Mixed mode oscillations due to the generalized Canard phenomenon

Brøns, Morten; Krupa, Martin; Wechselberger, Martin

doi:10.1090/fic/049/03

Cited by 128 publications

(344 citation statements)

References 0 publications

Supporting

Mentioning

333

Contrasting

Order By: Relevance

“…Based on geometric singular perturbation theory (GSPT) [4,8,10,17,18], we showed in that paper that the intrinsic dynamics of a model of intracellular calcium dynamics is mainly controlled by distinct time scales describing the (slow or fast) evolution of the model variables relative to each other. This inherent multiple time-scale structure provides us with a simple necessary condition for Class I and Class II dynamics by calculating a dimensionless quantity 1 representing the ratio of the different time scales involved.…”

Section: Multiple Time Scales In a Representative Modelmentioning

confidence: 99%

“…There is general agreement over the first steps of the process that results in calcium oscillations [11]: binding of an agonist to a cell-surface receptor initiates a series of reactions that ends in the formation of the intracellular second messenger, inositol (1,4,5)-trisphosphate (IP 3 ), which opens IP 3 receptors (which are also calcium channels) on the membrane of the endoplasmic reticulum (ER), leading to the release of calcium from that internal store. Oscillations in the cytoplasmic calcium concentration occur as calcium is alternatively released from, and pumped into, the ER.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Understanding anomalous delays in a model of intracellular calcium dynamics

Harvey

Kirk

Osinga

et al. 2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [16] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP 3 ) is applied to the cell.However, more detailed mathematical investigations [6] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of Class I, Class II and Hybrid models of calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems systems with three or more slow variables [19] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP 3 .Oscillations in the concentration of cytoplasmic free calcium are a signalling mechanism in practically every cell type, and control a wide variety of cellular functions, from secretion and movement to cell differentiation and gene 1 that a simple experiment (a single exogenous pulse of inositol trisphosphate, or IP 3 ) is able to distinguish between whether or not calcium oscillations in a cell are a result principally of a Class I or a Class II mechanism. However, more detailed mathematical studies of these models [6] have shown that the situation is more complex than appeared at first, and there are situations in which a Class I model can look remarkably like a Class II model, making interpretation of the experimental data ambiguous and difficult. Here we perform a detailed mathematical study of Class I and Class II models, investigate how they respond to a pulse of IP 3 , and provide a detailed explanation of why Class I and Class II models cannot necessarily be easily distinguished by the IP 3 pulse experiment. It turns out that this is related to the phenomenon of canards in systems with three slow variables, the theory of which has been developed recently [19].

show abstract

Section: Multiple Time Scales In a Representative Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Understanding anomalous delays in a model of intracellular calcium dynamics

Harvey

Kirk

Osinga

et al. 2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…Regimes in which the large oscillations alternate with small ones are known as "mixed-mode oscillations" [11]; one of the mechanisms of their onset is related to canards in systems with two and more slow variables [11,2,14]. Mixed-mode oscillations are believed to stand behind a variety of phenomena in neuronal dynamics and in physical chemistry (see e.g.…”

Section: Model Equations: Properties and Bifurcationsmentioning

confidence: 99%

“…The folded equilibrium becomes a node. According to [2,17], one of the reasons for the mixed-mode oscillations may lie in the complicated global topology of the invariant manifolds of the folded node. Slopes of eigenvectors of its Jacobian matrix are c z 0 /λ 1 ≈ 2/c and c z 0 /λ 2 ≈ εκ 1 /2 − G, respectively.…”

Section: Dynamics On the Slow Surfacementioning

confidence: 99%

See 1 more Smart Citation

On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

Zaks

2010

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

A Review of Multiple-Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods

Kristiansen

2023

Mathematics Online First Collections

View full text Add to dashboard Cite

In this paper, we review the most fundamental phenomena in multipletime scale systems as well as the Geometric Singular Perturbation Theory (GSPT) that can be used to analyze such systems. We put special emphasis on loss of normal hyperbolicity and illustrate how it relates to the different phenomena. Moreover, by working on lower dimensional model problems, we review the main technical tool of GSPT, the blowup method, and demonstrate -in line with recent research in this area -how it may provide detailed insight into problems of this kind.

show abstract

Mixed mode oscillations due to the generalized Canard phenomenon

Cited by 128 publications

References 0 publications

Understanding anomalous delays in a model of intracellular calcium dynamics

Understanding anomalous delays in a model of intracellular calcium dynamics

On Chaotic Subthreshold Oscillations in a Simple Neuronal Model

A Review of Multiple-Time-Scale Dynamics: Fundamental Phenomena and Mathematical Methods

Contact Info

Product

Resources

About