Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations

Glass, Leon; Sun, Jiande

doi:10.1103/physreve.50.5077

Cited by 59 publications

(43 citation statements)

References 23 publications

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“…Based on [5,20], we introduce the Poincaré oscillator here and summarize its properties. The oscillator can be described in polar coordinates as…”

Section: A Stochastic Poincaré Oscillatormentioning

confidence: 99%

“…The Poincaré oscillator and its variant are a member of a set of systems that are widely used in the analysis of neuronal or biological oscillators [4,9,[17][18][19][20][21][22][23][24][29][30][31]. Based on [5,20], we introduce the Poincaré oscillator here and summarize its properties.…”

Section: A Stochastic Poincaré Oscillatormentioning

confidence: 99%

“…For example, Glass and Sun [20] analyzed the dependence of the bifurcation structure of an impulse-driven nonlinear oscillator on the relaxation rate. For their analysis, they used a 2D extension of the phase transition curve for the nonlinear oscillator, which represents the phase shift due to a single isolated impulse.…”

Section: Introductionmentioning

confidence: 99%

“…The impulse-driven nonlinear oscillators are often used in the analysis of neuronal or biological oscillators [4,[17][18][19][20][21][22][23][24][25], and depending on the relaxation rate to the limit cycle, the state points of the nonlinear oscillator can be far from the limit cycle. For example, Glass and Sun [20] analyzed the dependence of the bifurcation structure of an impulse-driven nonlinear oscillator on the relaxation rate.…”

Section: Introductionmentioning

confidence: 99%

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Global dynamics of a stochastic neuronal oscillator

Yamanobe

2013

Phys. Rev. E

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Nonlinear oscillators have been used to model neurons that fire periodically in the absence of input. These oscillators, which are called neuronal oscillators, share some common response structures with other biological oscillations such as cardiac cells. In this study, we analyze the dependence of the global dynamics of an impulse-driven stochastic neuronal oscillator on the relaxation rate to the limit cycle, the strength of the intrinsic noise, and the impulsive input parameters. To do this, we use a Markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve, which describes the phase shift due to a single isolated impulse. Previously, we derived the Markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane. Here, we construct a Markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle. In both cases, the response of the stochastic neuronal oscillator to time-varying impulses is described by a product of Markov operators. Furthermore, we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density. Specifically, we analyze the dynamics of the number of spikes per unit time based on the properties of the Markov operators. Each Markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions. This allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator, which we show to be based on the dependence of the oscillator on past activity. Our analysis shows how the duration of the past neuronal activity depends on the relaxation rate, the noise strength, and the impulsive input parameters.

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“…Based on [5,20], we introduce the Poincaré oscillator here and summarize its properties. The oscillator can be described in polar coordinates as…”

Section: A Stochastic Poincaré Oscillatormentioning

confidence: 99%

Section: A Stochastic Poincaré Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global dynamics of a stochastic neuronal oscillator

Yamanobe

2013

Phys. Rev. E

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“…We then investigate the effect of a periodic modulation of the oscillator. Although many theoretical works have been devoted to periodically forced oscillators (Tomita et al 1977, Tomita and Kai 1979, Aronson et al 1986, Parlitz and Lauterborn 1987, Vance and Ross 1989, Glass and Sun 1994, Croisier et al 2009, to our knowledge, this is the first analytical study of the periodic forced Goodwin oscillator. We compare the entrainment performances when the free oscillator displays either self-sustained or damped oscillations.…”

Section: Introductionmentioning

confidence: 99%

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

2014

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The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

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