Role of inter-hemispheric connections in functional brain networks

Today the human brain can be modeled as a graph where nodes represent different regions and links stand for statistical interactions between their activities as recorded by different neuroimaging techniques. Empirical studies have lead to the hypothesis that brain functions rely on the coordination of a scattered mosaic of functionally specialized brain regions (modules or sub-networks), forming a web-like structure of coordinated assemblies (a network of networks. NoN). The study of brain dynamics would therefore benefit from an inspection of how functional sub-networks interact between them. In this paper, we model the brain as an interconnected system composed of two specific sub-networks, the left (L) and right (R) hemispheres, which compete with each other for centrality, a topological measure of importance in a networked system. Specifically, we considered functional scalp EEG networks (SEN) derived from high-density electroencephalographic (EEG) recordings and investigated how node centrality is shaped by interhemispheric connections. Our results show that the distribution of centrality strongly depends on the number of functional connections between hemispheres and the way these connections are distributed. Additionally, we investigated the consequences of node failure on hemispherical centrality, and showed how the abundance of inter-hemispheric links favors the functional balance of centrality distribution between the hemispheres.

: Numerical results from linear fits in both conditions and all bands. Here m and b are the statistics of the model y = mx + b with r 2 as coefficient of determination who measures how well the model predict the data (0 ≤ r 2 ≤ 1). Each macro columns represents results for each panel of Fig.S1.  Figure S1: Global and local centrality for all bands in EC and EO conditions. Global centrality u T is obtained from the complete matrix T , when all functional connections between hemispheres are maintained (horizontal axes); while local centrality u L,R is extracted from the hemisphere matrices L and R when hemispheres are disconnected (vertical axes). Frequency bands θ (blue), α (red), β (green), γ (yellow) are drew in triangle and circle markers for EC and EO conditions, respectively. Upper panel shows the EC condition for left (A) and right (B) hemispheres. Bottom panel: for left (C) and right (D) hemispheres in the EO condition. Table S1 for statistics of linear fits.
We also compute the centrality contrast and competition parameter for all bands in both conditions. Figure S2 shows the violin plots and means of these global features. We do not find statistical differences between each mean values and a zero mean distribution by means of Mann-Whitney U Tests. Summarized results are shown in Table S2. Slight differences in the hemispherical importance are observed when both the EC and EO conditions are compared. We observe positive values of centrality contrast in the left hemisphere during the EC condition for all bands. On the contrary, we report negative values of centrality contrast when individuals open their eyes, which indicates a slight imbalance of centrality between the hemispheres in this condition (see Fig. S2A and second column of Table S2).
Interestingly, results in Fig. S2B and Table S2) show that Ω L for α, β and γ are slightly negative. Since the competition parameter is defined taking the left hemisphere as reference, negative values of Ω L are consequence of an inter-hemispheric link distribution that slightly benefits the right hemisphere. Figure S2: Centrality contrast and Competition parameter. EC conditions (blue for centrality contrast and gray for Ω) and EO (orange for centrality contrast and yellow for Ω)) and four frequency bands: θ, α, β, γ. A. Average values of centrality contrast ( C L − C R ) over 54 subjects are indicated by the red circles. B. Average of the Competition Parameter Ω L for all subjects. According to values in each strategy: PP leads to C L max ≈ 0.7, the actual configuration gives C L = 0.49 and CC grants C R max ≈ 0.8. We obtain Ω L = −0.006, which reveals the real inter-hemispheric connectivity pattern as promoter of the centrality balance between both hemispheres.
A C B Figure S3: Example of reshuffling inter-hemispherical links. Three different configurations of the inter-hemispherical links. Intra-hemispherical connections are colored in grey, while blue is used for the inter-hemispherical links. Nodes' sizes and transparency are proportional to their local eigenvector centrality before connecting both hemispheres. The average α-EC network has an arbitrary threshold that maintains the 16% of the stronger links just for a better visualization purpose. A. Peripheral nodes are connected (PP strategy), resulting in optimal strategy for increasing left hemisphere's centrality. B. Actual distribution of inter-hemispheric connections, which leads to a balance of the centrality distribution. C. Right hemisphere's optimal strategy is obtained by connecting central nodes (CC strategy). Figure S4 shows the evolution of C L (note that C R = 1 − C L ) respect to the inter-hemispheric links. We distinguish between left-and right-dominant individuals based on the eigenvalue. The process of adding inter-hemispheric links shows a clear tendency: the hemisphere that initially has the "strongest" network (i.e., the higher λ 1 ) acquires a high amount of centrality when the number of inter-hemispherical links is low, but its centrality diminishes as the number of inter-links is increased. "Weak" hemispheres behave just in the opposite way.

Evolution of the left hemisphere centrality
We compute the largest eigenvalue λ 1 of L and R and call the "strong" ("weak") hemisphere that with the highest (lowest) λ 1 . We also distinguish between groups of people that are left-dominant when the eigenvalue λ 1 of L is higher than λ 1 of R, or right-dominant in the opposite case. Tab.S3 summarizes the percentage of each type of dominance according to the condition and frequency band.  Table S3: Hemisphere dominance is defined according to the largest eigenvalue λ 1 of the hemispheric connectivity matrix: the hemisphere with the highest λ 1 is the one that dominates over the other. Figure S4: Hemispherical centrality C L vs. the number of inter-hemispherical links for all subjects. Each subplot shows a different combination of a condition (EC or EO) and frequency band (θ, α, β and γ). Colors indicate whether the dominant hemisphere is the left (green) or the right (red). In each subplot, the region inside each curve includes the fifth and the 95th percentiles of C L and C R for all 54 subjects, with the average value plotted in dashed lines.

Robustness and resilience of all bands
We report the results of local impact vs local contribution in the three stages for all bands in both conditions. Tables with the slopes of linear fits between the local impact and local contribution are presented in Table S4 and S5. In Fig. S5 network damage is in the vertical axes and local contribution is in horizontal axes. The values of the three stages are shown for all bands and conditions.

Stage 3 Stage 2 Stage 1
Band  Band  Table S5: Correlation between the local impact and the local contribution (right hemisphere). Slopes of the linear fits of l R imp vs l R c for all bands, conditions and stages associated to Fig. S5 C y D. Note the increase of the slope from Stage 1 to Stage 3 as that described in the main text of the manuscript.  Figure S6 shows the topological distributions of averaged local impact nodes' positions ate three different stages. In this case, we focus only in α band. Local impact on centrality, clustering and shortest path Figures S7 and Fig. S8 show the behaviour of local impact on centrality, clustering ( l impc (i) ) and shortest path ( l imp d (i) ) respect to the local contribution l c (i) for all bands and both conditions. In both plots each condition, EC and EO, is represented with triangles and circles, respectively. Frequency bands are also differentiated: θ (blue), α (red), β (green) and γ (yellow). In contrast to the local impact on centrality, clustering and shortest path do not show important changes, no matter the stage considered.