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

Singular limit bifurcations at the lower fold  and their singular limit orbits in system (1) (, ). (a) (Type I) singular saddle-node homoclinic (SNIC) () together with a singular Bogdanov–Takens bifurcation (= singular BT/SNIC); (b) singular Andronov–Hopf bifurcation with incomplete family of canard cycles (); (c) family of (incomplete) canard cycles and family of singular saddle-node homoclinics of canard type (); (d) (Type II) singular Andronov–Hopf bifurcation and (complete) family of canard cycles ()
© Copyright Policy - OpenAccess
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4526515&req=5

Fig4: Singular limit bifurcations at the lower fold and their singular limit orbits in system (1) (, ). (a) (Type I) singular saddle-node homoclinic (SNIC) () together with a singular Bogdanov–Takens bifurcation (= singular BT/SNIC); (b) singular Andronov–Hopf bifurcation with incomplete family of canard cycles (); (c) family of (incomplete) canard cycles and family of singular saddle-node homoclinics of canard type (); (d) (Type II) singular Andronov–Hopf bifurcation and (complete) family of canard cycles ()

Mentions: On the other hand, an equilibrium from may cross or bifurcate at the lower fold , which is necessary to observe a bifurcation from an excitable to an oscillatory state. Assumptions 3 and 4 imply that the reduced flow on is either towards an equilibrium on or towards the lower fold (see Fig. 2 (right) and Fig. 4), another essential feature for an excitable/oscillatory system. Fig. 4


Neural Excitability and Singular Bifurcations.

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

Singular limit bifurcations at the lower fold  and their singular limit orbits in system (1) (, ). (a) (Type I) singular saddle-node homoclinic (SNIC) () together with a singular Bogdanov–Takens bifurcation (= singular BT/SNIC); (b) singular Andronov–Hopf bifurcation with incomplete family of canard cycles (); (c) family of (incomplete) canard cycles and family of singular saddle-node homoclinics of canard type (); (d) (Type II) singular Andronov–Hopf bifurcation and (complete) family of canard cycles ()
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig4: Singular limit bifurcations at the lower fold and their singular limit orbits in system (1) (, ). (a) (Type I) singular saddle-node homoclinic (SNIC) () together with a singular Bogdanov–Takens bifurcation (= singular BT/SNIC); (b) singular Andronov–Hopf bifurcation with incomplete family of canard cycles (); (c) family of (incomplete) canard cycles and family of singular saddle-node homoclinics of canard type (); (d) (Type II) singular Andronov–Hopf bifurcation and (complete) family of canard cycles ()
Mentions: On the other hand, an equilibrium from may cross or bifurcate at the lower fold , which is necessary to observe a bifurcation from an excitable to an oscillatory state. Assumptions 3 and 4 imply that the reduced flow on is either towards an equilibrium on or towards the lower fold (see Fig. 2 (right) and Fig. 4), another essential feature for an excitable/oscillatory system. Fig. 4

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