Greater accuracy and broadened applicability of phase reduction using isostable coordinates

Phase reduction explains chimera shape: When multibody interaction matters

We present an extension of the Kuramoto-Sakaguchi model for networks, deriving the second-order phase approximation for a paradigmatic model of oscillatory networks-an ensemble of nonidentical Stuart-Landau oscillators coupled pairwisely via an arbitrary coupling matrix. We explicitly demonstrate how this matrix translates into the coupling structure in the phase equations. To illustrate the power of our approach and the crucial importance of high-order phase reduction, we tackle a trendy setup of nonlocally coupled oscillators exhibiting a chimera state. We reveal that our second-order phase model reproduces the dependence of the chimera shape on the coupling strength that is not captured by the typically used first-order Kuramoto-like model. Our derivation contributes to a better understanding of complex networks' dynamics, establishing a relation between the coupling matrix and multibody interaction terms in the high-order phase model.

Phase and amplitude responses for delay equations using harmonic balance

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilizes the method of harmonic balance.

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.

Optimal entrainment for removal of pinned spiral waves

Cardiac fibrillation is caused by self-sustaining spiral waves that occur in the myocardium, some of which can be pinned to anatomical obstacles, making them more difficult to eliminate. A small electrical stimulation is often sufficient to unpin these spirals but only if it is applied during the vulnerable unpinning window. Even if these unpinning windows can be inferred from data, when multiple pinned spirals exist, their unpinning windows will not generally overlap. Using phase-based reduction techniques, we formulate and solve an optimal control problem to yield a time-varying external voltage gradient that can synchronize a collection of spiral waves that are pinned to a collection of heterogeneous obstacles. Upon synchronization, the unpinning windows overlap so that they can be simultaneously unpinned by applying an external voltage gradient pulse at an appropriate moment. Numerical validation is presented in bidomain model simulations. Results represent a proof-of-concept illustration of the proposed unpinning strategy which explicitly incorporates heterogeneity in the problem formulation and requires no real-time feedback about the system state.

Phase and frequency linear response theory for hyperbolic chaotic oscillators

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.

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.

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.

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

Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the van der Pol and Willamowski-Rössler oscillators with real or complex Floquet exponents is analyzed as examples. It is demonstrated that the proposed methods can achieve considerably faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.

Interspike interval correlations in neuron models with adaptation and correlated noise

The generation of neural action potentials (spikes) is random but nevertheless may result in a rich statistical structure of the spike sequence. In particular, contrary to the popular renewal assumption of theoreticians, the intervals between adjacent spikes are often correlated. Experimentally, different patterns of interspike-interval correlations have been observed and computational studies have identified spike-frequency adaptation and correlated noise as the two main mechanisms that can lead to such correlations. Analytical studies have focused on the single cases of either correlated (colored) noise or adaptation currents in combination with uncorrelated (white) noise. For low-pass filtered noise or adaptation, the serial correlation coefficient can be approximated as a single geometric sequence of the lag between the intervals, providing an explanation for some of the experimentally observed patterns. Here we address the problem of interval correlations for a widely used class of models, multidimensional integrate-and-fire neurons subject to a combination of colored and white noise sources and a spike-triggered adaptation current. Assuming weak noise, we derive a simple formula for the serial correlation coefficient, a sum of two geometric sequences, which accounts for a large class of correlation patterns. The theory is confirmed by means of numerical simulations in a number of special cases including the leaky, quadratic, and generalized integrate-and-fire models with colored noise and spike-frequency adaptation. Furthermore we study the case in which the adaptation current and the colored noise share the same time scale, corresponding to a slow stochastic population of adaptation channels; we demonstrate that our theory can account for a nonmonotonic dependence of the correlation coefficient on the channel's time scale. Another application of the theory is a neuron driven by network-noise-like fluctuations (green noise). We also discuss the range of validity of our weak-noise theory and show that by changing the relative strength of white and colored noise sources, we can change the sign of the correlation coefficient. Finally, we apply our theory to a conductance-based model which demonstrates its broad applicability.

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.

Resolving molecular contributions of ion channel noise to interspike interval variability through stochastic shielding

Molecular fluctuations can lead to macroscopically observable effects. The random gating of ion channels in the membrane of a nerve cell provides an important example. The contributions of independent noise sources to the variability of action potential timing have not previously been studied at the level of molecular transitions within a conductance-based model ion-state graph. Here we study a stochastic Langevin model for the Hodgkin-Huxley (HH) system based on a detailed representation of the underlying channel state Markov process, the "[Formula: see text]D model" introduced in (Pu and Thomas in Neural Computation 32(10):1775-1835, 2020). We show how to resolve the individual contributions that each transition in the ion channel graph makes to the variance of the interspike interval (ISI). We extend the mean return time (MRT) phase reduction developed in (Cao et al. in SIAM J Appl Math 80(1):422-447, 2020) to the second moment of the return time from an MRT isochron to itself. Because fixed-voltage spike detection triggers do not correspond to MRT isochrons, the inter-phase interval (IPI) variance only approximates the ISI variance. We find the IPI variance and ISI variance agree to within a few percent when both can be computed. Moreover, we prove rigorously, and show numerically, that our expression for the IPI variance is accurate in the small noise (large system size) regime; our theory is exact in the limit of small noise. By selectively including the noise associated with only those few transitions responsible for most of the ISI variance, our analysis extends the stochastic shielding (SS) paradigm (Schmandt and Galán in Phys Rev Lett 109(11):118101, 2012) from the stationary voltage clamp case to the current clamp case. We show numerically that the SS approximation has a high degree of accuracy even for larger, physiologically relevant noise levels. Finally, we demonstrate that the ISI variance is not an unambiguously defined quantity, but depends on the choice of voltage level set as the spike detection threshold. We find a small but significant increase in ISI variance, the higher the spike detection voltage, both for simulated stochastic HH data and for voltage traces recorded in in vitro experiments. In contrast, the IPI variance is invariant with respect to the choice of isochron used as a trigger for counting "spikes."

Optimizing deep brain stimulation based on isostable amplitude in essential tremor patient models

OBJECTIVE

Deep brain stimulation is a treatment for medically refractory essential tremor. To improve the therapy, closed-loop approaches are designed to deliver stimulation according to the system's state, which is constantly monitored by recording a pathological signal associated with symptoms (e.g. brain signal or limb tremor). Since the space of possible closed-loop stimulation strategies is vast and cannot be fully explored experimentally, how to stimulate according to the state should be informed by modeling. A typical modeling goal is to design a stimulation strategy that aims to maximally reduce the Hilbert amplitude of the pathological signal in order to minimize symptoms. Isostables provide a notion of amplitude related to convergence time to the attractor, which can be beneficial in model-based control problems. However, how isostable and Hilbert amplitudes compare when optimizing the amplitude response to stimulation in models constrained by data is unknown.

APPROACH

We formulate a simple closed-loop stimulation strategy based on models previously fitted to phase-locked deep brain stimulation data from essential tremor patients. We compare the performance of this strategy in suppressing oscillatory power when based on Hilbert amplitude and when based on isostable amplitude. We also compare performance to phase-locked stimulation and open-loop high-frequency stimulation.

MAIN RESULTS

For our closed-loop phase space stimulation strategy, stimulation based on isostable amplitude is significantly more effective than stimulation based on Hilbert amplitude when amplitude field computation time is limited to minutes. Performance is similar when there are no constraints, however constraints on computation time are expected in clinical applications. Even when computation time is limited to minutes, closed-loop phase space stimulation based on isostable amplitude is advantageous compared to phase-locked stimulation, and is more efficient than high-frequency stimulation.

SIGNIFICANCE

Our results suggest a potential benefit to using isostable amplitude more broadly for model-based optimization of stimulation in neurological disorders.

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.

Analysis of neural clusters due to deep brain stimulation pulses

Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson's disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, several theoretical studies of populations of neural oscillators stimulated by periodic pulses have suggested that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters and their stability properties, bifurcations, and basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system.

Quasi phase reduction of all-to-all strongly coupled λ-ω oscillators near incoherent states

The dynamics of an ensemble of N weakly coupled limit-cycle oscillators can be captured by their N phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as "quasiphase oscillators," evidencing the need of novel reduction strategies. We introduce, here, quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry (λ-ω systems). By applying QPR, we achieve a reduction to N+2 degrees of freedom: N phase oscillators interacting through one independent complex variable. This "quasi phase model" is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with N+2M degrees of freedom is obtained for coupling through the first M harmonics.

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.

Optimal open-loop desynchronization of neural oscillator populations

Deep brain stimulation (DBS) is an increasingly used medical treatment for various neurological disorders. While its mechanisms are not fully understood, experimental evidence suggests that through application of periodic electrical stimulation DBS may act to desynchronize pathologically synchronized populations of neurons resulting desirable changes to a larger brain circuit. However, the underlying mathematical mechanisms by which periodic stimulation can engender desynchronization in a coupled population of neurons is not well understood. In this work, a reduced phase-amplitude reduction framework is used to characterize the desynchronizing influence of periodic stimulation on a population of coupled oscillators. Subsequently, optimal control theory allows for the design of periodic, open-loop stimuli with the capacity to destabilize completely synchronized solutions while simultaneously stabilizing rotating block solutions. This framework exploits system nonlinearities in order to strategically modify unstable Floquet exponents. In the limit of weak neural coupling, it is shown that this method only requires information about the phase response curves of the individual neurons. The effects of noise and heterogeneity are also considered and numerical results are presented. This framework could ultimately be used to inform the design of more efficient deep brain stimulation waveforms for the treatment of neurological disease.

Stabilization of Weakly Unstable Fixed Points as a Common Dynamical Mechanism of High-Frequency Electrical Stimulation

While high-frequency electrical stimulation often used to treat various biological diseases, it is generally difficult to understand its dynamical mechanisms of action. In this work, high-frequency electrical stimulation is considered in the context of neurological and cardiological systems. Despite inherent differences between these systems, results from both theory and computational modeling suggest identical dynamical mechanisms responsible for desirable qualitative changes in behavior in response to high-frequency stimuli. Specifically, desynchronization observed in a population of periodically firing neurons and reversible conduction block that occurs in cardiomyocytes both result from bifurcations engendered by stimulation that modifies the stability of unstable fixed points. Using a reduced order phase-amplitude modeling framework, this phenomenon is described in detail from a theoretical perspective. Results are consistent with and provide additional insight for previously published experimental observations. Also, it is found that sinusoidal input is energy-optimal for modifying the stability of weakly unstable fixed points using periodic stimulation.

Phase-amplitude reduction far beyond the weakly perturbed paradigm

While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.

A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems

Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.

Recent advances in coupled oscillator theory

We review the theory of weakly coupled oscillators for smooth systems. We then examine situations where application of the standard theory falls short and illustrate how it can be extended. Specific examples are given to non-smooth systems with applications to the Izhikevich neuron. We then introduce the idea of isostable reduction to explore behaviours that the weak coupling paradigm cannot explain. In an additional example, we show how bifurcations that change the stability of phase-locked solutions in a pair of identical coupled neurons can be understood using the notion of isostable reduction. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Nonlinear phase coupling functions: a numerical study

Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Phase Models Beyond Weak Coupling

We use the theory of isostable reduction to incorporate higher order effects that are lost in the first order phase reduction of coupled oscillators. We apply this theory to weakly coupled complex Ginzburg-Landau equations, a pair of conductance-based neural models, and finally to a short derivation of the Kuramoto-Sivashinsky equations. Numerical and analytical examples illustrate bifurcations occurring in coupled oscillator networks that can cause standard phase-reduction methods to fail.

Synchronization of Electrically Coupled Resonate-and-Fire Neurons

Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with electrical coupling show a diversity of post-spike subthreshold fluctuations, often linked to subthreshold resonance, which are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with damped subthreshold oscillations and a range of post-spike voltage dynamics. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuous post-spike reset rule. We find that both spikes and subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau potential. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs.

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

A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

Isostable reduction of oscillators with piecewise smooth dynamics and complex Floquet multipliers

Phase-amplitude reduction is a widely applied technique in the study of limit cycle oscillators with the ability to represent a complicated and high-dimensional dynamical system in a more analytically tractable set of coordinates. Recent work has focused on the use of isostable coordinates, which characterize the transient decay of solutions toward a periodic orbit, and can ultimately be used to increase the accuracy of these reduced models. The breadth of systems to which this phase-amplitude reduction strategy can be applied, however, is still rather limited. In this work, the theory of phase-amplitude reduction using isostable coordinates is further developed to accommodate a broader set of dynamical systems. In the first part, limit cycles of piecewise smooth dynamical systems are considered and strategies are developed to compute the associated reduced equations. In the second part, the notion of isostable coordinates for complex-valued Floquet multipliers is introduced, resulting in one phaselike coordinate and one amplitudelike coordinate for each pair of complex conjugate Floquet multipliers. Examples are given with relevance to piecewise smooth representations of excitable cardiomyocytes and the relationship between the reduced coordinate system and the emergence of cardiac alternans is discussed. Also, phase-amplitude reduction is implemented for a chaotic, externally forced pendulum with complex Floquet multipliers and a resulting control strategy for the stabilization of its periodic solution is investigated.

Numerical phase reduction beyond the first order approximation

We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach.