A dynamic generative model can extract interpretable oscillatory components from multichannel neurophysiological recordings

Modern neurophysiological recordings are performed using multichannel sensor arrays that are able to record activity in an increasingly high number of channels numbering in the 100s to 1000s. Often, underlying lower-dimensional patterns of activity are responsible for the observed dynamics, but these representations are difficult to reliably identify using existing methods that attempt to summarize multivariate relationships in a post hoc manner from univariate analyses or using current blind source separation methods. While such methods can reveal appealing patterns of activity, determining the number of components to include, assessing their statistical significance, and interpreting them requires extensive manual intervention and subjective judgment in practice. These difficulties with component selection and interpretation occur in large part because these methods lack a generative model for the underlying spatio-temporal dynamics. Here, we describe a novel component analysis method anchored by a generative model where each source is described by a bio-physically inspired state-space representation. The parameters governing this representation readily capture the oscillatory temporal dynamics of the components, so we refer to it as oscillation component analysis. These parameters - the oscillatory properties, the component mixing weights at the sensors, and the number of oscillations - all are inferred in a data-driven fashion within a Bayesian framework employing an instance of the expectation maximization algorithm. We analyze high-dimensional electroencephalography and magnetoencephalography recordings from human studies to illustrate the potential utility of this method for neuroscience data.

Different Methods to Estimate the Phase of Neural Rhythms Agree But Only During Times of Low Uncertainty

Rhythms are a common feature of brain activity. Across different types of rhythms, the phase has been proposed to have functional consequences, thus requiring its accurate specification from noisy data. Phase is conventionally specified using techniques that presume a frequency band-limited rhythm. However, in practice, observed brain rhythms are typically nonsinusoidal and amplitude modulated. How these features impact methods to estimate phase remains unclear. To address this, we consider three phase estimation methods, each with different underlying assumptions about the rhythm. We apply these methods to rhythms simulated with different generative mechanisms and demonstrate inconsistency in phase estimates across the different methods. We propose two improvements to the practice of phase estimation: (1) estimating confidence in the phase estimate, and (2) examining the consistency of phase estimates between two (or more) methods.

Different methods to estimate the phase of neural rhythms agree, but only during times of low uncertainty

Rhythms are a common feature of brain activity. Across different types of rhythms, the phase has been proposed to have functional consequences, thus requiring its accurate specification from noisy data. Phase is conventionally specified using techniques that presume a frequency band-limited rhythm. However, in practice, observed brain rhythms are typically non-sinusoidal and amplitude modulated. How these features impact methods to estimate phase remains unclear. To address this, we consider three phase estimation methods, each with different underlying assumptions about the rhythm. We apply these methods to rhythms simulated with different generative mechanisms and demonstrate inconsistency in phase estimates across the different methods. We propose two improvements to the practice of phase estimation: (1) estimating confidence in the phase estimate, and (2) examining the consistency of phase estimates between two (or more) methods.

Gait Entrainment to Torque Pulses from a Hip Exoskeleton Robot

Robot-aided locomotor rehabilitation has proven challenging. To facilitate progress, it is important to first understand the neuro-mechanical dynamics and control of unimpaired human locomotion. Our previous studies found that human gait entrained to periodic torque pulses at the ankle when the pulse period was close to preferred stride duration. Moreover, synchronized gait exhibited constant phase relation with the pulses so that the robot provided mechanical assistance. To test the generality of mechanical gait entrainment, this study characterized unimpaired human subjects' responses to periodic torque pulses during overground walking. The intervention was applied by a hip exoskeleton robot, Samsung GEMS-H. Gait entrainment was assessed based on the time-course of the phase at which torque pulses occurred within each stride. Experiments were conducted for two consecutive days to evaluate whether the second day elicited more entrainment. Whether entrainment was affected by the difference between pulse period and preferred stride duration was also assessed. Results indicated that the intervention evoked gait entrainment that occurred more often when the period of perturbation was closer to subjects' preferred stride duration, but the difference between consecutive days was insignificant. Entrainment was accompanied by convergence of pulse phase to a similar value across all conditions, where the robot maximized mechanical assistance. Clear evidence of motor adaptation indicated the potential of the intervention for rehabilitation. This study quantified important aspects of the nonlinear neuro-mechanical dynamics underlying unimpaired human walking, which will inform the development of effective approaches to robot-aided locomotor rehabilitation, exploiting natural dynamics in a minimally-encumbering way.

A state space modeling approach to real-time phase estimation

Brain rhythms have been proposed to facilitate brain function, with an especially important role attributed to the phase of low-frequency rhythms. Understanding the role of phase in neural function requires interventions that perturb neural activity at a target phase, necessitating estimation of phase in real-time. Current methods for real-time phase estimation rely on bandpass filtering, which assumes narrowband signals and couples the signal and noise in the phase estimate, adding noise to the phase and impairing detections of relationships between phase and behavior. To address this, we propose a state space phase estimator for real-time tracking of phase. By tracking the analytic signal as a latent state, this framework avoids the requirement of bandpass filtering, separately models the signal and the noise, accounts for rhythmic confounds, and provides credible intervals for the phase estimate. We demonstrate in simulations that the state space phase estimator outperforms current state-of-the-art real-time methods in the contexts of common confounds such as broadband rhythms, phase resets, and co-occurring rhythms. Finally, we show applications of this approach to in vivo data. The method is available as a ready-to-use plug-in for the Open Ephys acquisition system, making it widely available for use in experiments.

Intrinsic Synchronization Analysis of Brain Activity in Obsessive-compulsive Disorders

Obsessive-compulsive disorder (OCD) is one of the neuropsychiatric disorders qualified by intrusive and iterative annoying thoughts and mental attitudes that are activated by these thoughts. In recent studies, advanced signal processing techniques have been favored to diagnose OCD. This research suggests four different measurements; intrinsic phase-locked value, intrinsic coherence, intrinsic synchronization likelihood, and intrinsic visibility graph similarity that quantifies the synchronization level and complexity in electroencephalography (EEG) signals. This intrinsic synchronization is achieved by utilizing Multivariate Empirical Mode Decomposition (MEMD), a data-driven method that resolves nonlinear and nonstationary data into their intrinsic mode functions. Our intrinsic technique in this study demonstrates that MEMD-based synchronization analysis gives us much more detailed knowledge rather than utilizing the synchronization method alone. Furthermore, the nonlinear synchronization method presents more consistent results considering OCD heterogeneity. Statistical evaluation using sample [Formula: see text]-test and [Formula: see text]-test has shown the significance of such new methodology.

The equal combination synchronization of a class of chaotic systems with discontinuous output

This paper investigates the equal combination synchronization of a class of chaotic systems. The chaotic systems are assumed that only the output state variable is available and the output may be discontinuous state variable. By constructing proper observers, some novel criteria for the equal combination synchronization are proposed. The Lorenz chaotic system is taken as an example to demonstrate the efficiency of the proposed approach.

Information flow in heterogeneously interacting systems

Motivated by studies on the dynamics of heterogeneously interacting systems in neocortical neural networks, we studied heterogeneously-coupled chaotic systems. We used information-theoretic measures to investigate directions of information flow in heterogeneously coupled Rössler systems, which we selected as a typical chaotic system. In bi-directionally coupled systems, spontaneous and irregular switchings of the phase difference between two chaotic oscillators were observed. The direction of information transmission spontaneously switched in an intermittent manner, depending on the phase difference between the two systems. When two further oscillatory inputs are added to the coupled systems, this system dynamically selects one of the two inputs by synchronizing, selection depending on the internal phase differences between the two systems. These results indicate that the effective direction of information transmission dynamically changes, induced by a switching of phase differences between the two systems.

Inhomogeneous stationary and oscillatory regimes in coupled chaotic oscillators

The dynamics of linearly coupled identical Lorenz and Pikovsky-Rabinovich oscillators are explored numerically and theoretically. We concentrate on the study of inhomogeneous stable steady states ("oscillation death (OD)" phenomenon) and accompanying periodic and chaotic regimes that emerge at an appropriate choice of the coupling matrix. The parameters, for which OD occurs, are determined by stability analysis of the chosen steady state. Three model-specific types of transitions to and from OD are observed: (1) a sharp transition to OD from a nonsymmetric chaotic attractor containing random intervals of synchronous chaos; (2) transition to OD from the symmetry-breaking chaotic regime created by negative coupling; (3) supercritical bifurcation of OD into inhomogeneous limit cycles and further evolution of the system to inhomogeneous chaotic regimes that coexist with complete synchronous chaos. These results may fill a gap in the understanding of the mechanism of OD in coupled chaotic systems.

Relaying phase synchrony in chaotic oscillator chains

We study the manner in which the effect of an external drive is transmitted through mutually coupled response systems by examining the phase synchrony between the drive and the response. Two different coupling schemes are used. Homogeneous couplings are via the same variables while heterogeneous couplings are through different variables. With the latter scenario, synchronization regimes are truncated with an increasing number of mutually coupled oscillators in contrast to homogeneous coupling schemes. Our results are illustrated for systems of coupled chaotic Rössler oscillators.

Nearest neighbors, phase tubes, and generalized synchronization

In this paper we report on the necessity of the refinement of the concept of generalized chaotic synchronization. We show that the state vectors of the interacting chaotic systems being in the generalized synchronization regime are related to each other by the functional, but not the functional relation as it was assumed until now. We propose the phase tube approach explaining the essence of generalized synchronization and allowing the detection and the study of this regime in many relevant physical circumstances. The finding discussed in this Brief Report provides great potential for different applications.

Routes to complex dynamics in a ring of unidirectionally coupled systems

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

Quasiperiodic forcing of coupled chaotic systems

We study the manner in which the effect of quasiperiodic modulation is transmitted in a coupled nonlinear dynamical system. A system of Rössler oscillators is considered, one of which is subject to driving, and the dynamics of other oscillators which are, in effect, indirectly forced is observed. Strange nonchaotic dynamics is known to arise only in quasiperiodically driven systems, and thus the transmitted effect is apparent when such motion is seen in subsystems that are not directly modulated. We also find instances of imperfect phase synchronization with forcing, where the system transits from one phase synchronized state to another, with arbitrary phase slips. The stability of phase synchrony for arbitrary initial conditions with identical forcing is observed as a general property of strange nonchaotic motion.

Phase synchronization analysis by assessment of the phase difference gradient

Phase synchronization is an important phenomenon of nonlinear dynamics and has recently received much scientific attention. In this work a method for identifying phase synchronization epochs is described which focuses on estimating the gradient of segments of the generalized phase differences between phase slips in an experimental time series. In phase synchronized systems, there should be a zero gradient of the generalized phase differences even if the systems are contaminated by noise. A method which tests if the gradient of the generalized phase difference is statistically different from zero is reported. The method has been validated by numerical studies on model systems and by comparing the results to those published previously. The method is applied to cardiorespiratory time series from a human volunteer measured in clinical settings and compared to synchrogram analysis for the same data. Potential problems with synchrogram analysis of experimental data are discussed.

Detection of synchronization between chaotic signals: An adaptive similarity-based approach

We present an adaptive similarity-based approach to detect generalized synchronization (GS) with n:m phase synchronization (PS), where n and m are integers and one of them is 1. This approach is based on the similarity index (SI) and Gaussian mixture model with the minimum description length criterion. The clustering method, which is shown to be superior to the closeness and connectivity of a continuous function, is employed in this study to detect the existence of GS with n:m PS. We conducted a computer simulation and a finger-lifting experiment to illustrate the effectiveness of the proposed method. In the simulation of a Rössler-Lorenz system, our method outperformed the conventional SI, and GS with 2:1 PS within the coupled system was found. In the experiment of self-paced finger-lifting movement, cortico-muscular GS with 1:2 and 1:3 PS was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area. The GS with n:m PS ( n or m=1 ) has been simultaneously resolved from both simulation and experiment. The proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science.

Intraburst versus interburst locking in networks of driven nonidentical oscillators

We investigate the effect of common periodic drive applied to mean-field coupled oscillators and observe a specific realization of synchronization for particular ranges of drive frequency. This synchronization occurs when the phase difference variability between a pair of oscillators on a given cycle is larger than that between consecutive cycles. This synchrony may have implications for neural systems, in which case the apparent locking between neurons based on the magnitude of their interspike intervals may not be consistent with their dynamical locking.

Two types of phase synchronization destruction

In this paper we report that there are two different types of destruction of the phase synchronization (PS) regime of chaotic oscillators depending on the parameter mismatch as well as in the case of the classical synchronization of periodic oscillators. When the parameter mismatch of the interacting chaotic oscillators is small enough, the PS breaking takes place without the destruction of the phase coherence of chaotic attractors of oscillators. Alternatively, due to the large frequency detuning, the PS breaking is accomplished by loss of the phase coherence of the chaotic attractors.

A novel approach to the detection of synchronisation in EEG based on empirical mode decomposition

Transient neural assemblies mediated by synchrony in particular frequency ranges are thought to underlie cognition. We propose a new approach to their detection, using empirical mode decomposition (EMD), a data-driven approach removing the need for arbitrary bandpass filter cut-offs. Phase locking is sought between modes. We explore the features of EMD, including making a quantitative assessment of its ability to preserve phase content of signals, and proceed to develop a statistical framework with which to assess synchrony episodes. Furthermore, we propose a new approach to ensure signal decomposition using EMD. We adapt the Hilbert spectrum to a time-frequency representation of phase locking and are able to locate synchrony successfully in time and frequency between synthetic signals reminiscent of EEG. We compare our approach, which we call EMD phase locking analysis (EMDPL) with existing methods and show it to offer improved time-frequency localisation of synchrony.

Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization

A new type of intermittent behavior is described to occur near the boundary of the phase synchronization regime of coupled chaotic oscillators. This mechanism, called ring intermittency, arises for sufficiently high initial mismatches in the frequencies of the two coupled systems. The laws for both the distribution and the mean length of the laminar phases versus the coupling strength are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown. We discuss how this mechanism is expected to take place in other relevant physical circumstances.

Information theoretic approach to quantify complete and phase synchronization of chaos

Based on an information theory approach we suggest a quantitative characteristic for evaluating the degree of chaotic synchronization. The proposed characteristic is tested for the cases of complete and phase synchronization of chaos. It is shown that this characteristic is stable with respect to the influence of small noise and nonlinear signal distortion.

Identification of coupling direction: application to cardiorespiratory interaction

We consider the problem of experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. We further develop the method introduced by Rosenblum and Pikovsky [Phys. Rev. E 64, 045202 (2001)], suggesting an alternative approach. Next, we consider another framework for identification of directionality, based on the idea of mutual predictability. Our algorithms provide directionality index that shows whether the coupling between the oscillators is unidirectional or bidirectional, and quantifies the asymmetry of bidirectional coupling. We demonstrate the efficiency of three different algorithms in determination of directionality index from short and noisy data. These techniques are then applied to analysis of cardiorespiratory interaction in healthy infants. The results reveal that the direction of coupling between cardiovascular and respiratory systems varies with the age within the first 6 months of life. We find a tendency to change from nearly symmetric bidirectional interaction to nearly unidirectional one (from respiration to the cardiovascular system).

Characterizing instantaneous phase relationships in whole-brain fMRI activation data

Typically, fMRI data is processed in the time domain with linear methods such as regression and correlation analysis. We propose that the theory of phase synchronization may be used to more completely understand the dynamics of interacting systems, and can be applied to fMRI data as a novel method of detecting activation. Generalized synchronization is a phenomenon that occurs when there is a nonlinear functional relationship present between two or more coupled, oscillatory systems, whereas phase synchronization is defined as the locking of the phases while the amplitudes may vary. In this study, we developed an application of phase synchronization analysis that is appropriate for fMRI data, in which the phase locking condition is investigated between a voxel time series and the reference function of the task performed. A synchronization index is calculated to quantify the level of phase locking, and a nonparametric permutation test is used to determine the statistical significance of the results. We performed the phase synchronization analysis on the data from five volunteers for an event-related finger-tapping task. Functional maps were created that provide information on the interrelations between the instantaneous phases of the reference function and the voxel time series in a whole-brain fMRI activation data set. We conclude that this method of analysis is useful for revealing additional information on the complex nature of the fMRI time series.

Detecting direction of coupling in interacting oscillators

We propose a method for experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. The technique is applicable to both noisy and chaotic systems that can be nonidentical or even structurally different. We introduce an index that quantifies the asymmetry in coupling.

Phase synchronization and suppression of chaos through intermittency in forcing of an electrochemical oscillator

External periodic forcing was applied to a chaotic chemical oscillator in experiments on the electrodissolution of Ni in sulfuric acid solution. The amplitude and the frequency (Omega) of the forcing signal were varied in a region around Omega=omega(0), where omega(0) is the frequency of the unforced signal. Phase synchronization occurred with increase in the amplitude of the forcing. For Omega/omega(0) near 1 the signal remained chaotic after the transition to the phase-locked state; for Omega/omega(0) somewhat farther from 1 the transition was to a periodic state via intermittency. The experimental results are supported by numerical simulations using a general model for electrochemical oscillations.

Enhanced phase synchrony in the electroencephalograph gamma band for musicians while listening to music

Multichannel electroencephalograph signals from two broad groups, 10 musicians and 10 nonmusicians, recorded in different states (in resting states or no task condition, with eyes opened and eyes closed, and with two musical tasks, listening to two different pieces of music) were studied. Degrees of phase synchrony in various frequency bands were assessed. No differences in the degree of synchronization in any frequency band were found between the two groups in resting conditions. Yet, while listening to music, significant increases of synchronization were found only in the gamma-frequency range (>30 Hz) over large cortical areas for the group of musicians. This high degree of synchronization elicited by music in the group of musicians might be due to their ability to host long-term memory representations of music and mediate access to these stored representations.

Phase synchronization with type-II intermittency in chaotic oscillators

We study the phase synchronization (PS) with type-II intermittency showing +/-2pi irregular phase jumping behavior before the PS transition occurs in a system of two coupled hyperchaotic Rossler oscillators. The behavior is understood as a stochastic hopping of an overdamped particle in a potential which has 2pi-periodic minima. We characterize it as type-II intermittency with external noise through the return map analysis. In epsilon(t)<epsilon<epsilon(c) (where epsilon(t) is the bifurcation point of type-II intermittency and epsilon(c) is the PS transition point in coupling strength parameter space), the average length of the time interval between two successive jumps follows <l> approximately exp(|epsilon(t)-epsilon|(2)), which agrees well with the scaling law obtained from the Fokker-Planck equation.

Synchronization of noisy systems by stochastic signals

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.

Synchronization in the human cardiorespiratory system

We investigate synchronization between cardiovascular and respiratory systems in healthy humans under free-running conditions. For this aim we analyze nonstationary irregular bivariate data, namely, electrocardiograms and measurements of respiratory flow. We briefly discuss a statistical approach to synchronization in noisy and chaotic systems and illustrate it with numerical examples; effects of phase and frequency locking are considered. Next, we present and discuss methods suitable for the detection of hidden synchronous epochs from such data. The analysis of the experimental records reveals synchronous regimes of different orders n:m and transitions between them; the physiological significance of this finding is discussed.