Mixed-Mode Oscillations with Multiple Time Scales

Desroches, Mathieu; Guckenheimer, John; Krauskopf, Bernd; Kuehn, Christian; Osinga, Hinke M.; Wechselberger, Martin

doi:10.1137/100791233

Cited by 507 publications

(671 citation statements)

References 223 publications

(210 reference statements)

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“…For this aim we used the bifurcation theory and the numerical bifurcation analysis to derive a separatix between EADs, i.e., mixed mode oscillations [30,31] with one large and one or more small oscillations, and no EADs in system (1), cf. Figure 9.…”

Section: Discussionmentioning

confidence: 99%

“…The authors used a time scale separation argument (not explicit) to identify the gating variable x as the slowest variable and then, they argued in principle that EADs are Hopf-induced, cf. [29][30][31], by considering a fast subsystem using x as bifurcation parameter. This approach is not applicable in our situation, since the gating variable f is much faster than x.…”

Section: Multiple Time Scalesmentioning

confidence: 99%

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Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

Erhardt

2018

Mathematics

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Abstract:In this paper, we study the dynamics of a certain Hodgkin-Huxley model describing the action potential (AP) of a cardiac muscle cell for a better understanding of the occurrence of a special type of cardiac arrhythmia, the so-called early afterdepolarisations (EADs). EADs are pathological voltage oscillations during the repolarisation or plateau phase of cardiac APs. They are considered as potential precursors to cardiac arrhythmia and are often associated with deficiencies in potassium currents or enhancements in the calcium or sodium currents, e.g., induced by ion channel diseases, drugs or stress. Our study is focused on the enhancement in the calcium current to identify regions, where EADs related to enhanced calcium current appear. To this aim, we study the dynamics of the model using bifurcation theory and numerical bifurcation analysis. Furthermore, we investigate the interaction of the potassium and calcium current. It turns out that a suitable increasing of the potassium current adjusted the EADs related to an enhanced calcium current. Thus, one can use our result to balance the EADs in the sense that an enhancement in the potassium currents may compensate the effect of enhanced calcium currents.

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Section: Discussionmentioning

confidence: 99%

Section: Multiple Time Scalesmentioning

confidence: 99%

Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

Erhardt

2018

Mathematics

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“…For instance there are various mechanisms, such as coherence resonance, by which the addition of noise to quiescent systems induces relatively regular oscillations [14,15,16], as well as more complicated dynamics such as mixed-mode oscillations [17]. Alternatively noise may suppress chaos [18].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Regular Grazing Bifurcations

Simpson¹,

Hogan²,

Kuske³

2013

SIAM J. Appl. Dyn. Syst.

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A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincaré map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude, ε. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms, or, if there exists an attracting period-n solution when ε = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1 n , for sufficiently small ε > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to √ ε, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.

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“…We observed greater radial sensitivity in relaxation than smooth oscillators in the nickel-oxide dissolution system. Generically, oscillators exhibit higher harmonics farther from the Hopf bifurcation [29,30], thus the radial model may aid description of these systems.…”

Section: Discussionmentioning

confidence: 99%

“…(2.1) appear. These higher harmonics cause the system to spend more time in certain values of the oscillation; these relaxation oscillators are sometimes called slow-fast oscillators [29]. The phase still increases uniformly by definition.…”

Section: Relaxation Oscillationsmentioning

confidence: 99%

Improving Reduced Variable Models for Complex Systems via Experiment

Blaha

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Populations of simple interacting oscillators can give rise to complex dynamics.Real-world examples of such systems abound in biology, chemistry, and physics.Reduced-variable models can describe these systems. The phase model is a particularly successful reduced-variable model which, in its simplest form, each element is represented by one variable, the phase of oscillation. The phase model can be constructed from observable variables, without specific knowledge of underlying phenomenological behavior. Such reduced variable models are easier to construct and more generalizable than models derived from underlying physics or chemistry.In this dissertation, we study the dynamics of a population of coupled electrochemical oscillators in order to modify the phase model to expand its applicability.Specifically, we develop a two-phase model, a phase and radius model, and models with network coupling.We analyze data from two coupled oscillators with a standard one-dimensional phase model and a newer two-dimensional phase model. The same quantity of data is needed for either model. The two-dimensional analysis reveals behaviors and coupling parameters in theoretical and experimental examples that the onedimensional analysis does not show. The two-dimensional model could be useful in systems that exhibit learning due to its ability to distinguish stimulation and reaction. We extend the Stuart-Landau model to describe clustering dynamics for higher harmonic oscillators. We present examples of asymmetrical clusters arising from networks of higher harmonic oscillators. We show that the amplitude of oscillation is functionally influenced by coupling; we believe this "amplitude coupling function" has not been previously described. This function can be constructed from the original time series with no additional measurements.Recently, combined phase and amplitude models have gained attention; we explore the conditions where a phase and amplitude model is useful. We show experimentally a change in cluster state due only to changes in coupling strength. Such a transition is not possible with a phase-only model. Simulations with the extended Stuart-Landau model match experimental results. We demonstrate a method for predicting the amplitude coupling function from the coupling. Prior to this study, the relationship between stimulation and response was not known for the amplitude. We suggest that the phase and amplitude model will be most useful (1) in modeling high-harmonic oscillations, and (2) where coupling exceeds the "weak coupling" approximation of the phase-only model. iv

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Mixed-Mode Oscillations with Multiple Time Scales

Cited by 507 publications

References 223 publications

Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

Bifurcation Analysis of a Certain Hodgkin-Huxley Model Depending on Multiple Bifurcation Parameters

Stochastic Regular Grazing Bifurcations

Improving Reduced Variable Models for Complex Systems via Experiment

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