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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

Sketch of bifurcation diagram in  parameter space for : cusp bifurcation and SN branches (black); Bautin bifurcation point (red) with Andronov–Hopf branches (sub = dashed/super = solid); and saddle-node of periodic orbit (SNPO) branch (green); small and large homoclinic ( and ) branches (blue); SNIC segment (blue) on the SN branch
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Fig8: Sketch of bifurcation diagram in parameter space for : cusp bifurcation and SN branches (black); Bautin bifurcation point (red) with Andronov–Hopf branches (sub = dashed/super = solid); and saddle-node of periodic orbit (SNPO) branch (green); small and large homoclinic ( and ) branches (blue); SNIC segment (blue) on the SN branch

Mentions: Figure 8 summarises all our observations for (compare with the singular limit bifurcation diagram in Fig. 5). Fig. 8


Neural Excitability and Singular Bifurcations.

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

Sketch of bifurcation diagram in  parameter space for : cusp bifurcation and SN branches (black); Bautin bifurcation point (red) with Andronov–Hopf branches (sub = dashed/super = solid); and saddle-node of periodic orbit (SNPO) branch (green); small and large homoclinic ( and ) branches (blue); SNIC segment (blue) on the SN branch
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4526515&req=5

Fig8: Sketch of bifurcation diagram in parameter space for : cusp bifurcation and SN branches (black); Bautin bifurcation point (red) with Andronov–Hopf branches (sub = dashed/super = solid); and saddle-node of periodic orbit (SNPO) branch (green); small and large homoclinic ( and ) branches (blue); SNIC segment (blue) on the SN branch
Mentions: Figure 8 summarises all our observations for (compare with the singular limit bifurcation diagram in Fig. 5). Fig. 8

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