Conceived and designed the experiments: BLC DHH DBC. Performed the experiments: BLC DHH. Analyzed the data: LRV BLC EP DBC. Contributed reagents/materials/analysis tools: LRV BLC EP DHH DBC. Wrote the paper: LRV DBC.
The authors have declared that no competing interests exist.
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
Connectomics, the generation and analysis of neuronal connectivity data, stands to revolutionize neurobiology just as genomics has revolutionized molecular biology. Indeed, since neuronal networks are the physical substrates upon which neural functions are carried out, their structural properties are intertwined with the organization and logic of function. In this paper, we report a nearcomplete wiring diagram of the nematode
Determining and examining base sequences in genomes
The neuronal network of the nematode
Despite a century of investigation
In this paper, we advance the experimental phase of the connectomics program
We advance the theoretical phase of connectomics
Our results should help investigate the function of the
Organization of the
The
The new version of the wiring diagram incorporates original data from White
The current wiring diagram is considered selfconsistent under the following criteria:
A record of Neuron
A record of gap junction between Neuron
Although the updated wiring diagram represents a significant advance, it is only about
The basic qualitative properties of the updated
The majority of sensory neuron and interneuron categories contain pairs of bilaterally symmetric neurons. Motor neurons along the body are organized in repeating groups whereas motor neurons in the head have four or sixfold symmetry. A large fraction of neurons send long processes to the nerve ring in the circumpharyngeal region to make synapses with other neurons
The neurons in
In the remainder of the paper, we describe and analyze the connectivity of gap junction and chemical synapse networks of
Within each category, neurons are in anteroposterior order. Among chemical synapse connections, small points indicate less than
Although gap junctions may have directionality, i.e. conduct current in only one direction, this has not been demonstrated in
Since chemical synapses possess clear directionality that can be extracted from electron micrographs, we represent the chemical network as a directed network with an asymmetric adjacency matrix,
Electron micrographs for
Although statistical measures that we investigate later in this paper provide significant insights, they are no substitute to exploring detailed connectivity in the neuronal network. As the number of connections between neurons is large even for relatively simple networks, such analysis requires a convenient way to visualize the wiring diagram. Previously, various fragments of the wiring diagram were drawn to illustrate specific pathways
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 nondirectional adjacency of neurons in the network.
The vertical axis in
The distance along the vertical coordinate corresponds roughly to the number of synapses from sensory to motor neurons—the signal flow depth of the network. Depending on the specific neurons considered, the distance from a sensory neuron to a motor neuron is
Neuronal position on the horizontal plane,
Thus,
For quantitative characterization, we first consider the gap junction network.
The gap junction network that we analyze consists of
To evaluate the significance of the number of neurons in the giant component, we compare it with those expected in random networks. We start with the ErdösRényi random network because its construction requires a single parameter, the probability of a connection between two neurons. An ErdösRényi random network with
A better comparison, however, can be made to random networks with degree distributions that match the degree distribution of the gap junction network
We may explore the utility of representing the wiring diagram as a threelayer network by grouping neurons by category (sensory neurons, interneurons, motor neurons). As shown in Tables 2A and 2B in
In this section, we analyze statistical properties of individual neurons and synaptic connections. To characterize the ability of individual neurons to propagate or collect signals, we compute the degree
To visualize the discrete degree distribution,
Neurons or connections with exceptionally high statistics are labeled. The tails of the distributions can be fit by a power law with the exponent
We perform a fitting procedure for the tail of the gap junction degree distribution
To characterize the direct impact that one neuron can have on another, we quantify the strength of connections by the multiplicity,
Finally, the sum of the multiplicities of all gap junction connections of a given neuron is called the number of terminals, or the nodal strength. The tail of the distribution of the number of synaptic terminals,
Identifying neurons that play a central or special role in the transmission or processing of information may also prove useful
Having described statistical properties of individual neurons and connections, such as the degree and multiplicity distributions, we now investigate properties that may describe the efficiency of signal transmission across the gap junction network. Traditionally
The geodesic distance,
A signal originating in one neuron in the giant component must cross
A second measure for signal propagation is the clustering coefficient
Small world networks have much higher clustering coefficient relative to random networks without sacrificing the mean path length. For the giant component of the gap junction network, the corresponding ratios are
Next we consider how quickly individual neurons reach all other neurons in the network. The normalized closeness of a neuron
Restricting to the gap junction giant component, the six most central neurons are AVAL, AVBR, RIGL, AVBL, RIBL, and AVKL. In addition to command interneuron classes AVA and AVB, these include RIBL and RIGL, both ring interneurons, and AVKL, an interneuron in the ventral ganglion of the head. The set of closeness central neurons mostly overlaps with the set of degree central neurons. The correlation between the two centrality measures does not extend to peripheral neurons, as the Spearman rank correlation coefficient
Global network properties discussed in the previous section characterize signal transmission while ignoring connection weights. As weights affect the effectiveness of signal transmission and vary among connections, we now analyze the weighted network by using linear systems theory. Although neuronal dynamics can be nonlinear, spectral properties nevertheless provide important insights into function. For example, the initial success of the Google search engine is largely attributed to linear spectral analysis of the World Wide Web
We characterize the dynamics of the gap junction network by the following system of linear differential equations, which follow from charge conservation
Assuming that gap junction conductance is greater than the membrane conductance, we temporarily neglect the last term and rewrite this equation in matrix form:
The system of coupled linear differential equations (6) can be solved by performing a coordinate transformation to the Laplacian eigenmodes. Since the Laplacian eigenmodes are decoupled and evolve independently in time, performing an eigendecomposition of initial conditions leads to a full description of the system dynamics. The survival function of the Laplacian eigenspectrum is shown in
(a). Survival function of the eigenvalue spectrum (blue). The algebraic connectivity,
What insight can be gained from inspection of the Laplacian eigenmodes? The gap junction network is equivalent to a network of resistors, where each gap junction acts as a resistor. The eigenmodes give intuition about experiments where a charge is distributed among neurons of the network and the spreading charge among the neurons is monitored in time. If the charge is distributed among neurons according to an eigenmode, the relative shape of the distribution does not change in time. The charge magnitude decays with a time constant specified by the eigenvalue. The smallest eigenvalue of the Laplacian is always zero, corresponding to the infinite relaxation time. In the corresponding eigenmode each neuron is charged equally.
If the charge is distributed according to eigenmodes corresponding to small eigenvalues, the decay is rather slow. Thus, these eigenmodes correspond to longlived excitation. The existence of slowly decaying modes often indicates that the network contains weakly coupled subnetworks, in which neurons are strongly coupled among themselves. The corresponding charge distribution usually has negative values on one subnetwork and positive values on the other subnetwork. Because of the relatively slow equilibration of charge between the subnetworks, such an eigenmode decays slowly.
As an example of slow equilibration implying a subnetwork that is strongly internally coupled, one might speculate that the eigenmode associated with
Another interesting example is the eigenmode associated with
These two examples demonstrate that spectral analysis can uncover circuits that have been described using experimental studies. The probability of a known functional circuit appearing in an eigenmode by chance is small (see
To prioritize further analysis of eigenmodes for biological significance, it may be advantageous to focus on the slow and sparse modes, where few neurons exhibit significant activity. We can quantify sparseness of normalized eigenmodes by the sum of absolute values of the eigenmode components, also known as the
The full set of eigenmodes of the connected component is shown in Figure 2 in
To understand timescales, one might wonder what the absolute values of decay constants for various eigenmodes are. Current knowledge of electrical parameters for
What is the effect of the dropped term corresponding to the membrane current in (6)? As this term would correspond to adding a scaled identity matrix to the Laplacian, the spectrum should uniformly shift to higher values by the corresponding amount. Thus, even the eigenmode corresponding to the zero eigenvalue would now have a finite decay time. Assuming the membrane conductance of about
In addition to highlighting groups of neurons that could be functionally related, spectral analysis allows us to predict, under linear approximation, the outcome of experiments that study the spread of an arbitrarily generated excitation in the neuronal network. Such excitation can be generated in sensory neurons by presenting a sensory stimulus
To predict the spread of activity, we may decompose the excitation pattern into the eigenmodes and, by taking advantage of eigenmode independence, express temporal evolution as a superposition of the independently decaying eigenmodes. The initial redistribution of charge would correspond to the fast eigenmodes, whereas the longterm evolution of charge distribution would be described by the slow eigenmodes.
Understanding propagation of neuronal activity in response to stimulation (either for the complete network or for ablation studies) may also be carried out directly in the time domain by stepping through the dynamics in (6) or more electrophysiologically realistic nonlinear dynamics. Predictions of experimental results would then be determined by stimulating and measuring exactly as in the experiment itself.
Several of the quantitative properties computed thus far measure global network structure or individual neuron properties. Now we analyze the frequency of various connectivity subnetworks among small local groups of neurons. Overrepresentation in the subnetwork distribution often displays building blocks of the network such as computational units
Overrepresented subnetworks are boxed, with the
Four neurons can be wired into 11 kinds of subnetworks; this distribution is shown in
Note that neurons participating in motifs also make connections with neurons outside of the motif, which are traditionally not drawn in putative functional circuits
Now we consider the chemical synapse network. Recall that due to structural differences between presynaptic and postsynaptic ends of a chemical synapse, electron micrographs can be used to determine the directionality of connections. Hence the adjacency matrix is not symmetric as it was for the gap junction network.
The network that we analyze consists of
There are two different definitions of connectivity for directed networks. A weakly connected component is a maximal group of neurons which are mutually reachable by possibly violating the connection directions, whereas a strongly connected component is a maximal group of neurons that are mutually reachable without violating the connection directions. The whole chemical synapse network is weakly connected and can be divided into a giant strongly connected component with
The random directed network corresponding to the chemical network is fully weakly connected, even when the degree distribution is taken into account (see
Since chemical synapses form a directed network, neuron connectivity is characterized by indegrees (the number of incoming connections) and outdegrees (the number of outgoing connections) rather than simply degrees. The joint distribution of indegrees and outdegrees is shown in
Neurons or connections with unusually high statistics are labeled. The tails of the distributions can be fit by a power law with exponents
The survival functions associated with the marginal distributions of indegrees and outdegrees are shown in
Multiplicity of connection,
As for the gap junction network, we can also study the distribution of number of synaptic terminals on a neuron. This involves adding the multiplicities of the connections, rather than just counting the number of pre or postsynaptic partners. The joint histogram (not shown) exhibits similar correlation as for the degree distribution, with Pearson correlation coefficient
As for the gap junction network, we can identify central neurons (cf.
In the strongly connected component, we can define the directed geodesic distance as the shortest path between two neurons that respects the direction of the connections. The distribution of the directed geodesic distance, Figure 1(b) in
Although there are several definitions of clustering for directed graphs in the literature
For directed networks, measures of incloseness and outcloseness may be defined using the average directed geodesic distance. In particular, the normalized incloseness is the average geodesic distance from all other neurons to a given neuron:
For the giant component of the chemical network, the most incloseness central neurons include AVAL, AVAR, AVBR, AVEL, AVER, and AVBL. All are command interneurons involved in the locomotory circuit; these neurons are also central in the gap junction network. The incloseness centrality of command interneurons may indicate that in the
The most outcloseness central neurons include DVA, ADEL, ADER, PVPR, AVJL, HSNR, PVCL, and BDUR. Only PVCL is a command interneuron involved in locomotion. The neuron DVA is an interneuron that performs mechanosensory integration; ADEL/R are sensory dopaminergic neurons in the head; and the other central neurons are interneurons in several parts of the worm. The outcloseness centrality of these neurons may indicate that signals can propagate efficiently from these neurons to the rest of the network and that they are in a good position for broadcast.
Although chemical synapses are likely to introduce more nonlinearities than gap junctions, linear systems analysis can provide interesting insights, especially in the absence of other tools. Such an approach has additional merit in
A major source of uncertainty in linear systems analysis of the chemical network is the unknown sign of connections, i.e. excitatory or inhibitory, due to the difficulty in performing electrophysiology experiments. We use a rough approximation that GABAergic neurons make inhibitory synapses, whereas glutamergic and cholinergic neurons form excitatory synapses
Similarly to the gap junction network, we write the system of linear differential equations for the chemical synapse network
To avoid redundancy we defer analyzing this system of differential equations to the next section, where we consider the combined system including both gap junctions and chemical synapses.
We also find subnetwork distributions for the chemical synapse network. Since the network is directed, there are many more possible subnetworks. In particular there are
Overrepresented subnetworks are boxed, with the
Overrepresentation of reciprocal
Having considered the gap junction network and the chemical synapse network separately, we also examine the two networks collectively. To study the two networks, one may either look at a single network that takes the union of the connections of the two networks or one may look at the interaction between the two networks.
First we look at a combined network, which is produced by simply adding the adjacency matrices of the gap junction and chemical networks together, while ignoring connection weights. Thus we implicitly treat gap junction connections as doublesided directed connections. This new network consists of
Naturally, the combined network is more compact than the individual networks. The mean path length
Turning to closeness centrality, the most inclose central neurons are AVAL/R, AVBR/L, and AVEL/R, as would be expected from the individual networks. The most outclose central neurons are DVA, ADEL, AVAR, AVBL, and AVAL, which include the top outclose neurons for both individual networks.
We can also calculate the degree distribution of this combined network. The Pearson correlation coefficient between the indegree and outdegree is
The neurons with the greatest degree centrality are AVAL and AVAR. As for the chemical synapse network, neuron AVAL has the largest indegree and AVAR has the second largest indegree, whereas AVAR has the largest outdegree and AVAL has the second largest outdegree (Figures 4(a) and 4(b) in
As for the chemical synapse network, the tail of the outnumber distribution was fit by a power law and the tail of the innumber distribution could not be fit satisfactorily. The tail of the outnumber distribution could also be fit by an exponential, albeit with lower likelihood. The multiplicity can be fit satisfactorily by a stretched exponential.
In this section we apply linear systems analysis to the combined network of chemical synapses and gap junctions taking into account multiplicities of individual connections. Due to our ignorance about the relative conductance of a single gap junction and of a single chemical synapse, we assume that they are equal. By combining equations (6) and (12) we arrive at:
We proceed to find a spectral decomposition for the combined network. To avoid trivial eigenmodes, we restrict our attention to the strongly connected component of the combined network containing
(a). Eigenvalues plotted in the complex plane. (b). The eigenmode associated with eigenvalue
What is the meaning of complex eigenvalues? The imaginary part of an eigenvalue is the frequency at which the associated eigenmode oscillates. The real part of an eigenvalue determines the amplitude of the oscillation as it varies with time. Eigenmodes that have an eigenvalue with a negative real part decay with time, whereas eigenmodes that have an eigenvalue with a positive real part grow with time. When examining the temporal evolution of the eigenmodes whose eigenvalues are shown in
As for the gap junction network alone, we can look for eigenmodes that may have functional significance. For example, the sixth eigenmode of the combined network,
Having the eigenspectrum of the combined network allows one to calculate the response of the network to various perturbations. By decomposing sensory stimulation among the eigenmodes and following the evolution of each eigenmode, one could predict the worm's response to the stimulation. A similar calculation could be done for artificial stimulation of the neuronal network, induced for example, using channelrhodopsin
As noted for the gap junction alone, the network may also be studied in the time domain directly by stepping through the dynamics in (13) or more electrophysiologically realistic nonlinear dynamics.
We have measured the structural properties of the combined network formed by adding together the adjacency matrices of the gap junction and chemical synapse network, however it is unclear how they interact. The two networks could be independent, or their connections could overlap more or less often than by chance.
To investigate how the two networks overlap, we look at local structure.
Durbin had found that chemical and gap junction networks are essentially independent when imposing physical adjacency restrictions
Why might the presence of connections in two networks either be correlated or anticorrelated? One possibility is that correlated connections simultaneously perform different functions
What are the different functions performed by chemical synapses and gap junctions that could lead to correlation? One possibility is that the two different functions are signinverting and noninverting coupling. Gap junctions are noninverting: higher potential in a neuron raises the potential in other gapjunctioncoupled neurons. Chemical synapses, on the other hand, may be either excitatory (noninverting) or inhibitory (inverting). When the likelihood computations are repeated considering only neuron pairs where the presynaptic neuron is known to be GABAergic
Another measure of the interaction between the two networks is the correlation between the degree sequences. The Pearson correlation coefficient between the gap junction degree and the chemical network indegree is greater than the Pearson correlation coefficient between the chemical network indegree and outdegree. The Pearson correlation coefficient between the gap junction degree and the chemical network outdegree is less than the Pearson correlation coefficient between the chemical network indegree and outdegree. This is shown in
gap/in  gap/out  in/out  email 

correlation coefficient 

avg. rand. perm. 
The two networks seem to primarily reinforce each other with correlated structure rather than augment each other with anticorrelated connections.
Although the reported wiring diagram corrects errors in previous work and is considered selfconsistent, one might wonder how remaining ambiguities and errors in the wiring diagram might affect the quantitative results presented. Furthermore there are connectivity pattern differences among individual worms; these individual variations may have similar effects on the analysis as errors and ambiguities.
For network properties that are defined locally, such as degree, multiplicity, and subnetwork distributions, clearly small errors in the measured wiring diagram lead to small errors in the calculated properties. For global properties such as characteristic path length and eigenmodes, things are less clear.
To study the robustness of global network properties to errors in the wiring diagram, we recalculate these properties in the wiring diagrams with simulated errors. We simulate errors by removing randomly chosen synaptic contacts with a certain probability and assigning them to a randomly chosen pair of neurons. Then, we calculate the global network properties on the ensemble of edited wiring diagrams. The variation of the properties in the ensemble gives us an idea of robustness.
First, we explore the robustness of the small world properties and the giant component calculations. We edit wiring diagrams by moving each gap junction contact with
Properties for the neuronal network from prior work in
Second, we characterize robustness for the linear systems analysis. Because of greater sensitivity of the eigenvalues to errors, we edit wiring diagrams by moving each gap junction contact with
The value
In addition to considering the effect of typical random edits, we can characterize the effect of worstcase errors on the eigenvalues using the
Electron micrographs of chemical synapses have a further ambiguity when more than one postsynaptic partner receives input at a release site. We treated such polyadic (send_joint) synapses no differently than other synapses, but one might alternatively determine multiplicity by counting such synapses at
Small deviations from equality when weighting gap junctions and chemical synapses to form the combined network yield similar spectral changes as the alternate quantitation of chemical synapses displayed in
We have presented a corrected and more comprehensive version of the neuronal wiring diagram of hermaphrodite
We proposed a convenient way to visualize the neuronal wiring diagram. The corrected wiring diagram and its visualization should help in planning experiments, such as neuron ablation.
Next, we performed several statistical analyses of the corrected wiring, which should help with inferring function from structure.
By using several different centrality indices, we found central neurons, which may play a special role in information processing. In particular, command interneurons responsible for worm locomotion have high degree centrality in both chemical and gap junction networks. Interestingly, command interneurons are also central according to incloseness, implying that they are in a good position to integrate signals. However, most command interneurons do not have highest outcloseness, meaning that other outcloseness central neurons, such as DVA, ADEL/R, PVPR, etc., are in a good position to deliver signals to the rest of the network.
Linear systems analysis yielded a principled methodology to hypothesize functional circuits and to predict the outcome of both sensory and artificial stimulation experiments. We have identified several modes that map onto previously identified behaviors.
Networks with similar statistical structural properties may share functional properties thus providing insight into the function of the
The tails of the degree and terminal number distributions for the gap, chemical and combined networks (with the exception of the innumbers) follow a power law consistent with the network being scalefree in the sense of Barabási and Albert
Several statistical properties of the
In addition, we found that motif frequencies in the chemical synapse network are similar to those in the mammalian cortex
To conclude the paper, let us note that our scientific development was not hypothesisdriven, but rather exploratory. Yet we hope that the reported statistics will help in formulating a theory that explains how function arises from structure.
This section describes the methods used to determine neuronal connectivity; see
We started assembling the wiring diagram by consolidating existing data from both published and unpublished sources. Using J. G. White
Next we incorporated R. M. Durbin's data for the anterior portion of the worm, reconstructed from animal
Studies based on green fluorescent protein (GFP) reporters mostly confirm the electron micrograph reconstructions described in MOW. A few differences between GFPstained neurons and White's work have been observed [Hobert O and Hall DH, unpublished]. Notably, the anterior processes of DVB and PVT could have been mistakenly switched in MOW
Most published works have focused on the neck and tail regions of
At the
We found that a large section on the dorsal side of the worm, from just anterior to the vulva to the preanal ganglion, was never electron micrographed at high power magnification. This dearth of imagery was why so many neurons were missing dorsal side reconstructions. Using original thin sections for the
From our compilation of wiring data, including new reconstructions of ventral cord motor neurons, we applied selfconsistency criteria to isolate neurons with mismatched reciprocal records. The discrepancies were reconciled by checking against electron micrographs and the laboratory notebooks of White
For a random network with
To apply this method to the weakly connected component of a directed network, we are interested in the undirected network formed by adding a connection between two neurons if there is a connection in either direction. For a random directed network with probability
Consider the ensemble of random networks with a given degree distribution
Using the computed
For the symmetrized chemical network, the generating function corresponding to the measured degree distribution is
The expected fraction of the network taken up by the giant component,
Continuing from the previous subsection, we find the derivative of the generating function
To find functional forms of the tails of various distributions, we follow the procedure outlined in
Let us bound the probability of finding an eigenmode that comprises a random set of neurons. Let
Now go through each eigenmode and add to a list all possible unordered
Additionally, we can compute the number of all unordered
Thus, if a random set of neurons was selected from all possible sets of neurons, the probability
Suppose we are interested in putative functional circuits of size
Suppose we know
The likelihood ratios shown in
We used the MATLAB package EigTool
Note that MATLAB code for computing several network properties is available at
The collected data is available from the WormAtlas
Algorithm for directed network drawing.
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Algebraic form of survival functions.
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Eigendecomposition.
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Supporting figures and tables.
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We thank John White and Jonathan Hodgkin for the generous donation of the MRC/LMB archival documents and experimental materials to Hall's laboratory at AECOM, without which this study would not have been possible. We also thank Markus Reigl for providing some of the software used in this study. We thank Sanjoy K. Mitter, Scott Emmons, Leon Avery, Mark Goldman, Cori Bargmann, Alexander Teplyaev, Shawn Lockery and Gonzalo de Polavieja for helpful discussions and for commenting on the manuscript.