Phase reduction and phase-based optimal control for biological systems: a tutorial

Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power series expansion contributes with additional higher-order multi-body (i.e. non-pairwise) interactions. This points to intricate multi-body phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling. II. MEAN-FIELD COMPLEX GINZBURG-LANDAU EQUATIONThe MF-CGLE consists of N diffusively coupled Stuart-Landau oscillators governed by N coupled (complex-valued)

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“…We anticipate that the results at second and third order in ǫ correspond to Eqs. (15) and (29) below.…”

Section: Systematic Phase Reductionmentioning

confidence: 97%

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

León

Pazó

2019

Phys. Rev. E

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“…Linearisation The response curves derived for the linearisation of a 2D focus in section 4 can be related to previously published expressions. In particular, the infinitesimal PRC for radial isochron clocks has been derived in [34], and has been recently included in [35] under the larger umbrella of general radial isochron clocks. The radial clock case (K(φ) = ω in [35]) perturbed along the first dimension agrees with our equation (17) for the case of a circular flow (see section 4.7).…”

Section: Discussionmentioning

confidence: 99%

“…In particular, the infinitesimal PRC for radial isochron clocks has been derived in [34], and has been recently included in [35] under the larger umbrella of general radial isochron clocks. The radial clock case (K(φ) = ω in [35]) perturbed along the first dimension agrees with our equation (17) for the case of a circular flow (see section 4.7). For this simple system, the asymptotic phase response is the same as the first order Hilbert phase response.…”

Section: Discussionmentioning

confidence: 99%

“…We are looking for φ max such that dX stim 1 dt = 0 at ωt = φ max . This give us e σ φmax ω [−ω(K stim a 1 + K stim b 1 ) sin φ max + ω(−K stim b 1 + K stim a 1 ) cos φ max + σ {(K stim a 1 + K stim b 1 ) cos φ max + (−K stim b 1 + K stim a 1 ) sin φ max }] = 0 (35) tan φ max = K stim (σa 1 − ωb 1 ) + K stim (σb 1 + ωa 1 ) K stim (σb 1 + ωa 1 ) + K stim (−σa 1 + ωb 1 )…”

Section: B Trajectory With Stimulationmentioning

confidence: 99%

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Phase dependence of response curves to stimulation and their relationship: from a Wilson-Cowan model to essential tremor patient data

Duchet

Weerasinghe

Cagnan

et al. 2019

Preprint

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AbstractEssential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Investigators have been looking at stimulating less, chiefly to reduce side effects. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor, once per period. In this study, we aim to reproduce the phase dependent effects of stimulation seen in patient data with a biologically inspired Wilson-Cowan model. To this end, we first analyse patient data, and conclude that half of the datasets have response curves that are better described by sinusoidal curves than by straight lines, while an effect of phase cannot be consistently identified in the remaining half. Using the Hilbert phase we derive analytical expressions for phase and amplitude responses to phase-dependent stimulation and study their relationship in the linearisation of a stable focus model, a simplification of the Wilson-Cowan model in the stable focus regime. Analytical results provide a good approximation for response curves observed in patients with consistent significance. Additionally, we fitted the full non-linear Wilson-Cowan model to these patients, and we show that the model can fit in each case to the dynamics of patient tremor as well as the phase response curve, and the best fits are found to be stable foci for each patients (tied best fit in one instance). The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. This can be partially explained by the relationship between the response curves in the model being compatible with what is found in the data. We also note that the non-linear Wilson-Cowan model is able to describe response to stimulation more precisely than the linearisation.

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“…The phase reduction theory has also been used in control and optimization of nonlinear oscillators [35]. For example, using the reduced phase equations, minimization of control power for an oscillator [36,37], maximization of the phase-locking range of an oscillator [38], maximization of linear stability of an oscillator entrained to a periodic forcing [39] and of mutual synchronization between two coupled oscillators [40,41], maximization of phase coherence of noisy oscillators [42], and phase-selective entrainment of oscillators [43] have been studied.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators

Kato

Nakao

2020

Phys. Rev. E

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Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase reduction theory recently developed for quantum nonlinear oscillators [1], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically

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Phase reduction and phase-based optimal control for biological systems: a tutorial

Cited by 96 publications

References 109 publications

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

Phase dependence of response curves to stimulation and their relationship: from a Wilson-Cowan model to essential tremor patient data

Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators

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