Bilateral Neuron Pairs Share Redundant Network Roles Despite Incomplete Connection Symmetry in Caenorhabditis elegans

ABSTRACT

Bilateral organization is a prominent feature of the C. elegans nervous system, yet bilateral neuron pairs differ substantially in their individual left–right connections. Whether, and how, such pairs nonetheless come to occupy similar roles in the network is not fully understood. Here we analyze the adult hermaphrodite and male C. elegans connectomes to examine bilateral redundancy using three complementary graph‐theoretic measures: connectivity similarity, which captures overlap in shared in‐ and out‐neighbours; motif‐fingerprint difference, which captures the difference in local motif participation (with lower differences indicating greater overlap); and path compensation, which quantifies the extent to which paired neurons make overlapping contributions to global communication. Across measures, bilateral pairs showed elevated redundancy relative to other neuron pairs. Null‐model comparisons further supported this finding, demonstrating that these patterns cannot arise from preserving symmetry alone: e.g., for connectivity similarity, randomizing the asymmetric edges while holding the symmetric ones reduced bilateral similarity, suggesting that bilateral asymmetry is itself what enables paired neurons to share neighbours and converge on similar network roles. Redundancy was not uniform across the nervous system: in the hermaphrodite it was more pronounced among interneurons than among sensory neurons across all three measures, while in the male this gradient was only significant for path compensation. Together, these results show that bilateral redundancy is a systematic multiscale principle of the C. elegans connectome, one that cannot be explained by bilateral symmetry alone.

We introduce graph‐theoretic measures to quantify bilateral symmetry and asymmetry in the C. elegans connectome. Our results show that symmetry supports functional redundancy and robustness, while structured asymmetry enables functional diversity across sensory, interneuron and motor systems.

Abbreviations

C. elegans

Caenorhabditis elegans

CS

connectivity similarity

JS

Jensen–Shannon

MF

motif fingerprint

PC

principal component

PCI

path compensation index

1. Introduction

Bilateral organization is a widespread feature of nervous systems, but biological bilateral symmetry is rarely exact. Such symmetry can support redundancy, robustness and coordinated dynamics by giving the two sides of a network overlapping structural roles (Tononi et al. 1999; Albert and Barabási 2002; Kitano 2004; Pecora et al. 2014). At the same time, nervous systems often retain left–right asymmetries that support lateralized processing and may increase cognitive capacity by reducing duplication of function across the two sides (Vallortigara and Rogers 2005; Rogers 2021). The resulting balance between redundancy and asymmetry is therefore a central question for understanding how nervous systems preserve robustness without simply duplicating every element.

The Caenorhabditis elegans connectome provides an unusually direct setting in which to study this balance. Its neurons are individually identified, its adult hermaphrodite and male connectomes have been comprehensively reconstructed and a large fraction of its neurons occur as left–right bilateral pairs (White et al. 1986; Cook et al. 2019). Yet these bilateral pairs are not mirror‐identical: many individual structural connections lack a contralateral counterpart and well‐studied sensory pairs such as ASE and AWC show functional left–right specialization (Hobert et al. 2002; Bauer Huang et al. 2007; Ortiz et al. 2009; Hobert 2014). Thus, C. elegans combines pronounced bilateral organization with structural and functional asymmetry within the same cell‐identified nervous system.

Previous analyses have shown that bilateral organization in C. elegans is high but incomplete, thus suggesting that exact mirror duplication is not the only organizing principle of the connectome (Reigl et al. 2004; Azulay et al. 2016; Brittin et al. 2021; Birari and Rabinowitch 2024). What remains less clear is whether the two members of a bilateral pair nonetheless occupy redundant network roles when redundancy is assessed beyond individual connections. In an imperfectly symmetric nervous system, redundancy may appear at different levels of abstraction: shared input–output partners, similar participation in local computational motifs, or alternative routes for global information flow. A multiscale analysis is therefore needed to determine whether incomplete connection symmetry still supports overlapping network roles.

Here we analyze the adult hermaphrodite and male C. elegans connectomes to test whether bilateral neuron pairs show graph‐theoretic redundancy despite incomplete connection‐level symmetry. We first quantify edge‐level bilateral symmetry as a baseline, then assess redundancy using three complementary descriptors: connectivity similarity, motif‐fingerprint difference and a path‐compensation index. Finally, we examine whether bilateral redundancy varies across sensory, motor and interneuron subsystems and across individual bilateral pairs. This framework asks not whether the C. elegans connectome is bilaterally organized, but whether this organization preserves redundant network roles in the presence of substantial asymmetry.

2. Results

We analyzed the adult hermaphrodite and male C. elegans connectomes (Cook et al. 2019), representing each as a directed binary graph in which neurons are nodes and synaptic connections as directed edges and gap junctions as reciprocal directed edges, disregarding connection type and strength (Methods 4.1). Following standard nomenclature (White et al. 1986), bilaterally paired neurons are labelled by their left and right members (e.g., ASEL, ASER), and the remaining, mostly midline neurons are treated as unilateral (Figure 1A). Analyzing both sexes allows us to differentiate the features shared across the nervous system from the sex‐specific ones.

FIGURE 1.

(A) The spatial configuration of the hermaphrodite C. elegans neurons. Bilaterally paired neurons are shown as filled circles (●) and unilateral neurons as plus signs (+). The sensory neurons are shown in blue, interneurons in green and motor neurons in orange. Note that around 69% of motor neurons are unilateral, mostly located along the centre of the body (i.e., sagittal plane). (B) Defining contralateral neurons, edges and symmetric edges. Contralateral neurons are denoted by an asterisk (*). The edges, (x, y*), (x*, y) and (z, u), have contralateral connections (x*, y), (x, y*) and (z, u), respectively. Meanwhile, the edge (x, z) has no contralateral connection. (C) Bilateral symmetry index for the hermaphrodite and male C. elegans connectomes. For each sex, the index is shown for the full network (dark blue‐grey) and for the bilaterally paired subnetwork with unilateral neurons excluded (red), as observed values (points) over the distribution of n = 1000°‐preserving randomized networks (violins; see Methods). z is the standardized deviation of the observed value from the null mean; all observed values exceed every null sample (empirical p < 0.001).

2.1. Quantifying Bilateral Symmetry as a Baseline

Bilateral symmetry of the C. elegans nervous system is well established: most neurons occur as left–right pairs (about two‐thirds: 198/302 in the hermaphrodite and 250/385 in the male), a paired organization documented from the original reconstructions (White et al. 1986) and since quantified at multiple scales (Brittin et al. 2021). We therefore do not set out to show that the connectome is bilaterally symmetric, but to recompute a simple edge‐level summary of this symmetry that serves as a baseline for the redundancy analyses that follow.

For each directed connection we ask whether its contralateral counterpart, the corresponding connection between the contralateral partners of the two neurons, is also present (Figure 1B; formal definitions in Methods). A connection whose counterpart is present is termed symmetric, one whose counterpart is absent, asymmetric. The bilateral symmetry index is the fraction of symmetric connections, equal to one for a perfectly symmetric network and zero for a fully asymmetric one; it can be read as the degree of left–right automorphism of the graph (Morone and Makse 2019). Computed on the two connectomes, the index was 0.68 (hermaphrodite) and 0.66 (male), far above the 0.18 ± 0.006 and 0.23 ± 0.005 of degree‐preserving randomized networks (n = 1000; z = 84 and 86; Figure 1C), confirming prominent bilateral symmetry consistent with prior reports. At the same time, both indices are well below one: roughly a third of connections (32%–34%) have no contralateral counterpart.

Because unilateral neurons are their own contralateral partner, edges among them are symmetric by construction; restricting the index to bilaterally paired neurons lowers it to 0.63 (hermaphrodite) and 0.56 (male) but leaves it far above the same degree‐preserving null (z = 63 and 69; Figure 1C), confirming the high symmetry is not an artefact of unilateral neurons. It is this incomplete symmetry, symmetric overall yet asymmetric in a substantial minority of connections, that the redundancy analyses below address.

2.2. Bilateral Pairs Show Elevated Connectivity Similarity That Does Not Depend on Symmetric Edges Alone

Connectivity similarity measures how far two neurons share the same input and output partners, capturing redundancy in the information each handles (Figure 2A; Methods 4.3). We computed it across two neighbourhood radii (one‐ and two‐step) and two partner sets (inputs alone and inputs and outputs combined), giving four variants (2 × 2). In each sex, bilateral pairs had substantially higher connectivity similarity than the overall population of neuron pairs in all four comparisons (custom shuffle test, p < 0.0001; Figure 2B). Relative to degree‐preserving null networks, this elevation was clear for the one‐step neighbourhoods in both sexes (Wilcoxon signed‐rank, observed pair vs. per‐pair null, p < 0.001), but attenuated for the two‐step neighbourhoods—markedly so in the hermaphrodite, where degree‐matched random networks reached comparably high similarity (Figure 2B).

FIGURE 2.

Bilateral neuron pairs share similar connectivity neighbourhoods. (A) Schematic of connectivity similarity (CS). CS quantifies the overlap between the connectivity neighbourhoods of two neurons (Jaccard index), computed separately for incoming connections (input) and for combined incoming and outgoing connections (input–output). One‐step CS considers directly connected neighbours, whereas two‐step CS additionally includes neighbours reached within two connections (compare the one‐step and two‐step neighbourhoods). (B) Connectivity similarity in the hermaphrodite (top row) and male (bottom row), for four neighbourhood definitions (one‐step input, one‐step input–output, two‐step input, two‐step input–output). In each panel, dots show individual bilateral pairs (Pair; horizontal bar, median), the grey distribution shows all neuron pairs (All) and the blue distribution shows a degree‐preserving null model (Null; 1000 randomizations per pair that preserve each neurons in‐ and out‐degree). Bilateral pairs are more similar than all pairs in every comparison (custom shuffle test, p < 0.0001). Relative to the degree‐preserving null, bilateral pairs are significantly more similar for the one‐step neighbourhoods in both sexes, whereas the effect is weaker for the two‐step neighbourhoods, where the null itself reaches high similarity (Wilcoxon signed‐rank test, observed pair vs. per‐pair null mean; *p < 0.05, **p < 0.01, ***p < 0.001; ns, not significant). n = 92 (hermaphrodite) and 119 (male) bilateral pairs. (C) Connectivity similarity as symmetric (blue) or asymmetric (red) edges are progressively randomized (breaking ratio 0%–100%), for the same neighbourhood definitions and sexes. Lines show the mean across bilateral pairs and shaded bands the 95% confidence interval. Randomizing symmetric edges lowers connectivity similarity more than randomizing asymmetric edges, most prominently for the one‐step neighbourhoods. An edge is classified as symmetric when its bilateral counterpart is also present in the partner neurons neighbourhood and as asymmetric otherwise. Within each sex, the vertical axes are matched across the one‐step panels and across the two‐step panels.

To examine the contribution of symmetric and asymmetric edges to this elevation in one‐step similarity, we held one edge class fixed while progressively randomizing the other (Methods 4.7). Randomizing the symmetric edges (while asymmetric edges retained) reduced one‐step connectivity similarity substantially (median reduction at full randomization ≈65% in the hermaphrodite and ≈75% in the male). Randomizing the asymmetric edges (while symmetric edges retained) also reduced it, by a smaller but consistent and statistically significant amount (median ≈17%–25% across the one‐step comparisons in both sexes; Wilcoxon signed‐rank, per‐pair asymmetric‐edge contribution > 0, p ≤ 5 × 10⁻⁸ in all four one‐step comparisons)—indicating that the specific arrangement of asymmetric connections, not only the symmetric backbone, contributes to bilateral connectivity similarity. The symmetric contribution nonetheless exceeded the asymmetric one in every comparison (per‐pair area‐under‐the‐curve, symmetric vs. asymmetric randomization, paired Wilcoxon, Holm‐adjusted p ≤ 9 × 10⁻¹³). For the two‐step neighbourhoods, connectivity similarity was largely unaffected by either perturbation and remained close to the degree‐preserving null. Consistent with this visual pattern, the symmetric‐edge and asymmetric‐edge randomization curves did not separate significantly across randomization levels in the male (mixed‐model condition × level interaction, n.s.; Figure 2C). This likely reflects substantial chance overlap of two‐step neighbourhoods in such a small network as C. elegans ; we therefore base the edge‐class comparison on the one‐step neighbourhoods.

That bilateral connectivity similarity does not depend on symmetric edges alone is consistent with earlier motif‐ and modularity‐level work, in which overlapping connectivity need not arise from mirror duplication (Reigl et al. 2004; Azulay et al. 2016). Our finding isolates the contribution of the asymmetric edges themselves: even with the symmetric backbone held fixed, their arrangement adds a significant component to bilateral redundancy.

2.3. Motif Fingerprints: Bilateral Pairs Are More Similar in Structural Motifs and More Strongly Functional Motifs

Connectivity similarity captures shared connection partners but not the local operations a neuron performs, so two neurons with overlapping inputs may still participate in different computations (e.g., synchronization, integration, or relay) (Sporns and Kötter 2004; Gollo et al. 2014). Network motifs, i.e., recurring local connection patterns (Figure 3A), capture the local circuit context in which a neuron is embedded, a structural feature often linked to function (Milo et al. 2002; Sporns and Kötter 2004), motivating us to compare motif participation between bilateral neurons.

FIGURE 3.

Bilateral pairs are more similar in motif participation, more strongly for functional than for structural motifs. (A) The 13 connected three‐node directed motifs (m³ ₁–m³ ₁₃). (B) Example bilateral pair: the left neuron (pink) participates in motif m₃ ³ and the right neuron (blue) in m₆ ³; four of the five connections are bilaterally symmetric and one (in m₆ ³) is asymmetric. (C) Computation of the motif‐fingerprint difference. For each neuron, the frequencies with which it participates in each motif form its motif fingerprint; the motif‐fingerprint difference between two neurons is the Jensen–Shannon divergence between their fingerprints (arrow). (D) Structural (upper) and functional (lower) motif‐fingerprint difference for three‐node and four‐node motifs, in the hermaphrodite and male. In each panel, points show individual bilateral pairs (Pair), violins show all neuron pairs (All) and the degree‐preserving null (Null) and open diamonds mark group means. Bilateral pairs have a lower motif‐fingerprint difference than all pairs in every comparison (custom shuffle test, p < 0.0001). Relative to the null, the structural motif‐fingerprint difference of bilateral pairs is higher (more different than chance) whereas the functional motif‐fingerprint difference is lower (more similar than chance); significance of Pair vs. Null is shown per panel (Wilcoxon signed‐rank, observed pair vs. per‐pair null mean; *p < 0.05, **p < 0.01, ***p < 0.001). Vertical axes are truncated at the upper bulk of each distribution to make the group means legible. n = 92 (hermaphrodite) and 119 (male) bilateral pairs.

Symmetric and asymmetric edges shape the local circuit context of a bilateral pair. For the pair in Figure 3B, for example, each neuron participates in motifs m₃ ³ and m₆ ³; four of the five connections are symmetric (symmetry ratio 0.8), with a single asymmetric connection in m₆ ³. Computational models suggest that these motifs can differ in dynamics, m₆ ³ supporting synchronized activity and m₃ ³ not (Gollo et al. 2014), so even a high symmetry ratio need not imply identical function. We note, however, that this dynamical distinction was derived in a Hodgkin–Huxley model and may not transfer directly to C. elegans . Thus, we use it not to make a claim about worm dynamics, but rather to motivate examining motif participation, capturing differences in local circuit context that a high symmetry ratio alone would miss.

We quantified motif participation as the Jensen–Shannon divergence between the motif‐frequency distributions of the two neurons (the motif‐fingerprint difference; Figure 3C; Methods 4.4), for three‐ and four‐node structural motifs (Milo et al. 2002, Sporns and Kötter 2004). Bilateral pairs had a lower structural motif‐fingerprint difference than the full set of neuron pairs (custom shuffle test, p < 0.0001; Figure 3D, upper panels), indicating participation in more similar structural motifs. Relative to a degree‐preserving null, the structural motif‐fingerprint difference of bilateral pairs was higher (i.e., apparently more different than chance); as we show below (Section 2.5). However, this comparison depends on the choice of null model: bilateral pairs appear more structurally different than expected under the unconstrained degree‐preserving null, but more structurally similar than expected once spatial embedding is accounted for. We therefore do not interpret the unconstrained‐null result as evidence of structural divergence.

Structural motifs define which connections are possible, but not all are functionally engaged at once. The functional motifs derived from a structural motif—i.e., the subsets obtainable by removing combinations of edges (Sporns and Kötter 2004)—capture this more directly; brain networks are thought to support diverse functional motifs from a limited structural repertoire (Sporns and Kötter 2004). For example, a feed‐forward structural motif with edges A → B, B → C and A → C contains four functional motifs over the same three nodes: the full feed‐forward pattern itself; the chain A → B → C (using A → B and B → C); the diverging pair A → B and A → C (A drives both B and C); and the converging pair A → C and B → C (both A and B drive C). Edge subsets that leave a node disconnected (for instance a single edge A → B, which isolates C) are fragmented and not counted. A neuron's functional‐motif fingerprint sums these functional motifs across all the structural motifs in which it participates. Because a single structural motif yields several functional motifs, two neurons can have similar functional fingerprints even when their exact structural motifs differ, which is what the functional motif‐fingerprint difference is designed to detect.

Bilateral pairs had a lower functional motif‐fingerprint difference than all neuron pairs (custom shuffle test, p < 0.0001; Figure 3D, lower panels), and, unlike the structural case, a lower functional motif‐fingerprint difference than the degree‐preserving null. Their functional motif participation was more similar than chance. Because removing the asymmetric edges would make the two sides' motif configurations identical, some functional similarity is expected; the key observation is that functional motif participation is conserved more strongly than structural participation. This ordering (not the apparent structural divergence, which is null‐dependent) persists under the spatially‐constrained null, where both exceed chance but functional more so (Section 2.5). We interpret this as a constrained form of degeneracy (that the bilateral neurons are wired similarly and their functional repertoires are more similar still) rather than divergent structure giving rise to similar function (see Discussion).

2.4. Bilateral Neuron Pairs Show Elevated Path Compensation

One potential consequence of having redundancy is the robustness of the nervous system (Dekker and Colbert 2004; Tononi et al. 1999; Albert and Barabási 2002). In particular, when two bilateral neurons (say A and B) are (at least partially) redundant regarding their functions, A can compensate for B's function (although not entirely as a backup) in case B is malfunctioning and vice versa (Kitano 2004). Graph‐theoretic methods for characterizing network organisation, such as modularity‐based community detection (Newman and Girvan 2004), laid the groundwork for analyzing complex networks, including connectomes. Building on these early approaches, our team and other research groups have used vulnerability analysis to probe the connectome's robustness, in this case as the network's capacity to compensate for the loss of its elements (Gol'dshtein et al. 2004; Kim et al. 2016; Mishkovski et al. 2011). However, such vulnerability analyses generally focus on the effects of removing individual nodes or links from a network, measuring the impact on the system's overall properties. While this approach provides valuable insight into the importance of individual components, it falls short of capturing the intricate interdependencies between pairs of neurons. Thus, we adapt a vulnerability‐based measure, path‐compensation index (PCI), to fill this analytical gap. This index quantifies the extent to which one neuron can compensate for another in the global information process when the latter is lost.

The PCI compares the decline in global efficiency (the sum of inverse shortest‐path lengths) when both neurons are removed together against the summed declines from removing each alone (Figure 4A; Method 4.5; Rubinov and Sporns 2010). It is near zero when the two contribute independently, positive when one provides an alternative pathway for the other (redundancy) and negative when removing one disrupts paths that run through the other (interdependence).

FIGURE 4.

Bilateral neuron pairs show high path‐compensation. (A) Schematic of the path‐compensation index (PCI). For a neuron pair in the network (x, red; y, blue), global efficiency is recomputed after removing neuron x alone, after removing neuron y alone and after removing both together. The PCI is the difference between the decline in global efficiency when both are removed and the sum of the declines when each is removed individually. (B) PCI for bilateral pairs (Pair, individual points), all neuron pairs (All, violin) and the degree‐preserving null (Null, violin) in the hermaphrodite and male. Open diamonds mark group means; the dotted line marks PCI = 0 (no net compensation). Bilateral pairs have a higher PCI than all pairs in both sexes (custom shuffle test, p < 0.0001). Relative to the null, bilateral PCI is higher in the hermaphrodite (Wilcoxon signed‐rank, observed pair vs. per‐pair null mean, p < 0.001) but not significantly in the male (p = 0.13). Vertical axis truncated at the upper bulk of the distributions. n = 92 (hermaphrodite) and 119 (male) bilateral pairs. Additional PCI matched‐control analyses are shown in Supplementary Figure S4 and summarized in Supplementary Table S6.

Bilateral neuron pairs had a significantly higher path‐compensation index than all neuron pairs in both sexes (custom shuffle test, p < 0.0001 for both; Figure 4B). Relative to a degree‐preserving null, this elevation was robust in the hermaphrodite (Wilcoxon signed‐rank, observed pair vs. per‐pair null mean, p < 0.001) but weaker in the male, where bilateral pairs exceeded the null on average yet not consistently across individual pairs (per‐pair p = 0.13).

To test whether this effect simply reflected comparison with unrelated neuron pairs or differences in the number of incident edges removed, we added matched‐pair controls (Supplementary Table S6; Supplementary Figure S4). For each bilateral pair, we compared the observed PCI with non‐homologous two‐neuron control pairs sampled from the same one‐step synaptic neighbourhood, from pairs matched for the number of directed incident edges removed and from pairs satisfying both constraints. In the hermaphrodite, bilateral pairs retained higher PCI under all three controls. Under the strict neighbourhood‐plus‐edge‐loss control, bilateral PCI remained higher than the matched‐control mean (bilateral mean PCI = 0.0306, control mean = 0.0067; one‐sided paired permutation p ≤ 1.0 × 10⁻⁴; Wilcoxon signed‐rank p = 0.0031). In the male, bilateral pairs also showed higher mean PCI than matched controls, including the strict combined control (bilateral mean PCI = 0.0239, control mean = 0.0057; one‐sided paired permutation p = 0.0019), but this effect was not significant by signed‐rank testing (p = 0.247). Thus, matched controls support path compensation most strongly in the hermaphrodite and suggest a weaker, subset‐driven effect in the male. Full statistics for all three matched controls are provided in Supplementary Table S6 and pair‐level observed‐versus‐control comparisons are shown in Supplementary Figure S4.

2.5. Bilateral Redundancy Is Robust to Spatial Embedding, Synapse Count and Dataset

We next tested whether the redundancy signatures established above survive three controls: the physical constraint that synapses can form only between adjacent neurons, the removal of weak connections and replication across independently mapped connectomes.

2.5.1. Spatial Embedding

Because synapses form only between physically adjacent neurons, an unconstrained degree‐preserving null permits physically impossible connections and may overstate how far the connectome departs from chance. We therefore repeated the key comparisons against a contactome‐constrained null, rewiring connections only between neuron pairs that share a membrane contact in the adult contactome (1000 randomizations; Methods 4.13, Supplementary Table S1, Supplementary Figure S1) (Brittin et al. 2021). Because this contactome constraint was based on adult hermaphrodite data, we applied this spatially constrained null to the hermaphrodite connectome only. This control therefore tests whether the hermaphrodite redundancy signatures can be explained by physical adjacency and should not be interpreted as a male‐specific spatial‐embedding control.

Because the contactome is itself bilaterally symmetric, this null is far more conservative: the connectivity‐similarity null baseline rose from 0.047 to 0.125 and the symmetry‐index null from 0.18 to 0.50. Even so, the bilateral symmetry index (z = 29), bilateral connectivity similarity (z = 12) and the functional‐motif similarity of bilateral pairs (z = −6.5) all remained well beyond the constrained null (see Methods 4.13). Bilateral path compensation likewise remained significant against this stricter null, although it weakened substantially under the spatial constraint—from z = 31.6 against the unconstrained null to z = 2.64 (p = 0.023)—indicating that most, but not all, of the bilateral path‐compensation advantage is attributable to physical proximity. The exception was the structural motif‐fingerprint difference: against the unconstrained null, bilateral pairs appeared structurally more different than chance (z = +5.7), but against the contactome‐constrained null they were structurally more similar than chance (z = −4.1). The apparent structural divergence of bilateral pairs therefore reflects comparison with physically implausible wiring; once spatial embedding is accounted for, bilateral pairs are structurally similar and their functional‐motif participation is conserved even more strongly (z_struct − z_func = +2.4 under the constrained null). In the hermaphrodite, bilateral connectivity similarity, edge‐level symmetry, functional‐motif similarity and path compensation are therefore robust to spatial embedding, although the path‐compensation effect is substantially weakened.

2.5.2. Synapse‐Count Thresholding

To check that these results are not driven by weak, low‐count connections, which are least reproducible across individuals, we recomputed all measures after retaining only connections supported by at least two, three, or five synaptic contacts (Methods 4.11; Supplementary Table S2, Supplementary Figure S2). Across thresholds and in both sexes, the bilateral symmetry index remained essentially unchanged (0.68–0.69 in the hermaphrodite, 0.65–0.66 in the male) while its degree‐preserving null declined, so the index stayed far above its null at every threshold. Bilateral pairs retained higher connectivity similarity, lower three‐node motif‐fingerprint difference and higher path compensation than the full set of neuron pairs at every threshold (all p < 0.001) and the motif effect strengthened with stricter thresholds.

2.5.3. Replication Across Connectomes

To test generalization, we repeated the analyses on the adult hermaphrodite connectome of Varshney et al. (2011) and eight individually reconstructed connectomes from Witvliet et al. (2021) spanning the L1, L2, L3 and adult stages. The adult connectome analyzed in the main text is included for reference, giving ten connectomes in total (Methods 4.12; Supplementary Table S3, Supplementary Figure S3). Each connectome was converted to a directed binary graph. Bilateral pairs were the bilaterally homologous neuron pairs present in that dataset and all remaining neuron pairs formed the comparison set. For each dataset we computed the bilateral symmetry index, connectivity similarity, structural and functional motif‐fingerprint differences and path‐compensation index, each against a degree‐preserving null and expressed every measure as a z‐score relative to its null; the structural/functional dissociation was summarized as the difference between the structural and functional z‐scores (z_struct − z_func). To compare effects across connectomes of different size, for the two measures defined by a bilateral‐versus‐all‐pairs contrast (connectivity similarity and path compensation), we additionally computed a network‐size‐independent effect size, Cohen's d = (mean_bilateral − mean_all‐pairs) / SD_all‐pairs, from the per‐pair distributions; the all‐pairs standard deviation (not the standard error of the mean) is used so that d does not scale with the number of pairs. The symmetry index, which has no all‐pairs counterpart, is reported as its value. For path compensation, pairs for which the index is undefined because no alternative path exists (‘path‐degenerate’ pairs) were excluded.

Across all ten connectomes, bilateral pairs showed the same direction of redundancy effects. One‐step connectivity similarity was consistently higher for bilateral pairs than for the all‐pairs background, with large effect sizes for both input similarity (Cohen'SD = 1.91–2.64) and input–output similarity (d = 2.24–3.67). Path compensation was also higher for bilateral pairs in every dataset, although with more variable magnitude (d = 0.55–2.74). Conversely, bilateral pairs showed lower motif‐fingerprint difference across all datasets, both for three‐node structural motifs (d = −1.00 to −0.67) and three‐node functional motifs (d = −1.01 to −0.48). Thus, the main redundancy signatures were reproduced across an independent adult hermaphrodite connectome and across developmental connectomes from L1 to adulthood.

2.6. Redundancy Varies Across Subsystems and Individual Neuron Pairs

We next asked how bilateral redundancy is distributed across the three neuronal classes: sensory, motor and interneurons (White et al. 1986). For each family of redundancy measure (the four connectivity‐similarity variants, the four motif‐fingerprint differences and the path‐compensation index; see previous sections), we summarized each pair by its first principal component (PC1); PC1 captured the large majority of the variance for the connectivity‐similarity and motif‐fingerprint families (81%–92%) and for the single path‐compensation index it is the index itself (Methods 4.9).

Comparing PC1 across classes reveals a consistent gradient across the three redundancy measures (Figure 5A–C). To assess these, for connectivity similarity and motif‐fingerprint difference we used one‐way ANOVA with Tukey post hoc comparisons; for the path‐compensation index, whose distribution is strongly right‐skewed and zero‐inflated with unequal group variances and therefore violates ANOVA assumptions, we used the rank‐based Kruskal–Wallis test with Dunn post hoc comparisons (Methods 4.9; corresponding ANOVA values are tabulated in Supplementary Table S5).

FIGURE 5.

Redundancy varies across neuronal subsystems and individual bilateral pairs. (A–C) Subsystem‐level redundancy. PC1 scores for each redundancy‐measure family are shown for sensory, interneuron and motor bilateral pairs: connectivity similarity (A), motif‐fingerprint difference (B) and path‐compensation index (C). Upper panels show the hermaphrodite connectome and lower panels show the male connectome. Violins indicate the distribution across bilateral pairs, points indicate individual pairs and open diamonds indicate group means. For motif‐fingerprint difference, the PC1 axis was sign‐flipped so that higher values indicate greater redundancy, consistent with the other measures. Class effects for connectivity similarity (A) and motif‐fingerprint difference (B) were tested by one‐way ANOVA with Tukey post hoc comparisons; for the path‐compensation index (C), which is non‐normally distributed, by Kruskal–Wallis with Dunn post hoc comparisons (Methods 4.9). Significance symbols indicate the sensory–interneuron comparison (*p < 0.05, **p < 0.01, ***p < 0.001, ns). Full ANOVA, Tukey post hoc and Kruskal–Wallis robustness statistics are reported in Supplementary Table S5. (D–F) Pair‐level redundancy ranking. Bilateral neuron pairs were ranked by redundancy PC1 for connectivity similarity (D), motif‐fingerprint difference (E) and path‐compensation index (F), with rank 1 indicating the most redundant pair. Each pair is coloured by its K‐means cluster (K = 3), corresponding to low, medium, or high redundancy. Known lateralized sensory pairs, ASE and AWC, are marked with triangles and command interneuron pairs, AVA and AVB, are marked with circles.

In the hermaphrodite, the class effect was significant for all three measures: connectivity similarity, F(2, 89) = 10.83, p = 6.16 × 10⁻⁵, η² = 0.196; motif‐fingerprint difference, F(2, 89) = 8.35, p = 4.76 × 10⁻⁴, η² = 0.158; and path‐compensation index, Kruskal–Wallis χ²(2) = 10.27, p = 0.0059, ε² = 0.11. The sensory–interneuron difference was significant in every case (connectivity similarity, Tukey p = 3.34 × 10⁻⁵; motif‐fingerprint difference, Tukey p = 6.44 × 10⁻⁴; path‐compensation index, Dunn p = 0.0041).

In the male, the gradient was present for path compensation but absent for the other two measures. Connectivity similarity and motif‐fingerprint difference showed no class effect under either parametric or nonparametric tests (connectivity similarity, F(2, 116) = 0.74, p = 0.482, Kruskal–Wallis p = 0.49; motif‐fingerprint difference, F(2, 116) = 0.47, p = 0.626, Kruskal–Wallis p = 0.53). The path‐compensation index, however, showed a significant class effect (Kruskal–Wallis χ²(2) = 15.77, p = 3.8 × 10⁻⁴, ε² = 0.13), with a significant sensory–interneuron difference (Dunn p = 4.9 × 10⁻⁴). The parametric ANOVA did not detect this effect (F(2, 116) = 2.57, p = 0.081) because the male PCI distribution is dominated by skew and a small number of high‐compensation outlier pairs that inflate within‐group variance; the rank‐based test, which is insensitive to these outliers, recovers the underlying gradient.

In sum, the sensory‐to‐interneuron increase in redundancy was robust across all three measures in the hermaphrodite, whereas in the male it was clear for path compensation but not for connectivity similarity or motif participation.

Beyond the subsystem averages, we ranked individual bilateral pairs. As a check that the measures recover known biology, we first sorted pairs into low‐, medium‐ and high‐redundancy groups by K‐means clustering (K = 3) on each measure family (higher ‘lateralization’ here denotes lower redundancy; Figure 5D–F). As expected, the functionally lateralized ASE and AWC pairs (Hobert et al. 2002) fell in the high‐lateralization group in both sexes, with one exception (connectivity similarity in the hermaphrodite) and the command interneurons AVA and AVB, established rich‐club hubs (Towlson et al. 2013), fell in the high‐redundancy group on connectivity similarity and motif‐fingerprint difference and ranked among the highest in path compensation. That these neurons land where prior physiology and network analysis would predict supports the measures as redundancy descriptors; with those specific findings not new, except as found under this measure.

Ranking all bilateral pairs by PC1 then extended this to the whole population (Figure 5D–F; full lists in Supplementary Table S4). The rankings were consistent across measures and both sexes and recapitulated the subsystem gradient at the level of individual pairs. The most redundant extreme comprised command and premotor interneurons: AVA and AVB ranked first or second on connectivity similarity and path compensation in both sexes, alongside AVE, PVC, RIM and RIC. The least redundant (most lateralized) extreme was dominated by sensory neurons, including the labial and cephalic classes (URA, URB, URY, IL1/IL2), the chemosensory AWA and AWB pairs and BAG; in the male, several sex‐specific pairs (the ray neurons R7A, R7B, R8A and PVD) also fell at this lateralized extreme. Thus, this ranking extends beyond a few well‐known neurons (e.g., AVA, AVB) to characterize the redundancy status of every bilateral pair, identifying which are most and least redundant across the whole population and showing that this ordering follows the same sensory‐to‐interneuron gradient seen at the subsystem level.

3. Discussion

Across three complementary measures and both sexes, bilateral neuron pairs in C. elegans occupied more similar network roles than the general population of pairs, despite differing substantially in their individual left and right connections. This redundancy held at three scales: shared input–output neighbourhoods, local motif participation and global path structure. Together, these findings suggest that the C. elegans nervous system preserves bilateral network‐level reliability not through exact mirror duplication of every connection, but by maintaining overlapping network roles across partially asymmetric wiring. It was robust to spatial embedding, synapse count and connectome dataset, and varied systematically across functional subsystems. We discuss, in turn, what this redundancy‐despite‐asymmetry implies (Section 3.1), how it relates to degeneracy and to the choice of null (Section 3.2), what it suggests about the design of sensory, interneuron and motor systems (Section 3.3), and how it sits within prior work (Section 3.4).

3.1. Bilateral Redundancy Despite Incomplete Symmetry

The three measures converge on the same conclusion from different starting assumptions and it is their agreement that is informative. Connectivity similarity asks whether two neurons draw on the same inputs and outputs, an overlap that predicts neural dynamics (Uzel et al. 2022); motif‐fingerprint difference asks whether they perform similar local operations regardless of partner identity; and path compensation asks whether either can substitute for the other in global information flow. Each measure ignores what the others retain, so each can be read as a different abstraction of bilateral symmetry. That bilateral pairs score as redundant on all three, in both sexes, indicates an overlap of network role that does not depend on any single definition of redundancy.

Indeed, this redundancy is not a simple function of how many connections are symmetric. For connectivity similarity, holding the symmetric edges fixed and randomizing the asymmetric ones shifted bilateral similarity away from its intact value (Section 2.2), so the particular arrangement of asymmetric connections, not just the symmetric backbone, contributes to it. We did not perform the equivalent decomposition for motif participation or path compensation and so restrict this claim to connectivity similarity rather than to bilateral redundancy in general.

The redundancy properties were largely shared between the sexes despite the male's additional sex‐specific circuitry (Cook et al. 2019): the symmetry index, the elevated connectivity similarity, lower motif‐fingerprint difference and higher path compensation of bilateral pairs all held in both and were robust to synapse‐count thresholding. The clearest difference was at the subsystem level: the sensory‐to‐interneuron increase in redundancy was significant in the hermaphrodite across all three redundancy measures, whereas in the male it was significant for path compensation (Kruskal–Wallis p < 0.001) but not for connectivity similarity or motif participation (Figure 5A–C). The male's extra sex‐specific neurons, many in the tail and serving mating, enlarge and diversify the sensory and interneuron populations and may blur a distinction that is sharper in the smaller, more stereotyped hermaphrodite. We therefore treat the connectivity‐ and motif‐based gradient as a hermaphrodite finding, while the path‐compensation gradient holds in both sexes.

3.2. Degeneracy and the Role of the Null

Degeneracy, the capacity of structurally different elements to perform the same function, can confer robustness without the cost of duplicating identical components (Kitano 2004; Tononi et al. 1999; Noppeney et al. 2004). Developmental programmes can generate distinct left–right terminal identities within otherwise bilateral neuron pairs, as exemplified by ASE and AWC sensory neurons (Hobert et al. 2002; Bauer Huang et al. 2007; Ortiz et al. 2009; Hobert 2014). Our measures are consistent with this broader picture: bilateral partners can retain overlapping network roles while differing in a substantial fraction of their individual connections. Whether this asymmetry amounts to genuine structural divergence, however, depends entirely on the null against which it is judged.

At face value, bilateral pairs participate in measurably different structural motifs yet very similar functional ones, which would suggest degeneracy in its strong sense: different structure producing similar function. The contactome‐constrained null overturns the structural half of this reading. Against an unconstrained null, bilateral pairs appear structurally more different than chance, but against physically plausible wiring they are structurally more similar than chance (Section 2.5); the apparent divergence was an artefact of comparing to impossible connections. What survives the stricter null and holds across all ten connectomes and across development, is the dissociation between the two motif levels: functional‐motif participation is conserved more strongly than structural‐motif participation. We therefore favour a constrained reading of degeneracy. Rather than structural divergence with functional convergence, the result is functional convergence that exceeds an already high structural and physical similarity: bilateral neurons are wired more similarly than physical adjacency requires and their functional repertoires are more similar still. This does not require the two neurons to be structurally divergent, only that their functional equivalence be maintained above and beyond their structural similarity.

Although our analysis is restricted to C. elegans , the results suggest a general conceptual framework in which bilateral organization can preserve overlapping network roles without requiring mirror‐identical wiring. Whether analogous principles support compensatory mechanisms in larger nervous systems remains an important question for future structural and dynamical studies.

3.3. Subsystem Gradient Suggests Diversified Input and Redundant Integration

The redundancy gradient across subsystems maps onto the connectome's basic sensory‐to‐motor architecture, in which sensory neurons feed one to three layers of interneurons, the last of which (the command interneurons) drive the motor neurons (Gray et al. 2005; Jarrell et al. 2012; Qian et al. 2011). Read against this S → I → M layout, the gradient suggests a simple design logic: asymmetric, lateralized input; symmetric, redundant integration; and unbiased output. We develop this as a hypothesis, noting that the underlying gradient was statistically clearest in the hermaphrodite (Section 3.1).

Lower redundancy in the sensory system is consistent with a strategy of diversifying the information an animal samples from its environment. The known lateralized sensory pairs ASE and AWC illustrate this pattern, but the conclusion rests on the broader subsystem‐level comparison across bilateral pairs rather than on these canonical examples alone. Broadening the range of detectable signals can improve detection and exploitation of resources, mates and predators (Arnqvist 2006; Ryan 1990; Jordan and Ryan 2015), so a less redundant, more lateralized sensory periphery carries a plausible adaptive advantage.

Higher redundancy in the interneuron system fits its role in integrating and relaying signals across the network (Hendricks et al. 2012; Ghosh et al. 2017; Gordus et al. 2015). Symmetric, redundant wiring increases resilience to noise, perturbation and partial damage (Dekker and Colbert 2004; Tononi et al. 1999; Albert and Barabási 2002) and it allows the diverse, asymmetric input arriving from sensory neurons to be combined into balanced signals that drive coordinated behaviour on both sides of the body. The command interneurons AVA and AVB, which were among the most redundant pairs, are consistent with this: both are central to robust locomotion control (Kato et al. 2015; Kaplan et al. 2018; Uzel et al. 2022).

The motor system fell between the two, but this placement should be read with caution: most motor neurons (about 69% in both sexes) are unilateral and lie near the midline and so are excluded from a pairwise redundancy analysis (Figure 1A). If these midline neurons are regarded as merged or not‐yet‐duplicated pairs, the motor system is better described as broadly unbiased than as intermediate, an organization compatible with the coordinated motor output required for forward undulatory locomotion (Wen et al. 2012; Zhen and Samuel 2015). The residual asymmetry in some motor pairs may relate to asymmetric motor behaviours such as male mating (Downes et al. 2012); because the motor results did not reach significance, we flag this only as a direction for future work.

3.4. Relation to Prior Analyses of Symmetry and Redundancy

Classical anatomical reconstructions already established bilateral organization as a defining feature of the C. elegans nervous system. Building on earlier posterior nervous system work by Hall, White's work described much of the nervous system as bilaterally organized and Durbin's thesis introduced terminology and developmental/anatomical concepts that underlie later connectome analyses (White et al. 1986; Durbin 1987; Hall 1977). The bilateral symmetry, contralateral pairing and incomplete left–right correspondence that we catalogue are therefore not new observations. Our contribution is instead to quantify whether bilateral partners retain redundant network roles across connectivity neighbourhoods, motif participation and global path structure, even when their detailed connectivity patterns differ.

Several graph‐theoretic studies do bear directly on this question. Sohn et al. (2011) used bilateral symmetry as an explicit constraint in community detection, requiring each bilateral pair to share a module and recovered hierarchically organized clusters matching known functional circuits, with command interneurons as inter‐module hubs. Although their reliance on bilateral co‐membership presupposes the shared network roles that we measure directly, their identification of command interneurons as hubs anticipates our finding that command‐interneuron pairs are among the most redundant.

Brittin et al. (2021) quantified bilateral organization across scales by combining the synaptic connectome with the contactome and synapse size. Our edge‐level symmetry analysis recovers the same qualitative signal—high but incomplete bilateral symmetry—and is therefore treated more as confirmation and extension rather than a new finding. Their multiscale analysis also motivates our contactome‐constrained and synapse‐count robustness controls.

Earlier motif‐ and module‐level analyses further show that bilateral organization is not reducible to exact mirror duplication. Reigl et al. (2004) showed that several overrepresented motifs in the C. elegans network cannot be explained by bilateral symmetry alone, while Azulay et al. (2016) showed that common‐neighbour structure reveals homogeneous, functionally interpretable circuit modules. As in any motif analysis, such over‐ and under‐representation must be interpreted against appropriate null models and spatial‐embedding constraints. Our contribution is not to re‐establish that symmetry is not the sole organizing principle, but to quantify with increasing precision and clarification, at the level of contralateral pairs and across three network scales, how redundancy is distributed given that organization.

Most directly, Birari and Rabinowitch (2024) examined how symmetry and asymmetry shape redundancy in the same connectome, but the two studies (theirs and ours) operationalize redundancy differently and are complementary rather than overlapping. They quantified asymmetry as the fraction of unpaired synapses within each neuron class and defined redundancy as parallel paths, contrasting it with reachability. We instead have asked whether the two members of a bilateral pair occupy similar network positions, quantifying redundancy as pairwise contralateral similarity across connectivity neighbourhoods, motif participation and global path structure. Despite these differences, both studies converge on a similar picture: substantial connection‐level asymmetry coexists with and does not abolish, bilateral redundancy. We extend this picture by showing that redundancy varies across functional subsystems and is preserved across both sexes, while leaving the reachability side of their framework as a natural direction for integration.

3.5. Limitations and Conclusion

Our analysis rests on two deliberate simplifications. First, we treated the connectome as a directed, binary graph, setting aside synapse number, strength and sign. The same abstraction collapses the polyadic architecture of C. elegans synapses (single presynaptic specializations contacting several postsynaptic partners) into independent pairwise edges, discarding a biological feature the network representation does not retain (Durbin 1987). Incorporating these would enable functional redundancy and robustness to be assessed at an even finer resolution. Second, our analyses were static, whereas neural function depends on dynamics and signal propagation that a structural graph captures only indirectly (Randi et al. 2023; Uzel et al. 2022). Structural redundancy of the kind we measure therefore bounds, but does not determine, functional redundancy and relating the two is a natural next analytical step.

In sum, the two members of a bilateral neuron pair in C. elegans tend to occupy redundant network roles even though their individual connections are only partly symmetric and this redundancy is graded along the sensory‐to‐interneuron axis. Whether the same balance of symmetry and asymmetry organizes larger nervous systems is an open question that the measures used here could help address.

4. Methods

4.1. Constructing Directed Graphs for Adult Hermaphrodite and Male C. elegans Connectomes

We analyzed the adult hermaphrodite and male C. elegans connectome datasets reported by Cook et al. (2019). The detailed depiction of connections within this connectome data encompasses various attributes, including the nature of connections (chemical synapses or gap junctions), their direction and their strength, as quantified by the number and sizes of connections. For the sake of this paper, we excluded the pharyngeal neurons from the connectome data for both sexes due to their distinction from the somatic nervous system.

We combined the network of gap junctions and chemical synapses to generate directed and binary networks for C. elegans . Chemical synapses are directed and were represented as an edge from the presynaptic to the postsynaptic neuron. Gap junctions (electrical synapses) are bidirectional and were represented as two reciprocal directed edges, one in each direction, between the connected neurons. Despite the differences between gap junctions and chemical synapses, our emphasis lies in concentrating on the process of information transfer, with a disregard, therefore, for the specific characteristics and intensities of these connections. Representing a gap junction as a reciprocal pair of edges does not by itself make it bilaterally symmetric: each gap‐junction edge is classified as symmetric or asymmetric by the same contralateral criterion used for chemical edges. For example, the gap junction between ASEL and AWCL is counted as symmetric only if the corresponding gap junction between ASER and AWCR is also present.

As a result, we were able to establish connectomes comprising 4916 connections among 280 neurons in hermaphrodites and 5459 connections among 362 neurons in male. Both sexes share 272 neurons, alongside sex‐specific neurons unique to each sex (eight in hermaphrodite and 90 in male). Additional details concerning individual C. elegans neurons, including their functions, were extracted from the wormatlas website (Altun and Hall 2015).

The C. elegans connectomes are graphically represented as $G=\left(V,E\right)$, where V represents the set of nodes (neurons $x,y,\dots$) and E represents the set of edges (connections $\left(x,y\right)$, …) among these nodes. The connectivity structure is succinctly summarized by the adjacency matrix A, wherein $A_{\mathit{ij}}=1$ if $\left(i,j\right)\in E$ which signifies a direct connection from neuron $i$ to neuron $j$, whereas $A_{\mathit{ij}}=0$ if $\left(i,j\right)\notin E$ which denotes the absence of such a connection. Self‐connections are treated as nonexistent ($\forall A_{\mathit{ii}}=0$).

4.2. Symmetry Index

The left–right symmetry of C. elegans neurons has been comprehensively categorized in cellular contexts through the efforts of White and his colleagues. By utilizing electron microscopy data to reconstruct the nervous system of C. elegans , White grouped neurons with analogous structures into distinct classes, denoting their symmetry through the final letter of the neuron's name as ‘L’ or ‘R’ (for instance, the AVA neuron class consisting of AVAL and AVAR). Roughly two‐thirds of C. elegans neurons (198 out of 302 neurons in hermaphrodites and 250 out of 385 neurons in males) exhibit bilateral symmetry. The remaining neurons are predominantly situated on or near the midline and lack contralateral counterparts.

Derived from these anatomical symmetry attributes in C. elegans , we can define contralaterality, marked as (*), as the bilateral counterpart of a given component. For instance, within a bilaterally symmetric neuron pair, if a neuron resides on the left side of the body, its contralateral partner in the pair would be positioned on the right side (e.g., x = ASEL is the contralateral neuron for x* = ASER). Unilateral neurons lack corresponding contralateral counterparts, thus resulting in defining contralateral neurons of unilateral neurons as the neurons themselves (for instance, the u = CA01 neuron serves as its own contralateral neuron, u* = u = CA01). The existence of contralateral neurons inherently introduces a contralateral edge between them. Specifically, we denote the contralateral edge by (*), to represent the connection between corresponding contralateral neurons.

$${\left(x,y\right)}^{*}=\left(x^{*},y^{*}\right)$$ (1)

An asymmetric edge is identified by its presence when the contralateral edge, which connects corresponding contralateral neurons, is absent within the network. Conversely, a symmetric edge is defined as a link that manifests when a contralateral edge exists between the corresponding contralateral neurons.

$$\left(x,y\right)=\left\{\begin{array}{c}\text{symmetric}\text{if}{\left(x,y\right)}^{*}\in E\\ \text{asymmetric}\text{if}{\left(x,y\right)}^{*}\notin E\end{array}\right.$$ (2)

4.3. Connectivity Similarity

The connectivity similarity measure is designed to determine the degree of resemblance between the sets of neighbouring nodes, facilitating the detection of potential symmetry in the connectivity identities. In this study, we focused on the two types of distinction of neighbourhoods: the 1‐step/2‐step neighbourhood and the input/output neighbourhood. The 1‐step/2‐step neighbourhood includes nodes that are connected to the given node by either one or two directed steps. Meanwhile, the input and output neighbourhood consist of nodes that provide inputs to the node under consideration and those that receive outputs from it, respectively. The neighbourhood of neuron $\mathcal{N}\left(x\right)$ is formally defined as follows:

$$\begin{array}{c}\begin{array}{c}{\mathcal{N}}_{\text{in},1}\left(x\right)=\left\{y\in V|{\mathrm{A}}_{\mathit{yx}}=1\right\},\\ {\mathcal{N}}_{\text{out},1}\left(x\right)=\left\{y\in V|{\mathrm{A}}_{\mathit{xy}}=1\right\},\end{array}\\ {\mathcal{N}}_{\text{in},2}\left(x\right)=\bigcup _{y\in {\mathcal{N}}_{\text{in},1}\left(x\right)}{\mathcal{N}}_{\text{in},1}\left(y\right),\\ {\mathcal{N}}_{\text{out},2}\left(x\right)=\bigcup _{y\in {\mathcal{N}}_{\text{out},1}\left(x\right)}{\mathcal{N}}_{\text{out},1}\left(y\right).\end{array}$$ (3)

With the definition of neighbourhood, k‐step input connectivity similarity (${\mathrm{CS}}_{\text{in},k}$) and k‐step input–output connectivity similarity (${\mathrm{CS}}_{\text{in}-\text{out},k}$) is defined as follows:

$$\begin{array}{c}{\mathrm{CS}}_{\text{in},k}\left(x,y\right)=\frac{\left|{\mathcal{N}}_{\text{in},k}\left(x\right)\cap {\mathcal{N}}_{\text{in},k}\left(y\right)\right|}{\left|{\mathcal{N}}_{\text{in},k}\left(x\right)\cup {\mathcal{N}}_{\text{in},k}\left(y\right)\right|},\\ {\mathrm{CS}}_{\text{in}-\text{out},k}\left(x,y\right)=\frac{\left|{\mathcal{N}}_{\text{in},k}\left(x\right)\cap {\mathcal{N}}_{\text{in},k}\left(y\right)\right|+\left|{\mathcal{N}}_{\text{out},k}\left(x\right)\cap {\mathcal{N}}_{\text{out},k}\left(y\right)\right|}{\left|{\mathcal{N}}_{i\mathrm{n},k}\left(x\right)\cup {\mathcal{N}}_{\text{in},k}\left(y\right)\right|+\left|{\mathcal{N}}_{\text{out},k}\left(x\right)\cup {\mathcal{N}}_{\text{out},k}\left(y\right)\right|}.\end{array}$$ (4)

4.4. Motif‐Fingerprint Difference

Motifs are defined as local connectivity patterns comprising a set of n nodes, without any fragmented components or nodes and they form the core elements of a network's architecture (Milo et al. 2002; Sporns and Kötter 2004). For instance, in the case of triads (motif size 3), there exist 13 potential configurations (see Figure 3), whereas for tetrads (motif size 4), the number of possible configurations rises to 199. Our investigation analyzed normalized motif fingerprints, mainly focusing on three and four‐node structural and functional motifs. The motifs fall into two main categories: structural and functional. Structural motifs are the local patterns of physical connections. In contrast, functional motifs include all possible non‐fragmented subgraphs of these local connectivity patterns found within structural motifs, thus representing the possible realization of elementary operation using given structural motifs.

Following (Sporns and Kötter 2004), we distinguish structural from functional motifs. A structural motif is an exact local connectivity pattern—a connected subgraph of n nodes with a specific set of edges. A functional motif is any connected subgraph that can be formed from a subset of the edges of a structural motif; a single structural motif therefore contains multiple functional motifs, corresponding to the different elementary connection patterns its wiring can support. For both motif types we used the same set of motif classes (13 for n = 3, 199 for n = 4) and computed, for each neuron, the frequency with which it participates in each class—counting exact occurrences for structural fingerprints and occurrences as a connected subset of the surrounding connectivity for functional fingerprints—using the corresponding structural and functional motif routines of the Brain Connectivity Toolbox (Rubinov and Sporns 2010).

The motif‐fingerprint difference measures the disparity between the motif probability distributions (be they structural or functional) in which a particular pair of neurons is involved. To determine the n‐node motif probability distributions for a specific neuron x (denoted as a probability vector ${\overrightarrow{p}}_x^n$, with each vector component indicating the probability of the k‐th motif configuration around neuron x), we first calculated the n‐node motif frequency around neuron x (${\overrightarrow{m}}_x^n$, where each vector component corresponds to the frequency of the k‐th motif configuration). This calculation was performed using the Brain Connectivity Toolbox (Rubinov and Sporns 2010). We then transformed this motif frequency into a probability distribution, applying Laplace (add‐one) smoothing to avoid the problem of zero probability. The formula for the n‐node motif probability distributions for neuron x is as follows:

$${\overrightarrow{p}}_x^n\left(k\right)=\frac{{\overrightarrow{m}}_x^n\left(k\right)+1}{{\sum }_{k=1}^{\left|{\overrightarrow{m}}_x^n\right|}\left[{\overrightarrow{m}}_x^n\left(k\right)+1\right]}$$ (5)

where ${\overrightarrow{m}}_x^n\left(k\right)$ and ${\overrightarrow{p}}_x^n\left(k\right)$ are the k‐th components of ${\overrightarrow{m}}_x^n$ and ${\overrightarrow{p}}_x^n$ (i.e., corresponding to the k‐th motif configuration), respectively. Then, the motif‐fingerprint difference between neuron x and neuron y was defined as the Jensen‐Shannon divergence ($D_{\mathrm{JS}}$) between the motif probability distributions of the given neurons x and y: $D_{\mathrm{JS}}\left({\overrightarrow{p}}_x^n\parallel {\overrightarrow{p}}_y^n\right)$. This same method was used to compute the four variants of motif‐fingerprint difference, specifically the three‐node structural, three‐node functional, four‐node structural and four‐node functional differences.

4.5. Path‐Compensation Index

We define path‐compensation index to quantify the extent to which one neuron can substitute for another in the overall process of information transmission, in cases where the latter is compromised (essentially, providing a redundant pathway). To calculate the path‐compensation index, we first define the vulnerability of a neuron set X to removal of a neuron set Y as the change in the shortest‐path structure among the neurons in X when the neurons in Y, together with their edges, are deleted from the graph. The formula for vulnerability is given as follows:

$${\mathrm{Vul}}_X\left(Y\right)={\mathrm{GE}}_X\left(V\right)-{\mathrm{GE}}_X\left(V-Y\right)$$ (6)

where ${\mathrm{GE}}_X\left(Z\right)$ is the sum of the inverses of the shortest paths between any two neurons in X when only edges connected between the neurons in Z are available for travelling paths. Thus, ${\mathrm{Vul}}_X\left(Y\right)$ represents the vulnerability of the paths (i.e., the extent of alteration required in the shortest paths) between any two neurons in X when the neurons in Y (as well as the edges connected with the neurons in Y) are removed from the graph G. Then, path‐compensation index (PCI) for neuron x and y is defined as the discrepancy between the vulnerability of the paths between any neurons in $V-\left\{x,y\right\}$ when both neurons are removed together (i.e., ${\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x,y\right\}\right)$) and the linear summation of the vulnerabilities of the paths between any neurons in $V-\left\{x,y\right\}$ when only one of them is removed individually (i.e., ${\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x\right\}\right)+{\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{y\right\}\right)$). Thus, PCI for neuron x and y is defined as follows:

$$\mathrm{PCI}\left(x,y\right)=\frac{{\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x,y\right\}\right)-\left({\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x\right\}\right)+{\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{y\right\}\right)\right)}{{\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x,y\right\}\right)}$$ (7)

where the value is normalized by ${\mathrm{Vul}}_{V-\left\{x,y\right\}}\left(\left\{x,y\right\}\right)$ to ensure that the PCI becomes one when the input–output connectivity similarity between neuron x and y is one (i.e., when neuron x and y connected with exactly identical neurons).

4.6. Null Models for Connectome (i.e., Random Networks)

In the current study, we employed the Maslov–Sneppen rewiring algorithm to generate null models that uphold the original network's degree distribution. Preserving each neuron's in‐ and out‐degree (only the identity of pre‐ and post‐synaptic partners is randomized) was pivotal in maintaining the inherent structural attributes of the network, even as we perturbed its fundamental connectivity arrangements. Subsequently, we subjected the network to approximately 1000 rewiring iterations for each connection. In total, we created 1000 null model networks to ensure robust statistical validation.

4.7. Symmetric‐Edge and Asymmetric‐Edge Randomization

Additionally, we utilized an alternative null model to examine the impact of both symmetric and asymmetric edges on redundancy measures within the network. The primary objective of this model was to determine whether network metrics predominantly derive from symmetric or asymmetric edges. To specifically investigate the influence of bilateral symmetry, we adjusted the proportions of asymmetric edges using the Maslov–Sneppen rewiring algorithm while ensuring that the remaining asymmetric edges and all symmetric edges were preserved. Conversely, we conducted a complementary procedure by manipulating the proportions of symmetric edges while retaining the remaining symmetric edges and all asymmetric edges. Notably, during the shuffling of symmetric edges, contralateral edges were rearranged concurrently.

This iterative process was carried out across a range of proportions, varying from 0 to 100% in increments of 10%. For each proportion, the corresponding fraction of edges (either symmetric or asymmetric) was randomly chosen and shuffled with maintenance of degree distribution, then combined with the remaining unselected (i.e., unshuffled) portion of the network. For instance, in the case of 30% symmetric shuffling, we randomly chose 30% of the symmetric edges and shuffled them using the Maslov–Sneppen rewiring algorithm. This created a new network comprised of the selected symmetric edges (the 30%). After this, the unselected symmetric edges and asymmetric edges were preserved and recombined with the shuffled selected network. This process was repeated to generate 100 randomized networks for each specified proportion.

4.8. Custom Shuffle Test for Comparison Between Bilateral and All Possible Neuron Pairs

When performing statistical tests between bilateral pairs and all possible neuron pairs, the network measure values might not be statistically independent, as required by certain conventional statistical tests. To address this, we used a specialized resampling technique that does not depend on specific assumptions about the underlying data distribution. In this custom shuffle test, we randomly selected neuron pairs in number equal to the original bilateral neuron pairs and computed the Welch's t statistics between this sampled pair group and all neuron pairs. This process was repeated 100,000 times. For the p‐value estimation, we calculated the proportion of these absolute t‐values that were equal to or greater than the original t‐value's absolute value, which was determined between the original bilateral neuron pairs and all neuron pairs.

4.9. Subsystem Analysis

For analyzing the difference of redundancy between subsystems (i.e., sensory, motor and interneurons), we extracted the first principal component (PC1) for each category of redundancy measures (connectivity similarity—four measures, motif‐fingerprint difference—four measures and path‐compensation index—one measure; detailed in the main text). Note that for the motif‐fingerprint difference measure, we applied a logarithmic transformation prior to principal component analysis (PCA) to correct the skewed distribution and restore normality in the data. In addition, before performing PCA, the graph‐theoretic measures were standardized, ensuring each measure had a mean of zero and a standard deviation of one, which guaranteed uniform variance across all variables. The choice to use the first principal component (PC1) was informed by its ability to represent the largest portion of variance in the dataset (detailed in the main text).

PC1 scores were compared across sensory, motor and interneuron bilateral pairs separately for each measure family and sex. For connectivity similarity and motif‐fingerprint difference, whose PC1 scores are approximately normally distributed, we used one‐way ANOVA with Tukey post hoc comparisons and report the F statistic, degrees of freedom, exact p values and η². For the path‐compensation index, which is strongly right‐skewed and zero‐inflated with unequal group variances (Levene p = 0.007 in the hermaphrodite and p = 0.08 in the male) and therefore violates the assumptions of one‐way ANOVA, we used the rank‐based Kruskal–Wallis test with Holm‐adjusted Dunn post hoc comparisons as the primary test and report χ² and ε² = H/(n − 1). Both parametric and nonparametric results for all families are reported in Supplementary Table S5; the two agreed for every comparison except the male path‐compensation index, where the skewed distribution renders ANOVA underpowered.

4.10. Clustering Bilateral Neuron Pairs

To distinguish each of the bilateral neuron pairs into high, medium, low lateralization group, we employed K‐means (K = 3) clustering analysis for each category of redundancy measures (connectivity similarity—four measures, motif‐fingerprint difference—four measures and path‐compensation index—one measure). Again, for the motif‐fingerprint difference measure, we applied a logarithmic transformation prior to the clustering analysis. Also, before conducting the clustering analysis, the graph‐theoretic measures were standardized, ensuring each measure had a mean of zero and a standard deviation of one, which guaranteed uniform scaling across all variables.

4.11. Synapse‐Count Thresholding

To assess robustness to weak connections, we constructed thresholded networks retaining only connections supported by at least k synaptic contacts (k = 2, 3, 5), in addition to the full network (k ≥ 1). For each thresholded network and each sex we recomputed the bilateral symmetry index, connectivity similarity, three‐node structural and functional motif‐fingerprint differences and the path‐compensation index, together with their degree‐preserving null distributions. Comparisons between bilateral pairs and all neuron pairs used the same custom shuffle test described in Section 4.8.

4.12. Additional Connectome Datasets

To test generalization, we repeated the analyses on the adult hermaphrodite connectome of (Varshney et al. 2011) and on eight individually reconstructed connectomes from (Witvliet et al. 2021) spanning the L1, L2, L3 and adult stages. Each connectome was converted to a directed binary graph as in Section 4.1, restricted to the bilaterally homologous neuron pairs present in that dataset. For each dataset we computed the bilateral symmetry index, connectivity similarity, structural and functional motif‐fingerprint differences and path‐compensation index with degree‐preserving nulls and expressed the motif results as z‐scores relative to the null. We summarized the structural/functional dissociation as the difference between the structural and functional z‐scores (z_struct − z_func). For each measure we computed a network‐size‐independent effect size, Cohen's d = (mean_bilateral − mean_all‐pairs) / SD_all‐pairs, from the per‐pair distributions in each connectome; the all‐pairs standard deviation (not the standard error of the mean) is used so that d does not scale with the number of pairs. For path compensation, pairs for which the index is undefined because no alternative path exists (‘path‐degenerate’ pairs) were excluded; results were essentially unchanged when they were retained.

4.13. Contactome‐Constrained Null

To account for spatial embedding, we used the adult membrane‐contact map (contactome) (Brittin et al. 2021), in which an undirected contact indicates that two neurons are physically adjacent and could in principle form a synapse. We generated a contactome‐constrained null by rewiring the connectome while (i) preserving each neuron's in‐ and out‐degree and (ii) permitting an edge only between neuron pairs that share a membrane contact (1000 randomizations). We recomputed the symmetry index, connectivity similarity, structural and functional motif‐fingerprint differences and path‐compensation index on the empirical connectome and expressed each as a z‐score relative to both the unconstrained degree‐preserving null and the contactome‐constrained null. For each measure, the null‐relative z‐score was calculated as z = (observed value − null mean) / null standard deviation, using the corresponding null distribution.

4.14. Matched Controls for PCI

To test whether elevated bilateral PCI reflected local circuit embedding or the number of incident edges removed, we performed matched‐control analyses for each bilateral pair. Three control distributions were generated. First, for the same‐neighbourhood control, we defined the local neighbourhood of a bilateral pair as the union of all presynaptic and postsynaptic partners of either member of the pair, excluding the bilateral pair itself. We then enumerated all unique two‐neuron control pairs within this neighbourhood. Second, for the edge‐loss‐matched control, we defined edge loss as the number of directed edges incident on either member of a two‐neuron pair when both neurons are removed, counting edges between the two removed neurons only once. Non‐homologous control pairs from the full network were sampled if their edge loss fell within ±20% of the bilateral pair's edge loss. Third, for the combined control, control pairs were required both to lie in the same one‐step neighbourhood and to satisfy the edge‐loss‐matching criterion. In all controls, true left–right homologous pairs were excluded from the control distributions.

For each bilateral pair and control condition, the control PCI was defined as the mean PCI across sampled control pairs. We compared observed bilateral PCI with the matched‐control mean using a one‐sided paired sign‐flip permutation test testing whether observed PCI exceeded control PCI. We used 10,000 group‐level permutations, with plus‐one correction for p‐value estimation; therefore, the smallest possible p‐value was 1.0 × 10⁻⁴. We also report one‐sided Wilcoxon signed‐rank tests as a conservative non‐parametric check of whether the effect was consistent across pairs. Pair‐level values and group‐level statistics are reported in Supplementary Table S6 and the observed‐versus‐control pairwise comparisons are visualized in Supplementary Figure S4.

Author Contributions

Pyeong Soo Kim: writing – review and editing, supervision, conceptualization. Youngjo Song: writing – review and editing, methodology, writing – original draft, visualization, investigation, software, formal analysis. Jerald D. Kralik: writing – review and editing. Jaeseung Jeong: writing – review and editing, supervision, conceptualization.

Funding

The authors have nothing to report.

Ethics Statement

This study used previously published, publicly available datasets and did not involve new experiments with human participants or animals. Therefore, ethical approval from an Institutional Review Board or Ethics Committee was not required.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Figure S1: Bilateral redundancy against unconstrained vs. contactome‐constrained nulls.

Figure S2: Supporting Information.

Figure S3: Supporting Information.

Figure S4: Matched controls for path compensation.

Table S1: Bilateral redundancy against an unconstrained versus a contactome‐constrained null. Each measure (adult hermaphrodite connectome) is compared with two degree‐preserving nulls: an unconstrained null, and a contactome‐constrained null that rewires only between neuron pairs sharing a membrane contact in the adult contactome (1000 randomizations each; Methods 4.13; Brittin et al. 2021). z = (observed—null mean) / null SD.

Table S2: Robustness of bilateral redundancy to synapse‐count thresholding. All measures were recomputed on networks retaining only connections supported by at least 1, 2, 3, or 5 synaptic contacts (Methods 4.11). For the redundancy measures, each cell gives the mean over bilateral pairs/the mean over all neuron pairs. The symmetry index is shown with the mean and 95% interval of its degree‐preserving null (n = 1000 randomizations).

Table S3: Generalization of bilateral redundancy across ten C. elegans connectomes. Each redundancy measure was recomputed on every connectome and summarized by the network‐size‐independent effect size Cohen's d = (mean_bilateral − mean_all‐pairs) / SD_all‐pairs (Methods 4.12). Positive d (CS, path‐compensation) and negative d (MF difference) both indicate bilateral pairs are more redundant than the all‐pairs background.

Table S4: Bilateral‐pair redundancy ranking—Hermaphrodite (PC1 of each measure family).

Table S5: Statistical tests for subsystem differences in redundancy PC1 scores. One‐way ANOVA compared PC1 scores across sensory, interneuron, and motor bilateral pairs, separately for each redundancy‐measure family and sex; Tukey HSD post hoc and Kruskal–Wallis non‐parametric checks are included. Significance: *p < 0.05, **p < 0.01, ***p < 0.001, ns = not significant.

Table S6: PCI matched‐control analysis.

Kim, P. S. , Song Y., Kralik J. D., and Jeong J.. 2026. “Bilateral Neuron Pairs Share Redundant Network Roles Despite Incomplete Connection Symmetry in Caenorhabditis elegans .” European Journal of Neuroscience 64, no. 2: e70629. 10.1111/ejn.70629.

Data Availability Statement

All data in this study have been made publicly available on Wormwiring (http://wormwiring.org/). The analysis in this article was conducted using Matlab (version 9.10), and functions from the Brain Connectivity Toolbox were employed (https://github.com/brainlife/BCT). The datasets generated and/or analyzed during the current study are available at https://github.com/gredada/2024_Celegans_Symmetry.