Features of spatial and functional segregation and integration of the primate connectome revealed by trade-off between wiring cost and efficiency

<div><p>The primate connectome, possessing a characteristic global topology and specific regional connectivity profiles, is well organized to support both segregated and integrated brain function. However, the organization mechanisms shaping the characteristic connectivity and its relationship to functional requirements remain unclear. The primate brain connectome is shaped by metabolic economy as well as functional values. Here, we explored the influence of two competing factors and additional advanced functional requirements on the primate connectome employing an optimal trade-off model between neural wiring cost and the representative functional requirement of processing efficiency. Moreover, we compared this model with a generative model combining spatial distance and topological similarity, with the objective of statistically reproducing multiple topological features of the network. The primate connectome indeed displays a cost-efficiency trade-off and that up to 67% of the connections were recovered by optimal combination of the two basic factors of wiring economy and processing efficiency, clearly higher than the proportion of connections (56%) explained by the generative model. While not explicitly aimed for, the trade-off model captured several key topological features of the real connectome as the generative model, yet better explained the connectivity of most regions. The majority of the remaining 33% of connections unexplained by the best trade-off model were long-distance links, which are concentrated on few cortical areas, termed long-distance connectors (LDCs). The LDCs are mainly non-hubs, but form a densely connected group overlapping on spatially segregated functional modalities. LDCs are crucial for both functional segregation and integration across different scales. These organization features revealed by the optimization analysis provide evidence that the demands of advanced functional segregation and integration among spatially distributed regions may play a significant role in shaping the cortical connectome, in addition to the basic cost-efficiency trade-off. These findings also shed light on inherent vulnerabilities of brain networks in diseases.</p></div

Trade-off between Multiple Constraints Enables Simultaneous Formation of Modules and Hubs in Neural Systems

<div><p>The formation of the complex network architecture of neural systems is subject to multiple structural and functional constraints. Two obvious but apparently contradictory constraints are low wiring cost and high processing efficiency, characterized by short overall wiring length and a small average number of processing steps, respectively. Growing evidence shows that neural networks are results from a trade-off between physical cost and functional value of the topology. However, the relationship between these competing constraints and complex topology is not well understood quantitatively. We explored this relationship systematically by reconstructing two known neural networks, Macaque cortical connectivity and <i>C. elegans</i> neuronal connections, from combinatory optimization of wiring cost and processing efficiency constraints, using a control parameter , and comparing the reconstructed networks to the real networks. We found that in both neural systems, the reconstructed networks derived from the two constraints can reveal some important relations between the spatial layout of nodes and the topological connectivity, and match several properties of the real networks. The reconstructed and real networks had a similar modular organization in a broad range of , resulting from spatial clustering of network nodes. Hubs emerged due to the competition of the two constraints, and their positions were close to, and partly coincided, with the real hubs in a range of values. The degree of nodes was correlated with the density of nodes in their spatial neighborhood in both reconstructed and real networks. Generally, the rebuilt network matched a significant portion of real links, especially short-distant ones. These findings provide clear evidence to support the hypothesis of trade-off between multiple constraints on brain networks. The two constraints of wiring cost and processing efficiency, however, cannot explain all salient features in the real networks. The discrepancy suggests that there are further relevant factors that are not yet captured here.</p> </div

Cost-efficiency trade-off in the primate cortical connectome and comparison to the generative model.

<p>(A) Wiring cost <i>l</i>_<i>p</i> (black bar) and graph path length <i>l</i>_<i>g</i> (white bar) in the real network and reconstructed networks with optimal wiring cost or efficiency, respectively, or their trade-off that maximally recovers the real connectivity. (B) Recovery rates <i>r</i>₁ (red) for all connected pairs and <i>r</i>₀ (blue) for all unconnected pairs in the real network, as functions of α (log scale). Inset: α on linear scale. The solid line with symbol represents the recovery rate by the trade-off model. The solid line without symbols denotes the corresponding results for random benchmarks. The dashed red or blue lines show the corresponding results by the extended generative model with fixed degrees. (C) and (D): <i>r</i>₁ and <i>r</i>₀ as a function of the spatial distance <i>x</i> between the pairs of areas in the real network, for α = 0.006 (black bars), α = 1 (white bars) in the cost-efficiency trade-off model, and in the extended generative model with fixed degrees (grey bars).</p

Properties of the reconstructed networks as functions of

<p><b>. </b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002937#s2" target="_blank">Results</a> are shown for the 1D model (left panel), Macaque cortical network (middle panel) and the <i>C. elegans</i> neuronal network (right panel). (A, B, C) and (normalized) of the reconstructed networks (with triangles or dots respectively), which are compared to those in the original networks in B and C. (D, E, F) , the number of hubs in the reconstructed networks vs. . Here a node with z-score of total degrees (input and output) larger than 2 is considered as a hub. (G, H, I) The average degree of the hubs in the reconstructed networks. (J, K, L) The probability of connections to the spatial nearest-neighbors. The results are obtained by averaging over 50 realizations of the reconstructed networks for each . In most cases the error bars are smaller than the symbol size. The dashed lines are the corresponding results for the real network.</p

The LDCs affect the functional segregation and integrations.

<p>(A) Hierarchical tree of the real <i>macaque</i> cortical network. The color of dots represents the corresponding function (visual: red, somatosensory: green, motor: blue, temporal: pink and frontal: black). The red stars indicate the 6 core special areas of LDCs, and the blue stars for the rest 6 LDCs. (B) Spatial distribution of different sub-trees in the real network mapped on the cortical surface using BrainNet Viewer [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref081" target="_blank">81</a>]. The LDCs in the sub-trees are shown as big balls with connections among each other. (C, D, E) Hierarchical modular organization of the cortical connectome, the reconstructed network at α = 0.006, and a randomized network (‘R-network’), respectively. Every pie in (C), (D) or (E) represents a hierarchical sub-tree (connectivity module) dominated by areas of a certain functional modality (matching ratio > 50%) in the corresponding network. The number of areas is listed next to the tree. The areas not involved in any hierarchical tree are combined into the “non-cluster” pie. The colors in the pie represent different functional systems (as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.g003" target="_blank">Fig 3B and 3A</a>), and the corresponding ratios of these systems in a given hierarchical sub-tree are listed. The original LDCs belonging to different sub-tree or non-cluster groups are shown by stars. These areas are grouped together by light solid lines with a diamond shape for the real network (C), by the shape of the circle in the reconstructed network (D), or by the shape of a hammer in the R-network (E). The arrows from the group of the special areas indicate the distribution of the links of this group to other sub-trees, with the ratio listed by bold fonts near the corresponding arrows.</p

Comparison of hierarchical modular organization of the real and surrogate networks.

<p>Hierarchy trees for the real network (A), reconstructed network (B) and R-network (C) where the unrecovered links of LDCs in the original network are rewired while keeping the wiring cost. Inset of B: random network rewired from the real network while keeping the degrees. (D) Modularity Q as a function of the threshold values for the trees in A, B and C (corresponding colors).</p

Comparison of reconstructed and original connectivity of <i>C. elegans</i> neuronal network.

<p>The left two plots (A and B) are for the original network. (A) Layout placement of 276 neurons and connections between them. (B) Adjacency matrix and the output () and input () degrees of the areas. The right four plots (C–F) show adjacency matrices and the degrees of areas in the reconstructed networks at various values of . (C) , (D) , (E) and (F) . The index of the neurons is the same for (C–F) and the names of the neurons are listed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002937#pcbi.1002937.s012" target="_blank">Table S2</a> of SI.</p

Modularity of original Macaque network and reconstructed networks.

<p>(A) Layout placement of 103 areas and the connections of the Macaque cortical network. (B) The two modules of the real network (open and filled circles) are compared to the two spatial clusters (blue and red). The corresponding modularity is . (C) As in (A), but for a reconstructed network at . The blue and red colors of the nodes represent the two spatial clusters. (D) The same as (B), but the modules are from the reconstructed network in (C), with . (E) Mismatch between the module partitions of the reconstructed and real networks. The mismatching areas are indicated by the pentagrams. (F) shows the fraction of mismatching areas between reconstructed and original networks in module partition, with respect to . The result did not include where the reconstructed networks did not show strong modularity. Blue bars are for the mismatching areas appearing in more than of all 50 realizations at each , while white bars are for the mismatching areas appearing at least once in the 50 realizations at the given . The dashed line represents the mismatching rate between the real networks and shuffled modules (see section “Matching between partitions” in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002937#s4" target="_blank"><i>Materials and Methods</i></a>).</p

The structural properties of LDCs.

<p>(A) Recovery rate <i>R</i>_recov for each area. (B) The Z-score of the recovery rate Z_<i>R</i> (<i>i</i>) of the reconstructed network (α = 0.006) when compared to the random benchmark networks. (C) The total wiring length <i>l</i>_<i>p</i> (<i>i</i>) of all the links of an area in the real network with respect to the corresponding total wiring length <i>l</i>_<i>pran</i> (<i>i</i>) (average) in random networks. (D) The Z-score of the areas sorted by the rank of total degree. The vertical dashed lines in (A-D) indicate the separation of the functional systems (visual (V): red; somatosensory (S): green; motor (M): blue; temporal (T): gray and frontal (F): black). The horizontal dashed lines in (B) and (D) indicate the range of Z-score in [− 1.65, 1.65]. The vertical dashed line in (D) separates the areas to the left with small degree and insignificant recovery. (E) Spatial location of LDCs on the <i>macaque</i> cortex using BrainNet Viewer [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref081" target="_blank">81</a>]. (F) The proportion of the links in each distance bin occupied by the six core areas (red bars) and the other six special areas (blue bars). The dashed line is a rough boundary between short-distance (< 30 mm) and long-distance (> 30 mm) links. Overall, the six core areas involve 38.1% and the 12 special areas involve 64.6% of long-distance links in the real network.</p

Comparison between the data of [45] and the present dataset.

<p>(A) Distribution of (binary) links with respect to distance between cortical areas for all the links in data of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>] (white bars) and the links overlapping with our data (black bars). (B) As in A, but for the distribution of the projection weights. The inset shows the average projection weight vs. distance. Here the gray bars are for the new links in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>] non-overlapping with our data. (C) The portion of the total projection weights within each distance bin in the data of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>] occupied by the two core LDC areas (5 and 46, red bars) and the other non-core LDC areas (2 and 7b, blue bars). (D) The total projection cost of each of the 24 targeted area in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>], with respect to that in the corresponding randomized networks, weighted l_<i>p</i>/<i>l</i>_<i>pran</i>, is compared to the corresponding l_<i>p</i>/<i>l</i>_<i>pran</i> from the present data (binary global network, similar to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.g004" target="_blank">Fig 4C</a>, but only concerning the afferent direction here). The red stars show the 2 areas of the data of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>] (5 and 46) appearing in the 6 core LDC areas in our data. The blue stars show the 2 areas of [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>] (2 and 7b) appearing in the non-core LDC areas. (E) Each bar corresponds to the weighted ratio l_<i>p</i>/<i>l</i>_<i>pran</i> for each of the 24 targeted areas in new dataset [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005776#pcbi.1005776.ref045" target="_blank">45</a>]. The first 4 bars refer to the 4 LDC areas in our data, which is ordered by the value of l_<i>p</i>/<i>l</i>_<i>pran</i>. The following 20 areas are also ordered by the l_<i>p</i>/<i>l</i>_<i>pran</i> values.</p