Nonlinear phase-amplitude reduction of delay-induced oscillations

Abstract

Spontaneous oscillations induced by time delays are observed in many real-world systems. Phase reduction theory for limit-cycle oscillators described by delay-differential equations (DDEs) has been developed to analyze their synchronization properties, but it is applicable only when the perturbation applied to the oscillator is sufficiently weak. In this study, we formulate a nonlinear phase-amplitude reduction theory for limit-cycle oscillators described by DDEs on the basis of the Floquet theorem, which is applicable when the oscillator is subjected to perturbations of moderate intensity. We propose a numerical method to evaluate the leading Floquet eigenvalues, eigenfunctions, and adjoint eigenfunctions necessary for the reduction and derive a set of low-dimensional nonlinear phase-amplitude equations approximately describing the oscillator dynamics. By analyzing an analytically tractable oscillator model with a cubic nonlinearity, we show that the asymptotic phase of the oscillator state in an infinite-dimensional state space can be approximately evaluated and non-trivial bistability of the oscillation amplitude caused by moderately strong periodic perturbations can be predicted on the basis of the derived phase-amplitude equations. We further analyze a model of generegulatory oscillator and illustrate that the reduced equations can elucidate the mechanism of its complex dynamics under non-weak perturbations, which may be relevant to real physiological phenomena such as circadian rhythm sleep disorders.

From local resynchronization to global pattern recovery in the zebrafish segmentation clock

“…The choice of a phase description for the cellular oscillator is motivated by its generality and by prior success in analyzing coupling in the segmentation clock ( Riedel-Kruse et al, 2007 ; Uriu et al, 2017 ; Herrgen et al, 2010 ). Core genetic components of the cellular oscillator have been identified and their dynamics can be described by detailed delay models ( Schröter et al, 2012 ; Lewis, 2003 ; Monk, 2003 ; Horikawa et al, 2006 ; Ay et al, 2013 ; Hirata et al, 2004 ; Jensen et al, 2003 ), but since we do not measure any of the components in these networks, the choice of phase oscillators captures the core oscillatory behavior without additional underconstrained parameters ( Kotani et al, 2012 ; Kotani et al, 2020 ). Thus, although we do not anticipate any qualitative differences between these modeling approaches, future work could include such a detailed description of oscillatory genes, potentially allowing a more direct connection to mutant conditions or imaging experiments.…”

“…We This agreement between experimental data and probability theory for the fraction of single defects indicates that recovery occurred independently between left and right PSMs, The choice of a phase description for the cellular oscillator is motivated by its generality and by prior success in analyzing coupling in the segmentation clock (19,28,43). Core genetic components of the cellular oscillator have been identified and their dynamics can be described by detailed delay models (12,14,15,17,(44)(45)(46), but since we do not measure any of the components in these networks, the choice of phase oscillators captures the core oscillatory behavior without additional underconstrained parameters (47,48). Thus, although we do not anticipate any qualitative differences between these modeling approaches, future work could include such a detailed description of oscillatory genes, The hallmark of these systems is an interplay between locally driven interactions and global morphological changes, pointing to a common principle of pattern dynamics within developing tissues.…”

Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

“…Moreover, though we considered only the optimization of the local linear stability of the phase-locked state, we may be able to the present methods to other optimization problems, such as those for global entrainment property 45 and phase-distribution control [42][43][44][45] . Finally, using the phase-amplitude reduction frameworks for time-delayed 56 and spatially-extended 57 systems, we would also be able to realize fast entrainment in such non-conventional infinite-dimensional systems via amplitude suppression.…”

“…Recent developments in the Koopman operator theory 49,50 have clarified that the deviations of the system state from the limit cycle can be characterized naturally by the amplitude variables, which are Koopman eigenfunctions associated with non-zero Floquet exponents of the system 51 . Phase-amplitude reduction theories that generalize the conventional phase-only reduction theory by including the amplitude variables have also been formulated 21,[52][53][54][55][56][57] . By using the resulting phase-amplitude equations, optimization methods have been proposed 18,55,58,59 , for example, for minimizing the control power applied to the oscillator with slow amplitude relaxation.…”

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

“…By using the asymptotic phase and amplitude functions, we can obtain a reduced description of limitcycle oscillators, which is useful for the analysis and control of synchronization dynamics of limit-cycle oscillators [18][19][20][21][22][23][24]. The theory can also be generalized to delay-differential systems [25] and spatially extended systems [26].…”