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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Circ Res. Author manuscript; available in PMC 2010 August 14.
Published in final edited form as:
PMCID: PMC2740370

Dynamical mechanism for subcellular alternans in cardiac myocytes



Cardiac repolarization alternans is an arrhythmogenic rhythm disturbance, manifested in individual myocytes as a beat-to-beat alternation of action potential durations and intracellular calcium transient magnitudes. Recent experimental studies have reported “subcellular alternans,” in which distinct regions of an individual cell are seen to have counter-phase calcium alternations, but the mechanism by which this occurs is not well understood. Although previous theoretical work has proposed a possible dynamical mechanism for subcellular alternans formation, no direct evidence for this mechanism has been reported in vitro. Rather, experimental studies have generally invoked fixed subcellular heterogeneities in calcium-cycling characteristics as the mechanism of subcellular alternans formation.


In this study, we have generalized the previously proposed dynamical mechanism to predict a simple pacing algorithm by which subcellular alternans can be induced in isolated cardiac myocytes in the presence or absence of fixed subcellular heterogeneity. We aimed to verify this hypothesis using computational modeling and to confirm it experimentally in isolated cardiac myocytes. Furthermore, we hypothesized that this dynamical mechanism may account for previous reports of subcellular alternans seen in statically paced, intact tissue.

Methods and Results

Using a physiologically realistic computational model of a cardiac myocyte, we show that our predicted pacing algorithm induces subcellular alternans in a manner consistent with theoretical predictions. We then use a combination of real-time electrophysiology and fluorescent calcium imaging to implement this protocol experimentally, and show that it robustly induces subcellular alternans in isolated guinea pig ventricular myocytes. Finally, we use computational modeling to demonstrate that subcellular alternans can indeed be dynamically induced during static pacing of 1-dimensional fibers of myocytes during tissue-level spatially discordant alternans.


Here we provide the first direct experimental evidence that subcellular alternans can be dynamically induced in cardiac myocytes. This proposed mechanism may contribute to subcellular alternans formation in the intact heart.

Keywords: Subcellular alternans, pattern-formation, arrhythmia, calcium-handling


Cardiac repolarization alternans is a disturbance in the normal rhythm of the heart that has been identified as a precursor to potentially lethal arrhythmias including ventricular tachycardia and ventricular fibrillation.1 Alternans is seen clinically as beat-to-beat alternations in the magnitude of the ECG T-wave, a manifestation of an underlying alternation in the duration of subsequent action potentials (APs) in individual myocytes.2 Using calcium-sensitive indicators, these action potential duration (APD) alternations are seen to coincide with beat-to-beat alternations in the magnitude of intracellular calcium (Ca) transients, either in phase (electromechanically (EM) concordant) or out-of-phase (EM-discordant) depending on the temperature and/or species.3 Although alternans has been mechanistically linked2, 46 to the formation of potentially fatal reentrant arrhythmias, significant uncertainty remains regarding its mechanism and effective therapy aimed at its treatment or prevention has yet to be developed.7 Although much recent experimental evidence has identified instabilities in intracellular Ca-cycling (i.e. sarcoplasmic reticulum (SR) calcium release/reuptake) as the primary cause of alternans, uncertainty remains regarding the contribution of membrane voltage (Vm) instability (due to insufficient time for recovery between excitations) to the total instability seen in alternating cells.1, 7

Alternans has been reported at multiple spatial scales, including spatially discordant alternans in tissue1 and counter-phase Ca-alternans in adjacent cells.8 Most recently, several studies have reported “subcellular alternans,” in which Ca-transients in distinct regions of an individual myocyte alternate out-of-phase. Importantly, the large gradients of calcium concentration formed between adjacent out-of-phase subcellular regions have been shown to predispose the cell to the formation of propagating waves of intracellular calcium release, which are thought to induce arrhythmogenic delayed afterdepolarizations.913 Subcellular alternans has been reported several times in intact tissue8, 14 as well as in isolated cells,10, 11, 15 but the cause of this phenomenon remains unclear.

Two mechanisms have been the most frequently proposed to account for the formation of subcellular alternans. The first and most prevalent attributes these subcellular differences in Ca-alternans either directly or indirectly to the presence of preexisting subcellular heterogeneity in calcium-cycling components along the length of the cell.911, 14 In one recent example, Aistrup et al. explained the subcellular alternans they observed in isolated rat hearts using a mathematical model containing significant heterogeneity in Ca-release properties in discrete regions along the length of the cell. In their model, a pacing protocol that reverses the phase of alternans (in both APD and Ca-transients) can induce subcellular alternans by reversing the phase of Ca-alternans in some regions earlier than others.14

A different hypothesis was proposed by Shiferaw and Karma, in which subcellular alternans results from a purely dynamical mechanism, in the absence of such preexisting subcellular heterogeneity.16 They proposed a well-known dynamical pattern-forming instability (a Turing instability) as a mechanism for subcellular alternans, wherein “a fluctuation that causes the amplitude of Ca-alternans to increase in a small region of the cell generates APD-alternans that in turn inhibits the growth of Ca-alternans far from that localized fluctuation.”16 Their theoretical analysis only applied, however, to EM-discordant myocytes, and has yet to be validated experimentally.

Here we extend this theory to predict a novel means of dynamically inducing subcellular alternans in EM-concordant myocytes and validate this hypothesis in a computational model, as well as in isolated guinea pig ventricular myocytes. Although physiological subcellular alternans may result from a combination of both dynamical and anatomical factors, here we present the first in vitro evidence that subcellular alternans can be dynamically induced without fixed heterogeneity. Furthermore, we hypothesize that this same dynamical mechanism may contribute to subcellular alternans seen during static pacing in intact tissue.

Alternans control

A body of literature has developed around the idea of terminating alternans for antiarrhythmic purposes.1 The most studied approach uses non-static pacing in which the pacing cycle length (T) is adjusted following each AP by an amount proportional to the existing APD-alternans magnitude (the difference in duration between the last two APs, An and An−1), according to


where g is the control gain.1719 Using this simple “alternans control” algorithm, the otherwise long DI following short APs is shortened and the normally short DI following long APs is lengthened. Because of a strong dependence of APD on the preceding DI (APD restitution), these continued cycle-length perturbations force the APDs of an alternating cell towards an unstable period-1 rhythm. As described below, we hypothesize that alternans control dynamically induces subcellular alternans in EM-concordant cells, thereby providing a controlled means to investigate its formation in real cardiac myocytes.


Detailed Materials and Methods are provided as a data supplement. Briefly, computational modeling was conducted in model cells in which a single myocyte consists of a chain of 75 individual sarcomeres (each with its own calcium concentrations) coupled by calcium diffusion (as in Ref. 14, 16). The membrane voltage of all sarcomeres is explicitly synchronized. 1-dimensional fiber simulations were conducted in a 400-cell cable of such model cells, paced from the left end. This model has the distinct advantage that it is easily tuned to exhibit either positive or negative Ca→Vm coupling as well as Vm-driven, Ca-driven, or Vm and Ca-driven (hereafter “mixed instability”) alternans (as described in data supplement). In all figures and movies, Ca-alternans is defined as Δc =(−1)n ·(cncn−1), where cn is the maximum [Ca2+]i of each sarcomere during beat n. APD-alternans is defined as ΔAPD =AnAn−1, where An is the APD of beat n.

Experimental studies were performed in isolated left ventricular myocytes from adult Hartley guinea pigs loaded with Fluo-4 AM fluorescent calcium indicator. Cells were paced at cycle lengths sufficiently fast to induce steady-state APD alternans (range 200 to 400ms), following which alternans control pacing (varying the cycle length in real-time using Eq. 1) and calcium imaging were initiated. All experiments were performed at room temperature.


Dynamical mechanism for subcellular alternans in EM-concordant myocytes

We hypothesized that alternans control can induce subcellular alternans in EM-concordant myocytes by a mechanism analogous to the dynamical instability previously predicted to cause subcellular alternans only in EM-discordant myocytes.16 Alternans control is predicted to force an alternating EM-concordant cell’s APDs towards their unstable period-1 rhythm while simultaneously amplifying any small spatial fluctuations in Ca-alternans magnitude into a marked subcellular alternans pattern, as follows (and seen in Fig. 1).

Figure 1
In the presence of fluctuations in Ca-alternans (Δc) between sarcomeres, alternans control pacing induces a marked subcellular alternans pattern in the EM-concordant model cell. Panels 1–5 are snapshots from Supplemental Video 2, showing ...

Central to our hypothesis is the notion that the large space constant of membrane voltage relative to calcium diffusion (and the length of a single cell) allows voltage to rapidly equilibrate on the scale of an individual myocyte, whereas the Ca-transients (and Ca-alternans magnitudes) can vary over much smaller distances.20 The effects of each local Ca-transient on Vm (through Ca-sensitive membrane currents), however, will quickly average with the effects of all others, meaning Vm (and APD) is influenced by the whole-cell average Ca-transient.

Given a perfectly homogeneous cell, the Ca-transients of each sarcomere are equal to the whole-cell average Ca-transient. In an EM-concordant cell, the magnitude and phase of APD-alternans (itself dependent on the whole-cell average Ca-transient, as above) will therefore be an accurate reflection of the local Ca-alternans of each sarcomere. In such a cell, the cycle-length perturbations of alternans control (which considers only APD-alternans magnitude) will appropriately control Ca-alternans at all points along the length of the cell, such that all points of the cell are predicted to converge on a period-1 Ca-transient rhythm (as seen in Supplemental Video 1).

In a real cell, however, fluctuations in calcium transients (and therefore Ca-alternans magnitude) exist between different subcellular regions due to, for example, the stochastic nature of calcium release, as well as any fixed subcellular heterogeneity in calcium release properties.20 In such a cell, the alternans control perturbations, a function of only the APD, will not be appropriately sized to control Ca-alternans at all points in the cell. If, for instance, some subcellular regions of the cell have Ca-alternans magnitude even slightly greater than the average Ca-alternans magnitude (and thus other parts have slightly less than the average) the cycle-length perturbations delivered by alternans control will be too small for control of the former regions and too large for the latter.

This is seen in Fig. 1 (and Supplemental Video 2), in which a model cell (described in Materials and Methods) is paced with alternans control. Note that in this simulation, for the sake of clarity a large, smooth gradient was added to initial Ca-concentrations (and thus Ca-alternans magnitude); similar results are seen with the incorporation of only small random noise (as was done in all other simulations in this study). When such a cell is far away from its period-1 rhythm, alternans control will force all regions of the cell toward their unstable period-1 Ca-transient rhythm (Fig. 1, panels 1–2). As control continues, however, some regions will have reached their period-1 Ca-rhythm (Ca-alternans magnitude = 0) before the whole-cell average Ca-transient is at a period-1 rhythm. Because the APD is still alternating (due to its dependence on the whole-cell average Ca-transients), alternans control will continue to perturb the cycle length, reversing the phase of Ca-alternans (changing the sign of Ca-alternans magnitude) in those regions that had the lowest initial Ca-alternans magnitude (Fig. 1, panel 3). For such regions of the cell, with Ca-alternans phase now of opposite-phase to that of the whole-cell average Ca-alternans (and thus the APD-alternans), the cycle-length perturbations delivered by alternans control will now amplify these counter-phase Ca-alternans. The magnitude of Ca-alternans in these regions will grow until the regions of opposite phase Ca-alternans balance, such that the whole-cell average Ca-alternans magnitude (and thus APD-alternans magnitude) will approach 0 (Fig. 1, panel 4). As this point is approached, alternans control will deliver progressively smaller perturbations to the cycle length (Eq. 1), approaching static pacing. Even with precisely static pacing, however, period-1 calcium dynamics are unstable (without such instability there would have been no alternans in the first place); Ca-alternans will therefore grow in magnitude at all points along the cell, regardless of phase (Fig. 1, panel 4–5), forming a marked subcellular alternans.

This mechanism predicts subcellular alternans will occur in EM-concordant myocytes during alternans control, but only in the presence of spatial variations in intracellular Ca concentrations (which can be satisfied by stochastic noise in initial conditions), and only if alternans is primarily driven by Ca-cycling instabilities.

Single Cell Modeling Results

Static pacing

When the model is tuned to exhibit EM concordance (positive Ca→Vm coupling), static pacing is unable to cause subcellular alternans at any pacing cycle length. The small degree of noise included in initial Ca-concentrations quickly smoothes out through Ca-diffusion between sarcomeres, which become approximately synchronized within 75 beats. In order to exhibit subcellular alternans (by supporting the previously proposed Turing mechanism16) during static pacing, it must be tuned to EM discordance (negative Ca→Vm coupling) and Ca-driven alternans, as in Ref. 16. Interestingly, with this parameter regime, the model never exhibits APD or whole-cell Ca-alternans. The Ca-transients of the two halves of the cell are always either at a period-1 rhythm or are exactly out-of-phase, making the whole-cell average Ca-transient a period-1 rhythm over all pacing cycle lengths (data not shown), inconsistent with most single-cell electrophysiology studies, in which APD alternans is seen with rapid pacing.

Alternans control induces subcellular alternans in an EM-concordant model cell

When the model is tuned to exhibit EM concordance, although subcellular alternans is never seen with static pacing, the model does undergo a more characteristic rate-dependent alternans bifurcation into a stable period-2 APD and whole-cell concordant Ca-alternans rhythm. As seen in Fig. 2 and Supplemental Video 3, alternans control forces the period-2 APD and whole-cell Ca-transients towards their unstable period-1 rhythm (Fig. 2A, black traces). Interestingly, this transition to a period-1 rhythm occurs simultaneously with a symmetry-breaking transition from whole-cell concordant Ca-alternans to subcellular alternans, in which the two halves of the cell are out-of-phase (causing a period-1 whole-cell average Ca-transient rhythm; Fig. 2A–B). Along the length of the model cell, Ca-alternans is seen to be largest at two terminal antinodes surrounding a single, central node with no Ca-alternans. Because Vm (and thus APD) is explicitly dictated by the ionic currents averaged over the length of the cell (in this model), the effects of calcium-sensitive membrane currents in each of the two halves average out to produce a period-1 APD. When control is released by terminating the infinitesimally small perturbations being applied to the static pacing interval, this transition is traversed in the opposite direction, causing a smooth return to the original whole-cell concordant Ca-alternans rhythm (not shown).

Figure 2
In the EM-concordant model, subcellular alternans occurs only when alternans control is applied. A, Alternans control is applied beginning with the fourth beat to an EM-concordant model cell, causing the APs (first black trace) and the whole-cell average ...

Subcellular alternans occurs in the EM-concordant model at all pacing cycle lengths at which alternans is seen, whenever alternans control is successful. As predicted, this effect is robust over a large range of alternans control gains above a low threshold ( g > 0.7 for most pacing cycle lengths) and is not seen without alternans control or in the absence of noise in initial-condition Ca-concentrations. Subcellular alternans was induced in this model whenever Ca-cycling dynamics were sufficiently unstable (both the mixed instability and Ca-driven models), but as predicted never occurred when alternans was due to Vm-instability. Subcellular alternans became unstable and reverted to whole-cell Ca-alternans with cessation of alternans control for all cycle lengths tested.

Isolated Cell Studies

To test our hypotheses in vitro, we applied alternans control to isolated guinea pig ventricular myocytes while imaging intracellular calcium. Subcellular alternans has previously been reported in intact guinea pig heart8 even though Guinea pig myocytes are well characterized as EM-concordant both as isolated cells22 and in tissue.23, 24 Isolated left ventricular cells (from all myocardial regions) were paced at cycle lengths sufficiently fast to induce steady-state alternans (range 200 to 400ms), after which alternans control was initiated. A total of 51 trials were conducted in 16 different cells (an average of 4 trials were performed in each cell). All recordings showed EM concordance throughout. Alternans control successfully suppressed alternans (in both APD and whole-cell average Ca-transient magnitude) in 45 of the 49 trials in which it was applied, and 42 of those 45 successful trials showed subcellular Ca-alternans (as in Fig. 3A–C, Supplementary Videos 5 and 6). Subcellular alternans was never seen without control, and none of the trials in which control failed showed subcellular alternans. This effect was seen at all cycle lengths tested (at which alternans occurred), independent of alternans control gain, and was robust to changes in patch-clamp configuration (perforated vs. ruptured patch) and dye affinity (Fluo-4 vs. Fluo-4 FF).

Figure 3
Alternans control induces subcellular alternans in isolated ventricular myocytes. A, Alternans control causes a steady-state, period-2 APD (first black trace) and whole-cell Ca-alternans (second black trace) to transition to a period-1 rhythm. Concurrently, ...

In 13 of 17 trials in which alternans control was released (static pacing resumed) during imaging, subcellular alternans transitioned back to whole-cell concordant alternans (as in Fig. 4, Supplementary Videos 5 and 6). In the 4 trials in which the subcellular alternans persisted after control was released, the APDs diverged to stable alternans after imaging had ended, implying that a longer imaging period would have captured a return to whole-cell concordant Ca-alternans as well.

Figure 4
During static pacing, subcellular alternans is unstable and transitions back to whole-cell concordant Ca-alternans. Alternans control was used to force a cell to near its period-1 APD rhythm (thereby inducing a 1-node subcellular alternans), at which ...

6 of the 16 cells tested showed more than 1 node (and more than 2 antinodes) of Ca-alternans during alternans control in one or more trials, in patterns similar to those seen in the mathematical model (as in Fig. 3A–C, Fig. S1 and Supplementary Video 6). The majority of cells showed a similar pattern and number of nodes in each trial, but 5 of the 16 cells exhibited different numbers of nodes in different trials at the same cycle length (as seen in Figure 3B,D).

1-Dimensional Fiber Simulations

Subcellular alternans is induced during static pacing of a simulated 1-dimensional fiber during tissue-level spatially discordant alternans

We further hypothesized that previous reports of subcellular alternans in statically paced, intact tissue (Ref. 8 and 14) may also be accounted for by our proposed dynamical mechanism. In particular, the cycle length perturbations induced by the alternans control protocol are directly analogous to the local activation times of cells away from the pacing site in whole heart given sufficiently steep conduction velocity (CV) restitution.21 We therefore predicted that given sufficiently steep CV restitution, subcellular alternans can be induced by the same dynamical mechanism in EM concordant tissue during static pacing.

To test this hypothesis, we implemented a cable of 400 model cells, each one itself a cable of 75 sarcomeres. The cells were tuned to be EM-concordant, with calcium-driven alternans (as described in the Supplemental Materials and Methods) and contained no fixed heterogeneity in Ca-release properties. The left-most five cells of the cable were statically paced for 300 beats at a slow cycle length (T=400ms, at which there is no alternans at any spatial scale) and then for an additional 200 beats at a faster cycle length (either 330ms or 320ms). When the pacing cycle length was decreased to T=330ms, conduction velocity was not steep enough to induce tissue-level spatially discordant alternans (SDA). In this case, alternans is seen in the Ca-transients of every sarcomere as well as the APD of every cell, but subcellular alternans is not observed (Fig. 5A). In contrast, when the pacing cycle length was instead decreased to 320ms, tissue-level spatially discordant alternans arises in both the APD and the whole-cell average calcium (Fig. 5B, top and middle panel). Additionally, over the course of the 200 beats at this faster cycle length, a prominent subcellular alternans transiently develops in the cells near the node of tissue-level spatially discordant alternans (Fig. 5B, bottom panel). Fig. 5C shows a pseudo-line scan image of a single cell near the node of SDA (the cell with the most significant subcellular alternans). The patterns of subcellular alternans formed were similar to those seen in previous experiments in intact rat heart (Ref. 8, 14), with nodes (dashed lines) migrating along the length of the cell as the simulation progressed until the entire cell has reversed phase of Ca-alternans. Note that in this model the membrane voltage is explicitly synchronized for all sarcomeres in each myocyte. Subcellular pattern formation in these simulations reflects a purely dynamical phenomenon.

Figure 5
Subcellular alternans is dynamically induced during static pacing of a simulated 1-dimensional fiber of EM-concordant myocytes when tissue-level spatially discordant alternans forms. A 1-dimensional fiber of model myocytes was stimulated (from the left) ...


Subcellular Ca-alternans has been reported multiple times in the intact heart as well as in isolated cardiac myocytes, but the mechanism for this striking phenomenon remains unclear. Although previous studies have invoked subcellular heterogeneities in Ca-cycling characteristics to explain the occurrence of subcellular alternans, we believe that the rich cardiac myocyte dynamics (which are known to contribute to alternans at the cellular and tissue level) cannot be discounted. An elegant theoretical study proposed a dynamical mechanism for subcellular alternans, 16 but it was believed relevant to only a subset of cardiac myocytes (EM-discordant cells) and has never been verified experimentally. We have shown that this same dynamical mechanism is predicted to cause subcellular alternans in the remaining subset of myocytes (EM-concordant cells) during alternans control pacing and have verified this prediction using computational modeling and in vitro electrophysiology. Importantly, this mechanism can induce subcellular alternans in the absence of subcellular heterogeneity in Ca-cycling characteristics. Although this mechanism can also induce subcellular alternans in the presence of such heterogeneity, the patterns seen experimentally do not suggest that such heterogeneity plays a major role (Supplemental Results; Fig. S1). Importantly, alternans control represents a controlled means of inducing this phenomenon in real cells, allowing in vitro confirmation for the first time.

Alternans control induces subcellular alternans in EM-concordant myocytes

When alternans control pacing was applied to an EM-concordant mathematical model, subcellular alternans was induced in a robust manner. Notably, subcellular alternans was induced in this manner without the need for fixed heterogeneity, thus predicting that a purely dynamical mechanism can induce subcellular alternans. Our in vitro studies similarly demonstrated that alternans control robustly induces subcellular alternans in (EM-concordant) isolated guinea pig ventricular myocytes. We believe that although there is undoubtedly fixed subcellular heterogeneity in real myocytes, our results support a primarily dynamical induction of subcellular alternans in our experiments and do not suggest the presence of heterogeneity of the type previously proposed to account for subcellular alternans formation.

Several experimental observations support these conclusions: (1) a change from static pacing to alternans control was sufficient to robustly induce a smooth transition to subcellular alternans. This agrees with our theoretical and mathematical modeling work, in which this pacing algorithm can induce subcellular alternans in cells by a purely dynamical mechanism. (2) Release of alternans control (resumption of static pacing) during subcellular alternans caused this rhythm to lose stability and return to a period-2 APD and whole-cell concordant Ca-alternans rhythm. It is important to note that successful alternans control (as was achieved prior to release in Figure 4) forces a cell towards its period-1 rhythm with progressively smaller perturbations to the cycle length, culminating in very small changes to the pacing interval around the target cycle length (nearly static pacing). As such, the transition from alternans control to static pacing is a smooth transition. (3) The nodes of subcellular alternans were distributed approximately symmetrically about the midline, as has been reported in several other studies.8, 10, 11, 15 In contrast, our mathematical modeling results indicate that in the presence of the large, fixed gradients of calcium-cycling components previously proposed to cause subcellular alternans (Fig. S1B), the pattern of subcellular alternans would be strongly dependent on the distribution of this underlying heterogeneity. We are unaware of any previous work that suggests the presence of the symmetric gradients of subcellular heterogeneity that would be required for the symmetrical pattern of subcellular alternans seen in our experimental studies (and those of others) to be caused by underlying heterogeneity alone. (4) The patterns of subcellular alternans were seen to vary in different trials of the same cell at the same cycle length (5/16 cells; as in Figure 3B,D). This is consistent with the dynamical model of Figure S1A but inconsistent with the predictions of the model when large, fixed gradients of calcium-cycling components were included (Figure S1B). Notably, with only a small degree of stochastic fixed heterogeneity in calcium-cycling components, the model was seen to exhibit different patterns in different trials with nodes symmetrically distributed about the midline, consistent with our experimental results. Future studies employing confocal imaging will be needed to precisely determine 3-dimensional cellular morphology and the location of nodal lines during subcellular alternans.

Notably, the occurrence of subcellular alternans during alternans control is only predicted when Ca-cycling instability is greater than instability due to membrane voltage dynamics (if this were not the case, all Ca-transients would alternate secondary to APD-alternans and would thus be synchronized). The occurrence of subcellular alternans in our guinea pig myocytes therefore supports the findings of several previous studies indicating that alternans is primarily due to Ca-cycling dynamics. Testing for subcellular alternans during alternans control pacing may represent a useful tool for determining the primary source of instability in other cell types/species as well as at different pacing cycle lengths.1

In intact tissue, CV restitution may induce subcellular alternans by this mechanism, even during static pacing

The dynamical induction of subcellular alternans in EM-concordant myocytes may also occur physiologically, and may contribute to rate-dependent subcellular alternans seen in (non-controlled) intact tissue.8, 14 In tissue, at sufficiently fast pacing rates the effect of steep conduction velocity (CV) restitution produces an effect similar to alternans control in the local activation times of cells distant from the pacing site. Specifically, if APD alternans occurs at the pacing site during static pacing, CV-restitution dictates that “long” APs will propagate away faster than “short” APs. Cells away from the pacing site will thus experience effectively non-static pacing, with the excitations following their “long” APDs arriving late (because they are carried by a more slowly traveling, “short” AP) and those following their “short” APDs arriving early (because they are carried by a faster, “long” AP). The further away from the pacing site, the greater the effect of propagation velocity on cycle length, and the stronger is this control-like effect. Indeed, these cycle length alternations can be strong enough to entirely eliminate APD-alternans (forming a period-1 node) and, even further away, reverse the phase of APD-alternans relative to that at the pacing site (forming tissue-level spatially discordant alternans). Notably, such cycle length alternations were observed concomitant with the subcellular alternans recently observed during static pacing of rat heart.14

Indeed, we found that when tissue-level spatially discordant alternans is induced by static pacing in a simulated 1-dimensional fiber of EM-concordant myocytes, subcellular alternans is dynamically induced in those cells adjacent to the tissue-level node (where cycle length alternations are the most pronounced). Although there are clearly many dynamical and anatomical differences between isolated cells and the intact heart, we believe that the proposed dynamical mechanism may contribute to the formation of physiologic subcellular alternans.

It is important to note that although subcellular alternans can be induced dynamically in isolated cells, the physiological formation of subcellular alternans is likely governed by a rich interplay of dynamical as well as anatomical factors. Still, we believe that our identification of a robust, dynamical mechanism by which subcellular alternans can be induced in isolated cardiac myocytes not only provides novel insights into the mechanism of this phenomenon in the real heart, but also will facilitate future studies into the formation and consequences of subcellular alternans and biological pattern formation.

Supplementary Material



We gratefully acknowledge Dr. Andrew Zygmunt and Dr. Lai-Hua Xie for valuable technical assistance, and Dr. Yohannes Shiferaw for providing model code.


This work was supported by NIH MSTP grant GM07739 (S.A.G.), NIH grant R01RR020115 (D.J.C.), and the Kenny Gordon Foundation.

Non-standard Abbreviations and Acronyms

Cytosolic calcium concentration
Action Potential
Action Potential Duration
Diastolic Interval
Conduction Velocity
Spatially Discordant Alternans
Membrane Voltage
Sarcoplasmic Reticulum
Cycle length
Target cycle length (during alternans control pacing)
Alternans control gain
Beat number
APD of beat n
Peak [Ca2+]i of the Ca-transient of beat n
Δ cn
Ca-alternans magnitude of beat n
Slope of the dependence of SR calcium release magnitude on SR calcium load
Degree of Ca-induced inactivation of L-type Ca-channels
Time constant of recovery from inactivation of L-type Ca-channels
Real-Time Experimental Interface
Peak concentration of the whole-cell average Ca-transient of beat n
Whole-cell average steady-state Ca-alternans magnitude
Difference in Ca-concentration between the largest and smallest subcellular Ca-transients of beat n
APD-alternans magnitude


Subject Codes: [132] Arrhythmias-basic studies, [136] Calcium cycling/excitationcontraction coupling




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