LR model

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 τ_min = 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 + I_st),
where C = 1 μF/cm² is the membrane capacitance, I_st is a stimulus current, and I_i is the sum of six ionic currents: I_Na, a fast sodium current; I_si, a slow inward current; I_K, a time-dependent potassium current; I_K1, a time-independent potassium current; 1_Kp, a plateau potassium current; and l_b, 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)/τ_y = α_y - (α_y + β_y)y,
where τ_y = l/(α_y + β_y) and y_∞ = α_y/(α_y + β_y). y represents any gating variable, τ, is its time constant, and y_∞, is the steady-state value of y. α_y and β_y are voltage-dependent rate constants. In addition, α_K1 and β_K1 of the I_K1 channel depend on extracellular potassium concentration.

Inward currents

Fast sodium current
I_Na = 23 m³hj (V - E_Na), E_Na = 54.4 mV.
m_∞(V), h_∞(V), j_∞(V) and 5 τ_m(V), τ_h(V)/30, τ_j(V)/100 (ms) are ploted below

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

Outward currents

Time-dependent potassium current
I_K = G_K X X_i (V - E_K), G_K = 0.282 ([K]_o/5.4)^1/2, [K]_o = 5.4 mM, E_K = -77 mV,
X_i =2.837{exp[0.04(V + 77)] - 1} / {(V + 77) exp[0.04(V + 35)]} for V > -100 mV and X_i = 1 for V < -100 mV,
d_∞(V), f_∞(V), x_∞(V) and τ_d(V)/50, τ_h(V)/1000, τ_x(V)/1000 (ms) are ploted below

Time-independent potassium current
I_K1 = G_K1 K1_∞ (V - E_K1), G_K1 = 0.6047 ([K]_o/5.4)^1/2,
E_K1 = (RT/F) In([K]_o / [K]_i) = -87.3 mV, [K]_i = 145 mM, RT/F = 26.52 mV,
Plateau potassium current
I_Kp= 0.0183 Kp (V - E_Kp), E_Kp = E_K1,
Kp = 1 / {1 + exp[(7.488 - V) / 5.98]}.
Background current
I_b = 0.03921 (V + 59.87).
Total time-independent potassium current
I_K1(T) = I_K1 + I_Kp + I_b.

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

α_d = 0.095 exp[-0.01(V - 5)] / {1 + exp[-0.072(V - 5)]},
β_d = 0.07 exp[-0.017(V + 44)] / {1 + exp[0.05(V + 44)]},
α_f = 0.012 exp[-0.008(V + 28)] / {1 + exp[0.15(V + 28)]},
β_f = 0.0065 exp[-0.02(V + 30)] / {1 + exp[-0.2(V + 30)]}.

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

α_K1 = 1.02 / {1 + exp[0.2385 (V - E_K1 - 59.215)]},
β_K1 = {0.49124 exp[0.08032 (V - E_K1 + 5.476)] + exp[0.06175 (V - E_K1 - 594.31)]} / {1 + exp[-0.5143 (V - E_K1 + 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