TNNP model. Temporal dynamics

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

I_Na/200 I_K1 I_to/5 I_Kr I_Ks I_CaL/5 I_NaCa I_NaK I_pCa I_pK I_bCa I_bNa
Na_i K_i/20 Ca_i10⁴ Ca_SS Ca_SR 100Iup 10Irel 100Ixfer 1000Ileak
The script makes 1000 it time steps dt.
fig.1

Calcium dynamics in the subspace Ca_SS, cytoplasm Ca_i and sarcoplasmic reticulum Ca_SR .

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
y_n+1 = y_n + (y_∞ - y_n ) 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 τ_m(-85) = 0.0011 ms and the maximal one is τ_f ~ 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 t_n to t_{n + 1} = t_n + Δt exactly
y_n+1 = y_∞ + (y_n - 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
y_n+1 ≈ y_∞ + (y_n - y_∞ )(1 - Δt/τ) = y_n + (y_∞ - y_n )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