The rich club of the C. elegans neuronal connectome

Abstract

There is increasing interest in topological analysis of brain networks as complex systems, with researchers often using neuroimaging to represent the large-scale organization of nervous systems without precise cellular resolution. Here we used graph theory to investigate the neuronal connectome of the nematode worm Caenorhabditis elegans, which is defined anatomically at a cellular scale as 2287 synaptic connections between 279 neurons. We identified a small number of highly connected neurons as a rich club (N = 11) interconnected with high efficiency and high connection distance. Rich club neurons comprise almost exclusively the interneurons of the locomotor circuits, with known functional importance for coordinated movement. The rich club neurons are connector hubs, with high betweenness centrality, and many intermodular connections to nodes in different modules. On identifying the shortest topological paths (motifs) between pairs of peripheral neurons, the motifs that are found most frequently traverse the rich club. The rich club neurons are born early in development, before visible movement of the animal and before the main phase of developmental elongation of its body. We conclude that the high wiring cost of the globally integrative rich club of neurons in the C. elegans connectome is justified by the adaptive value of coordinated movement of the animal. The economical trade-off between physical cost and behavioral value of rich club organization in a cellular connectome confirms theoretical expectations and recapitulates comparable results from human neuroimaging on much larger scale networks, suggesting that this may be a general and scale-invariant principle of brain network organization.

Figure 1.

Rich club of the C. elegans nervous system. a, The blue curve illustrates the rich club coefficient Φ(k) for the C. elegans neuronal network and the red curve is a randomized rich club curve, Φ_random(k), generated by averaging the rich club coefficients of 1000 random graphs at each value of k. The green curve is the normalized coefficient. Error bars on the Φ_random(k) and Φ_norm(k) curves are 1σ of the random graphs. Φ(k) ≥ Φ_random(k) + 1σ over the range 35 ≤ k ≤ 73, indicating that this is the rich club regime (highlighted in lightest gray). The more conservatively defined rich clubs of Φ(k) ≥ Φ_random(k) + 2σ and Φ(k) ≥ Φ_random(k) + 3σ are shaded darker grey (Table 1). b, A purely topological view of the rich club network. Nodes in yellow are located in the tail and those in red are located in the head. c, The rich club is shown in the context of the whole body of the animal. It only has components in the head and tail, which are enlarged to show the subset DVA and PVCL/R (tail, right) and the subset AVAL/R, AVBL/R, AVDL/R, and AVEL/R (head, left). Only synaptic connections between rich club neurons are shown.

Figure 2.

Motifs of the C. elegans network. a, The frequency of motif occurrence in the nematode network, compared with frequency of the same motif occurring in random networks, was defined in terms of interquartile deviances from the median and motifs were ranked in order of decreasing values. The motif that occurred with greatest (nonrandom) significance in the nematode network linked a pair of peripheral nodes via a series of local (L), feeder (F), and club (C) edges (denoted L-F-C-F-L). This indicates that many more of the shortest paths between peripheral neurons in the C. elegans network are mediated by the rich club than would be expected in a random network. b, The histogram shows the frequency distribution of the L-F-C-F-L motif in 1000 random networks. The frequency of the L-F-C-F-L motif in the nematode network is also shown; it is greater than the maximum frequency in the random network distribution, so it has p < 1/1000 = 0.001 under the null hypothesis that the frequency distribution of this motif is random in the nematode network. The top x-axis marks quartile deviances from the median, a nonparametric measure of distance from the central location of the random distribution. c, Construction of motifs from the shortest paths between pairs of neurons. As described in the key, rich club neurons are colored red and peripheral neurons are colored blue. An example of the frequently occurring motif L-F-C-F-L is given as a series of local (L), feeder (F) and club (C) connections to show how the topologically central rich club mediates many of the connections between topologically more peripheral neurons in the nematode nervous system. It is also illustrated anatomically within the head of the C. elegans network, where large bold nodes belong to the L-F-C-F-L motif and small pale nodes are in the rest of the network.

Figure 3.

The C. elegans rich club has higher nodal efficiency, betweenness, centrality, participation coefficient, and connection distance than the rest of the nervous system (the poor periphery). a–e, Box plots detailing the distributions of degree, betweenness centrality, average connection distance, nodal efficiency, and participation coefficient (a measure of intermodularity). For each metric, the rich club is compared with the poor periphery and with the network as a whole. f, Distribution of connection distances. Rich club connections have a bimodal distribution, including a relatively large proportion of the longest connection distances in the network and a majority of much shorter distance connections, feeder connections linking a peripheral node to a rich club node have intermediate probability of long connection distance, and local edges linking two peripheral nodes have the lowest probability of a long connection distance.

Figure 4.

Topological and spatial properties of the C. elegans nervous system are related: rich club neurons (red triangles) are distinguished from poor periphery neurons (blue circles) on all topological metrics. Rich club neurons tend to have higher degree (by definition), higher efficiency, higher betweenness, and higher participation coefficients than peripheral neurons. The connection distance of each neuron is the average of the physical distances between it and all of the other neurons to which it is synaptically connected in the network. Most rich club neurons have greater connection distance than most peripheral neurons, but some of the neurons with greatest connection distance are in the periphery.

Figure 5.

Neuronal birth times and key events in the development of C. elegans. Top (red bars), Number of rich club neurons born in each 5 min interval after fertilization. Bottom (dark blue bars), Birth times of the rest of the neurons in the C. elegans nervous system. The dashed vertical lines indicate when the animal begins to twitch, when it is first capable of coordinated movement, and when it hatches.