The relationship between two fast/slow analysis techniques for bursting oscillations

Teka, Wondimu; Tabak, Joël; Bertram, Richard

doi:10.1063/1.4766943

Cited by 53 publications

(22 citation statements)

References 36 publications

(67 reference statements)

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“…C → 0) decreases the time constant of V . Therefore, C is considered as the parameter of the dimensionless singular perturbation problem in this model [ 63 ]. Herein, at C ≈ 0, m ks and n are regarded as slow variables and V is the only fast variable in the model.…”

Section: Methodsmentioning

confidence: 99%

Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions

V-Ghaffari

Kouhnavard²,

Elbasiouny

2017

PLoS ONE

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Subthreshold oscillations in combination with large-amplitude oscillations generate mixed-mode oscillations (MMOs), which mediate various spatial and temporal cognition and memory processes and behavioral motor tasks. Although many studies have shown that canard theory is a reliable method to investigate the properties underlying the MMOs phenomena, the relationship between the results obtained by applying canard theory and conductance-based models of neurons and their electrophysiological mechanisms are still not well understood. The goal of this study was to apply canard theory to the conductance-based model of pyramidal neurons in layer V of the Entorhinal Cortex to investigate the properties of MMOs under antiepileptic drug conditions (i.e., when persistent sodium current is inhibited). We investigated not only the mathematical properties of MMOs in these neurons, but also the electrophysiological mechanisms that shape spike clustering. Our results show that pyramidal neurons can display two types of MMOs and the magnitude of the slow potassium current determines whether MMOs of type I or type II would emerge. Our results also indicate that slow potassium currents with large time constant have significant impact on generating the MMOs, as opposed to fast inward currents. Our results provide complete characterization of the subthreshold activities in MMOs in pyramidal neurons and provide explanation to experimental studies that showed MMOs of type I or type II in pyramidal neurons under antiepileptic drug conditions.

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Section: Methodsmentioning

confidence: 99%

Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions

V-Ghaffari

Kouhnavard²,

Elbasiouny

2017

PLoS ONE

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“…When dealing with such problems, there are always questions of which analysis is appropriate and how the different methods are related [52]. In this work, we have directly addressed how the classic and novel 2-timescale methods are related in the context of system (2.2).…”

Section: Delayed Hopf Bifurcation and Tourbillonmentioning

confidence: 99%

“…Moreover, M I 0 is the δ → 0 counterpart of the FSN I points M I δ , defined by (4.10), in the 1-fast/3-slow splitting (cf. [52]). We further note that, as far as (6.2) is concerned, the FSN II points (4.9) are codimension 2 bifurcations (in fact, they are special cases of FSN I points M I 0 ).…”

Section: Geometric Singular Perturbation Analysismentioning

confidence: 99%

“…In this section, we finally acknowledge the full 3-timescale structure of (2.2) and perform a geometric singular perturbation analysis. Analyses of 3-timescale problems are currently rare [25,28,29,52], and a rigorous theoretical framework has yet to be developed. Regardless, we demonstrate that our approach is effective for dealing with 3-timescale problems.…”

Section: The Distinction Between Mmo Typesmentioning

confidence: 99%

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Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Vo¹,

Bertram²,

Wechselberger³

2013

SIAM J. Appl. Dyn. Syst.

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Abstract. Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found in neural bursting models. In a recent model for the electrical activity and calcium signaling in a pituitary lactotroph, two types of pseudo-plateau bursts were discovered: one in which the calcium drives the bursts and another in which the calcium simply follows them. Multiple methods from dynamical systems theory have been used to understand the bursting. The classic 2-timescale approach treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems. A more novel and successful 2-timescale approach divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics. Both methods can be effective analytic tools, but there has been little justification for one approach over the other. In this work, we use the lactotroph model to demonstrate that the two analysis techniques are different unfoldings of a 3-timescale system. We show that elementary applications of geometric singular perturbation theory in the 2-timescale and 3-timescale methods provide us with substantial predictive power. We use that predictive power to explain the transient and long-term dynamics of the pituitary lactotroph model.

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“…Fast-slow dynamic analysis is the common method of studying multi-timescales coupling system and it is widely used in biology, physics, etc. (Teka et al, 2012;Bertram et al 2017;Upadhyay et al, 2017). For example, in the biology field, Belykh et al simulated the bursting and spiking dynamics of many biological cells and reveal the mechanism of different fast-slow dynamics (Belykh et al, 2000).…”

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confidence: 99%

Fast–slow dynamic behaviors of a hydraulic generating system with multi-timescales

Zhang¹,

Chen

Zhang³

et al. 2019

Journal of Vibration and Control

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Hydraulic generating systems are widely modeled in the literature for investigating their stability properties by means of transfer functions representing the dynamic behavior of the reservoir, penstock, surge tank, hydro-turbine, and the generator. Traditionally, in these models the electrical load is assumed constant to simplify the modeling process. This assumption can hide interesting dynamic behaviors caused by fluctuation of the load as actually occurred. Hence, in this study, the electrical load characterized with periodic excitation is introduced into a hydraulic generating system and the responses of the system show a novel dynamic behavior called the fast–slow dynamic phenomenon. To reveal the nature of this phenomenon, the effects of the three parameters (i.e., differential adjustment coefficient, amplitude, and frequency) on the dynamic behaviors of the hydraulic generating system are investigated, and the corresponding change rules are presented. The results show that the intensity of the fast–slow dynamic behaviors varies with the change of each parameter, which provides reference for the quantification of the hydraulic generating system parameters. More importantly, these results not only present rich nonlinear phenomena induced by multi-timescales, but also provide some theoretical bases for maintaining the safe and stable operation of a hydropower station.

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The relationship between two fast/slow analysis techniques for bursting oscillations

Cited by 53 publications

References 36 publications

Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions

Mixed-mode oscillations in pyramidal neurons under antiepileptic drug conditions

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Fast–slow dynamic behaviors of a hydraulic generating system with multi-timescales

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