TNNP model. Temporal dynamics

V dt it m h j xr1 xr2 xs r s d f f2 fcass

INa/200 IK1 Ito/5 IKr IKs ICaL/5 INaCa INaK IpCa IpK IbCa IbNa
Nai Ki/20 Cai104 CaSS CaSR 100Iup 10Irel 100Ixfer 1000Ileak
The script makes 1000 it time steps dt.

Calcium dynamics in the subspace CaSS , cytoplasm Cai and sarcoplasmic reticulum CaSR .

Integration schemes for the fastest m gate

Explicit Euler scheme

For the gate equation
    dy/dt = (y - y)/τ     (1)
the simple explicit Euler scheme is
    yn+1 = yn + (y - yn ) Δt/τ.     (2)
The scheme is stable if |1 - Δt/τ| ≤ 1 or if the time step Δt ≤ 2 τ (for large steps solutions oscillate and diverge).

The second-order Runge-Kutta method used in all one cell dynamics simulations before has the same stability condition.

Rush and Larsen integration scheme

For the TNNP model the minimal time is τm (-85) = 0.0011 ms and the maximal one is τf ~ 1200 ms. Therefore we have to use a lot of small time steps to get stable solution. Rush and Larsen proposed more accurate integration scheme. For y , τ = const one can integrate the gate equation (1) from tn to tn + 1 = tn + Δt exactly
    yn+1 = y + (yn - y ) e-Δt/τ.     (3)
In that case gate values are bunded 0 ≤ y ≤ 1 and reasonable for all Δt !

For slow gates with Δt << τ we can use e-Δt/τ ≈ 1 - Δt/τ and get
    yn+1 ≈ y + (yn - y )(1 - Δt/τ) = yn + (y - yn )Δt/τ.     (4)
It coinsides with the Euler formula (2) and we use (3) only for the fastest m gate.

Heart rhythms     updated 25 May 2012