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
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.