## Beeler-Reuter (BR) ionic model. Temporal dynamics

The model [1] incorporates two voltage- and time-dependent inward currents, the excitatory inward sodium current, iNa , and a secondary or slow inward current, is , primarily carried by calcium ions. A time-independent outward potassium current, iK , exhibiting inward-going rectification, and a voltage- and time-dependent outward current, ix , primarily carried by potassium ions, are further elements of the model.
dVm/dt = -(iK + ix + iNa + is - iexternal ),
d Ca /dt = -10 -7is + 0.07(10 -7 - Ca),
dy/dt = (y - y)/τy ,

where y is x, m, h, j, d or f gate (see the constant at the bottom).

Excitatory inward sodium current
iNa = (4 m3h j + 0.003)(Vm - 50)
m(V), h(V), j(V) and 20 τm(V), τh(V)/20, τj(V)/100 ms are ploted below

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

Slow inward current is
is = 0.09 d f (Vm - Es ),     Es = -82.3 - 13.0287 In Ca .
Outward potassium current ix
ix = x Ix ,     Ix = 0.8(exp[0.04(Vm + 77)] - 1) / exp[0.04(Vm + 35)].
d(V), f(V), x(V) and τd (V)/50, τf (V)/1000, τx(V)/1000 ms are ploted below

-100 ≤ V ≤ 50 (mV), 0 ≤ y ≤ 1.

Outward potassium current iK
iK = 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 τmin = 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
τy = 1/(αy + βy ),     y = αy /(αy + βy ),
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 αx 0.0005 0.083 50 0 0 0.057 1 βx 0.0013 -0.06 20 0 0 -0.04 1 αm 0 0 47 -1 47 -0.1 -1 βm 40 -0.056 72 0 0 0 0 αh 0.126 -0.25 77 0 0 0 0 βh 1.7 0 22.5 0 0 -0.082 1 αj 0.055 -0.25 78 0 0 -0.2 1 βj 0.3 0 32 0 0 -0.1 1 αd 0.095 -0.01 -5 0 0 -0.072 1 βd 0.07 -0.017 44 0 0 0.05 1 αf 0.012 -0.008 28 0 0 0.15 1 βf 0.0065 -0.02 30 0 0 -0.2 1

Heart rhythms     updated 26 May 2012