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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 singular limit bifurcation diagram in  parameter space: singular Bogdanov–Takens and saddle-node homoclinic (SNIC) at the origin (blue); singular Andronov–Hopf branch (red dashed) and cusp bifurcation + saddle-node branches (black); see Fig. 4 for the corresponding cases (a)–(d) along the singular AH branch
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Fig5: Sketch of singular limit bifurcation diagram in parameter space: singular Bogdanov–Takens and saddle-node homoclinic (SNIC) at the origin (blue); singular Andronov–Hopf branch (red dashed) and cusp bifurcation + saddle-node branches (black); see Fig. 4 for the corresponding cases (a)–(d) along the singular AH branch

Mentions: As can be seen in Fig. 4 for and , the layer problem of a type I neuron has a saddle-node bifurcation of equilibria at the lower fold . This allows for the construction of a singular homoclinic orbit Γ as follows: we start at the saddle-node equilibrium at the lower fold and concatenate a fast fibre of the layer problem that connects to the upper stable branch . Then we follow the reduced (slow) flow towards the upper fold where we concatenate a fast fibre at that connects back towards the lower attracting branch . Finally, we follow the reduced (slow) flow on towards the lower fold and hence end up at the saddle-node equilibrium. This homoclinic orbit is the singular limit representation of the SNIC indicated in Fig. 1. Hence for , we have identified a (global) singular SNIC bifurcation together with a (local) singular Bogdanov–Takens bifurcation. The unfolding of these singular bifurcations is done in Sect. 6. Figure 5 summarises all our singular limit observations. Fig. 5


Neural Excitability and Singular Bifurcations.

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

Sketch of singular limit bifurcation diagram in  parameter space: singular Bogdanov–Takens and saddle-node homoclinic (SNIC) at the origin (blue); singular Andronov–Hopf branch (red dashed) and cusp bifurcation + saddle-node branches (black); see Fig. 4 for the corresponding cases (a)–(d) along the singular AH branch
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Related In: Results  -  Collection

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Fig5: Sketch of singular limit bifurcation diagram in parameter space: singular Bogdanov–Takens and saddle-node homoclinic (SNIC) at the origin (blue); singular Andronov–Hopf branch (red dashed) and cusp bifurcation + saddle-node branches (black); see Fig. 4 for the corresponding cases (a)–(d) along the singular AH branch
Mentions: As can be seen in Fig. 4 for and , the layer problem of a type I neuron has a saddle-node bifurcation of equilibria at the lower fold . This allows for the construction of a singular homoclinic orbit Γ as follows: we start at the saddle-node equilibrium at the lower fold and concatenate a fast fibre of the layer problem that connects to the upper stable branch . Then we follow the reduced (slow) flow towards the upper fold where we concatenate a fast fibre at that connects back towards the lower attracting branch . Finally, we follow the reduced (slow) flow on towards the lower fold and hence end up at the saddle-node equilibrium. This homoclinic orbit is the singular limit representation of the SNIC indicated in Fig. 1. Hence for , we have identified a (global) singular SNIC bifurcation together with a (local) singular Bogdanov–Takens bifurcation. The unfolding of these singular bifurcations is done in Sect. 6. Figure 5 summarises all our singular limit observations. Fig. 5

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