Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards

DOI10.1137/17M1143411zbMath1404.37056OpenAlexW2784017543WikidataQ129520564 ScholiaQ129520564MaRDI QIDQ4686617

Vivien Kirk, Mathieu Desroches

Publication date: 4 October 2018

Published in: SIAM Journal on Applied Dynamical Systems (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1137/17m1143411

zbMATH Keywords

burstingmultiple time scalescanards

Mathematics Subject Classification ID

Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems (37C15) Bifurcations of limit cycles and periodic orbits in dynamical systems (37G15) Multiple scale methods for ordinary differential equations (34E13) Systems with slow and fast motions for nonlinear problems in mechanics (70K70) Relaxation oscillations for ordinary differential equations (34C26) Canard solutions to ordinary differential equations (34E17)

Related Items (8)

Complex periodic bursting structures in the Rayleigh-van der Pol-Duffing oscillator ⋮ Canard solutions in neural mass models: consequences on critical regimes ⋮ Idealized multiple-timescale model of cortical spreading depolarization initiation and pre-epileptic hyperexcitability caused by \(\mathrm{Na_V}1.1/ SCN1A\) mutations ⋮ A geometric analysis of the impact of large but finite switching rates on vaccination evolutionary games ⋮ Experimental and numerical study of the synchronization of canards in identical coupled canard-generating Bonhoeffer-van der Pol oscillators ⋮ Geometric blow-up for folded limit cycle manifolds in three time-scale systems ⋮ Mathematical birth of early afterdepolarizations in a cardiomyocyte model ⋮ Parabolic bursting, spike-adding, dips and slices in a minimal model

Uses Software

• AUTO
• AUTO-86
• AUTO-07P

Cites Work

• Unnamed Item
• Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle
• Local analysis near a folded saddle-node singularity
• Geometric singular perturbation theory for ordinary differential equations
• Topological and phenomenological classification of bursting oscillations
• Dynamical systems analysis of spike-adding mechanisms in transient bursts
• Spike-adding in parabolic bursters: the role of folded-saddle canards
• Food chain chaos due to junction-fold point
• Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points---Fold and Canard Points in Two Dimensions
• Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster
• Canard-Mediated (De)Synchronization in Coupled Phantom Bursters
• Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting
• Mixed-Mode Oscillations with Multiple Time Scales
• Folded Saddles and Faux Canards
• Understanding and Distinguishing Three-Time-Scale Oscillations: Case Study in a Coupled Morris--Lecar System
• Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example
• Singular Hopf Bifurcation in Systems with Two Slow Variables
• Chaotic Spikes Arising from a Model of Bursting in Excitable Membranes
• Existence and Bifurcation of Canards in $\mathbbR^3$ in the Case of a Folded Node
• Canard cycles and center manifolds
• Mixed-Mode Oscillations in a Multiple Time Scale Phantom Bursting System
• ON THREE-DIMENSIONAL DYNAMICAL SYSTEMS CLOSE TO SYSTEMS WITH A STRUCTURALLY UNSTABLE HOMOCLINIC CURVE. II
• Computing Slow Manifolds of Saddle Type
• Analysis of Interacting Local Oscillation Mechanisms in Three-Timescale Systems
• Fenichel Theory for Multiple Time Scale Singular Perturbation Problems
• NEURAL EXCITABILITY, SPIKING AND BURSTING
• Food chain chaos due to Shilnikov’s orbit
• Food chain chaos due to transcritical point
• Canards in \(\mathbb{R}^3\)

This page was built for publication: Spike-Adding in a Canonical Three-Time-Scale Model: Superslow Explosion and Folded-Saddle Canards