Mesoscopic organization reveals the constraints governing Caenorhabditis elegans nervous system

Abstract

One of the biggest challenges in biology is to understand how activity at the cellular level of neurons, as a result of their mutual interactions, leads to the observed behavior of an organism responding to a variety of environmental stimuli. Investigating the intermediate or mesoscopic level of organization in the nervous system is a vital step towards understanding how the integration of micro-level dynamics results in macro-level functioning. The coordination of many different co-occurring processes at this level underlies the command and control of overall network activity. In this paper, we have considered the somatic nervous system of the nematode Caenorhabditis elegans, for which the entire neuronal connectivity diagram is known. We focus on the organization of the system into modules, i.e., neuronal groups having relatively higher connection density compared to that of the overall network. We show that this mesoscopic feature cannot be explained exclusively in terms of considerations such as, optimizing for resource constraints (viz., total wiring cost) and communication efficiency (i.e., network path length). Even including information about the genetic relatedness of the cells cannot account for the observed modular structure. Comparison with other complex networks designed for efficient transport (of signals or resources) implies that neuronal networks form a distinct class. This suggests that the principal function of the network, viz., processing of sensory information resulting in appropriate motor response, may be playing a vital role in determining the connection topology. Using modular spectral analysis we make explicit the intimate relation between function and structure in the nervous system. This is further brought out by identifying functionally critical neurons purely on the basis of patterns of intra- and inter-modular connections. Our study reveals how the design of the nervous system reflects several constraints, including its key functional role as a processor of information.

Figure 1. Neuronal position and connectivity in the somatic nervous system of the nematode C. elegans indicating the different ganglia.

(A) Schematic diagram of C. elegans, indicating the different ganglia. (Inset) Schematic representation of connectivity between the neurons, partitioned into a strongly connected component (SCC), an in-component (IN), and an out-component (OUT). A directed path exists from any neuron in IN to any neuron in OUT through neurons in SCC, all of whose members can be reached from each other. The large SCC suggests that it is possible to transfer signals between almost all neurons of the network. The IN and OUT components have only and , respectively, of the 279 connected neurons in the somatic nervous system. (B, C) The connectivity matrix corresponding to the (B) Synaptic and (C) Gap-junctional connections between the somatic system neurons. In all figures, the partition symbols correspond to (G1) Anterior, (G2) Dorsal, (G3) Lateral, (G4) Ventral, (G5) Retrovesicular, (G6) Posterolateral, (G7) Preanal, (G8) Dorsorectal and (G9) Lumbar ganglion, and (G10) the Ventral cord.

Figure 2. Modular interconnectivity and decomposition according to neuron type.

(A) Matrix representing the average connection density between neurons occurring within modules and those in different modules. The figure indicates that neurons within a module are densely interconnected compared to the overall connectivity in the network. (B) The modules are decomposed according to the different neuron types comprising them. The figure shows that the modules are not simply composed of a single type of neuron.

Figure 3. Neuronal layout of the worm indicating cell body positions of each neuron.

The position of neuronal cell bodies along the longitudinal axis of the C. elegans body plan is shown, with the vertical offset and color indicating the module to which a neuron belongs. The mean and standard deviation of neuronal positions for each module is also indicated, suggesting relative absence of spatial localization in the modules.

Figure 4. Modular decomposition of neurons in different ganglia.

(A) Neurons belonging to different ganglia are decomposed according to their modular membership. The height of each bar in the histogram corresponds to the overlap between the ganglia and the modules, calculated as the fraction of neurons that are common to a particular ganglion and a specific module. (B) The matrix representing the average modular distance between the different ganglia, as calculated from the modular decomposition spectrum of each ganglion. The corresponding dendrogram indicates the closeness between different ganglia in the abstract 6-dimensional “modular” space. (C) The matrix of physical distances between the ganglia is shown for comparison with (B), calculated as the average distance between neurons belonging to the different ganglia. The corresponding dendrogram indicates the closeness between ganglia according to the geographical nearness of their constituent neurons in the nematode body. The difference indicates that the ganglia which are geographically close may not be neighbors in terms of their modular spectra.

Figure 5. Lineage distance between modules.

The matrix representing the average lineage distance between neurons occurring within the same module and those belonging to different modules. The figure indicates that neurons occurring in the same module have only a slightly lower lineage distance as compared to that between neurons occurring in different modules.

Figure 6. Trade-off between wiring cost and communication efficiency in the network.

The variation of communication efficiency, , as a function of the wiring cost, defined using either the “dedicated-wire” model (DW) or the “common-wire” model (CW), in the ensemble of random networks with degree sequence identical to the C. elegans neuronal network. The trend indicates a trade-off between increasing communication efficiency and decreasing wiring cost. The corresponding values for the empirical network are indicated by crosses for both DW and CW. The schematic figures shown above the main panel indicate the type of networks obtained in the limiting cases when only one of the two constraints are satisfied. In both curves, error bars indicate the standard deviations calculated for random realizations. We observe that the empirical network is suboptimal in terms of wiring cost and communication efficiency, suggesting the presence of other constraints governing the network organization.

Figure 7. Betweenness centrality and the average nearest neighbor degree as a function of the total degree of network.

(A) The average betweenness centrality, , and (B) the average nearest neighbor degree, of each node as a function of its total degree, . Betweenness centrality is a measure of how frequently a particular node is used when a signal is being sent between any pair of nodes in the network using the shortest path. In case of the Internet, BC of nodes increases with its degree which is sought to be linked with its information transport property. In C. elegans, although BC increases with degree, this increase is not significant when compared to the randomized version of the network. In the case of the relation between the average connectivity of nearest neighbors of a node with its total degree , we note that for both the Internet and protein interaction network, decreases with as a power law. This means that low connectivity nodes have high degree nodes as their neighbors and vice-versa. However, in the case of C. elegans, this relation is not very apparent and insignificant in comparison with the randomized version of the network. In both figures, error bars indicate the standard deviations calculated for random realizations. These results suggest that the C. elegans network forms a class distinct from the class of networks optimized only for signal propagation.

Figure 8. Modular decomposition of neurons in different functional circuits.

Neurons belonging to different functional circuits are decomposed according to their modular membership. The height of each bar in the histogram corresponds to the overlap between the modules and functional circuits (F1) mechanosensation, (F2) egg laying, (F3) thermotaxis, (F4) chemosensation, (F5) feeding, (F6) exploration and (F7) tap withdrawal. The overlap is measured in terms of the fraction of neurons common to a particular functional circuit and a specific module. The corresponding dendrogram represents the closeness between different functional circuits in the abstract 6-dimensional “modular” space.

Figure 9. The role of individual neurons according to their intra- and inter-modular connectivity.

(A) The within module degree -score of each neuron in the empirical neuronal network is shown against the corresponding participation coefficient . The within module degree measures the connectivity of a node to other nodes within its own module, while the participation coefficient measures its connectivity with neurons in the entire network. (B) The corresponding result for a randomized version of the C. elegans network where the degree of each neuron is kept unchanged is also shown. Neurons belonging to the different regions in the space are categorised as: (gray) R1: “ultraperipheral nodes”, i.e., nodes with all their links within their module, (blue) R2: “peripheral nodes”, i.e., nodes with most links within their module, (pink) R3: “nonhub connector nodes”, i.e., nodes with many links to other modules, (green) R4: “nonhub kinless nodes”, i.e., nodes with links homogeneously distributed among all modules, (yellow) R5: “provincial hubs”, i.e., hub nodes with the vast majority of links within their module, (red) R6: “connector hubs”, i.e., hubs with many links to most of the other modules, and (white) R7: “global hubs”, i.e., hubs with links homogeneously distributed among all modules. The neurons occurring as connector hubs are identified in the figure. Most of these neurons occur in different functional circuits indicating the close relation between functional importance and connectivity pattern of individual neurons. In addition, the neurons AVKL and SMBVL which are predicted to be functionally important are separately marked.