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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

Bifurcation diagram for fixed . The parabola shows the two equilibra near the fold  (the third equilibrium branch and second SN are not shown). The a-axis is not shown on scale, as the distance between  and  should be exponentially small. Keeping that in mind, an incomplete canard explosion is seen as a goes from  to
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Fig6: Bifurcation diagram for fixed . The parabola shows the two equilibra near the fold (the third equilibrium branch and second SN are not shown). The a-axis is not shown on scale, as the distance between and should be exponentially small. Keeping that in mind, an incomplete canard explosion is seen as a goes from to

Mentions: In system (15) under Assumptions1–5, for fixedandthere exists an unstable equilibrium on the middle branchbounded away from the lower fold. Furthermore, there exist functions\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < a_{\mathrm{snpo}}(\varepsilon ) < a_{\ell}(\varepsilon ) < a_{s}(\varepsilon ) < a_{c}(\varepsilon ) < a_{h}(\varepsilon ) < a_{\mathrm{sn}}^{+}(\varepsilon ) $$\end{document}0<asnpo(ε)<aℓ(ε)<as(ε)<ac(ε)<ah(ε)<asn+(ε)that all converge to zero in the singular limit (except) and for which the following holds (see also Fig. 6):


Neural Excitability and Singular Bifurcations.

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

Bifurcation diagram for fixed . The parabola shows the two equilibra near the fold  (the third equilibrium branch and second SN are not shown). The a-axis is not shown on scale, as the distance between  and  should be exponentially small. Keeping that in mind, an incomplete canard explosion is seen as a goes from  to
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4526515&req=5

Fig6: Bifurcation diagram for fixed . The parabola shows the two equilibra near the fold (the third equilibrium branch and second SN are not shown). The a-axis is not shown on scale, as the distance between and should be exponentially small. Keeping that in mind, an incomplete canard explosion is seen as a goes from to
Mentions: In system (15) under Assumptions1–5, for fixedandthere exists an unstable equilibrium on the middle branchbounded away from the lower fold. Furthermore, there exist functions\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < a_{\mathrm{snpo}}(\varepsilon ) < a_{\ell}(\varepsilon ) < a_{s}(\varepsilon ) < a_{c}(\varepsilon ) < a_{h}(\varepsilon ) < a_{\mathrm{sn}}^{+}(\varepsilon ) $$\end{document}0<asnpo(ε)<aℓ(ε)<as(ε)<ac(ε)<ah(ε)<asn+(ε)that all converge to zero in the singular limit (except) and for which the following holds (see also Fig. 6):

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