Phase-amplitude reduction far beyond the weakly perturbed paradigm

Optimal phase-based control of strongly perturbed limit cycle oscillators using phase reduction techniques

Phase reduction is a well-established technique for analysis and control of weakly perturbed limit cycle oscillators. However, its accuracy is diminished in a strongly perturbed setting where information about the amplitude dynamics must also be considered. In this paper, we consider phase-based control of general limit cycle oscillators in both weakly and strongly perturbed regimes. For use at the strongly perturbed end of the continuum, we propose a strategy for optimal phase control of general limit cycle oscillators that uses an adaptive phase-amplitude reduced order model in conjunction with dynamic programming. This strategy can accommodate large magnitude inputs at the expense of requiring additional dimensions in the reduced order equations, thereby increasing the computational complexity. We apply this strategy to two biologically motivated prototype problems and provide direct comparisons to two related phase-based control algorithms. In situations where other commonly used strategies fail due to the application of large magnitude inputs, the adaptive phase-amplitude reduction provides a viable reduced order model while still yielding a computationally tractable control problem. These results highlight the need for discernment in reduced order model selection for limit cycle oscillators to balance the trade-off between accuracy and dimensionality.

Insights into oscillator network dynamics using a phase-isostable framework

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviors under the variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bifurcations and dynamics of the network, which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators. In this work, we consider the phase-isostable network equations for an arbitrary but finite number of identical coupled oscillators, obtaining conditions required for the stability of phase-locked states including synchrony. For the mean-field complex Ginzburg-Landau equation where the solutions of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and, therefore, employ this to investigate the dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes). We observe qualitative correspondence between results from numerical simulations of the full system and the phase-isostable description demonstrating that in both small and large networks, the phase-isostable framework is able to capture dynamics that the first-order phase description cannot.

High-order phase reduction for coupled 2D oscillators

Phase reduction is a general approach to describe coupled oscillatory units in terms of their phases, assuming that the amplitudes are enslaved. The coupling should be small for such reduction, but one also expects the reduction to be valid for finite coupling. This paper presents a general framework, allowing us to obtain coupling terms in higher orders of the coupling parameter for generic two-dimensional oscillators and arbitrary coupling terms. The theory is illustrated with an accurate prediction of Arnold's tongue for the van der Pol oscillator exploiting higher-order phase reduction.

Phase-amplitude reduction and optimal phase locking of collectively oscillating networks

We present a phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems. The framework, which builds on the phase reduction method, takes into account not only the collective dynamics on the limit cycle but also deviations from it by introducing amplitude variables and using them with the phase variable. The framework allows us to study how networks react to applied inputs or coupling, including their synchronization and phase locking, while capturing the deviations of the network states from the unperturbed dynamics. Numerical simulations are used to demonstrate the effectiveness of the framework for networks composed of FitzHugh-Nagumo elements. The resulting phase-amplitude equations can be used in deriving optimal periodic waveforms or introducing feedback control for achieving fast phase locking while stabilizing the collective oscillations.

Control of coupled neural oscillations using near-periodic inputs

Deep brain stimulation (DBS) is a commonly used treatment for medication resistant Parkinson's disease and is an emerging treatment for other neurological disorders. More recently, phase-specific adaptive DBS (aDBS), whereby the application of stimulation is locked to a particular phase of tremor, has been proposed as a strategy to improve therapeutic efficacy and decrease side effects. In this work, in the context of these phase-specific aDBS strategies, we investigate the dynamical behavior of large populations of coupled neurons in response to near-periodic stimulation, namely, stimulation that is periodic except for a slowly changing amplitude and phase offset that can be used to coordinate the timing of applied input with a specified phase of model oscillations. Using an adaptive phase-amplitude reduction strategy, we illustrate that for a large population of oscillatory neurons, the temporal evolution of the associated phase distribution in response to near-periodic forcing can be captured using a reduced order model with four state variables. Subsequently, we devise and validate a closed-loop control strategy to disrupt synchronization caused by coupling. Additionally, we identify strategies for implementing the proposed control strategy in situations where underlying model equations are unavailable by estimating the necessary terms of the reduced order equations in real-time from observables.

Data-driven identification of dynamical models using adaptive parameter sets

This paper presents two data-driven model identification techniques for dynamical systems with fixed point attractors. Both strategies implement adaptive parameter update rules to limit truncation errors in the inferred dynamical models. The first strategy can be considered an extension of the dynamic mode decomposition with control (DMDc) algorithm. The second strategy uses a reduced order isostable coordinate basis that captures the behavior of the slowest decaying modes of the Koopman operator. The accuracy and robustness of both model identification algorithms is considered in a simple model with dynamics near a Hopf bifurcation. A more complicated model for nonlinear convective flow past an obstacle is also considered. In these examples, the proposed strategies outperform a collection of other commonly used data-driven model identification algorithms including Koopman model predictive control, Galerkin projection, and DMDc.

Isostables for Stochastic Oscillators

Thomas and Lindner [P. J. Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).PRLTAO0031-900710.1103/PhysRevLett.113.254101], defined an asymptotic phase for stochastic oscillators as the angle in the complex plane made by the eigenfunction, having a complex eigenvalue with a least negative real part, of the backward Kolmogorov (or stochastic Koopman) operator. We complete the phase-amplitude description of noisy oscillators by defining the stochastic isostable coordinate as the eigenfunction with the least negative nontrivial real eigenvalue. Our results suggest a framework for stochastic limit cycle dynamics that encompasses noise-induced oscillations.

Exploiting circadian memory to hasten recovery from circadian misalignment

Recent years have seen a sustained interest in the development of circadian reentrainment strategies to limit the deleterious effects of jet lag. Due to the dynamical complexity of many circadian models, phase-based model reduction techniques are often an imperative first step in the analysis. However, amplitude coordinates that capture lingering effects (i.e., memory) from past inputs are often neglected. In this work, we focus on these amplitude coordinates using an operational phase and an isostable coordinate framework in the context of the development of jet-lag amelioration strategies. By accounting for the influence of circadian memory, we identify a latent phase shift that can prime one's circadian cycle to reentrain more rapidly to an expected time-zone shift. A subsequent optimal control problem is proposed that balances the trade-off between control effort and the resulting latent phase shift. Data-driven model identification techniques for the inference of necessary reduced order, phase-amplitude-based models are considered in situations where the underlying model equations are unknown, and numerical results are illustrated in both a simple planar model and in a coupled population of circadian oscillators.

Data-driven phase-isostable reduction for optimal nonfeedback stabilization of cardiac alternans

Phase-isostable reduction is an emerging model reduction strategy that can be used to accurately replicate nonlinear behaviors in systems for which standard phase reduction techniques fail. In this work, we derive relationships between the cycle-to-cycle variance of the reduced isostable coordinates for systems subject to both additive white noise and periodic stimulation. Using this information, we propose a data-driven technique for inferring nonlinear terms of the phase-isostable coordinate reduction framework. We apply the proposed model inference strategy to the biologically motivated problem of eliminating cardiac alternans, an arrhythmia that is widely considered to be a precursor to more deadly cardiac arrhythmias. Using this strategy, by simply measuring a series of action potential durations in response to periodic stimulation, we are able to identify energy-optimal, nonfeedback control inputs to stabilize a period-1, alternans-free solution.

Data-driven inference of high-accuracy isostable-based dynamical models in response to external inputs

Isostable reduction is a powerful technique that can be used to characterize behaviors of nonlinear dynamical systems using a basis of slowly decaying eigenfunctions of the Koopman operator. When the underlying dynamical equations are known, previously developed numerical techniques allow for high-order accuracy computation of isostable reduced models. However, in situations where the dynamical equations are unknown, few general techniques are available that provide reliable estimates of the isostable reduced equations, especially in applications where large magnitude inputs are considered. In this work, a purely data-driven inference strategy yielding high-accuracy isostable reduced models is developed for dynamical systems with a fixed point attractor. By analyzing steady-state outputs of nonlinear systems in response to sinusoidal forcing, both isostable response functions and isostable-to-output relationships can be estimated to arbitrary accuracy in an expansion performed in the isostable coordinates. Detailed examples are considered for a population of synaptically coupled neurons and for the one-dimensional Burgers' equation. While linear estimates of the isostable response functions are sufficient to characterize the dynamical behavior when small magnitude inputs are considered, the high-accuracy reduced order model inference strategy proposed here is essential when considering large magnitude inputs.

Degenerate isostable reduction for fixed-point and limit-cycle attractors with defective linearizations

Isostable coordinates provide a convenient framework for understanding the transient behavior of dynamical systems with stable attractors. These isostable coordinates are often used to characterize the slowest decaying eigenfunctions of the Koopman operator; by neglecting the rapidly decaying Koopman eigenfunctions a reduced order model can be obtained. Existing work has focused primarily on nondegenerate isostable coordinates, that is, isostable coordinates that are associated with eigenvalues that have identical algebraic and geometric multiplicities. Current isostable reduction methods cannot be applied to characterize the decay associated with a defective eigenvalue. In this work, a degenerate isostable framework is proposed for use when eigenvalues are defective. These degenerate isostable coordinates are investigated in the context of various reduced order modeling frameworks that retain many of the important properties of standard (nondegenerate) isostable reduced modeling strategies. Reduced order modeling examples that require the use of degenerate isostable coordinates are presented with relevance to both circadian physiology and nonlinear fluid flows.

Analysis of input-induced oscillations using the isostable coordinate framework

Many reduced order modeling techniques for oscillatory dynamical systems are only applicable when the underlying system admits a stable periodic orbit in the absence of input. By contrast, very few reduction frameworks can be applied when the oscillations themselves are induced by coupling or other exogenous inputs. In this work, the behavior of such input-induced oscillations is considered. By leveraging the isostable coordinate framework, a high-accuracy reduced set of equations can be identified and used to predict coupling-induced bifurcations that precipitate stable oscillations. Subsequent analysis is performed to predict the steady state phase-locking relationships. Input-induced oscillations are considered for two classes of coupled dynamical systems. For the first, stable fixed points of systems with parameters near Hopf bifurcations are considered so that the salient dynamical features can be captured using an asymptotic expansion of the isostable coordinate dynamics. For the second, an adaptive phase-amplitude reduction framework is used to analyze input-induced oscillations that emerge in excitable systems. Examples with relevance to circadian and neural physiology are provided that highlight the utility of the proposed techniques.

Shape versus timing: linear responses of a limit cycle with hard boundaries under instantaneous and static perturbation

When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks, and stick-slip oscillators. In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory. The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape, respectively, for smooth systems. These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries. In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators. We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries: a nonsmooth kink in the asymptotic phase function, originating from the point at which the limit cycle smoothly departs the constraint surface, and propagating away from the hard boundary into the interior of the domain. Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations. By defining the "infinitesimal shape response curve" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations. In order to extract timing information, we also develop a "local timing response curve" (lTRC) that measures the timing sensitivity of a limit cycle within any given region. We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over global timing analysis given by the iPRC.

Global phase-amplitude description of oscillatory dynamics via the parameterization method

In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier-Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience.