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Multiscale graph theoretic framework

Procedure

A graph theoretic framework (Bullmore and Sporns, 2009, 2012; Sporns, 2010) was used to evaluate connectivity between brain regions. We construct a graph Gw = (V,E) in time window w, where V is the set of M nodes in the graph that represent brain electrodes, E is the set of edges that denote connections between the nodes. The $(u,v)th$ element of the adjacency matrix $aw(u,v)$ is given by the directed edge metric from node u to node v, defined as the change in DI from the baseline window. Thus, $aw(u,v)=I~w(u1N→v1N)−I~b(u1N→v1N)$, where b represents the baseline window and N the length of the time window. The edge metric captures the “change in DI from baseline” and represents the changes occurring because of the task dynamics. For each of the seven patients, a total of W(=52) brain graphs were obtained from 52 time windows (11 SA and 41 AA windows). To control for spurious edges, the adjacency matrices were thresholded using a combination of density thresholding and global thresholding techniques (van Wijk et al., 2010). Density thresholding for an adjacency matrix is a technique where the threshold is chosen such that the resulting network has a certain density of edges. Global thresholding for a group of graphs uses a single threshold value (T), and retains all the edges greater than the common threshold. In this work, density thresholding is done in one time window, to determine what T should be, and is then applied globally for all the time windows. Density thresholding was done on a graph corresponding to the window before articulation (AA: –256 to 0 ms), by retaining only the top 5% of the positive increases in DI values, among all pairs of nodes. The cutoff value determined the patient-specific threshold (T), which was then globally applied to all the remaining W – 1 windows. The window AA: –256 to 0 ms was chosen because of its high density of connections, to be very stringent and have a high value of threshold T. The results were found to be largely independent of the window chosen for density thresholding, as using a different window only changed the value of T slightly, which affected a few individual connections, but it was not sufficient to change the subsequent graph theoretical metrics. After thresholding, the elements of the adjacency matrix Aw(u,v) for window w are given by

and the edges now represent “increase in DI from baseline.” This thresholding technique retains values of DI that are much greater than baseline, and discards “decreases in DI from baseline.” This makes the adjacency matrices positive, allowing for the use and interpretation of most graph processing techniques. The final results are carefully interpreted within the framework of networks built out of increased information flow among brain regions. For each patient, the resulting time varying graphs were analyzed using a multiscale analysis procedure (Fig. 3) using metrics that shed light on the underlying process at various scales, coarse, intermediate, and fine (Fig. 4).

A, Coreness of a node is a measure of the node’s centrality in the network. While typically, a higher coreness value corresponds to a higher node degree, it is not always the case. For example, both nodes P and Q have a degree of 6, yet node P has a coreness value of 1, while node Q has a coreness value of 3. B, Louvain communities. C, Fine scale network measures given by in-degrees and out-degrees.

A, DI matrix in a window; x- and y-axes are the electrodes and the $(i,j)th$ element represents DI from the $jth$ to the $ith$ electrode. B, The DI matrix relative to the baseline (“change in DI” matrix) is obtained by subtracting the baseline DI matrix from the DI matrix of that window. C, The thresholded DI matrix, which only retains the top positive increases in DI, based on a threshold value T (described in Materials and Methods). D, W =52 “increase in DI” matrices were generated per patient. E, Multiscale graph analysis.

A patient with M electrodes has W thresholded adjacency matrices Aw, for each window w, of size M × M. The connection density of each matrix defined as the ratio of the number of connections (non-zero values) in the adjacency matrix, to the total number of possible connections M2-M. The results of one patient’s connection density versus W time windows is shown in Figure 5A.

A, Connection densities vary smoothly across time windows, with a local maximum occurring before articulation. B, Coreness of nodes heatmap identifies sets of nodes in the brain related to changes in connection density. The average coreness across nodes (shown in red below the heatmap), provides the same information as the coarse scale metric (in blue). C, The first row shows the maximal k-core network in windows where peaks were found in the coarse metric. Connections are shown in black lines. The max k-core network is a very strongly interconnected core of the graph. The second row shows the results of the Louvain analysis. The colors of directed lines only indicate that the nodes belong to the same community. Significant communities, Bonferroni corrected p <0.05 are shown.

K-cores analysis (Hagmann et al., 2008; Modha and Singh, 2010) is an intermediate scale graph theoretic metric, calculated using directed binarized graphs (Rubinov and Sporns, 2010a,b). K-cores of a graph are a set of connected components that remain, after all vertices of degree less than k have been removed, in an iterative manner. Coreness of node quantifies the highest k-core network a given node belongs to (Fig. 3A). The coreness values of the nodes were evaluated as follows (Shin et al., 2016): First, the 1-core network was identified by finding the isolated 0-degree nodes of the graph. These nodes were given a coreness value of 0, and then deleted from the network to reveal the 1-core network. Next, the 2-core network was identified, and the nodes deleted at this step were given the coreness value of 1. This process was repeated until every node was given a coreness value, until the largest k-core subnetwork of each graph Gw was found. The coreness of all nodes of a patient versus time windows has been plotted as a heatmap, as shown in Figure 5B. This revealed which set of nodes were involved in highly connected subnetworks, and in which time windows this occurred.

The Louvain algorithm (Reichardt and Bornholdt, 2006; Blondel et al., 2008; Ronhovde and Nussinov, 2009; Sun et al., 2009) is a fast, heuristic, agglomerative community detection algorithm, that finds the optimal partition structure of the nodes into communities, by maximizing a measure of partition quality; the modularity index Q (Newman and Girvan, 2004), example in Figure 3B. The Louvain algorithm in this work used an adapted modularity index suitable for directed weighted networks (Rubinov and Sporns, 2010a,b, 2011). It uses a greedy optimization phase which randomly selects the starting node, leads to an inherent variability of the Louvain algorithm. To overcome this, consensus clustering was done (Rubinov and Sporns, 2011; Lancichinetti and Fortunato, 2012; Dwyer et al., 2014), where the algorithm was run R = 100 times, and an R × R module allegiance matrix (Bassett et al., 2013, 2015) was created. The community detection algorithm was then run for a second time on this module allegiance matrix, which revealed the most robust partition of the data (Lancichinetti and Fortunato, 2012). The number of communities in each graph was determined by the output of the algorithm. The Louvain communities in three windows for a patient are shown in Figure 5C.

For a node $v∈V$, in-degrees $Inw(v)$ and out-degrees $Outw(v)$ provide a fine grained view of the graph structure, for time windows $w=1,2,. . .,W$. For weighted, directed networks, in-degrees of a node is defined as the sum of the weights of the edges entering that node. $Inw(v)=∑u∈VAw(v,u)$. Similarly, out-degrees of a node is defined as the sum of the weights of the edges leaving that node. $Outw(v)=∑u∈VAw(u,v)$. In-degrees is a measure of the sink strength of the nodes, while out-degrees measures source strength (Fig. 3C). The significant correlations between in-degrees and power, and out-degrees and power for all patients are shown in Figure 6 and Figure 7.

A, An inf-frontal gyrus pars opercularis electrode from patient 1, that shows positive correlation between hγ power and the network features. B, A negatively correlated orbital frontal cortex from patient 6. The coreness of nodes can be seen as a combined effect of in and out degrees. A detailed figure that shows the correlation coefficient of in/out degrees with other frequency bands is shown in Extended Data Figures 6-1, 6-2. A pictorial understanding of how in/out degrees correlates with power is shown in Extended Data Figure 6-3. R represents the correlation coefficient.

Every three-ring electrode on the brain denotes the correlation coefficient value of three feature spaces with power. The significant correlation coefficient of in-degrees and hγ power time-series are shown in the innermost circle, the correlation coefficient of out-degrees with power is denoted by the color of the middle ring, while the outer ring for each electrode’s color denotes the correlation coefficient of coreness of nodes with power. The absence of color in the outer ring, or the absence of either the middle or inner ring, denotes the lack of significant correlation in that electrode, with that feature space. This figure denotes the correlation coefficient calculated using the entire time-series, SA and AA time windows considered separately are shown in Extended Data Figure 7-1. The bar plot to the right shows the average percentage of electrodes that showed significant correlation for each feature space, after correcting for multiple comparisons (FDR, p < 0.05 for each feature space, per patient). It can be noted that most electrodes’ feature spaces show correlation in the same direction. Across patients, more electrodes have significant correlation of coreness of nodes feature space with power, than the other two feature spaces.

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