Ephaptic coupling and the source-sink effect of cardiac conduction

Abstract

Due to the electrotonic effect, under gap junction coupling, a critical number of excited cells is needed to elicit electrical conduction in cardiac tissue, a phenomenon called the source-sink effect. However, it is unclear how ephaptic coupling affects the source-sink effect. To address this issue, we carry out computer simulations to investigate the effects of ephaptic coupling and its interaction with gap junction coupling on conduction triggered by the following types of depolarization: 1) an external stimulation; 2) a delayed afterdepolarization, and 3) automaticity. Simulations were done using a one-dimensional cable with the presence of a cleft between two adjacent myocytes and junctional ion channel distributions. We show that: 1) under pure ephaptic coupling, only one stimulated cell is needed to elicit conduction, implying that there is no source-sink effect; 2) the minimum number of stimulated (or delayed afterdepolarization or oscillatory) cells needed to elicit a conduction increases as gap junction conductance increases, as predicted by the gap junction coupling theory; and 3) the dependence of the source-sink effect on the cleft width exhibits complex behaviors, which are mainly caused by self-attenuation of the junctional sodium current due to the ephaptic effect and self-enhancement of the junctional inward rectifier potassium current due to an increase of potassium concentration in the cleft. In conclusion, ephaptic coupling and cleft width play nontrivial roles in the source-sink effect of cardiac conduction and thus may be nontrivial for arrhythmogenesis under disease conditions in which gap junctions and cleft structure are altered.

Graphical Abstract

Adjacent cardiac myocytes are electrically coupled via gap junctions or through field changes in the cleft space of the intercalated disc. Na⁺ and K⁺ channels are densely localized in the junctional membranes of the intercalated disc (left panels). During an action potential, the membrane potential and Na⁺ and K⁺ concentrations are altered (middle panels), which in turn affect both depolarization and repolarization of the myocytes. These changes exert complex effects on the source–sink relationship in cardiac conduction, depending on the manner of depolarization. For example, the minimum number of cells required to elicit conduction depends non-monotonically on the width of the cleft and differs substantially among different depolarization mechanisms (right panels).

Introduction

One important feature of cardiac conduction is the so-called source-sink effect (Xie et al., 2010), i.e., a critical number of excited cells is needed to elicit an electrical conduction in cardiac tissue. This is because the unexcited cells surrounding the excited ones (source) are electrotonic sinks due to the electrotonic effect of gap junction coupling. Therefore, the number of source cells has to be large enough to overcome the sink effect for successful conduction. However, computational modeling and experimental studies have shown that gap junction coupling may not be necessary for cardiac conduction, which can occur via the so-called ephaptic coupling (Sperelakis & Mann, 1977; Kucera et al., 2002; Sperelakis, 2002; Mori et al., 2008; Lin & Keener, 2010; Adams et al., 2023). In ephaptic coupling, there is no direct current exchange between the coupled cells; the coupling of the cells is via a change in the electrical field in the cleft at the intercalated disc (ID). The requirements for ephaptic coupling are a narrow enough cleft and a high enough density of Na⁺ channels in the junctional membranes at the cleft. With ephaptic coupling, since there are no electronic currents between cells, the unexcited cells are not apparently sinks of the excited cells, and thus the source-sink effect is different from that of gap junction coupling. Moreover, even in the presence of normal gap junction coupling, “self-activation” and “self-attenuation” of the junctional sodium (Na⁺) current (I_Na) (Hichri et al., 2018; Otani et al., 2023a, b) or “self-enhancement” of the junctional inward rectifier potassium (K⁺) current (I_K1) would play non-trivial roles in the source-sink effect for conduction. In ephaptic coupling, opening of junctional Na⁺ channels in one cell lowers the voltage in the cleft, which causes self-activation of junctional Na⁺ channels of the adjacent cell. In the meantime, lowering the voltage in the cleft reduces the driving force of junctional I_Na, attenuating the I_Na amplitude. On the other hand, opening of K⁺ channels elevates the K⁺ concentration in the cleft, increasing the maximum conductance of junctional I_K1 to enhance I_K1.

The source-sink effect plays key roles in cardiac conduction, and the requirements are different for different functions of the heart. For example, for conduction from the sinoatrial nodal region to the atrium or conduction from Purkinje fibers to the working myocardium, a smaller sink effect would promote successful conduction. On the other hand, for atrial or ventricular conduction, one would expect enough sink effect to prevent ectopic excitations originating from regional early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs) or automaticity. Therefore, it is important to understand the source-sink effect on cardiac conduction in the presence of ephaptic coupling, or more broadly, the effects of ion channel localization on conduction. In this study, we use computer simulation to investigate the effects of ephaptic coupling, and its interaction with gap junction coupling, particularly the effects of the cleft and ion channel localization, on conduction and the source-sink effect. We simulate three depolarization scenarios: 1) conduction elicited by external stimulation; 2) conduction triggered by DADs; and 3) conduction caused by L-type calcium (Ca²⁺) current (I_Ca,L)-mediated automaticity. We show that in the absence of gap junction coupling, one only needs to stimulate a single cell to elicit a conduction in cardiac tissue, i.e., there is no source-sink effect when the coupling is purely ephaptic. When gap junction coupling is present, the two coupling mechanisms can work in synergy to promote conduction. The cleft width plays important roles in the source-sink effect via self-attenuation of the junctional I_Na and self-enhancement of the junctional I_K1, depending on the depolarization mechanism of the source.

Methods

One-dimensional (1D) cable model

We simulate a 1D cable model, with a cleft between two adjacent myocytes (Fig.1). In this model, each myocyte is divided into three segments, a lateral segment and two junctional segments (left and right). We assume that the intracellular resistance is zero and the intracellular potential is uniform inside the myocyte, denoted as ${\varphi }_i$. The potential in the cleft between two adjacent myocytes is denoted as ${\varphi }_c$. The potential of the bulk extracellular space is denoted as ${\varphi }_e$ and grounded, i.e., ${\varphi }_e=0$. Therefore, the transmembrane potential of the lateral membrane is: $V_l={\varphi }_i-{\varphi }_e={\varphi }_i$. The transmembrane potential of the junctional membrane is: $V_j={\varphi }_i-{\varphi }_c$. I_Na and I_K1 are distributed in the junctional and the lateral membranes, and all other currents are placed only in the lateral membrane. We denote the fraction of Na⁺ channels in the junctional membranes as $f_{Na}$, and thus that of the lateral membrane is $\left(1-f_{Na}\right)$. Therefore, the I_Na density in the lateral membrane is:

$$I_{Na}=\left(1-f_{Na}\right)G_{Na}{m}^{3}hj\left(V_l-E_{Na}\right)$$ (1)

with $m$, $h$, and $j$ being gating variables regulated by $V_l$. The I_Na density in one of the two junctional membranes is expressed as:

$$I_{Na}=\frac{f_{Na}}{2}{G}_{Na}{m}^{3}hj\left(V_j-E_{Na}\right)$$ (2)

with $m$, $h$, and $j$ being then regulated by $V_j$. Note that in Eq.1, $E_{Na}$ is a function of the Na⁺ concentration in the bulk extracellular space ([Na⁺]_o), but in Eq.2, it is a function of the Na⁺ concentration in the cleft ([Na⁺]_c), i.e., $E_{Na}=\frac{RT}{F}ln\frac{{\left[{Na}^{+}\right]}_c}{{\left[{Na}^{+}\right]}_i}$, where $F$ is the Faraday constant, $R$ the gas constant, and $T$ the temperature. Similarly, we assume that the fraction of I_K1 in the junctional membranes is $f_K$ and ($1-f_K$) in the lateral membrane. Then I_K1 in the lateral membrane is

$$I_{K1}=\left(1-f_K\right)G_{K1}\sqrt{\frac{{\left[K^{+}\right]}_o}{5.4}}R\left(V_l\right)\left(V_l-E_K\right)$$ (3)

and in one of the two junctional membranes is

$$I_{K1}=\frac{f_K}{2}{G}_{K1}\sqrt{\frac{{\left[K^{+}\right]}_c}{5.4}}R\left(V_j\right)\left(V_j-E_K\right)$$ (4)

where $E_K$ is a function of the K⁺ concentration in the cleft ([K⁺]_c). For control, we set $f_{Na}=f_K=0.9$ as their default values, i.e., 90% of the Na⁺ channels and I_K1 channels are in the junctional membranes. We will change $f_{Na}$ and $f_K$ to investigate the effects of the localization of these two types of channels on the source-sink effect.

Figure 1. Schematic diagram of the 1D cable model.

Top: Schematic plot of a 1D cable of electrically coupled cells. Middle: Schematic plot of the cleft (cyan), junctional membrane (red) and the currents going through the junctional and lateral membranes. $\varphi$ is the potential in different spaces as marked. I_gap is the gap junction current; I_Na is the Na⁺ current; I_K1 is the inward rectifier K⁺ current; I_c is the current from the bulk extracellular space to the cleft; and I_m is the total ionic current excluding I_Na and I_K1 through the lateral membrane of the cell. Bottom: Two-cell electrical circuit view for the 1D cable. I₁ is the total ionic current through the lateral membrane of the cell. I_j,L is the total ionic current through the junctional membrane at the left side of the cell. I_j,R is the total ionic current through the junctional membrane at the right side of the cell. The transmembrane potential of the lateral membrane is: $V_l={\varphi }_i-{\varphi }_e$. Since ${\varphi }_e=0$, then $V_l={\varphi }_i$. The transmembrane potential of the junctional membrane is: $V_j={\varphi }_i-{\varphi }_c$.

Using the Kirchhoff’s current law, i.e., the summation of the currents equals to zero, for the node inside the cell (red dot in the bottom panel in Fig.1), one has

$$C_l\frac{\text{d}{V}_l^k}{\text{d}t}+C_j\frac{\text{d}{V}_{j,L}^k}{\text{d}t}+C_j\frac{\text{d}{V}_{j,R}^k}{\text{d}t}+I_l^k+I_{j,L}^k+I_{j,R}^k+I_{\mathit{\text{gap}}}^k-I_{\mathit{\text{gap}}}^{k-1}=0$$ (5)

where $k=1,\dots ,N$ is the index of the cells. $V_l^k={\varphi }_i^k$ is the transmembrane potential of the lateral membrane. $V_{j,L}^k={\varphi }_i^k-{\varphi }_c^{k-1}$ is the transmembrane potential of the left junctional membrane of the cell and $V_{j,R}^k={\varphi }_i^k-{\varphi }_c^k$ is that of the right side. $I_l^k$ is the total lateral membrane current. $I_{j,L}^k$ and $I_{j,R}^k$ are currents through junctional membranes at the left and right side of the cell, respectively. $I_{\mathit{\text{gap}}}^{k-1}$ and $I_{\mathit{\text{gap}}}^k$ are the gap junction currents.

For the node in the cleft (green dot in the bottom panel in Fig.1), one has

$$-C_j\frac{\text{d}{V}_{j,R}^k}{\text{d}t}-C_j\frac{\text{d}{V}_{j,L}^{k+1}}{\text{d}t}-I_{j,R}^k-I_{j,L}^{k+1}-I_C^k=0$$ (6)

where $I_c^k$ is the current between the cleft and the bulk extracellular space.

Applying $V_l^k={\varphi }_i^k$, $V_{j,L}^k={\varphi }_i^k-{\varphi }_c^{k-1}$, and $V_{j,R}^k={\varphi }_i^k-{\varphi }_c^k$ to Eqs. 1 and 2, one transforms them into the following equations:

$$\left(C_l+2C_j\right)\frac{\text{d}{\varphi }_i^k}{\text{d}t}-C_j\frac{\text{d}{\varphi }_c^{k-1}}{\text{d}t}-C_j\frac{\text{d}{\varphi }_c^k}{\text{d}t}+I_l^k+I_{j,L}^k+I_{j,R}^k+I_{\mathit{\text{gap}}}^k-I_{\mathit{\text{gap}}}^{k-1}=0$$ (7)

$$C_j\frac{\text{d}{\varphi }_i^k}{\text{d}t}+C_j\frac{\text{d}{\varphi }_i^{k+1}}{\text{d}t}-2C_j\frac{\text{d}{\varphi }_c^k}{\text{d}t}+I_{j,R}^k+I_{j,L}^{k+1}+I_c^k=0$$ (8)

One can rewrite Eqs. 7 and 8 for all the cells and clefts into a vector form as

$$\mathit{C}\frac{d𝚽}{dt}=\mathbf{I}$$ (9)

where $\frac{d𝚽}{dt}$ is a ($2N-1$)-dimensional vector ($N$ cells plus $N-1$ clefts), i.e.,

$$\frac{d𝚽}{dt}=\left[\frac{d{\varphi }_i^1}{dt},\frac{d{\varphi }_c^1}{dt},\dots ,\frac{d{\varphi }_i^k}{dt},\frac{d{\varphi }_c^k}{dt},\dots ,\frac{d{\varphi }_i^{N-1}}{dt},\frac{d{\varphi }_c^{N-1}}{dt},\frac{d{\varphi }_i^N}{dt}\right]$$ (10)

$\mathit{C}$ is a $(2N-1)\times (2N-1)$ matrix composed of the lateral and junctional membrane capacitances and $\mathbf{I}$ is a ($2N-1$)-dimensional vector composed of all the ionic currents as described in Eqs.7 and 8. In numerical simulations, one first obtains the inverse matrix of $\mathit{C}$ and then solves the following differential equations numerically:

$$\frac{d𝚽}{dt}={\mathit{C}}^{-1}\mathbf{I}$$ (11)

The ionic current formulations were taken from the ventricular action potential (AP) model by Shannon et al (Shannon et al., 2004) or a later version of the Luo and Rudy model (Faber & Rudy, 2000). The gap junction current is described as

$$I_{\mathit{\text{gap}}}^k=g_j\left({\varphi }_i^{k+1}-{\varphi }_i^k\right)$$ (12)

where $g_j$ is the gap junction conductance. The ion flux between the cleft and the bulk extracellular space is composed of fluxes of Na⁺ and K⁺, i.e.,

$$I_c=I_{\text{Na},\text{c}}+I_{\text{K},\text{c}}$$ (13)

in which $I_{\text{Na},\text{c}}$ and $I_{\text{K},\text{c}}$ are formulated as in Ly and Weinberg (Ly & Weinberg, 2022), i.e.,

$$I_{\text{Na},\text{c}}=g_c{\varphi }_c\frac{{\left[{Na}^{+}\right]}_c-{\left[{Na}^{+}\right]}_e\mathit{\text{exp}}\left(-\frac{F}{RT}{\varphi }_c\right)}{\left({\left[{Na}^{+}\right]}_e+{\left[{\text{K}}^{+}\right]}_e\right)\left(1-\mathit{\text{exp}}\left(-\frac{F}{RT}{\varphi }_c\right)\right)}$$ (14)

$$I_{\text{K},\text{c}}=g_c{\varphi }_c\frac{{\left[K^{+}\right]}_c-{\left[K^{+}\right]}_e\mathit{\text{exp}}\left(-\frac{F}{RT}{\varphi }_c\right)}{\left({\left[{Na}^{+}\right]}_e+{\left[K^{+}\right]}_e\right)\left(1-\mathit{\text{exp}}\left(-\frac{F}{RT}{\varphi }_c\right)\right)}$$ (15)

where ${\left[{\text{Na}}^{+}\right]}_{\text{e}}$ and ${\left[{\text{K}}^{+}\right]}_{\text{e}}$ are the Na⁺ and K⁺ concentrations in the bulk extracellular space. $g_{\text{c}}$ is the conductance calculated as $g_{\text{c}}=\frac{8\pi w}{{\rho }_{\text{e}}}$ with $w$ the cleft width and ${\rho }_{\text{e}}=150\Omega \text{cm}$ the extracellular resistivity. The Na⁺ and K⁺ concentrations in the cleft are determined by

$$\frac{d{\left[{Na}^{+}\right]}_c^k}{dt}=\frac{I_{\text{Na},\text{R}}^k+I_{\text{Na},\text{L}}^{k+1}-I_{\text{Na},\text{c}}^k}{v_{\text{c}}F}$$ (16)

$$\frac{d{\left[K^{+}\right]}_c^k}{dt}=\frac{I_{\text{K}1,\text{R}}^k+I_{\text{K}1,\text{L}}^{k+1}-I_{\text{K},\text{c}}^k}{v_{\text{c}}F}$$ (17)

where $v_{\text{c}}=\pi r^{2}w$. The subscripts “R” and “L” indicate the right and left junctional membrane, respectively. The myocyte was modeled as a cylindrical cell of radius ($r$) and length ($l$), and thus the junctional membrane capacitance $C_j=\pi r^2{C}_m$ and the lateral membrane $C_l=2\pi rl{C}_m$ with $C_m=1\mu \text{F}/{\text{cm}}^2$. We set $r=11\mu \text{m}$ and $l=100\mu \text{m}$. We set the default value for $w$ as $w=20$ nm but clarified when $w$ was varied to investigate the effects of cleft width on the source-sink effect.

DADs and automaticity

To simulate DADs, we took the approach by Xie et al (Xie et al., 2010) in which instead of true spontaneous sarcoplasmic reticulum (SR) Ca²⁺ release, a commanded SR Ca²⁺ release was used. The spontaneous calcium release flux was defined as $J_{\mathit{\text{spon}}}=G_{\mathit{\text{spon}}}{g}_1{g}_2\left(\beta c_j-c_s\right),G_{\mathit{\text{spon}}}$ is the maximal conductance, $\beta$ denotes the submembrane-to-SR volume ratio, and $c_j$ and $c_s$ are the calcium concentrations in the junctional SR and submembrane compartments, respectively. Two time-dependent sigmoid functions $g_1=1/\left(1+\mathit{\text{exp}}\left(-\left(t-t_0\right)/{\tau }_1\right)\right.$ and $g_2=1/(1+\mathit{\text{exp}}\left(\left(t-t_0\right)/{\tau }_2\right)$ were used to shape the onset and termination of the spontaneous Ca²⁺ release, in which $t_0$ controls the onset time of SR Ca²⁺ release. We set $t_0=425\text{ms}$. The DAD amplitude was adjusted by modifying the maximal release conductance $G_{\mathit{\text{spon}}}$. The default value used in this paper is $G_{\mathit{\text{spon}}}=0.6{\text{ms}}^{-1}$. Other parameters are the same as in Xie et al (Xie et al., 2010).

Pacemaker-like activity was generated using the Luo and Rudy model (Faber & Rudy, 2000) by downgrading I_K1, following the approach by Silva and Rudy (Silva & Rudy, 2003).

Numerical methods

The differential equations (Eq.11) were solved using a forward Euler method with a time step 0.001 ms. Simulations were performed using CUDA with Nvidia GPUs. A computer code for the 1D cable can be downloaded from GitHub at: https://github.com/Mor5oN/source-sink.

Results

The source-sink effects on electrical conduction elicited by an external stimulus

We first investigated how cleft width and Na⁺ and K⁺ channel localization affect the source-sink effect for conduction elicited by an external stimulus. Fig.2A shows conduction velocity (CV) versus the cleft width $(w)$ for different gap junction coupling strength $\left(g_j\right)$. When $g_j>400\text{nS}$, CV increases monotonically with $w$ and saturates when $w$ is very large. When $g_j\le 400\text{nS}$, CV first increases then decreases with $w$ and then saturates when $w$ is very large (>100 nm). When $g_j=0$, there is no gap junction coupling but only ephaptic coupling, conduction exists for a small range of $w$ (from 17 nm to 27 nm, see also Fig.2E). This conduction behavior agrees closely with that shown in the cable model by Kucera et al (Kucera et al., 2002). The non-monotonic behavior is a consequence of the synergistic effect of the two coupling mechanisms. For example, for $g_j=20\text{nS}$ and $w=30\text{nm}$, CV is around 12 cm/s. However, for the same $g_j$ but with $w=1000\text{nm}$ (no ephaptic effect anymore), CV is much smaller, only 3 cm/s. For $w=30\text{nm}$ but $g_j=0$ (pure ephaptic coupling), no conduction is possible. Therefore, gap junction coupling and ephaptic coupling work in synergy to maintain normal CV. The dashed oval in Fig.2A marks where the two mechanisms work synergistically.

Figure 2. Conduction and source-sink effect under external stimulation.

A. CV versus $w$ for different $g_j$ as marked. The ephaptic coupling effect is weak or absent when $w$ is large (large $w$ is purely hypothetical for understanding the mechanism only since the cleft width in real cardiac tissue is less than 100 nm). B. Minimum number ($N_{\mathit{\text{min}}}$) of cells required for eliciting a conduction for a given $g_j$ and w. The grey boxes signify no conduction regardless of the number of stimulated cells. Stimulation is applied to cells in the middle of the cable. C. $N_{\mathit{\text{min}}}$ versus w for $g_j=0$ and $g_j=600\text{nS}$. D. $N_{\mathit{\text{min}}}$ versus G_Na for $g_j=0$ and $g_j=600\text{nS}$. Circles are for $w=20\text{nm}$ and triangles are for $w=200\text{nm}$. E. $N_{\mathit{\text{min}}}$ versus ${\text{I}}_{\text{stim}}$ for $g_j=0$ and $g_j=600\text{nS}$. Arrows in C-E indicate the last points beyond which a conduction cannot be elicited regardless of how many cells are stimulated.

To investigate the effects of cleft width and gap junction coupling on the source-sink effect, we sort out the minimum number ($N_{\mathit{\text{min}}}$) of stimulated cells required to elicit conduction in the cable, i.e., conduction cannot be elicited if $\left(N_{\mathit{\text{min}}}-1\right)$ cells or less are stimulated but can if $N_{\mathit{\text{min}}}$ or more cells are stimulated in the middle of the cable. Fig.2B shows $N_{\mathit{\text{min}}}$ versus $w$ and $g_j$. When the coupling is ephaptic alone ($g_j=0$), only one stimulated cell is needed to elicit conduction. As $g_j$ increases, more and more stimulated cells are needed. However, the dependence of $N_{\mathit{\text{min}}}$ on $w$ is not monotonic. Fig.2C shows the dependence for a finer increment of $w$ for $g_j=0$ and $g_j=600\text{nS}$. When $g_j=0$, $N_{\mathit{\text{min}}}$ is almost independent of $w$ ($N_{\mathit{\text{min}}}=1$ for $w$ from 17 to 26 nm and $N_{\mathit{\text{min}}}=2$ for $w=27\text{nm}$). When $w<17\text{nm}$ or $w>27\text{nm}$ (arrows), no conduction can be elicited regardless of the number of cells stimulated. When $g_j=600\text{nS}$, $N_{\mathit{\text{min}}}$ first decreases and then increases with $w$, no conduction can be elicited when $w<10\text{nm}$ (arrow). We also investigated the dependence of $N_{\mathit{\text{min}}}$ on G_Na (Fig.2D) and stimulation strength (${\text{I}}_{\text{sti}}$) (Fig.2E). When $g_j=0$, $N_{\mathit{\text{min}}}$ does not depend on these two parameters. But when $g_j=600\text{nS}$, $N_{\mathit{\text{min}}}$ decreases as G_Na or ${\text{I}}_{\text{sti}}$ increases. Interestingly, when $w=20\text{nS}$, even when $g_j=600\text{nS}$, conduction fails when $G_{Na}<8\text{mS}/{\text{cm}}^2$, the same as for $g_j=0$. When $w=200\text{nm}$, conduction is successful until $G_{Na}<3\text{mS}/{\text{cm}}^2$. This implies that even with normal gap junction coupling, I_Na conductance and cleft volume have nontrivial effects on conduction (this is also reflected by the impact of $w$ on CV when $g_j$ is large in Fig.2A).

In the current model, two types of ionic currents, I_Na and I_K1, are set in the junctional membrane. Here we investigate how the localization of the two currents affects conduction and the source-since effect. If we allocate almost all the I_K1 to the lateral membrane (due to the numerical algorithm for solving the differential equations, we have to retain 1% I_K1 in the junctional membrane), we saw a minor change in CV for the control (CV becomes slightly larger, compared with Fig.2A) (Fig.3A). If we allocate 50% I_Na in the junctional membrane (Fig.3B), the dependence of CV on $w$ becomes mild, i.e., the cleft width only has a small effect on CV. In this case, no conduction can be elicited when $g_j=0$, i.e., pure ephaptic coupling cannot facilitate conduction in the cable. We also evaluated $N_{\mathit{\text{min}}}$ versus $w$ under the two I_Na and I_K1 allocation conditions, plotted in Fig.3C. Re-allocating all I_K1 from the junctional membrane to the lateral membrane reduces $N_{\mathit{\text{min}}}$, but still exhibits a similar dependence on w. Allocating 50% I_Na to the junctional membrane causes $N_{\mathit{\text{min}}}$ to become independent of $w$. It is well known that under pure gap junction coupling, $CV\propto \sqrt{g_j}$ (red dashed line in Figs. 3 D and E) (Qu et al., 2014). The presence of the cleft with dense I_Na and I_K1 in the junctional membrane causes CV to deviate from this square root rule (Figs. 3 D and E). Specifically, when $w$ is very large or junctional I_Na is small, this rule can still roughly hold. When $w$ is small with dense I_Na in the junctional membrane, CV is attenuated and deviates from this rule.

Figure 3. Effects of I_Na and I_K1 localization on conduction properties.

A. CV versus w for different $g_j$ with 1% I_K1 in the junctional membrane. B. CV versus w for different $g_j$ with 50% I_Na in the junctional membrane. C. $N_{\mathit{\text{min}}}$ versus w for three combinations of I_Na and I_K1 localization as marked. D. CV versus $g_j$ for different w as marked for the case of 90% I_Na in the junctional membrane. E. Same as D but for 50% I_Na in the junctional membrane.

Although it is well-known that increasing $g_j$ facilitates conduction (Rohr et al., 1997; Morley et al., 2005; Xie et al., 2010), it is not well understood how the cleft affects the source-sink effect. In ephaptic coupling, the major change is in the potential in the cleft and thus the transmembrane potential of the junctional membrane ($V_j={\varphi }_i-{\varphi }_c$). Since $I_{Na}=G_{Na}{m}^{3}hj(V-E_{Na})$, changing ${\varphi }_c$ changes not only the activation (m) but also inactivation (hj) and the driving force $\left(V-E_{Na}\right)$. Furthermore, the change in [Na⁺]_c may also alter the reversal potential $\left(E_{Na}\right)$ to alter I_Na. To gain a better understanding, in Fig.4, we show the intracellular and cleft potentials, the ion concentrations in the cleft, and the total I_Na of the cell under different conditions. For pure ephaptic coupling ($g_j=0$, Fig.4A), the cleft potential (${\varphi }_c$) drops from 0 mV quickly to below −80 mV with small changes in [Na⁺]_c and [K⁺]_c. This quick decrease in ${\varphi }_c$ elevates the junctional transmembrane potential $V_j$ of the adjacent cell to above the threshold for I_Na activation, depolarizing the adjacent cell to result in conduction. Increasing $g_j$ to the normal value ($g_j=600\text{nS}$, Fig.4B), the time courses of ${\varphi }_c$, [Na⁺]_c, [K⁺]_c, and I_Na remain almost the same as in the case of $g_j=0$, but increases CV from 9 to 24 cm/s (Fig.2A) and $N_{\mathit{\text{min}}}$ from 1 to 7 (Figs.2 B and C). Under the same $g_j$, increasing $w$ from 20 to 200 nm (Fig.4C) reduces the amplitudes of ${\varphi }_c$, [Na⁺]_c, and [K⁺]_c, but causes the I_Na amplitude to more than double. The increase in I_Na causes CV to increase from 24 to 39 cm/s (Fig.2A), but there is almost no change in $N_{min}$ (Figs.2 B and C), i.e., increasing $w$ from 20 to 200 nm exhibits little effect on the source-sink effect. However, as shown in Figs.2 B and C, $N_{\mathit{\text{min}}}$ increases as $w$ decreases when $w$ is smaller than 20 nm. This is likely due to the fact that narrowing the cleft width decreases ${\varphi }_c$ further to more negative voltages, which causes further self-attenuation of I_Na. This further reduction in I_Na then increases $N_{\mathit{\text{min}}}$. This agrees with the results shown in Fig.2D, in which $N_{\mathit{\text{min}}}$ keeps roughly unchanged unless G_Na is reduced to very low levels. This may also explain how a 50% I_Na allocation in the junctional membrane eliminates the dependence of $N_{\mathit{\text{min}}}$ on w (Fig.3C) since ${\varphi }_c$ cannot be low enough to reduce I_Na to a level that impacts $N_{\mathit{\text{min}}}$.

Figure 4. Membrane potentials, ion concentrations, and I_Na during a stimulation elicited conduction.

Shown are $V_l$, ${\varphi }_c$, $V_j$, Na⁺ and K⁺ concentrations in the cleft, and total I_Na versus time for an external stimulus elicited conduction, recorded from cell 150 in a 200-cell cable. A. $g_j=0$ and $w=20\text{nm}$. B. $g_j=600\text{nS}$ and $w=20\text{nm}$. C. $g_j=600\text{nS}$ and $w=200\text{nm}$.

I_K1 localization plays a nontrivial role in DAD-triggered conduction

In cardiac tissue, one of the causes of ectopic excitations is via DAD-mediated triggered activity. Differing from the stimulated AP, there is a slow phase of depolarization before the triggered AP, which affects the activation of I_K1 and I_Na, and thus the formation of the ectopic conduction. For example, I_Na is smaller in the DAD-triggered AP than in a stimulated one, and I_K1 exhibits a very different time course (Fig.5A). Similar to the stimulated case, a minimum number $\left(N_{\mathit{\text{min}}}\right)$ of DAD cells is needed to elicit conduction (Fig.5B). In Fig.5C, we show $N_{\mathit{\text{min}}}$ versus w and $g_j$ (note: we placed the DAD cells at one end of the cable to reduce computation time. If they are in the center, $N_{\mathit{\text{min}}}$ doubles). Unlike in the stimulated case, the dependence of $N_{\mathit{\text{min}}}$ on $w$ and $g_j$ is more complex (Fig.5C). When $g_j=0$, a minimum of 2 DAD cells (4 DAD cells if placed in the middle of the cable) is needed to result in a conduction. This exists in a small range of $w$, and $N_{\mathit{\text{min}}}$ is independent of $w$ (see $g_j=0$ in Fig.5D). As $g_j$ increases, more and more DAD cells are needed for triggering conduction due to gap junction coupling-induced sink effect. However, the dependence of $N_{\mathit{\text{min}}}$ on $w$ is complex. First, for $40\le w\le 70\text{nm}$, no conduction can be elicited regardless of the number of DAD cells (grey boxes in Fig.5C or between the two arrows in Fig.5D). Second, for $g_j<600\text{nS}$, $N_{\mathit{\text{min}}}$ increases with $w$, but after passing the failing window, it decreases with $w$ and then saturates when $w$ is very large. When $g_j\ge 600\text{nS}$, $N_{\mathit{\text{min}}}$ first decreases and then increases before the failing window.

Figure 5. DAD-triggered conduction.

A. Top panel shows voltage versus time for a subthreshold DAD (blue) and a DAD-triggered action potential (red). Middle and bottom panels show I_Na and I_K1 for the two cases. B. Voltage traces from the first 30 cells recorded from a 100-cell cable showing a failed (left panel) and a successful conduction (right panel) with a difference in one DAD-cell. $g_j=600\text{nS}$ and w=20 nm. C. The minimum number ($N_{\mathit{\text{min}}}$) of DAD cells required for eliciting a conduction for a given $g_j$ and w. The grey boxes represent no conduction regardless of the number of DAD-cells. The DAD-cells are placed in one end of the 100-cell cable. D. $N_{\mathit{\text{min}}}$ versus w for different $g_j$ from the same simulations as in D but with smaller w increment.

To understand the causes of these complex behaviors, we first investigated the effects of Na⁺ and K⁺ channel localization on $N_{\mathit{\text{min}}}$. When I_K1 is removed from the junctional membrane (Fig.6A), $N_{\mathit{\text{min}}}$ no longer changes with $w$, indicating that I_K1 localization is the primary cause for the complex dependence. When I_Na is removed from the junctional membrane (Fig.6B), $N_{\mathit{\text{min}}}$ exhibits a biphasic dependence on $w$. Furthermore, there is no failing window as in Fig.5 when 90% I_Na is in the junctional membrane, indicating that the presence of I_Na in the junctional membrane is attenuating the DAD triggered conduction. This is the opposite of the stimulated case (see Fig.3C). To better understand the behavior, we performed another set of simulations using a short cable of five DAD cells (Figs. 6 C and D). We first fixed ${\text{G}}_{\text{spon}}$ at its control value (0.6) and measured the peak voltage for different $w$ (Fig.6C). Under this control value of ${\text{G}}_{\text{spon}}$, the DAD fails to trigger an action potential when $w$ is between 50 and 90 nm (peak voltage is around −60 mV). When we re-allocate either I_Na or I_K1 to the lateral membrane, the DAD can always trigger an AP for any $w>10\text{nm}$. In the second set of simulations (Fig.6D), we searched the ${\text{G}}_{\text{spon}}$ threshold (or the minimum ${\text{G}}_{\text{spon}}$ to trigger an AP) versus $w$ for different I_Na and I_K1 allocations. The dependence of ${\text{G}}_{\text{spon}}$ threshold on $w$ behaves the same way as that of $N_{\mathit{\text{min}}}$ on $w$ shown in Figs. 6 A and B. The failure of conduction for the range of $w$ is because ${\text{G}}_{\text{spon}}$ (=0.6, dashed line in Fig.6D) is below the threshold for a DAD to trigger an AP.

Figure 6. Effects of I_Na and I_K1 localization on DAD-triggered activity and conduction.

A. $N_{\mathit{\text{min}}}$ versus w when no I_K1 is in the junctional membrane for different $g_j$ as marked. B. $N_{\mathit{\text{min}}}$ versus w when no I_Na is in the junctional membrane for different $g_j$ as marked. C. Peak voltage $\left({\text{V}}_{\text{max}}\right)$ versus $w$ for different I_Na and I_K1 localizations. D. The ${\text{G}}_{\text{spon}}$ threshold for triggering an action potential versus w for different I_Na and I_K1 localizations. The dashed line is the control ${\text{G}}_{\text{spon}}$ used in current study. A and B were done using the same cable model as in Fig.5. C and D were done using a 5-cell cable of all DAD cells and data were recorded from the middle cell.

The simulation results in Fig.6 imply that: 1) localization of I_Na in the junctional membrane weakens a DAD to trigger conduction; 2) localization of I_K1 in the junctional membrane is the primary cause for the complex dependence of $N_{\mathit{\text{min}}}$ on $w$; and 3) the complex behavior is mainly caused by the effect of $w$ on the DAD threshold for triggering.

The reason that cleft localization of I_Na weakens DAD-triggering conduction is because of the self-attenuation of I_Na. As shown in Fig.7, when 90% I_Na is in the junctional membrane, ${\varphi }_c$ becomes highly repolarized during a DAD-trigged AP but does not when 1% I_Na is in the junctional membrane. When I_Na is localized in the junctional membrane, its self-attenuated results in a smaller I_Na than when I_Na is localized in the lateral membrane (Fig.7). This explains the failure of the DAD to trigger conduction in a certain range of $w$ when I_Na is mainly localized in the junctional membrane (Fig.5).

Figure 7. Cleft localization of I_Na attenuates DAD triggering APs by self-attenuation.

Shown are ${\varphi }_i$, ${\varphi }_c$, and I_Na versus time for a DAD just above the threshold (${\text{G}}_{\text{spon}}$ marked) for triggering an AP when 90% I_Na is in the junctional membrane (A) and 1% is in the junctional membrane (B).

On the other hand, the reason for the biphasic dependence of $N_{\mathit{\text{min}}}$ or the triggering threshold on $w$ is due to a biphasic dependence of I_K1 amplitude on $w$. As shown in Fig.8, although the peak K⁺ concentration in the cleft decreases monotonically with $w$ (black symbols in Fig.8D), the peak I_K1 exhibits a biphasic response (red symbols in Figs.8 D and F). The cause of this biphasic response can be understood as follows. The voltage threshold for a DAD to trigger an AP is roughly −60 mV regardless of the conditions (the DADs in Figs. 8 A–C are just below threshold for triggering APs). If one draws a family of I-V curves with different extracellular K⁺ concentrations and a vertical line at −60 mV (Fig.8E), the intersections of the vertical line with the I-V curves show a biphasic behavior (inset in Fig.8F), explaining the biphasic I_K1 response to [K⁺]_c or $w$.

Figure 8. Ion concentrations and I_K1 during a DAD.

A-C. Shown are ${\varphi }_c$, ${\varphi }_i$, Na⁺ and K⁺ concentrations in the cleft, and total I_K1 versus time for a subthreshold DAD just below the threshold (${\text{G}}_{\text{spon}}$ marked) for triggering an action potential for three different cleft widths with 1% I_Na in the junctional membrane. Simulations in 5-cell cable composed of DAD cells. Arrows indicate peak I_K1 during the DAD. D. Peak K⁺ concentration in the cleft and peak I_K1 of the whole cell versus $w$ during the just-below-threshold DAD. The condition is as in B and C but ${\text{G}}_{\text{spon}}$ is chosen to just below the threshold for each $w$ (just below the blue curve in Fig.6D). E. I-V curves of I_K1 for different extracellular K⁺ concentrations. F. Peak I_K1 of the whole cell versus peak K⁺ concentration (replot of data in D). Inset is a plot of E for V=-60 mV.

Conduction originating from I_Ca,L-mediated automaticity depends on I_K1 localization

Finally, we investigate the effects of cleft width and ion channel localization on automaticity-induced conduction. We used the Luo-Rudy model (Faber & Rudy, 2000); automaticity or pacemaker activity was induced by downgrading I_K1 (Silva & Rudy, 2003). We put the pacemaker cells at one end of the cable with the rest of the cable consisting of normal excitable cells. Once there are enough pacemaker cells, conduction can be initiated (Fig.9A). Fig.9B shows $N_{\mathit{\text{min}}}$ versus $w$ and $g_j$. No conduction is possible when $g_j=0$. $N_{\mathit{\text{min}}}$ increases as $w$ or $g_j$ increases. Re-allocating I_K1, but not I_Na, alters the $N_{\mathit{\text{min}}}$ dependence on $w$, indicating that I_K1 is responsible for the dependence of $N_{\mathit{\text{min}}}$ on $w$ (Fig.9C). Since there is no I_K1 in the pacemaker cells and I_Na is almost completely inactivated at the minimum potential (−60 mV) of the pacemaker cells, changing $w$ or I_Na and I_K1 localizations would not affect the pacemaker activity. The cause that affects the source-sink effect is the change in the resting potential of the excitable cells due to the change in K⁺ concentration in the cleft (Fig.9D). Specifically, as $w$ decreases, [K⁺]_c decreases, which in turn deceases the resting potential of the excitable cells. When the resting potential is lower, it is more difficult for the pacemaker cells to depolarize the excitable cells, resulting in the behavior that as $w$ increases, more pacemaker cells are needed for initiating conduction.

Figure 9. Automaticity induced conduction.

A. Space-time plot of voltage for an automaticity induced conduction. Red traces are from oscillatory cells and black traces are for normal excitable cells. B. The minimum number ($N_{\mathit{\text{min}}}$) of oscillatory cells required to initiate conduction in the cable versus $g_j$ and $w$. C. $N_{\mathit{\text{min}}}$ versus $w$ for three I_Na and I_K1 localizations, D. Resting potential (${\varphi }_{i,\mathit{\text{rest}}}$) of the normal excitable cells versus $w$. Inset shows [K⁺]_c versus $w$.

Discussion

In this study, we used computational modeling to investigate the source-sink effect of cardiac conduction in the presence of ephaptic coupling. We used three depolarization protocols to elicit conduction: 1) external stimulation; 2) DADs; and 3) I_Ca,L-mediated automaticity. The major findings are: 1) under pure ephaptic coupling (no gap junctions at all), only one stimulated cell is needed to elicit conduction; 2) the minimum number of stimulated (or DAD or oscillatory) cells needed to elicit a conduction increases as gap junction conductance increases, as predicted by the gap junction coupling theory; and 3) the dependence of the source-sink effect on the cleft width exhibit complex behaviors, which are mainly caused by self-attenuation of I_Na due to the ephaptic effect and self-enhancement of I_K1 due to the increase of K⁺ concentration in the cleft. Here we discuss how the cleft affects the source-sink effect and the implications and limitations for applying the simulation results to conduction and arrhythmias in cardiac tissue.

How does the cleft affect the source-sink effect?

Many simulation studies (Kucera et al., 2002; Mori et al., 2008; Lin & Keener, 2010; Lin & Keener, 2014; Tsumoto et al., 2014; Wei et al., 2016; Tveito et al., 2017; Weinberg, 2017; Hichri et al., 2018; Ivanovic & Kucera, 2021; Moise et al., 2021; Ivanovic & Kucera, 2022; Wei & Tolkacheva, 2022; Otani et al., 2023a; Morris et al., 2024) have been carried out to simulate the effects of ephaptic coupling on cardiac conduction, mostly focusing on CV, but how ephaptic coupling affects the source-sink effect has not been elucidated. Our current simulation study provides mechanistic insights into how ephaptic coupling and the cleft width impact the source-sink effect of cardiac conduction, as summarized below:

1. When conduction is elicited by an external stimulus, the major impacting factor of source-sink effect is I_Na self-attenuation via the ephaptic coupling-induced polarization of the cleft. During depolarization, the cleft potential ${\varphi }_c$ can become very negative, reducing the junctional transmembrane potential and thus the driving force of I_Na. However, although CV is tightly dependent on the I_Na strength (Qu et al., 2014), the minimum number of stimulated cells required to initiate a conduction is insensitive to the change of the cleft width unless it is very narrow (Fig.2). This is because $N_{\mathit{\text{min}}}$ becomes sensitive to the I_Na strength only when the I_Na strength is small enough (Fig.2D). For the same reason, when the portion of I_Na in the junctional membrane is reduced below a certain level, $N_{\mathit{\text{min}}}$ is no longer dependent on $w$ (Fig.3C) even though ${\varphi }_c$ still polarizes to negative voltages during depolarization. Under the external stimulation case, I_K1 also exhibits some effects on $N_{\mathit{\text{min}}}$ (Fig.3C), i.e., junctional localization of I_K1 reduces the source effect for conduction by elevating [K⁺]_c.

2. When conduction is triggered by a DAD, the major impacting factor is junctional I_K1. It is well-known that besides spontaneous Ca²⁺ release, I_K1 and I_NCX are the two currents playing critical roles in regulating the DAD amplitude and triggering APs (Liu et al., 2015; Liu et al., 2016; Ko et al., 2017). During a DAD, I_K1 is activated for a much longer duration than in the regular AP (Fig.5A). This brings in more K⁺ into the cleft, elevating [K⁺]_c to much higher levels (Fig.8). Elevating [K⁺]_c shifts the I-V curves. At the threshold voltage (around −60 mV) for the DAD to trigger an AP, the I_K1 amplitude exhibits a biphasic response (Fig.8E), which gives rise to a biphasic response of the Ca²⁺ release threshold for triggering AP (Fig.6D). This leads to a biphasic response of $N_{\mathit{\text{min}}}$ to the cleft width. Junctional I_Na localization exhibits a small effect on triggering AP and thus $N_{\mathit{\text{min}}}$. Specifically, junctional localization of I_Na weakens a DAD’s ability to trigger an AP, due to the self-attenuation of junctional I_Na (Fig.7).

3. When conduction originates from pacemaker cells in which the oscillations are I_Ca,L-driven, such as in the sinoatrial nodal cells, there is almost no I_Na and I_K1 during the oscillation, and thus neither the cleft width nor the I_Na and I_K1 localization affects the oscillatory behavior. On the other hand, the resting potential of the normal excitable cells is affected by the change of [K⁺]_c, which is affected by the cleft width and I_K1 localization. Because of this, it is easier for the pacemaking activity to conduct when the cleft is narrower due to a higher [K⁺]_c causing a higher resting potential of the normal excitable cells. Note that in our model, the pacemaker cells are already oscillatory when in isolation. As shown in a study by Poelzing et al (Poelzing et al., 2021), narrowing the cleft elevates [K⁺]_c, which can promote oscillations. In other words, narrowing the cleft can induce ectopic excitations in the heart if I_K1 is highly concentrated at the junctional membrane.

In summary, there are mainly two ways that the cleft can affect the source-sink effect of cardiac conduction. The first one is self-attenuation of I_Na caused by the field change in the cleft, which reduces the driving force to attenuate I_Na. The second one is via the change of K⁺ concentration in the cleft, which then affects I_K1. Since the change of [K⁺]_c is caused by I_K1 itself, this is a self-enhancement process mediated by the K⁺ concentration in the cleft, while in the case of I_Na, self-attenuation is mediated by the field change in the cleft. Increasing junctional I_Na weakens the source effect. Increasing junctional I_K1 can either promote or weaken the source by lowering or elevating the threshold for triggering AP but can also weaken the sink by elevating the resting potential.

Implications for cardiac conduction and arrhythmogenesis

The source-sink effect is an important property of cardiac conduction. Our simulation results show that cleft width and Na⁺ and K⁺ channel localization play nontrivial roles in the source-sink effect in addition to gap junction coupling. Whether these insights are applicable to conduction and arrhythmogenesis in real cardiac tissue depends on the cleft structure and its ion channel distributions. Experimental studies (Kucera et al., 2002; Rhett et al., 2011; Delmar & Liang, 2012; Rhett & Gourdie, 2012; Rhett et al., 2013; Veeraraghavan et al., 2015; Leo-Macias et al., 2016; Veeraraghavan et al., 2016; Raisch et al., 2018; Veeraraghavan et al., 2018; Mezache et al., 2020; Nielsen et al., 2023) have identified a narrow cleft, also called the perinexus, which has dense distributions of Na⁺ channels and inward rectifier K⁺ channels, providing structural evidence to support the theoretical results.

Our results show that in the absence of gap junction coupling, only a single stimulated cell is needed to initiate conduction (Fig.2), indicating that there is no source-sink effect or no sink effect from adjacent cells when the coupling is purely ephaptic. In the case of DAD-triggered conduction, due to the self-enhancement of junctional I_K1, more DAD cells are needed to initiate conduction under pure ephaptic coupling (Fig.5) than in the case of external stimulation. More interestingly, ephaptic coupling and gap junction coupling can work in synergy to facilitate conduction. For example, when $w$ is large (no ephaptic coupling) and gap junction conductance is small, although the source-sink effect is weak, CV is too small to maintain normal conduction (Fig.2A). However, for $w$ in the range of 20 to 50 nm (and thus with ephaptic coupling) and small gap junction conductance, the two coupling mechanisms work in synergy to maintain normal CV while still maintaining a weak source-sink effect. This type of favorable source-sink condition may play a role in conduction from the sinoatrial node to the atrium or conduction from the Purkinje fibers to the working myocardium. This is because in both the sinoatrial nodal region or the Purkinje-myocyte junctions, due to the source-sink effect, conduction is more prone to fail when gap junction coupling is strong (Rohr et al., 1997; Shaw & Rudy, 1997; Morley et al., 2005). On the other hand, if the gap junction conductance is too small, then conduction is too slow to be normal. When ephaptic coupling is present, the gap junction conductance can be small while CV is still normal, giving rise to a favorable source-sink condition to facilitate conduction in these specialized regions.

However, weak source-sink effects may promote arrhythmias in atrial and ventricular tissue. Note that in both DAD-triggered or the automaticity-induced conduction (Figs.5 and 9), when $w$ is small (strong ephaptic effect), the required number of DAD cells or automaticity cells is less than that needed when $w$ is large (weak or no ephaptic effect). Therefore, for the same gap junction coupling strength, the presence of ephaptic coupling allows DADs or automaticity to more easily elicit ectopic beats in the heart, promoting arrhythmias. This may be partially responsible for ventricular arrhythmias and sudden death in Cx43 knockout mice that still exhibit persistent normal conduction in the absence of or at very low gap junction conductance (Guerrero et al., 1997; Thomas et al., 1998; Morley et al., 1999; Eloff et al., 2001; Gutstein et al., 2001; Beauchamp et al., 2004; Danik et al., 2004; Rijen et al., 2004; Danik et al., 2008; Stein et al., 2009; Stein et al., 2011).

Limitations

There are several limitations and ambiguities in the current study that need to be clarified. First, we put 90% I_Na and 90% I_K1 in the junctional membrane, which may be unrealistic. Detailed micro-structures in the ID may also play important roles (Hichri et al., 2018; Ivanovic & Kucera, 2021, 2022; Jæger et al., 2023). However, it is challenging to scale up these micro-structural details into tissue-scale models due to computational limitations (Qu et al., 2025). Besides Na_v1.5 and K_ir2.1, experimental data have shown that other ion channels and pumps are densely localized at the ID region, including Na⁺-K⁺ pump (Chen et al., 2022), L-type calcium channel (De Smet et al., 2021), and Na⁺-Ca²⁺ exchange (Thomas et al., 2003). The localization of these channels in the ID region may also impact the source-sink effect, which remains to be elucidated. For example, in the current model, Na⁺ and K⁺ in the cleft are rebalanced via the currents between the cleft and the bulk extracellular space. In the presence of the ion pumps, rebalancing of the ions in the cleft will affect the magnitudes of I_Na and I_K1, impacting the source-sink effect. Moreover, these pumps are electrogenic, which will be impacted by the Na⁺ and K⁺ concentrations in the cleft and junctional membrane voltage, affecting repolarization. Another limitation of this study is that we did not investigate the source-sink effect on EAD-triggered conduction in the presence of ephaptic coupling. As shown in recent studies (Greer-Short et al., 2017; Nowak et al., 2020; Nowak et al., 2021; Wu et al., 2025), the presence of the late component of I_Na in the junctional membrane may lower Na⁺ concentration in the cleft during depolarization, which in turn reduces the amplitude of late I_Na by reducing the driving force of I_Na to suppress EADs. In other words, altering the cleft volume may alter action potential duration and the propensity to late I_Na-induced EADs and arrhythmias. However, it will be interesting to investigate the effect of junctional localization of I_Na and I_K1 on other mechanisms of EADs and the genesis of premature ventricular complexes due to action potential duration prolongation and EADs (Zhang et al., 2021).

Conclusions

Dense ion channel distributions in the junctional membrane of the IDs can have nontrivial effects on the source-sink effect of cardiac conduction. These effects are caused by I_Na self-attenuation via the ephaptic effect on the cleft potential and ${\text{I}}_{\text{K}1}$ self-enhancement via elevation of the cleft K⁺ concentration. The source-sink effects caused by ephaptic coupling, cleft width and ion channel localization may play nontrivial roles in cardiac conduction and arrhythmogenesis.

Key Points.

• Gap junction coupling is recognized as the primary mechanism of cardiac conduction, but ephaptic coupling may also play a nontrivial role.

• Source-sink effects are a key feature of cardiac conduction, and it is unclear how they are affected by ephaptic coupling.

• Computer simulations are used to investigate how source-sink effects are altered by ephaptic coupling and its interaction with gap junction coupling.

• Under pure ephaptic coupling, only one stimulated cell is needed to elicit conduction, implying that there is no source-sink effect.

• The minimum number of depolarized cells needed to elicit a conduction exhibits complex behaviors and depends on the manner of depolarization. These effects are mainly caused by self-attenuation of the junctional sodium current due to the ephaptic effect and self-enhancement of the inward rectifier potassium current due to the increase of potassium concentration in the cleft.

Data availability statement

Data in this study are from computer simulations. A standard computer code for the 1D cable can be downloaded from GitHub at: https://github.com/Mor5oN/source-sink.