Self-organised pacemaker

a k kT Gs Δt Δx D

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,, 1 July 2011, Volume 6, Issue 7, e21934

Heart rhythms     updated 19 June 2012