Scroll waves meandering in a model of an excitable medium

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Scroll waves meandering in a model of an
excitable medium

• Presenter Jianfeng Zhu
• Advisor Mark Alber

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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.

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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.

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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.

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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

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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.
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Numerical computation

• spiral waves and its core ( the white lines) in a
  2D excitable medium of elements.
  (hs0.6 and ht0.03)
• The light gray area represents the excited state
  of the tissue (egt0.6).

(Panfilov, et al., 2005)
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Extend the computations to 3D(128128128) 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)

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Filament under different time

• Filament dynamics at a0.18 at t 0s (a),
• t 1s (b) and t 3s (c)
• (aperiodic meandering)

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

• a0.18 with (black solid line)
• a0.18 with periodic boundary conditions (upper
• gray solid line)
• a0.15(lower gray solid line)
• a0.12(long dashed line) (quasi-2D meandering)
  

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3D meandering pattern depends on medium thickness

• (a) Relative filament length vs time for a0.18
  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)
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Discussion

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

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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.