Canonical models for torus canards in elliptic bursters

Area of Science:

• Computational Neuroscience
• Dynamical Systems Theory
• Mathematical Biology

Background:

• Elliptic bursting is a complex neuronal firing pattern.
• Torus canards are a class of solutions in dynamical systems that can mediate transitions between different behaviors.
• Understanding these transitions is crucial for modeling neuronal activity.

Purpose of the Study:

• To investigate the role of torus canards in the transitions to and from elliptic bursting dynamics.
• To identify the conditions under which classical and mixed-type torus canards appear.
• To develop a model that accurately captures these transition dynamics.

Main Methods:

• Analysis of elliptic bursting models, including Wilson-Cowan and Izhikevich-type models.
• Application of singular perturbation theory (ε=0) to predict dynamics for small ε > 0.
• Construction of singular orbits using slow, fast, and average slow flows.
• Introduction and analysis of a modified model (Leidenator) for elliptic bursting.

Main Results:

• Classical and mixed-type torus canards are identified at the onset and cessation of elliptic bursting.
• The difference between classical and mixed-type torus canards lies in the fast subsystem bifurcation approached (saddle-node of cycles vs. subcritical Hopf).
• A canonical model does not exhibit mixed-type torus canards due to a nongeneric transition; the modified Leidenator model does.

Conclusions:

• Torus canards play a significant role in the transitions associated with elliptic bursting.
• The Leidenator model provides a framework for studying elliptic bursting with both classical and mixed-type torus canards.
• Further research may explore connections between mixed-type torus canards and folded-saddle-node singularities in three-timescale systems.