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

Heteroclinic connections of canard type undergo a transition from headless canard to canard with head, from the jump-back canard homoclinic to the jump-away canard homoclinic
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Fig7: Heteroclinic connections of canard type undergo a transition from headless canard to canard with head, from the jump-back canard homoclinic to the jump-away canard homoclinic

Mentions: For in between the two homoclinic curves, the stable manifold connects in reverse time to an orbit that follows the large homoclinic for some time, but exiting at a time prior to the time needed to connect back to . Therefore, no limit cycles may appear. In fact, since we know there is an additional singularity on the middle branch , denoted by n, and assuming it is of node type, then the exponential gap between the two homoclinics is filled by canard curves along which canard-type heteroclinic connections appear between n and ; see Fig. 7. The transition from headless heteroclinic canard to heteroclinic canard with head can be seen as a natural continuation of the truncated canard explosion of the canard homoclinics. The proof of the presence of such heteroclinics is completely analogous to above. In particular, no limit cycles are present in this scenario. This proves part (8) and finishes the proof of the theorem. □ Fig. 7


Neural Excitability and Singular Bifurcations.

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

Heteroclinic connections of canard type undergo a transition from headless canard to canard with head, from the jump-back canard homoclinic to the jump-away canard homoclinic
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig7: Heteroclinic connections of canard type undergo a transition from headless canard to canard with head, from the jump-back canard homoclinic to the jump-away canard homoclinic
Mentions: For in between the two homoclinic curves, the stable manifold connects in reverse time to an orbit that follows the large homoclinic for some time, but exiting at a time prior to the time needed to connect back to . Therefore, no limit cycles may appear. In fact, since we know there is an additional singularity on the middle branch , denoted by n, and assuming it is of node type, then the exponential gap between the two homoclinics is filled by canard curves along which canard-type heteroclinic connections appear between n and ; see Fig. 7. The transition from headless heteroclinic canard to heteroclinic canard with head can be seen as a natural continuation of the truncated canard explosion of the canard homoclinics. The proof of the presence of such heteroclinics is completely analogous to above. In particular, no limit cycles are present in this scenario. This proves part (8) and finishes the proof of the theorem. □ Fig. 7

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