Distance-dependent consensus thresholds for generating group-representative structural brain networks

Abstract

Large-scale structural brain networks encode white matter connectivity patterns among distributed brain areas. These connection patterns are believed to support cognitive processes and, when compromised, can lead to neurocognitive deficits and maladaptive behavior. A powerful approach for studying the organizing principles of brain networks is to construct group-representative networks from multisubject cohorts. Doing so amplifies signal to noise ratios and provides a clearer picture of brain network organization. Here, we show that current approaches for generating sparse group-representative networks overestimate the proportion of short-range connections present in a network and, as a result, fail to match subject-level networks along a wide range of network statistics. We present an alternative approach that preserves the connection-length distribution of individual subjects. We have used this method in previous papers to generate group-representative networks, though to date its performance has not been appropriately benchmarked and compared against other methods. As a result of this simple modification, the networks generated using this approach successfully recapitulate subject-level properties, outperforming similar approaches by better preserving features that promote integrative brain function rather than segregative. The method developed here holds promise for future studies investigating basic organizational principles and features of large-scale structural brain networks.

Keywords: Complex networks; Connectome; Consensus; Group-representative; Wiring cost.

Figure 1.

Construction and superficial comparison of group-representative matrices. Group-representative connectivity matrices summarize subject-level network data (A) by retaining features that are consistently expressed across subjects. In most applications the features of interest are the edges between brain areas and their weights. The most straightforward approach for generating a group-representative matrix involves first constructing a consensus matrix (B), whose elements denote the fraction of all subjects in which edges are expressed. Group-representative matrices can be estimated by retaining all connections expressed in at least τ subjects and populating those connections with weights. Though this approach is common, it suffers from a number of shortcomings. In general, because the probability of observing any given short-range connection is greater than the probability of observing a long-range connections, short-range connections also appear more consistently across subjects. As a result, imposing a uniform consensus-based threshold across all elements of the consensus matrix will result in a group-representative matrix in which short-range connections are expressed with much greater frequency than any single subject (C). To circumvent this issue, we present a simple alternative approach. Briefly, this involves dividing all connections into m bins according to their length and, within each bin, retaining the connection that is most frequently expressed. This distance-dependent consensus-based thresholding approach results in networks with almost the exact same edge length distribution as the typical subject. We also show differences in the group-representative matrices generated using the distance-dependent and uniform consensus-based threshold (here, we choose τ for the uniform method such the resulting matrix has a number of connections equal to that of the average subject). (D) Connections present only in the uniform method are depicted in blue and those present only in the distance-depdendent method are shown in red. (E) We plot these same method-specific connections on the brain, and color them according to their length (in millimeters). (F) In general, the connections unique to the uniform method are short range (blue curve) while those unique to the distance-dependent method include long-distance connections.

Figure 2.

Comparing distributions across methods. We show cumulative distributions for (A) degree, (B) strength, (C) clustering coefficient, (D) betweenness centrality, and (E) connection length. Subject-level data are shown in gray. Superimposed on those distributions are curves associated with the four methods that we tested (in color). In panels (F)–(J) we show Kolmogorov-Smirnov (KS) statistics for each network measure, which compare cumulative distribution curves of methods with individual subjects.

Figure 3.

Comparing scalar network statistics. Here, we compare the performances of four different methods of group-representative brain networks to those of individual subjects. (A) Each bar represents the total number of binary connections for single subjects (gray), a uniform method with approximately the same number of connections as the average subject (bright red), a uniform method with a consensus threshold of τ = 0.5 (dark red), a “Simple” method that retains a connection if it is observed in even one subject (blue), and the distance-dependent method (black). Panels (B)–(D) show similar plots but for mean clustering, efficiency, and modularity. Panels (E)–(H) depict those same measures, but computed over weighted analogs of the binary networks. (I) For each measure shown (along with several others), we identified the method that was closest to that of the average across all subjects. In general, we find that the distance-dependent method consistently outperforms or performs comparably to the other tested methods, achieving rank 1 or 2 across all metrics.

Figure 4.

Comparing within-/between-RSN connectivity patterns. We compared different group-representative networks in terms of connection densities within and between canonical brain systems taken from Schaefer et al. (2017). (A) Inter-RSN connection density of the typical subject. (B) Inter-RSN connection densities for four different group-representative networks: (from left to right) uniform consensus method with same density as subjects, uniform consensus method with threshold set at τ = 0.5, “Simple” method, and distance-dependent method. (C) We show the correlation patterns of inter-RSN densities for each method (y-axis) with that of the subject average (x-axis). Of the methods compared here, the distance-dependent and the uniform method with same density as the typical subject performed the best. We compare these methods so as to better understand their differences. (D) Difference in inter-RSN connection density between distance-dependent and τ = 0.5 threshold methods. Blue colors indicate that connection density is greater in uniform method while red density indicates that connection density is greater in distance-dependent method. (E) We find that, on average, the uniform method results in weaker within-RSN density than the distance-dependent method, while the distance-dependent method has greater between-RSN density. (F) We show the observed difference in within- and between-RSN density and compare it against a null method. Here, we show the null distribution (blue) and the observed value (red). The null distribution was constructed by independently and randomly permuting rows/columns of each original connectivity matrix and reaggregating according to the RSN system labels. Then we compute the mean difference of within-/between-RSN densities.

Figure 5.

Comparing spatial distribution of hubs. We compare four measures of hubness: (A) betweenness centrality, (B) degree, (C) clustering coefficient, and (D) participation coefficient. Rather than compare raw values, which can fluctuate because of small differences in global network properties like total number of connections or weight, we compare ranked values of each measure and observe whether a node’s rank is smaller/greater under the distance-dependent or uniform method. Orange-colored nodes indicate that a node’s value is greater under the distance-dependent method than it is under the uniform method. Blue-colored nodes indicate the opposite. We then aggregated node-level differences in ranked measures by cognitive systems and compared the mean system-level values with those obtained under a null method. In panels (E)–(H) we show the z-scored system means. In general, large-magnitude z-scores indicate bigger greater system-level differences between the two methods.