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

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.

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.

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.

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

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.

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)

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)

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)

The length of the filament 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) (Panfilov, et al. 2005)

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)

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.

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 ,1996. L.Glass, Scroll waves in spherical shell geometries, Chaos, December 2001.