In our circuit model mimicking the anatomical gap junction network of C. elegans (details described in ‘Construction of our circuit model’ part of Methods) (Figure 1), we calculated all the transmission coefficient $T_{ij}$ of the wave signal for each of the 109,746 cell-pairs ${\displaystyle ⟨i,j⟩}$ between the total 469 cells (details described in ‘Calculation for the transmission coefficient’ part of Methods). Here, the transmission coefficient was a function of energy E of the incoming wave, and the E value was a function of the wavenumber k of the wave signal. And according to the wavenumber, E value was set up to exist in the range [–10, 10], and we calculated all the ${\displaystyle T_{ij}(E)}$ of the entire cell-pairs at E values of 801 points listed by equal intervals over the entire range ($\mathrm{\Delta }E=0.025$).

Figure 1

Download asset Open asset

First, the average transmission coefficient of all cell-pairs ${\displaystyle ⟨i,j⟩}$ for each E value, ${\displaystyle ⟨T_{ij}(E){⟩}_{Allijcell-pairs}}$, was obtained and drawn as a function of E (Figure 2A). The ${\displaystyle ⟨T(E)⟩}$ graph showed an incomplete but quite symmetrical form based on $E=0$. And on the real Y-axis scale, high ${\displaystyle ⟨T(E)⟩}$ values were broadly identified in these three ranges of the E values: [–1.575, –1.275], [–0.350, 0.350], and [1.275, 1.575]. Also on the log Y-axis scale, several relatively high ${\displaystyle ⟨T(E)⟩}$ values were locally identified in the form of peaks in the remaining ranges of the E values. Meanwhile, on the log Y-axis scale, the signal transmission of all cell-pairs almost disappeared in the E value range at both ends. In the field of solid-state physics, a threshold energy value in which electrical conductivity disappears in the energy band is called a ‘mobility edge’. We were able to define a threshold wavenumber, namely a signal mobility edge, even in the transmission of the wave signal of our circuit model. For example, considering that ${\displaystyle ⟨T(E)⟩={10}^{-7}}$ is a value that can be obtained when only one cell-pair out of all cell-pairs shows 1% transmission and all others are completely unable to transmit the wave, this 10^–7 value can be regarded as an extreme disappearance of signal transmission throughout the entire circuit. Thus, by taking this reference value, $E=\pm 5.650$ in our circuit model became a signal mobility edge. If the reference value is set more conservatively lower than now, the absolute value of E corresponding to the signal mobility edges will be larger than now.

Figure 2 with 1 supplement see all

Download asset Open asset

Next, the transmission coefficient of all individual cell-pairs ${\displaystyle T_{ij}(E=0.225)}$ was represented as a heatmap at the E value (or the corresponding wavenumber) where the ${\displaystyle ⟨T(E=0.225)⟩}$ value was the highest (Figure 2B). Here, we confirmed that not all cell-pairs transmit the wave signals with even transmission coefficients. Most of the strong transmissions were found in the cell-pairs of intra-body-wall muscles, and a small number of cells belonging to sensory neurons, motor neurons, and other organ cells also showed strong transmission in cell-pairs between them or to body-wall muscles. The rest of the cells except these cells were almost isolated separately without exchanging the wave signals with any cells even at this E value, which was the highest transmitted state on average.

In addition, the transmission coefficient of all individual cell-pairs ${\displaystyle T_{ij}(E=5.650)}$ was represented as a heatmap at the E value (or the corresponding wavenumber) which we defined as the signal mobility edge (Figure 2C). As expected, all cells were isolated separately without exchanging the wave signals with each other, and the entire circuit stop transmitting wave signals of this wavenumber.

Between the two E values discussed above ($E=0.225$ and $E=5.650$), the average transmission coefficient of all cell-pairs, ${\displaystyle ⟨T(E)⟩}$, did not simply increase or decrease, but fluctuated along the E value (Figure 2). Therefore, it was expected that the composition of the cell-pairs, which showed strong transmission, also changes according to the E value. To efficiently investigate the changes in the composition of the cell-pairs, instead of inspecting the ${\displaystyle T_{ij}(E)}$ heatmaps in all E values, we selected 31 specific E values and tried to analyze the composition of the cell-pairs only in these E values. We chose the specific E values on the following three criteria: (1) the ${\displaystyle ⟨T(E)⟩}$ value was the local maximum (the shape of a peak) in the ${\displaystyle ⟨T(E)⟩}$ graph, (2) the components of the ${\displaystyle T_{ij}(E)}$ showed greater than 0.5 at least one cell-pair, and (3) the E value was positive number (Figure 2—figure supplement 1). The reasons why we used these conditions are as follows: (1) we assumed that the peak occurred in the ${\displaystyle ⟨T(E)⟩}$ graph due to the emergence of new cell-pairs with strong transmission, (2) we set the reference value for strong transmission to be 0.5 or higher, and (3) the E values where peak occurred in the ${\displaystyle ⟨T(E)⟩}$ graph showed left–right symmetry based on $E=0$. (Additional explanations are in ‘Auxiliary paragraph 1’ of Supplementary Materials.)

The transmission coefficients of all individual cell-pairs were represented as a heatmap at each E value (or the corresponding wavenumber) for the 31 specific E values we selected (left panels in Figure 3, Figure 3—figure supplements 1–8). To closely observe the composition of cell-pairs with strong transmission, only cells (Appendix 1—tables 1–4) belonging to the cell-pairs with ${\displaystyle T_{ij}\left(E\right)>0.5}$ were selectively exhibited in these heatmaps, not for all 469 cells. To confirm the spatial status of the cell-pairs with strong transmission on the electrical synapse network model, a network diagram was prepared first. Since drawing all 469 cells as a network diagram was complicated and poorly visible, we presented nodes for only 173 cells that belonged to the cell-pairs with ${\displaystyle T_{ij}\left(E\right)>0.5}$ in the positive E values. Electrical synapses between cells were represented as edges, and the thickness of the edge proportional to our processed weights ($w_{ij}$) was used. (Additional explanations are in ‘Auxiliary paragraph 2’ of Supplementary Materials). On the prepared network diagram, we marked the cell-pairs with strong transmission that satisfies ${\displaystyle T_{ij}>0.5}$ (right panels in Figure 3, Figure 3—figure supplements 1–8).

Figure 3 with 8 supplements see all

Download asset Open asset

As expected above, the region of the cell-pairs with strong transmission changed depending on the value of E (or wavenumber), which we named ‘Wavenumber-Dependent Transmission Map (WDTM) of subthreshold waves on the electrical synapse network model’. However, it should be also noted that our WDTM was a result derived from a network where only electrical synapses were considered, not a general neural network where chemical and electrical synapses coexist.

We looked at the cell type of the cell-pairs that make up the 31 WDTMs (Figure 3, Figure 3—figure supplements 1–8). In the two E value ranges of [0, 0.350] and [1.275, 1.575], the high ${\displaystyle ⟨T(E)⟩}$ value was broadly identified (Figure 2—figure supplement 1), and many cell-pairs of intra-body-wall muscles were observed in the corresponding WDTMs of these E values (Figure 3A, C, Figure 3—figure supplements 1 and 2A, B, and Figure 3—figure supplement 4). Since the body-wall muscles consist of four muscle strands: dorsal/ventral-left/right strands, the cell-pair of intra-body-wall muscles can be divided into two cases: a cell-pair of intra-strand and a cell-pair of inter-strand. Following this classification, the cell-pairs of both intra- and inter-strand were found in the WDTMs of the E value range of [0, 0.350] (Figure 3A, Figure 3—figure supplements 1 and 2A, B), whereas only the cell-pairs of intra-strand were found in the WDTMs in the E value range of [1.275, 1.575] (Figure 3C and Figure 3—figure supplement 4). On the other hand, in the remaining WDTMs, a small number of cell-pairs within cells with less than 10 members were mostly found (Figure 3B, D, Figure 3—figure supplement 2C, D, Figure 3—figure supplement 3, and Figure 3—figure supplements 5–8), and the WDTMs of the two largest cases are shown in Figure 3B, D. The composition of cell-pairs with various combinations of cell types (sensory/inter/motor neurons, body-wall/sex-specific muscles, pharynx, and other organ cells) was shown in these remaining WDTMs.

When looking at each WDTM on the network diagram, it shows various patterns spatially. In particular, in the case of the body-wall muscles, the cell-pairs of intra-strand existed, ranging from short distance to the longer distance than the half of the full length of the strand (Figure 3A, C). Here, the ‘Distance’ was meant by the length of the shortest path along the network between two cells of a cell-pair. If two cells are directly connected to an edge, the distance between the two cells is one. In the WDTMs of the E value range of [1.275, 1.575], cell-pairs with regular patterned transmission of the body-wall muscles were observed through the network diagram (Figure 3—figure supplement 4). Only the cell-pairs of intra-strand were found in these WDTMs, and the configuration of the distance of these cell-pairs was different for each WDTM (multiple of 3 at $E=1.350$, multiple of 2 at $E=1.450$, multiple of 7 at $E=1.500$, and multiple of 4 at $E=1.575$).

On the other hand, when the cell-pairs of inter-strand of the body-wall muscles appeared in the WDTMs as shown in Figure 3A, B, the head mesodermal cell (hmc) seemed to play an important role in building a bridge between them. This would be recognized from the network diagram itself of black edges without WDTM, and hmc was strongly connected near the neck of the body-wall muscles (seventh and eighth muscles) for all four strands. In the WDTM of Figure 3A, there were many cell-pairs of inter-strand between dorsal-right and ventral-left strands, and there were also many cell-pairs between the ventral-left strand and hmc, which directly revealed the role of hmc as a bridge between the two body-wall muscle strands. In the WDTM of Figure 3B, it was shown that the role of hmc was indirectly revealed as the cell-pairs of inter-strand, which existed only among the neck of four body-wall muscle strands.

Additionally, we reaffirmed the important role of hmc while confirming the distance characteristics of the cell-pairs with strong transmission. Under the conditions used when preparing the network diagram (the transmission coefficient $T_{ij}\left(E\right)$ of a cell-pair was 0.5 or higher in the range of positive E values at least once), a total of 146 cell-pairs with strong transmission were identified excluding the cell-pairs of intra-body-wall muscles, and their distance was investigated (Appendix 1—table 5). (Additional explanations are in ‘Auxiliary paragraph 3’ of Supplementary Materials.) Most of the cell-pairs in Appendix 1—table 5 showed the kinds of short distances 1–3, which consisted of various cell types. On the other hand, the cell-pairs with long distances in Appendix 1—table 5 ranging from 4 to 17 were mainly pairs between hmc and body-wall muscles. The strong transmission between hmc and the tail muscle, which corresponds to the longest distance, passes through the intra-strand connections from the neck to the tail of body-wall muscles.

A few cell-pairs of strong transmission with long distance in Appendix 1—table 5 turned out to be not those between hmc and the body-wall muscles. In there, one inter neuron (AVBR) and two motor neurons (AS11 and VD04) formed a cell-pair with the body-wall muscles, respectively, and four sensory neurons (CEPDL, IL1R, IL1DR, and OLQDL) formed cell-pairs between them.

We looked at cell-pairs frequently appearing in all 31 WDTMs in order to identify the major hub cell-pairs that is commonly used for various wavenumbers. First, for each cell-pair ${\displaystyle ⟨i,j⟩}$, we calculated the average appearance rate in all 31 WDTMs, ${\displaystyle {⟨\theta \left[T_{ij}\left(E\right)-0.5\right]⟩}_{E\in \left\{31selectedE\right\}}}$, and represented this as a heatmap (Figure 4). Where, the ${\displaystyle \theta \left[x\right]}$ was a step function, which value was 1 if 𝑥 ≥ 0 or 0 if 𝑥 < 0. Thus, the average appearance rate was 1 in the case of a cell-pair that appeared in all 31 WDTMs, and 1/31 ≈ 0.03 in the case of a cell-pair that appeared only once. In this study, we declared a cell-pair as a major hub cell-pair if the average appearance rate of a cell-pair was 𝑥 < 0 or higher. The major hub cell-pairs were the cell-pairs of intra-strand of the body-wall muscles (Figure 4). As seen in the network diagram, the edges of each strand of the body-wall muscles from the neck (seventh or eighth muscle) to the tail (twenty-third or twenty-fourth muscle) were simply connected in a line, unlike the messy edges among most neurons in the network (Figure 3). In general, the complexity of connections among most neurons increase with the number of transmissible paths of wave propagation, and the sum of phase changes coming from various path-length differences causes the deconstructive interference. Therefore, the simple (or regular) connections in each strand of the body-wall muscles from the neck to the tail reduced the number of transmissible paths of wave signals, which suppressed deconstructive interference and thus resulted in the strong transmission for various wavenumbers.

Figure 4

Download asset Open asset

Also, the cell-pairs between hmc and the neck of the ventral-left strand of the body-wall muscles were also identified as major hub cell-pair (Figure 4). As seen in the network diagram, since the edges between hmc and the neck muscles were very strong compared to other edges (Figure 3), the relative contribution of wave signals coming from other cells could be almost ignored, and thus resulted in the strong transmission for various wavenumbers.

The cell-pairs among two sensory neurons (PVDL and PVDR) and three other organ cells (CANL, CANR, exc_cell) were also identified as major hub cell-pairs (Figure 4). The cell-pair between CANL and CANR was the most major hub cell-pair, with the highest average appearance rate of ${\displaystyle 7/31\approx 0.23}$. As seen in the network diagram, the three other organ cells were very strongly connected by the edges to each other, and the two sensory neurons were very strongly connected by the edges to both ALA inter neuron and hmc (Figure 3). Although there were cases where they were strongly connected by the edges to each other, such as between PVDL and CANL or between PVDR and CANR, they were not major hub cell-pair. Therefore, the strong edge shown on the network diagram was not a sufficient condition to implicate a major hub cell-pair.

We used the contents of the ‘hermaphrodite gap jn symmetric’ tab in the ‘SI 5 Connectome adjacency matrices, corrected July 2020.xlsx’ file of the C. elegans connectome dataset as data for the anatomical gap junction network (Cook et al., 2019). The 469 cell names and the gap junction weight ($g_{ij}$) of all cell-pairs connected by gap junction were provided in the tab, and the weights had a value distribution from a minimum of 1 to a maximum of 401 (Figure 1A). The weights were proportional to the spatial size of the corresponding gap junction, which we hypothesized that the larger the weight, the stronger the connection between the cells and thus the better transmit the wave signal to the connected cell.

Our circuit model described C. elegans’ anatomical gap junction network as follows (Figure 1B). First, each cell was treated as a node in our circuit. Each gap junction connected to a cell-pair was treated as an edge connecting two nodes in our circuit, and we inserted two virtual nodes (VNs) between those two nodes to be connected sequentially through VNs (Cell-VN-VN-Cell). Here, all edges inside our circuit were set to have the identical length a. The edge of Cell-to-VN (or VN-to-Cell) representing like axon or dendrite of neuron gave a weight of 1, and the edge of VN-to-VN representing a gap junction gave our processed weight (${\displaystyle w_{ij}=min\left\{1,g_{ij}/40\right\}}$). Our processed weight ($w_{ij}$) was proportional to the anatomical gap junction weight ($g_{ij}$) and normalized as a value between 0 and 1 but was fixed to 1 if ${\displaystyle g_{ij}>40}$. In our circuit model, the weight of the edge meant the transmission coefficient of the signal transmitted through this edge, so if the weight was 1 corresponded to complete transmission, and 0 corresponded to complete reflection (or unavailable edge), respectively. And within our circuit, the characteristic of the signal being spontaneously attenuated, and disappearing was excluded. Therefore, when a signal incoming from the outside to a node in our circuit was reflected to the outside at the same node and simultaneously transmitted to the outside at another node, the sum of the transmission and reflection coefficients to the outside always remained at 1 without loss.

We benefited from the method of calculating the transmission coefficient according to the energy E of the incoming wave when the electron probability wave incident on one point of the network lattice was detected on another point. This methodology, called the quantum percolation model, considered the propagation and interference of waves on the network lattice using the tight-binding Anderson Hamiltonian (Anderson, 1958; Chang et al., 1995; Meir et al., 1989; Shapir et al., 1982; Thomas and Nakanishi, 2016). In our methodology, we replaced the classical wave signals of complex number expression representing subthreshold membrane potential waves instead of the electron probability wave and replaced the network lattice with our circuit describing the electrical synapse network model built above. Then, the transmission coefficient calculated by the quantum percolation model told how well the subthreshold waves were transmitted from one cell i to another cell j while undergoing interference by multiple propagation paths in the electrical synapse network model. Here, the electrical synapses act as wave scatters that interrupt the propagation of the subthreshold waves from cell i to cell j.

In the quantum percolation model, the tight-binding Anderson Hamiltonian was defined as follows:

where the ${\displaystyle |m⟩}$ (also expressed as the ${\psi }_m$) is as the electron probability wave on the mth node, and the ${\displaystyle ⟨m|}$ means its complex conjugate ${\psi }_m^{*}$. We used ${\psi }_m$ as the classical wave signal observed on the mth node, which of complex number expression made it easy to express the phase shift of the wave and calculate the interference. The amplitude of this classical wave signal on the mth node was defined as the $⟨m|m⟩={\psi }_m^{*}{\psi }_m$. In the Hamiltonian, the ${ϵ}_m$ is the on-site energy, and we regarded it as the resting membrane potential of the mth cell and set it to 0, the identical value as the relative reference value of the resting membrane potential in all cells. The $〈mn〉$ represents a pair of nearest-neighbor sites in the Hamiltonian, we replaced it with a node-pair connected by the edges in our circuit. The $V_{mn}$ as a hopping matrix represents the entire structure of the network lattice in the Hamiltonian, which we implemented the structure of our circuit by setting it to $V_{mn}=1$ for a node-pair between cell and VN, and $V_{mn}=w_{mn}$ for that between two VNs. Since the total number of the cells was 469, and we inserted two VNs for every 1433 electrical synapses, the newly constructed tight-binding Anderson Hamiltonian for our circuit was represented as a matrix of 3335 by 3335.

By solving the time-independent Schrödinger equation ($\mathit{H}\overrightarrow{\psi }=E\overrightarrow{\psi }$) for the Hamiltonian of our circuit, we sought to obtain the reflection and transmission coefficients of the cell-pair between the IN-cell and the OUT-cell from the ${\psi }_{IN}$ of the IN-cell where the wave signal from the outside was incident and reflect, and the ${\psi }_{OUT}$ of the OUT-cell where the wave signal from our circuit transmitted to the outside. We defined ${\psi }_{IN}$ and ${\psi }_{OUT}$ as follows using complex numbers r and t with amplitudes and phases,

where the ${\displaystyle {\psi }_{IN}}$ was the sum of 1 and r corresponding to the incoming wave and the reflective wave, respectively, and the ${\psi }_{OUT}$ was t corresponding to the transmitted wave. The reason why the amplitude and the phase of the incoming wave on the IN-cell were 1 and 0, respectively, was because we set that as a reference amplitude and a reference phase in our circuit. The amplitude of the wave reflected from the IN-cell to the outside became the reflection coefficient with $R=r^{*}r$, and the amplitude of the wave transmitted from the OUT-cell to the outside became the transmission coefficient with $T=t^{*}t$, and $T+R=1$ had to be always satisfied, as mentioned above.

This methodology also assumed both an external injector and an external detector. The external injector injected the generated wave into the network lattice and received the reflective wave from the network lattice. The external detector received the transmitted wave from the network lattice. The waves on the external injector ${\varphi }_{Injector}$ and the external detector ${\varphi }_{Detector}$ were extended from the definitions of ${\psi }_{IN}$ and ${\psi }_{OUT}$ above, respectively, and determined as follows:

The external injector and detector were set to be connected to the IN-cell and OUT-cell by a perfect conducting wire with a length of a, respectively. The wave outgoing to the external injector (detector) from the IN-cell (OUT-cell) was determined by multiplying the phase difference of $e^{ika}$ to the wave on the IN-cell (OUT-cell), whereas the wave incoming to the IN-cell from the external injector was determined by multiplying the phase difference of $e^{-ika}$ to the wave on the IN-cell. Insert these terms into the time-independent Schrödinger equation for the Hamiltonian of our circuit, the following equation was derived describing the entire system consisting of our circuit and the external injector and detector,

where the E value on the right side representing the energy of the entire system was equivalent to the energy E of the incoming wave from the external injector.

In the quantum percolation model, the energy E of the incoming wave was defined as a tight-binding energy function as follows:

where $\gamma$ meant the binding energy between the nearest-neighbor atoms in the tight-binding model, but we had to set it to an arbitrary value. In this study, it was set to $\gamma =5$, and the decision process was described in the below section. Applying this energy function, the above equation for the entire system was expressed as the matrices as follows:

where m and n meant the 3333 remaining nodes except two nodes for IN-cell and OUT-cell. For a wavenumber k in the range of [${\displaystyle 0,\pi /a}$] (or an E value in the range of [${\displaystyle -2\gamma ,2\gamma }$]), all components of both the left-hand square-matrix and the right-hand column-vector were fully expressed as complex numbers, and then the r and t values in the left-hand column-vector were obtained by matrix multiplication of the inverse matrix of the left-hand square-matrix and the right-hand column-vector. From squared absolute the r and t values, the reflection and transmission coefficients of the cell-pair between IN-cell and OUT-cell at the E value (or the corresponding wavenumber k) were deduced, respectively.

To determine the appropriate $\gamma$ of the above energy function, we preliminarily calculated the transmission coefficient for all cell-pairs between the 469 cells under all four conditions of $\gamma$, and compared the average transmission coefficient graph as a function of the E value ${\displaystyle {⟨T_{ij}\left(E\right)⟩}_{Allijcell-pairs}}$ (Appendix 1—figure 1). Where four values of 1, 2.5, 5, and 10 were attempted for $\gamma$, and equally spaced a hundred values were used for the wavenumber k in the range of [${\displaystyle 0,\pi /a}$]. According to the above energy function, the minimum/maximum range of the E value differed as [${\displaystyle -2\gamma ,2\gamma }$], and as $\gamma$ was larger, the points of the $〈T\left(E\right)〉$ graph were placed sparsely on the X-axis of the E value. In each of the $〈T\left(E\right)〉$ graphs for these four conditions of $\gamma$, similarly high $〈T\left(E\right)〉$ values occurred in the vicinity of ${\displaystyle E=0,\pm 1.4}$. Based on this preliminarily estimation, we confirmed that the occurrence of high $〈T\left(E\right)〉$ values appeared as a function of the E value regardless of $\gamma$. Therefore, we considered $\gamma$ as just determining the minimum/maximum range of the E value.

The appropriate $\gamma$ we wanted in this study was large enough to provide a wide E value range to check the signal mobility edge in the $〈T\left(E\right)〉$ graph, and the appropriate $\gamma$ had to be small so that the points in the $〈T\left(E\right)〉$ graph did not become too sparse. It was the condition of $\gamma =5$ that satisfied this demand out of the four conditions of $\gamma$. Because the condition of $\gamma =2.5$ was not sufficient to confirm the signal mobility edge, and the condition of $\gamma =10$ had too sparse points along the X-axis.

In principle, the sum of transmission coefficient $T_{ij}\left(E\right)$ and reflection coefficient $R_{ij}\left(E\right)$ of any cell-pair at any E value always had to satisfy 1 because our circuit model did not consider spontaneous attenuation of the wave signal. However, in the calculating program we wrote and used for this study, the error occurred with the sum of the two coefficients exceeding 1 (by the tolerance was 10^–5) for a total of 5213 cell-pairs in three specific E values (45 cell-pairs at $E=-1.000$, 3836 cell-pairs at $E=-0.375$, and 1332 cell-pairs at $E=1.000$). We performed the calculations on a total of 109,746 cell-pairs for a total of 801 E values, so these errors correspond to 0.006% of the total calculation results. To eliminate the effect of these errors in this study, we post-corrected the two coefficients of the 5213 cell-pairs in the three specific E values in which the error occurred to ${\displaystyle T_{ij}\left(E\right)=0,R_{ij}\left(E\right)=1}$.

An electrical resistor network model was constructed that mimics C. elegans’ electrical synapse network, with the 469 cells replaced with electrical conducting nodes and the 1433 electrical synapses replaced with electrical resistors connecting these nodes. The effective resistance and its reciprocal, the effective conductance, when a direct current power source outside the electrical resistor network model contacted a cell by its positive pole, and another cell by its negative pole, respectively, were calculated (Klein and Randić, 1993). The effective conductance shows how well electric current without interference flows between the two cells contacted to the external power source.

We expressed the electric voltage at node i as ${\nu }_i$, the electric current flowing from node i to node j as ${\mu }_{ij}$, the resistance of the electrical resistor between node i and node j as ${\chi }_{ij}$, the conductance of that as ${\displaystyle {\omega }_{ij}\left(={\chi }_{ij}^{-1}\right)}$. The anatomical gap junction weights ($g_{ij}$) were used as the conductance value of the electrical resistors in the network, ${\omega }_{ij}=g_{ij}$, with arbitrary units.

By Ohm’s law, each electrical resistor in the network satisfies the following equation,

The sum of the electric currents entering each node by Kirchhoff’s current law satisfied the equation below,

where ${\displaystyle (+)}$ and ${\displaystyle (-)}$ meant a node in which the positive and negative poles of the external power source were in contact, respectively, and the capital letter I was the total electric current supplied by the external power source. When Ohm’s law above was substituted on the left-hand of this equation, it was expressed as ${\nu }_i\sum _{j\ne i}{\omega }_{ij}-\sum _{j\ne i}{\omega }_{ij}{\nu }_j$, and then the equation was able to express in matrix and vector as follows:

where ${\mathit{L}}^{\omega }$ was the Laplacian matrix of the electrical resistors’ conductance matrix ($\mathit{\omega }$), defined as $L_{ij}^{\omega }={\delta }_{ij}{\sum }_{\mathcal{l}}{\omega }_{i\mathcal{l}}-{\omega }_{ij}$, $\overrightarrow{\nu }$ was a column-vector with an electric voltage at all nodes as a component, and ${\displaystyle \overrightarrow{e_{(+)}}}$ ${\displaystyle (\overrightarrow{e_{(-)}})}$ was a unit column-vector of the identical dimension as $\overrightarrow{\nu }$, with only the component of the ${\displaystyle (+)}$ node (${\displaystyle (-)}$ node) having a value of 1 and all other components having a value of 0. By applying the pseudo-inverse matrix of the Laplacian matrix (${\mathit{L}}^{\omega +}$) on both sides of the equation, we obtained a particular solution as $\overrightarrow{\nu }=I{\mathit{L}}^{\omega +}\left(\overrightarrow{e_{(+)}}-\overrightarrow{e_{(-)}}\right)$.

The effective resistance applied between the positive and negative poles of the external power source was defined by Ohm’s law of the electric voltage difference between the ${\displaystyle (+)}$ and ${\displaystyle (-)}$ nodes and the total electric current, as follows:

Substituting here the particular solution of $\overrightarrow{\nu }$ obtained above was as follows:

As a result, we were able to obtain the effective resistance for any one cell-pair by calculating the pseudo-inverse matrix of the Laplacian matrix (${\mathit{L}}^{\omega +}$) from the electrical resistors’ conductance matrix ${\displaystyle (\mathit{\omega })}$. The effective conductance was defined as the reciprocal of the effective resistance, $G^{\text{eff}}={{(R}^{\text{eff}})}^{-1}$.

The 31 specific E values in this study were not fixed and were of a property that can change as a researcher adjusts the resolution of the E value. Perhaps the higher the resolution of the E value, the more specific E values will be found. And it was arbitrary that our reference value for strong transmission was 0.5, and this reference value can be raised or lowered depending on the researcher’s intention, and the lower the reference value, the greater the number of cell-pairs with strong transmission will be found.

Because of the omitted the rest of cells and their electrical synapses in the network diagram, some nodes in the diagram look like remote islands, but it should be remembered that the 173 cells were connected as one large cluster through both the drawn and the omitted electrical synapses.

Because the number of the cell-pairs of intra-body-wall muscles was overwhelmingly large and it was relatively easy to measure the distance compared to those of other cell-pairs, the investigation for the distance excluded them (e.g., the distance of two body-wall muscles in the same strand easily calculate from the given number in the name of the muscles).

Appendix 1—figure 1

Download asset Open asset

The funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.

We appreciate two Nobel Laureates, Prof. Erwin Neher and Prof. Kurt Wuthrich, for their insightful discussions on the impact of this work. We have benefited a lot from their suggestions in the course of this study. Funding: This work was supported in part by both pCoE program of DGIST [grant number: 23-CoE-BT-01] and a grant from iPT, Korea.

1. Sent for peer review: June 13, 2024
2. Preprint posted: June 16, 2024
3. Reviewed Preprint version 1: October 8, 2024
4. Reviewed Preprint version 2: December 3, 2024
5. Version of Record published: January 8, 2025

You can cite all versions using the DOI https://doi.org/10.7554/eLife.99904. This DOI represents all versions, and will always resolve to the latest one.

© 2024, Chang et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.