Structural properties of the Caenorhabditis elegans neuronal network

Abstract

Despite recent interest in reconstructing neuronal networks, complete wiring diagrams on the level of individual synapses remain scarce and the insights into function they can provide remain unclear. Even for Caenorhabditis elegans, whose neuronal network is relatively small and stereotypical from animal to animal, published wiring diagrams are neither accurate nor complete and self-consistent. Using materials from White et al. and new electron micrographs we assemble whole, self-consistent gap junction and chemical synapse networks of hermaphrodite C. elegans. We propose a method to visualize the wiring diagram, which reflects network signal flow. We calculate statistical and topological properties of the network, such as degree distributions, synaptic multiplicities, and small-world properties, that help in understanding network signal propagation. We identify neurons that may play central roles in information processing, and network motifs that could serve as functional modules of the network. We explore propagation of neuronal activity in response to sensory or artificial stimulation using linear systems theory and find several activity patterns that could serve as substrates of previously described behaviors. Finally, we analyze the interaction between the gap junction and the chemical synapse networks. Since several statistical properties of the C. elegans network, such as multiplicity and motif distributions are similar to those found in mammalian neocortex, they likely point to general principles of neuronal networks. The wiring diagram reported here can help in understanding the mechanistic basis of behavior by generating predictions about future experiments involving genetic perturbations, laser ablations, or monitoring propagation of neuronal activity in response to stimulation.

Figure 1. Adjacency matrices for the gap junction network (blue circles) and the chemical synapse network (red points) with neurons grouped by category (sensory neurons, interneurons, motor neurons).

Within each category, neurons are in anteroposterior order. Among chemical synapse connections, small points indicate less than synaptic contacts, whereas large points indicate or more synaptic contacts. All gap junction connections are depicted in the same way, irrespective of number of gap junction contacts.

Figure 2. The C. elegans wiring diagram is a network of identifiable, labeled neurons connected by chemical and electrical synapses.

Red, sensory neurons; blue, interneurons; green, motorneurons. (a). Signal flow view shows neurons arranged so that the direction of signal flow is mostly downward. (b). Affinity view shows structure in the horizontal plane reflecting weighted non-directional adjacency of neurons in the network.

Figure 3. Survival functions for the distributions of degree, multiplicity, and number of synaptic terminals in the gap junction network.

Neurons or connections with exceptionally high statistics are labeled. The tails of the distributions can be fit by a power law with the exponent for the degrees (a); for the multiplicity distribution (b); for the number of synaptic terminals (c). The exponents for the power law fits of the corresponding survival functions are obtained by subtracting one.

Figure 4. Linear systems analysis of the giant component of the gap junction network.

(a). Survival function of the eigenvalue spectrum (blue). The algebraic connectivity, , is and the spectral radius, , is . A time scale associated with the decay constant is also given. (b). Scatterplot showing the norm and decay constant of the eigenmodes of the Laplacian. The fastest modes from Figure 3 in Text S4 are marked in red. The sparsest and slowest modes, most amenable to biological analysis, are located in the lower-left corner of the diagram. (c). Eigenmode of Laplacian corresponding to (marked green in panel (b)). (d). Eigenmode of Laplacian corresponding to (marked cyan in panel (b)).

Figure 5. Subnetwork distributions for the gap junction network.

Overrepresented subnetworks are boxed, with the -value from the step-down min-P-based algorithm for multiple-hypothesis correction , () shown inside. (a). The ratio of the -subnetwork distribution and for the mean of a degree-preserving ensemble of random networks (). The counts for the particular random networks that appeared in the ensemble are also shown. (b). The ratio of the -subnetwork distribution and for the mean of a degree and triangle-preserving ensemble of random networks (). The counts for the particular random networks that appeared in the ensemble are also shown.

Figure 6. Degree distribution (a) and survival functions for the distributions of in-/out-degree, multiplicity, and in-/out-number of synaptic terminals in the chemical synapse network (b)–(f).

Neurons or connections with unusually high statistics are labeled. The tails of the distributions can be fit by a power law with exponents for in-degree (b); for out-degree (c); and for out-number (f). The exponents for the survival function fits can be obtained by subtracting one. The survival function of the multiplicity distribution for can be fit by a stretched exponential of the form where and (d). No satisfactory fit was found for the distribution of in-numbers (e).

Figure 7. Subnetwork distributions for the chemical synapse network.

Overrepresented subnetworks are boxed, with the -value from the step-down min-P-based algorithm for multiple-hypothesis correction , () shown inside. (a). The ratio of the -subnetwork distribution and the mean of a random network ensemble (). Realizations of the random network ensemble are also shown. (b). The ratio of the -subnetwork distribution and the mean of a random network ensemble (). Realizations of the random network ensemble are also shown.

Figure 8. Linear systems analysis for the strong giant component of the combined network.

(a). Eigenvalues plotted in the complex plane. (b). The eigenmode associated with eigenvalue (marked cyan in panel (c)). (c). Scatterplot showing the sparseness and decay constant of the eigenmodes. (d). Sparse and slow eigenmodes of the combined network (marked red in panel (c)). The real parts of the eigenmodes corresponding to , and are shown. The eigenmodes are labeled with neurons that take value above a fixed absolute value threshold. Neurons with negative values are in red, whereas neurons with positive values are in black.

Figure 9. Likelihood ratio for the possible chemical network doublets (horizontal axis) given the absence/presence of a gap junction between the two neurons (as indicated by the green marks).

Figure 10. The spectrum of the giant component of the combined network matrix (red), ε-disks around the spectrum (light blue), spectra of randomly edited networks (blue), and the ε-pseudospectrum (orange).

The value is used (the average spectral norm of the editing matrices was ). The spectrum of the giant component of the combined network matrix under an alternate quantitation of send_joint synapses is also shown (green).