BR model

Beeler-Reuter (BR) ionic model. Temporal dynamics

The model [1] incorporates two voltage- and time-dependent inward currents, the excitatory inward sodium current, i_Na, and a secondary or slow inward current, i_s, primarily carried by calcium ions. A time-independent outward potassium current, i_K, exhibiting inward-going rectification, and a voltage- and time-dependent outward current, i_x, primarily carried by potassium ions, are further elements of the model.
dVm/dt = -(i_K + i_x + i_Na + i_s - i_external ),
d Ca /dt = -10^-7i_s + 0.07(10^-7 - Ca),
dy/dt = (y_∞ - y)/τ_y ,
where y is x, m, h, j, d or f gate (see the constant at the bottom).

Excitatory inward sodium current
i_Na = (4 m³h j + 0.003)(Vm - 50)
m_∞(V), h_∞(V), j_∞(V) and 20 τ_m(V), τ_h(V)/20, τ_j(V)/100 ms are ploted below

here -100 ≤ V ≤ 50 (mV), 0 ≤ y ≤ 1. So the minimal times are τ_m(-85) ~ 0.012, τ_h(-85) ~ 1 ms.

Slow inward current i_s
i_s = 0.09 d f (Vm - E_s), E_s = -82.3 - 13.0287 In Ca .
Outward potassium current i_x
i_x = x I_x, I_x = 0.8(exp[0.04(Vm + 77)] - 1) / exp[0.04(Vm + 35)].
d_∞(V), f_∞(V), x_∞(V) and τ_d(V)/50, τ_f(V)/1000, τ_x(V)/1000 ms are ploted below

-100 ≤ V ≤ 50 (mV), 0 ≤ y ≤ 1.

Outward potassium current i_K
i_K = 0.35{4(exp[0.04(Vm + 85)] - 1) / (exp[0.08(Vm + 53)] + exp[0.04(Vm + 53)])
+ 0.2(Vm + 23) / (1 - exp[-0.04(Vm + 23)])}.

The model may be modifed by speeding up the calcium dynamics (i.e. d, f gates) by the mod value.

Vm(mV) dt(ms) it mod iNa/30 ix iK is

Ca x m h j d f
-100 ≤ Vm ≤ 50, 0 ≤ Es ≤ 150 (mV), 0 ≤ y ≤ 1. The script makes 1000 it time steps dt. It is stable for dt ≤ 2 τ_min = 0.025 ms (see Integration schemes for the fastest m gate).

[1] G. W. Beeler and H. Reuter Reconstruction of the action potential of ventricular myocardial fibres J. Physiol. 268 177 (1977)

Appendix
τ_y = 1/(α_y + β_y ), y_∞ = α_y /(α_y + β_y ),
α = (C1 exp[C2 (Vm + C3)] + C4 (Vm + C5)) / (exp[C6 (Vm + C3)] + C7),
the same equation as for α is used for β.

Const
(msec^-1) C1
(msec^-1) C2
(mV^-1) C3
(mV) C4
(mV.msec)^-1 C5
(mV) C6
(mV^-1) C7

α_x 0.0005 0.083 50 0 0 0.057 1

β_x 0.0013 -0.06 20 0 0 -0.04 1

α_m 0 0 47 -1 47 -0.1 -1

β_m 40 -0.056 72 0 0 0 0

α_h 0.126 -0.25 77 0 0 0 0

β_h 1.7 0 22.5 0 0 -0.082 1

α_j 0.055 -0.25 78 0 0 -0.2 1

β_j 0.3 0 32 0 0 -0.1 1

α_d 0.095 -0.01 -5 0 0 -0.072 1

β_d 0.07 -0.017 44 0 0 0.05 1

α_f 0.012 -0.008 28 0 0 0.15 1

β_f 0.0065 -0.02 30 0 0 -0.2 1

Heart rhythms updated 26 May 2012

Const (msec^-1)	C1 (msec^-1)	C2 (mV^-1)	C3 (mV)	C4 (mV.msec)^-1	C5 (mV)	C6 (mV^-1)	C7
α_x	0.0005	0.083	50	0	0	0.057	1
β_x	0.0013	-0.06	20	0	0	-0.04	1
α_m	0	0	47	-1	47	-0.1	-1
β_m	40	-0.056	72	0	0	0	0
α_h	0.126	-0.25	77	0	0	0	0
β_h	1.7	0	22.5	0	0	-0.082	1
α_j	0.055	-0.25	78	0	0	-0.2	1
β_j	0.3	0	32	0	0	-0.1	1
α_d	0.095	-0.01	-5	0	0	-0.072	1
β_d	0.07	-0.017	44	0	0	0.05	1
α_f	0.012	-0.008	28	0	0	0.15	1
β_f	0.0065	-0.02	30	0	0	-0.2	1