Action Potential Morphology Influences Intracellular Calcium Handling Stability and the Occurrence of Alternans

Abstract

Instability in the intracellular Ca²⁺ handling system leading to Ca²⁺ alternans is hypothesized to be an underlying cause of electrical alternans. The highly coupled nature of membrane voltage and Ca²⁺ regulation suggests that there should be reciprocal effects of membrane voltage on the stability of the Ca²⁺ handling system. We investigated such effects using a mathematical model of the cardiac intracellular Ca²⁺ handling system. We found that the morphology of the action potential has a significant effect on the stability of the Ca²⁺ handling system at any given pacing rate, with small changes in action potential morphology resulting in a transition from stable nonalternating Ca²⁺ transients to stable alternating Ca²⁺ transients. This bifurcation occurs as the alternans eigenvalue of the system changes from absolute value <1 to absolute value >1. These results suggest that the stability of the intracellular Ca²⁺ handling system and the occurrence of Ca²⁺ alternans are not dictated solely by the Ca²⁺ handling system itself, but are also modulated to a significant degree by membrane voltage (through its influence on sarcolemmal Ca²⁺ currents) and, therefore, by all ionic currents that affect membrane voltage.

INTRODUCTION

Cardiac electrical alternans appears to be a diagnostic indicator for vulnerability to arrhythmias (1–4). Recent evidence also demonstrates that electrical alternans can provide a suitable substrate for the initiation of reentry, suggesting a potential mechanistic link to the onset of arrhythmias (5–8). Recent research has focused on understanding the physiological mechanisms underlying alternans (9–19), and on developing new ways of eliminating alternans (15,20–25).

Particular interest has been paid to the role of the intracellular Ca²⁺ handling system in electrical alternans (26). This focus is due in part to experiments conducted in isolated rabbit ventricular myocytes, where it was demonstrated that the intracellular Ca²⁺ transient could alternate on a beat-to-beat basis during voltage clamping even though the applied voltage waveform was a train of identical action potentials (27). This demonstration, that Ca²⁺ alternans may occur independently of electrical alternans, appears to support the hypothesis that electrical alternans is caused by Ca²⁺ alternans (26). Based on this hypothesis, mathematical modeling has suggested that the intracellular Ca²⁺ handling system instability that leads to Ca²⁺ alternans during voltage clamping (i.e., in the absence of electrical alternans) can be explained by a steep and nonlinear dependence of Ca²⁺ release from the sarcoplasmic reticulum (SR) on SR Ca²⁺ load and diastolic Ca²⁺ concentration (19,28,29). Recent experimental results also point to the importance of the steep dependence of Ca²⁺ release on SR Ca²⁺ load and diastolic Ca²⁺ concentration (13,30).

Although the properties of the SR are important in understanding the regulation of intracellular Ca²⁺ concentration, other processes such as the passage of Ca²⁺ through voltage- and Ca²⁺-sensitive L-type Ca²⁺ channels (I_Ca) and Na⁺-Ca²⁺ exchangers (I_NaCa) also play a role in Ca²⁺ regulation. The voltage-sensitive gating of these sarcolemmal currents suggests that the morphology of the action potential should influence their activity and, therefore, the occurrence of Ca²⁺ alternans. Action potential clamp experiments conducted using rat and rabbit ventricular myocytes have shown that the morphology of the voltage waveform does have a marked influence on the shape of the intracellular Ca²⁺ transient (31). However, these experiments were conducted at excitation rates that were too slow for Ca²⁺ alternans to develop. In a mathematical model of intracellular Ca²⁺ handling that reproduces the rabbit ventricular myocyte action potential clamp experiments discussed above (29), and in recent experiments investigating Ca²⁺ alternans in rat myocytes during voltage clamping (13), it was demonstrated that the Ca²⁺-selective sarcolemmal currents alternate during Ca²⁺ alternans. These currents alternated despite being clamped by voltage waveforms that were identical on a beat-to-beat basis. However, the influence of different voltage waveforms on the behavior of the intracellular Ca²⁺ handling system and the incidence of Ca²⁺ alternans was not investigated in these studies. Recent experiments with epicardial and endocardial myocytes isolated from guinea pig hearts investigated the onset of Ca²⁺ alternans in response to two different action potential morphologies during action potential clamping (30). This study found that the excitation rate at which Ca²⁺ alternans was initiated during clamping did not depend upon which of the two action potentials were used to perform clamping (Figs. 2 and 4 in Wan et al. (30)). However, the influence of action potential morphology on Ca²⁺ handling system stability was not systematically investigated in these experiments.

FIGURE 2.

(A) Intracellular Ca²⁺ concentration, (B) Na⁺-Ca²⁺ exchange current, and (C) L-type Ca²⁺ channel current for two different values of β at a pacing period of T = 300 ms. When β = 2.0, all three components exhibit a stable period-2 alternans rhythm. For β = 0.9 alternans does not occur in any component, indicating that the period-1, nonalternating rhythm is stable.

FIGURE 4.

(A) Alternans eigenvalue () as a function of APD₅₀ at pacing periods T = 250 ms (black crosses), T = 320 ms (red circles), and T = 400 ms (blue triangles). For T = 250 ms and T = 320 ms, the transition from no alternans to alternans is characterized by a rapid change in with the switch from a very stable Ca²⁺ handling system to an unstable system occurring in response to a very small change in action potential morphology (as measured by the action potential duration at 50% repolarization [APD₅₀]). For T = 400 ms, the period-1 solution becomes much less stable in response to changes in action potential morphology, but does not become unstable. (B) The relationships between β and APD₅₀ for T = 250 ms (black line), T = 320 ms (red line), and T = 400 ms (blue line).

The influence of action potential morphology at any given excitation rate on the stability of the Ca²⁺ handling system and the development of Ca²⁺ alternans therefore remains poorly understood. Using the mathematical model that reproduces the action potential clamp experiments discussed above (29), we explored the stability of the Ca²⁺-handling system in response to different action potential morphologies. We demonstrate that the morphology of the action potential during voltage clamping is a significant factor in determining the stability of the Ca²⁺-handling system, and therefore plays an important role in determining whether Ca²⁺ alternans occurs. Given that the morphology of the action potential waveform during voltage clamping reflects the contribution of every membrane current, this suggests that every membrane current, and not just those of the Ca²⁺ handling system, influences the conditions under which alternans occurs.

MATERIALS AND METHODS

Mathematical model

Simulations were performed using a model of the intracellular Ca²⁺ handling system of a ventricular myocyte subjected to action potential voltage clamping (29). The model incorporates cytosolic, submembrane, junctional SR, and network SR Ca²⁺ concentrations, L-type Ca²⁺ (I_Ca), Na⁺-Ca²⁺ exchange (I_NaCa), and SR uptake (I_up) currents, Ca²⁺-induced Ca²⁺ release from the SR, and Ca²⁺ buffering. The membrane voltage was clamped according to

(1)

where T is the pacing period (in seconds), m denotes beat number, and β and x are parameters that characterize the morphology of the action potential. The parameter β allows the shape of the action potential in the plateau and repolarization phases to be varied while changing the action potential duration at 90% repolarization only slightly (if at all). β was varied between 0.8 and 2.0 in our simulations (β= 2.0 was the value used in the original modeling study (29)). The parameter x, which is the ratio of action potential duration to pacing period, varies with T according to

(2)

Stability analysis

The stability of the intracellular Ca²⁺ handling system was characterized using the “eigenmode” method (10), which was adapted to analyze the intracellular Ca²⁺-handling model used in this study. Briefly, the method involves: i), locating the period-1 (i.e., no alternans) solution for a given pacing period T (which may be inherently stable, as in situations where no Ca²⁺ alternans naturally occurs at that value of T, or inherently unstable, as in situations where Ca²⁺ alternans naturally occurs at that value of T); ii), systematically applying small perturbations to each state variable (i.e., Ca²⁺ concentrations, ion channel gating variables, etc.); and iii), calculating the eigenvalues of the system from the response of the state variables to the perturbations. As there are nine state variables in the model (V, [Ca²⁺]_s, [Ca²⁺]_i, [Ca²⁺]_j, [Ca²⁺]_j′, I_rel, f, q, [CaT]_i, and [CaT]_s; see Shiferaw et al. (29) for details), nine eigenvalues are calculated for each set of conditions. The alternans eigenvalue (), defined as the largest negative eigenvalue (10), characterizes the stability of the corresponding alternans eigenmode, and hence characterizes the stability of the period-1, nonalternating rhythm. If a stable, period-1, nonalternating rhythm will be present, whereas if the period-1 rhythm is unstable. Systems characterized by larger values of are less stable than systems characterized by smaller values of

RESULTS

Fig. 1 illustrates the effect of changing the value of β in Eq. 1 on the morphology of the action potential waveform applied during voltage clamping at a pacing period T = 300 ms. As Fig. 1 A illustrates, varying β markedly changes the slope of the plateau phase of the action potential (as indicated by the different action potential duration (APD) restitution curves calculated at 50% repolarization). Over this range of β-values the APD at 90% repolarization is practically identical (as indicated by the overlapping APD restitution curves at 90% repolarization).

FIGURE 1.

(A) Action potential morphology as a function of the parameter β (see Eq. 1) at a pacing period of T = 300 ms. (B) Action potential duration (APD) restitution curves for the three values of β in panel A, measured at both 50% (APD₅₀) and 90% (APD₉₀) repolarization. As panel B indicates, APD and diastolic interval (DI) measured at 90% repolarization are practically identical for the values of β shown.

Action potential morphology and Ca²⁺ alternans

Fig. 2 shows how changing the morphology of the action potential waveform at a given excitation rate influences the occurrence of Ca²⁺ alternans. As Fig. 2 A indicates, β = 2.0 at a pacing period T = 300 ms results in stable, period-2 Ca²⁺ alternans, whereas β = 0.9 at the same pacing period results in stable, period-1, nonalternating behavior. Fig. 2 also illustrates the activity of the Ca²⁺ currents I_NaCa (Fig. 2 B) and I_Ca (Fig. 2 C), with both currents alternating for β = 2.0 but neither current alternating for β = 0.9. Given that the intrinsic properties of the system (e.g., ionic current conductances, pump, and exchanger strengths) and the pacing period are identical for both values of β, Fig. 2 suggests that, at any given pacing period, the voltage waveform sensed by the sarcolemmal Ca²⁺ channels and exchangers (and not just the APD, diastolic interval (DI), or pacing period) is intimately involved in determining whether the intracellular Ca²⁺ handling system generates nonalternating or alternating Ca²⁺ transients.

Ca²⁺ handling system stability

The results in Fig. 2 demonstrate the effect of changing the morphology of the action potential waveform during voltage clamping at a single pacing period. Fig. 3 explores the effect of changing the morphology of the action potential on the stability of the Ca²⁺ handling system over a range of values of T. Simulations using three values of β (0.9, 1.5, and 2.0) are shown, with the bifurcation diagrams in Fig. 3 A depicting the peak value of [Ca²⁺]_i from two consecutive beats at steady state plotted as a function of T. The variety of β values reflects different combinations of ionic conductances that generate a range of action potential morphologies.

FIGURE 3.

(A) Bifurcation diagrams illustrating peak [Ca²⁺]_i from two consecutive beats as a function of pacing period for the three values of β shown in Fig. 1. Ca²⁺ alternans occurs over a range of pacing periods for β = 2.0 and β = 1.5, but alternans is absent at all pacing periods for β = 0.9. (B) Alternans eigenvalue () as a function of pacing period for the same values of β and T as in panel A. As panel B shows, for each value of β and T in panel A indicates the presence or absence of alternans, with the presence of alternans characterized by and the absence of alternans characterized by

Fig. 3 A shows that, for β = 1.5 and β = 2.0, peak [Ca²⁺]_i alternates over a range of pacing periods. For β = 2.0, alternans occurs between T = 320 ms and T = 200 ms, whereas for β = 1.5 alternans occurs between T = 340 ms and T = 180 ms. In contrast, there is no alternans for β = 0.9 at any value of T. This indicates that how the action potential morphology changes as a function of the pacing period is important in determining whether Ca²⁺ alternans will occur at any given value of T.

Fig. 3 B illustrates the alternans eigenvalue () calculated for the same values of β and T as in Fig. 3 A. Fig. 3 B shows that for every set of conditions (i.e., each combination of β and T) where Ca²⁺ alternans occurs in Fig. 3 A, whereas corresponds to the stable period-1 behavior seen in Fig. 3 A. Although indicates the absence or presence of alternans (i.e., the degree of stability of the system), Fig. 3 also shows that there is no correlation between and the magnitude of alternans (measured as the difference between the peak [Ca²⁺]_i from two consecutive beats in Fig. 3 A). For example, the period-1 solution for β = 1.5 becomes more unstable than the period-1 solution for β = 2.0 (indicated by the larger in Fig. 3 B), but the magnitude of alternans is always smaller for β = 1.5 than for β = 2.0. Therefore, alternans magnitude cannot be used as a measure of the degree of instability of the system.

Transition to Ca²⁺ alternans

As Fig. 3 B indicates, the nonalternating period-1 behavior becomes very stable (i.e., ) for action potential waveforms corresponding to β = 0.9 at pacing periods shorter than T = 300 ms. This is in stark contrast to the situation for β = 1.5, where the period-1 solution is unstable between T = 340 ms and T = 180 ms. At each pacing period over this range, the nonalternating period-1 solution becomes unstable at some value of β between 0.9 and 1.5. To explore the transition to instability, the value of β was varied at three different values of T, and for each value of β and T was calculated. The resulting values of shown in Fig. 4 A, are plotted as a function of the action potential duration at 50% repolarization (APD₅₀). For reference, the relationships between APD₅₀ and β for each value of T are illustrated in Fig. 4 B.

As Fig. 4 A illustrates, the stability of the nonalternating period-1 solution is highly sensitive to the morphology of the action potential (as measured by APD₅₀) at each pacing period, with the transition to instability resulting from small changes in APD₅₀ when T = 250 ms and T = 320 ms. For both of these values of T, the nonalternating period-1 rhythm reaches a maximal level of instability as APD₅₀ is lengthened, and in both cases the period-1 rhythm remains unstable until the last data point shown. At T = 400 ms, changing the morphology of the action potential is not sufficient to force the system to cross the stability boundary (). However, the system does become markedly less stable as the action potential morphology changes, with this reduction in stability similarly occurring in response to a very small change in morphology.

Stability sensitivity

Fig. 5 illustrates, for two action potential morphologies (corresponding to β = 1.0 and β = 2.0) and pacing periods (T = 250 ms and T = 333 ms, respectively), the sensitivity of to changes in the contribution of particular components of the Ca²⁺ handling system (changes that mimic, for example, the effects of pharmacological agents). Two of the components (u, v_up) are related to the sarcoplasmic reticulum, and two (g_Ca, g_NaCa) are related to the sarcolemmal Ca²⁺ currents. At baseline (corresponding to the 0% line) in both cases, indicating that the system is sitting on the stability boundary.

FIGURE 5.

Sensitivity of the transition to Ca²⁺ alternans to changes in the contribution of various components of the Ca²⁺ handling system. (A) β = 1.0, T = 250 ms; (B) β = 2.0, T = 333 ms. At baseline (i.e., 0% change) in both cases, the system is sitting on the stability boundary (i.e., ). Increasing or decreasing the contribution of any part of the Ca²⁺ handling system (i.e., moving to the left or the right along the axis) changes the stability of the entire system. u = slope of the sarcoplasmic reticulum (SR) Ca²⁺ release function; v_up = SR uptake pump strength; g_Ca = maximum conductance of the L-type Ca²⁺ channels; g_NaCa = Na⁺-Ca²⁺ exchanger strength.

Fig. 5 suggests that: i), changing any parameter in the model affects the stability of the Ca²⁺ handling system (i.e., changes the value of ) in the neighborhood of the transition from no alternans to alternans (i.e., in the neighborhood of the transition to the unstable period-1 solution); ii), the degree of sensitivity of the stability of the Ca²⁺ handling system is parameter dependent; and iii), particular parameter changes can have qualitatively different stability effects for different baseline states (e.g., increasing g_Ca or g_NaCa from baseline causes an increase in stability when β = 1.0, but a decrease in stability when β = 2.0).

Fig. 6 further explores the sensitivity of the system by systematically changing the four components studied in Fig. 5 while also varying the pacing period and action potential morphology. Fig. 6 illustrates the influence of increasing and decreasing by 20% the maximum conductance of the L-type Ca²⁺ channels (Fig. 6 A), the Na⁺-Ca²⁺ exchanger strength (Fig. 6 B), the SR uptake pump strength (Fig. 6 C), and the slope of the SR Ca²⁺ release function (Fig. 6 D) on the alternans eigenvalue over a range of pacing periods when β = 0.9 and β = 2.0. In both cases, the eigenvalues at baseline (i.e., 0% change) are identical to those shown in Fig. 3 B.

FIGURE 6.

Sensitivity of to +20% (▵) and −20% (∇) changes in the contribution of (A) maximum conductance of the L-type Ca²⁺ channels (g_Ca), (B) the Na⁺-Ca²⁺ exchanger strength (g_NaCa), (C) the SR uptake pump strength (v_up), and (D) the slope of the sarcoplasmic reticulum Ca²⁺ release function (u) for β = 0.9 (top panels) and β = 2.0 (bottom panels). Crosses indicate at baseline (identical to Fig. 3 B).

Fig. 6 reveals that when the action potential morphology is changed, the same percentage change in the contribution of a component of the Ca²⁺ handling system can have markedly different effects on the stability of the system as a function of pacing period. When β = 2.0, 20% changes in each of the four parameters in either direction result in fairly small changes to the stability of the system. However, when β = 0.9, the same Ca²⁺ handling system subjected to the same 20% changes but a different action potential morphology results in markedly different stability properties. For example, a reduction of the Na⁺-Ca²⁺ exchanger strength (Fig. 6 B) by 20% causes a very slight stabilization of the Ca²⁺ handling system compared to baseline for all pacing periods when β = 2.0. However, the same 20% reduction in I_NaCa for action potentials whose morphology is characterized by β = 0.9 destabilizes the system compared to baseline, with the amount of destabilization (as measured by the distance between the 0 and −20% alternans eigenvalues) varying widely as a function of pacing period.

Restitution curve slope

Fig. 3 indicates that how action potential morphology changes as a function of the pacing period is important in determining whether alternans will occur at a given value of T. To determine whether a relationship exists between the occurrence of Ca²⁺ alternans and the slope of the APD restitution curve, the slopes of the APD restitution curves characterizing the action potentials imposed during voltage clamping were calculated, and these slopes were plotted as a function of DI as shown in Fig. 7 A. The slopes of the restitution curves at 50% repolarization were used, because (as Fig. 1 B shows) the restitution curves at 90% repolarization for the three values of β used in Fig. 1 are identical. Fig. 7 B illustrates the alternans eigenvalues from Fig. 3 B at the same values of β as in Fig. 7 A, also plotted as a function of DI.

FIGURE 7.

(A) Slopes of the restitution curves from Fig. 1 B (50% repolarization). (B) for the same values of β as shown in panel A, plotted as a function of diastolic interval.

Fig. 7 shows that, for β = 2.0 and β = 1.5, Ca²⁺ alternans occurs under conditions corresponding to the APD restitution curve slope being at times <1 and at times >1. For β = 0.9, however, Ca²⁺ alternans is absent for all values of DI (as Fig. 7 B indicates), even for values of DI corresponding to a slope of the APD restitution curve >1. These results indicate that there is no correlation between the slope of the APD₅₀ restitution curve and the occurrence of Ca²⁺ alternans. Although not shown, no correlation was found between the slopes of the APD₉₀ restitution curves and the presence of Ca²⁺ alternans.

DISCUSSION

The results of this study indicate that, at any given excitation rate, the stability of the intracellular Ca²⁺ handling system is intimately linked to the temporal dynamics of membrane voltage. By changing the morphology of the action potential at a given excitation rate, qualitatively different behaviors of the Ca²⁺ handling system (either nonalternating or alternating Ca²⁺ transients) can be induced.

Whole-cell dependence of Ca²⁺ handling

The Ca²⁺ handling system in healthy cardiac myocytes possesses the ability to exquisitely control [Ca²⁺]_i (28). Normally, the amount of Ca²⁺ that enters a myocyte during each beat is exactly equal to the amount of Ca²⁺ that is pumped out, and the amount of Ca²⁺ that is released by the SR during each beat is equal to the amount that is returned to the SR (28). If a single beat provides insufficient time to allow this Ca²⁺ cycling process to reach steady state (as may occur at rapid excitation rates), the Ca²⁺ cycling process may instead reach steady state after every second beat. This will be manifested as an alternation in the shape of the Ca²⁺ transient on a beat-to-beat basis.

Certain elements of the Ca²⁺ handling system appear to be important in determining the conditions under which Ca²⁺ alternans occurs. Particular emphasis has recently been placed on the characteristics of the SR (32), with this emphasis stemming from insights both from experimental work (13) and from mathematical modeling (19,28,29). In this study we have instead focused on the influence of membrane voltage on the behavior of the Ca²⁺ handling system. We found that membrane voltage, which affects the system through the sarcolemmal currents I_Ca and I_NaCa, plays an important role in determining when Ca²⁺ alternans occurs.

Our results suggest that the conditions under which Ca²⁺ alternans occurs are governed not by any one part of the Ca²⁺ handling system, but by the entire interconnected system. From a dynamical systems perspective, this result is not surprising. The highly interconnected nature of the Ca²⁺ handling system means that every component must be considered when attempting to understand the stability properties of the system in the neighborhood of the transition from a stable to unstable period-1 solution. It is not sufficient to draw conclusions about the stability properties of the entire system by studying each component of the system in isolation, as this does not allow for the compensatory effects of other components. For this reason, techniques such as the “eigenmode” method are crucial for understanding complex physiological processes in their entirety (10). By taking into account every component of the model Ca²⁺ handling system with this method, we are able to analyze the stability properties of the system in the neighborhood of the transition to alternans, and thereby predict the occurrence of Ca²⁺ alternans.

Slope of the restitution curve

Several studies have suggested a relationship between the slope of the APD restitution curve and the occurrence of electrical alternans (33–39). These studies are based on the restitution hypothesis, which states that electrical alternans will occur when the slope of the APD restitution curve is >1. We sought to determine whether a relationship exists between the slope of the APD restitution curve during action potential voltage clamping and the occurrence of Ca²⁺ alternans.

As Fig. 7 indicates, there is no simple relationship between the slope of the APD restitution curve and the occurrence of Ca²⁺ alternans in this model. Even though the slopes of the three restitution curves shown in Fig. 1 all became >1 at short DIs, Ca²⁺ alternans did not uniformly occur for all action potential morphologies when the slopes of the respective APD restitution curves were >1. This finding is consistent with a number of recent studies that have found that the slope of the APD restitution curve is not an accurate predictor of the conditions under which alternans occurs (10,12,16,40,41). As our results indicate, the only accurate predictor of whether Ca²⁺ alternans will occur is the alternans eigenvalue.

Membrane currents and Ca²⁺ handling

During action potential voltage clamping, the predominant membrane currents influencing the dynamics of the Ca²⁺ handling system are Ca²⁺ currents (such as I_Ca and I_NaCa in the model used here). During unclamped action potentials, however, every membrane current in the myocyte contributes to shaping the membrane voltage as a function of time, and thus every membrane current contributes to the voltage that is sensed by the Ca²⁺-selective currents. Given that our results suggest that the morphology of the action potential plays a key role in determining the stability of the Ca²⁺ handling system, it follows that every membrane current in the myocyte (through its influence on membrane voltage) plays a role in modulating the stability of the Ca²⁺ handling system.

This has important implications for electrical alternans. If the hypothesis that intracellular Ca²⁺ alternans leads to electrical alternans is correct (26), the unclamped action potential would be expected to alternate when the Ca²⁺-selective currents such as I_Ca and I_NaCa begin to alternate. Therefore, a change in the contribution of any membrane current when the action potential is not clamped should alter the stability of the Ca²⁺ handling system and, in turn, should change the conditions under which Ca²⁺ alternans and, therefore, electrical alternans, occurs. This prediction is borne out in Fig. 8.

FIGURE 8.

Stability of the complete ionic model of the ventricular action potential, incorporating the Ca²⁺ handling system used in Figs. 1–7. Action potentials in the model were elicited by brief suprathreshold current injections at a pacing period of 365 ms. A change in the contribution of any of the membrane currents shown causes the morphology of the action potential to change (see insets a–d; APD₅₀ for a = 194 ms, b = 185 ms, c = 172 ms, d = 151 ms), which causes the stability of the system to change as well (as reflected in ). In some cases, the system becomes more stable, and in other cases the system moves into the unstable regime. It should be noted that action potentials a–d demonstrate that stability is not simply a function of APD, as the longer (a) and shorter (d) action potentials are more stable than the intermediate (b and c) action potentials. It is also worth noting that the stabilization seen as g_Kr increases is consistent with previous experimental findings demonstrating that increased I_Kr expression can suppress alternans (15). In the complete ionic model, u = 11.3 (same as for the voltage clamp model), γ = 0.7, and τ_f = 30 ms. g_Kr = maximum conductance of the fast component of the delayed rectifier potassium current; g_Ks = maximum conductance of the slow component of the delayed rectifier potassium current; g_to = maximum conductance of the transient outward potassium current; g_Na = maximum conductance of the fast sodium current; g_K1 = maximum conductance of the inward rectifier potassium current.

Fig. 8 was generated using a complete ionic model of the myocyte that incorporates the Ca²⁺ handling system used in this study (19). Instead of being clamped by a predetermined waveform, the membrane voltage in the complete ionic model varies dynamically in response to the contribution of a number of membrane currents including I_Na, I_Kr, I_Ks, I_to, and I_K1. During action potential voltage clamping, the Ca²⁺ handling system of the complete ionic model exhibits alternans (just as the action potential clamp model used elsewhere in this study does). When membrane voltage is not clamped, however, all components of the model alternate simultaneously, and becomes a measure of the stability of the complete interconnected system. In Fig. 8, the stability of the entire system, including the contribution of membrane currents, is shown.

The simulations in Fig. 8 are very similar to those in Fig. 3, where the morphology of the action potential during voltage clamping was changed directly and the effect on the stability of the system was determined. In Fig. 8, however, the morphology of the action potential is changed indirectly by varying the conductances of five sodium and potassium currents (to mimic, for example, the effects of pharmacological agents, or the effects of altered channel protein expression (15)). At baseline (i.e., at 0%), indicating that the system is sitting on the stability boundary. As predicted, changing the contribution of any membrane current at a given pacing period changes the stability characteristics of the system, and thereby changes the conditions under which Ca²⁺ alternans (and, therefore, electrical alternans) occurs. This suggests that every membrane current contributes to the stability of the Ca²⁺ handling system in the neighborhood of the transition from a stable to unstable period-1 solution and, hence, every membrane current contributes to establishing the conditions under which electrical alternans occurs.

Limitations

The results in this study were obtained using a mathematical model of the rabbit ventricular myocyte intracellular Ca²⁺ handling system. Although it is likely that there are quantitative differences between the model system and the human ventricular myocyte Ca²⁺ handling system, we expect the basic finding of this work—that the stability of the Ca²⁺ handling system is modulated by the morphology of the action potential and, therefore, by all membrane currents—to be valid. Although our results were obtained from simulated cardiac myocytes, this finding can, in principle, be tested experimentally.