Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales

DOI10.1186/2190-8567-1-9zbMath1259.92003OpenAlexW2142285825WikidataQ42008984 ScholiaQ42008984MaRDI QIDQ1941942

James Sneyd, Vivien Kirk, Martin Wechselberger, Wen-Jun Zhang

Publication date: 22 March 2013

Published in: The Journal of Mathematical Neuroscience (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1186/2190-8567-1-9

zbMATH Keywords

Hopf bifurcationsgeometric singular perturbation theoryquasi-steady-state reductionfirst Lyapunov coefficientphysiological model reduction

Mathematics Subject Classification ID

Neural biology (92C20) Dynamical systems in biology (37N25) Bifurcation theory for ordinary differential equations (34C23) General biology and biomathematics (92B05)

Related Items (5)

Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis ⋮ Effects of quasi-steady-state reduction on biophysical models with oscillations ⋮ Bifurcation and Spike Adding Transition in Chay–Keizer Model ⋮ Saddle slow manifolds and canard orbits in \(\mathbb{R}^{4}\) and application to the full Hodgkin-Huxley model ⋮ Complex oscillations in the delayed Fitzhugh-Nagumo equation

Uses Software

• AUTO-07P

Cites Work

• Dynamical probing of the mechanisms underlying calcium oscillations
• Local analysis near a folded saddle-node singularity
• Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations
• Geometric singular perturbation theory for ordinary differential equations
• Singular Hopf bifurcation in systems with fast and slow variables
• Relaxation oscillations in \({\mathbb R}^3\)
• Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model
• Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points---Fold and Canard Points in Two Dimensions
• Existence and Bifurcation of Canards in $\mathbbR^3$ in the Case of a Folded Node
• Singular Hopf Bifurcation to Relaxation Oscillations
• Chaos in the Hodgkin--Huxley Model
• Subcritical Elliptic Bursting of Bautin Type
• The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales
• Understanding anomalous delays in a model of intracellular calcium dynamics
• Relaxation oscillation and canard explosion
• Canards in \(\mathbb{R}^3\)

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