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Structural Properties of the Caenorhabditis elegans Neuronal Network

Lav R. Varshney, Beth L. Chen, Eric Paniagua, David H. Hall, Dmitri B. Chklovskii

arXiv:0907.2373v4q-bio.NC

TL;DR

The paper asks how a complete, reliable neuronal wiring diagram can support understanding of network function when existing connectomes are scarce and inconsistent. It assembles and analyzes updated C. elegans gap-junction and chemical-synapse networks, finding network dynamics that can guide experiments on neuronal activity and behavior.

  • Problem

    Complete synapse-level wiring diagrams are scarce, and existing C. elegans diagrams are incomplete, inaccurate, and not self-consistent.

  • Method

    The authors combine White et al.'s data with new electron-microscopy reconstructions and updates, then analyze network structure and signal propagation.

  • Results

    The eigenspectrum analysis characterizes neuronal-activity dynamics and should help predict and interpret responses to sensory or artificial stimulation and activity imaging.

  • Takeaways & Limitations

    The wiring diagram and its visualization can help design informative optical-ablation and genetic-inactivation experiments.

Abstract

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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.

INTRODUCTION

The paper addresses incomplete and inconsistent C. elegans wiring diagrams by assembling an updated near-complete network and analyzing its structure, signal propagation, and behavioral relevance.

  • Connectomics seeks to decode neuronal connectivity, but complete connectomes remain scarce and their functional significance is uncertain.
  • C. elegans is a tractable model because its 302 neurons are identifiable and its neuronal connections are stereotypical across animals.The reported reproducibility of connections exceeds 75%.
  • Earlier C. elegans wiring diagrams were incomplete, inconsistent, and affected by a major gap in ventral cord connectivity.Previous whole-network assemblies also made unjustified assumptions about whether reconstructed neurons represented others.
  • The authors report a near-complete wiring diagram built from White et al., new serial-section electron microscopy, and subsequent updates.The updated somatic network contains 279 analyzed neurons, 6393 chemical synapses, and 890 gap junctions.
  • They characterize local and global network properties, including multiplicity, terminal counts, and signal-propagation-related structure, using a hypothesis-generating systems-biology approach.The analyses treat gap junctions and chemical synapses separately before examining their combined network.
  • The proposed wiring-diagram visualization and eigenspectrum analysis are intended to guide ablation, genetic-inactivation, stimulation, and neuronal-imaging experiments.

3) Visualization:

The paper visualizes the neuronal network by separating signal-flow hierarchy from connectivity closeness, then characterizes gap-junction structure through component size, degree distributions, centrality, path length, and clustering. These analyses identify a highly clustered, small-world network with central neurons and heavy-tailed connection statistics.

  • Visualization: The visualization places sensory neurons above motor neurons to represent mostly downward chemical-synapse signal flow, while horizontal position reflects weighted non-directional adjacency.The affinity view uses connectivity in the combined chemical and electrical network rather than physical worm anatomy.
  • Visualization: The combined visualization separates motorneurons into ventral-cord and neck/tail lobes, with centrally located interneurons positioned to coordinate them.The two-lobe structure suggests partial autonomy of the two motor systems.
  • Gap Junction Network: The gap-junction network contains 279 neurons and 514 connections, with a 248-neuron giant component, two smaller components, and 26 isolated neurons.A degree-matched random network predicts a similarly sized giant component, whereas an Erdős-Rényi network predicts 271 neurons.
  • Distributions: The gap-junction degree ranges from 0 to 40 around a mean of 3.68, and its tail fits a power law with exponent γ = 3.14 rather than exponential decay.The result is consistent with a scale-free gap-junction network.
  • Distributions: Gap-junction multiplicity and terminal-count tails fit power laws with exponents γ = 2.76 and γ = 2.53, respectively.Multiplicity counts contacts between neuron pairs, while terminal count sums multiplicities across each neuron’s connections.
  • Small World Properties: Signals traverse 4.52 gap-junction connections on average in the giant component, versus 7.63 in an Erdős-Rényi network and 3.05 in a degree-matched random network.The measured network’s clustering coefficient is C = 0.21, compared with 0.0083 and 0.05 for the two random-network baselines.
  • Small World Properties: Because the gap-junction network is more clustered yet retains short average distances than Erdős-Rényi networks, it may be classified as a small-world network.The paper frames this classification as applying to the giant component.
  • Centrality: The six most closeness-central neurons are AVAL, AVBR, RIGL, AVBL, RIBL, and AVKL, largely overlapping the degree-central set.The set includes command interneurons, ring interneurons, and a ventral-ganglion interneuron.

4) Spectral Properties:

The paper models gap-junction dynamics with Laplacian eigenmodes, using their decay and sparsity to identify long-lived activity patterns and candidate functional circuits.

  • Model and spectral decomposition: The gap-junction network is modeled as coupled linear differential equations and analyzed through Laplacian eigendecomposition.Eigenmodes evolve independently, allowing the dynamics to be described from initial conditions.
  • Mode interpretation: Small-eigenvalue modes decay slowly and can indicate weakly coupled subnetworks with opposing activity patterns.Such modes correspond to long-lived excitation and may separate strongly coupled neuronal groups.
  • Candidate circuits: The λ3 eigenmode links tail chemosensory, interneuron, and egg-laying motor neurons against weakly coupled head neurons.The proposed interpretation is based on the spatial separation of neuron groups in the mode.
  • Candidate circuits: The λ13 eigenmode overlaps neurons in a previously described hub-and-spoke circuit for pheromone attraction, oxygen sensing, and social behavior.This overlap is consistent with a consensus-processing interpretation of that circuit.
  • Mode selection: Slow and sparse modes are prioritized for biological analysis because few neurons exhibit significant activity.Sparseness is quantified by the rectilinear norm, with smaller values indicating sparser modes.
  • Time scales: Assuming τ = 10ms, the slowest non-trivial mode has λ2 = 0.12 and an estimated decay time of about 83ms.Including an estimated membrane conductance shifts eigenvalues by 0.5 and reduces the slowest decay time to about 16ms.
  • Activity propagation: Spectral analysis can predict activity spread from sensory or artificial stimulation by superposing independently decaying eigenmodes.Fast modes describe initial charge redistribution, whereas slow modes describe long-term evolution.

5) Motifs:

The chemical network exhibits structured connectivity, central neurons, non-random motifs, and small-world organization, while its dynamical interpretation is limited by unknown synaptic signs.

  • Motifs: Gap-junction quadruplet motifs include overrepresented fan and diamond patterns, while the chemical network shows greater reciprocity and overrepresented densely connected triplets.Motif overrepresentation is interpreted as potentially functional rather than arising from proximity alone.
  • Basic structure: The chemical network contains 279 neurons and 2194 directed connections, with an adjacency structure suggestive of a three-layer architecture.Its subnetworks also contain many recurrent connections.
  • Connectivity organization: The chemical network has 237 neurons in its giant strongly connected component and is more segregated than a degree-matched random network.A strongly connected component as small as this is unlikely in the corresponding random network.
  • Degree and terminal distributions: In-degree and out-degree correlate at 0.52, while both have mean 7.86 connections and power-law tail exponents of 3.17 and 4.22.An exponential fit is ruled out for in-degree but not for out-degree.
  • Multiplicity: The mean multiplicity is 2.91 synapses per connection, and its distribution is better described by a stretched exponential with β = 0.36 and γ = 0.47.A power-law fit is not supported for multiplicity.
  • Small-world properties: The directed characteristic path length is 3.48 steps and the clustering coefficient is 0.079 versus an expected random value of about 0.018.The higher clustering supports classifying the chemical network as small-world.
  • Central neurons: Command interneurons including AVAL, AVAR, AVBR, AVEL, AVER, and AVBL have high in-closeness centrality and may efficiently integrate signals from diverse sources.These neurons are also central in the gap-junction network.
  • Limitation: Unknown excitatory or inhibitory signs are a major uncertainty in linear systems analysis of the chemical network.The analysis uses a rough neurotransmitter-based approximation for synaptic sign.

D. Full Network

The paper combines the gap-junction and chemical-synapse networks by unioning their adjacency matrices and examining the resulting directed connectivity.

  • Combined analysis: The two networks are first analyzed separately because their relative weights are unknown, then examined collectively.The combined analysis considers either a union network or interactions between the two networks.
  • Union network: The combined network has 279 neurons and 2990 directed connections, with one strongly connected component containing 274 neurons.Five neurons remain strongly isolated in the combined representation.

1) Single Combined Network:

The combined network is weakly connected, compact, and highly clustered, with degree and terminal distributions showing heavy-tailed structure. Linear systems analysis identifies eigenmodes that may relate network dynamics to behavior and stimulation responses.

  • The combined network is a single weakly connected component containing the reported isolated-neuron intersection.
  • Mean path length is L = 2.87, compared with L = 2.62 for a degree-matched random network, while clustering is C = 0.26 versus C = 0.10.
  • The combined network’s in-degree and out-degree correlate at 0.71, and both degree survival-function tails can be fit with power laws.
  • AVAL and AVAR have the greatest degree centrality, with AVBL/R next in both in-degree and out-degree.
  • The out-number tail fits a power law and alternatively an exponential, while multiplicity fits a stretched exponential.
  • The analysis assumes equal conductance for individual gap junctions and chemical synapses, restricts attention to a 274-neuron strongly connected component, and ignores the g_m V_i term.
  • The sixth eigenmode includes neurons involved in sinusoidal body movement, and the eigenspectrum can support predictions for sensory or artificial stimulation.

3) Interaction Between Networks:

Gap junction and chemical synapse connections are locally correlated, although most gap-junction-linked neuron pairs still lack chemical synapses. Their combined structure appears to reinforce rather than offset connectivity.

  • Chemical synapses are more likely to be absent, unidirectional, or bidirectional depending on gap-junction presence, indicating correlated networks.
  • The combined-network spectrum and eigenmodes are examined through linear systems analysis, including eigenvalues, sparseness, and decay constants.
  • The authors suggest overlapping inhibitory chemical synapses are not primarily counteracting excitatory gap junctions, but note neurotransmitter actions remain undetermined for many neurons.
  • The two networks appear to reinforce each other through correlated structure rather than augmenting each other through anticorrelated connections.

E. Robustness Analysis

Local network statistics are expected to be relatively stable under small wiring errors, whereas global properties—especially eigenvalues—are more sensitive. Random-edit analyses suggest reasonable robustness, but worst-case perturbations can strongly mix eigenmodes.

  • Local properties such as degree, multiplicity, and subnetwork distributions should change little under small wiring errors, unlike global path lengths and eigenmodes.
  • Robustness was assessed by randomly removing synaptic contacts and assigning them to randomly chosen neuron pairs before recalculating global properties.
  • Editing gap-junction contacts with 10% probability and chemical-synapse contacts with 5% probability produced global properties described as reasonably robust across 1000 edited networks.
  • The prior-work network corresponds to an editing distance of roughly 25.6% of synaptic contacts.
  • Eigenvalue robustness used smaller edit probabilities—1% for gap junctions and 0.5% for chemical synapses—because eigenvalues are more sensitive to errors.
  • The ε-pseudospectrum indicates that under worst-case perturbations most eigenmodes become mixed up, especially in chemical and combined networks.
  • Alternative counting of polyadic synapses changes multiplicity-sensitive spectra but not statistics that ignore multiplicity.

DISCUSSION

The study presents a corrected, self-consistent wiring diagram and uses statistical, centrality, motif, and spectral analyses to generate hypotheses about neuronal function. Several structural properties resemble those of mammalian cortex, while the conclusions remain exploratory.

  • The corrected wiring diagram and its visualization are intended to support experiments such as neuron ablation.
  • Statistical analyses of the corrected wiring are presented as tools for inferring function from structure.
  • Command interneurons have high degree centrality in both networks and high in-closeness, whereas other neurons have greater out-closeness.
  • Linear systems analysis identifies modes that map onto previously identified behaviors and supports predictions for sensory and artificial stimulation experiments.
  • The gap-junction, chemical-synapse, and combined networks may all qualify as small-world networks.
  • Degree and terminal-number tails generally follow power laws, although exponential fits can also describe some distributions and neither model fits every whole distribution.
  • Chemical-synapse multiplicity is well fit by a stretched exponential with a stretch factor of approximately 0.5, close to that reported for mammalian cortex.
  • Chemical-network motifs, including reciprocal pairs and fully connected triplets, are overrepresented similarly to motifs in mammalian cortex.

MATERIALS AND METHODS

The authors assembled a self-consistent wiring database by integrating published records, archival notebooks, new electron micrographs, and cross-checks to reconcile missing or mismatched connections.

  • Data acquisition: Published and unpublished records were manually entered with neuron identities, partners, and synaptic types into an electronic database.The starting point was White et al.’s The Mind of a Worm, supplemented with additional sources.
  • Data acquisition: 64 neurons had large wiring gaps or were missing, including 61 ventral-cord motor neurons, two excretory neurons, and RID.The mid-body region was especially under-reconstructed.
  • Data acquisition: New high-power electron micrographs were produced for a dorsal region that had never been imaged at sufficient magnification.This region accounted for many missing dorsal-side reconstructions.
  • Data integration: Connections were reconciled using electron micrographs, laboratory notebooks, posterior reconstructions, and self-consistency criteria.The reconciliation involved 561 synapses for 108 neurons.
  • Network analysis: For random networks matched to measured degree distributions, the expected giant component contained 251 chemical-network neurons and 278.9990 gap-junction-network neurons.The corresponding expected fractions were S = 0.90 and S = 0.999996, respectively.
  • Network analysis: For 20 known circuits of size 6, the probability that at least one appeared in gap-junction eigenmodes by chance was less than 1.4×10^-3.The calculation used a union bound.

APPENDIX A ALGORITHM FOR DIRECTED NETWORK DRAWING

The directed-network drawing method separates signal-flow direction from connectivity strength, placing neurons using an energy minimization and Laplacian-based coordinates.

  • Coordinate construction: Vertical coordinates encode signal-flow direction, while horizontal coordinates encode direction-independent connectivity strength.The two coordinate choices are made independently.
  • Coordinate construction: The vertical arrangement minimizes deviations between presynaptic and postsynaptic coordinates from a preferred separation of one.The arrangement is obtained by minimizing an energy function.
  • Coordinate construction: Horizontal coordinates are derived from Laplacian eigenmodes after normalizing the Laplacian by the number-of-terminals matrix.The second and third lowest eigenmodes are used.
  • Interpretation: The resulting visualization places each neuron near the weighted centroid of its neighbors, causing strongly coupled neurons to be colocated.This produces an aesthetically appealing drawing.

APPENDIX C FURTHER SPECTRAL PROPERTIES OF THE GAP JUNCTION NETWORK

Spectral analysis characterizes signal-propagation properties of the gap-junction network, showing reasonably strong algebraic connectivity but a large eigenratio relative to its optimum.

  • Eigenratio: The gap-junction network’s eigenratio is 1026, compared with a general lower bound of 41 for its maximum and minimum degrees.The network has maximum degree 40 and minimum degree 1.
  • Signal propagation: Because algebraic connectivity is fairly large, the gap-junction network also has a fairly large magnification coefficient.The magnification coefficient is associated with rapid signal transmission.
  • Algebraic connectivity: The gap-junction giant component has algebraic connectivity 0.12 versus an upper bound of 1.00.The deviation is attributed mainly to the network’s non-constant degree distribution.

APPENDIX D EIGENDECOMPOSITION

The eigendecomposition appendix formulates neuronal dynamics as a linear system whose responses can be decomposed into eigenmodes, each with a characteristic decay rate, while noting a scope limitation for directed networks.

  • Linear dynamics: Neuronal states are represented as a vector governed by coupled linear, constant-coefficient differential equations.The system can be written in matrix-vector form as dV/dt + LV(t) = M(t).
  • Eigenmodes: Eigendecomposition transforms the coupled system into coordinates associated with individual eigenmodes and eigenvalues.The transformed representation is intended to decouple the equations.
  • Scope: Schur modes are not considered for the chemical and combined networks, although they may provide additional insights for those directed networks.For the undirected gap-junction network, Schur decomposition is identical to eigendecomposition.
  • Eigenmodes: Each eigenmode has natural frequency -λ_i/τ, and larger eigenvalues produce faster decay when τ is fixed.The natural response is a superposition of distinct eigenmode dynamics.
  • Forced response: The forced response combines the response to an external perturbation with the natural response determined by the system’s eigenproperties.A forcing function can therefore alter the input-driven component while retaining eigenproperty-governed dynamics.

SUPPLEMENTAL MATERIAL

The supplemental material documents component structure, network comparisons, and eigenmode and degree-distribution analyses for the gap-junction and chemical networks.

  • Table S1 reports connected components of the gap junction network, including a single giant component and many isolated neurons.
  • Table S3 compares clustering coefficients and characteristic path lengths for the C. elegans gap-junction giant component with other small-world networks.
  • Table S4 summarizes chemical-synapse contacts between neuron categories and their percentages.
  • Table S5 catalogs strongly connected components of the chemical network, including a single giant component.
  • Supplemental figures show geodesic-distance distributions, gap-junction Laplacian eigenmodes, and power-law fits to combined-network degree-distribution tails.
  • Tables S6 and S7 compare structural properties of the biological gap-junction and chemical networks with randomly edited networks.
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