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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

Blow-up of the singular fold. ‘Birds-eye view’ of the upper blown-up sphere  in -space. Normal hyperbolicity of the manifolds  and  is gained at the equator, allowing one to extend them onto the blown-up sphere near singularities , respectively, . The two additional singularities represent connection to the fast fibre at
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Fig9: Blow-up of the singular fold. ‘Birds-eye view’ of the upper blown-up sphere in -space. Normal hyperbolicity of the manifolds and is gained at the equator, allowing one to extend them onto the blown-up sphere near singularities , respectively, . The two additional singularities represent connection to the fast fibre at

Mentions: The phase-directional rescaling chart. Here, we explain the dynamics near the equator of the blow-up sphere . When presenting a picture in blown-up -space, where the origin is replaced by (or blown-up to) a sphere, we can position the point of view from the top of the ε-axis; looking down on the -plane we see the spherical surface with as the interior of a circle , and the equator as the circle with the outer slow–fast dynamics around it; see Fig. 9. Fig. 9


Neural Excitability and Singular Bifurcations.

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

Blow-up of the singular fold. ‘Birds-eye view’ of the upper blown-up sphere  in -space. Normal hyperbolicity of the manifolds  and  is gained at the equator, allowing one to extend them onto the blown-up sphere near singularities , respectively, . The two additional singularities represent connection to the fast fibre at
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4526515&req=5

Fig9: Blow-up of the singular fold. ‘Birds-eye view’ of the upper blown-up sphere in -space. Normal hyperbolicity of the manifolds and is gained at the equator, allowing one to extend them onto the blown-up sphere near singularities , respectively, . The two additional singularities represent connection to the fast fibre at
Mentions: The phase-directional rescaling chart. Here, we explain the dynamics near the equator of the blow-up sphere . When presenting a picture in blown-up -space, where the origin is replaced by (or blown-up to) a sphere, we can position the point of view from the top of the ε-axis; looking down on the -plane we see the spherical surface with as the interior of a circle , and the equator as the circle with the outer slow–fast dynamics around it; see Fig. 9. Fig. 9

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