Mechanism for action potential alternans: The interplay between L-type calcium current and transient outward current

CREATIVE CONCEPTS

Mechanism for action potential alternans: The interplay between L-type calcium current and transient outward current Bruce Hopenfeld, PhD From the National Institutes of Health, National Heart, Lung and Blood Institute, Bethesda, Maryland. BACKGROUND The ionic mechanisms underlying action potential duration alternans are not established. OBJECTIVES The purpose of this study was to explore the mechanisms underlying action potential alternans. METHODS Computer simulations were performed using a model of a single ischemic myocyte. To emulate ischemia, extracellular potassium was raised to 10 mM, L-type calcium channel conductance was decreased, and the conductivity of the transient outward current Itowas varied. RESULTS Alternans occurred at basic cycle lengths between 350 and 1,800 ms. The alternans resulted from the interplay of the recovery kinetics of the calcium and transient outward current inactivation gates. Depending on the diastolic interval, the transient outward current was sufficiently strong and calcium current sufficiently weak to result in the abolition of the action potential plateau and thus in an abbreviated action potential. The inactivation and recovery kinetics of the inactivation gates were such that calcium current was relatively stronger than transient outward current after an abbreviated action potential. The subsequent action potential was long because calcium current was sufficiently large to restore the action potential plateau dome after the partial repolarization caused by the transient outward current. The long-short pattern repeated indefinitely. This alternans mechanism explains how 2:1 patterns can evolve into 3:1 patterns, as observed in at least one experiment, as ischemia progresses and calcium current diminishes. CONCLUSION Computer simulations and basic theory suggest that the interplay between L-type calcium and transient outward currents causes at least one type of alternans. KEYWORDS Ischemia; Myocyte; Electrical alternans (Heart Rhythm 2006;3:345–352) © 2006 Heart Rhythm Society. All rights reserved.

Introduction The action potential duration and amplitude of ischemic cells may alternate from beat to beat. This electrical asynchrony, which results in electrocardiographic T-wave alternans, is associated with arrhythmias.1 The mechanisms underlying alternans are not firmly established. Experiments by Lukas and Antzelevitch2 and Tachibana et al3 suggest that the transient outward current (Ito) plays a role in alternans. Specifically, both Lukas and Antzelevitch2 This research was supported by the Intramural Research Program of the National Heart Lung and Blood Institute Z01-HL004609 (principal investigator: Elliot McVeigh). Address reprint requests and correspondence: Dr. Bruce Hopenfeld, National Institutes of Health, National Heart, Lung and Blood Institute, 10 Center Drive, MSC 1061, Bethesda, Maryland 20892-1061. E-mail address: [email protected]. (Received August 23, 2005; accepted November 15, 2005.)

and Tachibana et al3 found that application of the Ito blocker 4-aminopyridine eliminated alternans. The goal of the present computer simulation study was to examine how Ito may contribute to alternans. More generally, a major aim of this study was to match the data of Lukas and Antzelevitch,2 who created ischemiclike conditions in isolated canine myocardial tissue and paced the tissue at varying basic cycle lengths (BCLs). They found that alternans occurred only when the BCL was between 600 and 1,800 ms. In this type of alternans, as shown in the top panel in Figure 1, every other action potential lacked a plateau. Similarly, in open chest pigs, recordings of transmembrane potentials by Downar et al4 in an ischemic area appeared to show an alternans pattern in which every other action potential lacked a plateau. This pattern was associated with ST-T wave alternans as recorded by extracellular electrograms in the ischemic region.4 In addition, Karagueuzian et al5 found that this al-

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doi:10.1016/j.hrthm.2005.11.016

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Heart Rhythm, Vol 3, No 3, March 2006 which recover from inactivation rapidly and slowly, respectively.13 For both sets of simulations, a previously published ventricular cell model by ten Tusscher et al14 was modified to reflect the changes in various ion channels wrought by acute ischemia, and to match the experiments of Lukas and Antzelevitch.2

Computer model outline

Figure 1 Different alternans patterns reported by Lukas and Antzelevitch2 (“Experiment”) at a basic cycle length (BCL) of 800 ms. The lower panel in each of the two sets shows the results of computer simulations at a BCL of 800 ms with the conductivity of “slow”-type transient outward current Ito,s set at 1.35 and 1.5 nS/pF, respectively, in the simulations associated with 2:1 and 3:1 alternans, respectively.

ternating plateau/nonplateau alternans occurred at relatively large BCLs (e.g., 1,000 ms). In some cases, Lukas and Antzelevitch2 observed a 3:1 alternans pattern, as shown in the third panel in Figure 1. Nearing and Verrier6 reported that a 3:1 pattern followed a 2:1 pattern in an open chest canine ischemia model. Increasingly complex patterns led to fibrillation.6 The present study presents an entirely new mechanism for 2:1 and 3:1 plateau/nonplateau alternans. This mechanism involves the interplay of the L-type calcium current (ICaL) and Ito. To the author’s knowledge, the interplay between Ito and ICaL has not been extensively studied as an alternans mechanism. In an experimental and modeling study, Yehia et al7 demonstrated that Ito contributed to Wenckebach-like rhythms in isolated rabbit ventricular cells. Yehia et al7 did not discuss the relationship between Ito and ICaL with respect to alternans. In a multicell model, Qu8 found that alternans can arise from heterogeneous cellto-cell coupling or action potential duration restitution, which is affected by calcium and potassium channel conductivity modifications. Other researchers have attributed alternans to the recovery kinetics of the calcium inactivation channel9 and calcium handling.10 –12 The present study departs from all of the aforementioned work by describing how 2:1 and 3:1 alternans can arise throughout a wide range of BCLs as a result of the interaction between Ito and ICaL.

Methods Basic plan Two different sets of simulations were run for two different types of Ito: a “fast” type (Ito,f) and a “slow” type (Ito,s),

The source code corresponding to the ten Tusscher model, which follows Hodgkin-Huxley formalism, is available at http://www-binf.bio.uu.nl/khwjtuss/HVM. In brief, according to the model, the change in transmembrane voltage (V) with respect to time is dV dt

⫽

Iion⫹Istim Cm

where Iion ⫽ ionic currents, t ⫽ time, Istim ⫽ applied stimulus current, and Cm ⫽ membrane capacitance per unit surface area. Iion ⫽ INa⫹Ito⫹ICaL⫹IK⫹Imisc. where INa ⫽ sodium current, IK ⫽ various potassium currents, and Imisc. ⫽ represents various pumps and exchangers. To simulate ischemia, the ATP-sensitive potassium current IKatp was added to Iion per Shaw and Rudy.15 IKatp maximal conductivity was set at 0.017 nS/pF, which corresponds to an intracellular ATP of 4 mM (a value in the midpoint of the ATP range chosen by Shaw and Rudy15). For the baseline ischemia simulations, extracellular potassium (Ko) was set at 10 mM. To estimate the sensitivity of the simulation results to changes in Ko, simulations were also performed with Ko set at 6 and 8 mM. Changes to ICaL, Ito, and INa are described in more detail below. The cell was stimulated at various BCLs for varying durations, as shown in the Results section.

Ito The calcium-insensitive type of Ito has two subcomponents: Ito,f and Ito,s, which recover from inactivation rapidly and slowly, respectively.13 In the ten Tusscher model, Ito,f and Ito,s are assigned to epicardial and endocardial cells, respectively. The kinetics of these currents were modified as little as possible from the baseline ten Tusscher parameters to produce spike-and-dome–type action potentials in ischemic conditions at relatively short BCLs.2 In the ten Tusscher model, Ito ⫽ Gito ⴱ s ⴱ r ⴱ 共V ⫺ Ek兲, where Gito ⫽ conductivity, s and r ⫽ inactivation and activation gates, respectively, V ⫽ transmembrane potential, and Ek ⫽ reversal potential for potassium. The subscripts f and s are used to denote the fast and slow Ito parameters, respectively. For the Ito,f simulations, the characteristics of the Ito,f channel were modified to enhance the activity of this current at the lower transmembrane potentials associated with the action potential upstroke of an

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Figure 2 Time constant and steady-state curves for ICaL (top panel) and slow Ito (Ito,s) (bottom panel) inactivation gates. In the time constant panels, changes from the standard ten Tusscher model (solid line) are shown as dashed lines. The fast and slow ICaL inactivation gates ff and fs have corresponding time constants ␶f,f and ␶f,s. The ICaL steady-state curve for both fs and ff (upper right panel) is identical to the standard ten Tusscher model curve. The steady-state curve for Ito,s (ss, lower right panel) is shifted ⫺5 mV with respect to the baseline ten Tusscher model.

ischemic cell. Specifically, Ito,f conductivity GIto,f was set at 0.68 nS/pF, 2.3 times the unmodified ten Tusscher model value. All of the Ito,f time constants and steady-state activation curves were shifted by ⫺5 mV. The activation curve was shifted by an additional ⫺5 mV. Peak Ito in the Ito,f simulations was ⬍8 pA/pF, approximately two thirds the peak Ito in the unmodified ten Tusscher model. The activation and inactivation curve shifts also were implemented for the Ito,s simulations. In addition, the Ito,s slow inactivation gate (ss) time constant was modified such that it inactivated more rapidly at higher transmembrane voltages and recovered from inactivation more slowly at the resting transmembrane voltage. The time constant and steady-state curves associated with the Ito,s inactivation gate (ss) are shown in Figure 2. The maximum Ito,s conductivity (GIto,s) was varied between 1.2 and 1.6 nS/pF. The minimum GIto,s value (1.2 nS/pF) was determined by the requirement that action potentials be abbreviated at long BCLs (e.g., 2,100 ms), in accordance with the data of Lukas and Antzelevitch.2 Peak Ito,s at a BCL of 2,100 ms was approximately 8 pA/pF and 8.8 pA/pF, respectively, for the minimum and maximum GIto,s values, respectively. These peak currents are less than the peak Ito in the unmodified ten Tusscher model.

L-type calcium channel According to the ten Tusscher model, ICaL ⫽ GCaL ⴱ f ⴱ f Ca ⴱ d ⴱ Vca共Vm兲 where GCaL ⫽ maximum ICaL conductance, f ⫽ inactivation

347 gate, d ⫽ activation gate, fCa ⫽ intracellular calcium inactivation gate, and Vca(Vm) ⫽ a Goldman-Hodgkin-Katz– type equation that determines the driving force on calcium. The ICaL channel was modified from the standard ten Tusscher model in order to match the experiments by Lukas and Antzelevitch.2 In these experiments, full action potentials occurred at relatively shorter BCLs (e.g., 400 ms) whereas abbreviated action potentials occurred at relatively longer BCLs (e.g. 2000 ms). Lukas and Antzelevitch2 hypothesized that enhanced Ito at long BCLs acted to abbreviate action potentials, which suggests that Ito increased relatively more than ICaL with increases in BCL. To match this pattern, a slow component ICaL,s was added to the ICaL channel so that ICaL ⫽ ICaL,s⫹ICaL,f, where ICaL,f ⫽ a “fast” channel and ICaL,s ⫽ a “slow” channel. This division of ICaL resulted in full action potentials at relatively shorter BCLs, as desired, due to rapid recovery of ICaL,f. Yet abbreviated action potentials occurred at longer BCLs, as required, when the amount of current carried by ICaL,s was relatively smaller than the amount of current carried by Ito,s. This ICaL split is consistent with data from Li et al,16 who found that the f gate recovered from inactivation with a time course best described by a biexponential curve, with fast and slow time constants (as is the case with Ito). At a resting transmembrane potential of ⫺80 mV, the time constants were 65 and 683 ms, respectively, whereas at a resting transmembrane potential of ⫺60 mV, the time constants were 164 and 697 ms, respectively.16 For the present simulations, 65 and 697 ms were selected as the fast and slow time constants for the recovery from inactivation of ICaL,s and ICaL,f, respectively. This selection likely overestimates the rate of ICaL recovery, because in the present simulations the resting transmembrane voltage was approximately ⫺70 mV, which is greater than the ⫺80 mV transmembrane potential at which Li et al16 estimated a “fast” recovery time constant of 65 ms. However, the shorter 65 ms time constant was chosen to maintain a relatively rapid ICaL recovery that is more in line with existing ventricular cell models. The time constant and steady-state curves associated with the ff and fs gates are shown in Figure 2. The conductance of the fast and slow ICaL channels was set equal to 0.000088 cm3␮F⫺1s⫺1 and 0.000044 cm3␮F⫺1s⫺1, respectively, for a total (ICaL,f⫹ICaL,s) conductance of 0.000132 cm3␮F⫺1s⫺1, a decrease of approximately 25% from the unmodified ten Tusscher model. This reduction reflects the decrease in ICaL availability associated with ischemia15. (At the baseline cell capacitance/unit area of 2.0 ␮F/cm2, a conductance of 0.000132 cm3␮F⫺1s⫺1 corresponds to a permeability of 0.00026 cm/s.) The 2:1 ratio of fast- to slowtype ICaL is consistent with the ICaL recovery curves described by Li et al.16 The slow and fast conductances were slightly decreased from the above values to simulate the reduction of ICaL availability as ischemia progresses.

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Results Ito,f

Figure 3 Steady-state action potentials for the Ito,f (top panel) and Ito,s simulations (bottom panel) at short and long basic cycle lengths.

INa In the ten Tusscher model, the amount of Ito increases with action potential upstroke velocity, which in turn depends on INa. INa is relatively smaller at the higher (less negative) cell resting potentials associated with ischemia. The corresponding decrease in action potential upstroke velocity decreases Ito. As mentioned, Ito conductivity was increased from the standard ten Tusscher model value to enhance Ito activity. Consistent with this change in conductivity, two steps were taken to relatively increase INa (and therefore Ito) in the baseline ischemia simulations. First, the decrease in maximum INa conductivity associated with ischemia15 was not implemented. Second, the time constant (␶j) associated with the sodium inactivation gate j (␶j) was decreased from the standard ten Tusscher value to enable the j gate to recover more completely from inactivation at relatively shorter BCLs, thereby relatively enhancing INa (and therefore Ito). To determine the effect of these parameters on alternans, simulations were also run with maximum INa conductivity decreased by 25%15 and with ␶j unaltered from the standard ten Tusscher model.

The left and right upper panels of Figure 3 show steadystate action potentials at BCLs of 250 and 800 ms, respectively. The 250 ms BCL action potential has a plateau but lacks the prominent “spike and dome” shown by the 800-ms action potential. Within a range of BCLs between approximately 350 ms and approximately 700 ms, 2:1 alternans occurred. Figure 4 shows alternans at a BCL of 400 ms. This pattern was stable for at least 2 minutes (greater length simulations were not performed). The top panel in Figure 5 shows action potential duration as a function of BCL. The action potential duration of full action potentials generally increased with increasing BCL, except that there was a slight dip for BCLs between 600 and 800 ms. The duration of abbreviated action potentials increased with increasing BCL due to the greater recovery of the slow ICaL (fs) gate at longer BCLs. The BCLs very close to the thresholds for 2:1 alternans exhibited long periods of alternans before a steady-state 1:1 pattern was reached. The top panel in Figure 6 shows how the values of the fs gate and the fast Ito inactivation gate (sf) related to alternans at a BCL of 400 ms. The sf gate recovered completely (to a value of 1) regardless of action potential morphology. After a full action potential, the fs gate had insufficient time to recover before the next depolarization. Thus, there was not enough ICaL to re-depolarize the cell after the phase 1 repolarization caused by Ito. An abbreviated action potential resulted, during which fs experienced relatively less inactivation because there was no plateau. Furthermore, the effective diastolic interval was relatively larger after an abbreviated action potential. The reduced inactivation and longer diastolic interval allowed the fs gate sufficient time to recover so that ICaL restored the plateau after the next phase 1 repolarization. The pattern repeated indefinitely.

Ito,s The left and right lower panels of Figure 3 show steadystate action potentials at BCLs of 400 and 2,100 ms, respectively, with GIto,s at 1.37 nS/pF. The 400 ms BCL

Implementation details Following ten Tusscher et al,14 the forward Euler method was used to integrate dV/dt, and the Rush and Larsen scheme17 was used to solve the Hodgkin-Huxley type equations for the model’s various gating variables. A variable time step was used: (1) 0.02 ms for the first 60 ms following a stimulus, (2) 0.04 ms between 60 and 150 ms following a stimulus, and (3) 0.1 ms after 150 ms following a stimulus. For a number of test cases, this variable time stepping scheme produced almost identical results as a uniform time step of 0.02 ms, which was used by ten Tusscher et al.14

Figure 4 400 ms.

Action potential alternans for the Ito,f simulations at

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Mechanism for action potential alternans

349 ing BCL due to the greater activity of Ito,s compared with ICaL,s with increases in BCL. Between BCLs of 1,000 and 1,500 ms and between BCLs of 1,800 and 2,100 ms, no stable pattern developed, at least over a period of 60 action potentials. Between BCLs of 1,500 and 1,800 ms, 3:1 alternans occurred, whereas at BCLs ⬎2,100 ms, all action potentials were abbreviated. Figure 7 shows a semi-quantitative summary of the Ito,s simulations. At lower BCLs, alternans was less likely to occur due to the relative lack of Ito,s. For any given BCL, alternans was more likely to occur as GIto,s was increased. At large BCLs and large GIto,s, there was no alternans because all action potentials were abbreviated. For most pairs of BCL and GIto,s values at which 2:1 alternans occurred, an increase in BCL and/or GIto,s resulted in 3:1 alternans. Between 3:1 alternans and fully abbreviated action potentials, unstable runs of varying patterns occurred. It is possible that these runs would have stabilized but that the

Figure 5 Action potential (AP) duration as a function of basic cycle length (BCL) for the fast (top panel) and slow (bottom panel) simulations, respectively. In the top panel, the bifurcations at BCLs of 350 and 600 ms indicate the occurrence of alternans. The dashed curve is the AP duration of the abbreviated APs. Alternans did not occur for BCLs greater than approximately 600 ms or less than approximately 350 ms. The bottom panel shows AP duration as a function of BCL with GIto,s ⫽ 1.37 nS/pF. At BCLs between 700 and 1,000 ms, 2:1 alternans occurred. Between BCLs of 1,000 and 1,500 ms and between 1,800 and 2,100 ms, no stable pattern developed, at least over a period of 60 APs. Between BCLs of 1,500 and 1,800 ms, 3:1 alternans occurred, whereas at BCLs greater than 2,100 ms, all APs were abbreviated, as indicated by the arrow.

action potential has a plateau, whereas the 2,100 ms BCL action potential lacks a plateau. Depending on the value of GIto,s and BCL, 2:1 or 3:1 alternans occurred, as well as other types of patterns. The second and fourth panels from the top in Figure 1 show 2:1 and 3:1 alternans at a BCL of 800 with GIto,s set at 1.35 and 1.5 nS/pF, respectively. These alternans patterns were stable for at least 1 minute (greater length simulations were not performed). The bottom panel in Figure 5 shows action potential duration as a function of BCL for GIto,s ⫽ 1.37nS/pF. Again, the action potential duration of full action potentials increased with increasing BCL. However, at BCLs between 700 and 1,000 ms, 2:1 alternans occurred, and the duration of the abbreviated action potentials decreased with increas-

Figure 6 Gates during alternans for the fast simulations (top panel) and slow simulations (bottom two panels). In the top panel, the values of the slow ICaL( fs ) (longer dashed line) and fast Ito( sf ) (solid line) inactivation gates are shown. The corresponding action potential shapes are superimposed. The middle and lower panels show slightly different types of 3:1 alternans patterns at relatively higher (middle panel) and relatively lower (lower panel) ICaL conductivities. In these panels, the values of the fs (longer dashed line) and slow Ito ss (solid line) inactivation gates are shown. In the middle panel, the points labeled P1, P2, and P3 correspond to the points with the same labels in Figure 8. In the lower panel, arrows point to the values of the fs and ss gates at the beginning of two successive cell depolarizations. See text for details.

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Heart Rhythm, Vol 3, No 3, March 2006 3:1 alternans. The difference between this pattern and the one shown in the middle panel of Figure 6 is that here, fs did not reach a larger value after the second abbreviated action potential, as shown in the figure (F1 ⬇ F2). Yet a full action potential occurred after the second abbreviated action potential because ss was relatively smaller (S2⬍S1) after the second abbreviated action potential. The common theme between the 3:1 patterns shown in the middle and bottom panels in Figure 6 is that, during abbreviated action potentials, fs recovered relatively more than ss because fs experienced relatively less inactivation due to the abbreviated or nonexistent plateaus associated with abbreviated action potentials. As expected, the magnitudes of the ICaL and Ito currents followed the patterns of the corresponding gating variables fs and ss.

Figure 7 Transition from full 1:1 action potentials to abbreviated 1:1 action potentials as a function of basic cycle length and conductivity of Ito,. The plot is semi-quantitative based on the simulation data. An increase in conductivity of Ito has similar effects as a decrease in conductivity of ICaL; thus, a decrease in ICaL,s conductivity shifts the entire plot down. The transition zones between different patterns generally resulted in unstable behavior.

simulation length (60 beats) was not sufficiently long to allow phase locking to occur. Decreasing GCaL,f and GCaL,s had essentially the same effect as increasing GIto,s, at least for the change from the baseline values to 8.6 and 4.0 cm3␮F⫺1s⫺110⫺5, respectively. This relatively small change in GCaL,f and GCaL,s resulted in the transition from 2:1 to 3:1 alternans at a BCL of 800 ms and GIto,s⫽1.37 nS/pF. The middle panel in Figure 6 shows how the values of the slow ICaL inactivation gate (fs) and slow Ito (ss) inactivation gate related to action potential duration at a BCL of 800 ms for the 3:1 alternans pattern shown in the bottom panel of Figure 1. During each action potential, the ss gate recovered to an intermediate value, which varied somewhat depending on whether the action potential was abbreviated or full. After a full action potential, the fs gate had insufficient time during the diastolic interval to achieve a value that was large enough to produce enough ICaL to offset Ito,s during the next action potential. Thus, the next action potential was abbreviated, which resulted in less inactivation of the fs gate because there was no plateau. However, fs was still too small during the next action potential, which was again abbreviated. The fs gate once again inactivated relatively less during the subsequent abbreviated action potential and recovered sufficiently during the subsequent diastolic interval, such that the next action potential was full. The pattern repeated indefinitely. The bottom panel in Figure 6 shows the fs and ss gates for a 3:1 alternans pattern at a BCL of 800 ms with GCaL,f and GCaL,s values set at 8.6 and 4.0 cm3␮F⫺1s⫺110⫺5, respectively, and GIto,s ⫽ 1.37 nS/pF. As mentioned earlier, this parameter set is very close to the threshold between 2:1 and

Sensitivity to extracellular potassium concentration and INa Holding all other parameters constant, decreases in INa led to decreases in Ito and therefore had the same effect on the alternans patterns as reductions in GIto,s. Decreases in INa reduced the maximum upstroke velocity of the action potential, which resulted in reduced Ito because the Ito inactivation gate had more time to decrease during these slower upstrokes. INa was reduced by (1) decreases in maximum INa conductivity, (2) implementation of the standard ten Tusscher model INa j gate compared with modified ischemia model, and (3) decreases in cell membrane resting potential, which were associated with decreases in extracellular potassium, which resulted in relatively greater inactivation of the INa h and j gates. As an example of the first two effects, at a BCL of 400 ms in the Ito,f simulations, a 25% decrease in maximum INa conductivity and implementation of the standard ten Tusscher model j gate reduced Ito,f sufficiently to abolish alternans; all action potentials had a plateau. An increase in GIto,f to 0.88 nS/pF (from the baseline value of 0.68 nS/pF) along with an increase in BCL to 500 ms restored alternans.

Discussion Alternans mechanism Alternans resulted from increasing cell resting transmembrane potential and decreasing ICaL activity. As mentioned in the Methods section, these changes likely occur during ischemia, during which alternans has been observed in vivo18 and in vitro.2 A minimum amount of Ito was also required to produce alternans. In the case of relatively higher (less negative) cell membrane resting potential, with a consequent relatively lower action potential amplitude, Ito influx brings cells relatively closer to the transmembrane potential threshold (approxi-

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Mechanism for action potential alternans

mately ⫺10 mV in the present simulations) for terminal “all-or-none” repolarization. At this threshold, a positive feedback repolarizing loop involving the rapid delayed rectifier current (IKr) activates to rapidly repolarize a cell. The exact value of this threshold depends on the magnitude of ICaL: less ICaL means that terminal repolarization is more likely to occur after Ito acts. If this threshold is crossed, an abbreviated action potential results, during which ICaL voltage inactivation gates decrease relatively less than Ito inactivation gates. Within certain parameter value ranges and within a certain range of BCLs, this asymmetric inactivation pattern results in relatively more ICaL compared with Ito during the subsequent action potential, such that the subsequent action potential has a plateau. During this plateau, ICaL voltage gates inactivate relatively more compared with the abbreviated action potential case, such that the subsequent action potential is abbreviated. This repeating pattern can be stable indefinitely.

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Figure 8 Restitution of the f gate in a stable 3:1 pattern. The points labeled P1, P2, and P3 correspond to the points with the same labels in the middle panel in Figure 6. See text for details.

f Gate kinetics The value of the f gate at the beginning of an action potential is a function of the value of the f gate at the beginning of the previous action potential and the change in the f gate between the previous action potential and the present action potential. In the case of 1:1 phase locking, the values of the f gate between successive cycles are identical. In the case of the simulations during which 2:1 alternans occurred, the value of f varied between successive cycles. As mentioned in the Results section, the f gate undergoes relatively less inactivation during an abbreviated action potential and recovers more during the ensuing diastolic interval, which is relatively longer following an abbreviated action potential. Consequently, the value of f (fn⫹1) at the upstroke of an action potential may be greater than its value fn at the beginning of the previous action potential, if the previous action potential cycle was abbreviated. Conversely, if the previous action potential had a plateau, it is possible that fn⫹1⬍fn. This bifurcation in the response of f can result in a stable oscillation pattern, as indicated in Figure 8, which shows a plot of fn⫹1 as a function of fn for a 3:1 pattern. (This type of plot is analogous to an action potential duration restitution plot.19) The points P1, P2, and P3 correspond to the points with the same labels shown in the middle panel in Figure 6. The horizontal axis in Figure 8 is divided into two sets: for values of fn less than a threshold value ft, an abbreviated action potential results, whereas a full action potential results for fn⬎ft. Starting at the point P1, which corresponds to a full action potential, the value of fn is approximately 0.95 and the value of fn⫹1 is 0.8. The new fn of 0.8 corresponds to point P2 and an abbreviated action potential. The fn⫹1 vs f curve is traversed to point P3, at which point fn ⫽ 0.95 and the curve is traversed to its beginning at P1. This pattern can repeat indefinitely. The fixed point (fn⫹1 ⫽ fn) is unstable because the fn⫹1 vs f curve has a very steep slope at this point.

The relative slopes of the two different portions of the fn⫹1 vs f curve and the value of fn at which action potential plateaus break down depend on various model parameters, most notably those that govern ICaL and Ito. The bifurcation in the fn⫹1 vs f curve can give rise to a variety of patterns, including 2:1 patterns and patterns involving a series of full action potentials followed by an abbreviated action potential. As mentioned in the Results section, the 3:1 pattern shown in the bottom panel in Figure 6 suggests that the most important quantity with regard to oscillations is not the f gate alone but the difference between the f and Ito inactivation (s) gates. Thus, Figure 8 may be reinterpreted as a relation between fn⫹1 ⫺ sn⫹1 and fn ⫺ sn.

Study limitations and conclusion The characteristics of the ICaL and Ito channels are complex, and the ten Tusscher model does not completely capture the behavior of these channels. Furthermore, the ten Tusscher model uses a relatively simple scheme for implementing calcium handling, which affects the activity of the ICaL calcium inactivation gate fCa. Because fCa modulates ICaL, simulations based on a model that includes a more detailed calcium handling and fCa scheme could yield different results than the present model. More generally, the changes in ionic currents caused by ischemia are not thoroughly understood. Furthermore, in a whole heart, electrical interaction between cells affects action potential duration and therefore alternans,8 whereas the present simulations involved a single cell. Nonetheless, the present model fits with the previously discussed experimental work and suggests that the interaction between ICaL and Ito may cause alternans during ischemia.

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