Limits...
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

Bifurcations found in (33) for  in the -parameter plane. Codimension-2: Bogdanov–Takens (BT), resonant homoclinic (Resonant) and saddle-node homoclinic (); codimension-1: saddle-node (SN), Andronov–Hopf (AH), saddle homoclinic ( and ), saddle-node of limit cycles (SNPO)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4526515&req=5

Fig10: Bifurcations found in (33) for in the -parameter plane. Codimension-2: Bogdanov–Takens (BT), resonant homoclinic (Resonant) and saddle-node homoclinic (); codimension-1: saddle-node (SN), Andronov–Hopf (AH), saddle homoclinic ( and ), saddle-node of limit cycles (SNPO)

Mentions: Let us mention, without proof, that the homoclinic saddle-loop bifurcation curve ( in Fig. 10) of the Bogdanov–Takens point (BT) at lies between the Andronov–Hopf curve (AH) and the parameter line () and tends towards this parameter line as it approaches infinity; see Fig. 10. This can be seen by studying (33) for near infinity [22]. Fig. 10


Neural Excitability and Singular Bifurcations.

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

Bifurcations found in (33) for  in the -parameter plane. Codimension-2: Bogdanov–Takens (BT), resonant homoclinic (Resonant) and saddle-node homoclinic (); codimension-1: saddle-node (SN), Andronov–Hopf (AH), saddle homoclinic ( and ), saddle-node of limit cycles (SNPO)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4526515&req=5

Fig10: Bifurcations found in (33) for in the -parameter plane. Codimension-2: Bogdanov–Takens (BT), resonant homoclinic (Resonant) and saddle-node homoclinic (); codimension-1: saddle-node (SN), Andronov–Hopf (AH), saddle homoclinic ( and ), saddle-node of limit cycles (SNPO)
Mentions: Let us mention, without proof, that the homoclinic saddle-loop bifurcation curve ( in Fig. 10) of the Bogdanov–Takens point (BT) at lies between the Andronov–Hopf curve (AH) and the parameter line () and tends towards this parameter line as it approaches infinity; see Fig. 10. This can be seen by studying (33) for near infinity [22]. Fig. 10

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