Meandering and Drifting Spiral Waves due to Inhomogeneities

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Meandering and Drifting Spiral Waves due to
Inhomogeneities
Andrew J. Foulkes, Vadim N. Biktashev Department
of Mathematical Sciences University of
Liverpool Peach Street Liverpool L69 7ZL
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Presentation Outline

• Introduction
• Meander of Spiral Waves
• Drift of Spiral Waves
• Meandering Spiral Waves which Drift
• Numerical Observations
• Further work

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Introduction

• Spiral Waves in excitable media cardiac tissue

Spiral Wave in a confluent layer of rat ventricle
cell culture1

• Rigidly rotating waves simplest type of spiral
  wave motion
• Barkleys Model

1Hwang, Yea and Lee, Regular and Alternant
Spiral Waves of Contractile Motion on Rat
Ventricle Cell Cultures, Physical Review
Letters, 92(19),198103-1, 2004
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Meander of Spiral Waves

• Quasiperiodic motion with two or more frequencies
  present.
• Study simple meander 2 frequencies.
• Wave changes shape as it evolves

Laboratory Frame of Reference
Moving Frame of Reference
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Meander of Spiral Waves (cont.)

• Theory published by Biktashev et al (1996) 2 who
  exploited the facts that meandering spiral waves
  are

• System of equations are invariant under Euclidean
  Symmetry
• Solution change shape as the wave evolves

2 V.N.Biktashev, A.V.Holden and E.V.Nikolaev,
Spiral Wave Meander and Symmetry of the Plane,
International Journal of Bifurcation and Chaos,
6(12A),p2433, 1996
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Meander of Spiral Waves (cont.)
Hopf Bifurcation
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Meander of Spiral Waves (cont.)
Numerical Solution
Analytical Solution
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Drift of Spiral Waves

• Motion of the spiral wave is around a moving
  point of rotation.

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Drift of Spiral Waves

• Perturbation problem3 the perturbation,
  ,moves the spiral.

• Represent this system in the moving frame of
  reference and extract the equations of motion
  from it by taking the scalar product of the
  system with the eigenfunctions of the adjoint
  operator (response functions).

3V.N.Biktashev and A.V.Holden, Resonant Drift of
Autowave Vortices in two dimensions and the
effects of Boundaries and Inhomogeneities,
Chaos, Solitons and Fractals, 5(3,4),p575, 1995
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Drift of Meandering Spiral Waves

• Aim Unify the theories of Meander and Drift
  both written independently and using different
  techniques.
• Theory of meandering spiral waves that drift is
  currently being developed (utilises results from
  Orbifold Theory, Group Theory, Lie Group Theory,
  Perturbation Theory and Hopf Bifurcation Theory).

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Numerical Observations

• Motivations

• How does the wave behave when meandering and
  drifting (tip patterns, velocities (both
  translational and angular), shape of wave)?
• What does the solution (Quotient System) look
  like?
• What happens when the wave passes through a point
  of resonance?

• Problem Noise in numerical data.
• Solution Tikhonov Regularisation to smooth out
  data together with the double sweep method.

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Numerical Observations (cont.)
Projection of Quotient System (full Euclidean
Group)
Tip trajectory
Quotient System (Translational Group)
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Results and Further Work

• Results
• RDS has been rewritten in moving frame of
  reference and a solvability condition created,
  from which we can extract the equations of
  motion.
• Numerical observations show interesting behaviour
  around a point of resonance, including a strange
  deviation from the trajectory which is not at the
  point of resonance.

• Further Work
• Finish completing theory of meandering spiral
  waves that drift.
• Study of Hypermeander.
• Development of a code to solve the quotient
  system.

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THANK YOU!
To you, the Audience, for listening
To Professor Biktashev, for his guidance and for
introducing me to such an interesting project.
To the University of Liverpool, for sponsoring me
to do this project