Dynamics of 1:2 Coordination: Generalizing Relative Phase to n:m Rhythms

Preparing to move: Setting initial conditions to simplify interactions with complex objects

Humans dexterously interact with a variety of objects, including those with complex internal dynamics. Even in the simple action of carrying a cup of coffee, the hand not only applies a force to the cup, but also indirectly to the liquid, which elicits complex reaction forces back on the hand. Due to underactuation and nonlinearity, the object's dynamic response to an action sensitively depends on its initial state and can display unpredictable, even chaotic behavior. With the overarching hypothesis that subjects strive for predictable object-hand interactions, this study examined how subjects explored and prepared the dynamics of an object for subsequent execution of the target task. We specifically hypothesized that subjects find initial conditions that shorten the transients prior to reaching a stable and predictable steady state. Reaching a predictable steady state is desirable as it may reduce the need for online error corrections and facilitate feed forward control. Alternative hypotheses were that subjects seek to reduce effort, increase smoothness, and reduce risk of failure. Motivated by the task of 'carrying a cup of coffee', a simplified cup-and-ball model was implemented in a virtual environment. Human subjects interacted with this virtual object via a robotic manipulandum that provided force feedback. Subjects were encouraged to first explore and prepare the cup-and-ball before initiating a rhythmic movement at a specified frequency between two targets without losing the ball. Consistent with the hypotheses, subjects increased the predictability of interaction forces between hand and object and converged to a set of initial conditions followed by significantly decreased transients. The three alternative hypotheses were not supported. Surprisingly, the subjects' strategy was more effortful and less smooth, unlike the observed behavior in simple reaching movements. Inverse dynamics of the cup-and-ball system and forward simulations with an impedance controller successfully described subjects' behavior. The initial conditions chosen by the subjects in the experiment matched those that produced the most predictable interactions in simulation. These results present first support for the hypothesis that humans prepare the object to minimize transients and increase stability and, overall, the predictability of hand-object interactions.

A flexible and accurate method to estimate the mode and stability of spontaneous coordinated behaviors: The index-of-stability (IS) analysis

Patterns of coordination result from the interaction between (at least) two oscillatory components. This interaction is typically understood by means of two variables: the mode that expresses the shape of the interaction, and the stability that is the robustness of the interaction in this mode. A potent method of investigating coordinated behaviors is to examine the extent to which patterns of coordination arise spontaneously. However, a prominent issue faced by researchers is that, to date, no standard methods exist to fairly assess the stability of spontaneous coordination. In the present study, we introduce a new method called the index-of-stability (IS) analysis. We developed this method from the phase-coupling (PC) analysis that has been traditionally used for examining locomotion-respiration coordinated systems. We compared the extents to which both methods estimate the stability of simulated coordinated behaviors. Computer-generated time series were used to simulate the coordination of two rhythmic components according to a selected mode m:n and a selected degree of stability. The IS analysis was superior to the PC analysis in estimating the stability of spontaneous coordinated behaviors, in three ways: First, the estimation of stability itself was found to be more accurate and more reliable with the IS analysis. Second, the IS analysis is not constrained by the limitations of the PC analysis. Third and last, the IS analysis offers more flexibility, and so can be adapted according to the user's needs.

Adapting relative phase of bimanual isometric force coordination through scaling visual information intermittency

Visual information plays an adaptive role in the relation between bimanual force coupling and error corrective processes of isometric force control. In the present study, the evolving distribution of the relative phase properties of bimanual isometric force coupling was examined by scaling within a trial the temporal feedback rate of visual intermittency (short to long presentation intervals and vice versa). The force error (RMSE) was reduced, and time-dependent irregularity (SampEn) of the force output was increased with greater amounts of visual information (shorter intermittency). Multi-stable coordination patterns of bimanual isometric force control were differentially shifted toward and away from the intrinsic dynamics by the changing the intermittency of visual information. The distribution of Hilbert transformed relative phase values showed progressively a predominantly anti-phase mode under less intermittent visual information to predominantly an in-phase mode with limited (almost no) visual information. Correlation between the hands showed a continuous reduction, rather than abrupt "transition," with increase in visual information, although no mean negative correlation was realized, despite the tendency towards an anti-phase distribution. Lastly, changes in both the performance outcome and bimanual isometric force coordination occurred at visual feedback rates faster than the minimal visual processing times established from single limb movement and isometric force protocols.

Philosophy for the rest of cognitive science

Cognitive science has always included multiple methodologies and theoretical commitments. The philosophy of cognitive science should embrace, or at least acknowledge, this diversity. Bechtel's (2009a) proposed philosophy of cognitive science, however, applies only to representationalist and mechanist cognitive science, ignoring the substantial minority of dynamically oriented cognitive scientists. As an example of nonrepresentational, dynamical cognitive science, we describe strong anticipation as a model for circadian systems (Stepp & Turvey, 2009). We then propose a philosophy of science appropriate to nonrepresentational, dynamical cognitive science.

The relevance of the cross-wavelet transform in the analysis of human interaction - a tutorial

This article sheds light on a quantitative method allowing psychologists and behavioral scientists to take into account the specific characteristics emerging from the interaction between two sets of data in general and two individuals in particular. The current article outlines the practical elements of the cross-wavelet transform (CWT) method, highlighting WHY such a method is important in the analysis of time-series in psychology. The idea is (1) to bridge the gap between physical measurements classically used in physiology - neuroscience and psychology; (2) and demonstrates how the CWT method can be applied in psychology. One of the aims is to answer three important questions WHO could use this method in psychology, WHEN it is appropriate to use it (suitable type of time-series) and HOW to use it. Throughout these explanations, an example with simulated data is used. Finally, data from real life application are analyzed. This data corresponds to a rating task where the participants had to rate in real time the emotional expression of a person. The objectives of this practical example are (i) to point out how to manipulate the properties of the CWT method on real data, (ii) to show how to extract meaningful information from the results, and (iii) to provide a new way to analyze psychological attributes.

Bimanual force control: cooperation and interference?

Three experiments were designed to determine the level of cooperation or interference observed from the forces generated in one limb on the forces exhibited by the contralateral limb when one or both limbs were producing a constant force (Experiment 1), one limb was producing a dynamic force while the other limb was producing a constant force (Experiment 2), and both limbs were producing dynamic force patterns (Experiment 3). The results for both Experiments 1 and 2 showed relatively strong positive time series cross correlations between the left and right limb forces indicating increases or decreases in the forces generated by one limb resulted in corresponding changes in the forces produced by the homologous muscles of the contralateral limb. Experiment 3 required participants to coordinate 1:1 and 1:2 rhythmical bimanual force production tasks when provided Lissajous feedback. The results indicated very effective performance of both bimanual coordination patterns. However, identifiable influences of right limb forces on the left limb force time series were observed in the 1:2 coordination pattern but not in the 1:1 pattern. The results of all three experiments support the notion that neural crosstalk is partially responsible for the stabilities and instabilities associated with bimanual coordination.

Visual multifrequency entrainment: can 1:2, 2:3, and 3:4 coordination occur spontaneously?

Complex patterns of interlimb coordination, such as multifrequency relationships of 1:2, 2:3, or 3:4, are difficult to perform intentionally without extensive practice. The current study investigated whether these patterns might nonetheless occur spontaneously given an appropriate balance between the movement frequencies, or oscillatory periods, of an individual's movements and a visual-environmental stimulus. In order to test this, participants swung a fixed-period wrist-pendulum while observing an oscillating computer-generated stimulus. Results indicated that at given differences in period, 1:2, 2:3, and 3:4 coordination patterns emerged between the participant and stimulus. This suggests that large period differences do not altogether prevent the emergence of rhythmic visuomotor coordination, but instead provide the opportunity for complex patterns of coordination to emerge spontaneously.

Learning to never forget-time scales and specificity of long-term memory of a motor skill

Despite anecdotal reports that humans retain acquired motor skills for many years, if not a lifetime, long-term memory of motor skills has received little attention. While numerous neuroimaging studies showed practice-induced cortical plasticity, the behavioral correlates, what is retained and also what is forgotten, are little understood. This longitudinal case study on four subjects presents detailed kinematic analyses of humans practicing a bimanual polyrhythmic task over 2 months with retention tests after 6 months and, for two subjects, after 8 years. Results showed that individuals not only retained the task, but also reproduced their individual "style" of performance, even after 8 years. During practice, variables such as the two hands' frequency ratio and relative phase, changed at different rates, indicative of multiple time scales of neural processes. Frequency leakage across hands, reflecting intermanual crosstalk, attenuated at a significantly slower rate and was the only variable not maintained after 8 years. Complementing recent findings on neuroplasticity in gray and white matter, our study presents new behavioral evidence that highlights the multi-scale process of practice-induced changes and its remarkable persistence. Results suggest that motor memory may comprise not only higher-level task variables but also individual kinematic signatures.

Effects of fatigue on synergies in a hierarchical system

We investigated the effect of fatigue produced by timed maximal voluntary contraction (MVC) of the index finger of one of the hands on performance in MVC and accurate cyclic force production tasks in right-handed subjects. Based on earlier studies, we hypothesized that fatigue would produce an increase in the indices of force-stabilizing synergies in both hands as well as between the hands in two-hand tasks. Synergies were defined as co-varied adjustments of commands to fingers (modes) across cycles that stabilized total force. Fatigue caused a significant reduction in the MVC of the exercised as well as the non-exercised hand. Indices of finger enslaving (lack of individuation) increased with fatigue in both hands, although the increase was significant in the exercised hand only. In contrast to the significant effects of fatigue on MVC forces performed by the non-exercised hand, there were no comparable transfer effects on the root mean square errors during accurate force production. During one-hand tasks, both hands showed high indices of force-stabilizing synergies. These indices were larger in the left hand. Fatigue led to a general increase in synergy indices. Exercise by the left hand had stronger effects on synergy indices seen in both hands. Exercise by the right hand showed ipsilateral effects only. Smaller effects of fatigue were observed on accuracy of performance of the force-down segments of the force cycles compared to the force-up segments. For the bimanual tasks, synergies were analyzed at two hierarchical levels, two-hand (four-finger) and within-a-hand (two-finger). An increase in the synergy index with fatigue was observed at the lower (two-finger) level of the hierarchy only. We interpret the lack of effects of fatigue at the upper (two-hand) level as a consequence of a trade-off between synergies at different levels of the hierarchy. The differences between the hands are discussed within the dynamic dominance hypothesis.

Increasingly complex bimanual multi-frequency coordination patterns are equally easy to perform with on-line relative velocity feedback

An experiment was conducted to determine whether multi-frequency continuous bimanual circling movements of varying difficulty (1:2, 2:3, 3:4, and 4:5) could be effectively performed following relatively little practice when on-line continuous relative velocity feedback is provided. The between-subjects results indicate extremely effective bimanual multi-frequency performance for all coordination patterns with relatively stable and continuous movements of both limbs. The findings suggest that the previous performance effects using Lissajous feedback with reciprocal movement can be extended to circling movements using on-line relative velocity feedback. Contrary to the long-held position that these coordination patterns result in increasing difficulty, we failed to find systematic relative velocity error, variability, or bias differences between the participants performing the various multi-frequency coordination patterns. Indeed, coordination error, variability, and biases were remarkably low for each of the tasks. The results clearly indicate the ease with which participants are able to produce bimanual coordination patterns typically considered difficult if not impossible when salient visual information is provided that allows the participants to detect and correct their coordination errors.

Visuospatial perception and navigation in Parkinson's disease

A shifted field of view, an altered perception of optic flow speed, and gait asymmetries may influence heading direction in Parkinson's disease (PD). PD participants (left body-side onset, LPD, n=14; right body-side onset, RPD, n=9) and Healthy Control participants (n=17) walked a virtual hallway in which the optic flow speeds of the walls varied. Three-dimensional kinematics showed participants veered away from the faster moving wall. Although veering normally occurs toward the side with smaller step length, in both LPD and RPD this bias was overridden by a shifted field of view, which caused veering in the opposite direction, toward the side of the brain with more basal ganglia damage.

Effects of Velocity and Limb Loading on the Coordination Between Limb Movements During Walking

The authors investigated the effects of velocity (increasing from 0.5 to 5.0 km/hr in steps of 0.5 km/hr) and limb loading on the coordination between arm and leg movements during treadmill walking in 7 participants. Both the consistency of the individual limb movements and the stability of their coordination increased with increasing velocity; the frequency coordination between arm and leg movements was 2:1 at the lower velocities and 1:1 at the higher velocities. The mass manipulation affected the individual limb movements but not their coordination, indicating that a stable walking pattern was preserved. The results differed qualitatively from those obtained in studies on bimanual interlimb coordination, implying that the dynamical principles identified therein are not readily applicable to locomotion.

Order parameter dynamics of body-scaled hysteresis and mode transitions in grasping behavior

Several experimental studies have shown that human grasping behavior exhibits a transition from one-handed to two-handed grasping when to-be-grasped objects become larger and larger. The transition point depends on the relative size of objects measured in terms of human body-scales. Most strikingly, the transitions between the two different behavioral 'modes' of grasping exhibit hysteresis. That is, one-to-two hand transitions and two-to-one hand transitions occur at different relative object sizes when objects are scaled up or down in size. In our study we approach body-scaled hysteresis and mode transitions in grasping by exploiting the notion that human behavior in general results from self-organization and satisfies appropriately-defined order parameter equations. To this end, grasping transitions and grasping hysteresis are discussed from a theoretical perspective in analogy to cognitive processes defined by Haken's neural network model for pattern recognition. In doing so, issues such as the exclusivity of grasping modes, biomechanical constraints, mode-mode interactions, single subject behavior and population behavior are explored.

On rhythmic and discrete movements: reflections, definitions and implications for motor control

At present, rhythmic and discrete movements are investigated by largely distinct research communities using different experimental paradigms and theoretical constructs. As these two classes of movements are tightly interlinked in everyday behavior, a common theoretical foundation spanning across these two types of movements would be valuable. Furthermore, it has been argued that these two movement types may constitute primitives for more complex behavior. The goal of this paper is to develop a rigorous taxonomic foundation that not only permits better communication between different research communities, but also helps in defining movement types in experimental design and thereby clarifies fundamental questions about primitives in motor control. We propose formal definitions for discrete and rhythmic movements, analyze some of their variants, and discuss the application of a smoothness measure to both types that enables quantification of discreteness and rhythmicity. Central to the definition of discrete movement is their separation by postures. Based on this intuitive definition, certain variants of rhythmic movement are indistinguishable from a sequence of discrete movements, reflecting an ongoing debate in the motor neuroscience literature. Conversely, there exist rhythmic movements that cannot be composed of a sequence of discrete movements. As such, this taxonomy may provide a language for studying more complex behaviors in a principled fashion.

Effect of expertise on butterfly stroke coordination

The aim of this study was to compare the arm-to-leg coordination in the butterfly stroke of three groups of male swimmers of varying skill (10 elite, 10 non-elite, and 10 young swimmers) at four race paces (400-m, 200-m, 100-m, and 50-m paces). Using qualitative video analysis and a hip velocity-video system (50 Hz), key events of the arm and leg movement cycles were defined and four-point estimates of relative phase were used to estimate the arm-to-leg coordination between the propulsive (pull and push of arms and downward movement of leg undulation) and non-propulsive phases (entry, catch, and recovery of arms and upward movement of leg undulation). With increasing race pace, the velocity, stroke rate, and synchronization between the arm and leg key points also increased, indicating that velocity and stroke rate may operate as control parameters. Finally, these changes led to greater continuity between the propulsive actions, which is favourable for improving the swim velocity, suggesting that coaches and swimmers should monitor arm-to-leg coordination.

One-to-one and polyrhythmic temporal coordination in bimanual circle tracing

The authors manipulated movement amplitude in a bimanual circle-tracing task to alter the natural tracing frequency of the arms. Participants (N = 14) traced different-diameter circles simultaneously with the two arms in either in-phase (0 degrees) or antiphase (180 degrees) coordination, using the index fingers or plastic styli. Movement amplitude altered the natural tracing frequency of the arms, as demonstrated by the following 2 findings: (a) The larger the difference in circle diameter, the larger was the shift from the fixed-point values of 0 degrees and 180 degrees, and the shift increased as movement frequency increased. Those results are consistent with the manipulation of delta omega in the bimanual pendulum paradigm. (b) Increasing movement frequency induced transitions from 1:1 to non-1:1 coordination, contrary to findings in previous investigations of polyrhythmic coordination. Tactile feedback played a minimal role in stabilizing bimanual coordination in the current tasks.

Influence of predominant patterns of coordination on the exploitation of interaction torques in a two-joint rhythmic arm movement

In this study we investigate the coordination between rhythmic flexion-extension (FE) and supination-pronation (SP) movements at the elbow joint-complex, while manipulating the intersegmental dynamics by means of a 2-degrees of freedom (df) robot arm. We hypothesized that constraints imposed by the structure of the neuromuscular-skeletal system would (1) result in predominant pattern(s) of coordination in the absence of interaction torques and (2) influence the capabilities of participants to exploit artificially induced interaction torques. Two experiments were conducted in which different conditions of interaction torques were applied on the SP-axis as a function of FE movements. These conditions promoted different patterns of coordination between the 2-df. Control trials conducted in the absence of interaction torques revealed that both the in-phase (supination synchronized with flexion) and the anti-phase (pronation synchronized with flexion) patterns were spontaneously established by participants. The predominance of these patterns of coordination is explained in terms of the mechanical action of bi-articular muscles acting at the elbow joint-complex, and in terms of the reflexes that link the activity of the muscles involved. Results obtained in the different conditions of interaction torques revealed that those neuromuscular-skeletal constraints either impede or favor the exploitation of intersegmental dynamics depending on the context. Interaction torques were indeed found to be exploited to a greater extent in conditions in which the profiles of interaction torques favored one of the two predominant patterns of coordination (i.e., in-phase or anti-phase) as opposed to other patterns of coordination (e.g., 90 degrees or 270 degrees). Those results are discussed in relation to recent studies reporting exploitation of interaction torques in the context of rhythmic movements.

Neuromuscular-skeletal origins of predominant patterns of coordination in rhythmic two-joint arm movement

The authors tested for predominant patterns of coordination in the combination of rhythmic flexion-extension (FE) and supination- (SP) at the elbow-joint complex. Participants (N=10) spontaneously established in-phase (supination synchronized with flexion) and antiphase (pronation synchronized with flexion) patterns. In addition, the authors used a motorized robot arm to generate involuntary SP movements with different phase relations with respect to voluntary FE. The involuntarily induced in-phase pattern was accentuated and was more consistent than other patterns. The result provides evidence that the predominance of the in-phase pattern originates in the influence of neuromuscular-skeletal constraints rather than in a preference dictated by perceptual-cognitive factors implicated in voluntary control. Neuromuscular-skeletal constraints involved in the predominance of the in-phase and the antiphase patterns are discussed.

Bimanual adaptation: internal representations of bimanual rhythmic movements

From tying your shoes and clipping your tie to the claps at the end of a fine seminar, bimanual coordination plays a major role in our daily activities. An important phenomenon in bimanual coordination is the predisposition toward mirror symmetry in the performance of bimanual rhythmic movements. Although learning and adaptation in bimanual coordination are phenomena that have been observed, they have not been studied in the context of adaptive control and internal representations-approaches that were successfully employed in the arena of reaching movements and adaptation to force perturbations. In this paper we examine the dynamics of the learning mechanisms involved when subjects are trained to perform a bimanual non-harmonic polyrhythm in a bimanual index finger tapping task. Subjects are trained in this task implicitly, using altered visual feedback, while their performance is continuously monitored throughout the experiment. Our experimental results indicate the existence of significant (p<<0.01) learning curves (i.e., error plots with significantly negative slopes) during training and aftereffects with a washout period after the visual feedback ceases to be altered. These results confirm the formation of internal representations in bimanual motor control. We present a simple, physiologically plausible, neural model that combines feedback and adaptation in the control process and which is able to reproduce key phenomena of bimanual coordination and adaptation.

Effector dynamics of rhythmic wrist activity and its implications for (modeling) bimanual coordination

To examine the role of the effector dynamics of the wrist in the production of rhythmic motor activity, we estimated the phase shifts between the EMG and the task-related output for a rhythmic isometric torque production task and an oscillatory movement, and found a substantial difference (45-52 degrees) between the two. For both tasks, the relation between EMG and task-related output (torque or displacement) was adequately reproduced with a physiologically motivated musculoskeletal model. The model simulations demonstrated the importance of the contribution of passive structures to the overall dynamics and provided an account for the observed phase shifts in the dynamic task. Additional simulations of the musculoskeletal model with added load suggested that particular changes in the phase relation between EMG and movement may follow largely from the intrinsic muscle dynamics, rather than being the result of adaptations in the neural control of joint stiffness. The implications of these results are discussed in relation to (models of) interlimb coordination in rhythmic tasks.

Task-effector asymmetries in a rhythmic continuation task

Variability in rhythmic movements has been interpreted as a signature of internal or peripheral noise processes. Grounded in an oscillator interpretation, this study hypothesized that period variability and drift arises from the asymmetry between target period and the limb's intrinsic dynamics. Participants synchronized to 7 target periods, swinging 1 of 3 pendulums in a continuation paradigm; 3 periods were longer, 3 shorter, and 1 identical to the preferred period. Results supported 5 predictions: Drift toward the preferred period was observed that scaled with the asymmetry. Variability was lowest for symmetry conditions and increased with the asymmetry. Variability decreased concomitant with the approach toward the preferred period. Periods exponentially approached the preferred period with positive autocorrelations up to 10 cycles.

Interlimb coordination in prosthetic walking: effects of asymmetry and walking velocity

The present study focuses on interlimb coordination in walking with an above-knee prosthesis using concepts and tools of dynamical systems theory (DST). Prosthetic walkers are an interesting group to investigate from this theory because their locomotory system is inherently asymmetric, while, according to DST, coordinative stability may be expected to be reduced as a function of the asymmetry of the oscillating components. Furthermore, previous work on locomotion motivated from DST has shown that the stability of interlimb coordination increases with walking velocity, leading to the additional expectation that the anticipated destabilizing effect of the prosthesis-induced asymmetry may be diminished at higher walking velocities. To examine these expectations, an experiment was conducted aimed at comparing interlimb coordination during treadmill walking between seven participants with an above-knee prosthesis and seven controls across a range of walking velocities. The observed gait patterns were analyzed in terms of standard gait measures (i.e., absolute and relative swing, stance and step times) and interlimb coordination measures (i.e., relative phase and frequency locking). As expected, the asymmetry brought about by the prosthesis led to a decrease in the stability of the coordination between the legs as compared to the control group, while coordinative stability increased with increasing walking velocity in both groups in the absence of a significant interaction. In addition, the 2:1 frequency coordination between arm and leg movements that is generally observed in healthy walkers at low walking velocities was absent in the prosthetic walkers. Collectively, these results suggest that both stability and adaptability of coordination are reduced in prosthetic walkers but may be enhanced by training them to walk at higher velocities.

Sinusoidal visuomotor tracking: intermittent servo-control or coupled oscillations?

In visuomotor tasks that involve accuracy demands, small directional changes in the trajectories have been taken as evidence of feedback-based error corrections. In the present study variability, or intermittency, in visuomanual tracking of sinusoidal targets was investigated. Two lines of analyses were pursued: First, the hypothesis that humans fundamentally act as intermittent servo-controllers was re-examined, probing the question of whether discontinuities in the movement trajectory directly imply intermittent control. Second, an alternative hypothesis was evaluated: that rhythmic tracking movements are generated by entrainment between the oscillations of the target and the actor, such that intermittency expresses the degree of stability. In 2 experiments, participants (N = 6 in each experiment) swung 1 of 2 different hand-held pendulums, tracking a rhythmic target that oscillated at different frequencies with a constant amplitude. In 1 line of analyses, the authors tested the intermittency hypothesis by using the typical kinematic error measures and spectral analysis. In a 2nd line, they examined relative phase and its variability, following analyses of rhythmic interlimb coordination. The results showed that visually guided corrective processes play a role, especially for slow movements. Intermittency, assessed as frequency and power components of the movement trajectory, was found to change as a function of both target frequency and the manipulandum's inertia. Support for entrainment was found in conditions in which task frequency was identical to or higher than the effector's eigenfrequency. The results suggest that it is the symmetry between task and effector that determines which behavioral regime is dominant.

Bouncing a ball: tuning into dynamic stability

Rhythmically bouncing a ball with a racket was investigated and modeled with a nonlinear map. Model analyses provided a variable defining a dynamically stable solution that obviates computationally expensive corrections. Three experiments evaluated whether dynamic stability is optimized and what perceptual support is necessary for stable behavior. Two hypotheses were tested: (a) Performance is stable if racket acceleration is negative at impact, and (b) variability is lowest at an impact acceleration between -4 and -1 m/s2. In Experiment 1 participants performed the task, eyes open or closed, bouncing a ball confined to a 1-dimensional trajectory. Experiment 2 eliminated constraints on racket and ball trajectory. Experiment 3 excluded visual or haptic information. Movements were performed with negative racket accelerations in the range of highest stability. Performance with eyes closed was more variable, leaving acceleration unaffected. With haptic information, performance was more stable than with visual information alone.

Coupling of breathing and movement during manual wheelchair propulsion

The hypothesis of this study was that stable coordination patterns may be found both within and between physiological subsystems. Many studies have been conducted on both monofrequency and multifrequency coordination, with a focus on both the frequency and phase relations among the limbs. In the present study, locomotor-respiratory coupling was observed in the maintenance of small-integer frequency ratios (2:1, 3:1, and 4:1) and in the consistent placement of the inspiratory phase just after the onset of the movement cycle during wheelchair propulsion. Level of experience and various motor and respiratory parameters were manipulated. Coupling was observed across levels of experience. Increases in movement frequency were accompanied by a shift to larger-integer ratios, suggesting that a single modeling strategy (e.g., the Farey tree; D. L. González & O. Piro, 1985) may be used for coordination both within the motor subsystem and between it and other physiological subsystems.

Deterministic variability and stability in detuned bimanual rhythmic coordination

We examined the effects of crossing different degrees of cooperation and competition on inphase and antiphase 1:1 frequency locked coordination of left- and right-hand-oscillated pendulums. Degree of cooperation was manipulated through the joint frequency of oscillation specified by a metronome (the higher the frequency, the weaker the cooperation), and degree of competition by length (and, therefore, preferred frequency) differences between the two pendulums (the greater the difference, the stronger the competition). Increasing competition was accompanied by either decreasing cooperation (for six participants) or increasing cooperation (for six different participants). On each trial, a participant attempted to produce a steady-state phase relation phi for a given combination of competition and cooperation. Numerical simulations of the extended Haken-Kelso-Bunz (HKB) equation were used to predict (a) the patterns of shift in phi from either 0 or pi radians due to the different competition-cooperation relations and (b) the patterns of variability in phi. It was expected that the HKB equation would be successful in respect to (a), which it was, but not in respect to (b). The observed failure to confirm (b) was expected from the variability due to the different nonharmonic dynamics of the component oscillators, a source of variability not included in the HKB equation. The experimental results together with simulations and analyses of the phase-plane trajectories of the component oscillators suggest the operation of deterministic in addition to stochastic variability in the phase relation of contralateral limbs.

Cognitive activity shifts the attractors of bimanual rhythmic coordination

The attractors of bimanual rhythmic coordination are given as the solutions of a motion equation in relative phase. How are those attractors affected by cognitive activity? In 3 experiments, participants (N = 10 in Experiments 1 and 2; N = 5 in Experiment 3) were required to produce in-phase or antiphase coordination while they either did or did not perform an information-reduction task. The average absolute deviation from in-phase (0 degrees ) and antiphase (180 degrees ) satisfying a particular parameterization of the motion equation was amplified by cognitive activity. That amplification of absolute phase shift was the same for both in-phase and antiphase coordination. Furthermore, the amplification (in degrees) increased linearly with the magnitude of cognitive activity (in bits). Cognitive activity had limited influence on the variability of relative phase and did not affect its average signed deviation. Collectively, the results suggest that cognitive activity produces a shift in the attractors of bimanual coordination dynamics that is directionally nonspecific and is independent of movement speed, detuning, and the in-phase-antiphase distinction.

Coordination between arm and leg movements during locomotion

To evaluate the contrasting dynamical and biomechanical interpretations of the 2:1 frequency coordination between arm and leg movements that occurs at low walking velocities and the 1:1 frequency coordination that occurs at higher walking velocities, the authors conducted an experiment in which they quantified the effect of walking velocity on the stability of the frequency and phase coordination between the individual limb movements. Spectral analyses revealed the presence of 2:1 frequency coordination as a constant feature of the data in only 3 out of 8 participants at walking velocities ranging from 1.0 to 2.0 km/h, in spite of the fact that the eigenfrequencies of the arms were rather similar across participants. The degree of interlimb coupling, as indexed by weighted coherence and variability of relative phase, was lower for the arm movements and for ipsilateral and diagonal combinations of arm and leg movements than for the leg movements. Furthermore, the coupling between all pairs of limb movements was found to increase with walking velocity, whereas no clear signs were observed that the switches from 2:1 to 1:1 frequency coordination and vice versa were preceded by loss of stability. Therefore, neither a purely biomechanical nor a purely dynamical model is optimally suited to explain these results. Instead, an integrative model involving elements of both approaches seems to be required.

Dynamics of 1:2 Coordination: Sources of Symmetry Breaking

Three asymmetries in the dynamics of 1:2 interlimb coordination were examined: the asymmetry in uncoupled frequencies, the asymmetry in coupled frequencies, and the left-right functional asymmetry of the body. In a bimanual 1:2 task, participants (N = 8) oscillated hand-held pendulums whose uncoupled frequencies were adjusted so that the first kind of asymmetry could be manipulated. For any given pendulum pair, the pendulum assuming the faster motion in the 1:2 coordination was oscillated in the right and the left hands. By assigning combinations of uncoupled eigenfrequencies and coupled task-specified frequencies across hands, the authors studied the interaction of all 3 asymmetries. The results confirm the appropriateness of generalized relative phase as a collective variable for 1:2 coordination. Additionally, they suggest that the generalized form of the detuning parameter represents the first asymmetry and that the coupling function expresses the second asymmetry. In 1:2 coordination, the body's functional asymmetry plays a limited role.

Dynamics of 1:2 Coordination: Temporal Scaling, Latent 1:1, and Bistability

The simplest interlimb multifrequency coordination of 1:2 can be performed at different speeds and in at least two different styles or modes. The effects of speed and mode (in-phase or antiphase) were evaluated in a bimanual 1:2 rhythmic task in which participants (N = 8) oscillated hand-held pendulums with identical or different uncoupled frequencies. A motion equation in relative phase that captures the asymmetries of components and task predicted the 1:2 coordination equilibria resulting from temporal scaling. According to the experimental results, both coordination modes proved to be equally stable. More detailed analyses of individual trials showed signs that the more fundamental 1:1 coordination intruded into the 1:2 coordination.