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Area of Science:

  • Computational biology
  • Mathematical modeling
  • Nonlinear dynamics

Background:

  • Excitable systems generate wave phenomena, crucial in biology and physics.
  • Approximating excitation wave initiation relies on linearization around unstable traveling waves.
  • Existing theories face challenges with translational invariance in slow-fast systems.

Purpose of the Study:

  • Investigate the asymptotic behavior of unstable traveling waves and eigenfunctions in FitzHugh-Nagumo systems.
  • Evaluate approximations of strength-extent curves in the slow-fast limit.
  • Develop improved heuristics to address limitations in predicting excitation wave initiation.

Main Methods:

  • Analysis of unstable traveling wave solutions and their linearizations.
  • Asymptotic analysis in the slow-fast limit for two-component excitable systems.
  • Testing approximations on four illustrative FitzHugh-Nagumo models, including degenerate cases.

Main Results:

  • Unstable traveling waves converge to critical nuclei in degenerate fast subsystems.
  • Asymptotics of left and right eigenspaces are distinct, with slow components persisting.
  • Previous heuristics for translational invariance negatively impact critical curve predictions.

Conclusions:

  • The slow component of the left eigenfunction significantly affects critical curve predictions.
  • Two novel heuristics, avoiding the problematic eigenfunction component, offer superior predictive accuracy.
  • These improved methods demonstrate enhanced utility, especially in asymptotic regimes.