Canonical models for torus canards in elliptic bursters

Baspinar, Emre; Avitabile, Daniele; Desroches, Mathieu

doi:10.1063/5.0037204

Cited by 5 publications

(4 citation statements)

References 53 publications

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“…This clearly resembles mixed-type canards as described in [51]. Very similar MTTCs have been reported in [47], where they evidently arise in the singular limit ε → 0. In particular Fig.…”

Section: Continuous Route To Bursting: Spike-adding Via Mixed-type-li...supporting

confidence: 80%

“…These invariant sets can correspond to equilibria but also to limit cycles. Following this definition, a particular type of canard can be found in elliptic bursters [47], which require at least a 2-fast 1-slow system. Here elliptic bursting can arise due to a subcritical Hopf-Bifurcation (in the fast subsystem) giving rise to unstable limit cycles, which stabilize via a fold bifurcation of cycles.…”

Section: Torus and Mixed-type Canardsmentioning

confidence: 99%

“…In particular Fig. 2 of [47] reports dynamics where a quasi-static motion of the full system along the attracting sheet of S 0 connects to a repelling set of limit cycles created by a subcritical Hopf bifurcation. In the NMSTP however, the understanding of mixed-type torus canards is more complex for various reasons and as we will show, they are only observed for intermediate timescale separations.…”

Section: Continuous Route To Bursting: Spike-adding Via Mixed-type-li...mentioning

confidence: 99%

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Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

2022

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We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that includes short-term synaptic plasticity. Neural mass models can mimic the collective dynamics of large-scale neuronal populations in terms of a few macroscopic variables like mean membrane potential and firing rate. The present one is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate and fire (QIF) neurons. Without synaptic dynamics, a periodic external current with slow frequency $$\varepsilon $$ ε can lead to burst-like dynamics. The firing patterns can be understood using singular perturbation theory, specifically slow–fast dissection. With synaptic dynamics, timescale separation leads to a variety of slow–fast phenomena and their role for bursting becomes inordinately more intricate. Canards are crucial to understand the route to bursting. They describe trajectories evolving nearby repelling locally invariant sets of the system and exist at the transition between subthreshold dynamics and bursting. Near the singular limit $$\varepsilon = 0$$ ε = 0 , we report peculiar jump-on canards, which block a continuous transition to bursting. In the biologically more plausible $$\varepsilon $$ ε -regime, this transition becomes continuous and bursts emerge via consecutive spike-adding transitions. The onset of bursting is complex and involves mixed-type-like torus canards, which form the very first spikes of the burst and follow fast-subsystem repelling limit cycles. We numerically evidence the same mechanisms to be responsible for bursting emergence in the QIF network with plastic synapses. The main conclusions apply for the network, owing to the exactness of the meanfield limit.

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Section: Continuous Route To Bursting: Spike-adding Via Mixed-type-li...supporting

confidence: 80%

Section: Torus and Mixed-type Canardsmentioning

confidence: 99%

Section: Continuous Route To Bursting: Spike-adding Via Mixed-type-li...mentioning

confidence: 99%

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Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

2022

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“…See e.g. [5,6,12,33,52] for more (predominantly numerical) important work on torus canards. The results in [56] were obtained using averaging theory, Floquet theory and (stationary) GSPT.…”

Section: Introductionmentioning

confidence: 99%

Geometric Blow-up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

Kuntz

Jelbart

Kuehn

2022

Preprint

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Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of 'stationary' multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularization known as blow-up. The theory for 'oscillatory' multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We show that the blow-up method can be applied to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift.MSC2020: 34D15, 34E10, 34E13, 34E15, 34E20, 34C45, 34C27

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Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

Jelbart,

Kuehn,

Kuntz

2023

J Nonlinear Sci

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Geometric singular perturbation theory provides a powerful mathematical framework for the analysis of ‘stationary’ multiple time-scale systems which possess a critical manifold, i.e. a smooth manifold of steady states for the limiting fast subsystem, particularly when combined with a method of desingularisation known as blow-up. The theory for ‘oscillatory’ multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We use the blow-up method to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale ‘semi-oscillatory’ systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale oscillatory counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our main results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift. We demonstrate the applicability of our results for systems with periodic forcing in the slow equation, in particular for a class of Liénard equations. Finally, we consider a toy model used to study tipping phenomena in climate systems with periodic forcing in the fast equation, which violates the conditions of our main results, in order to demonstrate the applicability of classical (two time-scale) theory for problems of this kind.

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Canonical models for torus canards in elliptic bursters

Cited by 5 publications

References 53 publications

Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

Bursting in a next generation neural mass model with synaptic dynamics: a slow–fast approach

Geometric Blow-up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems

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