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Neural Excitability and Singular Bifurcations.

De Maesschalck P, Wechselberger M - J Math Neurosci (2015)

Bottom Line: We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view.In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures.We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

View Article: PubMed Central - PubMed

Affiliation: Hasselt University, Agoralaan gebouw D, 3590, Diepenbeek, Belgium, peter.demaesschalck@uhasselt.be.

ABSTRACT
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

No MeSH data available.


Related in: MedlinePlus

Left: several branches of the critical manifold, with fold points  in between. Right: slow dynamics on the cubic. The cline  may intersect the cubic one or more times along the dotted part of this cubic
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Fig2: Left: several branches of the critical manifold, with fold points in between. Right: slow dynamics on the cubic. The cline may intersect the cubic one or more times along the dotted part of this cubic

Mentions: For each , the critical manifold S is cubic shaped and given as a graph , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=S_{a}^{-} \cup F^{-} \cup S_{r} \cup F^{+} \cup S_{a}^{+} , $$\end{document}S=Sa−∪F−∪Sr∪F+∪Sa+, with attracting outer branches , repelling middle branch , and folds . We also assume that the vertical fibres containing the two local folds , respectively, intersect the critical manifold one more time outside the fold points at points p, respectively, q; see Fig. 2. The two folds are assumed to be regular extremes of the graph . Fig. 2


Neural Excitability and Singular Bifurcations.

De Maesschalck P, Wechselberger M - J Math Neurosci (2015)

Left: several branches of the critical manifold, with fold points  in between. Right: slow dynamics on the cubic. The cline  may intersect the cubic one or more times along the dotted part of this cubic
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4526515&req=5

Fig2: Left: several branches of the critical manifold, with fold points in between. Right: slow dynamics on the cubic. The cline may intersect the cubic one or more times along the dotted part of this cubic
Mentions: For each , the critical manifold S is cubic shaped and given as a graph , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=S_{a}^{-} \cup F^{-} \cup S_{r} \cup F^{+} \cup S_{a}^{+} , $$\end{document}S=Sa−∪F−∪Sr∪F+∪Sa+, with attracting outer branches , repelling middle branch , and folds . We also assume that the vertical fibres containing the two local folds , respectively, intersect the critical manifold one more time outside the fold points at points p, respectively, q; see Fig. 2. The two folds are assumed to be regular extremes of the graph . Fig. 2

Bottom Line: We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view.In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures.We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

View Article: PubMed Central - PubMed

Affiliation: Hasselt University, Agoralaan gebouw D, 3590, Diepenbeek, Belgium, peter.demaesschalck@uhasselt.be.

ABSTRACT
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

No MeSH data available.


Related in: MedlinePlus