delay
Ymin
Ymax
it
fps
This script uses explicit Euler scheme. It makes it time steps per frame.
Full ring length is 1024×Δx.
u is the red curve, v is the blue one, Ta is
the green line, Taav level is shown by the black line.
Use Alt- r, s keys to control the script.
You can see here formation of a pacemaker in Aliev-Panfilov
model with non-oscillating kinetics. Pacemaking activity occurs because
the contraction of the medium (proportional to Ta(x) - Taav )
that follows a propagating wave of excitation subsequently stretches the medium
in the neighborhood of the initiation site. This stretch induces a depolarizing
stretch activated current Is that initiates a subsequent excitation wave.
Excitation-contraction coupling model
We consider 1D lattice that consists of material points located
at xi connected by springs. Similar to [1] all
springs follow Hooke's force-displacement relation and
may produce additional active contraction forces
fi+ =
c(xi+1 - xi - lo )/lo + Tai ≡
cδi + Tai ,
fi- =
-cδi-1 - Tai-1 ,
where lo , c are the spring length and stiffness (we put c = 1),
Tai is the value of variable Ta dTa/dt = ε(u) (kT u - Ta).
We consider the string with fixed length xn - xo = L.
As it is easy to check in that case
∑i=0n-1 δi = 0.
Following [1] elastostatics is assumed in this model, i.e. the stationary
deformations corresponding to each given configuration of active
forces and boundary conditions are computed. In mechanical equilibrium
fi+ + fi+ = 0, ⇒
δi - δi-1 =
-(Tai - Tai-1 ),
δi = Taav - Tai ,
Taav = ∑i Tai .
Physiological influence of contraction on cardiac tissue is given by a depolarising
stretch-activated current Is through stretch activated channels
Is = Gs ((1 + δ)1/2 - 1)(u - Es) ≅
0.5 Gs δ (u - Es) Gs = 1.5 and Es = 1 are the maximal conductance and reversal
potential of the stretch activated channels.
The stretch activated current is active only if δ > 0 (stretch).
(I have to set Gs = 0.5 to get 1D pacemaker).
[1] Louis D. Weise, Martyn P. Nash, Alexander V. Panfilov
"A Discrete Model to Study Reaction-Diffusion-Mechanics Systems" PLoS ONE, www.plosone.org, 1 July 2011, Volume 6, Issue 7, e21934
