Estimating the impact of structural directionality: How reliable are undirected connectomes?

Abstract

Directionality is a fundamental feature of network connections. Most structural brain networks are intrinsically directed because of the nature of chemical synapses, which comprise most neuronal connections. Because of the limitations of noninvasive imaging techniques, the directionality of connections between structurally connected regions of the human brain cannot be confirmed. Hence, connections are represented as undirected, and it is still unknown how this lack of directionality affects brain network topology. Using six directed brain networks from different species and parcellations (cat, mouse, C. elegans, and three macaque networks), we estimate the inaccuracies in network measures (degree, betweenness, clustering coefficient, path length, global efficiency, participation index, and small-worldness) associated with the removal of the directionality of connections. We employ three different methods to render directed brain networks undirected: (a) remove unidirectional connections, (b) add reciprocal connections, and (c) combine equal numbers of removed and added unidirectional connections. We quantify the extent of inaccuracy in network measures introduced through neglecting connection directionality for individual nodes and across the network. We find that the coarse division between core and peripheral nodes remains accurate for undirected networks. However, hub nodes differ considerably when directionality is neglected. Comparing the different methods to generate undirected networks from directed ones, we generally find that the addition of reciprocal connections (false positives) causes larger errors in graph-theoretic measures than the removal of the same number of directed connections (false negatives). These findings suggest that directionality plays an essential role in shaping brain networks and highlight some limitations of undirected connectomes.

Keywords: Connectome; Directionality; False positives; Graph theory; Hubs; Structural connectivity.

Figure 1.. The six connectomes analyzed in this study. Brain and connectome for three different parcellations of the macaque cortex (A) nodes N = 47 (Honey et al., 2007), (B) N = 71 (Young, 1993), and (C) N = 242 (Harriger et al., 2012), as well as three additional species including a (D) cat (Scannell et al., 1999), (E) mouse (Dong, 2008), and (F) C. elegans (White et al., ; Varshney et al., 2011). The connectomes represent connectivity matrices with rows and columns denoting brain regions (or nodes), and the elements within the matrices denoting the presence (filled) or absence (blank) of a connection between two regions. Unidirectional connections are highlighted in light blue (with the number of unidirectional connections stated below each connectome) and the nodal regions are arranged into modular communities. The bars below each connectome display the density of each network (A = 0.234, B = 0.15, C = 0.07, D = 0.308, E = 0.073, F = 0.063) and the proportion of unidirectional and bidirectional connections. The latter is segmented to display the proportion of unidirectional connections between modules (dark green: A = 0.123, B = 0.046, C = 0.238, D = 0.142, E = 0.304, F = 0.165) and within modules (light green: A = 0.117, B = 0.129, C = 0.255, D = 0.117, E = 0.404, F = 0.232) separately, as well as the proportion of bidirectional connections between modules (dark purple: A = 0.214, B = 0.236, C = 0.147, D = 0.21, E = 0.064, F = 0.147) and within modules (light purple: A = 0.547, B = 0.59, C = 0.359, D = 0.536, E = 0.229, F = 0.457).

Figure 2.. Structural connectome for the macaque N = 47 cortex and perturbed undirected variants, with an exemplar subnetwork. Subnetwork (top) encompassing the PITd region (white node) and neighboring nodes, the adjacency matrix (middle), and the entire network (bottom) for (A) macaque empirical connectome with the community modules outlined in red; (B) unidirectional connections of the connectome; (C) connectome with unidirectional connections removed (false negative network); (D) connectome with reciprocal connections added to unidirectional connections (false positive network); (E) connectome with one randomly selected reciprocal connection added to a unidirectional connection for each randomly selected unidirectional connection removed (density-preserving network). In each connectome, the connections linking PITd (dorsal posterior inferotemporal) to the rest of the network are colored orange.

Figure 3.. Graph-theoretic measures for a specific region of interest from each empirical and density-preserving connectome. (A) Empirical (blue) and density-preserving (red, an illustrative single trial with 100% of unidirectional connections altered) connectomes. Nodal regions are arranged into modular communities and the connections connecting the region of interest to the rest of the network in the empirical connectome are colored red. (B) Labels for each region of interest (top), and subnetworks of the local neighborhood around each region of interest (white node). (C) Graph-theoretic measures at the selected brain region for the empirical and density-preserving networks. Graph-theoretic measures are as follows: K = degree, C = clustering coefficient, and S = small-world index (Si→). *Normalized by the maximum value of that measure across all nodes in their respective network. PITd: dorsal posterior inferotemporal, A32: anterior cingulate area 32, 28m: medial entorhinal cortex, AAF: anterior auditory field, MOB: main olfactory bulb, VC05: ventral cord neuron 5.

Figure 4.. Identification of hubs, changes in graph-theoretic measures at the node level, and provincial/connecter hub classification. (A) Cortical areas of the macaque N = 47 connectome sorted by degree for the empirical and each perturbed network. Hubs are defined as nodes that have a total degree (in-degree plus out-degree) at least 1 standard deviation above the mean, and super hubs are defined as nodes that have a degree at least 1.5 standard deviations above the mean. The density-preserving results are from an illustrative single trial and show the standard deviation in degree for each node (over 1,000 trials). (B) Percentage of core, hub, and super-hub nodes across the perturbed networks of all six connectomes that retain correct classification according to their empirical connectome (as the mean over 1,000 trials). (C) Change in the participation index of each brain region from the empirical macaque N = 47 connectome to an illustrative case of the density-preserving network. (D) Identification and classification of hub nodes for the empirical (blue) macaque N = 47 connectome and an illustrative case of the density-preserving (red) network. The dotted line represents the hub definition based on the degree, and the dashed line represents the subclassification of hubs as either connector (Y > 0.35) or provincial (Y ≤ 0.35), based on the participation index. (E) Mean probability (across all connectomes over 1,000 trials) that hub nodes will cross over either, or both, of the threshold lines following density-preserving alterations in directionality, resulting in a classification that is inconsistent with the empirical connectomes. (A–E) Each perturbed network has 100% of unidirectional connections altered. Hub nodes are defined in the empirical network and retain the same definition in the perturbed networks.

Figure 5.. Nodal changes measured by the Rank-Shift Index. (A) The rank-shift index quantifies the change in the rank of nodes from the empirical connectome to the perturbed network when they are ordered by a particular graph-theoretic measure. More specifically, it calculates the sum of the difference between graph-theoretic values for each node in the empirical and perturbed matrices, divided by the maximum potential difference that could exist between these two networks (where a value of 0 indicates no change, and a value of 1 indicates the maximum change). See Methods for further explanation. (B) Rank-shift index of hub nodes across all perturbed networks, for each graph-theoretic measure. (C) Difference in the rank-shift index between the false negative and false positive networks for all nodes (left), and hub nodes (right). A positive value indicates that the false negative connections cause greater changes in the ranking of nodes, whereas a negative value indicates the same for false positive connections. (D) Rank-shift index for each graph-theoretic measure summed across all connectomes. (E) Rank-shift index values summed across all graph-theoretic measures for each density-preserving connectome. (B–E) Results correspond to the mean over 50 trials for which 5% of randomly selected unidirectional connections are modified in each perturbed network (error bars show the standard error of the mean). Graph-theoretic measures are as follows: K = degree, B = betweenness centrality, C = clustering coefficient, Y = participation index, and S = small-world index (Si→). M47: the macaque connectome with 47 nodes, M71: macaque N = 71, M242: macaque N = 242, C52: cat, M213: mouse, C279: C. elegans.

Figure 6.. Relative changes in mean graph-theoretic measures for perturbed networks. (A) Changes in mean graph-theoretic measures across all connectomes and each type of perturbed network. (B) Difference between the changes in mean graph-theoretic measures for the false negative and false positive networks. (C) Mean changes in graph-theoretic measures for each of the perturbed networks, summed across all connectomes. Two separate modularity inputs are used the participation index calculations for the perturbed networks: the consensus modularity of the empirical networks (light colors) and the new modularity assignments for each generated perturbed network (dark colors). (A–C) All results correspond to perturbed networks with 5% of randomly selected unidirectional connections modified. The results represent the mean of these networks over 50 trials, and describe the change in the mean graph-theoretic measure (from the empirical to perturbed network) normalized by the mean of the empirical network (error bars show the standard error of the mean). Graph-theoretic measures are as follows: K = degree, B = betweenness centrality, C = clustering coefficient, L = characteristic path length, G = global efficiency, Y = participation index, and S = small-world index (S^→, changes in this measure are presented as the mean over 1,000 trials). M47: the macaque connectome with 47 nodes, M71: macaque N = 71, M242: macaque N = 242, C52: cat, M213: mouse, C279: C. elegans.