Optimal Waveform for Fast Entrainment of Weakly Forced Nonlinear Oscillators

Zlotnik, Anatoly; Chen, Yifei; Kiss, István Z.; Tanaka, Hisaaki; Li, Jr-Shin

doi:10.1103/physrevlett.111.024102

Cited by 87 publications

(119 citation statements)

References 25 publications

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“…Furthermore, we find that the standard methods used to calculate PRCs in individual oscillators can produce misleading results when directly applied to populations of oscillators. This methodology could make control strategies such as [23], [24], and [25] more feasible for in vivo testing when the individual elements in the population are difficult to observe.…”

Section: Introductionmentioning

confidence: 99%

Determining individual phase response curves from aggregate population data

Wilson

Moehlis

2015

Phys. Rev. E

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Phase reduction is an invaluable technique for investigating the dynamics of nonlinear limit cycle oscillators. Central to the implementation of phase reduction is the ability to calculate phase response curves (PRCs), which describe an oscillator's response to an external perturbation. Current experimental techniques for inferring PRCs require data from individual oscillators, which can be impractical to obtain when the oscillator is part of a much larger population. Here we present a simple yet novel methodology to calculate PRCs of individual oscillators using an aggregate signal from a large homogeneous population. This methodology is shown to be accurate in the presence of inter-oscillator coupling and noise and can also provide a good estimate of an average PRC of a heterogeneous population. We also find that standard experimental techniques for PRC measurement can produce misleading results when applied to aggregate population data.

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Section: Introductionmentioning

confidence: 99%

Determining individual phase response curves from aggregate population data

Wilson

Moehlis

2015

Phys. Rev. E

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“…It should be noted that an optimal locking problem has been recently discussed for purely deterministic oscillations. There, the optimal condition was formulated as the maximal width of the Arnold's tongue (the synchronization region) or as the maximal stability of the locked state [3][4][5][6]. In our case there is an additional parameter, the noise intensity, and we will show that the optimal force profile depends on the noise amplitude.…”

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confidence: 99%

“…For the discussed above example of a bi-harmonic phase sensitivity function (21), all the optimal forcings can be found analytically, they are generally also bi-harmonic. The approach of [4] yields in this case…”

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confidence: 99%

“…While here we optimize the coherence in the presence of noise, in Refs. [3,4] purely deterministic criteria have been suggested. It is instructive to compare them with our approach in the limit of small noise.…”

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confidence: 99%

“…Suppose that the coupling function g(φ) has zeros at φ 1,2 (where φ 1 is the stable one), and extrema at φ 3,4 . In the approach of [4], the linear stability at the stable equilibrium |g ′ (x 1 )| is maximized. In the approach of [3], the width of the synchronization region ∼ |g(φ 3 ) − g(φ 4 )| is maximized.…”

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Maximizing Coherence of Oscillations by External Locking

Pikovsky

2015

Phys. Rev. Lett.

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We study how the coherence of noisy oscillations can be optimally enhanced by external locking. Basing on the condition of minimizing the phase diffusion constant, we find the optimal forcing explicitly in the limits of small and large noise, in dependence of phase sensitivity of the oscillator. We show that the form of the optimal force bifurcates with the noise intensity. In the limit of small noise, the results are compared with purely deterministic conditions of optimal locking. Autonomous self-sustained oscillations may be extremely regular (like, e.g., lasers) or rather incoherent (like many biological oscillators, e.g., ones responsible for cardiac or circadian rhythms). A usual way to improve the quality of oscillations is to lock (synchronize) them by an external pacing [1,2]. This is used in radiocontrolled clocks and in cardiac pacemakers; also circadian rhythms are nearly perfectly locked by the 24-hours day/night force.In this letter we address a question: which periodic force ensures, via locking, the maximal coherence of a noisy self-sustained oscillator? Of course, one has to fix the amplitude of the force, so the nontrivial problem is in finding the optimal force profile. We will treat this problem in the phase approximation [1], which is valid for general oscillators, provided the noise and the forcing are small. In this approximation the dynamics of the phase reduces to a noisy Adler equation [2,7], and the maximal coherence is achieved if the diffusion constant of the phase is minimal. It should be noted that an optimal locking problem has been recently discussed for purely deterministic oscillations. There, the optimal condition was formulated as the maximal width of the Arnold's tongue (the synchronization region) or as the maximal stability of the locked state [3][4][5][6]. In our case there is an additional parameter, the noise intensity, and we will show that the optimal force profile depends on the noise amplitude. Below we will also compare the limit of small noise with purely deterministic setups.Let us consider a self-sustained oscillator with frequency ω, its phase in presence of a small Gaussian white noise obeys the Langevin equationwhere β −1 is the noise intensity. A small periodic forcing with frequency Ω leads, in the first order in the force amplitude, to the following phase dynamics [1,7]:Here s(ϕ) is the phase sensitivity function (a.k.a. phase response curve), and f (Ωt) is the phase-projected force term. Our goal will be to find such a forcing f (·) that maximizes the coherence, i.e. minimizes the diffusion constant of the phase ϕ. This optimal force will depend on the phase sensitivity function s(·) and on the noise intensity β.As the first step we introduce the slow phase φ = ϕ − Ωt and perform the standard averaging over the period 2πΩ −1 [1, 2], this yieldswhereand we introduced the "potential"Let us consider a situation, where the mean frequency of oscillations is exactly that of the forcing; this means that the slow phase φ performs a random walk without a bias. Thi...

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