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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

Neural model (1): clines under variation of I which leads to the (singular limit) definition of : (Type I)  at a saddle-node bifurcation; (Type II)  at (singular) Andronov–Hopf bifurcation. In the transition from type II to type I, a cusp bifurcation appears as can be predicted by looking from the bottom pictures in this figure
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Fig3: Neural model (1): clines under variation of I which leads to the (singular limit) definition of : (Type I) at a saddle-node bifurcation; (Type II) at (singular) Andronov–Hopf bifurcation. In the transition from type II to type I, a cusp bifurcation appears as can be predicted by looking from the bottom pictures in this figure

Mentions: For all , system (4) can have one, two or three equilibria on , all of them located either on or on ; see Fig. 3. The number of equilibria and their exact locations depend on . Fig. 3


Neural Excitability and Singular Bifurcations.

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

Neural model (1): clines under variation of I which leads to the (singular limit) definition of : (Type I)  at a saddle-node bifurcation; (Type II)  at (singular) Andronov–Hopf bifurcation. In the transition from type II to type I, a cusp bifurcation appears as can be predicted by looking from the bottom pictures in this figure
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4526515&req=5

Fig3: Neural model (1): clines under variation of I which leads to the (singular limit) definition of : (Type I) at a saddle-node bifurcation; (Type II) at (singular) Andronov–Hopf bifurcation. In the transition from type II to type I, a cusp bifurcation appears as can be predicted by looking from the bottom pictures in this figure
Mentions: For all , system (4) can have one, two or three equilibria on , all of them located either on or on ; see Fig. 3. The number of equilibria and their exact locations depend on . Fig. 3

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