Stability of spatially discordant repolarization alternans in cardiac tissue

Abstract

Cardiac alternans, a period-2 behavior of excitation and contraction of the heart, is a precursor of ventricular arrhythmias and sudden cardiac death. One form of alternans is repolarization or action potential duration alternans. In cardiac tissue, repolarization alternans can be spatially in-phase, called spatially concordant alternans, or spatially out-of-phase, called spatially discordant alternans (SDA). In SDA, the border between two out-of-phase regions is called a node in a one-dimensional cable or a nodal line in a two-dimensional tissue. In this study, we investigate the stability and dynamics of the nodes and nodal lines of repolarization alternans driven by voltage instabilities. We use amplitude equation and coupled map lattice models to derive theoretical results, which are compared with simulation results from the ionic model. Both conduction velocity restitution induced SDA and non-conduction velocity restitution induced SDA are investigated. We show that the stability and dynamics of the SDA nodes or nodal lines are determined by the balance of the tensions generated by conduction velocity restitution, convection due to action potential propagation, curvature of the nodal lines, and repolarization and coupling heterogeneities. Our study provides mechanistic insights into the different SDA behaviors observed in experiments.

Spatially discordant alternans (SDA) in cardiac tissue is a spatiotemporal pattern in which the action potential duration exhibits a temporally period-2 and spatially out-of-phase behavior. SDA has been widely investigated in both computer modeling and experiments in the last two decades. The leading theory for SDA implies that conduction velocity restitution (CVR) is a parameter required for the formation and stability of SDA, which has been validated by experiments. However, there are also experimental observations that cannot be explained by this theory. On the other hand, SDA can be induced in cardiac tissue in the absence of CVR, called non-CVR-induced SDA. The non-CVR-induced SDA can be a candidate mechanism for the experimentally observed SDA that cannot be explained by the CVR-induced SDA mechanism or SDA in clinical settings where CVR may be absent. This study performs theoretical treatments to investigate the stability of the non-CVR induced SDA using models of amplitude equations and coupled map lattices, and the theoretical results are compared with those from the ionic model.

I. INTRODUCTION

Cardiac systems exhibit rich nonlinear dynamics.^1–3 Alternans, corresponding to a period-2 behavior, is the most widely investigated dynamical behavior in cardiac systems with a great number of publications in theoretical, computational, experimental, and clinical studies.^1,3–12 The importance of alternans is that pulsus alternans and electrical (or T-wave) alternans are widely known as precursors of lethal ventricular arrhythmias and sudden cardiac death.^4,5,10–12 A potential mechanism linking alternans to arrhythmias is spatially discordant alternans (SDA).^13–16 SDA is a dynamical phenomenon in cardiac tissue in which action potential duration (APD) exhibits a temporally period-2 but spatially out-of-phase (or antiphase) behavior. The border between two neighboring regions is called a node in a one-dimensional (1D) cable or a nodal line in a two-dimensional (2D) tissue (Fig. 1). The out-of-phase APD alternans results in large APD gradients in the nodal region, which are prone to conduction block and formation of spiral waves^13,15 or generation of arrhythmia triggers.^17–19

FIG. 1.

(a) Schematic diagrams of the two pacing protocols: local pacing (upper) and global pacing (lower). Arrows indicate the locations of pacing. (b) An APDR curve obtained from the ionic model using the S1S2 pacing protocol in which the cell is paced with an S1 pacing period T = 1000 ms and then a variable S1S2 interval is used to vary DI and thus the following APD (see the inset for definitions). (c) Bifurcation diagram showing APD vs T for the fast-pacing-induced APD alternans in a single cell. (d) Bifurcation diagram showing APD vs T for the I_to-induced APD alternans in a single cell. (e) An example of SDA node in a 1D cable of the ionic model with the I_to-induced APD alternans. Global pacing with T = 700 ms. Upper: 3D plot of time (t)–space (x)–voltage (V) for two consecutive beats. Lower: APD vs x for two consecutive beats. The intersection point forms a node. (f) An example of SDA nodal ring in a 2D tissue of the ionic model with the I_to-induced APD alternans. Global pacing with T = 700 ms. Red and blue are surface plots of APD from two consecutive beats. The intersection of the two surfaces forms a nodal ring.

The prevailing theory for the mechanisms of SDA is that SDA is a result of the interaction of cellular APD alternans and conduction velocity (CV) restitution (CVR).^15,20–22 Cellular APD alternans can be caused by instabilities originating from voltage,^7,8,23 intracellular calcium (Ca²⁺),^24–28 or the interaction of the two.^29–31 CVR is an action potential conduction property in which CV changes as the diastolic interval (DI) changes due to incomplete recovery of the sodium (Na⁺) current or changes of excitability.^{14,15,32–34} If there is no CVR, i.e., CV remains constant during conduction, then APD will alternate in-phase throughout the entire tissue, forming spatially concordant alternans (SCA). If there is CVR, then CV is variable along the conduction pathway, which will then cause variation in DI and APD along the conduction path, forming SDA^14,15 or promoting more complex dynamics.^32–34 Experimental evidence has been shown to support this theory.^35–39 However, there are other experimental results that do not support this theory.^39–41 Moreover, CVR may only be present under fast pacing rates^{13,14,35–42} at which Na⁺ current recovers incompletely. In many clinical settings, alternans is usually observed in normal or slow heart rates, such as in patients with long QT syndromes,^11,43–45 Brugada syndrome,^46–49 or heart failure.^50,51 At normal or slow heart rates, the Na⁺ current may fully recover at the end of each heartbeat and thus there is no CVR. Therefore, other mechanisms of SDA have to exist if SDA is indeed responsible for arrhythmogenesis in clinical settings.

Although previous theoretical analyses and computer simulations have shown that CVR is required for SDA formation, in a recent computer simulation study,⁵² we showed that SDA could be induced by heterogeneous initial conditions or tissue heterogeneities in the absence of CVR. We call the former as CVR-induced SDA and the latter as non-CVR-induced SDA. The non-CVR-induced SDA can be a candidate mechanism for the SDA observed in experiments that cannot be explained by the CVR-induced SDA mechanism, or for SDA in clinical settings where CVR is absent due to normal or slow heart rates. For the non-CVR-induced SDA, the stability and dynamics of the nodes or nodal lines are determined by repolarization heterogeneities, nodal line curvature, and pacing protocols. However, in the previous study,⁵² we carried out simulations of 1D cable and 2D tissue ionic models using the 1991 Luo and Rudy (LR1) action potential model⁵³ and the more physiologically detailed rabbit ventricular myocyte action potential model by Mahajan et al.,⁵⁴ which are too complex to be used for analytical treatments to rigorously elucidate the underlying mechanisms. For example, we showed that multiple nodes could be accommodated in a finite-length 1D cable, but it was unclear what determines the minimum node distance. We also showed that in homogeneous 2D tissue, the area inside a nodal ring decayed linearly with time, but it was unclear what determines this nodal ring dynamics.

Analytical work using an amplitude equation (AE) approach by Echebarria and Karma^21,55 has provided a rigorous understanding of the mechanism of the CVR-induced SDA; however, similar analyses for the non-CVR-induced SDA are still needed. In this study, we perform theoretical analyses and computer simulations of both simple and complex models to understand the mechanisms of non-CVR-induced SDA. We use this combined approach to reveal the roles of CVR, convection due to action potential propagation, curvature of the nodal lines, and repolarization and coupling heterogeneities in the formation, stability, and dynamics of SDA. We use three types of models: an AE model, a coupled map lattice (CML) model, and an ionic model. The AE model is a single partial differential equation (PDE) describing the spatiotemporal dynamics of the alternans amplitude, which allows us to perform rigorous analytical treatments. The CML model is composed of coupled iterated maps describing APD as a function of its value in the previous beat, which allows us to independently vary certain physiological parameters, such as CVR and APD restitution (APDR), to examine their roles in SDA dynamics. The ionic model is the traditional reaction-diffusion equation with cells described by the LR1 action potential model with modifications. The simulation results from the ionic model are compared with the predictions of the AE and CML models.

II. THEORETICAL AND COMPUTATIONAL MODELS

A. Pacing protocols

We use two pacing protocols [Fig. 1(a)], i.e., local pacing and global pacing, to investigate the SDA dynamics. In the local pacing protocol, the tissue is periodically paced from one end (1D cable), one side (2D tissue), or a point, which results in action potential conduction from the pacing site to the rest of the tissue. This is the traditional pacing protocol widely used in computer simulations. In the global pacing protocol, the cells in the tissue are paced simultaneously so that there is no action potential conduction. The advantage of this pacing protocol is that it allows us to perform certain analytical treatments of the AE model and the CML model. The physiological relevance of this pacing protocol is discussed in Sec. IV.

B. The AE model

The AE model consists of a single PDE describing the spatiotemporal evolution of the alternans amplitude, which was originally derived by Echebarria and Karma^21,55 for APD alternans dynamics in a 1D cable paced from one end. The equation is

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3-{\int }_0^x\frac{dy}{\mathrm{\Lambda }}\mathrm{\Delta }a(y)-\delta \frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }x}+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2},$$ (1)

where T is the pacing period and $t\equiv nT$, with n being the beat number. $\mathrm{\Delta }a(x,t)$ is the amplitude of alternans defined as $\mathrm{\Delta }a(x,t)={(-1)}^n[(a_{n+1}(x)-a_n(x)]/2$, in which the pre-factor ${(-1)}^n$ maintains the sign of $\mathrm{\Delta }a(x,t)$ during alternans. $a_n$ is the APD of the n^th beat [see Fig. 1(b) for an example of APD and APDR]. α and β, which are related to the slope of the APDR curve, are the parameters determining the amplitude of the alternans, Λ is a parameter related to the slope of the CVR curve, δ describes the strength of the convection effect (see the Appendix for a more detailed explanation of this term), and ξ describes the strength of diffusive coupling. The default parameters for the AE model are $\alpha =0.1$ and $\beta =0.0001\alpha$, and $\xi =0.235\text{cm}$. It will be mentioned if different values are used. More detailed information on definitions of these parameters and numerical simulations of Eq. (1) are presented in the Appendix.

C. The CML model

CML models have been widely used for investigating spatiotemporal dynamics of nonlinear systems.^56–59 In a previous study,⁶⁰ we developed a CML model to describe the spatiotemporal dynamics of APD in a 1D cable. Here, we use the same CML model to investigate the SDA dynamics in both 1D cable and 2D tissue. The CML model describes the spatiotemporal dynamics of APD denoted as $a_n(x)$ [Note: the AE model describes those of the APD alternans amplitude denoted as $\mathrm{\Delta }a(x,t)$]. Specific APDR and CVR functions are used for the CML model with ${\tau }_a$ controlling the slope of APDR and ${\tau }_{\theta }$ controlling the slope of CVR. A detailed description of the CML model and the default values of the parameters are presented in the Appendix.

D. The ionic model

Cardiac conduction is described by a PDE of membrane potential,² a reaction-diffusion type model. Besides the PDE, this type of model also includes many ordinary differential equations describing the gating variables of the ionic currents and variables for ion pumps. The mathematical formulations of the ionic currents and pumps as well as the number of variables are different in different action potential models. In this study, we use the LR1 action potential model⁵³ with modifications to generate two different types of APD alternans. The first type of APD alternans is the fast pacing-induced alternans [Fig. 1(c)], in which APD alternans occurs when the pacing rate is high (pacing period T is short). The second type of APD alternans, which can occur in a much wider range of heart rates [Fig. 1(d)], is caused by the presence of the transient outward K⁺ current (I_to). The examples of a node in a 1D cable [Fig. 1(e)] or a nodal ring in a 2D tissue [Fig. 1(f)] are generated from the second mechanism of alternans. Details of modifying the LR1 action potential model to generate the two types of alternans are described in our previous study.⁵² More detailed information on the LR1 model and the PDEs for 1D and 2D tissue models are presented in the Appendix.

III. RESULTS

A. SDA formation and dynamics in a homogeneous tissue

The goal of this section is to investigate how CVR, convection, and nodal line curvature affect the formation, stability, and dynamics of SDA in a homogeneous tissue. We first investigate node stability and node spacing when multiple nodes are present in a 1D cable in the absence of conduction. We then investigate node formation and stability in a 1D cable in the presence of conduction. Finally, we investigate the effects of curvature on nodal ring dynamics in a 2D homogeneous tissue.

1. Node stability and spacing in the absence of conduction in a 1D cable

Here, we investigate the node stability, the minimum cable length to accommodate a single node, and the node spacing in multi-node solutions in a homogeneous cable under global pacing. The SDA nodes are induced by heterogeneous initial conditions.

We first use the AE model to investigate SDA node dynamics. Under global pacing, there is no conduction, and thus the terms related to conduction are removed from the equation. Then, Eq. (1) becomes

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2},$$ (2)

which is simply the real Ginzburg–Landau equation⁶¹ or the bistable cable equation.^62,63 When $\alpha <0$, the only uniform solution is $\mathrm{\Delta }a(x)=0$, indicating that there is no alternans. When $\alpha >0$, there are three spatially uniform solutions: $\mathrm{\Delta }a(x)=0$ and $\mathrm{\Delta }a(x)=\pm \sqrt{\frac{\alpha }{\beta }}$. $\mathrm{\Delta }a(x)=0$ is unstable, and the eigenvalues of its Fourier modes can be obtained using Eq. (2), which is

$${\lambda }_k=\frac{\alpha }{T}-\frac{{\xi }^2}{T}{k}^2,$$ (3)

where k is the wave number. The other two solutions are spatially uniform alternans states, i.e., SCA, which are always stable since

$${\lambda }_k=-\frac{2\alpha }{T}-\frac{{\xi }^2}{T}{k}^2.$$ (4)

Note that for a larger α, the solution of $\mathrm{\Delta }a(x)=0$ is more unstable, but the SCA states $\left(\mathrm{\Delta }a(x)=\pm \sqrt{\frac{\alpha }{\beta }}\right)$ are more stable or the cellular alternans is more stable.

Besides the spatially uniform solutions (i.e., SCA), spatially non-uniform solutions (i.e., SDA) of Eq. (2) also exist. For an infinite-length cable $x\in (-\mathrm{\infty },+\mathrm{\infty })$, an analytical solution can be obtained as^62,63

$$\mathrm{\Delta }a(x)=\sqrt{\frac{\alpha }{\beta }}\text{tanh}\left(\sqrt{\frac{\alpha }{2{\xi }^2}}x\right),$$ (5)

$x=0$ (where $\mathrm{\Delta }a=0$) is the SDA node. Equation (2) is a symmetric bistable equation. If the bistable equation is asymmetric, the solution becomes^62,63 $\mathrm{\Delta }a(x,t)=\sqrt{\frac{\alpha }{\beta }}\text{tanh}\left[\sqrt{\frac{\alpha }{2{\xi }^2}}(x-ct)\right]$ with a nonzero c. This implies that the node drifts with a nonzero velocity c with any asymmetry of the bistable equation. Therefore, the solution [Eq. (5)] is stability neutral.

The exact solution [Eq. (5)] is obtained for an infinitely long cable.^62,63 However, for a finite-length cable, it is unclear if such exact solutions can be obtained. Moreover, Eq. (5) is a single-node solution, it is unclear if such exact solutions of multiple nodes can be obtained. Here, we use numerical simulations of the AE model to address the following question: can stable single- or multiple-node solutions exist in a finite length cable?

To initiate a node, we use a heterogeneous initial distribution of $\mathrm{\Delta }a(x,0)$. If there is no node in the initial condition [i.e.,$\mathrm{\Delta }a(x,0)>0$ or $\mathrm{\Delta }a(x,0)<0$ for all x], the system always approaches the SCA solution. When the cable is long enough, a node forms if the initial distribution exhibits a node independent of the magnitude of $\mathrm{\Delta }a(x,0)$ [Fig. 2(a)]. If the initial position of the node is exactly at the middle point, the node remains at this position in the whole simulated period, i.e., 10⁵ pacing beats [Fig. 2(b)]. For small α values [the alternans is less stable, see Eq. (4)], the node drifts away if the initial position is away from the center [Fig. 2(b)]. The node will eventually drift off the cable as long as the simulation time is long enough, indicating that the SDA solution is unstable. The drifting speed depends on α, and the node is less unstable for a larger α [Fig. 2(c)]. Therefore, for a very large α, the node appears to be stable within the 10⁵ pacing beats in a wide range of initial node positions [gray region in Fig. 2(d)]. The gray region increases as α increases. However, it is unclear whether the node is truly stable or it will become unstable for an even longer pacing time. Based on the following observations: (1) in the infinite-length cable, the exact solution [Eq. (5)] is stability neutral; (2) in a finite-length cable with small α [Figs. 2(b) and 2(c)], the node can stay without drifting only when the node is exactly at the middle point of the cable; and (3) increasing α slows the drifting speed, we conclude that in the case in Fig. 2(d), the node is likely very mildly unstable or at most stability neutral.

FIG. 2.

Node stability and spacing under global pacing in the AE model. (a) Plots of $\mathrm{\Delta }a(x)$ at different time points. Black: initial condition: $\mathrm{\Delta }a(x)=0.1\text{ms}$ for x ≤ 5 cm and $\mathrm{\Delta }a(x)=-0.1\text{ms}$ for x > 5 cm; red: n = 14; blue: n = 16; and green: n = 10⁵. α = 0.05 and T = 295 ms. (b) Node position vs n for a single node at the center of the cable (black line) and away from the center (colored lines). α = 0.05 and T = 295 ms. Inset shows the final node position away from the center (at 10⁵ beats) vs the initial position. Note that we only show node drift for nodes in the left side from the center, the same drift occurs for nodes in the right side since the system is symmetric. We also plot the data this way for the same type of panels in this figure and in Figs. 3 and 4. (c) Node position vs beat number for different α values (0.1, 0.039, 0.037, and 0.035 from black to green) for the same initial node position (0.0125 cm away from the middle of the cable). The inset shows the final node position (away from the middle point at 10⁵ beats) vs α. (d) Node position vs n for α = 1. Gray marks the region in which the node is stable within the 10⁵ beats. Nodes outside the gray region (colored lines) drift away or disappear. (e) Steady-state solutions $[\mathrm{\Delta }a(x)]$ in a 15 cm cable showing up to six nodes for α = 0.1. Black: 1 node; red: 2 nodes; blue: 3 nodes; green: 4 nodes; magenta: 5 nodes; and cyan: 6 nodes. The initial conditions are set as in (a) but with different spatial periodicities to obtain the different number of nodes. (f) L_min vs α from Eq. (6).

A cable can accommodate different numbers of nodes [Fig. 2(e)]. The maximum number of nodes in a fixed length cable depends on α and ξ. One can use Eq. (3) to estimate the maximum number of nodes that a cable can accommodate or the minimum length to exhibit one node. The maximum wave number of the unstable spatial modes ${\lambda }_k>0$ is $k_{\text{max}}=\sqrt{\frac{\alpha }{{\xi }^2}}$. This corresponds to the minimum cable length to exhibit a single node under the no-flux boundary condition, which is

$$L_{\text{min}}=\frac{\pi }{k_{\text{max}}}=\pi \sqrt{\frac{{\xi }^2}{\alpha }}.$$ (6)

We plot $L_{\text{min}}$ vs α from Eq. (6) in Fig. 2(f). Based on this relation, one can obtain the maximum number of nodes for a fixed length cable. For example, for $\alpha =0.1$, $L_{\text{min}}\approx 2.33\text{cm}$, a 15 cm cable can exhibit a maximum of six nodes [Fig. 2(e)] since $\frac{15}{L_{\text{min}}}=6.4<7$.

The node behaviors in the CML model (Fig. 3) are almost identical to those in the AE model. That is, the node stays without drifting in the entire simulation time only when it is in the middle point of the cable. Otherwise, it drifts with a drifting speed depending on the alternans stability (described by τ_a). The number of nodes that can be accommodated in a cable also depends on the alternans stability. Similarly, we can also predict the minimum cable length for a single node by calculating the eigenvalues of the Fourier modes of the unstable uniform state, which is⁶⁰

$${\lambda }_k=-\left(1-4{\sum }_{m=1}^M{w}_m\text{si}{\text{n}}^2\frac{\pi mk}{2L}\right)f^{\mathrm{\prime }},$$ (7)

where $f^{\mathrm{\prime }}$ is the derivative of the APDR function f at the unstable fixed point. The stability criterion for the discrete map is that the Fourier mode is unstable when ${\lambda }_k<-1$. Figure 3(f) shows the minimum cable length vs τ_a obtained using Eq. (7). For example, for ${\tau }_a=34\text{ms}$, $L_{\text{min}}\approx 2.225\text{cm}$ and thus for a 15 cm, the maximum number of nodes is 6, which agrees with the numerical simulation results in Fig. 3(e).

FIG. 3.

Node stability under global pacing in the 1D CML model. (a) Plots of alternans amplitude: $\mathrm{\Delta }a(x,n)={(-1)}^n[a(x,n)-a(x,n-1)]/2$, at different time points. Black: $\mathrm{\Delta }a(x,2)$; red: $\mathrm{\Delta }a(x,24)$; blue: $\mathrm{\Delta }a(x,28)$; and green: $\mathrm{\Delta }a(x,{10}^5)$. Initial condition: $d_1=99.9\text{ms}$ for x ≤ 5 cm and $d_1=100.1\text{ms}$ for x > 5 cm. Pacing period T = 225 ms. (b) Node position vs n for a single node at the center of the cable (black line) and away from the center (colored lines). ${\tau }_a=36\text{ms}$. The inset shows the final node position (at 10⁵ beats) vs the initial position. (c) Node position vs n for different ${\tau }_a$ values (25, 36.3, 36.35, and 36.4 ms from black to green) for the same initial position (0.0125 cm away from the middle of the cable). The inset shows the final node position (away from the middle point at 10⁵ beats) vs ${\tau }_a$. (d) Node position vs n for ${\tau }_a=25\text{ms}$. Gray marks the region in which the node is stable within the 10⁵ beats. Nodes outside the gray region (colored lines) drift away or disappear. (e) Steady-state solutions $[\mathrm{\Delta }a(x)]$ in a 15 cm cable showing up to six nodes for ${\tau }_a=34\text{ms}$. Black: 1 node; red: 2 nodes; blue: 3 nodes; green: 4 nodes; magenta: 5 nodes; and cyan: 6 nodes. The initial conditions are set as in (a) but with different spatial periodicities to obtain the different number of nodes. (f) L_min vs ${\tau }_a$ obtained using Eq. (7).

For the ionic model, we paced 500 beats in our previous study⁵² to observe the stability of the nodes. Here, we pace 10⁵ beats as for the AE and CML models. For both the fast-pacing-induced APD alternans [Fig. 4(a)] and I_to-induced APD alternans [Fig. 4(b)], we find that the node appears stable if its initial position is within a certain distance from the center of the cable. However, if the initial node position is outside this region, it drifts and disappears after enough pacing beats. We also find that for both cases, two nodes can be accommodated by the same cable [Figs. 4(c) and 4(d)]. These results are similar to those in Fig. 2(d) in the AE model and Fig. 3(d) in the CML model.

FIG. 4.

Node stability and spacing under global pacing in the 1D ionic model. (a) Node position vs n for fast-pacing-induced APD alternans [Fig. 1(c)]. T = 230 ms. Gray marks the region in which the node is stable within the 10⁵ beats. Nodes outside the gray region (colored lines) drift away or disappear. (b) Node position vs n for I_to-induced APD alternans [Fig. 1(d)]. T = 700 ms. Gray marks the region in which the node is stable within the 10⁵ beats. Nodes outside the gray region (colored lines) drift away or disappear. (c) The same as (a) but two nodes are initiated. (d) The same as (b) but two nodes are initiated.

Based on our results of the three types of models, we conclude that in homogeneous tissue with global pacing, the nodes are mildly unstable or stability neutral as the cable length approaches infinite. As shown in Fig. 2(d) [and Fig. 3(d)], increasing α (or decreasing ${\tau }_a$) makes the node less unstable (slower drifting). On the other hand, increasing α makes the cellular alternans more stable [see Eq. (4)]. Therefore, the node is less unstable if the cellular alternans is more stable.

2. Effects of CVR and convection on node formation and stability in a 1D cable

The effects of CVR on SDA formation have been widely studied. Here, we use the three types of models to perform a systematic investigation of the effects of CVR and convection on node formation and stability in a 1D cable with pacing from one end of the cable. We use both homogeneous and heterogeneous initial conditions to investigate the spontaneous formation of the SDA nodes and node stability.

Using the AE model [Eq. (1)], Echebarria and Karma^21,55 performed a stability analysis to investigate the instabilities leading to SDA. They showed that the interaction of CVR and convection gave rise to stable and traveling node solutions. Here, we first analyze a special case in which conduction is present but there is no CVR. This corresponds to the condition in which Na⁺ recovers quickly or the pacing period T is large so that CV does not change with DI. Under this condition, $\mathrm{\Lambda }\to \mathrm{\infty }$ [see Eq. (A3)], then Eq. (1) becomes

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3-\delta \frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }x}+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2}.$$ (8)

A traveling front solution of Eq. (8) for an infinite-length cable can be obtained as^62,63

$$\mathrm{\Delta }a(x,t)=\sqrt{\frac{\alpha }{\beta }}\text{tanh}\left[\sqrt{\frac{\alpha }{2{\xi }^2}}(x-ct)\right],$$ (9)

where $c=\frac{\delta }{T}$ is the velocity of the traveling front, i.e., the velocity of the node. Therefore, in a finite-length cable, without CVR, the SDA nodes initiated by heterogeneous initial conditions will always travel off the cable, leading to SCA. In other words, a node initiated by a heterogeneous initial condition in a homogeneous tissue will always be unstable, and the system can only have SCA solutions. This agrees with our previous simulation results of the ionic model⁵² that under local pacing and in the absence of CVR (i.e., slow pacing for I_to-induced alternans or EAD-induced alternans), the heterogeneous initial condition induced nodes are always unstable, which drift off the cable, leading to only SCA.

In the presence of CVR, exact solutions of Eq. (1) cannot be obtained. Based on the theoretical analysis of Echebarria and Karma,^21,55 the threshold for SDA is ${\alpha }_{th}={\xi }^2/\delta \mathrm{\Lambda }$. Equation (1) also predicts a steady-state traveling solution of SDA node with the threshold ${\alpha }_{th}=\frac{3}{2}{\left(\frac{\xi }{2\mathrm{\Lambda }}\right)}^{2/3}$, which is favored when $\mathrm{\Lambda }\ll {\xi }^4/{\delta }^3$. Here, we perform numerical simulations of Eq. (1) by scanning the δ-Λ parameter space for alternans behaviors [Fig. 5(a)], which are summarized below:

• (1)When δ and Λ are both large (blue), SDA nodes cannot be formed spontaneously. The heterogeneous initial condition induced nodes are unstable, drifting away from the pacing site, and finally disappear [see example “iii” shown in Fig. 5(a)], as predicted by Eq. (9). In this region, only SCA exists.
• (2)When δ and Λ are both small (red), SDA nodes form spontaneously from the distal end. The nodes are unstable, drifting toward the pacing site, but new nodes continuously form from the distal end, maintaining a stationary SDA pattern [see example “i” shown in Fig. 5(a)]. No SCA can exist in this region.
• (3)In the rest of δ and Λ parameter space (cyan), SDA nodes form spontaneously from the distal end and become stable after a transient process [see example “ii” shown in Fig. 5(a)]. The total number of nodes is determined by δ and Λ. No SCA can exist in this region.

FIG. 5.

Effects of CVR and convection on node formation and stability under local pacing. Shown for each model are a phase diagram (left panel) of alternans behaviors for convection strength (x-axis) and slope of CVR curve (y-axis) and node positions vs n from three locations marked in the phase diagram. Red: Traveling nodes in which new nodes form spontaneously from the right end, travel to the pacing site (left end) and disappear (example shown in panel i). Cyan: Stable nodes in which the nodes form spontaneously from the right end and become stable (example shown in panel ii). Blue: Nodes cannot form spontaneously and the initial condition induced nodes are unstable (example shown in panel iii); Orange: The initial condition induced nodes are stable but nodes cannot form spontaneously with homogeneous initial conditions. Gray: No alternans. (a) The 1D AE model (δ-Λ space). $\alpha =1$, T = 295 ms. (b) The 1D CML model (δ-τ_θ space). $a_0=30\text{ms}$, $d_0=50\text{ms}$, and T = 215 ms. (c) The 1D ionic model (G_Na-γ space) with fast-pacing-induced APD alternans. T = 230 ms.

To demonstrate the predictions of the AE model in the CML model, we scanned the δ-τ_θ parameter space for alternans behaviors [Fig. 5(b)]. τ_θ is the parameter controlling the slope of the CVR [see Eq. (A12) in the Appendix]. A larger τ_θ corresponds to a smaller Λ since the slope of the CVR curve is less steep. We observe the same three types of alternans behaviors: SCA only (blue), SDA with stable nodes (cyan), and SDA with moving nodes (red). However, there is a region where the alternans dynamics depend on initial conditions (orange). In this region, no node can be formed spontaneously with homogeneous initial conditions. In other words, under homogeneous initial conditions, only SCA can be observed. On the other hand, if one uses heterogeneous initiation conditions to induce SDA nodes, they remain stable due to competition between CVR and convection. Therefore, in this region, both SCA and SDA can occur depending on initial conditions, which is not predicted by the AE model.

We carry out the same simulations of the ionic model to demonstrate the predictions from the AE and CML models. We scan the recovery and maximum conductance (G_Na) of the Na⁺ current for alternans behaviors. Based on Echebarria and Karma,^21,55 $\delta =2D/c$ (c is CV), so we can change δ by changing c which is mainly determined by G_Na. We vary CVR by varying the recovery time constant ${\tau }_j$ of the Na⁺ channel by multiplying a factor γ, i.e., ${\tau }_j\to \gamma {\tau }_j$. A larger γ (thus a larger ${\tau }_j$) corresponds to a larger τ_θ in the CML model or a smaller Λ in the AE model. Again, we observe the same alternans behaviors as in the AE and CML models. Similar to the CML model, there is also a region in which both SCA and SDA can occur depending on initial conditions (orange). Note that changing G_Na or γ not just changes c or CVR, it also changes APDR and thus the cellular APD alternans.⁶⁴ This is the reason that the stable (no alternans) regions (gray) occur in the ionic model.

Therefore, in a homogeneous cable with local pacing, the node stability is determined by the competition of CVR and convection. The convection tends to move the node along the direction propagation while CVR tends to move the node in the opposite direction. When the effect of convection is too strong, the node is unstable. Since no node can spontaneously form in the cable under this condition, then only SCA occurs. When the effects of CVR is too strong, the node is also not stable. Since new nodes continuously form in the distal end, a stationary SDA pattern with traveling nodes forms. When the two effects are matched properly, the nodes become stable, resulting in a stable SDA pattern.

3. Nodal ring dynamics in a 2D homogeneous tissue

In this section, we investigate the nodal ring dynamics in a 2D homogeneous tissue model. In the previous study,⁵² we showed in computer simulations of the ionic model that curved nodal lines cannot be stable in a homogeneous tissue. Under global pacing, a nodal ring always shrinks and disappears. The total area inside the ring decays linearly with time. Here, we first use the AE model to analytically derive the nodal ring dynamics under both global pacing and local pacing protocols and then use numerical simulations of the CML and ionic models to demonstrate the theoretical predictions of the AE model.

a. Nodal ring dynamics under global pacing

We extend Eq. (2) into a 2D tissue model under global pacing, which becomes

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3+{\xi }^2\left(\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{y}^2}\right).$$ (10)

To investigate the nodal ring dynamics, we transform Eq. (10) to a polar coordinate system, i.e.,

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3+\frac{{\xi }^2}{r}\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }r}+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{r}^2}.$$ (11)

Although an exact solution of $\mathrm{\Delta }a(r,t)$ in the form of Eq. (9) is not available, one can transform Eq. (11) into Eq. (2) in a moving coordinate system with the following moving speed:^63,65–67

$$c=-\frac{{\xi }^2}{Tr}=-\frac{{\xi }^2}{T}\kappa ,$$ (12)

where $\kappa =1/r$ is the curvature of the ring. Equation (12) is a special form of the well-known eikonal-curvature equation for wave conduction in excitable media,^63,65–67

$$c=c_0-D\kappa ,$$ (13)

where $c_0$ is the CV of the planar wave and D the diffusion constant as in Eq. (A16) in the Appendix. Equation (12) is simply Eq. (13) with $c_0=0$. Since $c=\frac{dr}{dt}$, then one obtains

$$r^2=r_0^2-\frac{2{\xi }^2}{T}t,$$ (14)

where $r_0$ is the radius of the initial ring.

Equation (14) indicates that the area (A) inside the ring decays linearly with time, i.e., $A=\pi r^2=\pi r_0^2-\frac{2\pi {\xi }^2}{T}t$. Note that the node movement in the real tissue (or in the CML and ionic models) is not continuous but jumps discretely from beat to beat. If one substitutes $t=nT$, then $A=\pi r_0^2-2\pi {\xi }^{2}n$. Therefore, if one plots A against n, then A will decay with n in a slope,

$$\rho =-2\pi {\xi }^2.$$ (15)

To numerically validate the theoretical prediction, we carry out simulations using Eq. (10). A nodal ring is induced with a heterogeneous initial condition, which shrinks and disappears [Fig. 6(a)]. The area inside the ring decays exactly linearly with the beat number n. For the control value $\xi =0.235\text{cm}$, the predicted slope [Eq. (15)] is $\rho =-0.347\text{c}{\text{m}}^2/\text{beat}$. Numerical simulation using Eq. (10) gives rise to $\rho =-0.35\text{c}{\text{m}}^2/\text{beat}$, almost identical to the theoretical results.

FIG. 6.

Nodal ring dynamics in 2D homogeneous tissue under global pacing. (a) Nodal ring dynamics in the AE model. Left: Nodal rings at different time points showing a heterogeneous initial condition induced ring (black) shrinks and disappears. Right: A vs n. $\alpha =1$. (b) Nodal ring dynamics in the CML model. Left: Nodal rings at different time points showing a heterogeneous initial condition induced ring shrinks and disappears. Middle: A vs n. Right: the slope (ρ) of decay vs alternans amplitude (Δa). The slope is calculated as $\rho =\frac{\mathrm{\Delta }A}{\mathrm{\Delta }n}$ for n from 50 to 150. Different colors are for different $a_1$ values of the APDR function. Red: $a_1=130\text{ms}$; blue: $a_1=150\text{ms}$; cyan: $a_1=170\text{ms}$. For each $a_1$ value, different points are for different pacing period T, spanning the whole range of alternans. (c) Nodal ring dynamics in the ionic model. Left: Nodal rings at different time points. Middle: A vs n. Right: ρ vs Δa. The color dots are for the fast-pacing-induced alternans with G_K = 0.423 mS/cm² (red), 0.35 mS/cm² (blue), and 0.282 mS/cm² (cyan). Black dots are for I_to-induced alternans. For the same color, different dots are for different T. The tissue sizes for the AE, CML, and ionic models are the same, which is 12.5 × 12.5 cm².

Note that Eq. (12) or (14) is independent of α and β, which indicates that the nodal ring movement is independent of the properties of alternans. To examine the theoretical predictions of the AE model, we first carry out 2D tissue simulations using the CML model under global pacing. The same as the AE model, a nodal ring initiated by a heterogeneous initial condition shrinks and eventually disappears with A decaying linearly with n [Fig. 6(b)]. To examine whether the slope ρ depends on the alternans dynamics, we vary the alternans dynamics by using different $a_1$ in the APDR function [Eq. (A11) in the Appendix] and using different pacing period T spanning the alternans regime. Changing $a_1$ and T changes APD, alternans amplitude, and stability. We plot the calculated ρ against the alternans amplitude (Δa), which shows that ρ is independent of $a_1$ and T but affected by Δa [right panel in Fig. 6(b), the slope changes from −0.31 to −0.37 cm²/beat for Δa from 50 ms to 5 ms]. This seems to disagree with the theoretical prediction that the ring drift does not depend on alternans properties. We argue that this dependence is likely caused by the transient dynamics of the system since the transient to the steady state of alternans becomes longer as the system is closer to the bifurcation point. The alternans amplitude becomes smaller when the system is closer to the bifurcation point. Note that it only takes less than 200 beats for the ring to disappear in Fig. 6(b). The ring shrinks slower if the coupling strength is reduced, which can be done by reducing σ in the CML model. For example, when σ is reduced from 25 to 10, for the same initial ring size, it takes more than 1200 beats for the ring to disappear (thus we can drop more initial beats as transients for calculating the slope), and the dependence of ρ on Δa is much reduced [the slopes are between −0.049 and −0.051 cm²/beat for the same Δa range as in Fig. 6(b)]. This indicates that the transient is indeed the major cause of the dependence of the slope on alternans amplitude in the CML model. Therefore, the theoretical prediction of the AE model agrees well with the results of the CML model.

We then carry out simulations using the ionic model. As shown in our previous study,⁵² nodal rings in homogenous tissue always shrink and disappear and the area inside the ring decays linearly with time [an example shown in Fig. 6(c)]. We calculate ρ under different conditions similar to what we did for the CML model. We use both fast-pacing-induced alternans [color dots in Fig. 6(c)] and I_to-induced alternans [black dots in Fig. 6(c)]. For the fast-pacing-induced alternans case, we use different G_K to change APD and alternans dynamics. For each case, we pace the system using different pacing period T spanning the alternans regime. As shown in Fig. 6(c), we obtain ρ from −0.35 to −0.5 cm²/beat for the fast-pacing-induced alternans, but a steeper slope, from −0.9 to −1.5 cm²/beat for the I_to-induced alternans. Note that there is only a small variance of ρ within the fast-pacing-induced alternans but a large variance within the I_to-induced alternans. Moreover, there is a big difference between the two mechanisms of alternans.

The ionic model results seem to show that ρ exhibits a strong dependence on alternans dynamics, which disagrees with the results of the AE and CML models. This may be caused by the diffusion constants in the AE and CML models that describe the contribution of gap junction coupling in the entire pacing beat so that it may be a function of not only D but also APD as well as other properties of the system. For example, based on the estimation by Echebarria and Karma,^21,55 $\xi =\sqrt{D\times a_c}$ ($a_c$ is APD at the onset of alternans); therefore, ξ increases as APD increases. However, in the AE and CML models, the coupling strengths are set as constants. If the dependence of ξ on APD is taken into account, then the conclusion from the AE model that the nodal line movement is independent of alternans dynamics is somewhat misleading. However, this does not mean that the theoretical prediction is wrong. In other words, Eq. (15) may still be valid if ξ is a function of APD and other parameters (e.g., T). Since APD is longer in the case of I_to-induced alternans than the case of fast-pacing-induced alternans, ξ is larger based on the estimation by Echebarria and Karma. This may explain why we observed a steeper slope (more negative ρ values) in the case of I_to-induced alternans than in the case of fast-pacing-induced alternans. Moreover, the dynamical transient may further affect the slopes as discussed in the CML model. Nevertheless, the theoretical prediction that the area inside the nodal ring decays linearly with time is well conserved in all three types of models.

b. Nodal ring dynamics under local pacing from the tissue center

Here, we study a case in which pacing is applied to a point in a 2D tissue. We assume that the pacing rate is slow so that there is no CVR engagement. In this case, the nodal line is affected by curvature and convection. Since curvature generates a tension to cause the ring to shrink, and convection generates a tension to cause the node to move away from the pacing site, the two tensions then compete to give rise to different nodal ring movements.

In the polar coordinate system, we set the pacing site to be the origin. Then, we can describe this by adding a convection term into Eq. (11), which becomes

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3-\delta \frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }r}+\frac{{\xi }^2}{r}\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }r}+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{r}^2}.$$ (16)

Similarly, the eikonal-curvature equation for Eq. (16) is

$$c=\frac{\delta }{T}-\frac{{\xi }^2}{Tr}.$$ (17)

From Eq. (17), one can predict that when $r>\frac{{\xi }^2}{\delta }$, $c=\frac{dr}{dt}>0$, the nodal ring expands. When $r<\frac{{\xi }^2}{\delta }$, $c=\frac{dr}{dt}<0$, the nodal ring shrinks. In other words, if the initial radius of the ring $r>\frac{{\xi }^2}{\delta }$, the ring expands, otherwise it shrinks. Replacing $c=\frac{dr}{dt}$ in Eq. (17), one obtains the solution as

$$\frac{T}{{\delta }^2}(\delta \text{r}-{\xi }^2)+\frac{T{\xi }^2}{{\delta }^2}\text{ln}(\delta r-{\xi }^2)+C=t,$$ (18)

where C is a constant determined by the initial condition.

To examine this theoretical prediction, we carry out numerical simulations of the 2D ionic model with pacing from the tissue center. We use the I_to-induced APD alternans so that the pacing period T is large enough to avoid CVR engagement. Agreeing with the theoretical prediction, when the initial ring size is smaller than the critical size [black circle in Fig. 7(a)], the ring shrinks. When the initial ring size is larger than the critical size, the ring expands. Moreover, if we plot the trace of Eq. (18) with properly chosen δ and ξ values [Fig. 7(b)], the theoretical result matches very well with the numerical simulation result of the ionic model until the ring becomes very large where the boundary effects become strong.

FIG. 7.

Nodal ring dynamics in 2D homogeneous tissue with center pacing. (a) Nodal rings showing expanding (red) and shrinking (blue) rings from the simulations of the ionic model. The black nodal ring is the critical ring separating the two types of nodal ring behaviors. I_to-induced APD alternans with T = 700 ms. Pacing is applied in the center of the tissue. Arrows indicate the directions of the nodal line movement. Tissue size is 12.5 × 12.5 cm². (b) Area inside the ring vs n. Dots are the results from the ionic model shown in (a), and the solid lines are the plots of theoretical results of Eq. (18) for δ = 0.38 cm and ξ = 0.8 cm. Red is for the expanding ring, and blue is for the shrinking ring.

B. Node stability and dynamics in heterogeneous tissue models

The goal of this section is to investigate the effects of heterogeneities on SDA node dynamics. In the previous study,⁵² we showed in the ionic model that in the case of fast-pacing-induced APD alternans, the node drifted from the long APD region (larger Δa) to the short APD region (smaller Δa). This agrees with the simulation results and theoretical predictions of the AE model in a previous study by Echebarria and Karma.⁶⁸ However, in the case of I_to-induced APD alternans, the node drifted from the short APD region (smaller Δa) to the long APD region (larger Δa). These results are reproduced in Figs. 8(a) and 8(b). We cannot explain the seemly contradictory results using computer simulations of the ionic model. Here, we answer this question using the AE model. We also investigate the effects of gap junction coupling heterogeneity on node drift.

FIG. 8.

Node drift induced by heterogeneous repolarization and cell coupling. (a) Node drift caused by repolarization heterogeneity in the ionic model with fast-pacing-induced APD alternans [Fig. 1(c)]. Upper panel: APD in space for two consecutive beats. The numbers indicate the APD differences (161 ms and 128 ms) of the two beats at the two ends of the cable. Lower panel: Node position vs n. Open arrow indicates the direction of drift. T = 230 ms. $G_K=0.38\text{mS/c}{\text{m}}^2$ in the first half and $G_K=0.423\text{mS/c}{\text{m}}^2$ in the second half of the cable. (b) Node drift caused by repolarization heterogeneity in the ionic model with the I_to-induced APD alternans [Fig. 1(d)]. Upper panel: APD in space for two consecutive beats. The numbers indicate the APD differences (266 ms and 255 ms) of the two beats at the two ends of the cable. Lower panel: Node position vs n. T = 700 ms. $G_K=0.282\text{mS/c}{\text{m}}^2$ in the first half and $G_K=0.44\text{mS/c}{\text{m}}^2$ in the second half of the cable. (c) Node drift in the AE model for $\alpha =0.1$ and $\beta =0.0001g(x)$. $g(x)=0.05+\frac{0.1}{1+e^{-\frac{x-3.125}{0.375}}}$ which increases with x. (d) Node drift in the AE model for $\alpha =g(x)$ and $\beta =0.0001g(x)$. $g(x)$ is the same as in (c). (e) Node drift induced by heterogeneous cell coupling. Node position vs n obtained from the ionic model under global pacing. $D(x)=0.0005+\frac{0.001}{1+e^{-\frac{x-5}{0.375}}}$, which is a function increasing with x. Fast-pacing-induced APD alternans with T = 230 ms.

1. Node drift induced by repolarization heterogeneities in a 1D cable

We hypothesize that the node drift is determined by heterogeneities in both alternans amplitude and stability. To incorporate and separate the effects of the two heterogeneities, we numerically simulate the AE model by setting α and β as functions of space x, i.e.,

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha (x)\mathrm{\Delta }a-\beta (x)\mathrm{\Delta }{a}^3+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2}.$$ (19)

Note that the stability of alternans is determined by α [see Eqs. (3) and (4)] and the amplitude of alternans is determined by α/β. We alter α and β in space separately or together to alter the heterogeneities of alternans amplitude and stability as described under the following two conditions:

• (1)Spatially homogeneous α but heterogeneous β. Based on Eqs. (3) and (4), in this case, the alternans stability remains the same, but the SCA amplitude over space becomes heterogeneous, i.e., $\mathrm{\Delta }a(x)=\sqrt{\frac{\alpha }{\beta (x)}}$. The simulation result of Eq. (19) shows that the SDA node drifts from the larger amplitude region to the smaller amplitude region [Fig. 8(c)].
• (2)Spatially heterogeneous α and β, but $\frac{\alpha (x)}{\beta (x)}=constant$. In this case, the alternans stability changes over space but the SCA amplitude remains constant, i.e., $\mathrm{\Delta }a(x)=\sqrt{\frac{\alpha (x)}{\beta (x)}}=constant$. The simulation result of Eq. (19) shows that the SDA node drifts from the more stable region to the less stable region [Fig. 8(d)].

Therefore, the node drifts from the larger alternans amplitude region toward the smaller amplitude region or from the more stable alternans region to the less stable alternans region.

The insights from the AE model may provide an explanation to the seemingly contradictory results seen in the ionic model. In the case of fast-pacing-induced APD alternans, increasing G_K shortens APD and reduces the alternans magnitude and stability, causing the node drifting from the larger amplitude region to the smaller amplitude region. In the case of I_to-induced alternans, increasing G_K also reduces the alternans amplitude; however, increasing G_K may result in a more stable alternans, causing the node drifting from the smaller amplitude region to the large amplitude region. However, we cannot separate the effects of alternans amplitude and alternan stability gradients in the ionic model, and this cannot pinpoint whether this explanation is plausible.

2. Node drift induced by gap junction coupling heterogeneities in a 1D cable

Besides repolarization heterogeneities, gap junction coupling heterogeneity may also cause the SDA node to drift. Assume ${\xi }^2={\xi }^2(x)$, then the AE [Eq. (2)] becomes

$$\begin{array}{rl}T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}& =\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3+\frac{\mathrm{\partial }}{\mathrm{\partial }x}{\xi }^2(x)\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }x}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3\\ & +\frac{\mathrm{\partial }{\xi }^2(x)}{\mathrm{\partial }x}\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }x}+{\xi }^2(x)\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2}.\end{array}$$ (20)

If one assumes that $\frac{\mathrm{\partial }{\xi }^2(x)}{\mathrm{\partial }x}=\gamma \ll 1$ so that ${\xi }^2(x)$ can be treated as a constant. Following the same argument, we can obtain that the node drift velocity is $c\approx -\frac{\gamma }{T}$, which implies that the node drifts from the stronger coupling region to the weaker coupling region. We carry out computer simulation of the ionic model [Eq. (A17) in the Appendix] with a gradient of gap junction coupling strength, which confirms this prediction [Fig. 8(e)].

3. Nodal line dynamics in 2D tissue due to tension competition

In 2D tissue, the tensions generated by CVR, convection, curvature of the nodal line, and tissue heterogeneities compete to determine the nodal line stability and dynamics. Therefore, the nodal line dynamics will depend on the pacing protocol, the geometry of the heterogeneity, and the initial position of the nodal line. In the previous study,⁵² we have shown examples of the nodal line dynamics caused by the interactions of the effects of nodal line curvature and repolarization gradients. Here, we show more examples of nodal line dynamics in 2D tissue due to the tensions generated by convection, curvature, repolarization gradient, and CVR.

We first show the nodal line dynamics caused by the interactions of convection, curvature, and repolarization gradient in the absence of CVR. Figures 9(a)–9(c) show nodal line dynamics in a homogeneous 2D tissue under different pacing protocols. The APD alternans is induced by I_to with T = 700 ms so that there is no CVR engagement. A horizontal nodal line is generated by a heterogeneous initial condition. When the tissue is paced globally [Fig. 9(a), Multimedia view], the nodal line is stable as the node in the 1D cable [see Fig. 4(b)]. When the tissue is paced from the left side [Fig. 9(b), Multimedia view], the nodal line is also stable. Although there is conduction and thus convection in the x-direction, there is no conduction and thus no convection in the y-direction. In other words, the cells along a line in the y-direction are excited simultaneously, equivalent to be globally paced. Therefore, under this pacing protocol, the nodal line dynamics are the same as under global pacing. When the tissue is paced from the top [Fig. 9(c), Multimedia view], the nodal line moves downward and eventually off the tissue boundary due to the tension generated by the convection, leading to SCA in the tissue.

FIG. 9.

Nodal line dynamics in 2D tissue models with different pacing protocols. (a) Dynamics of a heterogeneous initial condition induced horizontal nodal line in a homogeneous tissue under global pacing. I_to-induced APD alternans as in Fig. 1(d) with T = 700 ms. The tissue size is 10 × 10 cm². The initial nodal line is induced the same way as the node is initiated in Fig. 4. (b) The same as (a) but pacing from the left side as indicated by the red arrows. (c) The same as (a) but pacing from the top as indicated by the red arrows. Colored lines in (a)–(c) are nodal lines at different time points as marked by the beat # on the right. (d) Nodal line dynamics in a heterogeneous tissue under global pacing. I_to-induced APD alternans as in Fig. 1(d) with T = 700 ms. The tissue size is 10 × 10 cm². Heterogeneity is done by setting G_K = 0.282 mS/cm² in the light gray region and G_K = 0.44 mS/cm² in the dark gray region. Colored arrows indicate the directions of the tensions generated by the APD gradient (blue), curvature (cyan), and convection (magenta). The top, middle, and bottom panels plot the nodal lines at different time points as marked by the beat #. (e) The same as (d) but pacing from the upper-left corner as indicated by the red arrow. (f) The same as (d) but pacing from the lower-right corner as indicated by the red arrow. Multimedia views: https://doi.org/10.1063/5.0029209.1; https://doi.org/10.1063/5.0029209.2; https://doi.org/10.1063/5.0029209.3; https://doi.org/10.1063/5.0029209.4; https://doi.org/10.1063/5.0029209.5; https://doi.org/10.1063/5.0029209.6 Download video file ^{(274.1KB, mp4)} DOI: 10.1063/5.0029209.1 Download video file ^{(385.8KB, mp4)} DOI: 10.1063/5.0029209.2 Download video file ^{(186.7KB, mp4)} DOI: 10.1063/5.0029209.3 Download video file ^{(978KB, mp4)} DOI: 10.1063/5.0029209.4 Download video file ^{(1.1MB, mp4)} DOI: 10.1063/5.0029209.5 Download video file ^{(764.7KB, mp4)} DOI: 10.1063/5.0029209.6

Figures 9(d)–9(f) show nodal dynamics in a heterogeneous tissue under different pacing protocols. The heterogeneity is generated by setting a larger G_K in the sector area than in the rest of the tissue. Based on the 1D cable simulation in Fig. 8(b), a node in the heterogeneous region drifts toward the smaller G_K area. The heterogeneous initial condition gives rise to two nodal lines in the tissue [see upper panels in Figs. 9(d)–9(f)]. Under global pacing [Fig. 9(d), Multimedia view], the nodal line in the gradient region is stable but the one in the sector area disappears. This is because the nodal line in the gradient region is subjected to two tensions, one by the gradient and one by the curvature, which are balanced to maintain the nodal line stability. The nodal line in the sector region is only subjected to the tension caused by the curvature. When the tissue is paced from the upper-left corner [Fig. 9(e), Multimedia view], there is an additional tension caused by convection for both nodal lines. The tension generated by the gradient can still be balanced with the tensions generated by the curvature and convection to result in a stable nodal line in the gradient region. The nodal line in the sector region drifts downward and disappears. However, when the tissue is paced from the lower-right corner [Fig. 9(f), Multimedia view], the tension generated by the convection changes the direction. This change breaks the balance, driving both nodal lines to drift upward and disappear, leading to SCA in the tissue.

We then show examples of nodal line dynamics caused by interactions of CVR with other effects. To engage CVR, we use the fast-pacing-induced APD alternans [Fig. 1(c)]. For a 2D homogeneous tissue, pacing from one side of the tissue results in a straight nodal line, which becomes stable in less than 100 beats [see Fig. 5(c) for CVR-induced node formation in the ionic model]. Here, we show nodal line dynamics in a homogeneous tissue paced from the left side with an initially curved nodal line. We use a heterogeneous initial condition to generate a horizontal nodal line that is set the same way as in Fig. 9(a). If the tissue is paced globally, the nodal line remains stable. When the tissue is paced from the left side, the action potential propagates from the left to the right. The engagement of CVR generates a nodal line in the vertical direction, which fuses with the horizontal one to form a curved nodal line [Fig. 10(a), Multimedia view]. This curved nodal line moves slowly toward the lower-left corner. At around 110 beats, another curved nodal line forms, moving leftward with time. At around 550 beats, the lower curved nodal line disappears, leaving a single nodal line in the tissue. After 770 beats, the nodal line becomes a vertically straight line and stabilizes, reaching the steady state. If the CVR effect is enhanced by slowing the Na⁺ channel recovery, it goes a similar process but takes a much shorter time to reach the final steady state [Fig. 10(b), Multimedia view]. As indicated by the arrows in the left panel in Fig. 10(a), the nodal line is subjected to the tensions caused by CVR, convection, and curvature. Without curvature, the tensions generated by CVR and convection are balanced to give rise to stable straight nodal lines. The presence of curvature generates an additional tension that causes the nodal line to drift, which causes the nodal line to either disappear or become a vertically straight line.

FIG. 10.

Nodal line dynamics in a homogeneous tissue with an initial curved nodal line. Shown are nodal lines at different time points as marked above each panel. Fast-pacing induced APD alternans with the parameters the same as in Fig. 1(c). The Na⁺ channel recovery time constant ${\tau }_j$ is altered by a pre-factor γ to alter CVR. (a) $\gamma =1.35$. (b) $\gamma =2.5$. Red arrows indicate the pacing locations. Colored arrows on the left panel in (a) indicate the tensions generated by CVR (blue), convection (magenta), and curvature (cyan). Pacing period T = 250 ms. Tissue size is 10 × 10 cm². Multimedia views: https://doi.org/10.1063/5.0029209.7; https://doi.org/10.1063/5.0029209.8 Download video file ^{(7.7MB, mp4)} DOI: 10.1063/5.0029209.7 Download video file ^{(3.5MB, mp4)} DOI: 10.1063/5.0029209.8

IV. SUMMARY AND DISCUSSION

Previous simulation and theoretical studies^15,20–22 have shown that SDA formation and stability require the engagement of CVR, which is supported by experiments.^35–39 However, other experimental results^39–41 do not support this theory and other mechanisms of SDA have been proposed.^13,69,70 In a previous simulation study,⁵² we showed that SDA could be formed by heterogeneous initial conditions, repolarization heterogeneities, or premature ventricular complexes. The stability of the nodes or nodal lines depends on the balance of the tensions generated by nodal line curvature and repolarization heterogeneities, providing theoretical explanation to the SDA behaviors observed in experiments^39–41 that cannot be explained by CVR. However, our previous study is a numerical simulation of the ionic model, rigorous theoretical treatments on node stability and dynamics are lacking due to the complexity of the model. In this study, we use the AE model, the CML model, as well as the ionic model to perform a more rigorous investigation of SDA node formation, stability, and dynamics in cardiac tissue, combining analytical and numerical treatments. Our major findings are as follows:

• (1)In homogeneous tissue and in the absence of conduction (global pacing), single-node or multiple-node SDA solutions always exist as long as the system is greater than a critical size. The nodes in a 1D homogeneous cable are stability neutral or mildly unstable. A nodal ring in 2D homogeneous tissue always shrinks with a moving speed proportional to its curvature.
• (2)In homogeneous tissue and in the presence of conduction but no engagement of CVR, such as slow pacing, the SDA solutions exist but the initial condition induced nodes drift off the tissue due to convection, leaving only stable SCA.
• (3)In homogeneous tissue and in the presence of conduction with the engagement of CVR, SDA nodes form spontaneously. The competition of CVR and convection determines the node dynamics.
• (4)In heterogeneous tissue with either repolarization or coupling gradients, an SDA node will drift away from the gradient region when CVR and convection are absent.
• (5)In real cardiac tissue, the effects of CVR, convection, curvature, and heterogeneities may work in synergy or compete each other to determine the SDA dynamics. As shown in simulations of 2D tissue models in the previous⁵² and the current studies, the nodal line dynamics also depend on the pacing protocol, the geometry of the heterogeneity, and the initial position of the nodal line. One effect may become dominant over others under certain conditions. For example, for fast local pacing, CVR is the dominant factor, but for slow pacing, the other factors become important.

The use of models with different complexities and combining rigorous analytical treatments with computer simulations provide a solid and cohesive understanding of the mechanisms of SDA in cardiac tissue. The mechanistic insights may provide a plausible interpretation of different SDA behaviors observed in experiments and help postulate the potential roles of SDA in cardiac arrhythmogenesis in clinical settings.

Although the use of models with different complexities is very useful for us to elucidate the underlying mechanisms of SDA, each model has its advantages and disadvantages (caveats). Specifically, the AE model is extremely simple comparing to the ionic model, and thus it can be used for analytical treatments. However, it is derived for the behaviors close to the onset of alternans, with both continuous time and space. In real systems, alternans is a temporally discretized behavior. Moreover, it cannot be used for more complex APD dynamics, such as higher periodicity and chaos. The CML model is still very simple compared to the ionic models. It can be used for certain rigorous theoretical analyses. It also has the advantage of using the APDR and CVR functions to simulate the spatiotemporal APD dynamics. The ionic model simulates the actual membrane potential conduction and the detailed physiological process, but it is complex and limited for analytical treatments. It is also computationally much more tedious.

We only investigated 1D and 2D tissue models, but the real heart is a 3D organ, which may exhibit additional effects. For example, in 3D tissue, the SDA nodes form nodal surfaces. Following the same method as for the nodal ring in 2D tissue, one obtains the node drifting velocity for a nodal sphere in a homogeneous 3D tissue as $c=-\frac{2{\xi }^2}{T}\kappa$, which is two times of the nodal ring drifting velocity as shown in Eq. (12). Similarly, Eq. (17) becomes $c=\frac{\delta }{T}-\frac{2{\xi }^2}{Tr}$ in homogeneous 3D tissue. Moreover, the heart contracts, which may affect the SDA dynamics via electromechanical coupling. For example, previous simulation studies have shown that this coupling affects spiral wave dynamics.^71,72 The effects of electromechanical coupling on SDA need to be investigated in future studies.

In this study, we use two pacing protocols, local pacing and global pacing, to investigate the nodal line dynamics in 1D cable and 2D tissue models. The rationale for using these two pacing protocols is to distinguish the contributions of CVR and convection from those of others, such as APD gradient and nodal line curvature. Although in most of the computer simulation studies of 1D cable models, the local pacing protocol has been used. However, the heart is a 3D organ, and the conduction of the action potential follows a certain pathway. Therefore, the nodal line (or nodal surface) dynamics may depend on the pacing protocol. For example, as shown in Fig. 9, the dynamics of an initial condition induced nodal line is different under a different pacing protocol. When the nodal line is along the direction of propagation [Fig. 9(b)], its dynamics is the same as that under global pacing although there is conduction in the tissue. The reason is that the cells on a vertical line in the 2D tissue are excited simultaneously, equivalent to being paced globally. As shown in the simulation results of 2D tissue models in Figs. 9 and 10 in this study and in our previous study,⁵² the nodal line dynamics depends on the pacing protocol. Therefore, it needs to be cautious in extending the conclusions from the 1D cable to 2D or 3D tissue.

Finally, in this study, we only investigate the voltage-driven alternans. It is well known that alternans can originate from Ca²⁺ cycling instabilities.^24–28 Computer simulations show that Ca²⁺ instabilities may also play important roles in the formation of SDA in cardiac tissue,^73–75 which requires more rigorous analyses to elucidate the underlying dynamical mechanisms. Our current approach can be extended to treat this problem.

V. CONCLUSIONS

Using a combination of models with different levels of complexity, we investigate analytically and computationally the mechanisms of SDA dynamics in cardiac tissue. We show that the SDA stability and dynamics in cardiac tissue are determined by the balance of tensions generated by CVR, convection, and repolarization and coupling heterogeneities, as well as the nodal line curvature. Depending on specific conditions, one or more of the effects may be dominant over the others. For example, under fast heart rates and along the direction of conduction, CVR-induced SDA is the dominant mechanism, while for slow heart rates or tissue layers transverse to the direction of conduction, the non-CVR-induced SDA mechanism becomes important. The mechanistic insights from this study may provide a plausible interpretation of different SDA behaviors observed in experiments and help postulate the potential roles of SDA in cardiac arrhythmogenesis in clinical settings.

SUPPLEMENTARY MATERIAL

See the supplementary material for still images and captions for multimedia movies corresponding to Figs. 9 and 10.

APPENDIX: MATHEMATICAL MODELS

1. The AE model

The AE model describing APD alternans dynamics in a 1D cable derived by Echebarria and Karma^21,55 is (for convenience, we changed some of the notations of parameters and variables used in their original papers)

$$T\frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }t}=\alpha \mathrm{\Delta }a-\beta \mathrm{\Delta }{a}^3-{\int }_0^x\frac{dy}{\mathrm{\Lambda }}\mathrm{\Delta }a(y)-\delta \frac{\mathrm{\partial }\mathrm{\Delta }a}{\mathrm{\partial }x}+{\xi }^2\frac{{\mathrm{\partial }}^2\mathrm{\Delta }a}{\mathrm{\partial }{x}^2},$$ (A1)

where T is the pacing period and $t\equiv nT$ with n being the beat number. $\mathrm{\Delta }a(x,t)$ is the amplitude of alternans defined as $\mathrm{\Delta }a(x,t)={(-1)}^n[(a_{n+1}(x)-a_n(x)]/2$, in which the pre-factor ${(-1)}^n$ maintains the sign of $\mathrm{\Delta }a(x,t)$ during alternans. $a_n$ is the APD of the n^th beat. $\alpha =f^{\mathrm{\prime }\mathrm{\prime }}(T-T_c)/2$ and $\beta ={f{}^{\mathrm{\prime }\mathrm{\prime }}}^2/4-f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}/6$ with the derivatives evaluated at the bifurcation point from no alternans to alternans in a single cell. f is the function describing APDR, i.e.,

$$a_{n+1}=f(d_n),$$ (A2)

where $d_n$ is the DI of the n^th beat. Figure 1(b) shows an APDR curve and the definitions of $a_n$ and $d_n$.

Λ in Eq. (A1) is a parameter describing the effect of CVR, which is given as

$$\mathrm{\Lambda }=c^2/2{\theta }^{\mathrm{\prime }},$$ (A3)

in which c is the CV and ${\theta }^{\mathrm{\prime }}$ is the slope of the CVR curve evaluated at the bifurcation point. θ is the function describing CVR, i.e., CV as a function of DI,

$$c_{n+1}=\theta (d_n).$$ (A4)

When there is no CVR, i.e., CV is not changing with DI, ${\theta }^{\mathrm{\prime }}=0$, and thus $\mathrm{\Lambda }\to \mathrm{\infty }$, then the integral term vanishes in Eq. (A1).

δ in Eq. (A1) describes the strength of the convection effect, which was estimated as $\delta =2D/c$ in which D is the diffusion constant of voltage describing the coupling of myocytes in the ionic model [see Eq. (A16)]. Note that without the integral term, Eq. (A1) becomes the classical convection-diffusion equation for heat, energy, or mass transfer in physical systems with the first derivative term describing the effect of convection.⁷⁶ The convection effect is caused by the motion of the flow in a fluid, which can cause convective instabilities.⁶¹ Similarly, in cardiac conduction, the action potential propagation will exhibit an effect on APD and its dynamics, which is similar to the convection effect in heat transfer in a fluid. In the CML model, we also add a convection term [see Eqs. (A7) and (A8)]. The effect of convection on the nodal line dynamics is clearly demonstrated in the simulations of the ionic model shown in Figs. 7 and 9.

ξ in Eq. (A1) is the diffusion constant for Δa, which is related to D and APD as $\xi =\sqrt{D\times a_c}$ with $a_c$ being the APD at the bifurcation point.

Besides analytical treatments, we also carry out numerical simulations of the AE model using a simple Euler method with $\mathrm{\Delta }x=0.0125\text{cm}$ and $\mathrm{\Delta }t=0.02\text{ms}$.

2. The CML model

a. The 1D CML model for global pacing

Under global pacing, there is no conduction, and thus every cell in the cable has the same excitation period, which is just the pacing period T. Therefore, the DI and APD of a cell satisfy the following relationship:

$$d_n(i)=T-a_n(i),$$ (A5)

where i is the cell index in the cable and n is the beat number. Based on our previous formulation,⁶⁰ $a_n(i)$ is determined as follows:

$$a_n(i)=f[d_{n-1}(i)]+\epsilon {\sum }_{k=-M}^M{w}_k[f[d_{n-1}(i+k)]-f[d_{n-1}(i)]],$$ (A6)

where f is the function describing the APDR as described in Eq. (A2). ɛ is a parameter describing the coupling strength, and M is the maximum length of diffusive coupling. w_k describes the distance-dependent coupling strength, which is a Gaussian function, i.e., $w_k=\frac{e^{-k^2/2{\sigma }^2}}{\sqrt{2\pi }\sigma }$. We use ɛ = 1, σ = 25, and M = 100. No-flux boundary conditions are used. Details of the CML model and the boundary conditions are presented in Wang et al.⁶⁰

b. The 1D CML model for local pacing

Under local pacing, action potential conducts in the cable from pacing site and thus the effects of conduction need to be included. The CML model becomes⁶⁰

$$d_n(i)=T+{\sum }_{k=1}^{i-1}\left(\frac{\mathrm{\Delta }x}{\theta [d_n(k)]}-\frac{\mathrm{\Delta }x}{\theta [d_{n-1}(k)]}\right)-a_n(i)+b_n(i).$$ (A7)

The second term on the right-hand side is the time difference of excitation caused by the difference in conduction (due to CVR) of two consecutive beats. θ is the CVR function as described in Eq. (A4). $\mathrm{\Delta }x=0.0125$ cm is the cell length. $a_n(i)$ is determined by Eq. (A6). The last term in Eq. (A7), $b_n(i)$, is a term describing the effect of convection due to action potential propagation, which is formulated as

$$b_n(i)=\delta {\sum }_{k=1}^M{w}_k[f[d_{n-1}(i)]-f[d_{n-1}(i-k)]].$$ (A8)

This convection term is similar to the one in the AE model [Eq. (A1)], and we used the same symbol δ in both models to describe the strength of the convection effect.

c. The 2D CML model for global pacing

The 1D CML model can be easily extended to a 2D CML model for global pacing. The equations are

$$d_n(i,j)=T-a_n(i,j)$$ (A9)

and

$$\begin{array}{rl}{a}_n(i,j)& =f[d_{n-1}(i,j)]+\epsilon {\sum }_{k=-M}^M{\sum }_{l=-M}^M{w}_{k,l}[f[d_{n-1}(i+k,j+l)]\\ & -f[d_{n-1}(i,j)]],\end{array}$$ (A10)

where $w_{k,l}$ is the weight of coupling strength, i.e., $w_{k,l}=\frac{e^{-(k^2+l^2)/2{\sigma }^2}}{2\pi {\sigma }^2}$. No-flux boundary conditions are used. The parameters are the same as for the 1D CML model. However, it is nontrivial to extend the 1D CML model with local pacing into a 2D CML model with local pacing. This is because it is difficult to define conduction in the lattice when the wavefront becomes curved. Therefore, we only simulate the 2D CML model with global pacing.

d. The APDR and CVR functions

In the CML model, we use the following APDR function:

$$f(d_n)=a_0+a_1/\left(1+e^{-\frac{d_n-d_0}{{\tau }_a}}\right),$$ (A11)

where $a_0$, $a_1$, $d_0$, and ${\tau }_a$ are parameters. f is a sigmoidal function with the minimum value $a_0$, maximum $a_0+a_1$, and the middle point at $d_n=d_0$. ${\tau }_a$ is the parameter changing the slope of the APDR curve without changing the maximum APD. We use $a_0=50\text{ms}$, $a_1=150\text{ms}$, $d_0=100\text{ms}$, and ${\tau }_a=30\text{ms}$ as the default parameters and describe if different values are used.

CVR is described by the following function:

$$\theta (d_n)=\{\begin{array}{c}{\theta }_0\left(1-0.6e^{-\frac{d_n-10}{{\tau }_{\theta }}}\right),d_n\ge 10\text{ms,}\\ 0,d_n<10\text{ms,}\end{array}$$ (A12)

where ${\theta }_0=0.5\text{mm/ms}$ is the maximum CV. ${\tau }_{\theta }$ is the parameter determining the slope of the CVR curve. We set ${\tau }_{\theta }=50\text{ms}$ as the control value.

3. The ionic model

The LR1 action potential model is an eight-variable model. The specific mathematical formulations of the ionic currents and differential equations are referred to the original publication.⁵³ Here, we give a brief description of the LR1 model. The membrane voltage (V) is described by

$$\frac{dV}{dt}=-(I_{ion}+I_{stim})/C_m,$$ (A13)

where C_m= 1 μF/cm² is the membrane capacitance, I_stim is the stimulus current density, and $I_{ion}$ is the total ionic current density consisting of different types of ionic currents, i.e.,

$$I_{ion}=I_{Na}+I_{si}+I_K+I_{K1}+I_{Kp}+I_b,$$ (A14)

where $I_{Na}$ is the Na⁺ current. $I_{si}$ is called the slow inward current, which is known as the L-type Ca²⁺ current. $I_K$, $I_{K1}$, and $I_{Kp}$ are K⁺ currents; $I_b$ is a background current. The individual current is described by a Hodgkin-Huxley type of formulation, i.e., $I_m=G{x}^n{y}^k(V-E_m)$, where G is the maximum conductance and $E_m$ is the reversal potential. x is the activation gating variable and y is the inactivation gating variable with n and k being integers. Both x and y obey the following type of differential equation: $\frac{dx}{dt}=(x_{\mathrm{\infty }}-x)/{\tau }_x$, where $x_{\mathrm{\infty }}$ and ${\tau }_x$ are functions of V. For example, the formulation of I_K is $I_K=G_{K}x{x}_1(V-E_K)$. We alter G_k in space to model the repolarization heterogeneities. To model the I_to-induced APD alternans, an I_to is added to Eq. (A13) as described previously.⁵²

a. Homogenous 1D cable and 2D tissue models

The governing PDE for voltage (V) in the 1D cable is

$$\frac{\mathrm{\partial }V}{\mathrm{\partial }t}=-\frac{I_{ion}+I_{stim}}{C_m}+D\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2},$$ (A15)

where D = 0.001 cm²/ms. I_ion is the total ionic current density [Eq. (A13)] described by the LR1 model. I_stim is the stimulus current density, which is a 2 ms duration and −25 μA/cm² pulse applied periodically with a pacing period T.

The governing PDE for V in the isotropic 2D tissue model is

$$\frac{\mathrm{\partial }V}{\mathrm{\partial }t}=-\frac{I_{ion}+I_{stim}}{C_m}+D\left(\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{y}^2}\right).$$ (A16)

No-flux boundary conditions are used for both 1D and 2D tissue models.

b. Heterogeneous tissue models

Pre-existing repolarization heterogeneities are modeled by altering the maximum conductance (G_K) of the time-dependent K⁺ current (I_K) in space. Details of the G_K distributions are described in the figure legend of each case. The effect of heterogeneous cell coupling is also investigated by using a spatial function for the diffusion constant. For this case, the PDE for V becomes

$$\frac{\mathrm{\partial }V}{\mathrm{\partial }t}=-\frac{I_{ion}}{C_m}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}D(x)\frac{\mathrm{\partial }V}{\mathrm{\partial }x},$$ (A17)

in which D(x) is a function of space.

c. Numerical methods

A forward Euler method with $\mathrm{\Delta }x=\mathrm{\Delta }y=0.0125\text{cm}$ and $\mathrm{\Delta }t=0.01\text{ms}$ was used for numerical simulations of the differential equations. APD and DI [see Fig. 1(b)] are defined by a voltage threshold of −72 mV.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.