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CH Luo and Y Rudy

repolarization, and their interaction

A model of the ventricular cardiac action potential. Depolarization,

ISSN: 1524-4571

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1501

Original Contributions

A Model of the Ventricular Cardiac

Action Potential

Depolarization, Repolarization, and Their Interaction

Ching-hsing Luo and Yoram Rudy

A mathematical model of the membrane action potential of the mammalian ventricular cell is

introduced. The model is based, whenever possible, on recent single-cell and single-channel data

and incorporates the possibility ofchanging extracellular potassium concentration [K].. The fast

sodium current, 'Nap is characterized by fast upstroke velocity (Vma.=400V/sec) and slow recovery

from inactivation. The time-independent potassium current, IKI, includes a negative-slope phase

and displays significant crossover phenomenon as [K], is varied. The time-dependent potassium

current, IK, shows only a minimal degree of crossover. A novel potassium current that activates

at plateau potentials is included in the model. The simulated action potential duplicates the

experimentally observed effects ofchanges in [K], on action potential duration and rest potential.

Physiological simulations focus on the interaction between depolarization and repolarization

(i.e., premature stimulation). Results demonstrate the importance of the slow recovery of INa in

determining the response of the cell. Simulated responses to periodic stimulation include

monotonic Wenckebach patterns and alternans at normal [K]0, whereas at low [K]0 nonmono-

tonic Wenckebach periodicities, aperiodic patterns, and enhanced supernormal excitability that

results in unstable responses ("chaotic activity") are observed. The results are consistent with

recent experimental observations, and the model simulations relate these phenomena to the

underlying ionic channel kinetics. (Circulation Research 1991;68:1501-1526)

In the late 1970s, two models of the electrical

activity of cardiac cells were formulated based

on the formalism introduced by Hodgkin and

Huxley.1 McAllister et a12 developed a model of the

cardiac Purkinje fiber action potential. Subsequently,

Beeler and Reuter3 published a model of the electri-

cal activity of the mammalian ventricular myocyte

(referred to as the B-R model in the present paper).

The B-R model was based on experimental data that

were available at the time from voltage-clamp stud-

ies. These data were subject to limitations in avail-

able voltage-clamp techniques and their application

to multicellular preparations of cardiac muscle.4 In

addition, the concentrations of ions in the extracel-

lular clefts of the multicellular preparations were

unknown.

With the development of single-cell and single-

channel recording techniques in the 1980s, the limi-

From the Department of Biomedical Engineering, Case West-

ern Reserve University, Cleveland, Ohio.

Supported by grant HL-33343 from the National Heart, Lung,

and Blood Institute, National Institutes of Health, and by a

fellowship from the Ministry of Education, Republic of China.

Address for correspondence: Yoram Rudy, PhD, Department of

Biomedical Engineering, Case Western Reserve University,Wick-

enden Building, Room 505, Cleveland, OH 44106.

Received July 10, 1990; accepted January 23, 1991.

tations of voltage-clamp measurements were over-

come and the intracellular and extracellular ionic

environments could be controlled. The data from

single-channel recordings provide the basis for a

quantitative description of channel kinetics and

membrane ionic currents. In 1985, DiFrancesco and

Noble5 developed a model of the Purkinje fiber

action potential based on available single-cell and

single-channel data. Our goal is to incorporate,

whenever possible, recent experimental information

that have accumulated since the formulation of the

B-R model into the formulation of a modified model

(referred to as the L-R model in the text) of the

mammalian ventricular action potential. The work

presented here constitutes the first phase of this

effort. In this paper we formulate the fast inward

sodium current and the outward potassium currents.

We use the model to investigate phenomena that are

dominated by these currents and are only minimally

influenced by the slow inward current. Therefore, we

retain the B-R formulation of the slow inward cur-

rent to support the plateau of the action potential.

This paper focuses on the depolarization and repo-

larization phases of the action potential and on phe-

nomena that involve interaction between these pro-

cesses. These include supernormal excitability,

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Circulation Research

Vol 68, No 6June 1991

Wenckebach periodicity, and aperiodic response of the

cell to periodic stimulation. These phenomena were

observed recently in single ventricular cells and in

Purkinje fibers.67 In the B-R model, the extracellular

concentrations of ions are fixed. Because changes in

extracellular potassium concentration exert a strong

effect on the time course of repolarization,8 9 we intro-

duce in the model the dependence of the potassium

currents on potassium concentration. We also incorpo-

rate a negative-slope characteristic of the time-inde-

pendent potassium current10'll and a novel potassium

channel that activates at plateau potentials.'2 The fast

sodium current is characterized by fast upstroke veloc-

ity (Vm)13 and slow recovery from inactivation,'4 a

property that strongly influences the response of the

cell to premature stimulation.

In addition to the study of mechanisms that

determine the behavior of the single cell, an accu-

rate model of the action potential is important to

simulation studies of propagation of excitation in

cardiac tissue. Our studies of propagation15x6 were

limited by the inability to simulate important situ-

ations of physiological and clinical significance such

as the effects of elevated extracellular potassium

concentration, an important aspect of ischemia. In

addition, our model simulations of reentry'7 dem-

onstrated the importance of the interaction be-

tween depolarization and repolarization in the in-

duction and maintenance of reentrant arrhythmias.

The need for an accurate representation of this

interaction to further elucidate mechanisms under-

lying abnormal propagation and arrhythmogenesis

provided yet another motivation for the develop-

ment of the action potential model presented here.

Methods

The general approach is based on a numerical

reconstruction of the ventricular action potential by

using Hodgkin-Huxley-type formalism.' The rate of

change of membrane potential (V) is given by

dV/dt= -(1/C) (Ii+Is,)

where C is the membrane capacitance, 'St is a stimu-

lus current, and Ii is the sum of six ionic currents: INa,

a fast sodium current; Is,, a slow inward current; IK,a

time-dependent potassium current; 1K1, a time-inde-

pendent potassium current; 1Kp,a plateau potassium

current; and lb, a time-independent background cur-

rent. The ionic currents are determined by ionic

gates, whose gating variables are obtained as a solu-

tion to a coupled system of eight nonlinear ordinary

differential equations. The ionic currents, in turn,

change V, which subsequently affects the ionic gates

and currents. The differential equations are of the

form

(1)

dy/dt=(yx -y)/ry

(2)

where

and

y +=ac(a\+f3)

y represents any gating variable,

stant, and y. is the steady-state value of y. a, and /3

are voltage-dependent rate constants. In addition,

aKI and I3K1 of the IK1 channel depend on extracellular

potassium concentration.

The integration algorithm used to solve the differ-

ential equations is based on the hybrid methods

introduced by Rush and Larsen'8 and Victorri et al.9

Briefly, the algorithm uses an adaptive time step that

is always smaller than At

vals of relatively slow changes in V (AV.AVmin=0.2

mV), At is set equal to AVmaxV,where AVVmsx=0.8

mV. For time intervals of fast changes

(AV.lAVmax), At is set equal to AVmin/V. If this At

results in AV.AVmax, At is reduced until the condi-

tion AV<AVmax is met. During the stimulus, a fixed

time step (0.05 or 0.01 msec) is used to minimize

variability in the stimulus duration caused by the time

discretization procedure.2t)

Rate constants of ionic gates were obtained by

parameter estimation with an adaptive nonlinear least-

squares algorithm developed by Dennis et ali2" All

computer programs were coded in FORTRAN 77 (Mi-

crosoft, Seattle), and all simulations were implemented

(double precision) on a Macintosh IIcx computer.

r,is its time con-

=

msec. For time inter-

in V

Fornulation of Equations for Ionic Currents

All ionic currents are computed for

membrane. Membrane capacity is set at 1 ALF/cm.22

The formulation

adjusted to 37°C by using a Q1(, adjustment factor.

Ionic concentrations for standard preparations are

[K1.=5.4 mM, [K]i=145 mM,1' [Na]i=18 mM,1323

[Na]o=140 mM, and [Ca]0=1.8 mM. [Ca]i varies

during the action potential; we set [Ca]i=2x104mM

as an initial value under standard conditions. We

assume that a short-term stimulation does not appre-

ciably affect the ionic environment of the cell under

normal conditions and, therefore, the ionic concen-

trations (except [Ca]i) do not change dynamically in

our simulations. The complete set of equations for all

ionic currents is provided in Table 1.

INa,: Fast sodium current. The model of the fast

sodium channel incorporates both a slow process of

recovery from inactivation and adequate maximum

conductance that results in a realistic rate of mem-

brane depolarization. We adopt the activation (m)

and inactivation (h) parameters of Ebihara and

Johnson4 (we refer to the Ebihara-Johnson model of

INa as the E-J model). The formulation of these

parameters is based on data from cardiac cells (chick-

en embryo) and results in a realistic rate of depolar-

ization (Vmax=300 V/sec). However, it does not in-

clude the property of slow recovery. Following the

methods of Beeler and Reuter,3 we incorporate a

slow inactivation gate (j) to represent this slow

process. As suggested by Haas et al,24 the steady-

1 cm- of

is based on experimental data

ry=l/(ay+ y)

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Luo and Rudy A Model of the Ventricular Cardiac Action Potential

1503

TABLE 1.

Fonnulations of Ionic Currents

Inward currents

Fast sodium current

INa=23 * m3*h.j * (V-ENa)

For V>-40 mV

ah=aj=0.0, 13h=l/(0.13{1+exp[(V+10.66)/-11.1J})

,6j=0.3* exp(-2.535* 10-7 V)/{1+exp[-0.1(V+32)]}

For V<-40 mV

ah=0.135 *exp[(80+V)/-6.8], /h=3.56 exp(0.079V)+3.1

aj=[-1.2714*105' exp(0.2444V)-3.474

Pj=0.1212* exp(-0.01052V)/{1 +exp[-0.1378(V+40.14)J}

For all range of V

am=0.32(V+47.13)/{1-exp[-0.1(V+47.13)]}, 1l3m=0.08 exp(-V/11)

Slow inward current

I,=0.09 d f(V-EBi), E,i=7.7-13.0287 1n([Ca]i)

ced=0.095* exp[-0.01(V-5)]/{1+exp[-0.072(V-5)]}

1d=0.07* exp[-0.017(V+44)]/{1+exp[0.05(V+44)]}

af=0.012* exp[-0.008(V+28)]/{1 +exp[0.15(V+28)J}

,Bf=0.0065 exp[-0.02(V+30)]/{1+exp[-0.2(V+30)]}

Calcium uptake: d([CaJi)/dt=- 10-4 I±+0.07(10-4_[Ca]i)

Outward currents

Time-dependent potassium current

IK=GK X * Xi * (V-EK), UK=0-.282 *jKJJ5.4

Xj=2.837*{exp[O.04(V+77)]-1}/{(V+77) * exp[0.04(V+35)]} for V> -100 mV andXj=1 for V<-100 mV

ax=0.0005* exp[0.083(V+50)]/{1+exp[0.057(V+50)]}

13x=0.0013 exp[-0.06(V+20)]/{1+exp[-0.04(V+20)J}

Time-independent potassium current

IK1=GK1 K1. *(V-EKI), GK1=0.6047 V[K]J5.4

aK,=1.02/{1+exp[0.2385 * (V-EK1-59.215)]}

,S3K1={0.49124 exp[0.08032* (V-EK1+5.476)I+exp[0.06175 (V-EK1-594.31)]}/

{1+exp[-0.5143* (V-EKl+4.753)I}

Plateau potassium current

IKP=0-0183 * Kp * (V-EKp), EKp=EK

Kp= /{1+exp[(7.488-V)/5.98]}

Background current

1b=0.03921 (V+59.87)

Total time-independent potassium current

IK1(T)=K1+IKp+ Ib

IK, time-dependent potassium current (/uA/cm'); IK, fully activated potassium current (/LA/cm') (IK=IK/X); IK1,

time-independent potassium current (pA/cm'); 1Kp, plateau potassium current (gA/cm'); 'b, background leakage

current (gA/cm'); IK1(T), total time-independent potassium current (gA/cm') (IK1(T)=IK1+IKP+Ib); INa, fast sodium

current (gA/cm'); ILj,slow inward current (giA/cm'); V, membrane potential (mV); V, time derivative of V (V/sec);

Vnax, maximum rate of rise of V (V/sec); Ei, reversal potential of ion i (mV); G,, maximum conductance of channel i

(mS/cm2); [A]0, [A],, extracellular and intracellular concentrations of ion A, respectively (mM); m, h, j, activation gate,

fast inactivation gate, and slow inactivation gate of INa; d, f, activation gate and inactivation gate of1,j; X, Xi, activation

gate and inactivation gate of IK; Kl, inactivation gate ofIKI; Y0, steady-state value of activation (inactivation) gate y;ay,

fly,opening and closing rate constants of gate y (msec-1); ry,time constant of gate y (msec).

105 exp(0.35V)

10`5 exp(-0.04391V)] (V+37.78)/{1+exp[0.311 * (V+79.23)]}

state values of j (ja) are obtained by setting jmha,

where ha is from the E-J model. The time constant of

j (t)is setequalto therjof the B-R model. The rate

constants ac and ,B1 are obtained by using the param-

eter estimation procedure mentioned above.21 The

sodium current is

INa=GNa* m3 h

j (V-ENa)

(3)

where GNa is the maximum conductance of the so-

dium channel (23 mS/cm2)425; ENa is the reversal

potential of sodium [ENa=(RT/F)

and m, h, and j are obtained as solutions to Equation

2 with the appropriate rate constants. Note that ENa

computed with the E-J model

[Na]i=40 mM in chicken embryo heart cells.25 In our

model we set ENa=54.4 mV based on [Na]i=18 mM

in mammalian ventricular cells.13'23

IA: Slow inward current. Representation of I1i is the

same as in the B-R model. The formulation

provided in Table 1.

iIn([Na]0/[Na]l)];

is 29 mV since

is

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Vol 68, No 6June 1991

-100 -80 -60 -40 -20

0

20

40

V (mV)

FIGURE 1.

vated time-dependent potassium current (I) current (denoted

IKin thefigure) for [K] =5.4, 50, and 150 mM. Note minimal

crossover between curves of different [K]0 and strong inward

rectification.

Current-voltage (I-V) curves of the fully acti-

IK: Time-dependentpotassium current. Inpatch-clamp

experiments, Shibasaki26 showed that 1) the IK channel

is controlled by a time-dependent activation gate (X)

and a time-independent inactivation gate (Xi), neither

of which depends on [K]O; and 2) the single-channel

conductance is proportional to

equation suggested by Shibasaki:

[K]0. We use the

1K=GK-X-Xi * (V-EK)

(4)

and introduce the [K]O dependence through

GK=0.282.

[K]0/5.4

and

EK=(RT/F) * In ([K]o+PRNaK [Na]0

k[K]i+PRNaK' [Na]li

where PRNaK=0.01833 is the Na/K permeability ra-

tio.27,28 For this value of PRNaK and[K])=5.4 mM, the

computed EK is -77 mV, a result that is consistent

with the measurements of Beeler and Reuter.3 Also,

note that GK=0.282 mS/cm2 for [K]0=5.4 mM. This

value is obtained from the fully activated current

IK=IK/X (B-R model,3 Figure 1) for V=-100 mV, a

potential at which Xi=1.

Xi introduces the inward rectification property of

IK. It is obtained from the B-R expression for IK by

factoring out GK- (V-EK) at [K]O=5.4 mM. The

formulation is provided in Table 1. To verify that this

formulation correctly introduces the [K]0 depen-

dence of IK we plotted the computed fully activated

current (!K) for[K]O=5.4, 50, and 150 mM (Figure 1).

The behavior is consistent with the experimental

observations of McDonald and Trautwein,29 Matsu-

ura et al,27 and Shibasaki.26 Note the strong inward

rectification and the minimal crossover between

curves of different [K]0.

IKI: Time-independent potassium current. Sakmann

and Trube,1011using patch-clamp techniques, dem-

onstrated two important properties ofIKI: 1) square

V (mV)

FIGURE 2.

pendentpotassium current (IK1) for different[K], (indicated in

the figure in mM). Note the large degree of crossover between

curves ofdifferent [K]O, strong inward rectification, negative

slope of the I-V curves, and zero-cuirrent contribution at high

potentials.

Current-voltage (I-V) curves of the time-inde-

root dependence of single-channel conductance on

[K]O; 2) high selectivity for potassium

-Nernst potential of potassium). Kurachi30 identi-

fied an inactivation gate (K1) of the IKI channel. This

gate, in addition to its dependence on membrane

potential, depends on EKI and therefore on [K]0 KI

closes at high potentials and therefore IK1 has no

contribution at this range. In addition, the time

constant of KI is small (TKI=0.7 msec at V=-50 mV

for [K]0=5.4 mM), so it can be approximated by KI,

(its steady-state value).

Based on these findings, we formulate the

current as follows:

(i.e., EKI

IKI

1KI=GKI

. KIX

.(V-EKI)

(5)

where

EKI=(RT/F)

.In([K]0/[K]j)

KI x

aKl/(aKI+13K1)

GK1=0.6047 \[K]0/5.4

(GK1=0.6047at[K]0=5.4 mM10,31)

To verify that this formulation correctly introduces

the [K]O dependence of IKI we plotted the computed

current-voltage curve for [K]0= 10, 20, 40, 75, and 150

mM (Figure 2). The results resemble the single-

channel measurements of Kurachi (see Figure 12B of

Reference 30). Note the strong inward rectification,

the crossover between curves of different [K]o, the

zero contribution at high potentials, and the negative

slope over a certain potential range. These charac-

teristics reflect the voltage and [K]o dependence of

the KI gate.

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