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1 Research Center, Montreal Heart Institute, Montreal, Quebec H1T 1C8; Départements de 2 Physiologie and 4 Médecine, Université de Montréal, Montreal, Quebec H3C 3J7; and 3 Department of Pharmacology, McGill University, Montreal, Quebec H3G 1Y6, Canada
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ABSTRACT |
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The mechanisms underlying many important properties of the human atrial action potential (AP) are poorly understood. Using specific formulations of the K+, Na+, and Ca2+ currents based on data recorded from human atrial myocytes, along with representations of pump, exchange, and background currents, we developed a mathematical model of the AP. The model AP resembles APs recorded from human atrial samples and responds to rate changes, L-type Ca2+ current blockade, Na+/Ca2+ exchanger inhibition, and variations in transient outward current amplitude in a fashion similar to experimental recordings. Rate-dependent adaptation of AP duration, an important determinant of susceptibility to atrial fibrillation, was attributable to incomplete L-type Ca2+ current recovery from inactivation and incomplete delayed rectifier current deactivation at rapid rates. Experimental observations of variable AP morphology could be accounted for by changes in transient outward current density, as suggested experimentally. We conclude that this mathematical model of the human atrial AP reproduces a variety of observed AP behaviors and provides insights into the mechanisms of clinically important AP properties.
action potential morphology; action potential rate dependence; transient outward current; L-type calcium current; sodium/calcium exchanger
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INTRODUCTION |
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THE HUMAN ATRIAL ACTION POTENTIAL (AP) and ionic currents that underlie its morphology are of critical importance to our understanding of the electrical properties of atrial tissues in normal and pathological conditions. AP shape and underlying ionic current densities are frequency dependent and may be modulated by specific pharmacological agents (33). Their properties are known to change during the course of atrial arrhythmias or as a result of other cardiac pathologies (32, 39). In addition, APs have shown significant variability in morphology, ranging from triangular APs with no sustained plateaus to long APs with definite spike-and-dome morphology, even when recorded from cells isolated from the same atrial sample (4, 33, 54). There is a need to integrate the information obtained from measurements of single currents to understand complex current interactions during the AP and their role in controlling AP morphology.
Until recently, limited data were available on the various currents underlying the AP. New findings have revealed important properties and significant interspecies differences. For example, the transient outward current (Ito), present in human and rabbit atrial cells (8, 16, 25, 47), has been shown to recover from inactivation at least two orders of magnitude faster in humans than in rabbits (20, 47). A novel delayed rectifier current (IKur) has been identified in human atrial cells (1, 18, 41, 48, 53) and is different from the sustained current (Isus) observed in rabbit atrial cells (12). IKur also differs from delayed rectifiers in other species that show kinetically similar characteristics (3, 6, 29). Additional data have been obtained on the classic delayed rectifier currents (IKr and IKs) (54, 55), the time-independent inward rectifier current (IK1) (30, 31), the fast Na+ current (INa) (42), the L-type Ca2+ current (ICa,L) (35) and its inactivation by voltage and intracellular Ca2+ (26, 49), and the Na+/Ca2+ exchanger current (INaCa) (4).
Models of atrial cells based on animal data only have been published (14, 36, 40, 60). Although these models have provided valuable insights into the mechanisms underlying AP generation in these species, the existence of significant interspecies differences and the amount of human data available indicate a need for an AP model based specifically on direct measurements of human atrial currents. To address this issue, we developed a mathematical model of the AP based on ionic current data obtained directly in human atrial cells. When human data were insufficient to characterize a given atrial current fully, we supplemented with animal data and referred to existing published models of atrial and ventricular APs (14, 36, 37, 40, 60). The most recent published models of mammalian cardiac cells are those of Luo and Rudy (37) (LR2 model) and Linblad et al. (36) (LMCG model), based on measurements from guinea pig ventricular cells and rabbit atrial cells, respectively. Our model builds mostly on the work of Luo and Rudy to develop a working model of the human atrial AP.
Our primary goal was to develop a useful model of the AP from which we could gain insights into experimental observations made on human atrial cells and tissues and make predictions regarding the behavior of these cells under previously untested conditions. Throughout our description of the model, we clearly indicate the sources of data used to derive model parameters, how the parameters were obtained, and whether they were derived directly from data or modified to improve our representation of the AP. We then use the model to investigate the mechanism of experimentally measured AP rate dependence, the changes in AP morphology in the presence of pharmaceutical blockers of the ICa,L and the Na+/Ca2+ exchanger, and the variability in experimentally observed AP morphology.
Glossary
| V | Transmembrane potential |
| R | Gas constant |
| T | Temperature |
| F | Faraday constant |
| Cm | Membrane capacitance |
| AP | Human atrial action potential |
max |
Maximal AP upstroke velocity (dV/dtmax) |
| APA | AP amplitude (peak AP voltage minus diastolic voltage at onset of AP) |
| APO | AP overshoot (peak AP voltage >0 mV) |
| APD | AP duration |
| APD50 | APD measured from AP onset to 50% of total APA repolarization |
| APD90 | APD measured from AP onset to 90% of total APA repolarization |
| AF | Atrial fibrillation |
| I-V | Current-voltage |
| Q10 | Temperature adjustment factor, k(T ) = k(T0)Q(T T0)/1010
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SR
Sarcoplasmic reticulum
JSR
Junctional SR, SR release compartment
NSR
Network SR, SR uptake compartment
Vcell
Cell volume
Vi
Intracellular volume
Vup
SR uptake compartment volume
Vrel
SR release compartment volume
[X ]o
Extracellular concentration of ion X
[X ]i
Intracellular concentration of ion X
Cmdn
Calmodulin, sarcoplasmic Ca2+ buffer
Trpn
Troponin, sarcoplasmic Ca2+ buffer
Csqn
Calsequestrin, JSR Ca2+ buffer
[Ca2+]rel
Ca2+ concentration in release compartment
[Ca2+]up
Ca2+ concentration in uptake compartment
[Ca2+]Cmdn
Ca2+-bound calmodulin concentration
[Ca2+]Trpn
Ca2+-bound troponin concentration
[Ca2+]Csqn
Ca2+-bound calsequestrin concentration
EX
Equilibrium potential for ion X
Iion
Total ionic current
Ist
Stimulus current
x
Forward rate constant for gating variable x
x
Backward rate constant for gating variable x
x
Time constant for gating variable x
x
Steady-state relation for gating variable x
INa
Fast inward Na+ current
gNa
Maximal INa conductance
m
Activation gating variable for INa
h
Fast inactivation gating variable for INa
j
Slow inactivation gating variable for INa
IK1
Inward rectifier K+ current
gK1
Maximal IK1 conductance
Ito
Transient outward K+ current
gto
Maximal Ito conductance
KQ10
Q10-based temperature adjustment factor
oa
Activation gating variable for Ito
oi
Inactivation gating variable for Ito
IKur
Ultrarapid delayed rectifier K+ current
gKur
Maximal IKur conductance
ua
Activation gating variable for IKur
ui
Inactivation gating variable for IKur
IKr
Rapid delayed rectifier K+ current
gKr
Maximal IKr conductance
xr
Activation gating variable for IKur
IKs
Slow delayed rectifier K+ current
gKs
Maximal IKs conductance
xs
Activation gating variable for IKs
ICa,L
L-type inward Ca2+ current
gCa,L
Maximal ICa,L conductance
d
Activation gating variable for ICa,L
f
Voltage-dependent inactivation gating variable for ICa,L
fCa
Ca2+-dependent inactivation gating variable for ICa,L
Ip,Ca
Sarcoplasmic Ca2+ pump current
Ip,Ca(max)
Maximal Ip,Ca
INaK
Na+-K+ pump current
INaK(max)
Maximal INaK
fNaK
Voltage-dependence parameter for INaK
[Na+]o-dependence parameter for INaK
Km,Na(i)
[Na+]i half-saturation constant for INaK
Km,K(o)
[K+]o half-saturation constant for INaK
INaCa
Na+/Ca2+ exchanger current
INaCa(max)
INaCa scaling factor
Km,Na
[Na+]o saturation constant for INaCa
Km,Ca
[Ca2+]o saturation constant for INaCa
ksat
Saturation factor for INaCa
Voltage-dependence parameter for INaCa
Ib,Na
Background Na+ current
gb,Na
Maximal Ib,Na conductance
Ib,Ca
Background Ca2+ current
gb,Ca
Maximal Ib,Ca conductance
Irel
Ca2+ release current from the JSR
krel
Maximal Ca2+ release rate for Irel
u
Activation gating variable for Irel
v
Ca2+ flux-dependent inactivation gating variable for Irel
w
Voltage-dependent inactivation gating variable for Irel
Fn
Sarcoplasmic Ca2+ flux signal for Irel
Iup
Ca2+ uptake current into the NSR
Iup(max)
Maximal Ca2+ uptake rate for Iup
[Ca2+]up(max)
Maximal Ca2+ concentration in NSR
Itr
Ca2+ transfer current from NSR to JSR
tr
Ca2+ transfer time constant
Iup,leak
Ca2+ leak current from the NSR
[Cmdn]max
Total calmodulin concentration in myoplasm
[Trpn]max
Total troponin concentration in myoplasm
[Csqn]max
Total calsequestrin concentration in JSR
Km,Cmdn
Ca2+ half-saturation constant for calmodulin
Km,Trpn
Ca2+ half-saturation constant for troponin
Km,Csqn
Ca2+ half-saturation constant for calsequestrin
Vrest
Resting membrane potential
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MODEL DESCRIPTION |
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General
We model the cell membrane as a capacitor connected in parallel with variable resistances and batteries representing the ionic channels and driving forces. The time derivative of the membrane potential V (with the assumption of an equipotential cell) is given by
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(1) |
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(2) |
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A modified Euler method is used to integrate Eq. 1 with a fixed
time step
t = 0.005 ms (see APPENDIX,
Numerical Integration). All simulations were performed on a
local workstation (SGI Indigo 2XZ R4400) using double-precision
arithmetic. A description of the model current formulations and their
general mathematical representations are given below. A detailed
listing of all equations and parameters can be found in the
APPENDIX.
Membrane Currents
Fast Na+ current. The model implements INa as a modification of the widely used Ebihara-Johnson model (15) proposed by Luo and Rudy (37). The current is given by
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(3) |
max). We use
gNa = 7.8 nS/pF, which is ~1.3 times the
temperature-adjusted value (with the assumption of
Q10 = 3) reported by Sakakibara et al. Figure
2 shows the steady-state activation and
inactivation curves, as well as time constants, used in the model. Data
from Sakakibara et al., shifted by the amounts described above and temperature corrected, are overlaid on the model curves. For activation time constants, temperature-adjusted data from Schneider et al. are
shown. The peak current-voltage (I-V )
relationship for INa is displayed in Fig.
3. No experimental I-V
curve was included, because available data (19, 42, 46), as a result of
equal intra- and extracellular Na+ concentrations, display
a nonphysiological reversal potential for INa close
to 0 mV.
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Inward rectifier K+ current. The model implements IK1 on the basis of available current data and resting membrane resistance measurements. The current is given by
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(4) |
for a 100-pF cell). We decreased
IK1 conductance to compensate for this
using gK1 = 0.09 nS/pF. When the latter is used,
input resistance calculated for a voltage step from
80 to
90 mV
is ~174 M
, closer to experimentally recorded values (e.g., ~150
M
in Ref. 52). However, there is a large range of measured values
for the input resistance of human atrial cells that may reflect
differences in the isolation procedure and its effect on
isolation-sensitive currents such as IK1. The
I-V relationship of IK1 is shown in
Fig. 4.
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Transient outward and ultrarapid rectifier K+ currents. The model implements Ito and IKur on the basis of data from our laboratory (20, 53). The currents are given by
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(5) |
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(6) |
80 mV. The traces of
Ito + IKur show typical rapid inactivation of Ito followed by a sustained outward
current carried by IKur. The combined model
current traces are overlaid with scaled experimental recordings from
Wang et al. (53).
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Rapid and slow delayed rectifier K+ currents. The model implements the two components of the classic delayed rectifier, IKr and IKs, on the basis of data from our laboratory (54, 55). The currents are given by
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(7) |
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(8) |
x(r) and
x(s) on V. Current amplitudes were
adjusted to match AP morphology. The I-V relationships
of both currents, along with scaled experimental I-V
curves from Wang et al., are shown in Fig. 3. Figure
8 displays traces for
IKr and IKs in response to
3,000-ms voltage-clamp steps from a holding potential of
80 mV. The
model traces are overlaid with scaled experimental recordings from Wang
et al. The combined model IK trace shows rapid and
slow activation phases linked to activation of IKr
and IKs, respectively.
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L-type Ca2+ current. The model implements the slow inward Ca2+ current on the basis of previous models and data from human atrial cells (35, 37, 40, 49). The current is given by
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(9) |
f,Ca = 2 ms)
is incorporated into the fCa gate, as reported by
Friedman et al. (21, 22) (see SR Ca2+ storage and
release), to better reproduce the time course of Ca2+-dependent inactivation of ICa,L.
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80 mV. The current traces exhibit rapid
activation, followed by two distinct inactivation time scales. Rapid
inactivation mediated by intracellular Ca2+ inactivates a
large fraction of the current soon after the onset of the voltage
step and is followed by a slower voltage-dependent inactivation.
The model response after a voltage-clamp step to 0 mV is compared with
a scaled averaged experimental recording from Sun et al. in Fig.
10B.
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Na+-K+ pump. Our formulation of the Na+-K+ pump follows the model of Luo and Rudy (37). The amplitude of the pump current [INaK(max)] is adjusted to maintain stable intracellular ion concentrations at rest (37, 59). The pump current is given by
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(10) |
Na+/Ca2+ exchanger. The exchanger current follows the revised formulation of Luo and Rudy (37). The exchanger current is given by
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(11) |
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Background Ca2+ and Na+ currents. The Ca2+ leak current is adjusted to maintain a stable [Ca2+]i at rest. The Na+ leak current is adjusted along with INaK to maintain stable [Na+]i and [K+]i at rest. Both approaches have been used extensively in previous AP models (37, 59). The background currents are given by
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(12) |
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(13) |
Ca2+ pump current. A sarcolemmal Ca2+ pump is included in the model to maintain [Ca2+]i at physiological levels. The pump current formulation (37, 40) is given by
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(14) |
SR Ca2+ storage and release. SR Ca2+ uptake and release are implemented using a two-compartment model, as reported by Luo and Rudy (37). Intracellular Ca2+ is taken up into an SR uptake compartment [network SR (NSR), 5.52% of cell volume] coupled to a release compartment [junctional SR (JSR), 0.48% of cell volume]. Uptake is [Ca2+]i dependent and includes a leak from the uptake compartment back into the intracellular space. The main currents (in mM/ms) involved in SR Ca2+ handling are given by
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(15) |
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(16) |
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(17) |
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Myoplasmic and SR Ca2+ buffers. Ca2+ buffering within the cytoplasm is mediated by troponin and calmodulin. Ca2+ buffering in the release compartment is mediated by calsequestrin. In particular, calsequestrin is essential in providing sufficient Ca2+ stores for release within the limited JSR volume. Buffers are considered at equilibrium throughout our simulations, and incoming Ca2+ in the myoplasm and JSR compartments is instantaneously equilibrated into free and buffer-bound fractions (see APPENDIX for computational methods). Buffer concentrations and binding constants are given in Table 1 and follow the models of Luo and Rudy (37) and Rasmusson et al. (40).
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RESULTS |
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Resting Properties
The model has a stable resting potential near
81 mV. All
intracellular concentrations are stable at rest with
[Ca2+]i = 0.1 µM,
[Na+]i = 11.2 mM, and
[K+]i = 139.0 mM. As stated earlier,
the resting input resistance measured as the current change from
80
to
90 mV is ~174 M
. The first 100-ms period of Fig.
14 shows the balance of membrane currents
in diastole during stimulation at 1,000 ms. During the diastolic phase
the steady state involves a balance between the pump and
exchanger currents, the background currents, and
IK1.
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Model AP
Figure 14 shows a model AP generated during stimulation at 1,000 ms in response to 2-ms pulses of 2-nA amplitude (twice diastolic threshold). The time course of membrane currents during the AP is also shown in Fig. 14. AP duration (APD) at 50 and 90% repolarization (APD50 and APD90), APA, AP overshoot (APO), and
max are reported in
Table 3. The AP generated with the control
parameters exhibits a spike-and-dome morphology commonly observed in
human atrial AP recordings. This morphology is similar to
experimentally recorded spike-and-dome APs reported by Benardeau et al.
(4) and Wang et al. (53). Intracellular Ca2+ dynamics
during the AP are shown in Fig. 15.
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AP Rate Dependence
Rate dependence of APD and AP refractoriness is an essential property of atrial cells that is central to our understanding of excitability and propagation patterns in the atria, particularly during reentrant arrhythmias. Table 3 summarizes the main properties of the model APs at various pacing periods. Figure 16 displays model APs, after 12 s of pacing, for various stimulation periods ranging from 1,000 to 300 ms. There is no significant increase in APD at >1,000 ms. This is in agreement with data from Fermini et al. (20) that showed no change in AP morphology and APD on changing the stimulation period abruptly from 10 to 1 s. Larger relative rate-dependent changes in APD are measured with APD50 than with APD90, as in experimental preparations (20, 32). Figure 16 also shows APs obtained in a tissue preparation at stimulation periods of 1,000, 600, and 300 ms (56). The experimentally observed changes in APD and AP morphology with stimulation rate are qualitatively similar to those observed in the model.
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The mechanism underlying APD changes may be investigated by examining the detailed current traces for APs at stimulation periods of 1,000 and 300 ms depicted in Fig. 16. The traces reveal that an increase in stimulation rate conspicuously reduces ICa,L. We inspected the time course of gating variables for ICa,L, IKr, and IKs to quantify their role in rate-dependent AP changes. We first consider a comparison of the slow gating variables f, fCa, xr, and x2s at the onset of the AP for stimulation periods of 1,000 and 300 ms. These slow variables are most susceptible to alteration at fast rates. For IKr and IKs, we found that incomplete deactivation at 300 ms resulted in an increase of 0.05 for xr and 0.004 for x2s compared with their values at 1,000 ms. This compares with reductions of 0.2 for f and 0.16 for fCa. The maximal conductances of ICa,L and IKs are comparable, whereas the maximal conductance of IKr is four times smaller. Hence, the changes in the f and fCa gate result in larger changes in current at the faster rate, pointing to ICa,L as an important determinant of AP rate dependence. However, the absolute magnitude of current changes with rate must be considered relative to the amplitude of other currents activating at a similar stage during the AP. For this reason, a small rate-dependent change in IK amplitude may have significant effects during late repolarization, when IK is the dominant repolarizing current.
To better ascertain the relative importance of IK and ICa,L in AP rate adaptation, we use a simulation protocol where the rate-induced changes in a specific current are identified and implemented individually to determine its specific contributions to AP rate dependence. We begin with a simulation of pacing at 1,000 ms that is interrupted just before the onset of a stimulus. Before continuation of the simulation and application of the scheduled stimulus, the values of the gating variables for the current under study are substituted with their values just before a stimulus during pacing at 300 ms. When the simulation is continued and the scheduled stimulus is applied, the resulting AP displays the rate-adaptation properties associated with the current under study, the state of which has been reset to its value during rapid pacing. We have carried out this procedure to examine the rate-dependent effect of ICa,L (via the d and f gating variables), IKr (via the xr gating variable), IKs (via the xs gating variable), and [Ca2+]i (via [Ca2+]i, [Ca2+]up, [Ca2+]rel, and the u, v, and w gating variables) on the AP.
Figure 17 displays the APs after 12 s of pacing at 1,000 and 300 ms, along with the APs arising from isolating the rate-adaptation effects of ICa,L, IK (IKr + IKs), and ICa,L + IK. We note that when the effect of either IKr or IKs alone was examined, the observed change in APD90 was ~50% of the total reduction seen with both currents combined (IK in Fig. 17). Also, the effect on APD90 of [Ca2+]i and ICa,L combined was similar to that of ICa,L alone, which is why the results for [Ca2+]i were not included in Fig. 17. The limited role of [Ca2+]i at fast pacing rates may reflect the opposing effects of a higher resting [Ca2+]i and a lower Ca2+ transient amplitude on ICa,L inactivation. Our first observation is that the effect of the rate-dependent decrease in available ICa,L alone is to lower the plateau potential and to significantly accelerate early repolarization. The lower plateau level reduces IK activation, slows terminal repolarization, and results in no net change in total APD. Second, we observe that the effect of the rate-dependent increase in available IK alone is to lower the plateau potential and to compensate for the effect of the reduced plateau level on IK activation, leaving the rate of terminal repolarization unchanged. The effect of IK rate adaptation alone explains less than one-half of the total rate-dependent reduction in APD90. Third, the effect of ICa,L and IK combined is synergistic, lowering the plateau potential and accelerating late repolarization relative to the effect of ICa,L alone. Changes in these two currents together are sufficient to account for the total rate-dependent AP shortening observed on reducing the pacing period from 1,000 to 300 ms. Hence, we conclude that ICa,L and IK act together to produce the observed rate-dependent AP shortening.
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ICa,L and Rate Dependence
Li and Nattel (35) showed that rate dependence can be abolished in isolated human atrial myocytes by blocking ICa,L. To compare these experimental data with the behavior of the model, we evaluated the effects of a strong decrease in L-type Ca2+ channel conductance on model AP rate dependence. Figure 18A shows model APs after 12 s of pacing with stimulation at 5,000 and 300 ms with a 90% reduction in ICa,L conductance. Figure 18B shows corresponding experimental results from Li and Nattel in which nifedipine was used to block ICa,L. In the model and experiments, block of ICa,L abolishes AP rate dependence. Although we have shown that ICa,L and IK play a role in rate adaptation, it appears that when ICa,L is strongly inhibited, as in Fig. 18, the plateau level is lowered to the point where IK activation is greatly reduced and can no longer contribute to rate adaptation.
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Block of INaCa and Role in APD
Recently, Benardeau et al. (4) investigated the role of Na+/Ca2+ exchange in AP morphology and APD. In particular, they examined the effect of INaCa block by Li+ on AP morphology. Their results revealed a shortening of APD and transient hyperpolarization after the AP in the presence of Li+. Figure 19 shows control and INaCa-blocked (90% decrease in maximum exchanger activity compared with control) model APs with stimulation at 1 s. Model APs display both of the changes observed in experimental preparations (see Fig. 7A in Ref. 4), i.e., AP shortening and hyperpolarization. Inspection of the current records shows that the decrease in APD may be related to two mechanisms: 1) an increase in Ca2+-dependent ICa,L inactivation caused by an increase in resting [Ca2+]i and Ca2+ transient amplitude and 2) a reduction in depolarizing exchanger current during the late phase of the AP. The observed membrane hyperpolarization is due to a reduced inward exchanger current at rest, causing a negative shift of the resting membrane potential.
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Variability in AP Morphology
Using the model, we attempted to simulate changes in current density that could explain the changes in human atrial AP morphology seen experimentally by many investigators (1, 4, 17, 24, 50, 51, 54). It has been suggested that variations in the relative density of the transient outward current may contribute to some of the AP variability (17, 54). We concentrated on three currents: Ito, IKur, and ICa,L. Figure 20 shows the result of varying the density of each of these three currents, from one-tenth to three times the control conductance value, on AP morphology. Figure 20, A-D, shows a continuum of AP morphologies after 12 s of pacing at a stimulation period of 1,000 ms for various modifications of the current densities. For ICa,L, we observe an increase in APD90 as ICa,L conductance increases. There is little change in APD90 as IKur conductance decreases. For Ito, we observe an increase in APD90 as Ito conductance decreases for large conductances, then a decrease in APD90 as Ito conductance decreases for small conductances. Of the three currents we have investigated, it appears that only the Ito variations can explain the whole spectrum of AP morphologies observed experimentally (4, 33, 54). This ranges from rectangular APs with a short plateau at the lowest Ito conductance values to shorter triangular APs at high Ito conductance values, through long spike-and-dome morphologies at intermediate Ito conductance values. Both IKur and ICa,L alone can explain a change from a spike-and-dome to a triangular AP shape, but a reduction in Ito is required in the model to produce the high plateau and absence of notch characteristic of a rectangular AP (54). In addition, the relation between Ito and AP morphology in the model parallels the relative changes in Ito observed experimentally in the AP classification of Wang et al. (54).
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DISCUSSION |
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We have developed a mathematical model of the human atrial AP. Whenever possible, we used data directly measured on human atrial cells to derive model parameters. When human data were not available to completely characterize a given current, we relied on existing AP models based mainly on guinea pig ventricular (37) and rabbit atrial (36) data. The model yields AP morphologies that are consistent with a variety of experimental observations and gives potential insights into their underlying ionic mechanisms.
Model Current Formulations Compared with Experimental Data
Fast Na+ current.
INa is a large, rapid inward current that has
proved difficult to measure accurately using voltage-clamp techniques.
Data from isolated human atrial cells are available elsewhere (19, 42).
However, these data for steady-state activation and especially inactivation of the current are shifted to negative potentials compared
with other human (7, 46) and animal (57) data. According to the
steady-state inactivation curves in these studies, ~90% of the
available INa would be inactivated at resting
potentials positive to
80 mV. In conjunction with the small maximum
conductances measured in these human cells (40 pA/pF maximum current in
Ref. 19 compared with 200 pA/pF in Ref. 57 for rabbit atrial cells), there would be insufficient current available to generate the large
max (167 V/s in Ref. 56) and
nanoampere-range current amplitudes typical of the AP upstroke. In
their model, Lindblad et al. (36) use the temperature correction
suggested by Colatsky (10) (+3 mV/10°C) to shift the model curves to
more positive potential. In the case of the human data reported by Feng
et al. (19) and Sakakibara et al. (42), the resulting shift of
~5-10 mV still leaves an inactivation half-maximal
potential (V1/2) well negative of
the value observed in other mammalian cells and insufficient to
generate the required fast inward current during the AP.
60 and +90 mV). The model displays fast
inactivation time constants in this range for potentials positive to
40 mV, but at more negative potentials (between
80 and
40 mV),
fast inactivation time constants increase in the model (to a maximum of
25 ms at about
65 mV). The rapid kinetics of fast inactivation over
a wide voltage range reported by Schneider et al. may be due in part to
their use of a fitting procedure allowing for overlap of activation and
inactivation processes. It is unclear whether fast inactivation slows
down in the study of Schneider et al. at potentials negative to
60 mV. They do not provide any fast time constants for recovery from inactivation at potentials between
95 and
60 mV, making it
difficult to determine the value of
h at these
potentials. On the basis of their recovery from inactivation data,
there appear to be three time constants involved, whereas the data of
Sakakibara et al. suggest two time constants. Given the difficulties in
interpreting the data in these studies within the context of the simple
formulation of Eq. 3, we chose to formulate the kinetics of
INa exactly as in the work of Luo and Rudy (37),
which nevertheless matches experimental data quite well, as shown in
Fig. 2.
Transient outward current. Ito has been shown to differ substantially in its faster recovery kinetics in humans than in other mammalian species (20, 47). On the basis of voltage-clamp current measurements, we found that Ito activation was best fit using a cubed activation gate. Activation and inactivation were fit to data from Wang et al. (55). A single inactivation gate was included in the model. In some current traces, slow inactivation can be seen, but this was attributed to slow inactivation of IKur and was not included directly in our formulation of Ito. The simulated current recordings shown in Fig. 6 indicate that the model reproduces the essential properties of experimental recordings of Ito + IKur (55).
Ultrarapid delayed rectifier current or sustained outward current. IKur has been shown to play an important role in AP current balance at plateau potentials in human atrial myocytes (1, 47, 55). On the basis of data from Wang et al. (55), we found that IKur had activation kinetics similar to those of Ito but activated at more negative potentials and displayed only partial slow inactivation. Those features are included in our formulation of the model. As with Ito, changes in IKur density can have considerable effects on AP morphology, as shown in Fig. 20.
L-type Ca2+ current.
ICa,L has been studied extensively in human atrial
tissue (9, 32, 35, 39, 49). Its complex inactivation kinetics, mediated
in part by intracellular Ca2+, are only partially
understood (49). This has made modeling the current more difficult and
highly dependent on an accurate formulation of intracellular
Ca2+ homeostasis and its control by the SR system (see
SR Ca2+ handling and buffering). Our
formulation follows that of Luo and Rudy (37) and includes
voltage-dependent activation and inactivation gates as well as a
Ca2+-dependent inactivation gate. We have made
modifications to the original formulation of Luo and Rudy on the basis
of available experimental data (35, 49). However, as discussed by Luo
and Rudy, fundamental questions regarding the complex voltage- and Ca2+-driven inactivation of ICa,L
remain, from an experimental and a modeling standpoint. For example,
data from Sun et al. (49) suggest that strictly voltage-dependent
inactivation of ICa,L is incomplete even for very
long inactivation pulses. This suggests that steady-state inactivation
( f
) saturates at an inactivation fraction greater than zero (0.45 measured in Ref. 49). As reported by
Luo and Rudy, we found that implementing such a relationship gave rise
to uncharacteristic results, such as large ICa,L
window currents observed on reuptake of Ca2+ during long
APs. Another example can be found in the data of Sun et al., where an
apparent second Ca2+-dependent inactivation time constant,
on the order of 300-400 ms, could be observed during inactivation
of ICa,L. This is not accounted for in the model,
and its origin has not been clarified experimentally. It may be that
additional dynamical complexity in ICa,L
inactivation arises as a result of a more complex time course of the
Ca2+ transient. This is in agreement with data from Hatem
et al. (26) that point to rapid and slow phases during the rise of the
Ca2+ transient in human atrial myocytes. These factors are
related to limitations in our modeling of the SR Ca2+
handling, which represents a simplified formulation of a temporally and
spatially complex intracellular process (see SR
Ca2+ handling and buffering and
Limitations).
Delayed rectifier current. IK has been characterized by Wang et al. (54) in human atrial cells. As in other mammalian species (44, 45), IK in human myocytes is the sum of a rapid and a slow component. We adopted the current formulation of Zeng et al. (61) fit to human atrial data (54). Steady-state activation and inactivation curves from human atrial cells are similar to those in other species (e.g., rabbit atria in Ref. 45). As pointed out by Zeng et al., there is conflicting evidence regarding the activation and deactivation kinetics of both currents. Wang et al. concluded that activation was well fit with a single time constant for both currents; however, they measured extremely slow activation time constants for IKs. When included in the model, these slow kinetics limit the role of IKs during the APO. Recent evidence from experiments on rabbit and guinea pig ventricular myocytes suggests fast and slow time constants for activation and deactivation of IKr and IKs (27, 43). Fast time constants for activation of IKs reached a maximum of 330 ms at +10 mV and 37°C. We account for possible faster kinetics of IKs in the model by using time constants that are one-half the values measured by Wang et al., putting the maximum time constant for IKs activation at ~500 ms at +10 mV. By use of this formulation, IKs plays a more significant role in repolarization during the AP but remains much less important than IKr at the lower plateau potentials typical of the human atrial AP.
SR Ca2+ handling and buffering.
SR Ca2+ handling and buffering involve uptake, storage, and
release of Ca2+ from various cellular compartments (58). We
followed the formulation of Luo and Rudy (37) but included a
modification to the release mechanism suggested by Friedman et al. (21,
22). Release of Ca2+ from the JSR compartment depends on
the flux of Ca2+ through the sarcolemmal and SR release
channels. This formulation aims to reflect the close coupling of L-type
Ca2+ channels in the sarcolemma and JSR Ca2+
release channel (34). For closely juxtaposed channels, one can expect
that true local Ca2+ concentration in the vicinity of the
release channel is more accurately reflected by local Ca2+
flux than by bulk cellular Ca2+ concentration. The
formulation is also adapted to afford a greater sensitivity of the
release mechanism to flux through the L-type Ca2+ channel,
as opposed to Ca2+ entry through the
Na+/Ca2+ exchanger (22). Release is triggered
when the computed Ca2+ flux exceeds a predefined threshold
through the opening of an activation gate u. Inactivation is
mediated through one flux-dependent (v) and one
voltage-dependent (w) inactivation gate. An added advantage of
this formulation is that it is completely analytic and does not require
identification of qualitative trigger events such as the occurrence of
max.
Inward rectifier K+ current.
IK1 plays a major role in the late repolarization
phase of the AP and in determining resting membrane potential and
resistance. As stated earlier, there is considerable variability in the
measured input resistance of isolated human atrial cells (1, 52), with
resting potentials often displaced to positive potentials compared with
multicellular tissue preparations (1, 52, 54, 56). Our formulation of
IK1 was adjusted to reproduce the slow late
repolarization rate observed in human atrial AP recordings. In doing
so, we were able to produce a realistic resting membrane resistance of
180 M
(52) and a resting potential characteristic of healthy human
atrial multicellular preparations of around
80 mV (56). The model
I-V relationship for IK1 (Fig. 4)
has a shape characteristic of direct recordings in human atrial cells
(31), but the specific parameters were adjusted to match the AP
characteristics described above.
Behavior of the Model AP
Control model AP. The control model AP exhibits a spike-and-dome morphology. As discussed below, several AP morphologies have been recorded from human atrial cells. In the model the transient outward current produces the initial phase 1 repolarization after the INa-driven upstroke. On inactivation of Ito, ICa,L, which remains after rapid Ca2+-dependent inactivation, causes a slight depolarization of the membrane and produces the dome of the AP. This net inward current is eventually countered by the slowly activating outward IK, resulting in slow repolarization. Slow late repolarization is a balance between deactivating IK, time-independent IK1, and opposing inward current generated by the Na+/Ca2+ exchanger during the late phase of the AP.
Na+/Ca2+ exchanger. The Na+/Ca2+ exchanger plays an important role in determining APD. As in experiments, we have shown that the exchanger is partly responsible for producing the slow late repolarization typical of human atrial APs. Block of the exchanger resulted in a significant reduction in APD90 (30% in Fig. 19). Because of the slow repolarization rate and small current amplitudes during late repolarization, APD90 is sensitive to small changes in late AP currents such as IK, IK1, and INaCa. This may explain some of the variability in APD90 measurements from human atrial cells and tissues (5, 23, 24, 32, 33).
Rate dependence of model AP. Rate dependence of the model AP is described in Fig. 16 and 17 and Table 3. The model is consistent with experimental data in several respects. First, it exhibits little rate dependence at frequencies <1 Hz, consistent with data from Fermini et al. (20) and Le Grand et al. (32). The model predicts greater relative AP shortening at APD50 than at APD90. This is seen in the data of Le Grand et al., even though their APD90 values are somewhat greater than those of the model. Comparing the top panels of Fig. 16, we observe that in experimental recordings and model simulations the AP waveforms at various rates tend to converge at some positive potential before complete recovery. Late repolarization is slowed at faster rates, in part because of a reduced activation of IK in APs with shorter, more negative plateaus that opposes the rate-dependent increase in IK availability. This tends to reduce rate adaptation of the model AP measured using APD90 and is a direct consequence of the sensitivity of late AP repolarization to small changes in IK, IK1, and INaCa, as discussed above.
Variability in AP morphology. Variability in AP morphology is often observed in recordings of human atrial APs. Early observations of APs from isolated human atrial tissue demonstrated a considerable variability in their morphology (17, 24, 50, 51). Some cells showed little phase 1 repolarization and a positive plateau; others exhibited a prominent phase 1 with a "spike-and-dome" morphology (17, 24, 50). It was suggested that age-related differences may be due to changes in Ito (17). Recently, there have been attempts to define various types of APs on the basis of their morphological characteristics and/or underlying ionic current densities (1, 4, 54). Wang et al. (54) identified three "types" of APs on the basis of morphology, ranging from type 1, a rectangular AP with a positive plateau, through type 2, a spike-and-dome AP with a plateau at ~0 mV, to type 3, a triangular AP with little plateau. They associated these changes in morphology with a measured increase in the relative density of Ito to IK from type 1 to type 3 APs. Benardeau et al. (4) recently characterized two AP types in their isolated human atrial cell preparations: type A APs are spike-and-dome APs with a high plateau similar to Wang's type 1 or 2 APs; type B APs were triangular APs similar to Wang's type 3. In an attempt to identify the source of differences in AP morphology between types A and B, these investigators found no discernible difference in ICa,L density between the two cell types. All three AP types have been recorded from human atrial multicellular preparations with fine-tipped microelectrodes (38).
The control model AP is typical of the type 1 APs described previously (38, 54). Our investigation of the role of Ito, IKur, and ICa,L in AP morphology in Fig. 20 reveals that, in our model, variations in Ito alone are able to generate many of the variations in AP morphology observed experimentally. AP shapes range from a triangular morphology (type 3 in Refs. 38 and 54 and type A in Ref. 4) for large Ito, through a spike-and-dome morphology (type 1 in Refs. 38 and 54) at intermediate Ito conductances, to a rectangular morphology (type 2 in Refs. 38 and 54 and type B in Ref. 4) for small Ito amplitudes. In contrast, variations in IKur and ICa,L alone cannot reproduce the rectangular morphology observed only in the presence of a small Ito. The role of Ito as an important determinant of AP morphology is supported by current measurements (54) showing that changes in AP morphology types (1, 2, and 3) were accompanied by changes in the relative amplitude of Ito consistent with the results of Fig. 20. An increase in Ito density was also shown to accompany the change from rectangular to spike-and-dome AP morphology observed in young vs. adult human atrial myocytes (17). Although the amplitude of Ito appears to be an important determinant of AP morphology, it is clear that it cannot be considered independently of other membrane currents and that AP morphology will depend on the relative amplitudes of all currents involved.Clinical Relevance
Atrial fibrillation (AF) is the most common sustained arrhythmia in clinical practice and that for which treatment remains the most problematic (38). The AP model provides insights into the basic mechanisms underlying a variety of important clinical determinants of AF. Impaired atrial APD accommodation to rate has been observed in patients with AF (5) and likely underlies the flat refractory period-heart rate relationship of patients susceptible to AF (2). Our model points to rate-dependent ICa,L inactivation combined with incomplete IK deactivation as the basic mechanism of APD accommodation to rate. Work by Li and Nattel (35) suggests that inhibition of ICa,L alone is sufficient to abolish rate adaptation of the human atrial AP, and it has been shown that human myocytes from dilated atria have a substantially reduced ICa,L (32, 39). In the present study, strong ICa,L reduction alone was sufficient to abolish rate adaptation of the model AP (Fig. 18). Although IK and ICa,L contribute to model AP rate adaptation, important reductions in ICa,L lower plateau height to the point where IK is greatly reduced and AP accommodation is virtually abolished. The model explains how strong reductions in ICa,L, as caused by atrial disease in humans or by exposure experimentally to ICa,L blockers, can largely abolish AP rate adaptation and contribute to the susceptibility to AF.Another property strongly associated with the occurrence of AF is AP heterogeneity (38). The present model indicates that variations in plateau currents (particularly Ito) over the experimentally measured range can account for much of the AP variability reported in human atria. These mechanistic insights hold the promise of an understanding of how drugs that alter ionic currents can affect these important AP properties and, thus, may help in the development of improved therapeutic approaches to treating atrial reentrant arrhythmias like AF.
Potential Limitations
Although we have tried to develop a model that is closely based on experimental data, limitations exist because of availability of data and considerable variability in existing experimental data. Limitations of this work are similar to those discussed for recent ionic models (36). The following issues should be considered in evaluating the model.Data from human atrial cells and tissues typically display considerable variability, with a wide range of current densities having direct consequences on AP shape, rate dependence, and response to pharmacological intervention. The use of averaged data from selected preparations does not take this into consideration and introduces possible complications in comparing model output with experimental AP recordings under various experimental conditions. This is in part why we have considered carefully the effect of varying specific current conductances on AP morphology. In addition, the temperature dependence of current kinetics and amplitude is often poorly characterized and must be approximated, with the help of available data, during the course of model development.
When possible, we have attempted to show experimental data along with corresponding simulated behavior. However, we note that, for voltage-clamp current recordings and experimental I-V curves, there may be significant bias in the measured current amplitudes. Selection factors such as the ability to record measurable currents and the viability of cells contribute to bias, especially with currents that are difficult to measure or display significant variability, such as ICa,L or IK. These amplitude-dependent selection factors are unlikely to bias the measured kinetics of a given current. Because of bias in current amplitude measurements, it is rare that exact current densities reported in experimental studies can be incorporated directly into a model and produce a realistic AP morphology. When necessary, current densities from experimental data were adjusted in the model to improve overall AP properties. The same limitations apply to direct comparisons of current recordings to corresponding model output. Such comparisons can be misleading unless one selects a current trace with an amplitude that exactly matches the amplitude used in the model or unless a scaling factor is used to match the peak current amplitudes. This is why we chose to present scaled experimental data for current recordings and I-V curves, with the scaling factor selected to match the maximum current amplitudes in the experiments to those in the model. Whenever such scaling is carried out, we ensured that the corresponding scaling factor was explicitly given to allow comparison of current magnitudes between model and experimental data.
Limited data are available on intracellular Ca2+ dynamics, Ca2+ storage and release kinetics from the SR, and its role in the inactivation of ICa,L. We have used a modified version of the intracellular Ca2+-handling scheme proposed by Luo and Rudy (37). This produces a realistic intracellular Ca2+ transient in response to suprathreshold Ca2+ flux into the cell. However, recent experimental results suggest that Ca2+-dependent ICa,L inactivation may involve complex local phenomena requiring an additional level of model complexity (49), beyond an accurate representation of the average Ca2+ transient. This is also important in view of the sensitive dependence of the model AP and its rate dependence on the dynamics of late repolarization, controlled in part by the Ca2+-dependent INaCa.
Our formulation of the delayed rectifier current represents a compromise between a simple current representation and available experimental evidence that IKr and IKs may display complex, multiexponential deactivation kinetics (27, 43). Slow deactivation of these currents may have important effects on late repolarization, refractoriness, and AP rate dependence. Whenever possible, we have tried to select the simplest formulation that captures the essential features of available experimental data. It is possible that the complex kinetics of some currents, such as IK, require more detailed current formulations involving several channel states not readily modeled using a simple Hodgkin-Huxley formalism.
We have chosen not to consider time-varying extracellular ion concentrations in our model at this stage. Certain currents, such as IK1, are known to depend on extracellular ion concentrations. Accumulation of ions in the extracellular cleft space may be an important modulator of ionic currents and AP characteristics at fast pacing rates. These phenomena may be taken into account by modifying appropriate current formulations and including dynamic extracellular cleft space ion concentrations in the model.
Finally, our model was adjusted to produce stable ionic concentrations at rest. In response to periodic stimulation, ionic balance is disturbed and slow changes in intracellular ionic concentrations occur in the model. Initial transients due to kinetic rate adaptation of the currents dissipate over a few seconds. However, small changes in ionic concentrations can take several minutes to develop and result in slow changes in the morphology of the AP. These changes are dependent on the nature and the formulation of the stimulus. The long-term behavior of the model depends on whether the charge applied through the stimulus is carried by an ion, the balance of which is explicitly included in the model. The choice of the ion selected to carry the stimulus charge also affects the outcome. We have chosen to simulate the model without the stimulus having a direct effect on the ion concentrations. The sensitivity of long-term ionic concentrations to stimulus formulation and frequency is a property of ionic models that attempt to preserve a detailed ionic balance (25a, 50a). When discussing results under periodic pacing, we have chosen to present model results after initial rapid transients have dissipated, after 12 s of pacing from rest. The AP morphology at this point is independent of the form and implementation of the stimulus and takes into account the kinetic adaptation of the currents at the specified pacing frequency. However, it does not include the effect of concentration changes occurring on a slower time scale in the model. This is also true of the experimental data on which the model is based. Most frequency-dependent data are collected with the assumption that preparations stabilize within seconds of the onset of stimulation. A detailed investigation of the long-term dynamics of the model and its dependence on the formulation of the stimulus is left for further study.
| |
APPENDIX. MODEL FORMULATION |
|---|
|
|
|---|
Some fractional equations require evaluation of a limit to determine their values at membrane potentials for which their denominator is zero.
Differential Equations
Instantaneous equilibration of Ca2+ with buffers is assumed in all cases
|
(18) |
|
|
(19) |
|
(20) |
|
(21) |
|
(22) |
|
(23) |
|
|
(24) |
|
(25) |
|
(26) |
|
(27) |
Equilibrium Potential
|
(28) |
Fast Na+ Current
|
(29) |
|
|
(30) |
|
|
(31) |
|
(32) |
|
(33) |
|
(34) |
Time-Independent K+ Current
|
(35) |
Transient Outward K+ Current
|
(36) |
|
|
(37) |
|
|
(38) |
|
|
(39) |
|
|
(40) |
Ultrarapid Delayed Rectifier K+ Current
|
(41) |
|
(42) |
|
|
(43) |
|
|
(44) |
|
|
(45) |
|
|
(46) |
Rapid Delayed Outward Rectifier K+ Current
|
(47) |
|
|
(48) |
|
|
(49) |
Slow Delayed Outward Rectifier K+ Current
|
(50) |
|
|
(51) |
|
|
(52) |
L-Type Ca2+ Current
|
(53) |
|
|
(54) |
|
|
(55) |
|
(56) |
Na+-K+ Pump Current
|
(57) |
|
(58) |
|
(59) |
Na+/Ca2+ Exchanger Current
|
(60) |
Background Currents
|
(61) |
|
(62) |
Ca2+ Pump Current
|
(63) |
Ca2+ Release Current From JSR
|
(64) |
|
(65) |
|
|
(66) |
|
|
(67) |
|
(68) |
Transfer Current From NSR to JSR
|
(69) |
|
(70) |
Ca2+ Uptake Current by the NSR
|
(71) |
Ca2+ Leak Current by the NSR
|
(72) |
Ca2+ Buffers
|
(73) |
|
(74) |
|
(75) |
Numerical Integration
At time step p, the updated value of a time-dependent variable is given by
|
(76) |
= V, any time-dependent ionic concentration, and
|
(77) |
|
| |
ACKNOWLEDGEMENTS |
|---|
The authors thank Drs. Jianlin Feng, Normand Leblanc, Gui-Rong Li, Hui Sun, and Zhiguo Wang for help with experimental data and Jean Perrault and Charles Dupont for the use of computer equipment.
| |
FOOTNOTES |
|---|
This work was funded by grants from the Medical Research Council of Canada (S. Nattel) and the Natural Sciences and Engineering Research Council (M. Courtemanche). M. Courtemanche is supported by a Chercheur-Boursier Scholarship from the Fonds de Recherche en Santé du Québec.
Address for reprint requests: M. Courtemanche, Research Center, Montreal Heart Institute, 5000 E. Belanger St., Montreal, PQ, Canada H1T 1C8.
Received 29 September 1997; accepted in final form 5 February 1998.
| |
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