Detecting the harmonics of oscillations with time-variable frequencies

A Robust Deep Neural Network for Rolling Element Fault Diagnosis under Various Operating and Noisy Conditions

This study proposes a new intelligent diagnostic method for bearing faults in rotating machinery. The method uses a combination of nonlinear mode decomposition based on the improved fast kurtogram, gramian angular field, and convolutional neural network to detect the bearing state of rotating machinery. The nonlinear mode decomposition based on the improved fast kurtogram inherits the advantages of the original algorithm while improving the computational efficiency and signal-to-noise ratio. The gramian angular field can construct a two-dimensional image without destroying the time relationship of the signal. Therefore, the proposed method can perform fault diagnosis on rotating machinery under complex operating conditions. The proposed method is verified on the Paderborn dataset under heavy noise and multiple operating conditions to evaluate its effectiveness. Experimental results show that the proposed model outperforms wavelet denoising and the traditional adaptive decomposition method. The proposed model achieves over 99.6% accuracy in all four operating conditions provided by this dataset, and 93.8% accuracy in a strong noise environment with a signal-to-noise ratio of -4 dB.

Multiscale Time-resolved Analysis Reveals Remaining Behavioral Rhythms in Mice Without Canonical Circadian Clocks

Circadian rhythms are internal processes repeating approximately every 24 hours in living organisms. The dominant circadian pacemaker is synchronized to the environmental light-dark cycle. Other circadian pacemakers, which can have noncanonical circadian mechanisms, are revealed by arousing stimuli, such as scheduled feeding, palatable meals and running wheel access, or methamphetamine administration. Organisms also have ultradian rhythms, which have periods shorter than circadian rhythms. However, the biological mechanism, origin, and functional significance of ultradian rhythms are not well-elucidated. The dominant circadian rhythm often masks ultradian rhythms; therefore, we disabled the canonical circadian clock of mice by knocking out Per1/2/3 genes, where Per1 and Per2 are essential components of the mammalian light-sensitive circadian mechanism. Furthermore, we recorded wheel-running activity every minute under constant darkness for 272 days. We then investigated rhythmic components in the absence of external influences, applying unique multiscale time-resolved methods to analyze the oscillatory dynamics with time-varying frequencies. We found four rhythmic components with periods of ∼17 h, ∼8 h, ∼4 h, and ∼20 min. When the ∼17-h rhythm was prominent, the ∼8-h rhythm was of low amplitude. This phenomenon occurred periodically approximately every 2-3 weeks. We found that the ∼4-h and ∼20-min rhythms were harmonics of the ∼8-h rhythm. Coupling analysis of the ridge-extracted instantaneous frequencies revealed strong and stable phase coupling from the slower oscillations (∼17, ∼8, and ∼4 h) to the faster oscillations (∼20 min), and weak and less stable phase coupling in the reverse direction and between the slower oscillations. Together, this study elucidated the relationship between the oscillators in the absence of the canonical circadian clock, which is critical for understanding their functional significance. These studies are essential as disruption of circadian rhythms contributes to diseases, such as cancer and obesity, as well as mood disorders.

Suppression of Motion Artifacts Caused by Temporally Recurring Tracer Distributions in Multi-Patch Magnetic Particle Imaging

Magnetic particle imaging is a tracer based imaging technique to determine the spatial distribution of superparamagnetic iron oxide nanoparticles with a high spatial and temporal resolution. Due to physiological constraints, the imaging volume is restricted in size and larger volumes are covered by shifting object and imaging volume relative to each other. This results in reduced temporal resolution, which can lead to motion artifacts when imaging dynamic tracer distributions. A common source of such dynamic distributions are cardiac and respiratory motion in in-vivo experiments, which are in good approximation periodic. We present a raw data processing technique that combines data snippets into virtual frames corresponding to a specific state of the dynamic motion. The technique is evaluated on the basis of measurement data obtained from a rotational phantom at two different rotational frequencies. These frequencies are determined from the raw data without reconstruction and without an additional navigator signal. The reconstructed images give reasonable representations of the rotational phantom frozen in several different states of motion while motion artifacts are suppressed.

Unraveling nonlinear electrophysiologic processes in the human visual system with full dimension spectral analysis

Natural sensory signals have nonlinear structures dynamically composed of the carrier frequencies and the variation of the amplitude (i.e., envelope). How the human brain processes the envelope information is still poorly understood, largely due to the conventional analysis failing to quantify it directly. Here, we used a recently developed method, Holo-Hilbert spectral analysis, and steady-state visually evoked potential collected using electroencephalography (EEG) recordings to investigate how the human visual system processes the envelope of amplitude-modulated signals, in this case with a 14 Hz carrier and a 2 Hz envelope. The EEG results demonstrated that in addition to the fundamental stimulus frequencies, 4 Hz amplitude modulation residing in 14 Hz carrier and a broad range of carrier frequencies covering from 8 to 32 Hz modulated by 2 Hz amplitude modulation are also found in the two-dimensional frequency spectrum, which have not yet been recognized before. The envelope of the stimulus is also found to dominantly modulate the response to the incoming signal. The findings thus reveal that the electrophysiological response to amplitude-modulated stimuli is more complex than could be revealed by, for example, Fourier analysis. This highlights the dynamics of neural processes in the visual system.

Variable Preloading Force in an Archetypal Oscillator

In this paper, the influence of the time variable preloading force on the vibration of an archetypal oscillator is investigated. The oscillator is modeled as a slider-string system which is mathematically described with a second order nonlinear differential equation with time variable parameters. An approximate procedure for solving the equation is introduced. It is based on the exact solution of the pure nonlinear equation in the form of the Ateb function. The obtained result gives the vibration amplitude and phase variation of the oscillator depending on the preloading force variation. Based on this result, the procedure for regulation of the preloading force as the function of the required vibration amplitude decrease is developed. It is concluded that the preloading force may be used as a control parameter of the oscillator.

Nonlinear mode decomposition: a noise-robust, adaptive decomposition method

The signals emanating from complex systems are usually composed of a mixture of different oscillations which, for a reliable analysis, should be separated from each other and from the inevitable background of noise. Here we introduce an adaptive decomposition tool-nonlinear mode decomposition (NMD)-which decomposes a given signal into a set of physically meaningful oscillations for any wave form, simultaneously removing the noise. NMD is based on the powerful combination of time-frequency analysis techniques-which, together with the adaptive choice of their parameters, make it extremely noise robust-and surrogate data tests used to identify interdependent oscillations and to distinguish deterministic from random activity. We illustrate the application of NMD to both simulated and real signals and demonstrate its qualitative and quantitative superiority over other approaches, such as (ensemble) empirical mode decomposition, Karhunen-Loève expansion, and independent component analysis. We point out that NMD is likely to be applicable and useful in many different areas of research, such as geophysics, finance, and the life sciences. The necessary matlab codes for running NMD are freely available for download.

Generalized chronotaxic systems: time-dependent oscillatory dynamics stable under continuous perturbation

Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in Phys. Rev. Lett. 111, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincaré oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems.

Inverse approach to chronotaxic systems for single-variable time series

Following the development of a new class of self-sustained oscillators with a time-varying but stable frequency, the inverse approach to these systems is now formulated. We show how observed data arranged in a single-variable time series can be used to recognize such systems. This approach makes use of time-frequency domain information using the wavelet transform as well as the recently developed method of Bayesian-based inference. In addition, a set of methods, named phase fluctuation analysis, is introduced to detect the defining properties of the new class of systems by directly analyzing the statistics of the observed perturbations.We apply these methods to numerical examples but also elaborate further on the cardiac system.

Chronotaxic systems with separable amplitude and phase dynamics

Until recently, deterministic nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies were not recognized as such and have often been mistreated as stochastic. These systems, named chronotaxic, were introduced in Phys. Rev. Lett. 111, 024101 (2013). In contrast to conventional limit cycle models of self-sustained oscillators, these systems posses a time-dependent point attractor or steady state. This allows oscillations with time-varying frequencies to resist perturbations, a phenomenon which is ubiquitous in living systems. In this work a detailed theory of chronotaxic systems is presented, specifically in the case of separable amplitude and phase dynamics. The theory is extended by the introduction of chronotaxic amplitude dynamics. The wide applicability of chronotaxic systems to a range of fields from biological and condensed matter systems to robotics and control theory is discussed.

Chronotaxic systems: a new class of self-sustained nonautonomous oscillators

Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks.

Characterizing an ensemble of interacting oscillators: the mean-field variability index

We introduce a way of characterizing an ensemble of interacting oscillators in terms of their mean-field variability index κ, a dimensionless parameter defined as the variance of the oscillators' mean field r divided by the mean square of r. Based on the assumption that the overall mean field is the sum of a very large number of oscillators, each giving a small contribution to the total signal, we show that κ depends on the mutual interactions between the oscillators, independently of their number or spectral properties. For purely random phasors, or a noninteracting ensemble of oscillators, κ converges on 0.215. Interactions push κ in different directions: lower where there is interoscillator phase coherence, tending to zero for complete phase synchronization, or higher for amplitude synchronization or intermittent synchronization. We calculate κ for several different cases to illustrate its utility, using both numerically simulated data and electroencephalograph signals from the brains of human subjects while awake, while anesthetized, and while undergoing an epileptic fit.

Local cooling reduces skin ischemia under surface pressure in rats: an assessment by wavelet analysis of laser Doppler blood flow oscillations

The objectives of this study were to investigate the effects of local cooling on skin blood flow response to prolonged surface pressure and to identify associated physiological controls mediating these responses using the wavelet analysis of blood flow oscillations in rats. Twelve Sprague-Dawley rats were randomly assigned to three protocols, including pressure with local cooling (Δt = -10 °C), pressure with local heating (Δt = 10 °C) and pressure without temperature changes. Pressure of 700 mmHg was applied to the right trochanter area of rats for 3 h. Skin blood flow was measured using laser Doppler flowmetry. The 3 h loading period was divided into non-overlapping 30 min epochs for the analysis of the changes of skin blood flow oscillations using wavelet spectral analysis. The wavelet amplitudes and powers of three frequencies (metabolic, neurogenic and myogenic) of skin blood flow oscillations were calculated. The results showed that after an initial loading period of 30 min, skin blood flow continually decreased under the conditions of pressure with heating and of pressure without temperature changes, but maintained stable under the condition of pressure with cooling. Wavelet analysis revealed that stable skin blood flow under pressure with cooling was attributed to changes in the metabolic and myogenic frequencies. This study demonstrates that local cooling may be useful for reducing ischemia of weight-bearing soft tissues that prevents pressure ulcers.

Testing for time-localized coherence in bivariate data

We present a method for the testing of significance when evaluating the coherence of two oscillatory time series that may have variable amplitude and frequency. It is based on evaluating the self-correlations of the time series. We demonstrate our approach by the application of wavelet-based coherence measures to artificial and physiological examples. Because coherence measures of this kind are strongly biased by the spectral characteristics of the time series, we evaluate significance by estimation of the characteristics of the distribution of values that may occur due to chance associations in the data. The expectation value and standard deviation of this distribution are shown to depend on the autocorrelations and higher order statistics of the data. Where the coherence value falls outside this distribution, we may conclude that there is a causal relationship between the signals regardless of their spectral similarities or differences.