Meander of Spiral Waves

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Meander of Spiral Waves

• By Andy Foulkes
• Under the supervision of Prof. V. N. Biktashev

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• Outline
• A simple PDE model
• What is a Spiral Wave?
• Definition of the tip of the Spiral Wave
• Rigidly Rotating Waves
• Meander of Spiral Waves
• Drifting due to Inhomogeneities
• A system of ODEs to describe the tip motion
• Conclusion Further Work

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A Simple PDE Model
Dwight Barkley (1990) - simplified model of the
FitzHugh-Nagumo. No Flux boundary conditions.
Study Local Dynamics to see how system evolves
(no diffusion).
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• Nullclines

• Fixed Points

• Typical Phase Portrait

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• Nullclines

• Fixed Points

• Typical Phase Portrait

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• Nullclines

• Fixed Points

• Typical Phase Portrait

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• Nullclines

• Fixed Points

• Typical Phase Portrait

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• Nullclines

• Fixed Points

• Typical Phase Portrait

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Spiral Waves
Action potentials corresponding to a typical
trajectory
u v
x-ct
Translate this information to a 2-D plane.
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Spiral Wave Evolution
y
y
x
x
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Spiral Wave Evolution (cont.)
y
y
x
x
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Spiral Wave Tip
Location of the tip - needed to see exact motion
of the spiral wave. There are 2 main definitions
1st Def
, 2nd Def
For Barkleys model, we use the 2nd definition

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Rigidly Rotating Waves
EZSpiral program developed by Barkley to study
motion of tip. The program uses several numerical
methods. Free Licence available to download and
amend.
A Rigidly Rotating Wave (RW) is a periodic
solution. The tip of the RW traces out a perfect
circle
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A Rigidly Rotating Wave
a0.85 , b0.1 ,eps0.01
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Meandering Spiral Waves
Outward facing petals a0.7, b0.1, eps0.01
Inward facing petals a1.0, b0.225, eps0.01
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Meander of Spiral Waves
Meandering Waves (aka Compound or Modulated
Rotating Waves (MRWs)), occur in a 2 frequency
quasiperiodic system. The trajectory of the tip
of MRW is a projection of motion on a 2-Torus to
a 2-D plane. Flower-like patterns are traced
out by the tip of the wave.
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Parametric Portrait
Parametric portrait used EZSpiral to carry out
numerical work. Parameter kept constant -
Green Blue
regions of MRWs
boundaries between Spiral Waves and no waves
Red
Grey
Excitable region
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Drifting due to Inhomogeneities
Parameters constant up to now Drifting due to
Inhomogeneities parameters are spatial
dependent Amended EZSpiral code parameter
is now defined as
With parameter constant rotations around
fixed point With parameter dependent on x
point of rotation moves along a straight line
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Drifting due to Inhomogeneities
Drifting RW
Drifting MRW
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The ODE model
The motion of the tip can be studied using a
system of ODEs. Numerical evidence - transition
from RW to MRW is via a Supercritical Hopf
Bifurcation (Barkley 1990). Barkley (1994)
system of ODEs in which a) there is a
Supercritical Hopf Bifurcation present, and b)
the system is invariant under Euclidean
Symmetry. System is a priori but can shown both
RW and MRW Patterns, and the transition between
them.
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The ODE Model (cont.)
where
Using the following change of variables
We get
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A simple C code was constructed which solved
these 5 equations using the Forward Euler
Method. Same patterns observed as those in the
PDE model.
Above 3 pictures - same set of data points,
different time intervals. Large period of time -
trajectory forms dense orbits on the 2-Torus.
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Conclusion

• Meander patterns are produced in a PDE system,
  with the same patterns being simulated by a
  system of ODEs.

Future Work

• Study of Drifting Spiral Waves due to
  inhomogeneities Meandering.
• Production of a system of ODEs to describe the
  tip of the Meandering Wave that drifts due to
  inhomogeneities.

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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