BR Notes:
Vrest: = -80 mV: It was simplified here to allow the three ionic current to be totally
distinct sections of code for the purposes of building modules for reconstructing cell models.
BR77 fig 2 is reproduced on plot "AP_Ca" plot 2 (lower)
BR77 fig 3 is reproduced on plot "Current" plot 2 (right upper)
BR77 fig 4 is reproduced on plot "Current" plot 1 (left upper) and plot "AP_Ca" (upper)
BR77 fig 5 h*j product is reproduced on plot "AP_Ca" plot 1 (upper)
Conditions need to be changed using the Vclamp conditions described
BR77 fig 6 time course of f is reproduced on plot "AP_Ca" plot 1 (upper)
of x1 is reproduced on plot "current" plot 3 (lower left)
BR77 fig 7: Upper: Change gbCa +/- 10% for the BR variable gbars.
Change ixscale +/- 10% for BR ibar xi.
BR77 fig 8: Parset BRfig8
The influence of current during the early plateau phase is more evident in the Purkinje fibre than in the ventricular myocyte. To mimic the
Purkinje fibre, double ixscale and halve iK1scale. For Iext, use fgen_8, to select the Pulse2 set up to give 2 pulses: the first is the same as
the usual stimulus pulse, while the second (fgen_8.Pulse2.p2Amplitude) is varied from -0.04 to higher levels in 0.02 steps using the loops.
Greater depolarization during the plateau now shortens the AP.
Restore the orig parameter set: Runtime, Model params; then reset
ixscale to 0.0008 and iK1scale to 0,00035. or alternatively go to:
Par Set; Load project parameter set; select BR from popup window.
BR77 fig 12.Par set BRfig12. Effects of tetrodotoxin shutting down iNa.
Turn off stimulus, setting Iext = 0 to see spontaneous APs.
Diminish or knock out gbNa = 0 (_ block iNa). Increase the leak
Na conductance, gNaC by 8-fold
Other Figures need to be added using Vclamp.
JSim v1.1 import nsrunit; unit msec = 0.001 sec; unit conversion on;
// MODEL NUMBER: 0078
// MODEL NAME:BeelerReuter77
// SHORT DESCRIPTION: Cardiac Action Potential with Ca, K, and Na currents. Beeler-Reuter 1977 paper.
math BR77 { realDomain t msec; t.min=0; t.max=600; t.delta=0.04; // time
private real Vu = 1 mV; // non-dimensionalizing V by V/Vu
private real msm1 = 1 msec^(-1); // to balance units in the tau's
//PARAMETERS: General:
real Cm=1 uF/cm^2, clamp_flag = 0, pulsamp=0.05;
//VARIABLES: General:
real V(t) mV, VV(t) mV, Vrest = -80 mV, Iion(t) mA/cm^2; //Vrest = VK + 0.015*VNa in BR77 paper
extern real Iext(t) mA/cm^2,// pulse1, start=0 pulseAmp-1, duration=1. repeat=600
Vclamp(t) mV;
// Calcium Parameters and Variables: ************************************************ Ca
real Cao=2 mM, Cai0=2e-4 mM, tauCa = 150 msec,// for Ca pumps
Caconst= 1e-4 mM*msec^(-1)*cm^2/mA; /*=molar flux per volume per current*/
real VCa(t) mV,
iCa(t) mA/cm^2, Cai(t) mM, d(t),f(t),taud(t) msec,dinf(t),tauf(t) msec,finf(t),
gbCa =0.09 mS/cm^2;
// INIT COND: Calcium
when (t = t.min) { d=0.003; f=0.9999; Cai=Cai0;}
//ALGEBRA for Calcium time constants and steady state functions for gating variables
taud=1/(msm1*((0.095*exp(-0.01*(V/Vu-5)))/(1+exp(-0.072*(V/Vu-5)))
+0.07*exp(-0.017*(V/Vu+44))/(exp(0.05*(V/Vu+44))+1)));
dinf=taud*msm1*(0.095*exp(-0.01*(V/Vu-5)))/(1+exp(-0.072*(V/Vu-5)));
tauf=1/(msm1*((0.012*exp(-0.008*(V/Vu+28)))/(1+exp(0.15*(V/Vu+28)))
+0.0065*exp(-0.02*(V/Vu+30))/(exp(-0.2*(V/Vu+30))+1)));
finf=tauf*msm1*(0.012*exp(-0.008*(V/Vu+28)))/(1+exp(0.15*(V/Vu+28)));
VCa=-(30 mV)*log(Cai/Cao); iCa = gbCa*d*f*(V-VCa);
// ODEs for Ca:
Cai:t = -iCa*Caconst - (Cai - Cai0)/tauCa;
d:t = (dinf-d)/taud;
f:t = (finf-f)/tauf;
// Na Parameters and Variables: ***************************************************** Na
real Nao = 143 mM, Nai = 12 mM, VNa =-(60 mV)*log(Nai/Nao), gNaC = 0.003 mS/cm^2,
gbNa = 4 mS/cm^2;
real iNa(t) mA/cm^2,
m(t),h(t),j(t),taum(t) msec, minf(t),tauh(t) msec,hinf(t),tauj(t) msec,jinf(t);
// INIT COND: Sodium
when (t = t.min) { m=0.0104;h=0.989;j=0.976;}
//ALGEBRA for Sodium time constants and steady state functions for gating variables
taum=1/(msm1*((-1*(V/Vu+47))/(exp(-0.1*(V/Vu+47))-1)+40*exp(-0.056*(V/Vu+72))));
minf=taum*msm1*(-1*(V/Vu+47))/(exp(-0.1*(V/Vu+47))-1);
tauh=1/(msm1*(0.126*exp(-0.25*(V/Vu+77))+(1.7/(1+exp(-0.082*(V/Vu+22.5))))));
hinf=tauh*msm1*0.126*exp(-0.25*(V/Vu+77));
tauj=1/(msm1*((0.055*exp(-0.25*(V/Vu+78)))/(1+exp(-0.2*(V/Vu+78)))
+0.3/(exp(-0.1*(V/Vu+32))+1)));
jinf=tauj*msm1*(0.055*exp(-0.25*(V/Vu+78)))/(1+exp(-0.2*(V/Vu+78)));
iNa= (gbNa*m^3*h*j + gNaC)*(V-VNa);
// ODEs for Sodium:
m:t = (minf-m)/taum;
h:t = (hinf-h)/tauh;
j:t = (jinf-j)/tauj;
// K parameters and Variables: ******************************************************** K
real Ko = 5 mM, Ki = 150 mM, VK = -(60 mV)*log(Ki/Ko),
iK1scale = 0.00035 mA/cm^2, ixscale = 0.0008 mA/cm^2;
real x1(t),taux1(t) msec,x1inf(t), ix1(t) mA/cm^2, ibx1(t) mA/cm^2, iK1(t) mA/cm^2;
// INIT COND: Potassium:
when (t = t.min) { x1=0.0054;}
//ALGEBRA for Potassium time constants and steady state functions for gating variables
iK1 = iK1scale*(4*(exp(0.04*(V-VK)/Vu)-1)/(exp(0.08*(V/Vu+53))
+exp(0.04*(V/Vu+53))) +0.2*(V/Vu+23)/(1-exp(-0.04*(V/Vu+23))));
ibx1 = ixscale*(exp(0.04*(V-Vrest)/Vu)-1)/exp(0.04*(V/Vu+35));
ix1 = ibx1*x1;
taux1=1/(msm1*((0.0005*exp(0.083*(V/Vu+50)))/(1+exp(0.057*(V/Vu+50)))
+0.0013*exp(-0.06*(V/Vu+20))/(exp(-0.04*(V/Vu+20))+1)));
x1inf=taux1*msm1*(0.0005*exp(0.083*(V/Vu+50))/(1+exp(0.057*(V/Vu+50))));
// ODEs for Potassium:
x1:t = (x1inf-x1)/taux1;
// INITIAL CONDITIONS: General:
when (t=t.min) { VV=if(clamp_flag=0) Vrest else Vclamp;}
// ALGEBRA for General:
Iion = iNa+iK1 +ix1 +iCa;
//ODEs, General:
VV:t = if (clamp_flag=0) -(Iion-Iext*pulsamp)/Cm else (0 mV/s);
V = if (clamp_flag=0) VV else Vclamp;
} //..............................................................//end of BR77 code
/*
DETAILED DESCRIPTION:
Beeler-Reuter 1977, the first action potential model accounting for calcium currents.
The Na current, iNa, is a variant of the Hodgkin Huxley Na current, modified by
having a second inactivation variable j, The potassium current iK1 is inward rectifier.
The ix1 current represents a summation of other K currents in the dog
heart, and is somewhat empirical. The code is overly compact, to keep it on one page.
Notation is provided after the program code. The Na and K concentrations are fixed,
but Cai is variable: Ca influx with each excitation is counterbalanced by a slow
"efflux" represented by an exponential return (time constant tauCa) toward the
initial Cai.
TERMS USED:
> > Caconst: Conversion factor from Ca current, iCa, to change in concentration, Cai
> > Cai: Intracellular cytosolic average (mixed) concentration of free calcium ion, mM
> > Cai0: Value of Cai at t = ) sec; used as initial conditions for the calculations, mM
> > Cao: Extracellular average (mixed) free calcium ion concentration, mM
> > Cm: Plasmalemmal membrane capacitance of cardiomyocyte, uF/cm^2
> > Iext: Externally driven transmembrane current, mA/cm^2
> > Iion: Transmembrane current, the sum of the individual ionic channel and pump currents, mA/cm^2
> > Ki: Intracellular cytosolic average (mixed) concentration of free potassium ion, mM
> > Ko: Extracellular average (mixed) free potassium ion concentration, mM
> > Nai: Intracellular cytosolic average (mixed) concentration of free sodium ion, mM
> > Nao: Extracellular average (mixed) free sodium ion concentration, mM
> > VCa: Transmembrane Nernst potential for calcium, mV
> > VK: Transmembrane Nernst potential for potassium, mV
> > VNa: Transmembrane Nernst potential for sodium, mV
> > V: Transmembrane voltage for cardiomyocyte, mV
> > VV: Internal representation of V, for choosing Vlamp versus Iext
> > Vclamp: Voltage to which transmembrane voltage is driven using a current driver
> > Vrest: Resting transmembrane potential
> > Vu: Reference voltage, 1 mV, used to cancel units for dimensionless calculations, e.g. exp
> > clamp_fla: Flag to choose using voltage clamp versus current stimulus
> > d: Activation variable for the slow inward Ca current, iCa; dimensionless, 0 < d < 1
> > dinf: Value for d at infinite time
> > f: Inactivation variable for the slow inward Ca current, iCa; dimensionless, 0 < f < 1
> > finf: Value for f at infinite time
> > gNaC: Passive constant leak conductance for Na, mS/cm^2
> > gbCa: Maximum channel conductance for Ca, when d = f = 1, mS/cm^2
> > gbNa: Maximum channel conductance for Na, when m = h = j = 1, mS/cm^2
> > h: Inactivation variable for the Na current, iNa; dimensionless, 0 < h < 1
> > hinf: Value for h at infinite time
> > iCa: Current through Ca channel, mA/cm^2, Calcium
> > iK1: Current through K1 channel, mA/cm^2, Potassium
> > iNa: Current through Na channel, mA/cm^2, Sodium
> > ix1: Current through x1 channel, mA/cm^2. mainly potassium
> > ibx1: Current through x1 channel, mA/cm^2, mainly potassium, if x1 = 1.
> > ixscale Scalar for ix1 current, mA/cm^2, to adjust for sensitivity assessment
> > j: Second inactivation variable for the Na current, iNa; dimensionless, 0 < j < 1
> > jinf: Value for j at infinite time
> > m: Activation variable for the Na current, iNa; dimensionless, 0 < m < 1
> > minf: Value for m at infinite time
> > msm1: Constant, ms^(-1), used to make dimensionless time in exponentiation
> > pulsamp: Amplitude of current pulse, mA/cm^2
> > tauCa: Time constant for intracelleluar Ca to return to normal initial level, ms
> > taud: Time constant for conductance variable d, ms
> > tauf: Time constant for conductance variable f, ms
> > tauh: Time constant for conductance variable h, ms
> > tauj: Time constant for conductance variable j, ms
> > taum: Time constant for conductance variable m, ms
> > taux1: Time constant for conductance variable ibx1, ms
> > x1: Voltage- and time-dependent conductance variable, dimensionless. 0 < x1 < 1
> > x1inf: Value for ibx1 at infinite time, dimensionless
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: Cardiac electrophysiology, ionic currents, voltage clamp, dog canine heart,
ventricular fibre, fiber action potential, cardiomyocyte, Nernst potential,
time and voltage-dependent time constants, cell physiology,
REFERENCES:
Beeler GW Jr and Reuter H. Reconstruction of the action potential of ventricular
myocardial fibres. J Physiol (Lond) 268: 177-210, 1977.
Others related:
Bassingthwaighte JB and Reuter H. Calcium movements and excitation-contraction
coupling in cardiac cells. In: Electrical Phenomena in the Heart, edited by
DeMello WC. New York: Academic Press, Inc., 1972, pp 353-395.[Defines variables d and f kinetics]
Bassingthwaighte JB, Beeler GW, Sidell PM, Reuter H, and Safford RE. A model
for calcium movements and excitation-contraction coupling in cardiac cells.
In: Regulation and Control in Physiological Systems, edited by Iberall AS and
Guyton AC. Rochester, New York: Internat. Proc. Conf.Amer.Physiol.Soc,1973, pp 36-38.
[Implements model of 1972 paper]
Beeler GW Jr and Reuter H. Voltage clamp experiments on ventricular myocardial
fibres. J Physiol 207: 165-190, 1970.
Beeler GW Jr and Reuter H. Membrane calcium current in ventricular myocardial
fibres. J Physiol 207: 191-209, 1970.
Beeler GW Jr and Reuter H. The relation between membrane potential, membrane
currents, and activation of contraction in ventricular myocardial fibres.
J Physiol 207: 211-229, 1970.
REVISION HISTORY:
Original Author : JBB Date: 08/12/10
Revised by : BEJ Date: 08/18/10
Revision: 1) Update format of comments.
2) description of revision
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2010 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send an email
with the citation and, if possible, a PDF file of the paper to: staff@physiome.org.
*/