Driven by Pacemaker 

Spiral State 

Disordered State 

Notes
 a, b, eta are parameters of the equation.
 delta is a "small" parameter that sets when a simplified iteration scheme is used that is good only for small values of one of the fields (the scheme is used when the value is less than delta). Increasing delta to 10^{2} increases the speed at the cost of faithfulness to the original equations.
 amp is the amplitude of a sinusoidal forcing at one corner with frequency 0.5
 speed is a benign way of increasing the speed at the cost of jerkiness of the plotting: the iteration is done speed times between each plot update.
The system size is 32 along each edge in the first two frames and 64 along each edge in the third. The mesh size h for the finite difference code is 0.5 and the time step dt=0.1 h^{2}=0.025.
Dwight Barkley [Physica D49, 61 (1991)] modified the standard FitzHughNagumo type model specifically to increase the numerical iteration speed. See the reference for the equations and other parameter values to try. The three frames show different states at the same parameter values (except in the first frame an external oscillating drive is applied to one corner and the system size is twice as large in the last frame).
FitzHughNagumo equations have been used as crude models of biological excitable systems such as the heart muscle. The frames then would then crudely correspond in the ventricle to
 the naturally functioning muscle driven by signals from the atrioventricular node to give a regular beating;
 a selfsustained "reentrant" state which is thought perhaps to manifest itself as tachycardia arythmia;
 a disordered "reentrant" state which might correspond to (fatal if not corrected!) fibrillation.
The demonstrations here are very crude, so don't take the displayed results too seriously. However there is alot of really interesting work along these lines with much more realistic models. For recent papers see the focus issue of Chaos, volume 8 No. 1 (1998) or a recent paper by Qu et al. in the American Journal of Phsyiology 276 H269 (1999) (see the cover picture of this volume too).
Last modified Friday, May 7, 1999
Michael Cross