Slow–fast n-dimensional piecewise linear differential systems

Prohens, R.; Teruel, Antonio E.; Vich, Catalina

doi:10.1016/j.jde.2015.09.046

Cited by 33 publications

(38 citation statements)

References 21 publications

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“…Moreover, when δ → 0, it tends to the critical manifold y = −x + 2. It follows that Π a provides a canonical choice of an attracting Fenichel slow manifold S a,δ [13][14][15][16], finishing the proof. Now that we have defined the attracting and repelling Fenichel slow manifolds for system (8), it is possible to define the primary and secondary canards, as follows.…”

Section: Number Of Small Oscillations Passage Through Primary and Sementioning

confidence: 85%

“…From this equation, taking into account that the value of X for which the flow of fast system (1) enters the extra zone can be approximated by X min , (beginning of phase I), we find c n (X min ) given in (16). However, not every n ≥ 0 corresponds to an orbit connecting to S r,δ .…”

Section: Theorem 2 Consider Systemmentioning

confidence: 99%

“…It follows that Π r provides a canonical choice of a repelling Fenichel slow manifold S r,δ [13][14][15][16].…”

Section: Number Of Small Oscillations Passage Through Primary and Sementioning

confidence: 99%

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Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

Fernández-García

Krupa

Clément

2016

Physica D: Nonlinear Phenomena

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International audienceIn this work, we analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse

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Section: Number Of Small Oscillations Passage Through Primary and Sementioning

confidence: 85%

Section: Theorem 2 Consider Systemmentioning

confidence: 99%

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Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

Fernández-García

Krupa

Clément

2016

Physica D: Nonlinear Phenomena

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“…In fact, when C tends to zero, the limit cycle orbit tends to a limit set composed by two line segments on the critical manifold (slow subsystem) plus two more line segments on the stratified flow (fast subsystem), see Figure 4. We refer the reader to Prohens et al (2016) for an overview on piecewise slow-fast dynamics terminology.…”

Section: Quantitative Analysis Of the Limit Cycle Periodmentioning

confidence: 99%

“…Let q M,L and q M,R be the intersection points of the central slow manifold with v = a/2 and v = (1+a)/2, respectively (see Figure 5 for a representation of these points). Since the distance between the slow and critical manifolds of system (1) is of O(C), see Prohens et al (2016), the distance between q L and q M,L is also of O(C). Moreover, the distance between q R and q M,R is also of O(C).…”

Section: An Approximation Of the Period Of The Limit Cyclementioning

confidence: 99%

Estimation of Synaptic Conductance in the Spiking Regime for the McKean Neuron Model

Guillamón¹,

Prohens²,

Teruel³

et al. 2017

SIAM J. Appl. Dyn. Syst.

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In this work, we aim at giving a first proof of concept to address the estimation of synaptic conductances when a neuron is spiking, a complex inverse non-linear problem which is an open challenge in neuroscience. Our approach is based on a simplified model of neuronal activity, namely a piecewise linear version of the FitzHugh-Nagumo model. This simplified model allows a precise knowledge of the non-linear f-I curve by using standard techniques of non-smooth dynamical systems. In the regular firing regime of the neuron model, we obtain an approximation of the period which, in addition, improves previous approximations given in the literature up-to-date. By knowing both this expression of the period and the current applied to the neuron, and then solving an inverse problem with a unique solution, we are able to estimate the steady synaptic conductance of the cell's oscillatory activity. Moreover, the method gives also good estimations when the synaptic conductance varies slowly in time.

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Piecewise-Linear (PWL) Canard Dynamics

Desroches¹,

Fernández-García

Krupa

et al. 2018

Understanding Complex Systems

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In this chapter we gather recent results on piecewise-linear (PWL) slowfast dynamical systems in the canard regime. By focusing on minimal systems in R 2 (one slow and one fast variables) and R 3 (two slow and one fast variables), we prove the existence of (maximal) canard solutions and show that the main salient features from smooth systems is preserved. We also highlight how the PWL setup carries a level of simplification of singular perturbation theory in the canard regime, which makes it more amenable to present it to various audiences at an introductory level. Finally, we present a PWL version of Fenichel theorems about slow manifolds, which are valid in the normally hyperbolic regime and in any dimension, which also offers a simplified framework for such persistence results.

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Slow–fast n-dimensional piecewise linear differential systems

Cited by 33 publications

References 21 publications

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

Estimation of Synaptic Conductance in the Spiking Regime for the McKean Neuron Model

Piecewise-Linear (PWL) Canard Dynamics

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