Order error in the calculation of continuous relative phase

The aims of this study were to demonstrate “order error” in the calculation of continuous relative phase (CRP) and to suggest two alternative methods—(i) constructing phase-plane portraits by plotting position over velocity; and (ii), the Hilbert transform—to rectify it. Order error is the change of CRP order between two degrees of freedom (e.g., body segments) when using the conventional method of constructing phase-plane portraits (i.e., velocity over position). Both sinusoidal and non-sinusoidal simulated signals as well as signals from human movement kinematics were used to investigate order error and the performance of the two alternative methods. Both methods have been shown to lead to correct results for simulated sinusoidal and non-sinusoidal signals. For human movement data, however, the Hilbert transform is superior for calculating CRP.     

Quantifying Neural Oscillatory Synchronization: A Comparison between Spectral Coherence and Phase-Locking Value Approaches

Synchronization or phase-locking between oscillating neuronal groups is considered to be important for coordination of information among cortical networks. Spectral coherence is a commonly used approach to quantify phase locking between neural signals. We systematically explored the validity of spectral coherence measures for quantifying synchronization among neural oscillators. To that aim, we simulated coupled oscillatory signals that exhibited synchronization dynamics using an abstract phase-oscillator model as well as interacting gamma-generating spiking neural networks. We found that, within a large parameter range, the spectral coherence measure deviated substantially from the expected phase-locking. Moreover, spectral coherence did not converge to the expected value with increasing signal-to-noise ratio. We found that spectral coherence particularly failed when oscillators were in the partially (intermittent) synchronized state, which we expect to be the most likely state for neural synchronization. The failure was due to the fast frequency and amplitude changes induced by synchronization forces. We then investigated whether spectral coherence reflected the information flow among networks measured by transfer entropy (TE) of spike trains. We found that spectral coherence failed to robustly reflect changes in synchrony-mediated information flow between neural networks in many instances. As an alternative approach we explored a phase-locking value (PLV) method based on the reconstruction of the instantaneous phase. As one approach for reconstructing instantaneous phase, we used the Hilbert Transform (HT) preceded by Singular Spectrum Decomposition (SSD) of the signal. PLV estimates have broad applicability as they do not rely on stationarity, and, unlike spectral coherence, they enable more accurate estimations of oscillatory synchronization across a wide range of different synchronization regimes, and better tracking of synchronization-mediated information flow among networks.     

Changes in movement variability and task performance during a fatiguing repetitive pointing task

Changes in neuromuscular strategies employed with fatigue during multi-joint movements are still poorly understood. Studies have shown that motor variability of individual joints increases when performing upper limb tasks to fatigue, while movement parameters related to the task goal remain constant. However, how the inter-limb coordination and its variability change during specific movement phases with fatigue is still unclear. The aim of this study was to assess the effects of neck-shoulder fatigue on shoulder and elbow kinematic variabilities, shoulder-elbow coordination and its variability, and endpoint characteristics during different phases of a forward pointing movement. Nineteen healthy young adults continuously performed a repetitive pointing task until fatigue (Borg rating of 8/10). Changes in elbow-shoulder coordination through the movement were assessed using the continuous relative phase and statistical nonparametric mapping methods. At the end of the task, muscle fatigue was evidenced by significant increases in anterior deltoid (+13%) and biceps brachii (+30%) activity. Shoulder horizontal abduction, elbow flexion variability and shoulder-elbow coordination variability were increased with fatigue at different moments of the movement cycle (shoulder: during the first 17% and most of the second half movement, elbow: from 73% to 91%, coordination: almost the whole movement). However, movement timing errors and endpoint spatial variability were mostly preserved, even with fatigue. We showed that increased variability with fatigue is not only observed in the fatigued joint (shoulder), but also in the elbow and shoulder-elbow coordination, and may have a goal of preserving global task performance.     

Variability in kinematic coupling assessed by vector coding and continuous relative phase

Variability in the spatio-temporal coordination of human movement kinematics is often assessed by vector coding and continuous relative phase (CRP). To facilitate appropriate comparisons between the findings of studies that have used different techniques to assess variability, the purposes of this study were: (1) to determine if both vector coding and CRP behave according to dynamical systems theories on variability and state space transitions; and (2) to determine if trends in coordination variability during movement are consistent when using either vector coding or CRP. We present both a theoretical case (the Lorenz Attractor) and two experimental cases (rearfoot–forefoot coupling during overground walking for 22 subjects; the effect of treadmill speed on thigh-leg coupling for five subjects). In the theoretical case, variability quantified by CRP agreed with dynamical systems theory on state space transitions more so than variability quantified by vector coding. In experimental cases, this distinction was less clear, although CRP appeared to be a more conservative metric for variability. The magnitudes (all p<0.001) and timings (all p<0.04) of peaks in variability during the stance phase of overground walking depended on whether vector coding or CRP was used for two couplings. Similar distinctions were observed for peaks during the stride cycle of treadmill locomotion (all effect sizes >2.8). However, changes in the average variability during the stride cycle as speed increased were consistent for both methods (all effect sizes <0.2). The results suggest that comparisons between the findings of studies that have quantified variability using CRP and those that have used vector coding should be made with caution.     

Dynamic cerebral autoregulation under sinusoidal gravitational loading

Dynamic cerebral autoregulation (CA) has been studied previously using spectral analysis of oscillations in arterial blood pressure (ABP) and cerebral blood flow velocity (CBFV). The dynamics of the CA can be modeled as a high-pass filter. The purpose of this study is to compare CA of blood pressure oscillations induced by gravitational loading to CA during resting conditions. We subjected twelve healthy subjects to repeated sinusoidal head-up (0 degrees - 60 degrees) tilts at several set frequencies (0.07 to 0.25 Hz) on a computer controlled tilt table while we recorded ABP (Finapres) and CBFV (transcranial Doppler ultrasound). We fitted the data sets to a high-pass filter model and computed an average time constant (T). Our results show similar phase leads of CBFV to ABPbrain in the rest recording and in sinusoidal tilting, in the studied frequency range. The transfer function gain of the resting spectra increased with increasing frequency, the gain of the tilting spectra did not. Fitting the phase responses of both data sets to a high pass filter model yielded similar time constants.     

Accuracy, stability, and corrective behavior in a visuomotor tracking task: a preliminary study

Visuomotor tracking tasks have been used to elucidate the underlying mechanisms that allow for the coordination of a movement to an environmental event. The main purpose of the present study was to examine the relationship between accuracy and stability of tracking performance and the amount of corrective movements that emerge for various coordination patterns in a unimanual visuomotor tracking task. Participants (N = 6) produced rhythmic elbow flexion-extension motions and were required to track an external sinusoidal signal at five different relative phases, 0°, 45°, 90°, 135°, and 180°. Differential accuracy and stability were found among the five tracking patterns with the 0° relative phase pattern being the most accurate and stable pattern. Corrective movements were correlated with changes in accuracy only for the 0° relative phase pattern, with more corrections emerging for less accurate performance. The amount of corrective movements decreased as the stability of tracking performance increased for the 0°, 45°, and 135° patterns. For the 90° and 180° tracking patterns, the amount of corrective movements was not correlated with pattern accuracy or pattern stability. The results demonstrate that corrective behaviors are an important motor process in maintaining the stability of stable perception-action coordination patterns, while offering little benefit for unstable perception-action patterns.     

Resonant frequencies of arms and legs identify different walking patterns

The present study is aimed at investigating changes in the coordination of arm and leg movements in young healthy subjects. It was hypothesized that with changes in walking velocity there is a change in frequency and phase coupling between the arms and the legs. In addition, it was hypothesized that the preferred frequencies of the different coordination patterns can be predicted on the basis of the resonant frequencies of arms and legs with a simple pendulum model. The kinematics of arms and legs during treadmill walking in seven healthy subjects were recorded with accelerometers in the sagittal plane at a wide range of different velocities (i.e., 0.3-1. 3m/s). Power spectral analyses revealed a statistically significant change in the frequency relation between arms and legs, i.e., within the velocity range 0.3-0.7m/s arm movement frequencies were dominantly synchronized with the step frequency, whereas from 0.8m/s onwards arm frequencies were locked onto stride frequency. Significant effects of walking speed on mean relative phase between leg and arm movements were found. All limb pairs showed a significantly more stable coordination pattern from 0.8 to 1.0m/s onwards. Results from the pendulum modelling demonstrated that for most subjects at low-velocity preferred movement frequencies of the arms are predicted by the resonant frequencies of individual arms (about 0.98Hz), whereas at higher velocities these are predicted on the basis of the resonant frequencies of the individual legs (about 0.85Hz). The results support the above-mentioned hypotheses, and suggest that different patterns of coordination, as shown by changes in frequency coupling and phase relations, can exist within the human walking mode.     

Parametric stabilization of biological coordination: a theoretical model

In human coordination studies information from the environment may not only pace rhythmic behavior, but also contribute to the observed dynamics, e.g. aphenomenon known as anchoring in the literature. For the paradigmatic caseof bimanual coordination we study these contributions mathematically and develop a model of the interaction between the limb's intrinsic dynamics and environmental signals from a metronome in terms of oscillator equations. We discuss additive versus multiplicative metronomeimpact and show the latter to be more appropriate.Our model describes single limb-metronome interaction, as well as multilimb-metronome interaction. We establish a parametricstabilization term which preserves the characteristicsof bimanual coordination and additionally explains the varyingstability of movement under different metronome conditions, the frequency dependence of the amplitudes of finger movements, anchoring phenomena andgeometries of phase space trajectories. Predictions of our model are tested against experimental observations.     

Coordination dynamics: (in)stability and metastability in the behavioural and neural systems

For more than 20 years, coordination dynamics have provided research on human movement science with new views about the nonlinear relationships between behavioral and neural dynamics. A number of studies across various experimental settings including bimanual, postural or interpersonal coordination, and also coordination between movements of a limb and an external event in the environment revealed the self-organized nature of human coordination. Here we review an extensive body of literature - in the human movement science and the neuroscience fields - that has investigated the coordination dynamics of brain and behavior when individuals are involved in two rhythmic coordination patterns: synchronization (on-the-beat movements) and syncopation (in-between beats movements). When the frequency of movement approaches 2 Hz, the syncopation mode is destabilized and synchronization is spontaneously adopted. The abrupt change between the two patterns illustrates a phenomenon known as non-equilibrium phase transition. Phase transitions offer a novel entry point into the investigation of pattern formation (and dissolution) at both the behavioral and the cerebral levels as they illustrate the loss of stability of the system. Brain imaging methods (MEG, EEG and fMRI) were used to reveal the neural signatures of (in)stability underlying the differences between behavioral coordination patterns, and pointed at the role of self-organization and metastability principles in brain functioning. Relationships between behavioral and brain dynamics can therefore be investigated within a unified empirical and theoretical framework.     

Relation of spatial and temporal frequencies to vernier acuity

Vernier thresholds were measured with a pair of vertical sinusoidal gratings of one and a half cycles as targets. The amplitude was weighted by a one-dimensional Gaussian and contrast was set one log unit above contrast threshold. The vernier thresholds were estimated with the method of constant stimuli. Temporal frequency effects were introduced by movement of the vernier targets. It was found that vernier thresholds expressed in phase angle were unchanged in the effective range of spatial frequencies provided that the temporal frequency and the visibility were unchanged, and that thresholds deteriorated by increasing the temporal frequency. It is suggested that the detection of relative phase may be involved in the discrimination of vernier offsets and that it may be mediated by a sustained unit. Three possible types of mechanisms, edge-localization processes, orientation-selective units and phase-sensitive units, were considered in relation to vernier acuity.     

Relative phase dynamics in perturbed interlimb coordination: stability and stochasticity

 Various stability features of bimanual rhythmic coordination, including phase transitions, have been modeled successfully by means of a one-dimensional equation of motion for relative phase obeying a gradient dynamics, the Haken-Kelso-Bunz model. The present study aimed at assessing pattern stability for stationary performance and estimating the model parameters (a, b, and Q) for the stochastic extension of this model. Estimates of a and b allowed for reconstruction of the potential defining the gradient dynamics. Two coordination patterns between the forearms (in-phase, anti-phase) were performed at seven different frequencies. Model parameters were estimated on the basis of an exponential decay parameter describing the relaxation behavior of continuous relative phase following a mechanical perturbation. Variability of relative phase and relaxation time provided measures of pattern stability. Although the predicted inverse relation between pattern stability and movement frequency was observed for the lower tempo conditions, it was absent for the higher tempos, reflecting the influence of task constraints. No statistically significant differences in stability were observed between the two coordination modes, indicating the influence of intention. The reconstructed potential reflected the observed stability features, underscoring the adequacy of the parameter estimations. The relaxation process could not be captured adequately by means of a simple exponential decay function but required an additional oscillatory term. In accordance with previous assumptions, noise strength Q did not vary as a function of movement frequency. However, systematic differences in Q were observed between the two coordination modes. The advantages and (potential) pitfalls of using stationary performance of single patterns to examine the stability features of a bistable potential were discussed. Received: 12 July 1999 / Accepted in revised form: 14 April 2000     

Hilbert phase entropy imaging of fMRI time series

Functional magnetic resonance imaging (fMRI) data-processing methods in the time domain include correlation analysis and the general linear model, among others. Virtually, many fMRI processing strategies utilise temporal information and ignore or pay little attention to phase information, resulting in an unnecessary loss of efficiency. We proposed a novel method named Hilbert phase entropy imaging (HPEI) that used the discrete Hilbert transform of the magnitude time series to detect brain functional activation. The data from two simulation studies and two in vivo fMRI studies that both contained block-design and event-related experiments revealed that the HPEI method enabled the effective detection of brain functional activation and the distinction of different response patterns. Our results demonstrate that this method is useful as a complementary analysis, but hypothesis-constrained, in revealing additional information regarding the complex nature of fMRI time series.     

Estimating Brownian motion dispersal rate, longevity and population density from spatially explicit mark-recapture data on tropical butterflies

1.?We develop a Bayesian method for analysing mark-recapture data in continuous habitat using a model in which individuals movement paths are Brownian motions, life spans are exponentially distributed and capture events occur at given instants in time if individuals are within a certain attractive distance of the traps. 2.?The joint posterior distribution of the dispersal rate, longevity, trap attraction distances and a number of latent variables representing the unobserved movement paths and time of death of all individuals is computed using Gibbs sampling. 3.?An estimate of absolute local population density is obtained simply by dividing the Poisson counts of individuals captured at given points in time by the estimated total attraction area of all traps. Our approach for estimating population density in continuous habitat avoids the need to define an arbitrary effective trapping area that characterized previous mark-recapture methods in continuous habitat. 4.?We applied our method to estimate spatial demography parameters in nine species of neotropical butterflies. Path analysis of interspecific variation in demographic parameters and mean wing length revealed a simple network of strong causation. Larger wing length increases dispersal rate, which in turn increases trap attraction distance. However, higher dispersal rate also decreases longevity, thus explaining the surprising observation of a negative correlation between wing length and longevity.     

Modeling inter-human movement coordination: synchronization governs joint task dynamics

Human interaction partners tend to synchronize their movements during repetitive actions such as walking. Research of inter-human coordination in purely rhythmic action tasks reveals that the observed patterns of interaction are dominated by synchronization effects. Initiated by our finding that human dyads synchronize their arm movements even in a goal-directed action task, we present a step-wise approach to a model of inter-human movement coordination. In an experiment, the hand trajectories of ten human dyads are recorded. Governed by a dynamical process of phase synchronization, the participants establish in-phase as well as anti-phase relations. The emerging relations are successfully reproduced by the attractor dynamics of coupled phase oscillators inspired by the Kuramoto model. Three different methods on transforming the motion trajectories into instantaneous phases are investigated and their influence on the model fit to the experimental data is evaluated. System identification technique allows us to estimate the model parameters, which are the coupling strength and the frequency detuning among the dyad. The stability properties of the identified model match the relations observed in the experimental data. In short, our model predicts the dynamics of inter-human movement coordination. It can directly be implemented to enrich human-robot interaction.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment.     

Quantification of six herbicide metabolites in human urine

We developed a sensitive, selective and precise method for measuring herbicide metabolites in human urine. Our method uses automated liquid delivery of internal standards and acetate buffer and a mixed polarity polymeric phase solid phase extraction of a 2 mL urine sample. The concentrated eluate is analyzed using high-performance liquid chromatography-tandem mass spectrometry. Isotope dilution calibration is used for quantification of all analytes. The limits of detection of our method range from 0.036 to 0.075 ng/mL. The within- and between-day variation in pooled quality control samples range from 2.5 to 9.0% and from 3.2 to 16%, respectively, for all analytes at concentrations ranging from 0.6 to 12 ng/mL. Precision was similar with samples fortified with 0.1 and 0.25 ng/mL that were analyzed in each run. We validated our selective method against a less selective method used previously in our laboratory by analyzing human specimens using both methods. The methods produced results that were in agreement, with no significant bias observed.     

Frequency dependence of the action-perception cycle for postural control in a moving visual environment: relative phase dynamics

When standing human subjects are exposed to a moving visual environment, the induced postural sway displays varying degrees of coherence with the visual information. In our experiment we varied the frequency of an oscillatory visual display and analysed the temporal relationship between visual motion and sway. We found that subjects maintain sizeable sway amplitudes even as temporal coherence with the display is lost. Postural sway tended to phase lead (for frequencies below 0.2 Hz) or phase lag (above 0.3 Hz). However, we also observed at a fixed frequency, highly variable phase relationships in which a preferred range of phase lags is prevalent, but phase jumps occur that return the system into the preferred range after phase has begun drifting out of the preferred regime. By comparing the results quantitatively with a dynamical model (the sine-circle map), we show that this effect can be understood as a form of relative coordination and arises through an instability of the dynamics of the action-perception cycle. Because such instabilities cannot arise in passively driven systems, we conclude that postural sway in this situation is actively generated as rhythmic movement which is coupled dynamically to the visual motion. Received: 7 September 1993/Accepted in revised form: 2 May 1994     

The cerebellum and initiation of movement: the stretch reflex

Studies of the stretch reflex in decerebrate cats indicate a phase advance of peak sinusoidal tension in steady-state cycles between 0.1 and 10 Hz. This phase advance is reduced in acute and chronic cerebellectomy, as shown in previous investigations. Also, the augmentation of muscle peak tension in initial sinusoidal stretch cycles at 0.5-5 Hz has been found to be reduced during the time of reflex and motor instability in the several months following cerebellar ablation. This report shows the increased amplitude and phase lead of integrated electromyographic activity in initiating sinusoidal stretch cycles in the decerebrate cat. These reflex aspects are demonstrated in relation to the discharge of neurons in the dorsal spinocerebellar tract and of cerebellar cortical Purkinje cells in initial sinusoidal cycles. The intensity and phase advance of the discharge in dorsal spinocerebellar tract neurons is altered little, but these features are usually increased in Purkinje cells during initial stretches compared to continuous cycling. In terms of overall motor control, these findings are compatible with concepts of movement control, modulated by the cerebellum, in which the discharge of antagonist motor neurons is regulated in concert with that of agonist muscles upon initiation and termination of movement.