Synchronization of spatially discordant voltage and calcium alternans in cardiac tissue

The heart is an excitable medium which is excited by membrane potential depolarization and propagation. Membrane potential depolarization brings in calcium (Ca) through the Ca channels to trigger intracellular Ca release for contraction of the heart. Ca also affects voltage via Ca-dependent ionic currents, and thus, voltage and Ca are bidirectionally coupled. It has been shown that the voltage subsystem or the Ca subsystem can generate its own dynamical instabilities which are affected by their bidirectional couplings, leading to complex dynamics of action potential and Ca cycling. Moreover, the dynamics become spatiotemporal in tissue in which cells are diffusively coupled through voltage. A widely investigated spatiotemporal dynamics is spatially discordant alternans (SDA) in which action potential duration (APD) or Ca amplitude exhibits temporally period-2 and spatially out-of-phase patterns, i.e., APD-SDA and Ca-SDA patterns, respectively. However, the mechanisms of formation, stability, and synchronization of APD-SDA and Ca-SDA patterns remain incompletely understood. In this paper, we use cardiac tissue models described by an amplitude equation, coupled iterated maps, and reaction-diffusion equations with detailed physiology (the ionic model) to perform analytical and computational investigations. We show that, when the Ca subsystem is stable, the Ca-SDA pattern always follows the APD-SDA pattern, and thus, they are always synchronized. When the Ca subsystem is unstable, synchronization of APD-SDA and Ca-SDA patterns depends on the stabilities of both subsystems, their coupling strengths, and the spatial scales of the initial Ca-SDA patterns. Spontaneous (initial condition-independent) synchronization is promoted by enhancing APD instability and reducing Ca instability as well as stronger Ca-to-APD and APD-to-Ca coupling, a pattern formation caused by dynamical instabilities. When Ca is more unstable and APD is less unstable or APD-to-Ca coupling is weak, synchronization of APD-SDA and Ca-SDA patterns is promoted by larger initially synchronized Ca-SDA clusters, i.e., initial condition-dependent synchronization. The synchronized APD-SDA and Ca-SDA patterns can be locked in-phase, antiphase, or quasiperiodic depending on the coupling relationship between APD and Ca. These theoretical and simulation results provide mechanistic insights into the APD-SDA and Ca-SDA dynamics observed in experimental studies.

Figure 1

Spatially discordant action potential duration (APD) and Ca alternans in a rabbit heart. (a) An image of a rabbit heart and the optical mapping area. (b) Simultaneous recordings of voltage and Ca from three different sites marked in (a). (c) Nodal lines (white) in APD alternans map. (d) Nodal lines (white) in the Ca alternans map. Modified from Hayashi et al. [44].

Figure 2

Schematic diagrams of cell-to-cell coupling and Ca and voltage coupling. (a) A schematic diagram of coupling between cells and coupling between voltage and Ca in a chain of cardiac cells. (b) Ca–to–action potential duration (APD) coupling. Increasing Ca (black dashed) can either lengthen APD (positive Ca-to-APD coupling) or shorten APD (negative Ca-to-APD coupling). (c) APD-to-Ca coupling. Lengthening APD in the first beat causes shortening of diastolic interval (DI), which may result in either a smaller Ca (positive APD-to-Ca coupling) or a larger Ca (negative APD-to-Ca coupling) in the second beat. See main text for more detailed description.

Figure 3

Electromechanically concordant and discordant spatially discordant alternans (SDA). (a) An example of an electromechanically concordant SDA in which action potential duration (APD) and Ca alternate in-phase. Top: APD vs cell no. for two consecutive beats. Middle: Peak Ca vs cell no. for the same two consecutive beats. Bottom: Alternans amplitude of APD ($\mathrm{\Delta }a$) and Ca ($\mathrm{\Delta }c$) calculated from the two beats. (b) Same as (a) but for an electromechanically discordant SDA in which APD and Ca alternate in antiphase. (a) and (b) are simulation results of the coupled map lattice (CML) model (see the Appendix) with the following parameters: $T=250$ ms, $\gamma =0.002, {\sigma }_R=0.8, {\tau }_a=40$, and $\beta =4$ for (a) and $T=250$ ms, $\gamma =-0.002, {\sigma }_R=0.8, {\tau }_a=38$, and $\beta =4.6$ for (b).

Figure 4

Synchronization of Ca–spatially discordant alternans (SDA) to action potential duration (APD)-SDA patterns in the amplitude equation (AE) model without Ca-to-APD coupling ($\gamma =0$). (a) Plots of function $f\left(\mathrm{\Delta }c\right)$ [Eq. (12)] for different $\sigma \mathrm{\Delta }a$ values. (b) Space-time plots of $\mathrm{\Delta }c$ for $\sigma =0.1$ (left) and $\sigma =0.5$ (right) for a one-node APD-SDA as indicated. (c) Space-time plots of $\mathrm{\Delta }c$ for $\sigma =0.1$ with two noise strengths: $D=500$ (left) and $D=1000$ (right). (d) $S_{vc}$ vs noise strength $D$. For each $D$, 20 simulations with different random initial conditions are carried out with the corresponding $S_{vc}$ plotted. In the simulations in (b)–(d), the initial conditions for Ca-SDA are spatially random in which $\mathrm{\Delta }c$ is a binary number, randomly chosen as either −100 or 100. $\mathrm{\Delta }a$ is a one-node SDA as indicated in (b). $\alpha =0.5$ and $\rho =0.5$.

Figure 5

Synchronization of action potential duration (APD)–spatially discordant alternans (SDA) to Ca-SDA in the amplitude equation (AE) model without APD-to-Ca coupling ($\sigma =0$). (a) Schematic plot of synchronization of an APD-SDA (blue) to a Ca-SDA (red). Arrows indicate that an initial APD-SDA node away from the Ca-SDA node drifts toward the Ca-SDA node. (b) Simulation of the AE model showing the scenarios in (a). (c) Synchronization of APD-SDA to a Ca-SDA pattern when the spatial scale of Ca-SDA pattern is large. (d) Same as (c) but the spatial scale of Ca-SDA pattern is small. (e) $S_{vc}$ vs the spatial scale ($l$) of the Ca-SDA pattern. $\alpha =0.5, \rho =0.5$, and $\gamma =0.5$. In the simulations in (c)–(e), the initial conditions are spatially random SDAs in which $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ are binary numbers, randomly chosen as either −100 or 100 with spatial segmentation length $l$. $l=80$ cells for (c), and $l=10$ cells for (d). In (e), 20 random initial conditions are used for each $l$.

Figure 6

Spatiotemporal Ca and action potential duration (APD) dynamics in the amplitude equation (AE) model for $\gamma \sigma >0$. (a) Stability boundaries (colored lines) in the ρ-α plane for different spatial modes of the linear stability analysis of the steady state. Shown are the boundary for $k=0$ (red), 4 (blue), and 50 (green). $\gamma =0.5$ and $\sigma =0.5$. (b) Steady-state $\mathrm{\Delta }a$ (black) and $\mathrm{\Delta }c$ (red) in space for beat no. 100 with 1 node in both $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$. (c) Same as (b) but with 20 nodes in both $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$. (d) Space-time plots of $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ with an initial condition of $\mathrm{\Delta }a=0$ and random $\mathrm{\Delta }c$ in which $\mathrm{\Delta }c$ is a binary number, uniformly chosen as either −100 or 100. $\gamma =0.5$ and $\sigma =0.5$. (e) Same as (d) but with $\sigma =0.1$. (f) $S_{vc}$ vs σ. For each σ, 20 $S_{vc}$ values from different initial conditions as in (c) and (d) are plotted. $\gamma =0.5$. (g) Color map of $S_{vc}$ vs α and ρ. For each set of α and ρ, one random initial condition (spatial scale $l=1$ cell) is simulated. $\gamma =0.5$ and $\sigma =0.3$. The white line is the synchronization boundary predicted by Eq. (13) under $\gamma =0$, showing that positive Ca-to-APD coupling ($\gamma >0$) enhances spontaneous synchronization. (h) Same as (g) but with a larger spatial scale of the initial conditions: $l=20$ cells.

Figure 7

Spatiotemporal Ca and action potential duration (APD) dynamics in the amplitude equation (AE) model for $\gamma \sigma <0$. (a) Stability boundaries (colored lines) in the ρ-α plane for different spatial modes of the linear stability analysis of the steady state. Shown are the boundaries for $k=0$ (red), 4 (blue), and 50 (green). The spatial mode is stable inside (in the lower-left direction of) the solid lines. $\gamma =-0.5$ and $\sigma =0.5$. (b) Steady-state $\mathrm{\Delta }a$ (black) and $\mathrm{\Delta }c$ (red) showing an electromechanically concordant spatially discordant alternans (SDA) in which $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ are in-phase. $\alpha =0.6$ and $\rho =-0.8$, marked by the circle in (a). (c) Steady-state $\mathrm{\Delta }a$ (black) and $\mathrm{\Delta }c$ (red) showing an electromechanically discordant SDA in which $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ are antiphase. $\alpha =-0.7$ and $\rho =0.5$, marked by the square in (a). (d) Space-time color-scale plots of $\mathrm{\Delta }a$ (left) and $\mathrm{\Delta }c$ (middle) showing a quasiperiodic SDA in which $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ are not phase-locked but quasiperiodic (right). $\alpha =0.6$ and $\rho =-0.3$, marked by the diamond in (a). (e) A uniform initial condition leads to a stable steady state (left, no alternans), but a nonuniform (one-node) initial condition leads to multinode electromechanically discordant SDA (right). $\alpha =-0.5$ and $\rho =0.4$, marked by the triangle in (a). (f) A one-node initial condition leads to a quasiperiodic SDA (left), but a multinode initial condition leads to a multinode electromechanically discordant SDA. $\alpha =0.6$ and $\rho =0.4$, marked by the pentagon in (a). In (e) and (f), the upper panels are color-scale plots of $\mathrm{\Delta }a$, and the lower panels are those of $\mathrm{\Delta }c$.

Figure 8

Color maps of $S_{vc}$ vs σ and γ in the amplitude equation (AE) model. (a) Purely random initial conditions of $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ (spatial scale $l=1$ cell). (b) Random initial conditions with a spatial scale of $l=20$ cells. $\alpha =0.5$ and $\rho =0.5$. Note that in the upper-left and lower-right quadrals, $\gamma \sigma <0$, the spatially discordant alternans (SDA) dynamics is quasiperiodic, $S_{vc}$ measured at the last beat depends on the status the SDA pattern, which varies from beat to beat. Therefore, the $S_{vc}$ values in these two quadrals exhibit a randomlike pattern in the color map. In the upper-right (electromechanically concordant) and lower-left (electromechanically discordant) quadrals, the SDA patterns are phase-locked and stable.

Figure 9

Spatiotemporal Ca and action potential duration (APD) dynamics in the coupled map lattice (CML) model for $\gamma >0$. (a) Stability boundary in the ${\tau }_a$-β plane for $k=0$ (or a single cell). (b) $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ vs cell no. recorded from beat no. 999 and no. 1000 with a single node. (c) Same as (b) but with multiple nodes. (d) $\mathrm{\Delta }a$ and $\mathrm{\Delta }c$ vs $n$ with a random initial condition. (e) Same as (d) but with a weaker coupling, ${\sigma }_R=0.3$. (f) $S_{vc}$ vs ${\sigma }_R$. 20 different random initial conditions are used for each ${\sigma }_R$. (g) $S_{vc}$ vs $T$. 20 random initial conditions are used for each $T$. Arrows mark the $T$ for the transition between desynchronized and synchronized SDA patterns. In (f) and (g), the cases of $S_{vc}=1$ correspond to SCA. (h) $a$ and $c$ vs $T$ for a single cell. The default parameters are $T=270$ ms, ${\tau }_a=40, \beta =4, \gamma =0.002$, and ${\sigma }_R=0.4$.

Figure 10

Spatiotemporal Ca and action potential duration (APD) dynamics in the coupled map lattice (CML) model for $\gamma <0$. (a) Phase diagram showing different Ca and APD dynamics vs ${\tau }_a$ and β in a single cell. White: stable steady state; cyan: electromechanically concordant alternans; green: electromechanically discordant alternans; and red: quasiperiodicity. $\gamma =-0.002, {\sigma }_R=0.8$, and $T=250$ ms. (b) Steady-state $\mathrm{\Delta }a$ (black) and $\mathrm{\Delta }c$ (red) showing an electromechanically concordant spatially discordant alternans (SDA). ${\tau }_a=24$ and $\beta =5.7$, marked by the circle in (a). (c) Steady-state $\mathrm{\Delta }a$ (black) and $\mathrm{\Delta }c$ (red) showing an electromechanically discordant SDA. ${\tau }_a=38$ and $\beta =4.6$, marked by the square in (a). (d) Space-time color-scale plots of $\mathrm{\Delta }a$ (left) and $\mathrm{\Delta }c$ (middle) showing a quasiperiodic SDA (right). ${\tau }_a=24$ and $\beta =5.3$, marked by the diamond in (a). (e) A uniform initial condition leads to a stable steady state (left, no alternans), but a nonuniform (one-node) initial condition leads to multinode electromechanically discordant SDA (right). ${\tau }_a=32$ and $\beta =4.6$, marked by the triangle in (a). (f) A one-node initial condition leads to a quasiperiodic SDA (left), but a multinode initial condition leads to a multinode electromechanically discordant SDA. ${\tau }_a=27$ and $\beta =4.5$, marked by the pentagon in (a). In (e) and (f), the upper panels are color-scale plots of $\mathrm{\Delta }a$, and the lower panels are those of $\mathrm{\Delta }c$.

Figure 11

Action potential duration (APD)–spatially discordant alternans (SDA) and Ca-SDA patterns in a one-dimensional (1D) cable of the rabbit ventricular cell model. (a) Bifurcation diagrams showing $a$ and peak $c_i$ vs the pacing period $T$. (b)–(f) Steady-state APD-SDA and Ca-SDA patterns for different initial conditions. The spatial scales of random initial conditions in (d)–(f) are $l=50$ cells, $l=10$ cells, and $l=1$ cell, respectively. $T=180$ ms. Random initiation conditions are set by random initial values of the sarcoplasmic reticulum (SR) Ca load.

Figure 12

Action potential duration (APD)–spatially discordant alternans (SDA) and Ca-SDA nodal ring dynamics in two-dimensional (2D) tissue of the amplitude equation (AE) model. (a) Schematic plots to illustrate the effects of Ca-SDA nodal ring (red) and curvature on APD-SDA nodal ring (dashed blues). The blue arrows indicate the force of the Ca-SDA nodal ring, and the green arrows indicate the force of the curvature on the APD-SDA ring. (b) The critical ring radius ($r_c$) vs σ determined via computer simulations of the AE model [Eqs. (20) and (21)]. $\alpha =0.5, \rho =0.5$, and $\gamma =0.1$. (c) Ring radius $r$ vs beat no. for different initial ring radius for $\sigma =0.1$. Left: The initial APD-SDA nodal ring (blue) radius $r>r_c$. Right: The initial APD-SDA nodal ring (green) radius $r<r_c$. (d) Ring radius $r$ vs beat no. for different initial ring radius for $\sigma =0.3$. Left: The initial APD-SDA nodal ring (blue) radius $r>r_c$. Right: The initial APD-SDA nodal ring (green) radius $r<r_c$. Simulations are done in a 2D tissue of $800\times 800$ cells.

Figure 13

Synchronization of action potential duration (APD)–spatially discordant alternans (SDA) and Ca-SDA patterns when both APD and Ca are unstable. Simulations are done in the amplitude equation (AE) model with $\alpha =0.5, \rho =0.5$, and $\gamma =0.1$. Tissue size is 800 × 800 cells. (a) $S_{vc}$ vs σ for different spatial scales of the initial condition with random block sizes: 1 × 1 cell (black), 10 × 10 cells (red), 20 × 20 cells (blue), 50 × 50 cells (green), and 100 × 100 cells (magenta). Note: Only one random initial condition is used for each σ for the calculation of $S_{vc}$. When $\sigma <{\sigma }_c\approx 0.19, S_{vc}$ depends strongly on the block size of the initial condition. (b) Example APD-SDA (left) and Ca-SDA (right) patterns for $\sigma =0.3$. The initial condition block size is 100 × 100 cells. The calculated synchronization index is $S_{vc}=0.93$. (c) Example APD-SDA (left) and Ca-SDA (right) patterns for $\sigma =0.1$. The initial condition block size is 100 × 100 cells. The calculated synchronization index is $S_{vc}=0.75$. In (b)–(c), the white lines are the nodal lines ($\mathrm{\Delta }a=0$ or $\mathrm{\Delta }c=0$).

Figure 14

Action potential duration (APD)–spatially discordant alternans (SDA) and Ca-SDA dynamics in two-dimensional (2D) tissue when alternans is driven by Ca alone. Simulations are done in the amplitude equation (AE) model with $\alpha =-0.5, \rho =0.5$, and $\gamma =0.1$. Tissue size is 800 × 800 cells. (a) Steady-state APD-SDA (black) and Ca-SDA (red) nodal rings. $\sigma =0.1$. (b) Ring radius vs beat no. showing the time evolution for the APD-SDA (blue and green) and Ca-SDA (red) nodal rings shown in (a). (c) $S_{vc}$ vs σ for different spatial scales of the random initial condition. Block sizes: 1 × 1 cell (black), 10 × 10 cells (red), 20 × 20 cells (blue), 50 × 50 cells (green), and 100 × 100 cells (magenta). (d) Example APD-SDA (upper) and Ca-SDA (lower) patterns for $\sigma =0.1$. The initial condition block size is 160 × 160 cells. The calculated synchronization index is $S_{vc}=0.99$. (e) Example APD-SDA (upper) and Ca-SDA (lower) patterns for $\sigma =0.1$. The initial condition block size is 20 × 20 cells. The calculated synchronization index is $S_{vc}=0.54$. In (d) and (e), the white lines are the nodal lines ($\mathrm{\Delta }a=0$ or $\mathrm{\Delta }c=0$).

Figure 15

Action potential duration (APD)–spatially discordant alternans (SDA) and Ca-SDA dynamics in a two-dimensional (2D) tissue of the rabbit ventricular cell model. Tissue size is 800 × 800 cells and $T=180$ ms. (a) Steady-state APD-SDA (black) and Ca-SDA (red) nodal rings. (b) Ring radius vs beat no. showing the time evolution of the APD-SDA (blue) and Ca-SDA (red) nodal rings. (c) Example APD-SDA (upper) and Ca-SDA (lower) patterns. The initial condition block size is 100 × 100 cells. The calculated synchronization index is $S_{vc}=0.93$. (d) Example APD-SDA (upper) and Ca-SDA (lower) patterns. The initial condition block size is 50 × 50 cells. The calculated synchronization index is $S_{vc}=0.76$. (e) Example APD-SDA (upper) and Ca-SDA (lower) patterns. The initial condition block size is 20 × 20 cells. The calculated synchronization index is $S_{vc}=0.66$. In (c)–(e), the white lines are the nodal lines ($\mathrm{\Delta }a=0$ or $\mathrm{\Delta }c=0$). Random initiation conditions are set by the random initial values of the sarcoplasmic reticulum (SR) Ca load and the gating variable of ${\mathrm{I}}_{\mathrm{Ks}}$.