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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

Bifurcation diagrams of the canonical model (1) together with ‘frequency–current’ (f–I) plots: (Type I) ; SNIC bifurcation near  where the frequency approaches zero; (Type II) ; supercritical singular Andronov–Hopf bifurcation near ; the subsequent canard explosion is clearly visible; note the small frequency band for the relaxation oscillation branch
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Fig1: Bifurcation diagrams of the canonical model (1) together with ‘frequency–current’ (f–I) plots: (Type I) ; SNIC bifurcation near where the frequency approaches zero; (Type II) ; supercritical singular Andronov–Hopf bifurcation near ; the subsequent canard explosion is clearly visible; note the small frequency band for the relaxation oscillation branch

Mentions: A first answer to the question of the neuron’s computational properties was given by Hodgkin [1] in the 1940s, who identified three basic types (classes) of excitable axons distinguished by their different responses to injected steps of currents of various amplitudes. Type I (class I) axons are able to integrate the input strength of an injected current step, i.e. the corresponding frequency–current (f–I) curve is continuous (see Fig. 1). Type II (class II) axons have a discontinuous f–I curve because of their inability to maintain spiking below a certain frequency. The frequency band of a type II neuron is very limited and, hence the frequency is relatively insensitive to the strength of the injected current. It appears that type II neurons resonate with a preferred frequency input. Type III (class III) axons will only fire a single or a few action potentials at the onset of the injected current step, but are not able to fire repetitive action potentials like type I and type II neurons. Type III neurons are able to differentiate, i.e. they are able to encode the occurrence of a ‘change’ in the stimulus. Such phasic firing (versus tonic or repetitive firing) identifies these type III neurons as slope detectors [2]. Obviously, the f–I curve is not defined for type III neurons. Fig. 1


Neural Excitability and Singular Bifurcations.

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

Bifurcation diagrams of the canonical model (1) together with ‘frequency–current’ (f–I) plots: (Type I) ; SNIC bifurcation near  where the frequency approaches zero; (Type II) ; supercritical singular Andronov–Hopf bifurcation near ; the subsequent canard explosion is clearly visible; note the small frequency band for the relaxation oscillation branch
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig1: Bifurcation diagrams of the canonical model (1) together with ‘frequency–current’ (f–I) plots: (Type I) ; SNIC bifurcation near where the frequency approaches zero; (Type II) ; supercritical singular Andronov–Hopf bifurcation near ; the subsequent canard explosion is clearly visible; note the small frequency band for the relaxation oscillation branch
Mentions: A first answer to the question of the neuron’s computational properties was given by Hodgkin [1] in the 1940s, who identified three basic types (classes) of excitable axons distinguished by their different responses to injected steps of currents of various amplitudes. Type I (class I) axons are able to integrate the input strength of an injected current step, i.e. the corresponding frequency–current (f–I) curve is continuous (see Fig. 1). Type II (class II) axons have a discontinuous f–I curve because of their inability to maintain spiking below a certain frequency. The frequency band of a type II neuron is very limited and, hence the frequency is relatively insensitive to the strength of the injected current. It appears that type II neurons resonate with a preferred frequency input. Type III (class III) axons will only fire a single or a few action potentials at the onset of the injected current step, but are not able to fire repetitive action potentials like type I and type II neurons. Type III neurons are able to differentiate, i.e. they are able to encode the occurrence of a ‘change’ in the stimulus. Such phasic firing (versus tonic or repetitive firing) identifies these type III neurons as slope detectors [2]. Obviously, the f–I curve is not defined for type III neurons. Fig. 1

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