Desynchronization of stochastically synchronized chemical oscillators

Snari, Razan; Tinsley, Mark R.; Wilson, Dan; Faramarzi, Sadegh; Netoff, Theoden I.; Moehlis, Jeff; Showalter, Kenneth

doi:10.1063/1.4937724

Cited by 18 publications

(12 citation statements)

References 50 publications

(64 reference statements)

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“…Consequently, the amplitude of allowable perturbations is limited by the size of the Floquet multipliers [8]; stable orbits with Floquet multipliers with magnitude close to 1 can only admit relatively small perturbations without the risk of being driven away from the limit cycle over time. In many applications [5,[9][10][11], however, the efficacy of a given control strategy is directly related to the magnitude of allowable perturbations. Furthermore, for unstable periodic orbits, (2) alone cannot adequately describe the long term behavior of (1), rendering it unusable.…”

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confidence: 99%

Isostable reduction of periodic orbits

2016

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The well-established method of phase reduction neglects information about a limit-cycle oscillator's approach towards its periodic orbit. Consequently, phase reduction suffers in practicality unless the magnitude of the Floquet multipliers of the underlying limit cycle are small in magnitude. By defining isostable coordinates of a periodic orbit, we present an augmentation to classical phase reduction which obviates this restriction on the Floquet multipliers. This framework allows for the study and understanding of periodic dynamics for which standard phase reduction alone is inadequate. Most notably, isostable reduction allows for a convenient and self-contained characterization of the dynamics near unstable periodic orbits.

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confidence: 99%

Isostable reduction of periodic orbits

2016

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“…Provided the magnitude of u p is sufficiently small and perturbations transverse to the periodic orbit decay rapidly, it is well established that the resulting phase response curve can be used to predict the behavior of arbitrary perturbations using the phase reduced equation \. \theta = \omega + z(\theta )u(t) [10], a strategy which has been successfully applied in many experimental settings [21], [27], [38]. A similar strategy for direct measurement of the terms in the OPR has been proposed here.…”

Section: 2mentioning

confidence: 99%

An Operational Definition of Phase Characterizes the Transient Response of Perturbed Limit Cycle Oscillators

Wilson¹,

Ermentrout²

2018

SIAM J. Appl. Dyn. Syst.

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Phase reduction is an essential tool used in the study of perturbed limit cycle oscillators. Usually, phase is defined with respect to some asymptotic limit, often making phase reduction's ability to predict transient behavior during the approach to a limit cycle inadequate. In this work, we present an operational notion of phase, defined with respect to a distinct feature of the oscillator, for example, the timing of a periodically firing neuron's action potential. We derive relationships between standard and operational phase reduction in order to make comparisons between these two approaches. In theoretical and numerical examples, we show how operational phase reduction accurately predicts both the asymptotic and transient behavior due to perturbations without relying on the notion of higher order phase response curves which are valid only for specific perturbations. Furthermore, we develop a strategy for direct measurement of the necessary terms of the operational phase reduction when the full dynamical equations are unknown.

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“…We therefore developed an oscillator array that permits the simultaneous perturbing and coupling of over 1000 oscillators. (Our previous studies of photochemically coupled BZ oscillators have been limited to approximately 40 oscillators [19][20][21][22][23].) In the oscillator array, the photosensitive oscillators are catalyst-loaded beads of about 200 μm in diameter, each of which is fixed on a film of polydimethylsiloxane (PDMS), which is then positioned in an open reactor that is continually replenished with a catalyst-free BZ solution.…”

Section: Methodsmentioning

confidence: 99%

Echo Behavior in Large Populations of Chemical Oscillators

et al. 2016

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Experimental and theoretical studies are reported, for the first time, on the observation and characterization of echo phenomena in oscillatory chemical reactions. Populations of uncoupled and coupled oscillators are globally perturbed. The macroscopic response to this perturbation dies out with time: At some time τ after the perturbation (where τ is long enough that the response has died out), the system is again perturbed, and the initial response to this second perturbation again dies out. Echoes can potentially appear as responses that arise at 2τ; 3τ; ... after the first perturbation. The phase-resetting character of the chemical oscillators allows a detailed analysis, offering insights into the origin of the echo in terms of an intricate structure of phase relationships. Groups of oscillators experiencing different perturbations are analyzed with a geometric approach and in an analytical theory. The characterization of echo phenomena in populations of chemical oscillators reinforces recent theoretical studies of the behavior in populations of phase oscillators [E. Ott et al., Chaos 18, 037115 (2008)]. This indicates the generality of the behavior, including its likely occurrence in biological systems.

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Desynchronization of stochastically synchronized chemical oscillators

Cited by 18 publications

References 50 publications

Isostable reduction of periodic orbits

Isostable reduction of periodic orbits

An Operational Definition of Phase Characterizes the Transient Response of Perturbed Limit Cycle Oscillators

Echo Behavior in Large Populations of Chemical Oscillators

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