Variations in ionic concentrations over 10 minutes of constant pacing at cycle lengths (CLs) of 1000, 500, and 350 ms for the NM. Note that different scales are used for each concentration and each CL to indicate ongoing adjustments to concentrations appropriately.

Important transmembrane currents during the first 350 ms of an AP for the CM, NM, and NMf at CLs of 1000, 500, and 350 ms. All currents are shown for the last AP obtained during 30 seconds of pacing at the given CL. Currents at a CL of 1000 ms are shown in bold. Note that some traces have been rescaled as indicated.

Issues associated with calculating restitution curves. A–C. Calculation of steady-state APD restitution curves using either a non-varying fixed voltage threshold (dashed) or CL-dependent threshold associated with APD₉₀ (solid) for (A) the CM, (B) the NM, and (C) the NMf. The differences between the two curves are not substantial at long DIs but become significant at shorter DIs. In addition, the shapes of the curves are modified by using a constant voltage threshold for APD calculation, which makes the curves for the NM and NMf have larger regions of negative slope when not using CL-dependent APD₉₀. Alternans in the CM are shown using filled circles. D–E. Calculation of steady-state APD restitution curves using the NM. When ionic concentrations vary freely (D), drift in these concentrations does not allow the model to reach steady-state even after 60 seconds of pacing. Consequently, action potentials for a given CL continually decrease over time, and a steady-state restitution curve cannot be obtained, even after several minutes (see Figure 1). When ionic concentrations are fixed in the NMf (E), steady-state is reached within 30 seconds of pacing, and little variation is seen with only 10 seconds of pacing.

Dynamics of DADs in the NM with O_TMgMg clamped and with the buffering variable O_C clamped (dashed). Pacing was maintained at a CL of 350 ms for 30 s, after which pacing was discontinued (time = 0). A. Voltage behavior for the two cases: clamping O_C prevents DADs following cessation of pacing. B. The initial depolarization when O_TMgMg is held constant is due to I_NaCa, which responds to an increase in [Ca²⁺]_i by increasing in magnitude. The upstroke of the DAD-induced AP therefore is slow early on, when it is controlled by I_NaCa, and then more rapid as the voltage reaches the threshold for I_Na activation. When O_C is held constant instead, I_NaCa does not depolarize the cell. C. Early calcium release from the SR occurs when O_TMgMg is clamped but not when O_C is clamped. The release current in this case is of a smaller magnitude but for a longer time and begins well before the AP upstroke. D. The gradual depolarization following the last paced beat when O_TMgMg is clamped coincides with a slow increase in [Ca²⁺]_i, which leads to the increase in the magnitude of I_NaCa. The calcium transient is much smaller when O_C is clamped instead, and it does not increase following cessation of pacing. E. The initial increase in [Ca²⁺]_i following cessation of pacing when O_TMgMg is clamped occurs when the sign of d[Ca²⁺]_i changes from negative to positive; here the plotted range of d[Ca²⁺]_i has been reduced to illustrate this occurrence. When O_C is clamped instead, no sign change occurs until the beginning of the upstroke of a paced beat. The sign of d[Ca²⁺]_i depends directly on the balance of three transmembrane currents, three intracellular calcium currents, and the rate of change of three calcium buffering variables, as well as indirectly on a number of other variables, all of which allows for numerous possibilities for enhancing or suppressing DADs.

Important transmembrane currents and the calcium transient [Ca²⁺]_i during the first 80 ms of an action potential (AP) for the CM, NM, and NMf at CLs of 1000, 500, and 350 ms. All currents are shown for the last action potential (AP) obtained during 30 seconds of pacing at the given CL. Currents at a CL of 1000 ms are shown in bold. Note that some traces have been rescaled as indicated.

A–C. Rate dependence of AP morphology and duration at CLs of 1000 ms (solid), 500 ms (dashed), and 350 ms (dotted) for (A) the CM, (B) the NM, and (C) the NMf. Each AP is the last in a 30-second interval of pacing at the specified CL. Only the CM exhibits significant rate dependence of AP duration (APD) and shape, switching from spike-and-dome morphology at long CLs to triangular morphology at short CLs, whereas the NM and NM-f exhibit significant rate dependence of resting membrane potential (RMP), which increases with decreasing CL. D. Resting and peak membrane potentials as a function of CL for the CM (solid), NM (long dashes), and NMf (short dashes). As CL decreases, all of the models show an increase in RMP along with a decrease followed by an increase in peak membrane potential. E–G. Restitution of APD₉₀ for (E) the CM, (F) the NM, and (G) the NMf. Steady-state restitution curves (solid lines) obtained after pacing for 30 seconds at each CL are shown along with S1–S2 restitution curves obtained after 30 seconds of pacing at S1 CLs of 1000 ms (long dashes), 500 ms (short dashes), and 350 ms (dash-dots). The CM shows increasing APDs in S1–S2 curves with decreasing S1 CL at short DIs, as does the NMf at long CLs. The NMf also has a slightly biphasic steady-state restitution curve, as does the CM at longer DIs than shown here. Alternans occur only in the CM and are shown using filled circles. Note that although different maximum and minimum values of APD are shown, the scaling is the same for the three panels. Stimulus is twice diastolic threshold and is applied for 3 ms here and throughout.

Delayed afterdepolarizations (DADs) and auto-oscillatory behavior in the NM induced after pacing for 30 s at a given CL and then ceasing pacing (at time = 0). A. DADs induced after pacing at a CL of 350 ms under normal conditions (dotted) and with the buffering variable O_TMgMg held constant (solid). Under normal conditions, the afterdepolarization activity died out within 17 s. In both cases, the DAD frequency decreased over time. B. Activity induced under normal conditions after pacing at CLs of 350 ms (dashed), 700 ms (dotted), and 1050 ms (solid). DADs are suppressed entirely at the 1050 ms CL and are spaced farther apart in time for 700 ms than for 350 ms. C. Activity induced after pacing at a CL of 350 ms under normal conditions (dashed), with O_TMgMg held constant (dotted), and with the buffering variable O_TC held constant (solid). D. Activity induced after pacing at a CL of 350 ms under normal conditions (dashed), with the time rate of change of the buffer occupancy dO/dt multiplied by 4 (dotted), and with I_NaCa multiplied by 0.1 (solid). E. Activity induced after pacing at a CL of 350 ms under normal conditions (dashed), with with I_Kr multiplied by 500 (dotted), and with I_K1 multiplied by 10 (solid).

Action potentials and restitution curves obtained using modifications to current magnitudes suggested to improve agreement between the two models, as proposed by Nygren et al. (2001) and Syed et al. (2005). A–D. Action potentials after 30 seconds of pacing at CLs of 1000, 500, and 350 ms for (A) the CM-N, (B) the NM-C, (C) the NMf-C, and (D) the NMf-C2; compare with Figure 4A–C. Although the modifications have some effects on the overall APs, the rate dependence of APD, AP morphology, and RMP are different compared to the original NM and CM the model variants seek to reproduce. E–H. Restitution of APD₉₀ for (E) the CM-N, (F) the NM-C, (G) the NMf-C, and (H) the NMf-C2; compare with Figure 4E–G. Steady-state restitution curves (solid lines) obtained after pacing for 30 seconds at each CL are shown along with S1–S2 restitution curves obtained after 30 seconds of pacing at S1 CLs of 1000 ms (long dashes), 500 ms (short dashes), and 350 ms (dash-dots). Gray lines indicate the steady-state restitution curve of the model each variant seeks to emulate (the NM for E, and the CM for F–H). Applying the inverse of the proposed changes to the original CM (D) eliminates nearly all memory from the system; restitution curves obtained using steady-state and S1–S2 protocols are nearly identical, with almost no dependence on S1 CL. However, the restitution curve still differs from the restitution curves of the NM and NMf, and instead is similar to the steady-state restitution curve obtained without modifications. Using the proposed modifications to the original NM (F) minimally affects the restitutions curves; APDs and the minimum DI are slightly decreased. However, decreasing the S1 CL results in shorter APDs, unlike the original CM, where longer APDs occur. Furthermore, all APDs, especially the smallest APDs during steady–state pacing, are significantly shorter than for the original CM. Application of the proposed modifications to the NMf (G) eliminates the biphasic portion of the restitution curve and steepens the curve overall. However, the S1-dependence of the S1–S2 protocols remains inverted to that of the original CM, and APDs overall remain shorter. For the NMf-C2, S1-S2 restitution curves cannot be obtained for all S1 CLs over all diastolic intervals for the NMf-C2 due to pronounced DADs. For the steady-state curve, APD₉₀ remains longer for the CM than for the NMf-C2.

Modified restitution curves for the original CM and the NMf. A. Modifications to currents in the original NMf can produce a steeper steady-state restitution curve over a broader range of DIs. In addition, these modifications remove the extremely short APDs at extremely short DIs and eliminate the biphasic region of the steady-state restitution curve. Although the restitution curve is steepened, its slope is only greater than one for DIs smaller than 45ms. See text for details. B. Modifications to currents in the original CM can reverse the dependence of S1–S2 restitution curves on S1 CL at short DIs. In the present arrangement, APDs decrease with decreasing S1 CL as well as with decreasing DI.

Action potentials and restitution curves after 30 seconds of pacing using the current magnitude modifications of Courtemanche et al. (1999) to simulate AF-induced electrophysiological changes. A–C. Action potentials at CLs of 1000, 500, and 350 ms are shown for (A) the CM-AF, (B) the NM-AF, and (C) the NMf-AF. Agreement among the models is substantially greater under these conditions, although the RMPs in the CM-AF generally remain lower than in the NM-AF and NMf-AF. D–F. Restitution of APD₉₀ for (D) the CM-AF, (E) the NM-AF, and (F) the NMf-AF. Steady-state restitution curves (solid lines) obtained after pacing for 30 seconds at each CL are shown along with S1–S2 restitution curves obtained after 30 seconds of pacing at S1 CLs of 1000 ms (long dashes), 500 ms (short dashes), and 350 ms (dash-dots). The restitution curves obtained using steady-state and S1–S2 protocols for the CM-AF, like those for the CM-N, are nearly identical, indicating that nearly all memory has been eliminated. The modifications hardly affect the original NM’s restitution; if anything, action potentials are slightly prolonged for the NM-AF. For the NMf-AF, the APDs are decreased slightly, but the overall shapes of the curves remain quite similar to those obtained without modifications.

Action potentials and restitution curves obtained by interchanging the 12 transmembrane current formulations used by the models. A–C. Action potentials after 30 seconds of pacing at CLs of 1000, 500, and 350 ms for (A) the CM-X, (B) the NM-X, and (C) the NMf-X. Although the rate dependence of RMP follows the transmembrane current descriptions, the rate dependence of APD and AP morphology appear to be determined primarily by calcium handling. D–F. Restitution of APD₉₀ for (D) the CM-X, (E) the NM-X, and (F) the NMf-X. Steady-state restitution properties including curve flatness and biphasic segments appear to be determined by calcium handling and other ionic concentrations, while the memory properties as evidenced by S1–S2 restitution curves appear more closely tied to the transmembrane current descriptions. Note that the plot scales are varied. G–I. Steady-state restitution of APD₉₀ for (G) the CM-X, (H) the NM-X, and (I) the NMf-X, plotted together with steady-state restitution curves of the NM and CM. Note that the plot scales are varied.