## Luo-Rudy (LR) ionic model. Temporal dynamics

V(mV) dt(ms)   it mod iNa/30 iK iK1 iKp is ib

Ca x m h j d f
-100 < V, Es < 50 (mV), 0 < y < 1. The script makes 1000×it time steps dt. It is stable for dt ≤ 2 τmin = 0.011 ms (see Integration schemes for the fastest m gate).

The rate of change of membrane potential V is given by [1]
dV/dt = - (1/C) (Ii + Ist ),
where C = 1 μF/cm2 is the membrane capacitance, Ist is a stimulus current, and Ii is the sum of six ionic currents: INa , a fast sodium current; Isi , a slow inward current; IK , a time-dependent potassium current; IK1 , a time-independent potassium current; 1Kp , a plateau potassium current; and lb , a time-independent background current. The ionic currents are determined by ionic gates, whose gating variables are obtained as a solution 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
dy/dt = (y - y)/τy = αy - (αy + βy )y,
where   τy = l/(αy + βy ) and   y = αy /(αy + βy ). y represents any gating variable, τ, is its time constant, and y, is the steady-state value of y. αy and βy are voltage-dependent rate constants. In addition, αK1 and βK1 of the IK1 channel depend on extracellular potassium concentration.

## Inward currents

Fast sodium current
INa = 23 m3hj (V - ENa ),   ENa = 54.4 mV.
m(V), h(V), j(V) and 5 τm(V), τh(V)/30, τj(V)/100 (ms) are ploted below

here -100 ≤ V ≤ 50 (mV), 0 ≤ y ≤ 1. The minimal times are τm (-85) ~ 0.0055,   τh (20) ~ 0.14 ms.
Slow inward current
Isi = 0.09 d f (V - Esi ),   Esi = 7.7 - 13.0287 1n([Ca]i ),
Calcium uptake:   d([Ca]i )/dt = -10 -4 Isi + 0.07(10 -4 - [Ca]i ).

## Outward currents

Time-dependent potassium current
IK = GK X Xi (V - EK ),   GK = 0.282 ([K]o /5.4)1/2,   [K]o = 5.4 mM,   EK = -77 mV,
Xi =2.837{exp[0.04(V + 77)] - 1} / {(V + 77) exp[0.04(V + 35)]}   for V > -100 mV and   Xi = 1   for V < -100 mV,
d(V), f(V), x(V) and τd (V)/50, τh (V)/1000, τx (V)/1000 (ms) are ploted below

Time-independent potassium current
IK1 = GK1 K1 (V - EK1 ),   GK1 = 0.6047 ([K]o /5.4)1/2,
EK1 = (RT/F) In([K]o / [K]i ) = -87.3
mV,   [K]i = 145 mM,   RT/F = 26.52 mV,
Plateau potassium current
IKp= 0.0183 Kp (V - EKp ),   EKp = EK1 ,
Kp = 1 / {1 + exp[(7.488 - V) / 5.98]}
.
Background current
Ib = 0.03921 (V + 59.87).
Total time-independent potassium current
IK1(T) = IK1 + IKp + Ib.

Appendix
For V ≥ -40 mV
αh = αj = 0.0,   βh = l / (0.13{1 + exp[(V + 10.66)/-11.1]}),
βj = 0.3 exp(-2.535⋅10 -7V) / {1 + exp[-0.1(V + 32)]}
.
For V < -40 mV
αh = 0.135 exp[(80 + V)/-6.8],   βh = 3.56 exp(0.079V) + 3.1⋅105exp(0.35V),
αj = [-1.2714⋅105exp(0.2444V) - 3.474⋅10 -5exp(-0.04391V)](V + 37.78) / {1 + exp[0.311(V + 79.23)]},
βj = 0.1212 exp(-0.01052V) / {1 + exp[-0.1378(V + 40.14)]}
.
For all range of V
αm = 0.32(V + 47.13) / {1 - exp[-0.1(V + 47.13)]},   βm = 0.08 exp(-V/11)
.

αd = 0.095 exp[-0.01(V - 5)] / {1 + exp[-0.072(V - 5)]},
βd = 0.07 exp[-0.017(V + 44)] / {1 + exp[0.05(V + 44)]},
αf = 0.012 exp[-0.008(V + 28)] / {1 + exp[0.15(V + 28)]},
βf = 0.0065 exp[-0.02(V + 30)] / {1 + exp[-0.2(V + 30)]}
.

αx = 0.0005 exp[0.083(V + 50)] / {1 + exp[0.057(V + 50)]},
βx = 0.0013 exp[-0.06(V + 20)] / {1+exp[-0.04(V + 20)]}
.

αK1 = 1.02 / {1 + exp[0.2385 (V - EK1 - 59.215)]},
βK1 = {0.49124 exp[0.08032 (V - EK1 + 5.476)] + exp[0.06175 (V - EK1 - 594.31)]} / {1 + exp[-0.5143 (V - EK1 + 4.753)]}
.

[1] C. Luo and Y. Rudy   "A Model of the Ventricular Cardiac Action Potential. Depolarization, Repolarization, and Their Interaction"   Circulation Research 68 1501 (1991)

Heart rhythms     updated 27 May 2012