Specificity and robustness of long-distance connections in weighted, interareal connectomes

Abstract

Brain areas' functional repertoires are shaped by their incoming and outgoing structural connections. In empirically measured networks, most connections are short, reflecting spatial and energetic constraints. Nonetheless, a small number of connections span long distances, consistent with the notion that the functionality of these connections must outweigh their cost. While the precise function of long-distance connections is unknown, the leading hypothesis is that they act to reduce the topological distance between brain areas and increase the efficiency of interareal communication. However, this hypothesis implies a nonspecificity of long-distance connections that we contend is unlikely. Instead, we propose that long-distance connections serve to diversify brain areas' inputs and outputs, thereby promoting complex dynamics. Through analysis of five weighted interareal network datasets, we show that long-distance connections play only minor roles in reducing average interareal topological distance. In contrast, areas' long-distance and short-range neighbors exhibit marked differences in their connectivity profiles, suggesting that long-distance connections enhance dissimilarity between areal inputs and outputs. Next, we show that-in isolation-areas' long-distance connectivity profiles exhibit nonrandom levels of similarity, suggesting that the communication pathways formed by long connections exhibit redundancies that may serve to promote robustness. Finally, we use a linearization of Wilson-Cowan dynamics to simulate the covariance structure of neural activity and show that in the absence of long-distance connections a common measure of functional diversity decreases. Collectively, our findings suggest that long-distance connections are necessary for supporting diverse and complex brain dynamics.

Keywords: communication; complex networks; connectome; wiring cost.

Fig. 1.

Connectivity (Top) and Euclidean distance (Bottom) matrices for (A) mouse, (B) Drosophila, (C) macaque, (D) human (low-resolution), and (E) human (high-resolution) connectome data.

Fig. 2.

Network distance dependence. (A) Edge weight versus distance. (B) Cosine similarity versus distance. (C) Frequency of edge weights across all connections (gray) and long-distance connections (color; top 25% longest connections). (D) Frequency of connection lengths (gray) and lengths of existing connections (color).

Fig. 3.

Regional connection length profiles relate to functional specificity in the human network dataset. (A) Schematic illustrating connection length profiles for two example nodes. The orange node makes mostly short- and midrange connections, while the blue node exhibits some long-distance ($>$100 mm) connections. (B) Connection length distributions for a low- and a high-resolution human network dataset. Nodes are ordered according to the cluster to which they were assigned using a $k$-means algorithm. (C) Interquartile range of brain areas’ connection length distributions. (D) Interquartile range plotted for each functional system: visual (VIS), temporal + precuneus (T+P), dorsal attention (DAN), somatomotor (SMN), salience (SAL), default mode (DMN), frontal (FR), control (CONT), limbic (LIM), and subcortex (SUB; applies only to the low-resolution dataset). (E) Similarity of $k$-means partitions with functional system labels. (F) Standardized ($z$ score) overlap of clusters with functional systems. Circle size indicates the absolute value of the $z$ score, and color indicates the sign of the $z$ score. Large orange circles indicate that areas within pairs of systems were more likely to be coclustered based on their connection length distributions than expected by chance (permutation model).

Fig. 4.

Shortest path use in weighted interareal networks. (A) The fraction of total connections used in shortest paths. The total fraction is shown in gray; the long-distance fraction (top 25% length) is shown in color. The x axis represents the edge strength-to-distance parameter, $\alpha$. Larger values of $\alpha$ increase the relative strength of already-strong connections compared with weak connections. (B) Edge length distributions of connections participating in shortest paths. Gray curves show the mean distribution under a permutation-based null model; colored curves show $\alpha =\mathrm{1,2,3,4}$. (C) Percent change in weighted clustering coefficient as a result of removing different fractions of long and short connections. (D) Percent change in weighted characteristic path length as a result of removing different fractions of long and short connections. Note: bar plots in D are shown with $\alpha =\frac{1}{3}$. At larger values of $\alpha$, long-distance connections play no role in shortest path structure and removing them leads to no change in the weighted characteristic path length.

Fig. 5.

Similarity of long- and short-range connectivity profiles. (A) Schematic of processing pipeline for assessing the similarity of long- and short-range inputs. The network depicted is the human, low-resolution network, thresholded to a binary density of $\rho =0.25$ and with subcortical areas removed for visualization purposes only. For the empirical analysis, we performed no thresholding and retained all areas in all computations. All connections incident upon area $i$ are identified and their lengths are tabulated. Node $i$’s neighbors are then classified as either nearby or distant. Note: The same distance threshold was applied uniformly to all brain areas. Separately, the connectivity profiles of nearby versus distant neighbors are summed. The summed profiles, which represent possible inputs to node $i$ from its neighbors, are compared with one another using the cosine similarity measure. This process results in a single similarity score for each area (node). We compare these scores against a null distribution obtained by randomly reclassifying neighbors as nearby versus distant. (B) Cumulative distributions of area-level $z$ scores for each network. The different panels represent variation of the threshold for classifying neighbors as nearby versus distant. From Left to Right, nearby (distant) neighbors were those connected by the top (bottom) 5%, 10%, 20%, and 25% of connection lengths.

Fig. 6.

Redundancy of long-range connectivity. (A) Example empirical and randomized networks thresholded to retain the 25% longest connections (Leftmost). We then compute the pairwise cosine similarity between areas’ long-range connectivity profiles (Rightmost). (B) Pairwise similarity measures are averaged for the empirical and randomized networks. We repeat this process using four different definitions of “long range”: 5%, 10%, 20%, and 25% longest connections as defined by Euclidean distance between regional center of mass.

Fig. 7.

Linearized dynamics and participation coefficient. (A) Schematic illustrating analysis pipeline. The structural connectivity matrix is used to constrain a linearization of Wilson–Cowan dynamics, which results in an estimated covariance (FC) matrix. We compute network modules ahead of time, and based on those modules and on the simulated covariance structure, we compute brain areas’ functional participation coefficients. (B) Schematic illustrating the concept of the participation coefficient ($\mathcal{P}$). An area with high participation coefficient (red connections) forms connections to many different modules, while the connections of an area with low participation coefficient (green connections) are largely restricted to a single module. An area’s participation coefficient can be interpreted as a measure of its connectional diversity. When computed using a covariance matrix, an area’s participation coefficient measures its functional diversity. (C) Change in mean participation coefficient after removing different percentages of a network’s longest and shortest connections. The mean participation coefficient of the intact network is depicted as a red dashed line. Note that removing long-distance connections consistently reduces the mean participation coefficient (colored bars), indicating a decrease in functional diversity. Removing short-range connections (gray bars) has the opposite effect. Note that for the low- and high-resolution human datasets, we computed the participation coefficient with respect to structural and functional modules.