Scroll waves meandering in a model of an excitable medium

1. Scroll waves meandering in a model of an excitable medium Presenter: Jianfeng Zhu Advisor: Mark Alber

2. Introduction • Scroll waves are three-dimensional(3D) vortices which are extensions of spiral waves (2D) that occurs in a variety of excitable media. • A scroll wave is usually characterized by its filament, which is an extension into three dimensions of the notion of the core of the spiral wave.

3. Excitable medium • Excitable medium is the systems which have the ability to propagate signals without damping. e.g. forest is an excitable medium when the forest fire travels as a wave. • Passive medium : the wave propagation is characterized by a gradual damping of signal amplitude due to friction. e.g. sound waves passing through the air.

4. Heart as an excitable medium • Heartbeat is a wave that passes across the heart muscle. A small triggering impulse can lead to a large response (an electrical discharge across the cell membranes, together with a contraction of the heart muscle). • A piece of heart tissue can be triggered by the excitation of a neighboring piece of tissue, which is the basis for the wave action • Heart undergoes scroll waves when the heart is malfunctioning, such as cardiac arrhythmia and fibrillation in the ventricles of the heart.

5. Aliev-Panfilov Model • e: membrane potential • r: conductance of the slow inward current • The parameters are related to the key characteristics of the cardiac tissue, such as the shape of the action potential, refractoriness and the restitution of action potential duration. • we choose and a is varied between 0.12 and 0.18

6. Action potential • The cardiac action potential is the electrical activity of the individual cells of the electrical conduction system of the heart. The cardiac action potential has five phases.

7. Numerical computation • spiral waves and its core ( the white lines) in a 2D excitable medium of elements. (hs=0.6 and ht=0.03) • The light gray area represents the excited state of the tissue (e>0.6). (Panfilov, et al., 2005)

8. Extend the computations to 3D(128*128*128) domain • Copy the 2D spiral wave pattern to all layers of our numerical grid in z-direction. • Shift the whole 2D spiral wave for each (z) slice of the system in the x-direction as -- thickness of the medium (12.7mm)

9. Filament under different time • Filament dynamics at a=0.18 at t =0s (a), t =1s (b) and t = 3s (c) (aperiodic meandering) (Panfilov, et al. 2005)

10. The length of the filament (Panfilov, et al. 2005) • a=0.18 with (black solid line) • a=0.18 with periodic boundary conditions (upper gray solid line) • a=0.15(lower gray solid line) • a=0.12(long dashed line) (quasi-2D meandering)

11. 3D meandering pattern depends on medium thickness • (a) Relative filament length vs time for a=0.18 and the medium thickness of 12.7 mm (solid line), 3.1 mm ( gray line) and 2.5 mm ( dashed line). • (b) Filament meandering for the medium of 3.1 mm thick (periodic meandering). (Panfilov, et al. 2005)

12. Discussion • We find three types of meandering of a scroll wave filament:quasi-2D, periodic and aperiodic meandering in a model of cardiac tissue. • Different meanderings depend on parameter settings and thickness of the medium.

13. References • A.V.Panfilov, Scroll waves meandering in a model of an excitable medium, Physical Review E 72,022902(2005) • A.V.Panfilov, A simple Two-variable Model of Cardiac Excitation, Chaos, Solitons and Fractals Vol.7,No,3,pp.293-301,1996. • L.Glass, Scroll waves in spherical shell geometries, Chaos, December 2001.