strengths, and degrees are rescaled by the average weight, average strength, or average degree of the respective layer.

Fig 5. Comparison of the MHW connectome of UL Subject 14 with random surrogates. Left column: CP-WR. Right column: CR-GRW. In all the graphs, the dots correspond to empirical data and the curves to the random surrogates. A, B) Complementary cumulative weight distributions. Inset in B: complementary cumulative degree distributions. C, D) Complementary cumulative strength distributions. Inset in C: exponent of the s∼kη relationship. E, F) Normalized density of intermodular connections versus percent of edges considered by global thresholding. The standard deviation of the null model ensemble is represented by the shadowed area. In the CR-GRW, weight and strength have been rescaled by the average weight and strength of the corresponding layer.