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Neural Excitability and Singular Bifurcations.

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Summary

This study defines neural excitability using geometric singular perturbation theory. It links Type I excitability to singular Bogdanov-Takens/SNIC bifurcations and Type II to singular Andronov-Hopf bifurcations, explaining transitions.

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Area of Science:

  • Computational Neuroscience
  • Dynamical Systems Theory
  • Mathematical Biology

Background:

  • Neural models exhibit complex behaviors, including excitability, crucial for information processing.
  • Slow/fast neural models possess an inherent singular nature that influences their dynamics.
  • Existing frameworks for analyzing excitability in these models are often limited.

Purpose of the Study:

  • To analyze neural excitability in 2D slow/fast neural models using geometric singular perturbation theory.
  • To define and classify excitability types based on singular bifurcations.
  • To elucidate the role of canards in the unfolding of these bifurcations.

Main Methods:

  • Application of geometric singular perturbation theory to 2D slow/fast neural models.
  • Identification and analysis of singular bifurcations, specifically Bogdanov-Takens/SNIC and Andronov-Hopf.
  • Investigation of the role of canard trajectories in bifurcation phenomena.

Main Results:

  • Type I excitability is characterized by a novel singular Bogdanov-Takens/SNIC bifurcation.
  • Type II excitability is associated with a singular Andronov-Hopf bifurcation.
  • Canards are identified as key structures in understanding the unfolding of these singular bifurcations.

Conclusions:

  • A complete analysis of neural excitability in 2D slow/fast models is provided through the lens of geometric singular perturbation theory.
  • The study establishes a clear link between specific singular bifurcations and distinct types of neural excitability.
  • The findings offer a unified framework for understanding excitability and transitions between types in neural systems.