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Rotating waves in simple scalar excitable media: approximations and numerical solutions.

B Ermentrout1, B I S van der Ventel2

  • 1Department of Mathematics, University of Pittsburgh, Pittsburgh, USA. bard@pitt.edu.

Journal of Mathematical Biology
|March 30, 2016
PubMed
Summary

This study investigates rotating waves in annular regions using semi-analytical methods and a phase model. The research focuses on calculating radial phase shifts, finding spiral twist varies non-monotonically with excitability near bifurcations.

Keywords:
Excitable mediaPhase equationsRotating wavesTheta model

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Area of Science:

  • Nonlinear dynamics
  • Mathematical modeling
  • Computational physics

Background:

  • Rotating waves are crucial in various scientific fields.
  • Understanding their behavior in confined geometries is complex.
  • Phase models offer simplified yet insightful approaches.

Purpose of the Study:

  • To analyze rotating waves in a 2D annular region.
  • To develop and compare semi-analytical methods for predicting wave behavior.
  • To determine the radial phase shift and its relation to spiral dynamics.

Main Methods:

  • Utilized a one-variable phase model derived from saddle-node invariant circle (SNIC) bifurcation.
  • Derived asymptotic expressions for the scalar dispersion relationship.
  • Compared approximation methods against direct numerical simulations of the governing nonlinear partial differential equation.

Main Results:

  • Developed approximation methods based on decomposing the solution into a base function and perturbation terms.
  • Successfully derived expressions for the radial phase shift.
  • Observed that the total twist of the spiral is not a monotonic function of excitability.

Conclusions:

  • The radial phase shift is a key factor in understanding rotating wave dynamics.
  • Maximum spiral twist occurs near the transition between excitable and oscillatory behaviors.
  • Semi-analytical methods provide valuable approximations for complex nonlinear systems.