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Turing patterns in a network-reduced FitzHugh-Nagumo model.

Timoteo Carletti1, Hiroya Nakao2

  • 1naXys, Namur Institute for Complex Systems, University of Namur, Namur B5000, Belgium.

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Summary

This study simplifies the FitzHugh-Nagumo model to a single component on networks, revealing conditions for Turing pattern emergence in networked dynamical systems.

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Area of Science:

  • Mathematical modeling
  • Network dynamics
  • Chemical kinetics

Background:

  • The FitzHugh-Nagumo model is a foundational two-component system for studying excitable phenomena.
  • Network effects on dynamical systems, particularly chemical reactions, are complex and require simplified models.
  • Understanding pattern formation in spatially extended systems is crucial in various scientific fields.

Purpose of the Study:

  • To reduce a two-component FitzHugh-Nagumo model to a single-component model on general networks.
  • To investigate the emergence of Turing patterns in the reduced single-component model.
  • To analyze the conditions for instability of homogeneous states in both original and reduced models.

Main Methods:

  • Adiabatic elimination of a fast variable to reduce model complexity.
  • Analysis of networked dynamical systems on multigraphs using local and nonlocal Laplace matrices.
  • Study of homogeneous state instability conditions.

Main Results:

  • A single-component model with long-range connections is derived from the FitzHugh-Nagumo model.
  • Turing patterns are shown to emerge in both the original and the reduced models.
  • Conditions for the instability of homogeneous states are identified for the reduced model.

Conclusions:

  • The reduction of the FitzHugh-Nagumo model provides a simplified yet effective framework for studying pattern formation on networks.
  • The derived single-component model captures essential dynamics, including Turing pattern formation.
  • The methodology is generalizable to other slow-fast systems, highlighting the peculiarity of the FitzHugh-Nagumo model.