Asymmetry in synaptic connectivity balances redundancy and reachability in the Caenorhabditis elegans connectome

Summary

The brain is overall bilaterally symmetrical, but also exhibits considerable asymmetry. While symmetry may endow neural networks with robustness and resilience, asymmetry may enable parallel information processing and functional specialization. How is this tradeoff between symmetrical and asymmetrical brain architecture balanced? To address this, we focused on the Caenorhabditis elegans connectome, comprising 99 classes of bilaterally symmetrical neuron pairs. We found symmetry in the number of synaptic partners between neuron class members, but pronounced asymmetry in the identity of these synapses. We applied graph theoretical metrics for evaluating Redundancy, the selective reinforcement of specific neural paths by multiple alternative synaptic connections, and Reachability, the extent and diversity of synaptic connectivity of each neuron class. We found Redundancy and Reachability to be stochastically tunable by the level of network asymmetry, driving the C. elegans connectome to favor Redundancy over Reachability. These results elucidate fundamental relations between lateralized neural connectivity and function.

Graphical abstract

Highlights

• •The brain is overall symmetrical, but exhibits also considerable asymmetry
• •The C. elegans connectome shows stochastic asymmetry in synaptic connectivity
• •Synaptic lateralization balances a tradeoff between Redundancy and Reachability
• •In the C. elegans connectome Redundancy is favored over Reachability

Natural sciences; Biological sciences; Neuroscience; Systems neuroscience

Introduction

Different patterns of synaptic connectivity enable specific modes of neural circuit function. For example, convergent connections may promote the integration of neural information,^1,2 and hierarchical connectivity may afford multi-layered processing.³ Perhaps the most prominent gross feature of brain structure is its overall bilateral symmetrical organization,⁴ evident in its division, in many species, into left and right hemispheres, and the bilateral duplication of entire brain regions, neuron types, and synaptic connections. What are the functional implications of bilateral symmetry in the nervous system?

In many animals, symmetrical brain structure corresponds to bilateral symmetry of the body (e.g., left-right limbs, eyes, and lungs), enabling walking and running, the manipulation of large objects, stereoscopic vision, and spatial sound localization. Brain symmetry also provides full sensorimotor coverage from both sides of the body, for better detection of threats⁵ and opportunities.⁶ At the neural level, symmetrical duplication of brain regions into left-right equivalents (e.g., two hippocampi, left and right basal ganglia, etc.) may contribute to robust signaling, reducing noise and uncertainty, overcoming errors or functional deficits, and additively increasing output strength.

At the same time, asymmetry also prevails,^7,8 such as in organ position (e.g., heart and liver), behavior (e.g., mothers cradling babies predominantly on their left side⁹), brain connectivity¹⁰ (e.g., greater regional interconnectivity in the right vs. left hemisphere¹¹), and brain function (e.g., speech vs. prosody¹²). Such lateralization could enhance specialization, by maximizing the use of brain resources and allocating separate functions to distinct parallel left-right regions.¹³ A lateralized brain reduces the need for left-right coordination, which saves energy, shortens reaction times, and eliminates potential cross-interferences between left and right circuits. Lateralization may also facilitate multi-tasking through parallel left-right processing.¹⁴ Bilateral compartmentalization can thus enable more efficient and less interruptible brain operation, and has been suggested to enhance cognitive abilities.^5,15,16 Conversely, diminished asymmetry has been associated with several disease conditions.¹⁷ The tradeoff between robustness, additivity, and increased coverage, afforded by symmetrical neural organization, on the one hand, and specialization, compartmentalization and parallel processing, enabled by brain lateralization, on the other hand, epitomizes a fundamental interplay between structure and function in the brain.

In order to better understand the functional impact of symmetrical vs. asymmetrical neuronal organization on brain operation, we analyzed the connectome, the map of all synaptic connections, of the nematode worm Caenorhabditis elegans, focusing on lateralization in chemical synaptic connectivity, and in particular, on the differences in synaptic partners between the left and right halves of the nervous system. The adult hermaphrodite C. elegans nervous system contains 198 (out of 302) bilaterally symmetrical neurons,¹⁸ belonging to distinct neuron classes whose left and right members (typically, just two neurons) are clearly labeled.¹⁹ We framed the question of network symmetry vs. asymmetry in terms of Redundancy, the selective reinforcement of specific synaptic paths between distinct neuron classes through parallel alternative connections, and Reachability, the extent and diversity of connectivity of each neuron class to other neuron classes. We applied graph theoretical metrics for formalizing and comparing between Redundancy and Reachability, making it possible to resolve the functional balance between network symmetry and asymmetry in several connectome networks as well as in simulated networks.

Results

The number of synaptic contacts of each neuron is consistent across connectome networks

In order to examine asymmetrical features of synaptic connectivity in C. elegans, we constructed connectome networks from 3 available connectome datasets of adult C. elegans (Table 1; see STAR methods). We labeled these networks CeH0 (a corrected and updated version of the first published C. elegans connectome²⁰), and CeH2 and CeH3 (more recently mapped connectomes²¹). CeH2 and CeH3 networks do not include neurons from the body and tail, and thus contain fewer neurons ($N=180$) than CeH0 ($N=280$). We thus generated, for the purpose of analysis, a subnetwork of CeH0 containing only the neurons included in the CeH2 and CeH3 networks, and named it CeH1. In addition, we restricted most of our analysis to chemical synapses (networks CeH1c, CeH2c, and CeH3c), for which more details exist compared to electrical synapses, and we considered only the presence or absence of connections, ignoring their weights (i.e., the number of synaptic contacts). This enabled us to concentrate on the basic structure of the network, avoiding a prominent source of variance and complexity and the difficulty to relate anatomical counts of synaptic contacts with functional synaptic efficacy.

Table 1.

List of C. elegans connectome networks

Network	Sex	Age	Note	Nodes	Edges	${\mathit{f}}^{\mathit{d}}$	Source (published data filenames)
CeH0c	⚥	Adult		280	2895	0.363	“SI 5 Connectome adjacency matrices, corrected July 2020.xlsx”20
CeH0e	“	“		275	1065	0.499	“
CeMc	♂	“		358	3195	0.447	“
CeMe	“	“		317	1168	0.532	“
CeH1c	⚥	Adult		180	2007	0.125	CeH0c
CeH2c	“	“		180	1669	0.104	“witvliet_2020_7 adult.xlsx”21
CeH3c	“	“		180	1633	0.101	“witvliet_2020_8 adult.xlsx”21
CeH1Dc	⚥	Adult	Directed	180	2520	0.156	CeH0c
CeH2Dc	“	“	“	180	1933	0.120	“witvliet_2020_7 adult.xlsx”21
CeH3Dc	“	“	“	180	1931	0.120	“witvliet_2020_8 adult.xlsx”21
CeL1c	“	L1		161	615	0.375	“witvliet_2020_1 L1.xlsx”21
CeL2c	“	L2		173	1155	0.337	“witvliet_2020_5 L2.xlsx”21
CeL3c	“	L3		174	1174	0.271	“witvliet_2020_6 L3.xlsx”21
CePc	“	Adult	Pharynx	20	108	0.262	“cne24932-sup-0004-supinfo4.csv”32
DmFc	♀	1st instar larva	CNS	2952	96527	0.022	“science.add9330_data_s2.csv”; “all-all_connectivity_matrix.csv”35

Italicized networks are used throughout most of the study. Non-italicized networks used in Figure 8.

As in many other nervous systems,²² the number of synaptic connections of each neuron in C. elegans varies according to neuron type. This number is known as the neuron’s degree. Comparing the degree, $d_i$, of each neuron, $\mathsf{i}$, across the 3 connectome networks revealed similar degree distributions (Figure 1A), and strong correlations between the networks (Figure 1B), suggesting that the degree of a neuron is to a large extent, a fixed feature of neuronal connectivity.

Figure 1.

Left-right symmetry in neuron chemical synapse degree

(A) Individual neuron degree, $d_i$, within each connectome network. Dark bars indicate average values.

(B) Comparison of $d_i$ between connectome networks. $\rho$, Spearman rank correlation. n = 180, p < 0.0001 for all plots.

(C) Comparisons between left, $d_{k^L}$, and right, $d_{k^R}$, neuron class members for each connectome network. Slopes fitted from simple linear regression. n = 83, p < 0.0001 for all plots. Dotted diagonal is the identity line.

(D) ${\Delta }_k^d$, the Euclidean distance of each $\left(d_{k^L},d_{k^R}\right)$ coordinate from the symmetry identity line (in C). Inset shows an example neuron class (black dot). Dark bars indicated average values.

Bilateral neuron pairs exhibit symmetry in the number of synaptic contacts

Given the typical degree of each individual neuron (Figure 1A), we asked whether the left and right neurons within each neuron class may show similarity in degree. We examined for each neuron class, $k$, the relations between the degree of the left and right members of that class, $k^L$ and $k^R$. We found a strong correlation between $d_{k^L}$ and $d_{k^R}$ (Figure 1C), with most values lying close to the identity line, representing perfect symmetry (Figure 1C). This result suggests that in addition to the preserved degree of each individual neuron (Figure 1B), bilateral neurons within neuron classes tend to maintain degree symmetry. The similar degree of each neuron class member may represent a property of that neuron class.

To evaluate the modest differences in degree that nevertheless exist between left and right class members, we calculated ${\Delta }_k^d=\frac{\sqrt{2}}{2}\left(d_{k^L}-d_{k^R}\right)$, the Euclidean distance of each $\left(d_{k^L},d_{k^R}\right)$ coordinate from the symmetry identity line, as a measure of deviation from class degree symmetry (Figure 1D, inset). ${\Delta }_k^d$ showed a comparable distribution between networks, centered close to zero (Figure 1D), implying no particular overall left or right bias. In addition, ${\Delta }_k^d$ values showed no correlation between networks (Figure S1A), suggesting that the deviations from perfect class symmetry might not be fixed. Finally, ${\Delta }_k^d$ seemed to not be strongly dependent on the total degree of the neuron pair, $d_k$ (Figure S1B), further supporting variable rather than systematic deviations between $d_{k^L}$ and $d_{k^R}$. Together, these findings suggest that neuron degree is a consistent and innate property of each neuron class shared by its members, reflecting a fundamental quantitative aspect of network symmetry, including a small stochastic deviation.

Bilateral neuron pairs show asymmetry in synapse composition

The left-right resemblance in the number of synaptic connections within each neuron class raises the question of whether the composition of these synapses is symmetrical as well. To address this, we examined the synaptic partners of the left versus right members of each neuron class. We considered synapses linking between two contralateral neurons from one class and two contralateral neurons from another class as paired synapses. All other synapses that cannot be matched in this way are unpaired synapses. For example, in the small model network (Figure 2A; Table 2), neuron DL (the left member of neuron class D) participates in 2 synapses, DL-CL and DL-GL. Its contralateral class member neuron, DR (the right member of neuron class D) partakes in 3 synapses, DR-AR, DR-CR, and DR-FR. Since CL and CR are contralateral neurons of the same neuron class, C, the DL-CL synapse can be said to be paired with the DR-CR synapse (Figure 2A, dashed connectors). All other 3 synapses of the D neuron class are unpaired (Figure 2A, continuous connectors). The fraction of unpaired synapses of this neuron class is $\frac{3}{5}$ (3 unpaired synapses out of a total of 5 synapses for the entire class).

Figure 2.

Asymmetry in chemical synaptic composition

(A) Small model network consisting of 8 bilateral neuron pairs. Dashed connectors represent paired synapses. Continuous connectors are unpaired synapses. Darkened connectors correspond to the examples in the text.

(B) Fraction of unpaired synapses, $f_k^u$, in the different connectome networks. Shown also is the average (black bar), $f^u$, for each network.

(C) Comparison of $f_k^u$ values between connectome networks. $\rho$, Spearman rank correlation. n = 83, p < 0.0001 for all plots.

(D) Comparison between $n_{k^L}^u$ and $n_{k^R}^u$, the number of unpaired synapses of the left vs. right member of each neuron class, for each connectome network. $\rho$, Spearman rank correlation. n = 83 for all plots. p = 0.0004 for CeH1c and p < 0.0001 for CeH2c and CeH3c networks.

Table 2.

List of random networks and the figures, in which they appear

Network	Nodes	Classes	Edges	${\mathit{f}}^{\mathit{d}}$	Construction method	Figures
Model	16	8	19	0.158	Manual	Figures 2A, 4A, 5A–5C, 5E, and 5F
Small1	40	20	80	0.103	NetworkX: erdos_renyi_graph	Figure 4B
SR	180	68	1770	0.110	NetworkX: gnm_random	Figures 6C, 6E, and 6F
ER	180	68	2027	0.126	NetworkX: erdos_renyi_graph	Figures 6C and 6F
DCB	180	68	1994	0.124	Custom	Figures 6C and 6F
Small2	40	20	39–156	0.05–0.2	NetworkX: gnm_random	Figures 7E and S3

In general, we define the fraction of unpaired synapses of neuron class $k$ as $f_k^u=\frac{n_k^u}{d_k}$, where $n_k^u$ = $n_{k^L}^u+n_{k^R}^u$ is the sum of unpaired synapses of the left and right members of neuron class $k$. An $f_k^u$ value close to 0, indicates a neuron class with high symmetry in synaptic composition. Values close to 1 represent high asymmetry in synaptic composition. Computing the fraction of unpaired synapses, $f_k^u$, in the different connectome networks revealed a large spread of values, with comparable averages, $f^u=\frac{1}{N^c}\Sigma f_k^u$ (Figure 2B), where $N^c$ is the number of neuron classes. Not even one neuron class in any network was perfectly symmetrical ($f_k^u=0$). $f_k^u$ values showed a significant (Spearman rank test) correlation between networks (Figure 2C), suggesting that the extent of asymmetry in synaptic composition of each neuron class is a conserved feature of that neuron class. We also found a significant correlation (Spearman rank test) between $n_{k^L}^u$ and $n_{k^R}^u$ in all networks (Figure 2D). Taken together, our results show that bilateral members of a neuron class exhibit asymmetry in the composition of a subset of their synapses, and this subset shows similarity in size in the two neurons.

Stochasticity contributes to asymmetry in synaptic composition

Asymmetry in synaptic composition (Figure 2) could stem from predetermined genetic²³ or biomechanical²⁴ factors, or from ongoing adjustments by activity-dependent²⁵ or stochastic²⁶ processes. To distinguish between these possibilities, we examined the consistency of all chemical synaptic connections across the 3 connectome networks (Figure 3A). We reasoned that consistent synapses are more likely to be predetermined than adjusted. The average portion of all chemical synapses present in all 3 networks was surprisingly low, ∼24% (Figure 3A, left). Out of these, the share of consistent paired synapses was significantly higher, ∼39% (Figure 3A, middle; Z = 9.51, p < 0.00001). In contrast, the average portion of unpaired synapses included in all networks was only ∼2.5% (Figure 3A, right), significantly smaller than the fraction of paired synapses (Z = 19.67, p < 0.00001). This finding suggests that unpaired synapses are particularly variable across individuals, and are therefore more likely to be determined by stochastic or experience-dependent processes than paired synapses.

Figure 3.

Asymmetry in chemical synaptic composition is stochastically driven

(A) Venn diagrams showing the number of all (left), paired (middle), and unpaired (right) chemical synapses in each connectome network and the number of such synapses shared by all three networks (center of each diagram).

(B) Comparison between $n_i^u$, the number of unpaired synapses of each individual neuron, and $d_i$, the neuron’s degree for each connectome network. $\rho$, Spearman rank correlation. Slopes fitted from simple linear regression. n = 166, p < 0.0001 for all plots.

To extend our analysis, we examined the relation between $n_i^u$, the number of unpaired synapses of each individual neuron, and $d_i$, the neuron’s degree. These two variables showed significant correlation (Spearman rank test) in all networks (Figure 3B), indicating that the number of unpaired synapses of each neuron is proportional to the neuron’s degree. This proportion can be estimated from the slopes in Figure 3B, whose values across networks were quite similar, suggesting a consistent fixed fraction of unpaired synapses in C. elegans neurons. These results point at a possible stochastic process underlying the choice of unpaired synapses, presumably tuned by a global fixed probability for the formation of unpaired synapses in each neuron (reflected in the rather consistent ratio between $n_i^u$ and $d_i$ in Figure 3B), as opposed to an experience-dependent process, expected to show much more heterogeneity across neurons (since plasticity is presumably selective, affecting only specific neurons and synapses²⁷). The similarity in neuron degree (Figure 1C) and in the number of unpaired synapses (Figure 2D), between left and right class members suggests that within each individual neuron, a fixed portion of unpaired synapses is governed by a seemingly stationary stochastic process. Importantly, this finding is an observation about the connectome makeup rather than a hypothesis about how synapses are formed. Synaptogenesis is a complex process involving many factors that we have not considered.

Symmetry in synapse composition has a non-stochastic component

As we have found, paired (symmetrical) synapses appear to be more consistent than unpaired synapses across connectome networks (Figure 3A), suggesting that they are less driven by stochasticity than unpaired synapses. We sought to study the extent to which randomness may nevertheless guide symmetry in synaptic composition through stochastic formation of paired synapses. To this end we considered network directionality, the arrangement of chemical synapses so that transmission occurs only in one direction (from presynaptic to postsynaptic neuron).

A completely random directed network is expected to contain more unpaired synapses than an equivalent undirected network, maintaining all synaptic connections, but ignoring their directionality. This is because the directed network offers more opportunities for unpaired synapses to occur. For example, in the undirected small model network (Figure 4A, left), the AL-BL synapse has a contralateral counterpart, AR-BR, and is therefore considered a paired synapse (dashed connector). If directionality is added to the network, then an AL→BL connection, for instance (Figure 4A, right), now has a 50% chance of being unpaired instead of paired (AL→BL would pair with AR→BR, but not with BR→AR), reducing the relative number of paired connections and increasing asymmetry. To illustrate, we generated a series of small random undirected networks containing $N=40$ neurons and $d=80$ synapses, and symmetrized or desymmetrized them (see the following text) so that their average fraction of unpaired synapses, $f^u$, was 0.00, 0.34, or 0.84 (Table 2). We derived corresponding directed versions of these networks, by assigning random directionality to all their synapses. In order to assess overall network asymmetry, we computed $f^u$ for each undirected and equivalent directed network. As we had postulated for such stochastic networks, $f^u$ was substantially higher in the directed networks compared to the undirected ones (Figure 4Bi), with the gap between undirected and directed $f^u$ decreasing as overall network asymmetry increased (Figure 4Bii). The average $f_k^u$ correlation between undirected and directed networks (with mean undirected $f^u=0.34$) was moderate (Figure 4Biii), and the average slope of the linear regressions constrained by these values was considerably smaller than 1 (Figure 4Biii), consistent with greater asymmetry of directed compared to undirected random networks.

Figure 4.

Symmetry in chemical synaptic composition is not strongly stochastic

(A) Left, small undirected model network consisting of 8 bilateral neuron pairs. Dashed connectors represent paired synapses. Continuous connectors are unpaired synapses. Darkened connectors point to an example connection between the A and B neuron classes. Right, several possible directed connections between the A and B neuron classes. The top two options represent paired connections, and the bottom two unpaired connections.

(B) Small undirected and matched directed random networks consisting of 40 neurons and 80 interconnecting synapses grouped according to their average fraction of undirected unpaired synapses, $f^u=0.00,0.34,0.84$. (i) $f^u$ compared between undirected and directed versions of each network. (ii) Difference in $f^u$ between directed and undirected versions of each network. n = 20. One-way ANOVA p < 0.0001. Post hoc Dunnett’s multiple comparisons test, p < 0.0001 for both comparisons. (iii) Results of comparison of $f_k^u$ between undirected and directed versions of each network (with average undirected $f^u=0.34$). Shown are the Spearman rank correlation coefficient, $\rho$, values and the slope as fitted by a linear regression, for each network instance. Error bars represent 95% confidence intervals. Each dot represents a different network.

(C) Fraction of unpaired synapses, $f^u$, in directed vs. undirected connectome networks.

(D) Comparison of the fraction of neuron class unpaired synapses, $f_k^u$, between undirected and directed connectome networks. $\rho$, Spearman rank correlation coefficient. Slope as fitted by a linear regression. n = 83, p < 0.0001 for all plots.

In order to assess stochasticity in the specification of C. elegans paired synapses, we thus compared $f^u$ values between the original directed connectome networks and corresponding networks, constructed to be undirected. A large decrease in $f^u$ (as in the purely random networks; Figure 4B) would indicate a strong stochastic influence on the emergence of paired synapses. In reality, the calculated differences in $f^u$ values between directed and undirected connectome networks were rather minor (compare Figures 4C and 4Bi), and the fraction of unpaired synapses within each neuron class, $f_k^u$ (Figure 4D), and within each individual neuron, $f_i^u$ (Figure S2), showed strong correlations between the undirected and directed connectomes with slopes close to 1 (compare with Figure 4Biii). These results provide further evidence for regulated rather than random specification of symmetrical synapses, or at least, tight coordination in synaptic choice between neuron class members. This conclusion is particularly relevant to the C. elegans nervous system, whereby many neurites both receive synaptic inputs and send synaptic outputs, enabling synaptic communication in both directions, providing more options for connectivity than the number actually observed.

Redundancy and reachability reflect a tradeoff between network symmetry and asymmetry

The $f^u$ values calculated from the C. elegans chemical connectomes reveal a considerable level of asymmetry in synaptic connectivity (Figure 4C). What are the implications of such asymmetry? To begin to address this question, we sought to identify specific network properties that capture key differences between symmetry and asymmetry. To gain a better grasp of the issue, we turn once again to small example networks, this time ranging from fully symmetrical (Figure 5A; $f^u=0.0$) to fully asymmetrical (Figure 5C; $f^u=1.0$). We accompany each detailed individual neuron network with a collapsed version of that network, showing the connections between its neuron classes (Figures 5A–5C). We assume that the members of each neuron class perform similar or related functions. We note that bilateral members of half (53.6%) of all neuron classes are interconnected by at least one chemical or electrical synapse (in the most detailed connectome network, CeH0), potentially coupling their activity. As can be seen, the same number of synapses is distributed very differently in symmetrical vs. asymmetrical networks. In the symmetrical network (Figure 5A, right) fewer classes are connected to each other compared to the asymmetrical network (Figure 5C, right). However, each inter-class connection is stronger, comprising more synapses (Figure 5A, right). This tradeoff in synapse distribution may reflect an important balance between symmetrical and asymmetrical network arrangements. We frame this tradeoff in terms of Redundancy vs. Reachability.

Figure 5.

Redundancy and Reachability in a small model network and in the C. elegans connectomes

(A–C) Small model networks consisting of 8 bilateral neuron pairs interconnected by 19 undirected synapses. Dashed connectors represent paired synapses. Continuous connectors are unpaired synapses. Each network is presented at the individual neuron level (left) and at the neuron class level (right). The number of connectors between each two neuron classes indicates the number of synapses connecting between their individual neuron members. The networks range from (A) fully symmetrical ($f^u=0$.00), through (B) intermediate ($f^u=0.42$), to (C) fully asymmetrical ($f^u=1.00$).

(D) Mean shortest path length ($SP$) connecting any two neurons (left) or any two neuron classes (right) in the various C. elegans connectome networks. n = 180 individual neurons, n = 83 neuron classes of bilateral neuron pairs, and n = 14 neuron classes of single non-bilateral neurons.

(E) Redundancy, $R^n$, and Reachability, $S^n$, plotted against the fraction of unpaired synapses, $f^u$, of the three model networks (A-C).

(F) Mean shortest path length ($SP$) connecting any two neurons in the small model networks (A–C).

(G) Redundancy, $R^n$, and Reachability, $S^n$, plotted against the fraction of unpaired synapses, $f^u$, of the undirected C. elegans connectome networks. Dotted lines indicate the $f^u$ of each network.

We refer to Redundancy as an abundance of synaptic connections linking specific neuron classes. Redundancy could selectively enhance or refine particular routes of information flow in the circuit, and when necessary, serve as back up in response, for example, to weakened signaling or neural damage. For a given number of synapses in a network, greater symmetry in synaptic composition should lead to increased Redundancy (owing to duplication).

We denote by Reachability the number of distinct neuron classes that can be reached by synaptic connections from each neuron class. Since asymmetrical networks include more diverse synaptic connections than symmetric networks (at the expense of duplication), increased asymmetry should raise the number of interconnected neuron classes, thus enhancing Reachability.

Redundancy and Reachability vary as a function of network symmetry vs. asymmetry

In order to formally characterize Redundancy and Reachability, we first introduce several useful terms. In a given network, a path of length $n$ is a sequence of $n+1$ distinct neurons connected by $n$ synapses. For example, in Figure 5B (left), CR-HL-GL-FL (blue path) is a path of length $n=3$ comprising 4 neurons. We distinguish between a neuron path, composed of a sequence of individual neurons, and a class path, composed of a sequence of neuron classes (disregarding the identity of individual neuron members of each class), in the example, C-H-G-F is a class path (Figure 5B, right).

We define a realizable class path as a class path, for which at least one corresponding neuron path exists in the network. For example, in Figure 5C (right), the class path B-F-G (purple path) is not realizable, since there is no corresponding neuron path (Figure 5C, left) that can instantiate this class path. In contrast, A-C-H (orange path) is realizable by two neuron paths: AL-CR-HR and AR-CR-HR. The theoretical maximal number of neuron paths that could instantiate a particular class path, $j$, of length $n$ in an undirected (see the following text) network is $M_j^n=m_1·m_2·...·m_{n+1}$, where $m_k$ denotes the number of neuron members belonging to neuron class $k$ (in the case where all neuron classes in class path $j$ are bilateral neuron pairs, then $M_j^n=2^{n+1}$). Since neural networks are rarely, if ever, complete (i.e., including synapses between every possible pair of neurons), the actual number of neuron paths, ${\mu }_j^n$, that realize a specific class path, $j$, of length $n$ should be typically much smaller than the maximal possible value $M_j^n$. For example, as shown previously, the 2 realizable neuron paths for class path A-C-H are few relative to $M_{A-C-H}^2=2^3=8$. Notably, many class paths lack any corresponding neuron paths whatsoever, and thus are not realizable (e.g., class path B-F-G in Figure 5C, described previously).

Thus, the evaluation of network Redundancy and Reachability relies on path analysis. This may be considerably simplified if performed in undirected rather than directed networks. Such an approximation can be justified by the similarities we have found between undirected and directed C. elegans connectome networks with respect to symmetry and asymmetry (Figure 4C and 4D). In addition, we find the mean shortest path length ($SP$) connecting any two neurons, and especially any two neuron classes, to be comparable between directed and undirected C. elegans connectome networks (Figure 5D), and to covary within each network (e.g., a network with smaller undirected $SP$ shows also smaller directed $SP$ compared to other networks). Thus, the removal of directionality constraints may result in quantitative changes to certain network properties. However, the qualitative effects of this approximation on network paths seem to be minor.

We formally define network Redundancy, $R^n=\frac{1}{N^n}\Sigma \frac{{\mu }_j^n}{M_j^n}$, as the average number, ${\mu }_j^n$, of neuron paths of length $n$ that realize each class path, $j$, in the network, where $N^n$ is the number of realizable class paths of length $n$. Importantly, only realizable class paths are included in the calculation. In addition, ${\mu }_j^n$ is normalized (before averaging) by the maximal possible number of neuron paths, $M_j^n$, that could constitute the class path. $R^n$ may thus range between 0 and 1. However, since only realizable class paths are considered, and the network contains a minimal number of synaptic connections, the actual minimal $R^n$ value will be higher than 0. To illustrate the relations between $R^n$ and network symmetry vs. asymmetry, we derive $R^1$ for the model networks in Figure 5. In the fully symmetrical network (Figure 5A; $f^u=0$) all realizable class paths of length $n=1$ have 2 corresponding neuron paths (double connectors in Figure 5A, right) out of a maximum of $M_j^1=2^2=4$. Thus, for this network, $R^1=0.50$ (Figure 5E). A larger $R^1$ value would have been obtained had the network consisted of more synapses (see the following text). As asymmetry increases (Figure 5B; $f^u=0.42$) Redundancy decreases, $R^1=0.40$ (Figure 5E), and is lowest, $R^1=0.32$ for the fully asymmetric configuration (Figures 5C and 5E; $f^u=1$). For longer path lengths of $n>1$, overall $R^{n>1}$ levels drop, and $R^n$ compared to $R^1$ shows decreased dependence on $f^u$ (Figure 5E). This is because $M_j^n$ grows exponentially with $n$, whereas the extent of duplication does not change. Therefore $R^1$ may serve as a more sensitive measure of network redundancy compared to the use of longer path lengths.

We define network Reachability, $S^n=\frac{{\Sigma \nu }_k^n}{N^c\left(N^c-1\right)}$, as the average number, ${\nu }_k^n$, of neuron classes that can be reached from each neuron class, $k$, by a realizable class path of length, $n$. $N^c$ is the number of all neuron classes. ${\Sigma \nu }_k^n$ is normalized by the maximal possible number of classes that could be reached from each neuron class, $N^c-1$. The values of $S^n$ may range from 0 to 1, but are in practice, higher than 0, since it can be assumed that each neuron in the network has at least one synaptic contact, enabling minimal reachability. To demonstrate how $S^n$ relates to network symmetry vs. asymmetry we compute $S^1$ for the model networks in Figure 5. In the symmetrical network (Figure 5A; $f^u=0.00$), the average number of neuron classes reachable from each neuron class via a realizable class path of length $n=1$ is $S^1=0.32$ (Figure 5E). As asymmetry increases (Figure 5B; $f^u=0.42$) so does Reachability, $S^1=0.43$ (Figure 5E), and is highest, $S^1=0.54$, for the fully asymmetric network (Figure 5C and 5E; $f^u=1.00$). For paths of length $n>1$, $S^n$ values generally rise (Figure 5E). This is because the average shortest path length of each of these networks is close to 3 (Figure 5F), so that paths of length $n=2$ already enable reaching many neuron classes. Indeed, the relatively lower SP of the intermediate network (Figure 5F; $f^u=0.42$) shows also higher $S^2$ and $S^3$, compared to the fully symmetrical ($f^u=0.0$) or fully asymmetrical ($f^u=1.0$) networks. Therefore, similarly to $R^1$, we find also $S^1$ to be advantageous in capturing network reachability in a sensitive and straightforward manner.

The C. elegans connectome is obviously more complex than a small model network, differing from it in topology and other aspects. We calculated $R^n$ and $S^n$ for the undirected connectome networks (Figure 5G). Notably, even within the small range of $f^u$ values of the different networks, $R^n$ covaried negatively, and $S^n$ covaried positively with $f^u$ (Figure 5G), similar to the model network (Figure 5E). Here in particular, it was obvious that $R^1$ and $S^1$ are the most informative, as $R^2$ and $R^3$ had smaller values, with diminishing slopes (−0.35 and −0.27, respectively, relative to −0.43 for $R^1$), and $S^2$ and $S^3$ saturated with values equal to or close to 1.0 for all $f^u$. These results demonstrate the sensitivity and applicability of direct ($n=1$) Redundancy and Reachability in the analysis of connectome symmetry vs. asymmetry in C. elegans.

The level of asymmetry in the connectome may favor Redundancy over Reachability

What could the $f^u$ values of the C. elegans chemical synaptic connectome tell about the balance in these networks between Redundancy and Reachability? To address this question, it is useful to first appreciate how $R^1$ and $S^1$ are related to $f^u$, and how sensitive these measures are to differences in $f^u$ values. A linear dependency, for example (Figure 6A), would imply that smaller $f^u$ values, as occurs in the connectome (Figures 4C and 5G) should correspond to a preference for Redundancy at the expense of Reachability. In this (linear) case, the tradeoff between $R^1$ and $S^1$ will depend also on their respective slopes. For instance, a steep $R^1$ slope in conjunction with a shallow $S^1$ slope would indicate that Redundancy ($R^1$) can be gained or lost with little impact on Reachability ($S^1$). Alternatively, a non-linear relationship between $R^1$ or $S^1$ and $f^u$ (e.g., Figure 6B) may entail other preferences and tradeoffs balanced by the level of network symmetry vs. asymmetry.

Figure 6.

A symmetrical bias of the connectomes favors Redundancy over Reachability

(A) Hypothetical linear dependency between $R^1$ and $S^1$, and $f^u$ with varying slopes.

(B) Hypothetical exponential dependency between $R^1$ and $S^1$, and $f^u$ with varying exponents.

(C) Redundancy, $R^1$, and Reachability, $S^1$, plotted against the fraction of unpaired synapses, $f^u$, for various large random networks designed to vary in $f^u$ from 0.0 to 1.0.

(D) Redundancy, $R^1$, and Reachability, $S^1$, plotted against the fraction of unpaired synapses, $f^u$, for the three C. elegans connectome networks systematically modified to vary in $f^u$ from 0.0 to 1.0. Dotted lines indicate actual $f^u$ value for each network.

(E) Redundancy, $R^1$, and Reachability, $S^1$, for 3 sets of 200 simple random (SR) networks with 3 different levels of asymmetry ($f^u=0.3,0.6,0.9$).

(F) Slopes of the $R^1$ and $S^1$ vs. $f^u$ plots from (C and D) as fitted by linear regression.

To determine how $R^1$ and $S^1$ are governed by $f^u$, we generated several random networks similar in dimension to the connectome networks (Table 2; see STAR Methods): SR, a simple random network; ER, an Erdős–Rényi random network^28,29; and DCB, a random network with initial neuron degree values, $d_i$, and a restricted synaptic partner pool based on the connectome and “contactome” (list of all neuronal adjacencies^30,31; see STAR Methods). We paired the neurons in the networks to form bilateral neuron classes, as in the connectome. We devised an algorithm (see STAR Methods) for symmetrizing or desymmetrizing the network, based on iterative replacement of random synaptic connections, to obtain a series of networks with $f^u$, ranging from complete symmetry ($f^u=0$) to full asymmetry ($f^u=1$). We computed for each such series of networks $R^1$ and $S^1$ at fixed $f^u$ steps. Plotting $R^1$ and $S^1$ against $f^u$ (Figure 6C) revealed a highly linear dependency of both $R^1$ and $S^1$ on $f^u$ (Figure 6C). We found a similar linear relationship also when we subjected the original connectome networks to symmetrization and desymmetrization (Figure 6D). Such linearity is suggested also by the comparison of $R^1$ and $S^1$ between the 3 connectome networks (Figure 5G).

To further examine the strength of the association between $R^1$, $S^1$ and $f^u$, We generated 200 random instances of the SR network, which were symmetrized or desymmetrized to obtain networks with particular $f^u$. The resulting $R^1$ and $S^1$ values were restricted to a considerably narrow range (Figure 6E), emphasizing the tight relations between Redundancy, Reachability, and network symmetry vs. asymmetry.

We performed linear regression to each of the $R^1$ and $S^1$ curves and attained the fitted slopes (Figure 6F). While the $R^1$ slopes were similar in magnitude to $S^1$ in the connectome-generated networks (Figures 6D and 6F), the random networks showed smaller $R^1$ slopes compared to $S^1$ (Figures 6C and 6F). This difference could stem from particular topological features distinctive of the real connectome networks. The similar slopes of $R^1$ and $S^1$ (Figures 6D and 6F) position the putative crossover point between Redundancy and Reachability in the middle between symmetry and asymmetry ($f^u\approx 0.5$). Since we find C. elegans $f^u$ values to be somewhat biased toward 0 (Figure 6D, dotted lines), and $R^1$ and $S^1$ are anti-correlated, vary monotonically and linearly with $f^u$ and have similar slopes (Figures 6D and 6F), this suggests that the layout of the C. elegans connectome may prioritize Redundancy over Reachability, so that relatively fewer but stronger inter-class connections are preferred over broader but weaker connectivity.

Paired and unpaired synapses play distinct role in tuning Redundancy and Reachability

As we have shown, paired synapses inherently reinforce specific class paths through duplication, thus promoting Redundancy. However, unpaired synapses may also contribute to Redundancy. For example (Figure 5B), the two unpaired synapses, EL-HR and ER-HR, provide two alternative connections for realizing the E-H class path, and thus contribute to Redundancy despite being unpaired. In the extreme case, of a fully asymmetrical network ($f^u=1$), all synapses are unpaired, and so any Redundancy in such a network is exclusively due to these unpaired connections. To directly probe the particular contribution of paired synapses to Redundancy, we calculated $R^{p,1}$, a metric similar to $R^1$, but comprising only paired synapses (with the average, however, taken for all realizable class paths, as in $R^1$, regardless of whether they include paired or unpaired synapses; see STAR Methods). We plotted $R^{p,1}$ for networks of different $f^u$ values for the three connectomes (Figure 7A). As expected, in fully symmetrical ($f^u=0$) networks, $R^{p,1}=R^1$, since Redundancy in this case is entirely established by paired synapses. In contrast, $R^{p,1}$ equaled 0 for fully asymmetrical ($f^u=1$) networks, since Redundancy is entirely due to unpaired synapses. Calculating the fraction of $R^{p,1}$ out of the total $R^1$, $f_{R1}^p$, for the actual connectome $f^u$ values (Figure 7A, dotted lines), revealed that paired synapses contribute a substantial portion of Redundancy to the network (∼65%), but unpaired synapses participate as well (∼35%; Figure 7B).

Figure 7.

The influence of relative network degree on Redundancy and Reachability

(A) Redundancy due exclusively to paired synapses, $R^{p,1}$, plotted against the fraction of unpaired synapses, $f^u$, for the three C. elegans connectome networks systematically modified to vary in $f^u$ from 0.0 to 1.0. Dotted lines indicate actual $f^u$ value for each network.

(B) $f_{R1}^p$, fraction of Redundancy due only to paired synapses, $R^{p,1}$, out of total Redundancy, $R^1$.

(C) Reachability due exclusively to unpaired synapses, $S^{u,1}$, plotted against the fraction of unpaired synapses, $f^u$, for the three C. elegans connectome networks systematically modified to vary in $f^u$ from 0.0 to 1.0. Dotted lines indicate actual $f^u$ value for each network.

(D) $f_{S1}^u$, fraction of Reachability due to unpaired synapses, $S^{p,1}$, out of total Reachability, $S^1$.

(E) Redundancy, $R^1$, and Reachability, $S^1$, of a series of small random networks (40 neurons) with varying number of synapses ($f^d=0.05,0.1,0.125,0.2$) and fraction of unpaired synapses, $f^u$. Dotted lines indicated C. elegans connectome networks $f^d$ values.

(F) Redundancy, $R^1$, Reachability, $S^1$, and fraction of unpaired synapses, $f^u$, plotted against C. elegans connectome networks $f^d$ values.

We also evaluated the relative contribution of unpaired synapses to Reachability. To this end we calculated $S^{u,1}$ similarly to $S^1$, considering for each neuron class only connected partner classes that are linked via unpaired synapses (Figure 7C). Obviously, for fully symmetrical networks, comprising exclusively paired synapses ($f^u=0$), $S^{u,1}=0$, and in fully asymmetrical networks lacking any paired synapses ($f^u=1$), $S^{u,1}$ equals $S^1$. The ratio of $S^{u,1}$ to $S^1$, $f_{S1}^u$, for the connectome $f^u$ values was rather high (average ∼60%; Figure 7D), indicating that Reachability is largely based on unpaired synapses, but also paired synapses have an appreciable share (∼40%).

Network degree affects the dependency of Redundancy and Reachability on network symmetry

Network degree, $d$ (the total number of synapses in the network), may affect the relations between asymmetry, Redundancy, and Reachability. To examine this, we studied the effects of varying $f^d=\frac{d}{D}$, the ratio between $d$ and $D=\frac{N\left(N-1\right)}{2}$ (the maximal possible degree of a complete network, where all neurons are connected to all other neurons) on $R^1$ and $S^1$. On the one hand, extremely sparse networks with low $f^d$ are likely to be inherently asymmetrical since they present fewer opportunities for synaptic pairing. This would lead to low Redundancy, but also low Reachability and moderate dependency on $f^u$. The opposite is true for very dense networks, which should be distinctly symmetrical with high, close to constant, Redundancy and Reachability. To examine intermediate $f^d$ values, we constructed a series of small (40 neuron) undirected SR networks with variable $f^d$ (Table 2; Figure 7E; Figure S3), including the $f^d$ values of the C. elegans connectome networks (Figure 7E, dashed lines; Figure 7F). Interestingly, within this range, $R^1$ showed weak dependency on $f^d$, compared to $S^1$, which increased considerably with $f^d$, for a broad range of $f^u$ values (Figure 7E). This difference between seemingly scale-independent $R^1$ and scale-dependent $S^1$ stems from two distinct effects of network degree within intermediate values. First, although a relatively larger $f^d$, implying more synapses, produces more realizable class paths, the number of alternative neuron paths that realize these paths remains similar, unaffecting overall Redundancy. Second, a denser network increases the possibilities for each neuron class to connect to other classes, thus enhancing Reachability. Therefore, in theory, in the C. elegans connectome, Reachability ($S^1$) could have been larger without requiring an increase in asymmetry ($f^u$) or a decrease in Redundancy ($R^1$), if the total number of synapses were larger. In reality, it appears that a smaller network degree is favored over increased Reachability.

Different connectomes vary mostly in asymmetry and Reachability

To conclude our investigation, we wished to expand our analysis and probe additional available connectome networks (Figure 8A; Table 1). First, we examined the full corrected original C. elegans connectome, CeH0c, from which we had derived CeH1c in the preceding analyses.²⁰ We found CeH0c to be comparable to the other networks that we have examined (Figure 8B), and used this network for further analyses. Overall comparison between various connectomes (shaded connectomes in Figure 8A) revealed a relatively broad distribution of symmetry vs. asymmetry (Figure 8C; $f^u$) and Reachability (Figure 8C; $S^1$). Redundancy, in contrast, showed a much narrower range of values (Figure 8C; $R^1$), in accordance with our finding that Redundancy is less sensitive to network configuration than Reachability (Figure 7E). The average contribution of paired synapses to Redundancy and unpaired synapses to Reachability was similar across connectomes (Figure 8C; $f_{R1}^p$ and $f_{S1}^u$, respectively).

Figure 8.

Different connectomes vary mostly in asymmetry and Reachability and less so in Redundancy

(A) List of additional C. elegans connectomes analyzed from different sexes, brain region, type of synapse (chemical vs. electrical), and developmental stage.

(B) Comparison between the complete C. elegans chemical connectome network, CeH0c, and the connectome networks used throughout the study, CeH1c, CeH2c, and CeH3c.

(C) Distribution of $f^u$, $f^d$, $R^1$, $S^1$, $f_{R1}^p$, and $f_{S1}^u$, for the connectome networks shaded in (A). Error bars indicate mean ± standard deviation.

(D) Comparison of asymmetry, Redundancy, and Reachability between hermaphrodite (⚥; CeH0c) and male (♂; CeMc) chemical connectome networks.

(E) Comparison of asymmetry, Redundancy, and Reachability between chemical (CeH0c and CeMc) and electrical (CeH0e and CeMe) connectome networks.

(F) Comparison of asymmetry, Redundancy, and Reachability between main (CeH0c) and pharynx (CePc) chemical connectome networks.

(G) Comparison of asymmetry, Redundancy, and Reachability between different developmental (CeL1c, CeL2c, and CeL3c) and adult (CeH0c) chemical connectome networks.

(H) Comparision of asymmetry, Redundancy, and Reachability between chemical connectome networks of different species, adult C. elegans hermaphrodites (CeH0c), and first instar female Drosophila melanogaster larva (DmFc).

Comparing hermaphrodite (⚥; CeH0c) and male (♂; CeMc) chemical connectome networks, revealed greater asymmetry in males (Figure 8D; higher $f^u$), and consequently, reduced Redundancy (Figure 8D; lower $R^1)$. However, the increased asymmetry in the male network was not sufficient to enhance Reachability, which was also lower compared to the hermaphrodite network (Figure 8D; lower $S^1$). This was likely due to a difference in relative connectivity between the networks (Figure 8D; lower male $f^d$), with the male connectome appearing sparser, an effect that can substantially reduce Reachability (Figure 7E). We observed a similar relationship between chemical and electrical connectome networks both in hermaphrodites and in males (CeH0c and CeMc vs. CeH0e and CeMe). Electrical synapses, which are less abundant than chemical synapses, showed more asymmetry, lower Redundancy, but also lower Reachability (Figure 8E).

In C. elegans, in addition to the main somatic connectome, a small (20 neuron) separate pharyngeal connectome regulates the pharynx, the feeding apparatus of the nematode.³² We found dramatic differences between the main (CeH0c) and pharyngeal (CePc) chemical networks (Figure 8F). The pharyngeal network is considerably denser in its relative number of synaptic connections (Figure 8F; higher $f^d$), resulting in greater symmetry (Figure 8F; lower $f^u$) and thus, more Redundancy (Figure 8F; higher $R^1$), but also more Reachability (Figure 8F; higher $S^1$), likely due to the high volume of connectivity. These features may be suited to the particular task of the pharyngeal network, tasked with coordinating food intake.

We also compared network lateralization between different C. elegans larval development stages,²¹ L1 through L3 (CeL1c, CeL2c, and CeL3c). We found a gradual increase in symmetry with development (Figure 8G; decreasing $f^u$), followed by a return to higher asymmetry in adulthood (Figure 8G; $f^u$). At the same time, the developing networks exhibited remarkably stable Redundancy (Figure 8G; $R^1$), but a steady increase in Reachability (Figure 8G; $S^1$), reminiscent perhaps of the progressive interconnectedness appearing in the human brain as it matures.^33,34

Finally, we wished to compare Reachability and Redundancy in the C. elegans (Ce) connectome to another species. To this end, we examined the recently attained 1^st instar larva female Drosophila melanogaster (fruit fly) chemical connectome network³⁵ (Figure 8H, Dm). The fly connectome showed higher asymmetry, $f^u$, than that of C. elegans, and a lower synaptic density, $f^d$ (Figure 8H). Interestingly, Redundancy was comparable between fly and worm (Figure 8H; $R^1$), whereas Reachability was much lower in the fly connectome compared to C. elegans (Figure 8H; $S^1$), reminiscent of the low Reachability in C. elegans early larvae (Figure 8G; L1).

Discussion

We have shown how the relative level of symmetry vs. asymmetry in synaptic connectivity may settle an inherent tradeoff between Redundancy and Reachability: the selective boosting of specific class paths vs. the expansion and diversification of neuron class connections. We identify a possible tuning principle between symmetry and asymmetry, and thus between Redundancy and Reachability. We find that while contralateral neuron members of each neuron class display symmetry in the number of synaptic contacts they possess (consistent with observed structural symmetry^30,31), the identity of a subset of these synapses, the unpaired synapses, varies between left and right. The number of such unpaired synapses in each neuron is proportional to that neuron’s total number of synapses, and their composition appears to be stochastic. Thus, each neuron may be allocated a certain number of synaptic partners, to which it can randomly link, independent of its contralateral class member, contributing in this way to overall network asymmetry. This allowance of random synaptic connections within each neuron could enhance Reachability by presenting new possibilities for connectivity.

At the same time, the paired synapses in each neuron show much less stochasticity in their identity, with possible coordination between left and right neuron class members. These synapses are a source of network symmetry, supporting Redundancy (in addition to many other features, including broader sensorimotor coverage, better threat detection, robust signaling, and increased output strength) by the duplication of selected connections, reinforcing these synaptic paths. Synapse formation and specification involve complex interactions of genetic,²³ biomechanical,²⁴ activity-dependent,²⁵ and stochastic²⁶ processes. One prediction of our model is that these factors may segregate according to synaptic subpopulation, with paired synapses tending to be prespecified, while unpaired synapses being more stochastic. Evidently, many additional factors may account for neuronal and synaptic lateralization.⁸

Our study focused mostly on the C. elegans hermaphrodite connectome, and in particular on chemical synaptic connectivity. However, the principles we have derived seem to hold both in artificially constructed networks and in other C. elegans connectomes, such as in the male, larval, and pharyngeal networks, and also in the D. melanogaster 1^st instar larva chemical connectome, which we found to resemble in its Redundancy and Reachability that of C. elegans larvae. It is notable that although nematodes lack any clear anatomical lateralization in their body, eliminating the need to control left and right extremities or movement, they still exhibit a largely bilateral symmetrical nervous system, with a majority of neurons paired into left-right classes.¹⁹ This suggests that bilateral symmetry may be a deeply conserved feature of the nervous system.⁸ For a compact system like the C. elegans connectome, the duplication of individual synapses can have considerable impact. This is evident, for example, in the clear behavioral impact of genetically inserting even one synaptic connection to the C. elegans network.³⁶ However, even in a much more complex systems, two hippocampi rather than one, or a doubling of the number of olfactory sensory neuron inputs converging onto the olfactory glomeruli, should increase Redundancy.

A key premise underlying our analyses is that neurons pertaining to the same class share similar or related functions. This is often taken as an implicit assumption in the study of C. elegans and other neural circuits. Notably, left/right differences in molecular composition have been identified in several C. elegans neuron classes,³⁷ and functional lateralization, in particular, has been extensively characterized in two specific chemosensory neuron classes, ASE^38,39,40 and AWC.^41,42 However, despite the differences between the left and right ASE and AWC neurons, for example, both members of these neuron classes are involved in similar tasks and play related roles in the circuit. In addition, as we have noted, at least half of C. elegans neuron classes show chemical and/or electrical synaptic coupling between their bilateral members. Therefore, it is reasonable to assume that analogous information is sent or received by the distinct synaptic contacts of each class member, justifying the neuron classes as a unit for analysis of lateralization.

In more complex systems neuron classes comprise many more than just one bilateral pair of neurons, and the number of synaptic contacts of each neuron can be several orders of magnitude larger than in C. elegans. Thus, in such systems an entire brain region could be considered equivalent to a C. elegans neuron class. A classic example is the human inferior frontal gyrus (Broca’s area). In the dominant (typically left) hemisphere, this brain region is responsible for language comprehension and speech production,⁴³ as opposed to the analogous contralateral region that controls prosody. These bilateral functional disparities are associated with left-right dissimilarities in circuit structure and wiring.^44,45 Despite the dramatic differences between these two regions, they are nevertheless bilaterally symmetrical in their overall anatomy and position, they are interconnected by commissural fibers, and ultimately, are both involved in language-related processes. Thus, larger, more complex connectomes may also be amenable to analysis using the Redundancy vs. Reachability framework that we have developed.

In order to properly grasp the impact of lateralization on neuronal structure-function relations, connectome maps must label bilaterally equivalent neurons or structures and resolve the differences and similarities in synaptic connectivity between them, as in the C. elegans connectome. In any case, synaptic connectivity should not be assumed to be symmetrical, and such assumptions are unsuitable for validating proper connectome mapping. In addition, our study highlights the importance of lateralization at the microscopic connectome resolution, complementing much of the previous work on brain asymmetry, which has focused on gross brain measures such as volume, thickness and surface area.⁴⁶

As we have shown, it is possible to adapt standard concepts and tools from graph theory to the special case of neural networks, accounting for the grouping of neurons into functional classes with bilateral structure. To this end, it is important to distinguish between standard graph paths connecting individual neurons, and realizable class paths that capture an interesting aspect of the interplay between structure and function in neural networks. As noted, our analysis is based on several apparent simplifications, focusing on undirected, unweighted, unsigned exclusively chemical connections. However, the conceptual framework we developed could readily apply to more detailed network representations, enabling further investigation into the impact of synaptic lateralization on salient network properties. Moreover, features like Redundancy and Reachability point at potential functional implications, and at the possibilities afforded by network configuration. In reality, however, neural information flow and dynamics may be much more complex and not necessarily compatible with physical connectivity.⁴⁷ When the full complexity of neural networks is considered, effective network function might diverge from that suggested by synaptic connectivity alone. For example, broad long-distance neuropeptide signaling^48,49 may boost Reachability even in a relatively symmetrical network. Concomitantly, various forms of synaptic plasticity continuously modify the connectivity map. Does plasticity act symmetrically or unilaterally? Can it shift the balance between Redundancy and Reachability or is their impact restricted to a fixed setpoint? A recent study showed a striking switch in lateralization of synaptic connections in the C. elegans gustatory circuit following salt learning.⁵⁰ Probing the interplay between lateralization and network function prompts many such essential questions. In general, structural and functional connectomes may differ substantially,⁵¹ and it remains challenging to infer one from the other, especially due to functional connectivity being considerably more dynamic and complex than structural connectivity. This applies also to structural vs. functional Redundancy and Reachability. Progress in this direction could be achieved by mapping functional connectivity, with left-right labeling, under distinct conditions, and studying the extent to which lateralization, Redundancy and Reachability vary.

The broader impact of Redundancy and Reachability, as we have framed these concepts, on overall brain function and, in particular, on such capacities as perception, action, decision making, learning and more, is challenging to discern. Changes in brain lateralization have been shown to be associated with cognitive performance.⁵² It can generally be postulated that Redundancy may be especially important for proper sensory or motor function, where robustness and fault tolerance are crucial. Redundancy in synaptic connectivity between specific pairs of neurons may also facilitate learning at the neuronal level.⁵³ At the same time, Reachability may facilitate the integration, distribution and coordination of neural information, presumably required for more complex brain operation. Interestingly, disease conditions such as schizophrenia have been associated with abnormal connectivity,¹⁷ including a decrease in the level of asymmetry normally observed in healthy brains,⁵⁴ and increased segregation in network topology,⁵⁵ which could be interpreted as reduced Reachability. Such coupling between reduced asymmetry and Reachability is reminiscent of our findings on the relationship between lateralization and network properties.

Limitations of the study

Our analysis did not take into account such important properties of the connectome as the weight (number or efficacy) or sign (excitatory vs. inhibitory) of each synapse. Much of this information is not yet fully resolved. In particular, it could be difficult to define paired vs. unpaired synapses in a weighted network (do paired synapses have to have the exact same weight?) and to incorporate synaptic weights in Redundancy computation (is weighted Redundancy equal to unweighted Redundancy?). As more connectome data become available such important factors could more easily be incorporated into connectome structural analysis.

Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Ithai Rabinowitch (ithai.rabinowitch@mail.huji.ac.il).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •Analysis and simulation results are available as supplemental information.
• •All original code has been deposited at Github and is publicly available as of the date of publication. DOIs are listed in the key resources table.
• •Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

The study was funded by Israel Science Foundation grant 1465/20 and by the Horizon Europe, PathFinder European Innovation Council Work Program under grant agreement No 101098722. Views and opinions expressed in this article are those of the authors only and do not necessarily reflect those of the European Union or European Innovation Council and SMEs Executive Agency (EISMEA). Neither the European Union nor the granting authority can be held responsible for them.

Author contributions

Conceptualization, V.S.B and I.R.; methodology, V.S.B and I.R.; software, V.S.B.; formal analysis, V.S.B and I.R.; investigation, V.S.B and I.R.; writing – original draft, I.R.; writing – review & editing, V.S.B and I.R.; visualization, I.R.; supervision, I.R.; funding acquisition, I.R.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
Code for connectome lateralization analysis	Custom code combining functions from NetworkX	Github: https://github.com/RabinowitchLab/connectome_asymmetry Zenodo: https://doi.org/10.5281/zenodo.10778067

Method details

Neural networks used in the study

Our study included three types of neural networks, generated and analyzed using the NetworkX library in Python. (1) C. elegans connectome networks (Table 1), based on datasets curated to a large part by the Emmons lab (https://wormwiring.org) and the Drosophila melanogaster (fruit fly) 1^st instar larva female chemical connectome.³⁵ (2) Random neural networks (Table 2), related to the C. elegans connectome networks. (3) Symmetrized or desymmetrized networks.

Connectome networks

The connectome networks used in this study are listed in Table 1. The italicized networks are the main ones featured in most of the work. The rest of the networks are used in Figure 8. Unless otherwise specified, all connectome networks were constructed as undirected and unweighted networks, including only chemical synapses. All non-neuronal cells and their synaptic connections were removed. For each network, except for CePc, we also excluded the pharyngeal neurons. Finally, for our analysis, we used the single largest component, the largest connected subgraph within the entire graph, of every connectome dataset, to avoid split networks with disconnected components.

To enable comparison and correlation between different adult C. elegans connectome networks, we reduced CeH0c, to a smaller subnetwork, CeH1c, containing the same neurons as CeH2c and CeH3c. Directed CeH1Dc, CeH2Dc and CeH3Dc networks maintained the original directionality assignments of the edges, and allowed for reciprocal edges between the nodes.

The D. melanogaster network was generated from the neuron pairs whose both left and right neurons had been identified in the dataset. These neurons were mapped onto the connectivity matrix of all neurons, to generate a subnetwork consisting of only left-right neuron pairs. The network was generated as undirected and unweighted. For our analysis, each pair of left-right neurons was considered as a single class.

Random networks

The random networks generated for the study are listed in Table 2. First, the Model network variants, used for demonstration purposes, were composed manually. Second, SR networks, as well as the Small2 networks were generated using the ‘gnm_random’ function in NetworkX. This procedure accepts two arguments, n, the number of nodes and m, the number of edges. For the SR networks, n equaled 180, and m equaled the average number of edges in the connectome networks (CeH1c, CeH2c and CeH3c), 1,770. The nodes were labeled according to the connectome classes (some of which have more than 2 members). To construct the Small2 networks with variable $f^d$, we varied the number of edges, symmetrized or desymmetrized (see below) the networks and selected networks with specific $f^u$ values.

Third, the ER (Erdős–Rényi) network was generated using the ‘erdos_renyi_graph’ function in NetworkX. The procedure accepts two arguments, n, the number of nodes and p, the probability of edge creation between any two nodes. n equaled 180, and p equaled the relative degree, $f^d=0.125$, of the undirected C. elegans connectome network CeH1c. To generate the set of undirected vs. directed Small1 networks, we first constructed 20 networks using the ‘erdos_renyi_graph’ function with $n=40,p=0.1$. We applied symmetrization and desymmetrization (see below) to derive 3 sets of 20 networks each with a different average $f^u$ value for each set. We then created for each undirected network an equivalent directed network by randomly assigning directionality to every edge. This was done using the following algorithm:

Algorithm 1: Convert undirected to directed.

• 1.Create an empty directed network using the NetworkX DiGraph function with same nodes as the undirected network. This ensures that every edge added will be directed.
• 2.Select an edge from the undirected network linking between nodes A and B.
• 3.For the same nodes in the directed network, randomly add an edge in either the A→B or A←B direction.
• 4.Repeat for all the edges in the undirected network to generate a directed graph with the same number of nodes and edges as the original undirected network.

Finally, the DCB (degree connectome-based) network was generated as a random network with a degree distribution similar to that of the connectome network CeH1c. First, we created a network with 180 nodes and no edges. Then for each node, edges were formed with random partners out of a list combining the partners of that node in the CeH1c network and in the ‘contactome’, the physical adjacency between each two neurons in the connectome^30,31 (we used the source file ‘cel_n2u_nr_adj.csv’³⁰). This process was repeated until each node reached its target degree.

As in the connectome networks, when generating the random networks, we only accepted network outputs where all the nodes belonged to the single largest component (the largest connected subgraph within the entire graph).

Symmetrized and desymmetrized networks

In order to construct networks ranging from fully symmetrical to fully asymmetrical, we developed an algorithm for symmetrizing or desymmetrizing an initial network (Figure S4). The algorithm is based on iterative addition and removal of edges in the network to progressively reduce or increases $f^u$, while maintaining network degree. We applied this algorithm to initial networks from Table 1 and 2, and obtained in this manner sequences of networks with incremental symmetry/asymmetry levels.

Algorithm 2: Network symmetrization / desymmetrization.

Define $L_{f^u}$, a list of target $f^u$ values.

• 1.Input a network for symmetrization (desymmetrization).
• 2.
  Create/update:

  • •
    $L_r$, a list of all candidate edges, r, for removal from the current network.
  • •
    $L_a$, a list of all candidate edges, a, for addition to the current network.
• 3.Compute and record current network u$f_c$.
• 4.Randomly select edge a from $L_a$ and add to network, and randomly select edge r from $L_r$ and remove from network.
• 5.
  Compute the new $f^u$ for the modified network.

  • A.
    If $f^u$ = 0 ($f^u$ = 1)

    • •
      Save network.
    • •
      Stop.
  • B.
    If $f^u$ < $f_c^u$ ($f^u$ > u$f_c$)

    • •
      Add r to $L_a$; remove a from $L_a$; add a to $L_r$ (Step 2).
    • •
      Set new network as current network, and set $f_c^u=f^u$ (Step 3).
    • •
      If $f^u\in L_{f^u}$, save network.
    • •
      Go to Step 4.
  • C.
    If $f^u$ ≥ $f_c^u$ ($f^u$ ≤ u$f_c$)

    • •
      Undo changes.
    • •
      Go to Step 4.

Computing symmetry/asymmetry metrics

We developed custom algorithms for computing symmetry and asymmetry metrics. All functions are from NetworkX. First, in order to compute the Redundancy metric, $R^n$, it is necessary to derive ${\mu }_j^n$, the number of neuron paths of length $n$ that realize each class path, $j$. The computation considers all neuron classes, including those with more than 2 bilateral neuron pair members. ^★ The ${\mu }_j^n$ algorithm is also used for deriving Redundancy based exclusively on paired synapses, $R^{p,1}$ (in the special case of $n=1$). In this case, only neuron classes comprising a bilateral pair of neurons are included. Thus, $R^{p,1}$ represents a lower bound for the contribution of paired synapses to Redundancy.

Algorithm 3: Rn Redundancy (μjn and 1Rp,).

• 1.
  For each pair of neurons belonging to different classes:

  • a.
    Use the has_path function to determine whether a path between them exists.
  • b.
    Use the all_simple_paths function to find all paths of length n between them.
• 2.Convert each neuron path to a class path by replacing individual neurons with their neuron class.
• 3.Keep only class paths between distinct classes.
• 4.Count the number of occurrences of each class path to obtain ${\mu }_j^n$.

★ To derive $R^{p,1}$, count only those class paths in Step 4 that are composed of paired synapses (for $n=1$ path lengths).

Second, to compute the Reachability metric, $S^n$, it is necessary to first derive ${\nu }_k^n$, the number of neuron classes that can be reached from each neuron class, $k$, by a realizable class path of length, $n$. This computation includes also neuron classes with more than 2 bilateral neuron pair members. ^★ The ${\nu }_k^n$ algorithm is also used for deriving Reachability based exclusively on unpaired synapses, $S^{u,1}$ (in the special case of $n=1$). In this case, only classes composed of 2 bilateral neuron pairs are considered.

Algorithm 4: Sn Reachability (νkn and 1Su,).

• 1.For each neuron class, k, list all neurons connected to its individual neuron members.
• 2.Repeat Step 1 n times, progressing along the neuron path, using depth-first search.
• 3.Combine reachable neuron paths to class paths, to obtain for each neuron class the number of realizable class paths stemming from it, ${\nu }_k^n$.

★ To derive $S^{u,1}$, perform Step 1 only ($n=1$), eliminating connections made by paired synapses.

Quantification and statistical analysis

All statistical details can be found in the relevant figure legends, figures and results, including the statistical tests used, exact value of n, and the definition of center, and dispersion.

Published: August 13, 2024