The influence of intercalated disk nanostructure on local ionic currents and cardiac conduction

Abstract

The intercalated disk (ID) is a structurally heterogeneous junctional complex essential for synchronized cardiac conduction and contraction. Previous computational models have investigated the influence of ID structure on cardiac conduction. However, most have relied on oversimplified geometries and uniformly distributed ion channels, limiting their ability to capture nanoscale heterogeneity. In this study, we expand our previous finite element mesh framework to produce a more physiologically realistic representation of the ID, incorporating spatially heterogeneous gap junctions and multiple ion and ionic current dynamics. We systematically quantify the impact of key structural and electrophysiological features on conduction by generating a comprehensive library of 384 ID mesh configurations and simulating tissue-level conduction for both strong and reduced gap junctional coupling. Further, we employed a multilayer perceptron neural network approach to quantify gradient-based sensitivity analysis, enabling a systematic quantification of the relative influence of geometric and nanostructural factors on ID and cleft dynamics, as well as tissue-level conduction across multiple regimes. In particular, sensitivity analysis revealed that gap junctional coupling, cleft geometry, and nanostructure heterogeneity are the dominant determinants of cleft potential, sodium current synchronization, and conduction velocity. We identify that membrane separation of the ID interplicate and plicate regions can exhibit context-dependent influences on conduction, either enhancing or slowing, depending on gap junctional coupling. Collectively, these findings highlight the regime-dependent roles of ID ultrastructure and establish a quantitative framework that links nanoscale ID morphology to tissue-scale cardiac conduction.

Significance

Cardiac conduction critically depends on the nanoscale organization of the cell-cell junction, the intercalated disk (ID). We employed a refined 3D finite element mesh model of the ID nanostructure, integrating key ion channels and gap junctions and their nonuniform distributions, thus facilitating the quantification of the relative role of these properties on cardiac conduction. Building upon these high-fidelity simulations, we further applied machine learning algorithms to systematically evaluate the relative contributions of distinct ID structural and electrophysiological features on ID dynamics and conduction. This combined modeling and neural-network-based analysis framework provides new mechanistic insights into how ID nanostructural heterogeneity regulates conduction.

Introduction

The intercalated disk (ID) forms the specialized interface between adjacent cardiac myocytes, ensuring both mechanical integrity and electrical connectivity required for coordinated contraction. Although historically considered a passive structure responsible for transmitting force through mechanical junctions and electric current via gap junctions (GJs), recent evidence indicates that the ID actively participates in regulating cardiac excitability and conduction (1,2). Notably, the ID is structurally heterogeneous, comprising highly folded plicate regions and flatter interplicate regions, each with distinct molecular compositions and biophysical properties. At the nanoscale level, key electrogenic proteins, in particular voltage-gated sodium (Na⁺) channels (Na_V1.5), are not uniformly distributed across the cell membrane but instead cluster within specialized ID nanodomains, such as the GJ-adjacent perinexus and mechanical-junction-associated adhesion-excitability nodes (3). We and others have shown that these microdomains provide a structural basis for ephaptic coupling, a form of electrical communication in which extracellular potential changes in narrow intercellular clefts influence ion channel dynamics on adjacent cells (4,5,6), in addition to the more canonical GJ-mediated coupling governing electrical conduction in the heart (2,7,8). The degree of GJ coupling is a key determinant of conduction velocity and electrical synchrony in the myocardium (9). Although often treated as distinct mechanisms, our recent work suggests that gap junctional and ephaptic coupling operate in a complementary and interdependent manner to maintain robust electrical conduction across a wide range of physiological and pathological conditions (10,11).

Detailed ultrastructural analyses have revealed that the ID exhibits an intricate architecture (12). Scanning electron microscopy of myocardial tissue shows that IDs form extensive, zig-zagging interdigitations with prominent plicate regions, especially in Purkinje fibers and working myocytes (13). A recent study demonstrated that the complex tortuosity of IDs, characterized by tightly intermingled plicate (“folded”) and interplicate (“flat”) regions, plays a functional role in modulating ephaptic coupling (14). Moreover, ion channels localized within the ID play a significant role in regulating conduction. Na_V1.5 channels preferential concentration at IDs, colocalized with GJs (3,15), forming specialized nanodomains called the perinexus, membrane regions in which the intercellular membrane spacing is dramatically reduced (16,17,18,19).

ID abnormalities, notably involving GJ proteins such as connexin 43 (Cx43), have profound implications for cardiac function at both the electrical and mechanical levels. Loss or mislocalization of Cx43 disrupts electrical coupling between cardiomyocytes, leading to anisotropic conduction and regions of slow conduction, which significantly raising the susceptibility to reentrant arrhythmias. For instance, studies in murine models with Cx43 downregulation demonstrated conduction velocity (CV) slowing and increased arrhythmic risk, particularly under stress or abnormal ionic conditions (20). Moreover, abnormal intercellular adhesion and electromechanical uncoupling at the ID contribute to arrhythmogenesis beyond conduction delay, including increased dispersion of repolarization, premature depolarizations, and even structural remodeling of the myocardium. This is exemplified in arrhythmogenic cardiomyopathy, where desmosomal and connexin defects co-occur and precipitate both arrhythmic events and sudden cardiac death (21). Further, emerging evidence supports that ID dysfunction may underlie the pathological mechanisms underlying various cardiac diseases (22,23). The structural disorganization of IDs, characterized by widened intercellular clefts and altered ion channel organization, has been associated with diverse cardiac pathologies, including atrial fibrillation, dilated cardiomyopathy, and inflammatory cardiomyopathy (24,25,26). Thus, ID abnormalities are associated with conduction disturbances, arrhythmogenesis, and mechanical dysfunction that ultimately compromise cardiac performance and patient outcomes (27).

Mathematical models have been previously developed to investigate the role of the ID in cardiac conduction. Early modeling efforts were based on the simplified single-cleft framework, which omitted structural details of the ID and assumed uniform ion channel distributions (5). To capture the functional complexity of the ID, work from our group and others incorporated structural heterogeneity, enabling more detailed simulations of nanoscale electrical interactions between cells (6,11,14,28,29,30). Although these previous studies have made significant progress in modeling intercellular electrical conduction at the ID, many of these efforts are based on simplified geometries and limited electrophysiological complexity (i.e., neglecting nanoscale cleft width variations, subdomain organization of plicate and interplicate regions, heterogeneous GJs conductance, and clustered distribution of ion channels). Building on these advances, we recently developed a multiscale modeling approach that integrated the properties of the nanoscale ID geometry, based on measurements from electron microscopy, first developing an ID representation using a finite element model (FEM) mesh and subsequently reducing the FEM representation to an electrical network, which in turn is integrated into an electrophysiological cardiac tissue model. Importantly, this approach enabled investigation of how ID nanostructure regulates cardiac conduction properties, focusing specifically on the dependence on interplicate intermembrane separation (d_ip) (10). Although our prior work incorporated a novel level of ID nanostructure, we previously did not consider two key levels of detail: the heterogeneity of local GJ distribution within the ID and fluxes of all ions within the cleft (i.e., Ca²⁺ and K⁺), nor did we investigate the influence of additional nanostructural properties on cardiac conduction.

In this work, we extend our previous FEM framework by incorporating multiple ionic currents and ion flux dynamics within the ID, in addition to the spatial variation of GJ conductances. We systematically investigate the effects of nanoscale ID structure on local ID ionic currents and cardiac conduction. Additionally, we quantify the relative contributions of key ID structural features to electrophysiological measures and conduction properties by employing a neural-network-based sensitivity analysis, highlighting the most influential geometrical and biophysical properties governing ID-mediated currents and conduction. Key findings highlight that the dependence of plicate intermembrane distance (d_p) and d_ip on CV is further influenced by GJ coupling strength (G_gap): for conditions of strong GJ coupling, increasing d_p or d_ip increases CV, whereas for reduced GJ coupling, there is a complex interdependence between d_p and d_ip regulating CV. Further, we find that key statistical properties of the ID nanostructure are also predictive of cardiac conduction under conditions of both strong and reduced GJ coupling. Critically, this integrative modeling approach bridges the gap between ID nanostructure and tissue-scale conduction.

Materials and methods

Mesh generation and simulation

In our previous work (10), we developed a pipeline to investigate the role of ID nanoscale structural heterogeneity in modulating cardiac conduction. Briefly, we developed a 1D tissue model that incorporates detailed ID structural features to predict their influence on conduction. A 3D FEM mesh representing the ID extracellular space is constructed using a parametrically defined algorithm, based on TEM measurements of overall diameter, plicate and interplicate region lengths, intermembrane separation, and GJ distribution (Figs. 1 and S1). To perform tissue-scale simulations, the resulting 3D FEM mesh is reduced to an equivalent 100-node electrical network, which is then integrated into a 1D tissue model (Fig. 2 A). We also integrate detailed ion channel distribution within the ID for sodium (NaV1.5), potassium (Kir2.1) channels, and the sodium-potassium ATPase (NKA) derived from superresolution microscopy data (12), defining the Na_area, Kir_area, and NKA_area, respectively, for each of the 100 ID nodes. Full details of the FEM mesh generation, equivalent electrical circuit reduction, and coupling to the tissue-scale circuit are presented in our prior work (see Figs. 3 and S1 in (10)).

Figure 1.

3D FEM model representation of the intercalated disk (ID). (A) Detailed 3D FEM representation showing embedded reduced network topology within the ID structure. Baseline morphological parameters: plicate distance d_p = 10 nm, interplicate distance d_ip = 10 nm, fold amplitude A_f = 90 nm. Scale bar indicates ID diameter d = 10 μm. (B) Parametric variation of 3D FEM models demonstrating morphological heterogeneity across different ID diameters (d) and interplicate heights (h_ip).

Figure 2.

Schematic of tissue and reduced ID and cleft electrical network models. (A) Schematic of the tissue model electrical circuit. (B) Representative reduced electrical network model discretized from the 3D FEM model shown in Fig. 1A. Plicate nodes (red), interplicate nodes (blue), center node (black), and node 22 (arbitrarily selected, magenta) serve as exemplars to demonstrate key network metrics: minimum weighted path to center node ${\sigma }_{\text{center}}^{\min }$ (orange path), minimum weighted path to bulk region ${\sigma }_{\text{bulk}}^{\min }$ (green path), and Euclidean distance to center node d_center (black dashed line). (C) Example of action potential propagation in the 1D tissue. (D) Action potential upstrokes in the 1D tissue, with center cells 25 and 26 highlighted.

Figure 3.

Structure of the neural network model. The network consists of an input layer (${\mathbb{R}}^m$), two hidden layers with 256 and 128 neurons respectively, and an output layer (${\mathbb{R}}^n$). ${\varphi }_j^i$ denotes the activation function of the j^th neuron in the i^th layer, and bⁱ represents the bias vector of the i^th layer. Input and output dimensions, m and n respectively, differ based on the specific neural network analysis performed, as described in the main text.

In addition, we expand on our previous work by incorporating several new critical model details: 1) the electrical circuit model includes a resistor network representing the conductivity on the intracellular side of the ID membrane—mirroring the network in the extracellular cleft—to model the nanoscale structure on both sides of the ID membrane. 2) The electrical circuit directly integrates the GJ spatial distribution, based on the location of GJs within the FEM mesh, as described in more detail in the supporting material. 3) We incorporate a dynamic representation of cations Na⁺, K⁺, and Ca²⁺ and generic anion A ⁻ to account for electrodiffusion and electroneutrality in the intercellular cleft space, based on recent model predictions of electrodiffusion cleft fluxes (31). Full electrical circuit equations and electrodiffusion fluxes are provided in the supporting material, with additional numerical methods and details provided in our prior work (10). Finally, we systematically investigate ID mesh heterogeneity by generating a library of 384 meshes, exploring all combinations of the ID mesh geometric parameters values in Table 1 for diameter (d), interplicate distance (d_ip), plicate distance (d_p), fold amplitude (A_f), and interplicate height (h_ip) (see representative visualizations in Figs. 1 and S1).

Table 1.

Key ID Mesh and Model Notation, Measures, and Values

Key ID Mesh and Model Notation, Measures, and Values
Symbol	Definition	Values	References
d	Diameter (μm)	5, 10, 15	(10,12)
dip	Interplicate distance (nm)	10, 30, 50, 70	(10,12)
dp	Plicate distance (nm)	10, 30, 50, 70	(10,12)
Af	Fold amplitude (nm)	0, 90, 180, 360	(10,12)
hip	Interplicate height (μm)	3.2, 6.4	(10,12)
dcenter	Euclidean distance to center node (μm)	–
${\sigma }_{center}^{\mathit{min}}$	Minimum weighted path to center node (nS)	–
σlocal	Local weighted connectivity (nS)	–
${\sigma }_{bulk}^{\mathit{min}}$	Minimum weighted path to bulk region (nS)	–
Naarea	Normalized Na+ channel area	–
Kirarea	Normalized K+ channel area	–
NKAarea	Normalized Na/K-ATPase area	–
Ggap	Gap junction conductance (nS)	73.5, 735	(32,33,34,35,36,37,38,39,40,41,42)

The table summarizes the symbols, definitions, and parameter values.

FEM-based electrical networks are subsequently integrated into a 1D cable model composed of 50 ventricular myocytes, as described in our prior work (10). The 1D cable was paced in cells 1–5 at a 1000-ms basic cycle length for 10 beats using an adaptive time-step integration method. A detailed mathematical derivation of the updated FEM cable model is provided in the supplemental methods (Eq. (S1)). The O’Hara-Rudy model of human ventricular myocyte electrophysiology was used to describe ionic currents (43). An example of action potential propagation is shown in Fig. 2 C and D. All numerical simulations were conducted using MATLAB (44) on the high-performance computing infrastructure provided by the Ohio Supercomputer Center (45). We note that with the modifications above, the tissue simulation numerical methods, including the numerical discretization scheme and voltage, ionic currents, and cleft concentration operator splitting, are performed as we have previously described (see Eq. (S8) and “Tissue simulation numerical methods” section in the Supporting Material of (10)). Validation of the numerical method convergence is described in supplemental methods and illustrated in Figs. S2 and S3. Tissue simulation and sensitivity analysis code is provided in the GitHub repository github.com/SHWeinberg/ID_Cleft_Model_2025.

We simulate conditions of strong and reduced gap junctional coupling, with total gap junctional conductance values (i.e., summed conductance distributed to all GJs within the ID) of G_gap set to 735 nS and 73.5 nS, respectively. To assess the influence of ID nanostructure on electrophysiological behavior, we analyze the ID and cleft dynamics in the middle of the tissue, the 25^th cell pair (highlighted in Fig. 2 C and D). For each FEM mesh, we define ID and cleft features by performing topological network analysis on the corresponding reduced electrical network (Fig. 2 B): this reduced network is formally a connected, weighted, undirected geometric graph with cycles, ensuring that the shortest weighted path between any two nodes can be defined. Accordingly, we identified the center node for each mesh. We calculated the shortest weighted paths from all nodes to the center node and to the closest bulk node, denoted as ${\sigma }_{\text{center}}^{\min }$ and ${\sigma }_{\text{bulk}}^{\min }$, respectively. To characterize spatial organization, the Euclidean distance from each node to the center node (d_center) was also computed. Finally, the local weighted connectivity (σ_local) was quantified as the sum of edge weights connected to each individual node.

Specifically, these metrics provide a quantitative link between the complex 3D geometry of the ID and its corresponding electrical properties. The Euclidean distance to the center node (d_center) quantifies a node’s radial position, quantifying the distance from the ID center, in contrast with the ID periphery, which is electrically coupled to the bulk extracellular space. The minimum weighted path to the bulk (${\sigma }_{\text{bulk}}^{\min }$) represents the electrical resistance along the most conductive pathway from any given node within the intercellular cleft to the bulk extracellular space, which governs the distribution of ionic currents from the confined cleft space, thus determining the magnitude of local cleft potential gradients. Finally, the minimum path to the center node (${\sigma }_{\text{center}}^{\min }$) quantifies the electrical connectivity within the internal structure of the ID, providing a measure of how electrically accessible or isolated a node is from the ID geometric center. Together, these parameters translate the static architecture of the ID into functionally relevant electrical characteristics (Fig. 2 B). These network metrics were selected based on their ability to capture centrality, spatial accessibility, and local connectivity within the ID and cleft, which we hypothesize influence the dynamics of the ID and cleft.

Neural network sensitivity analysis

We employed a feed-forward multilayer perceptron (MLP; (46)) to capture the intrinsic nonlinearity of ID and cleft dynamics, utilizing steady-state input features mapped to corresponding output responses. The architecture of the MLP is illustrated in Fig. 3. Specifically, the input features include mesh geometric parameters (ID diameter (d), interplicate distance (d_ip), plicate distance (d_p), fold amplitude (A_f), and interplicate height (h_ip)); either local or distribution measurements of the ID nanoscale structure topological features, described above (σ_local, ${\sigma }_{center}^{\mathit{min}}$, ${\sigma }_{bulk}^{\mathit{min}}$, and d_center); and local ID channel area distribution (Na_area, Kir_area, and NKA_area). The output variables comprise either ID and cleft electrophysiological dynamical measurements (peak cleft potential ϕ, peak pre- and post-junctional sodium current $I_{Na}^{pre/post}$, time of peak pre- and post-junctional sodium current t_pre/post and their difference Δt, and peak cleft ionic Na⁺, K⁺, Ca²⁺, and A⁻ concentration, S_Na, S_K, S_Ca, and S_A) or CV under different conditions. In total, nine MLPs were trained: mesh and nanostructure properties versus ID and cleft dynamics, mesh properties versus CV, and mesh nanostructure distribution measures versus CV, each evaluated under three data subsets: all data (both strong and reduced GJs coupling), strong coupling only, and reduced coupling only.

Previous studies have demonstrated that partial derivative analysis is a reliable approach for interpreting neural networks, exhibiting robustness across different training conditions and network architectures (47,48), described as follows. For a single step of an MLP, where the output of the l^th layer y^l is updated from the (L-1)^th layer with activating function ϕ^l, weight matrix W^l, and bias terms b^l, the update rule is given by

$${\mathbf{y}}^l={\varphi }^l\left({\mathbf{W}}^l{\mathbf{y}}^{l-1}+{\mathbf{b}}^l\right)$$ (1)

Letting z^l = W^ly^l−1+b^l and applying the chain rule, we obtain

$$\frac{\partial {\mathbf{y}}^l}{\partial {\mathbf{y}}^{l-1}}=\frac{\partial {\mathbf{y}}^l}{\partial {\mathbf{z}}^l}·\frac{\partial {\mathbf{z}}^l}{\partial {\mathbf{y}}^{l-1}}$$ (2)

Thus, $\frac{\partial {\mathbf{z}}^l}{\partial {\mathbf{y}}^{l-1}}={\mathbf{W}}^l$ and further the Jacobian matrix of the activation function is defined as ${\mathbf{J}}^l=\frac{\partial {\mathbf{y}}^l}{\partial {\mathbf{z}}^l}$, which takes the following form:

$${\mathbf{J}}^l=\left[\begin{array}{cccc}\frac{\partial {\varphi }_1^l}{\partial z_1^l}\left(z_1^l\right)& \frac{\partial {\varphi }_2^l}{\partial z_1^l}\left(z_1^l\right)& \cdots & \frac{\partial {\varphi }_{n_l}^l}{\partial z_1^l}\left(z_1^l\right)\\ \frac{\partial {\varphi }_1^l}{\partial z_2^l}\left(z_2^l\right)& \frac{\partial {\varphi }_2^l}{\partial z_2^l}\left(z_2^l\right)& \cdots & \frac{\partial {\varphi }_{n_l}^l}{\partial z_2^l}\left(z_2^l\right)\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial {\varphi }_1^l}{\partial z_{n_l}^l}\left(z_{n_l}^l\right)& \frac{\partial {\varphi }_2^l}{\partial z_{n_l}^l}\left(z_{n_l}^l\right)& \cdots & \frac{\partial {\varphi }_{n_l}^l}{\partial z_{n_l}^l}\left(z_{n_l}^l\right)\end{array}\right]$$ (3)

Extending this to the entire network and all layers, the total derivative of the output y with respect to the input x is given by

$$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}=\frac{\partial {\mathbf{y}}^L}{\partial {\mathbf{y}}^{L-1}}·\frac{\partial {\mathbf{y}}^{L-1}}{\partial {\mathbf{y}}^{L-2}}\cdots \frac{\partial {\mathbf{y}}^1}{\partial \mathbf{x}}=\prod _{l=1}^L{\mathbf{J}}^l{\mathbf{W}}^l$$ (4)

Thus, the partial derivative product of terms in Eq. (4) defines a sensitivity analysis of each network output relative to each input, based on the fit MLP. In practice, these partial derivatives can be efficiently computed using automatic differentiation tools, such as the autograd module in PyTorch. Further, to mitigate the uncertainty caused by nonunique models, we applied fivefold validation and averaged the resulting partial derivatives across the five folds.

Results

Dynamics of the ID and cleft

We first examine how variations in cleft nanostructural properties affect ID membrane and cleft dynamics by performing simulations across 384 meshes under two distinct GJ conductances, G_gap, of 735 nS and 73.5 nS, for a total of 768 tissue simulations. Fig. 4 illustrates four examples selected from the full set of 384 meshes (with baseline parameters: diameter d = 5 nm, interplicate height h_ip = 3.2 μm, fold amplitude A_f = 90 nm), for narrow and wide plicate distance (d_p), narrow and wide interplicate distance (d_ip), and both GJ conductance (G_gap) levels. For each of these eight examples, we plot the cleft potential (ϕ_cleft, purple), sodium currents ($I_{\text{Na}}^{\text{pre}}$, blue; $I_{\text{Na}}^{\text{post}}$, orange), transmembrane potential (ϕ_axial, gray), and CV.

Figure 4.

ID membrane and cleft dynamics for different ID properties and gap junction conductances. (A and B) Dynamics for G_gap of (A) 735 nS and (B) 73.5 nS. Top row of each subpanel: cleft potential ϕ_cleft (purple). Bottom row: prejunctional sodium current $I_{\text{Na}}^{\text{pre}}$ (blue) and postjunctional sodium current $I_{\text{Na}}^{\text{post}}$ (red). Transmembrane potential ϕ_axial is shown in gray as background, with two highlighted traces corresponding to the two cells adjacent to the measured cleft. Colored bold lines represent the average traces. Each subpanel shows results under different interplicate distances d_ip and plicate distances d_p (as labeled). (C) Conduction velocity (CV) corresponding to cases in (A) and (B) for G_gap of (a) 735 nS and (b) 73.5 nS. Baseline ID geometric parameters: diameter d = 5μm, interplicate height h_ip = 3.2μm, fold amplitude A_f = 90 nm.

For strong GJ coupling, cleft depolarization was more synchronized between adjacent cells, reducing the temporal dispersion in ϕ_cleft and similarly synchronizing sodium current (Fig. 4 A). Notably, peak $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$ are more aligned in time. In contrast, for reduced GJ coupling, both ϕ_cleft and sodium currents exhibited greater temporal spread, reflecting impaired synchronization (Fig. 4 B). To understand the influence of individual structural parameters, we examined the effects of varying d_p and d_ip independently. The strong and reduced GJ coupling cases share some similarities. For example, under both GJ coupling conditions, increasing either d_p or d_ip resulted in less negative ϕ_cleft peaks and increased magnitudes of $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$ (thick lines). However, investigation of individual potentials within the cleft and individual $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$ (thin lines) reveals significant temporal spread. Narrow d_p and d_ip tend to result in less synchronous activation within the ID and cleft.

The interplay between $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$ synchronization and magnitude due to d_p and d_ip variation results in a complex dependence on CV, further depending on GJ coupling strength (Fig. 4 C). Stronger GJ coupling supported faster conduction across all geometries, compared with reduced GJ coupling. Further, CV increased with increasing either d_p or d_ip individually. In contrast, for reduced GJ coupling, CV decreased with increasing d_p. However, increasing d_ip increased conduction for narrow d_p whereas slowing conduction for wide d_p.

Influence of ID nanostructure and GJ coupling on ID dynamics and CV

We next more systematically vary ID nanostructural properties, varying d_p, d_ip, and A_f over 96 meshes and both GJ coupling levels, measuring the peak ϕ_cleft (Fig. 5) and CV (Fig. 6) for the corresponding 192 simulations. We find that peak ϕ_cleft exhibits similar qualitative trends under both strong and reduced GJ coupling conditions. For both conditions, peak ϕ_cleft is less negative (i.e., closer to zero) with increasing either d_p and d_ip; that is, wider cleft geometry reduces peak hyperpolarization. Plicate fold amplitude A_f exhibits a more subtle influence: increasing A_f results in marginally more negative peak ϕ_cleft. However, the relationship between peak ϕ_cleft and mesh diameter d is less straightforward. For example, for d_p and d_ip of 70 nm (upper right box in each panel), increasing d results in more negative ϕ_cleft. In contrast, for d_p and d_ip of 10 nm (lower left box in each panel), increasing d results in less negative ϕ_cleft. This suggests a nonmonotonic relationship between diameter and cleft dynamics. Finally, reduced GJ coupling (c.f., top and bottom rows in Fig. 5) results in a less negative peak ϕ_cleft, a consequence of the asynchronous activation of $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$.

Figure 5.

Peak cleft potential as a function of plicate and interplicate distances for different ID mesh geometries. Heatmaps display the peak cleft potential ϕ_cleft for different combinations of plicate distance (d_p) and interplicate distance (d_ip). (A) and (C) show results with no fold amplitude (A_f = 0 nm), and (B) and (D) includes folds with amplitude A_f = 90 nm. G_gap for top row (A and B) of 735 nS and bottom row (C and D) of 73.5 nS. Cleft diameter d is varied within each subpanel as indicated. Color intensity encodes the magnitude of peak cleft potential, with more negative values indicated by darker shades.

Figure 6.

Conduction velocity (CV) as a function of plicate and interplicate distances for different ID mesh geometries. Heatmaps display the peak cleft potential ϕ_cleft for different combinations of plicate distance (d_p) and interplicate distance (d_ip). (A) and (C) show results with no fold amplitude (A_f = 0 nm), and (B) and (D) include folds with amplitude A_f = 90 nm. G_gap for top row (A and B) of 735 nS and bottom row (C and D) of 73.5 nS. Cleft diameter d is varied within each subpanel as indicated. Color intensity encodes the CV magnitude.

We next plot CV for the corresponding 192 simulations (Fig. 6), illustrating pronounced relationship differences between the strong and reduced GJ coupling conditions. As expected, CV is notably larger for stronger GJ coupling, with CV ranging from 25 to 45 cm/s for strong GJ coupling and from 12 to 20 cm/s for reduced GJ coupling. For strong GJ coupling, as in Fig. 4, CV increases for both increasing d_p and d_ip, for all values considered for d and A_f (Fig. 6, top row), indicating that expanded clefts facilitate faster conduction.

In contrast, for reduced GJ coupling, CV exhibits more complex dependencies (Fig. 6, bottom row). Depending on the specific values of mesh diameter d, increasing d_p results in an increasing, decreasing, or inverted U (increasing then decreasing) CV, generally depending on d_ip, and similar trends are observed with increasing d_ip depending on d_p. For both GJ coupling conditions, CV generally slows for increasing mesh diameter d, whereas there is minimal dependence on fold amplitude A_f. Collectively, we find that for strong GJ coupling, cleft hyperpolarization trends in Fig. 5 mirror CV trends, for which more negative peak ϕ_cleft values are associated with slower CV. However, for reduced GJ coupling, the properties governing CV are more complex, so we next investigate this systematically using a neural network sensitivity analysis approach.

Neural network sensitivity analysis of ID and cleft dynamics and CV

We next quantify the influence of different ID geometric factors on ID and cleft dynamics, using a neural network model fit to simulation parameters and measures (Fig. 3). This facilitates the systematic assessment of the sensitivity of ID and cleft measures (Fig. 7) and CV (Fig. 8) to specific ID nanoscale geometric features. As noted above, we simulate two data sets under both strong and reduced GJ coupling. We perform this sensitivity analysis either with all simulation results or two separate analyses for both strong and reduced GJ coupling. Thus, this analysis incorporates measures from all 384 meshes and 768 simulations. Further, for output measures of ID and cleft dynamics that spatially vary within the ID or cleft, respectively, this analysis integrates measures from all 100 nodes within each mesh for each simulation, and thus, it utilizes 76,800 input/output data points. Moreover, as described in the materials and methods section, the credibility of the partial derivative neural-network-based sensitivity analysis critically depends on the fitting accuracy. We quantify the training and testing loss values of the neural network models (Table S1) and plot predicted versus true fitting results (Fig. S4), which collectively support the robustness of the sensitivity analysis.

Figure 7.

Neural network sensitivity analysis of ID and cleft dynamics. Each panel presents the sensitivity of the neural network outputs, ID, and cleft dynamic measures (x-axis) on each input ID mesh and local property features (y-axis). Sensitivity analysis is based on the (a) full simulation data set of both G_gap values and based on the simulation data subset for G_gap of (b) 735 nS and (c) 73.5 nS. Color denotes the sensitivity sign and magnitude, with red and blue indicating a positive and negative influence, respectively.

Figure 8.

Neural network sensitivity analysis of ID mesh properties on conduction velocity (CV). Each panel presents the sensitivity of the neural network output, CV (x-axis), on each ID mesh input feature (y-axis). Sensitivity analysis is based on the (a) full simulation data set of both G_gap values and based on the simulation data subset for G_gap of (b) 735 nS and (c) 73.5 nS. Color denotes the sensitivity sign and magnitude, with red and blue indicating a positive and negative influence on CV, respectively.

In Fig. 7, we specifically quantify the sensitivity of the spatially variable cleft and ID dynamics, specifically the cleft voltage and ion concentration peak magnitudes and both the ID sodium current peak magnitudes and timing, on mesh ID properties (d, d_ip, d_p, A_f, h_ip), spatially variable local nanostructure measures (d_center, ${\sigma }_{center}^{\mathit{min}}$, σ_local, ${\sigma }_{\text{bulk}}^{\min }$, Na_area, Kir_area, NKA_area), and tissue GJ coupling (G_gap). We find that most ID and cleft dynamical measures have similar sensitivities across all three analyses, with generally stronger sensitivity for the analyses for specific GJ coupling levels (Fig. 7 b and c). Inclusion of G_gap in the analysis illustrates a strong negative relationship with t_pre, t_post, and Δt, consistent with increasing GJ coupling driving a shorter delay and more synchronization across sodium current timing (Fig. 7 a), as shown in Fig. 4.

Across all analyses, cleft potential (ϕ) and sodium currents ($I_{\text{Na}}^{\text{pre}}$, $I_{\text{Na}}^{\text{post}}$) were found to be particularly sensitive to variations in mesh diameter (d), interplicate distance (d_ip), plicate distance (d_p), Euclidean distance to center node (d_center), and local conductivity (${\sigma }_{\text{bulk}}^{\min }$). Increasing the mesh diameter (d) leads to a broader spatial distribution of transmembrane current and thus of the potential hyperpolarization across the cleft. As a result, ϕ becomes more negative with increasing d. Increases in either d_p or d_ip increase the conductivity within the cleft and thus result in less negative ϕ, which is also consistent with the positive sensitivity between σ_local and ϕ. Since the periphery of the ID is electrical ground, both d_center and ${\sigma }_{bulk}^{\mathit{min}}$ exhibit a strong dependence on ϕ: closer to the periphery, ϕ is less hyperpolarized, resulting in positive and negative sensitivities on d_center and ${\sigma }_{\text{bulk}}^{\min }$, respectively. As the sodium currents exhibit strong dependence on the local cleft potential ϕ, such that diminished cleft hyperpolarization leads to a reduction in the driving force for sodium influx, we find that $I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$ generally exhibit the opposite sensitivity dependencies as ϕ. Further, as might be expected, increases in the local sodium density (Na_area) directly increase the magnitude of the local sodium currents. Collectively, these results highlight that ID and cleft geometry regulate sodium currents indirectly by modulating the cleft potential, with additional regulation via channel density.

We next perform a similar sensitivity analysis on CV (Fig. 8). We note that, as CV is a global output measure of function and spatially variable within the ID or cleft, this analysis is limited to globally defined ID mesh properties. Combining data sets from both strong and reduced GJ coupling conditions (Fig. 8 a), we find that G_gap is quantified as the most influential factor, consistent with GJ coupling exhibiting a dominant role in determining CV (as in Figs. 4 C and 6). However, the sensitivity of CV to structural properties differed substantially between the strong and reduced GJ coupling. Under strong coupling conditions (Fig. 8 b), CV exhibited strong positive sensitivity to d_p and d_ip, whereas both A_f and interplicate height h_ip showed a more moderate negative influence. In contrast, under reduced GJ coupling conditions (Fig. 8 c), d_p and d_ip on CV exhibit a negative sensitivity, with a weaker dependence on d_ip compared with d_p. This analysis is consistent with the context-dependent modulation of CV in Fig. 6. For both conditions, d_p exhibits a stronger sensitivity (i.e., a darker color) compared with d_ip. Interestingly, we also find that the relative sensitivity magnitudes for d_p and d_ip are fairly similar, comparing between strong and reduced GJ coupling, consistent with ID structural modulation impacting CV to a similar relative extent independent of GJ coupling. Note this is consistent with the relative changes in CV values in Fig. 6, that is, both strong and reduced GJ coupling exhibiting an approximately 40% range between the fastest and slowest CV measures for the same ID mesh properties.

We further investigate how CV depends on the properties of the ID nanostructure, performing the same neural network model sensitivity analysis based on distribution measures of the ID mesh, specifically the maximum, minimum, mean, and standard deviation of the previously defined spatially varying nanoscale properties (Fig. 9). Similar to Fig. 8 a, across data, G_gap is the most influential factor in determining CV. Also consistent with the analysis of mesh properties in Fig. 8, several distribution measures exhibit the opposite sensitivities between strong GJ coupling and reduced GJ coupling, such as mean(σ_local) and $std\left({\sigma }_{center}^{\mathit{min}}\right)$. In contrast, max(d_center) and mean(d_center) both exhibit strong negative sensitivities for both GJ coupling conditions.

Figure 9.

Neural network sensitivity analysis of ID nanostructural properties on conduction velocity (CV). Each panel presents the sensitivity of the neural network output, CV (x-axis), on each input ID nanostructural distribution measurement features (y-axis). Sensitivity analysis is based on the (a) full simulation data set of both G_gap values and based on the simulation data subset for G_gap of (b) 735 nS and (c) 73.5 nS. Color denotes the sensitivity sign and magnitude, with red and blue indicating a positive and negative influence on CV, respectively.

Finally, to demonstrate the predictive accuracy of this neural network analysis approach, we evaluated the model performance by generating 20 ID meshes with parameters randomly sampled within ranges for d, d_p, d_ip, A_f, and h_ip, as in Table 1. The six CV-trained neural networks were then used to predict CVs for tissues with these 20 ID meshes, and the predictions were then compared against the ground-truth CV obtained from direct tissue simulation (Fig. 10). Overall, the neural network trained on ID mesh properties outperformed the one based solely on mesh nanostructure, and the inclusion of G_gap substantially improved prediction accuracy. Furthermore, prediction performance for strong GJ coupling was generally superior to that with reduced GJ coupling. However, all neural networks predicted CV with high accuracy, with the mean absolute relative errors generally at most 6% or lower.

Figure 10.

Validation of neural network predictions for conduction velocity (CV) using ID mesh properties and mesh nanostructure distribution models. Each panel illustrates predicted versus true CV, based on predictions from the neural network model trained on for all data (left) and G_gap of 735 nS (middle) or 73.5 nS (right), for 20 test cases generated by randomly sampling ID mesh parameters (d, d_p, d_ip) within physiologically reasonable ranges. Predictions from each set of networks were obtained and averaged using fivefold cross-validation and compared against the corresponding true CV values from tissue simulations incorporating the ID meshes. Coefficients of determination (R²) and mean absolute percentage error (MAPE) are indicated in each panel.

Discussion

In this study, we performed a comprehensive and systematic quantitative analysis of the relationship between ID heterogeneity and cardiac electrical conduction, focusing on two key aspects: 1) how multiple geometric and nanostructural features influence conduction and 2) how these effects vary between strong and reduced GJ coupling regimes. Overall, we find that under strong GJ coupling (high G_gap), varying certain geometric features, such as increasing d_p and d_ip, significantly enhanced the synchrony of potentials in the intercellular cleft and consequently the pre- and postjunctional sodium currents and, thus, enhanced conduction. In contrast, for reduced GJ coupling (low G_gap), although the same structural changes often slowed conduction, the relationship in fact exhibits a complex interdependence, in which the dependence of CV on interplicate and plicate membrane expansion each influences the other. Thus, ID nanostructure exerts distinct influences on cardiac conduction depending on the regime.

Beyond elucidating these complex relationships, our systematic analysis gives rise to physiological hypotheses regarding the functional role of ID structural plasticity. We hypothesize that the variable ID nanostructure provides additional “knobs” or “dials” to dynamically tune cardiac conduction in a manner that depends on the prevailing cell-cell coupling regime. Although GJ coupling strength exhibits the greatest sensitivity (Fig. 8), ID nanostructure and ion channel organization provide multiple additional and potentially orthogonal dimensions for conduction regulation: For conditions with strong GJ coupling, an expanded cleft is advantageous, reducing cleft hyperpolarization and mitigating self-attenuation. Conversely, for reduced GJ coupling, ID structural modulation that enhances ephaptic coupling can compensate for the loss of direct communication. Critically, in both coupling regimes, conduction can be modified by both overall ID properties (Fig. 8) and local nanostructure (Fig. 9).

Previous modeling efforts have established important groundwork in elucidating the role of the ID in cardiac conduction. In particular, prior computational studies demonstrate that preferential localization of Na⁺ channels within the ID can preserve conduction for conditions of reduced GJ coupling via enhanced ephaptic interactions in the narrow cleft (3,4,5). Although identifying a functional role of ephaptic coupling, these early studies generally oversimplified the ID structural details and ion channel distribution. Later models were developed based on a more detailed finite element mesh representation of the ID structure (10,14,28). Ivanovic and Kucera developed a high-resolution 3D FEM of two longitudinally adjacent cardiomyocytes to investigate sodium channel clusters situated within narrowed perinexi. However, the high computational cost constrained the investigation to a simplified two-cell configuration and simplified ion channel representation (28). A subsequent study from these authors incorporated structural tortuosity into the two-cell configuration (14). Our prior work developed the ID nanoscale geometrical pipeline based on transmission electron microscopy data, which could be subsequently reduced and integrated into tissue-scale simulation; however, our initial approach did not account for GJ heterogeneity or the contributions of other ion channels, such as K⁺ channels and the Na/K-ATPase within the ID (10).

Thus, the model employed in this study represents a refined version of the 3D finite element mesh framework, incorporating previously omitted ion channels and GJ heterogeneity, and we performed a systematic investigation of the heterogeneous behavior of the ID across a wide range of ID structural properties. Further, sensitivity analysis based on a multilayer perceptron (MLP) neural network enabled quantification of the impact of individual structural parameters on electrophysiological properties and CV. Notably, our study is the first, to our knowledge, to comprehensively examine multiple parametrically defined ID structural properties, in the setting of both strong and reduced GJ coupling, and integrate machine learning interpretation within a single framework, thereby offering a more refined and systematic understanding of the ID structure-function relationship.

Our group has previously demonstrated that GJ and ephaptic coupling operate in tandem to maintain robust conduction under varying conditions (6,10). For strong GJ coupling conditions, where resistive gap junctional current predominates, cleft widening both enhances sodium current synchronization and also mitigates self-attenuation (i.e., the reduction in sodium current driving force from cleft hyperpolarization (5)). In contrast, for reduced GJ coupling, where ephaptic mechanisms dominate, wider clefts can both reduce self-attenuation and facilitate more uniform lateral spread of depolarizing currents within the ID cleft space, improving the synchronization of activation between cells, leading to potentially conduction enhancing or slowing, depending on the ID region. Similar regime-dependent reversals have been reported in both experimental and computational studies (27,28), supporting the notion that the same structural perturbation can either impair or enhance synchronization depending on the underlying conduction mode.

Our comprehensive parameter sweeps also provide a mechanistic framework for interpreting known variations in ID nanostructure between cardiac chambers and in disease states. We previously illustrated that significant structural differences exist between atrial and ventricular IDs, with atrial IDs exhibiting a smaller diameter compared with ventricular IDs and featuring wider intercellular cleft distances (d_p, d_ip) (12). In the context of this study (Table 1), ID meshes with a smaller diameter and wider clefts are thus more representative of atrial tissue, whereas those with a larger diameter and narrower clefts are more comparable to ventricular tissue. Our findings suggest that these chamber-specific architectures may reflect a mechanism for functional tuning in different conduction regimes. The structurally complex and tightly packed ventricular ID appears optimized for robust conduction under the strong GJ coupling typical of the ventricles. Conversely, the inherently wider clefts of atrial IDs may render this chamber more vulnerable to conduction slowing when GJ coupling is compromised, as in pathologies such as atrial fibrillation, which is associated with further ID remodeling and cleft expansion (24). Our model predicts that in a reduced GJ coupling state, this pathological widening would severely impair conduction, linking chamber-specific nanostructure, the differential susceptibility to arrhythmias (such as the atrial predisposition to arrhythmia), and distinct electrophysiological properties.

Further, our framework allows for the dissection of concurrent alterations that may occur in disease, simulating specific combinations of perturbed ID structure and reduced GJ coupling often occurring in diseases. Sensitivity analysis can also highlight the extent to which ID structural disruption or GJ coupling loss is the more critical driver of conduction slowing in a specific pathological context. Further, this approach can prioritize research avenues. For example, our finding that d_p is a dominant factor regulating conduction suggests that experimental models targeting proteins that regulate the plicate region structure may yield more significant electrophysiological consequences than those affecting other ID features. Furthermore, model predictions can explain seemingly contradictory experimental results: if different studies report conflicting effects of a particular structural change, our framework suggests that the baseline level of GJ coupling could be an underlying explanatory variable, thus providing a quantitative map to guide the interpretation of new experimental data.

Although the geometric parameters discussed above describe micrometer-scale features of the ID membrane and cleft, the nanostructural organization within this space, specifically the spatial distribution and variability of cleft conductances (e.g., ${\sigma }_{bulk}^{\mathit{min}}$, σ_local) and the membrane area occupancy of key ion channels such as Na⁺ channels, K⁺ channels, and Na/K-ATPase, play a critical role in governing conduction. Our sensitivity analysis reinforces this concept, identifying both electrical and geometric metrics, such as ${\sigma }_{bulk}^{\mathit{min}}$ and d_center, as critical modulators of local cleft dynamics (Fig. 7). The physical implication of these metrics is direct: ${\sigma }_{bulk}^{\mathit{min}}$, representing the electrical path from within the cleft to the bulk, dictates cleft potentials and thus local potential gradients. These potentials and gradients in turn are directly and bidirectionally coupled to the electrical driving force for sodium influx, modulating the magnitude and synchronization of the local sodium currents ($I_{\text{Na}}^{\text{pre}}$ and $I_{\text{Na}}^{\text{post}}$). Similarly, nodes geometrically closer to the periphery (i.e., larger d_center) generally are more directly coupled to the bulk space, similarly modifying cleft potential and sodium currents. Thus, the sensitivity analysis establishes a direct link between the tortuosity of the nanostructure and the mechanisms governing ephaptic coupling and thus conduction. Further, these nanoscale electrical properties govern the fine-scale modulation of local electric fields and current distributions, such that the sensitivity analysis (Fig. 9) reveals distinct patterns of influence under strong versus reduced GJ coupling, suggesting that variations in channel distribution heterogeneity may either reinforce or counteract the effects of ID and cleft geometry. Sensitivity analysis highlights several nanostructural parameters that strongly modulate CV (Fig. 9), with distinct directionality and magnitude under strong and reduced GJ coupling conditions (e.g., distribution measures of ${\sigma }_{center}^{\mathit{min}}$, ${\sigma }_{bulk}^{\mathit{min}}$, σ_local, and d_center). Interestingly, sensitivity analysis of all data combined identifies G_gap as the most influential factor. Importantly, as CV is a tissue-scale measure, the distribution measures over the mesh nanoscale properties inevitably do not fully capture the relationships between ID nanoscale structure and conduction. As such, this analysis should be viewed as an initial attempt to predict CV from ID nanostructure and to explore the underlying mechanisms. Notably, as shown in Fig. 10, prediction accuracy for randomly generated mesh property data exceeds that obtained from mesh nanostructure data.

The limitations in our development of the ID and cleft FEM mesh, noted in our prior study (10), are inherent in this study as well. A key limitation lies in the physical formulation of the ionic dynamics within the nanoscale cleft. Although our model incorporates nanometer-scale geometric detail, it does not solve the full Poisson-Nernst-Planck (PNP) equations. Consequently, our approach assumes local electroneutrality within the cleft and does not explicitly resolve phenomena such as the formation of the Debye layer at the membrane-cleft interface. As demonstrated by Jaeger and Tveito in their detailed modeling of the intercellular cleft, solving the complete PNP system can reveal significant deviations from electroneutrality within a few nanometers of charged surfaces, leading to substantial local electric fields and ion concentration changes not captured by simpler models (31). Additionally, our FEM mesh generation does not capture the natural membrane widening that occurs in the GJ-adjacent membrane region, the perinexus. Using high-resolution, detailed modeling of adjacent cells, Ivanovic and Kucera have nicely illustrated that the width of both the perinexus and the surrounding membrane space both modulates local sodium currents and thus the efficacy of cell-cell electrical transmission (28). However, the decision to use a reduced electrical network model was a necessary trade-off; solving the full PNP equations and representing detailed perinexal structure across the entire complex geometry of the ID for tissue-level simulations remains computationally prohibitive. Therefore, although our model represents nanometer-scale resolution in its geometry, nanoscale detail is not fully captured in the underlying ionic dynamics and local membrane structure.

Expanding in spatial scale, the model formulation is also a simplified approximation of the intrinsically complex architecture of whole heart tissue, despite including significant subcellular detail in its representation (10). Specifically, the model is a one-dimensional representation of cardiac tissue, which cannot fully capture the three-dimensional structure and behavior of the heart. This point is especially important in pathological settings, where phenomena such as the lateralization of GJs fundamentally alter conduction patterns that a 1D model cannot capture. A primary goal for our future work is to extend this framework by integrating our detailed ID network model into 2D and 3D tissue simulations, to investigate how ID nanostructure and pathological remodeling, such as GJ lateralization, govern anisotropic conduction and contribute to arrhythmogenesis in a more physiological model. However, we emphasize that these simplifications were necessary to perform the comprehensive and systematic quantitative framework for linking ID structural heterogeneity, spanning both micrometer-scale cleft geometry and nanometer-scale subcleft organization, to key electrophysiological outcomes. By integrating finite element modeling with neural network-based sensitivity analysis, we were able to delineate the relative importance of individual structural parameters under distinct GJ coupling regimes, offering mechanistic insights into how the same geometric or nanostructural change can exert opposite effects depending on the prevailing mode of conduction.

Our model also does not explicitly incorporate mobile or stationary ionic buffers within the cleft. To our knowledge, buffering capacity in this nanoscale extracellular space has not been measured. It has been established that Ca²⁺ in the cleft can bind with cell adhesion molecules from the cadherin families (49), and prior work has shown extracellular Ca²⁺-dependent modulation of ID membrane separation (27,50). However, the dynamics of Ca²⁺-adhesion protein binding-associated ID separation have not been measured. As such, it is not clear whether the beat-to-beat dynamics of cleft Ca²⁺ are an additional mechanism of ID structural regulation. Nevertheless, inclusion of these additional regulatory pathways for ID structure would be a valuable model refinement for future work, in particular in the setting of dysfunction Ca²⁺ handling.

A limitation of our sensitivity analysis is that the MLP neural network was fit based on steady-state (time- and space-independent) conditions. However, ID and cleft dynamics and cardiac conduction are complex spatiotemporal processes. As this initial approach does not capture the temporal dependencies inherent in this system, our analysis utilized peak values in Fig. 7 and aggregate calculations for mesh nanostructure in Fig. 9. Developing more advanced algorithms capable of modeling such complex behavior and predicting ID and cleft dynamics, in addition to cardiac conduction, is an important avenue for future work. Finally, it is important to note that although the present study primarily relies on computational modeling to elucidate how nanoscale variations in ID structure modulate local ionic currents and conduction, recent experimental evidence strongly supports the feasibility of the predicted ephaptic mechanisms. Using dual cell patch-clamp experiments, Waaben and colleagues recently have demonstrated for the first time that action potentials can be transmitted between isolated myocytes purely through ephaptic coupling, specifically in the absence of measurable gap junctional current (51). As an important and critical next step, while technically challenging, the application of experimental perturbations, such as adhesion disrupting peptides (15), applied in concert with similar patch-clamp studies, could be used to test and validate computational predictions of the present study.

Conclusion

In conclusion, although the present formulation inevitably simplifies the complex, three-dimensional architecture of cardiac tissue and the nanoscale of the myocyte junctional geometry, this study nonetheless advances the understanding of ID structure-function relationships and the relative role in physiological and pathological conduction. The analytical approach developed here serves as a foundation for future work aimed at incorporating full 3D tissue models, dynamic cleft behaviors, and advanced spatiotemporal learning algorithms to capture the inherently nonlinear nature of cardiac conduction. Ultimately, such refinements may enable more accurate prediction of conduction disturbances in disease and guide the design of targeted interventions at the nanoscale.

Acknowledgments

This project was supported by funding from the National Institutes of Health R01HL169610 (S.H.W.) and R01HL165751 (S.H.W.) and American Heart Association Career Development Award 25CDA1436439 (N.M.).

Author contributions

R.S., N.M., and S.H.W. designed the research. R.S. and N.M. carried out all simulations and analyzed all data. R.S., N.M., and S.H.W. wrote the article.

Declaration of interests

The authors declare no competing interests.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the authors used ChatGPT in order to improve the readability of the manuscript text. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

Editor: Eleonora Grandi.