Guided graph spectral embedding: Application to the C. elegans connectome

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Article

Reference

Guided graph spectral embedding: Application to the C. elegans connectome

PETROVIC, Miljan, et al.

Abstract

Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions—for example, based on wavelets and Slepians—that can be applied to filter signals defined on the graph. In this work, we take inspiration from these constructions to define a new guided spectral embedding that combines maximizing energy concentration with minimizing modified embedded distance for a given importance weighting of the nodes. We show that these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows us to put the focus on particular nodes and tune the trade-off between global and local effects. Following the derivation of our new optimization criterion, we exemplify the methodology on the C. elegans structural connectome. The results of our analyses confirm known observations on the nematode's neural network in terms of functionality and importance of cells. Compared with Laplacian embedding, the [...]

PETROVIC, Miljan, et al . Guided graph spectral embedding: Application to the C. elegans connectome. Network Neuroscience , 2019, vol. 3, no. 3, p. 807-826

DOI : 10.1162/netn_a_00084 PMID : 31410381

Available at:

http://archive-ouverte.unige.ch/unige:138715

Disclaimer: layout of this document may differ from the published version.

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SUPPORTIVE INFORMATION

Results of k-means clustering

In Supplementary Fig. S4, we present the proposed embedding from Figs. 4-6 and clusters of nodes with cooperation weight 1 derived by the k-means approach with 20 repetitions. Dimensionality of the

considered data points was set to 2, i.e. entries of the two Slepian eigenvectors were used for clustering – the second and the third. The Silhouette method (Rousseeuw, 1987) was used to estimate the optimal number of clusters as the one which produces the minimal number of negative silhouette values. Convex hulls of each found cluster are represented by dashed black lines. The exact lists of neurons assigned to each cluster, for the three investigated cell types, are provided in Supplementary Tables 1, 2 and 3.

Table 1. Sensory neurons of theC. elegans. Columns correspond to clusters derived by optimized k-means.

33

C1 C2 C3 C4 C5 C6 C7

AVM CEPDR AFDL PHAL ADEL ADFR ADFL PDEL CEPVR AFDR PHAR ADER ADLR ADLL PDER IL1DR ASEL PHBL ALMR ASGL ALA PHCL IL1R ASER PHBR CEPDL ASGR ALML

PHCR IL1VR ASIL CEPVL ASHR ALNL

PLML IL2DR ASIR IL1DL ASKL ALNR

PLMR IL2R AWAL IL1L ASKR AQR

PVM IL2VR AWAR IL1VL AWBL ASHL

OLLR AWCL IL2DL AWBR ASJL

OLQVR AWCR IL2L ASJR

URXR IL2VL BAGL

OLLL BAGR

OLQDL FLPL

OLQDR FLPR

OLQVL PLNL

URYDL PLNR

URYDR PQR

URYVL PVDL

URYVR PVDR

SDQL SDQR URXL

Evaluation of The Clustering

In order to evaluate the inspected clusters of sensory neurons (Fig. 4B), interneurons (Fig. 5B) and motoneurons (Fig. 6B), we used statistical testing of communities (clusters). In all three cases, the nodes with importance m

i

= 0 are considered as one additional cluster. We use the Newman-Girvan modularity as statistic (M. E. J. Newman, 2006). A vector of nodal assignments to clusters expresses its goodness of fit to the underlying adjacency matrix through the value of modularity Q. It is calculated as:

Petrovic, M., Bolton, T.A.W., Preti, M.G., Liégeois, R., & Van De Ville, D.

(2019). Supporting information for “Guided graph spectral embedding:

application to the c. elegans connectome.” Network Neuroscience, 3(3),

807–826. https://doi.org/10.1162/netn_a_00084

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Table 2. Interneurons of theC. elegans. Columns correspond to clusters derived by optimized k-means.

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C1 C2 C3 C4 C5 C6

AVBL AIML AIBL AIAL ADAL AVAL AVBR AIMR AIBR AIAR ADAR AVAR

AVG AVFL AINL AVEL AVDL

AVJL AVFR AINR AVER AVDR

AVJR AVHL AIYL AVKL LUAL

BDUL AVHR AIYR AVKR LUAR

BDUR PVQL AIZL DVA PVCL

PVPR PVQR AIZR DVC PVCR

RIFL AUAL PVPL PVR

RIFR AUAR PVT PVWL

RIAL RICL PVWR

RIAR RICR

RIBL RIGL

RIBR RIGR

RIH RIPL

RIR RIPR

RIS RMGL RMGR URBL URBR

Q = 1 2w

X

N

i,j

([A

bin

]

i,j

d

i

d

j

2w )

Ci,Cj

,

where N is the number of nodes, w is the total strength of edges in the graph, A

bin

is the graph binary adjacency matrix, d

i

denotes the degree of the i

^th

node, is the Kronecker delta function, and C

i

denotes the cluster to which the i

^th

node belongs.

In Supplementary Fig. S8, we present the results of the statistical approach for the case of sensory (red plots), inter- (grey plots) and motoneurons (green plots). The modularity values for the assignments to clusters as found by k-means clustering are marked with the dashed lines and labeled with Q

sensory

, Q

inter-

, and Q

moto-

(Supplementary Fig. S8B). For the number of clusters estimated by the Silhouette method, we generated 999 random assignment vectors and calculated Q each time, in order to build a null distribution (Supplementary Fig. S8A).

As Q

sensory

, Q

inter-

, and Q

moto-

are above the corresponding distributions of modularity for random assignments, we conclude that the found clustering is significant. Since these modularity values are strictly greater than all other Q values for random assignments, and, consequently, from any chosen percentile of the calculated distributions, the test rejects the null hypothesis that the chosen clustering is random at even very small significance levels. Finally, we note that the distribution of Q in the case of interneurons is slightly closer to the corresponding value of Q

inter-

than in the case of sensory or

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Table 3. Motoneurons of theC. elegans. Columns correspond to clusters derived by optimized k-means.

35

C1 C2 C3 C4 C5 C6 C7

AS10 DD04 DD05 DA04 AVL AS11 AS01 AS06 VA07 VA08 DB03 DA08 DA09 AS02 AS07 VB06 VA09 DB04 DB07 DD06 AS03 AS08 VB07 VB08 DD02 PVNR DVB AS04 AS09 VD07 VB09 DD03 RID PDA AS05 DA07 VD08 VD10 VA06 RIML PDB DA01

DB05 VD09 VB02 RIMR VA11 DA02

DB06 VB03 RIVL VA12 DA03

HSNL VB04 RIVR VB10 DA05

PVNL VB05 RMDDL VB11 DA06

RMHL VC01 RMDDR VD12 DB01

RMHR VC02 RMDL VD13 DB02

SABD VC03 RMDR DD01

SABVL VD02 RMDVL HSNR

SABVR VD03 RMDVR VA01

SIADL VD04 RMED VA02

SIADR VD05 RMEL VA03

SIAVL VD06 RMER VA04

SIAVR RMEV VA05

SIBDL RMFL VB01

SIBDR RMFR VC04

SIBVL SAADL VD01

SIBVR SAADR

SMBDR SAAVL

URADL SAAVR

VA10 SMBDL

VC05 SMBVL

SMBVR SMDDL SMDDR SMDVL SMDVR URADR URAVL URAVR VD11

motoneurons. This can be expected, since interneurons are more strongly connected to other cell types, and thus, do not impose as strong communities as for the clusters formed from sensory or motoneurons.

Supplementary Figures

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-0.4 -0.2 0 0.2 0.4

0 0.4

0.8 W = 100

W = 150 W = 200 W = 279

0 0.2 0.4 0.6 0.8 1

Normalized index (k/W)

Energy conc. μ

0 0.4 0.8

1.2 W = 100

W = 150 W = 200 W = 279

0 0.2 0.4 0.6 0.8 ¹

Normalized index (k/W)

Mod. emb. dist."

-0.4 -0.2 0 0.2 0.4

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4

Slep. 28 (μ= 1)

Slep. 40 (μ = 0.7699)

Energy concentra!on criterion

W = 100 W = 150 W = 200 W = 279

Slep. 20 (μ= 1) Slep. 15 (μ= 1)

Slep. 10 (μ= 0.9964)

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4

-0.4 -0.2 0 0.2 0.4

Slep. 90 (μ = 0.0106) Slep. 60 (μ = 0.8156)Slep. 135 (μ = 2.63 10-4) Slep. 80 (μ = 0.98) Slep. 112 (μ = 1)

Slep. 60 (μ = 0.4122) Slep. 90 (μ = 0.3436) Slep. 120 (μ = 0.2929) Slep. 167 (μ = 0)

Slep. 251 (μ = 0)

W = 100 W = 150 W = 200 W = 279

Slep. 180 (μ = 0)

Slep. 28 (ξ= 0)

Slep. 40 ("= 0.2723)

Slepian 20 (ξ= 0) Slep. 15 (ξ = 1.549 10^-4)

Slep. 10 (ξ= 0.0058)

Slep. 90 ("= 0.7853) Slep. 60 (" = 0.2538)Slep. 135 (" = 0.8999) Slep. 80 (" = 0.2282) Slep. 112 (" = 0)

Slep. 60 (ξ = 0.5349) Slep. 90 (ξ = 0.5928) Slep. 120 (ξ = 0.7208) Slep. 167 (ξ = 0.8789)

Slep. 251 (" = 1.183)

Slep. 180 (" = 1.0212)

Modified embedded distance criterion

A B

C

D

Figure S1. For energy concentration (A) and modified embedded distance (B) criteria, eigenspectra at bandwidthW= 100,150,200,279, as depicted by increasingly darker blue or purple shades, respectively. Yellow and orange vertical bars map locations of the eigenspectra at which Slepian vectors are shown (CandD). They are displayed for increasing bandwidth going from left (W = 100) to right (W = 279, full bandwidth). For energy concentration (C), the first row illustrates two Slepian vectors mapping the start of the spectrum (normalized indices of0.1— strongly concentrated inS— and0.4— still concentrated, but less for lower bandwidth). The second row denotes two Slepian vectors from the second half of the spectrum (normalized indices of0.6— mildly concentrated inSusing a smaller bandwidth — and0.9— not concentrated at all). Visualizations are similar for modified embedded distance (D), but in this case, low eigenvalues imply either non-concentrated (e.g., X axis, first row of plots) or mildly concentrated but low localized spatial frequency Slepian vectors (for instance, Y axis, first row of plots,W = 200), while high eigenvalues relate to high localized spatial frequency Slepian vectors (see Y axis, second row of plots). SeeVan De Ville et al. (2017a)for another preliminary analysis of the dataset from the modified embedded distance viewpoint.µand⇠values of the shown Slepian vectors are provided in parentheses on each axis.

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-0.4 0 0.4 0.8 1.2

0 0.2 0.4 0.6 0.8 1

Normalized index (k/W)

N e w c ri te ri o n

ζ

-0.1 0 0.1 0.2

-0.2 0 0.1 0.2

-0.2 0 0.2 0.4

-0.4 -0.2 0 0.2 0.4

S le p . 2 (

ζ

= 0 .6 5 2 4 )

Slep. 1 (ζ = 0.8249)

0 0.04 0.16

0 0.1 0.2

0

S le p . 3 (

ζ

= 0 .5 8 5 4 ) S le p . 3 (

ζ

= 6 0 1 7 )

Slep. 2(ζ = 0.6524) Slep. 2 (ζ = 0.6616) Slep. 2 (ζ = 6754)

S le p . 3 (

ζ

= 0 .6 2 0 8 )

S le p . 3 (

ζ

= 0 .6 1 9 8 )

Sle p. 1 4 0 (

ζ

= 0 )

-0.1

-0.1 -0.2

0 0.1 0.2

-0.1

-0.1 -0.2

0 0.1 0.2

-0.1

-0.1 -0.2

ζ

= 0 )

Slep. 56 (ζ = 0.0892)

0.6

0 0.1 0.2

Slep. 223 (ζ = -0.1253)

0 0.1 0.2 -0.1

-0.2 0.3

Linear Quadra!c Order 10 Full

0.7 0.9

0.5 0.1 0.3

A

B

Figure S2. Eigenspectrum of the newly developped⇣criterion (A), with vertical bars highlighting the locations of the spectrum at which four pairs of Slepian vectors were sampled for display (B, from first to fourth row as respectively depicted by yellow, orange, brown and red color codes). Results obtained with linear, quadratic and order 10 approximations, as well as from a full computation ofM L^1/2ML^1/2, are respectively shown from left to right.⇣ values of the shown Slepian vectors are provided in parentheses on each axis.

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-0.2 -0.1 0 0.1 0.2

ADAL

ADEL ADFR AIAL

AIBL AIYL

AIZL ASEL ASER

ASGR ASIR

AUAL AUAR AWCR

RIAL RIAR

RMER AVHL

DA03 HSNL

PHAL PVQL PVQR

PVR VA09

-0.2 -0.1 0 0.1 0.2

ADFL AFDL AIAL

AIAR

AINL AIYL AIYR ASEL ASER

ASGR ASIL ASIR

AUAR AWAR

AWCL AWCR

CEPVR IL1DL IL1R IL1VR RIAR

URAVR ADER ASHR ASJL

ASKL

AVAR AVFL

AVFR

AVM PHAL PHBR

PVR

-0.2 -0.1 0 0.1 0.2

ADER AIZR

ALMR ASGR

ASKR AWAR

CEPDR CEPVR

IL1DR IL1R

IL1VR IL2DR

IL2R IL2VR OLLR

OLQDR OLQVR RIH

RIPR URBR

URXR

ADEL ADLL

AFDL AFDR

AIBL AIYL

AIZL

ALML ASEL ASER

ASGL

ASHL ASIL

ASIR

AWAL

AWBL AWCR

CEPVL

IL1DL IL1L

IL1VL IL2DL

IL2VL OLLL OLQDL

OLQVL RIPL URXL

URYDL

-0.4 -0.2 0 0.2

ADEL ADFL ADFR AFDL AIAL AIAR

AIBR AIYL AIZL ASEL ASER

AUAR

CEPVR IL1R RIAL RIAR

RIH RIR

RMEL RMER AVAL

AVAR AVBL AVDR

AVFL HSNL

HSNR PVQL

PVR VA09

-0.4 -0.2 0 0.2

ADEL AVEL AVER

AVKL RIVR VB01

ADLL AVDL

DA09 DD01 DD02

DD03

DD04

DD05 DD06

DVB PDA PVDL VA07

VA08

VA09 VB05

VB06 VB07

VB08 VB09 VC03

VD01

VD10 VD02 VD03

VD04

VD06

VD07 VD08

VD09

-0.4 -0.2 0 0.2

AS11 AS02

AS09 AVAR AVDL

AVL DA01 DA02

DA08 DA09 DB07 DD01

DD06 DVB HSNR

PDA PDB RID

VA11 VA12 VB10

VB11 VD11

VD13 AVBL

DB03 DD02

DD03

DD04

DD05

PVDL VA07

VA08

VA09 VB05

VB06 VB07

VB08 VB09

VC01

VD10

VD02 VD04

VD07 VD08

VD09

-0.2 -0.1 0 0.1 0.2

ADFL AIAL

AIAR

AIBL

AIBR AINL

AINR AIYL AIYR

AIZL

AIZR ASER ASGL

AUAL AUAR

AVEL AVER AVKL BAGL

DVA PVT

RIAL RIAR RIBL RIBR RICL RICR

RIGL RIGR

RIH RIR

RIS AIML

AIMR

ASJL ASKL

AVAL AVAR AVBL AVBR

AVDL AVDR AVFL

AVFR

AVG AVHL

AVHR

AVJL AVJR

BDUL HSNL

LUAL LUAR

PLMR

PVCL

PVCR PVNL

PVPL PVQL

PVQR

PVR

PVWL PVWR RIFL RIFR

VA12

-0.2 -0.1 0 0.1 0.2

ADAL ADFR

AFDR AIAL

AIAR

AINL AINR AIYL

AIYR AIZL

AIZR ASGR

ASHL ASHR

ASIL

AUAL AUAR

AVEL AVKL AWCR

CEPDL RIAL

RIAR RIBL

RIBR RIGL

RIH RIPL RIVL AIML

AIMR ASKL

AVAL AVAR AVDL AVDR AVFL AVFR

AVG AVHL AVHR

AVJL

DVA HSNL

LUAR PHAL

PHAR

PVCL PVCR

PVQL

PVR

-0.2 -0.1 0 0.1 0.2

ADFL ADFR AIAL

AIAR AIBL AIBR AINL AIYL

AIZL ASEL

AUAR

CEPDR RIAL

RIAR RIBR

RIH

RMEL RMER AVAR

AVBR AVDR

AVFL

DD05 HSNL

HSNR

PHAL

PVCR PVQL

PVR

-0.2 0 0.2 0.4

-0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2

-0.2 0 0.2 0.4

Slep. 2 (ζ = 0.8226) Slep. 2 (ζ = 0.5525) Slep. 2 (ζ = 0.4247)

S le p . 3 ( ζ = 0 .7 2 9 6 ) S le p . 3 ( ζ = 0 .4 9 0 1 ) S le p . 3 ( ζ = 0 .3 6 5 8 )

Slep. 2 (ζ = 0.8226) Slep. 2 (ζ = 0.6366) Slep. 2 (ζ = 0.6131)

S le p . 3 ( ζ = 0 .7 2 9 6 ) S le p . 3 ( ζ = 0 .6 2 4 1 ) S le p . 3 ( ζ = 0 .5 6 9 5 )

Slep. 2 (ζ = 0.8226) Slep. 2 (ζ = 0.5907) Slep. 2 (ζ = 0.4912)

S le p . 3 ( ζ = 0 .7 2 9 6 ) S le p . 3 ( ζ = 0 .5 1 9 3 ) S le p . 3 ( ζ = 0 .3 7 1 4 )

M

_I

= 1

M

_S

= 1 M

_M

= 1 M

_S

= 1 M

_M

= 1

A

B

C

Figure S3. Start (left column), intermediate (middle column) and end (right column) representations of sensory neuron (A), interneuron (B) or motoneuron (C) trajectories, respectively, setting cooperation weights for other neuron types to1,0.5or0. Cells are labeled according toVarshney et al. (2011). The start representation is the same across cases, since thenM=Iand the problem boils down to the eigendecomposition of the adjacency matrixA, or equivalently of the LaplacianL=I Ahighlighted in Fig.1.

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A B C

Figure S4. Clusters derived by repeated k-means clustering of the focused nodes in the case of sensory neurons (A), interneurons (B) or motoneurons (C).

The optimal number of clusters was estimated with the Silhouette approach. Nodes constituting the border of each cluster’s convex hull are connected by a dashed black line to visualize the clusters.

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-0.2 -0.1 0 0.1 0.2 0.3

ADFR

ADLL ADLR

AFDL AFDR ASEL

ASER ASGL

ASGR

ASHR

ASIL ASIR ASKL ASKR

AWAL AWAR

AWBL AWBR

AWCL AWCR CEPDR

CEPVR

IL1DR IL1R

IL1VL IL1VR

IL2R IL2VR

OLLR

OLQVL OLQVR

URXR

0 0.1 0.2 0.3

-0.2 -0.1 0 0.1 0.2

ADAL ADAR AIAR

AIBL AIBR

AIML AIMR AINL

AINR

AIYL AIYR

AIZR AUAR

AVAL AVAR

AVBL AVBR AVDR AVEL AVER

AVFL AVFR

AVHL AVJL AVJR PVCL PVNL

PVPL PVPR

PVQL PVR

PVT

PVWL RIBL RIBR

RICR

RIFL

RIFR RIGL RIGR RIR

RIS

RMGL

RMGR

0 0.1 0.2

-0.2 -0.1 0 0.1 0.2 0.3

AS11

AS02

AS06 AS09

AVL

DA02 DA08

DA09

DB01 DB07

DD01 DD02

DD03

DD04 DD05 DD06

DVB PDA

PDB

PDER PHCL PHCR

PVDL PVNR

RID VA11

VA12

VA07

VA08 VA09 VB10 VB11

VB02 VB05

VB06 VB07

VB08 VB09

VC03

VD10 VD11

VD12 VD13

VD02

VD06 VD07 VD08

VD09

0 0.1 0.2 0.3

-0.2 -0.1 0 0.1 0.2

AS06

DD02 DD03

DD04

PVNR

RID VA06

VA07 VA08

VB05 VB06

VB07

VD04 VD05

VD06 VD07 VD08

DA03

DD01

DD05 RIVR RMDL

RMDR

RMED RMEL RMER RMEV

RMFL

VA01 VB01

VB08 VD01

VD10 VD02

VD03 VD09

0 0.1 0.2 0.3

-0.1 -0.2 -0.2

-0.1 0 0.1 0.2 0.3

ADAL AFDR

AIAL AIAR

AIBL AIML

AINL

AINR AIYL AIYR

AIZL ASGL ASGR

ASKL AUAL

AUAR

AVDL

AVDR AVJL AWAR

BDUR HSNR

LUAL LUAR PVCL

PVCR

RIFL RMGL ADAR

AIMR

AIZR ASKR

AVHR AVKL

AVKR DVC

PVPL PVPR

PVQL PVQR

PVT

RIAR

RICL RICR

RIFR

RIGL RIGR

RIML RIPR

RMFR RMGR

URBR

0 0.1 0.2 0.3 -0.1

-0.2 -0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

ADAL ADAR

AS10 AVAL AVAR

AVDR

AVER AVFL AVFR BDUL

FLPL OLQDR PVR

PVT PVWL

RIFL

RIFR RIR AIBL

AIBR

AIML AIMR

AVBL AVBR

AVDL

AVEL

AVJR

LUAL LUAR PVPL

PVWR RIBL RICL

RIPL RIPR

SABD

0 0.1 0.2 0.3

-0.1 -0.2 -0.3 -0.4

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

AFDR AINR

ASER

ASGR ASHL

ASIL

ASJL ASKR AWAL

AWBR

ADAL

AFDL

AIZL

ASEL

ASGL ASHR ASJR

ASKL

AWAR

AWBL AWCR CEPVR

OLQVL

0 0.2 0.4

-0.2 -0.4

-0.4 -0.2 0 0.2 0.4 0.6

AS11 AS05 DD04

DD06

VA07 VB03

VB06 VC03

VD12 DB01

PDA PDB

VA09 VB05

VB07 VD07

VD08

0 0.2 0.4

-0.2 -0.4

Slep. 278 (ζ = -0.2986) Slep. 278 (ζ = -0.2493) Slep. 278 (ζ = -0.3815)

Slep. 279 (ζ = -0.3071) Slep. 279 (ζ = -0.2911) Slep. 279 (ζ = -0.3964)

Slep. 4 (ζ = 0.3623) Slep. 4 (ζ = 0.3057) Slep. 4 (ζ = 0.5126)

Slep. 5 (ζ = 0.3335) Slep. 5 (ζ = 0.2802) Slep. 5 (ζ = -0.4206)

Slep. 1 (ζ = 0.5386) Slep. 1 (ζ = 0.6303) Slep. 1 (ζ = 0.7352)

Slep. 2 (ζ = 0.4912) Slep. 2 (ζ = 0.4247) Slep. 2 (ζ = -0.6131)

-0.2 -0.1 0 0.1 0.2 0.3 0.4

ASGL ASHL

ASJL ASJR

ASKL ASKR

CEPVR HSNL

PDEL PDER

PHAL PHAR PHBL PHBR PHCL PLML

PVM PVQL ADFL

ASEL ASER ASIL

ASIR AWAL

AWCL AWCR IL1DL OLQVL

URXL

0 0.1 0.2 0.3

-0.1

-0.2 0.4

A B C

Figure S5. Focussing on (A) sensory neurons (red), (B) interneurons (black) or (C) motoneurons (green), two-dimensional visualization using alternative sets of Slepian vectors: first and second (first row), fourth and fifth (second row), or last two (third row). Cells are labeled according toVarshney et al. (2011).

⇣values of the shown Slepian vectors are provided in parentheses on each axis.

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-0.05 0 0.05

0.1 0.15 0.2 0.25

ADAL ADEL

AINL AINR

AIZL AUAL

AUAR AVFL

AVFR AVHL AVHR

PHBR PVQL PVQR

VC04

-0.05 0 0.05 0.1 0.15

AS11 PDB RID RMDVR

SAAVR SMBVL DD01

DD02 DD03 DD04 VC01 VC02 VC03

VC04 VC05

VD01 VD02 VD04 VD08

-0.2 -0.1 0 0.1 0.2

ADER ADFR AIAR

AINL AIYR

ALNR ASGR

ASJR ASKR AWAR

CEPDR

CEPVR IL1DR IL1R

IL1VR IL2R

IL2VR OLLR

OLQVR RIH

RMEL

URXR URYVR

ADEL ADLL

AFDR

AIBL AIYL

ALML ASEL ASER

ASGL

ASHL ASIL

ASIR

ASKL AWAL

AWBL AWCL

AWCR

CEPDL CEPVL

IL1L

IL1VL

IL2L IL2VL OLLL

OLQDL

OLQVL

RIPL URYDL URYVL

-0.04 0 0.04 0.08

AIBR AS11

AS02

AS05 AVAL AVDL DA02

DB01 DB03

DD01

DD05 PVDL

VA01 VA03 VB01

VB02

VB08 VB09 VC01 VC02

VD01

VD10 VD02 VD03 VD04 VD05

VD09 ALA AS06 AVBL AVG DB04 DD03

DD04

PVNR VA06

VA07 VA08

VA09 VB05 VB06

VB07

VC03

VD06

VD07

VD08

-0.2 -0.1 0 0.1 0.2

ADFR AIAL

AIAR AIBL

AINL AINR AIYL

AIYR AIZR ASEL ASER ASGL

AUAL AUAR

AVEL AVER AVKL

DVA DVC PVT

RIBL RIBR RIH AIML

AIMR ASJL

ASKL

ASKR

AVAL AVAR AVBL

AVDL AVDR AVFL AVFR

AVG AVHL

AVHR

AVJL

AVJR BDUR

DB05 HSNL

LUAL LUAR

PHAL

PQR

PVCL

PVCR PVPR PVQL

PVQR

PVR PVWL

PVWR RIFL RIFR

VA12

-0.16 -0.12 -0.08 -0.04 0 0.04

IL1DL IL1DR

IL1L IL1R IL1VL

IL1VR IL2R OLLL

OLLR RMED

RMEV URXR

ALML

ALMR AVDL

AVM

FLPL PVR

-0.2 -0.1 0 0.1 0.2 0.3 -0.2 -0.1 0 0.1 0.2 0.3

-0.2 -0.1 0 0.1 -0.1 0 0.1

-0.2 -0.1 0 0.1 0.2

-0.2 -0.1 0 0.1 0.2 0.3

-0.3

Slep. 2 (ζ = 0.514) Slep. 2 (ζ = 0.819)

Slep. 2 (ζ = 0.7479) Slep. 2 (ζ = 0.892)

Slep. 2 (ζ = 0.8518) Slep. 2 (ζ = 0.4425)

S le p . 3 ( ζ = 0 .4 0 0 9 ) S le p . 3 ( ζ = 0 .7 1 2 3 ) S le p . 3 ( ζ = 0 .7 9 4 2 )

S le p . 3 ( ζ = 0 .6 7 1 3 ) S le p . 3 ( ζ = 0 .4 2 5 3 ) S le p . 3 ( ζ = 0 .7 3 9 9 )

Chemical synapses Gap junc!ons

A

B

C

Figure S6. Separate two-dimensional visualizations when only considering chemical synapses (left column) or gap junctions (right column) for sensory neurons (A), interneurons (B) or motoneurons (C). Cells are labeled according toVarshney et al. (2011).⇣values of the shown Slepian vectors are provided in

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(11)

-0.2 -0.1 0 0.1 0.2

AIAL

AIAR

AIBL

AIBR AINL AINR AIYL

AIYR AIZL

AIZR ASER ASGL

AUAL AUAR

AVEL AVER AVKL DVA PVT

RIAL RIAR

RIBL RIBR RIGL RIGR

RIH RIR

RIS AIML

AIMR

ASKL

AVAL AVAR AVDL

AVDR AVFL

AVFR

AVG AVHL

AVHR

AVJL AVJR

LUAL LUAR PVCL

PVCR PVQL

PVQR

PVR

PVWL PVWR RIFL RIFR

VA12

-0.4 -0.2 0 0.2

AS11 AS02

AS09 AVL DA02

DA08 DA09 DD01

DD06

DVB PDA

PDB RID SABD

VA12 VB11 VD12

VD13 DB03

DD02

DD03

DD04

DD05

PVDL VA07

VA08

VA09 VB05

VB06 VB07

VB08 VB09

VC02

VD10

VD02 VD03

VD07 VD08

VD09

-0.2 -0.1 0 0.1 0.2

ADER ADFR AIAR ASGR

ASJR ASKR AWAR

CEPDR CEPVR

IL1DR IL1R

IL1VR IL2DR

IL2R IL2VR OLLR

OLQDR OLQVR RIH

RIPR URXR

ADEL ADFL

AFDL AFDR AIAL AIYL

ASEL ASER

ASGL ASIL AWAL

AWBL AWCL AWCR

CEPVL

IL1DL IL1L

IL1VL IL2DL

IL2L OLLL OLQDL

OLQVL RIPL URYDL

X position [mm]

-1 -0.8 -0.6 -0.4 -0.2 0

Anterior Posterior

0.4 0.2 0.3 0

-0.2 -0.1 0.1

0.4 0.2 0.3 0

-0.2 -0.1 0.1 -0.2 -0.1 0 0.1 0.2

Slep. 2 (ζ = 0.4247) Slep. 2 (ζ = 0.6131)

Slep. 2 (ζ = 0.4912)

S le p . 3 ( ζ = 0 .3 7 1 4 ) S le p . 3 ( ζ = 0 .3 6 5 8 ) S le p . 3 ( ζ = 0 .5 6 9 5 )

A B C

Figure S7. Two-dimensional visualizations for sensory neurons (A), interneurons (B) or motoneurons (C) when representing each neuron as a function of its position along the X direction. A value of 0 indicates the location of the nerve ring, and positional data was retrieved fromVarier and Kaiser (2011). ⇣ values of the shown Slepian vectors are provided in parentheses on each axis.

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10

(12)

A B

Modularity

Qsensory = 0.1809

Qinter- = 0.1127 Qmoto- = 0.18

Null distribution

Figure S8. Testing of clustering assignments when the focus is on sensory neurons (red), interneurons (grey) or motoneurons (green). For each case, all nodes of other types are considered as one additional cluster. The null distributions of the modularity of random assignments to clusters are given on the left side of the plot (A).The dashed straight lines on the right represent values of modularityQfor the clusters in Fig. S4 derived by the k-means approach (B). The x-axis is broken at0.06for better visualization.

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