Modeling rhythmic interlimb coordination: The roles of movement amplitude and time delays

Rhythmic interlimb coordination is characterized by attraction to stable phase and frequency relations. Sudden transitions between the resulting coordination patterns, which are observed when movement frequency is gradually increased, have been modeled at two formally related levels: a potential function and a system of coupled oscillators. At the latter level of the model, two alternative derivations resulted in different predictions with respect to the way in which movement frequency and amplitude affect pattern stability. Our recent results contradicted the prevailing version of the model, which predicts that the influence of movement frequency is fully mediated by the associated changes in amplitude. Although the results could be reconciled with the alternative derivation of the model in which time delays (possibly related to neurophysiological delays) were incorporated, the absence of amplitude-mediated effects on pattern stability challenges both versions of the model. It is argued that by studying coordination dynamics at the level of the potential function as well as at the level of coupled oscillators, new insights into the way in which control parameters influence pattern dynamics may be obtained. This may open up ways to link the coordination dynamics to specific characteristics of the movements of the limbs and the way in which they interact.PsycINFO classification: 2300; 2330     

Realism in energy transition processes: an example from Bohmian quantum mechanics

In this paper we study in details a system of two weakly coupled harmonic oscillators from the point of view of Bohm’s interpretation of quantum mechanics. This system may be viewed as a simple model for the interaction between a photon and a photodetector. We obtain exact solutions for the general case. We then compute approximate solutions for the case where one oscillator is initially in its first excited state (a single photon) reaching the other oscillator in its ground state (the photodetector). The approximate solutions represent the state of both oscillators after the interaction, which is not an eigenstate of the individual hamiltonians for each oscillator, and therefore the energies for each oscillator do not exist in the Copenhagen interpretation of Quantum Mechanics. We use the approximate solutions that we obtained to compute Bohmian trajectories and to study the energy transfer between the oscillators. We conclude that, even using the Bohmian view, the energy of each individual oscillator is not well defined, as the nonlocal quantum potential is not negligible even after the coupling is turned off.     

Patterns of Human Interlimb Coordination Emerge from the Properties of Non-Linear,Limit Cycle Oscillatory Processes

The present article represents an initial attempt to offer a principled solution to a fundamental problem of movement identified by Bernstein (1967), namely, how the degrees of freedom of the motor system are regulated. Conventional views of movement control focus on motor programs or closed-loop devices and have little or nothing to say on this matter. As an appropriate conceptual framework we offer Iberall and his colleagues’ physical theory of homeokinetics first elaborated for movement by Kugler, Kelso, and Turvey (1980). Homeokinetic theory characterizes biological systems as ensembles of non-linear, limit cycle oscillatory processes coupled and mutually entrained at all levels of organization. Patterns of interlimb coordination may be predicted from the properties of non-linear, limit cycle oscillators. In a set of experiments and formal demonstrations we show that cyclical, two-handed movements maintain fixed amplitude and frequency (a stable limit cycle organization) under the following conditions: (a) when brief and constantly applied load perturbations are imposed on one hand or the other, (b) regardless of the presence or absence of fixed mechanical constraints, and (c) in the face of a range of external driving frequencies from a visual source. In addition, we observe a tight phasic relationship between the hands before and after perturbations (quantified by cross-correlation techniques), a tendency of one limb to entrain the other (mutual entrainment) and that limbs cycling at different frequencies reveal non-arbitrary, sub-harmonic relationships (small integer, subharmonic entrainment). In short, all the above patterns of interlimb coordination fall out of a non-linear oscillatory design. Discussion focuses on the compatibility of these results with past and present neurobiological work, and the theoretical insights into problems of movement offered by homeokinetic physics. Among these are, we think, the beginnings of a principled solution to the degrees of freedom problem, and the tentative claim that coordination and control are emergent consequences of dynamical interactions among non-linear, limit cycle oscillatory processes.     

Explanatory limitations of the HKB model: incentives for a two-tiered model of rhythmic interlimb coordination

The HKB model for rhythmic interlimb coordination has highlighted the importance of coordinative stability and loss of stability, and introduced, with this focus, a new set of explanatory constructs. However, the phenomenological character of both parts of this model (i.e., the potential and the associated system of coupled oscillators) precludes an understanding of how the observed stability characteristics are related to more specific (e.g., biomechanical and neurophysiological) aspects of the movement system. A two-tiered model (involving a distinction between 'neural' and 'effector' dynamics) is discussed that offers handles for addressing such underpinnings of the identified coordination dynamics. The promise of the model in this regard is illustrated by two recent studies showing how explicit accounts of the effector dynamics may help disclose why (and how) particular properties of the peripheral system affect the overall coordination dynamics.     

Oscillator mechanisms in the human motor system: investigating their properties using the aftercontraction effect

It is proposed that the human motor system is organized to use hardware and/or software non-linear oscillator mechanisms, the output of these oscillators being responsible for driving the limbs via signals to muscle groups. Following earlier theoretical development, it is argued that these muscle groupings act as a unit and themselves are likely to behave as a non-linear system. The attributes of non-linear oscillators are many, and they are potentially significant for the explanation of motor behavior. This paper reviews and presents recent experiments that investigated the properties of muscular aftercontraction. The basic finding shows that subsequent to a period of moderate strain against a fixed surface the treated limb exhibits prolonged involuntary molar oscillations in the plane of the treatment. These results provide for the presence of driving oscillator mechanisms in the human motor apparatus. The mechanisms show generality of action in that directed attention can lead to oscillation of untreated limbs. Overall, the experiments showed that the movements exhibited the mutual interaction, synchronization, and preservation of phase relationships that are fundamental properties of non-linear oscillators. the picture that emerges is that these mechanisms can drive involuntary movements that are richly patterned: like slow versions of voluntary movements. The aftercontraction phenomenon proves to be an excellent tool for research on the oscillatory substrate of human motor organization.     

Adequate formalization

This article identifies problems with regard to providing criteria that regulate the matching of logical formulae and natural language. We then take on to solve these problems by defining a necessary and sufficient criterion of adequate formalization. On the basis of this criterion we argue that logic should not be seen as an ars iudicandi capable of evaluating the validity or invalidity of informal arguments, but as an ars explicandi that renders transparent the formal structure of informal reasoning.     

Chaos in Human Rhythmic Movement

Rhythmic movements typical of locomotory actions are usually modeled as limit cycle dynamics, and their deviations from pure periodicity are attributed to stochastic physiological noise. In the present study, the dynamics of human rhythmic movements were found to contain more than the 2 dynamically active variables expected from limit cycle dynamics; the number depended upon the size of the limb oscillator. Observed positive Lyapunov exponents and fractal attractor dimensions indicated that the gross variability of human rhythmic movements may stem largely from low-dimensional chaotic motion on strange attractors.     

Patterns of human interlimb coordination emerge from the properties of non-linear, limit cycle oscillatory processes: theory and data

The present article represents an initial attempt to offer a principled solution to a fundamental problem of movement identified by Bernstein (1967), namely, how the degrees of freedom of the motor system are regulated. Conventional views of movement control focus on motor programs or closed-loop devices and have little or nothing to say on this matter. As an appropriate conceptual framework we offer Iberall and his colleagues' physical theory of homeokinetics first elaborated for movement by Kugler, Kelso, and Turvey (1980). Homeo kinetic theory characterizes biological systems as ensembles of non-linear, limit cycle oscillatory processes couple and mutually entrained at all the levels of organization. Patterns of interlimb coordination may be predicted from the properties of non-linear, limit cycle oscillators. In a set of experiments and formal demonstrations we show that cyclical, two-handed movements maintain fixed amplitude and frequency ( a stable limit cycle organization) under the following conditions: (a) when brief and constantly applied load perturbations are imposed on one hand or the other, (b) regardless of the presence or absence of fixed mechanical constraints, and (c) in the face of a range of external driving frequencies from a visual source. In addition, we observe a tight phasic relationship between the hands before and after perturbations (quantified by cross-correlation techniques), a tendency of one limb to entrain the other (mutual entrainment) and that limbs cycling at different frequencies reveal non-arbitrary, sub-harmonic relationships (small integer, subharmonic entrainment). In short, all the above patterns of interlimb coordination fall out of a non-linear oscillatory design. Discussion focuses on the compatibility of these results with past and present neurobiological work, and the theoretical insights into problems of movement offered by homeokinetic physics. Among these are, we think, the beginnings of a principled solution to the degrees of freedom problem, and the tentative claim that coordination and control are emergent consequences of dynamical interaction among non-linear, limit cycle oscillatory processes.     

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.     

A novel two-body sensor system to study spontaneous movements in infants during caregiver physical contact

Spontaneous movements, which refer to repetitive limb movements in the absence of any external stimulus, have been found to be reflective of neurodevelopmental status during infancy. These movements are modulated by both individual and environmental factors, including physical contact (holding) with the caregiver. However, it is a challenge to measure spontaneous movements during physical contact because infant-generated movements become coupled with caregiver-generated movements in such contexts. Here, we propose the use of a novel two-body sensor system to distinguish infant-generated movements in the presence of physical contact with the caregiver. Data from seven typically developing infants and their caregivers were recorded during different simulated home activities, which involved different combinations of physical interaction, caregiver’s movement and infant positions. The two-body sensor system consisted of two wearable accelerometers – one placed on the infant’s arm and one on the caregiver’s arm, and we developed a Kalman-filter based algorithm to isolate the infant-generated movements. In addition, video was recorded for qualitative analysis. Results indicated that spontaneous movement activity was higher when there was no physical contact with caregiver. When there was physical contact, spontaneous movements were increased when the caregiver was still and when the infant was held horizontally. These results show that the novel two-body sensor system and the associated algorithms were able to isolate infant-generated movements during physical contact with the caregiver. This approach holds promise for the automated long-term tracking of spontaneous movements in infants, which may provide critical insight into developmental disorders.     

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 consistent 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.     

A linked muscular activation model for movement generation and control

A new model for movement control is presented which incorporates characteristics of impulse-variability and mass-spring models. Movements in the model were controlled with phasic torque impulses in agonist and antagonist muscles and a tonic agonist torque. Characteristics of the phasic agonist and antagonist torque profiles were based on observed properties of movement-related EMGs and muscle isometric torques. Variability of the phasic impulses depended on impulse magnitude as in impulse-variability models. The model therefore predicted a speed-accuracy tradeoff for limb movement. The time of onset and magnitude of the antagonist torque depended on the magnitude of the preceding agonist torque as indicated in studies of movement-related EMGs. This led to the new concept of linkage between the agonist and antagonist muscle forces which was shown to be important for reducing variability of fast movements. Progressive development of linkage during practice could explain the previous findings of decreased movement variability with practice coupled with increased variability of movement-related EMGs. It was concluded that an inherently variable motor system deals with the variability associated with generation of large muscle forces by linking the forces produced by opposing muscles. In this way, variability in net joint torques and in movements can be decreased without the need for the nervous system to closely regulate the individual torques.     

A Linked Muscular Activation Model for Movement Generation and Control

A new model for movement control is presented which incorporates characteristics of impulse-variability and mass-spring models. Movements in the model were controlled with phasic torque impulses in agonist and antagonist muscles and a tonic agonist torque.

Characteristics of the phasic agonist and antagonist torque profiles were based on observed properties of movement-related EMGs and muscle isometric torques. Variability of the phasic impulses depended on impulse magnitude as in impulse-variability models. The model therefore predicted a speed-accuracy tradeoff for limb movement. The time of onset and magnitude of the antagonist torque depended on the magnitude of the preceding agonist torque as indicated in studies of movement-related EMGs. This led to the new concept of linkage between the agonist and antagonist muscle forces which was shown to be important for reducing variability of fast movements. Progressive development of linkage during practice could explain the previous findings of decreased movement variability with practice coupled with increased variability of movement-related EMGs.

It was concluded that an inherently variable motor system deals with the variability associated with generation of large muscle forces by linking the forces produced by opposing muscles. In this way, variability in net joint torques and in movements can be decreased without the need for the nervous system to closely regulate the individual torques.     

Stabilisation of bimanual coordination through visual coupling

Eight right-handed participants performed a bilateral circle tracing task in symmetric or asymmetric patterns. Circle tracing was performed in synchrony with an auditory metronome and a visual display at, or comfortably below, each participant's transition frequency. The visual display consisted of a row of five light-emitting diodes (LEDs) arranged between the two circles (hands). Bimanual pattern stability was examined under conditions where the direction of illumination of the visual stimuli was compatible or incompatible with the hand direction. Symmetric patterns maintained stability for both movement rates whereas asymmetric patterns exhibited loss of stability at the transition frequency. Spontaneous reversals in circling direction occurred predominantly (94%) through the nondominant hand. Laterality effects were also evident in the aspect ratio (circularity of trajectories) and limb frequency variation, particularly in asymmetric patterns at the transition frequency. Compatibility between the stimulus direction and circling direction served to: stabilise symmetric patterns; stabilise asymmetric patterns by delaying the onset of transition; and stabilise the individual limb dynamics when the direction of the dominant side was compatible with the visual stimulus. The data from this multisegmental task lend support to a model of coupled oscillators whereby the coupling strength is anisotropic between the dominant and nondominant side, and lend further support for an account of manual asymmetries by way of a preferential perception–action coupling through the dominant limb. PsycINFO Classification: 2320     

Proprioceptive perception of phase variability

Previous work has established that judgments of relative phase variability of 2 visually presented oscillators covary with mean relative phase. Ninety degrees is judged to be more variable than 0 degrees or 180 degrees, independently of the actual level of phase variability. Judged levels of variability also increase at 180 degrees. This pattern of judgments matches the pattern of movement coordination results. Here, participants judged the phase variability of their own finger movements, which they generated by actively tracking a manipulandum moving at 0 degrees, 90 degrees, or 180 degrees, and with 1 of 4 levels of Phase Variability. Judgments covaried as an inverted U-shaped function of mean relative phase. With an increase in frequency, 180 degrees was judged more variable whereas 0 degrees was not. Higher frequency also reduced discrimination of the levels of Phase Variability. This matching of the proprioceptive and visual results, and of both to movement results, supports the hypothesized role of online perception in the coupling of limb movements. Differences in the 2 cases are discussed as due primarily to the different sensitivities of the systems to the information.     

An analysis of air-stepping in normal infant vervet monkeys

Limb movements during air-stepping were analyzed in three neonatal vervet monkeys over a three-week period. The movements had similar temporal organization both across animals and across time. For example, the duration of both the hind and the forelimb cycle equaled about 500 ms, with hind limb return strokes lasting much longer than the hind limb power strokes. Furthermore, there were clear indications of both intra- and interlimb coordination. Specifically, all the joints of a limb tended to flex and extend simultaneously, and contralateral and ipsilateral limb pairs had an average phase relationship of approximately 50% of cycle duration. Despite a qualitative similarity between limb movements during air-stepping in the neonates and overground locomotion in older animals, there were notable differences both in temporal relationships and joint displacement patterns. Finally, there appeared to be important similarities between air-stepping in these monkeys and stepping in newborn humans. Most notably, both tended to disappear after a limited period. The implications of these similarities, as well as the overall results, are discussed in relation to the understanding of the development of locomotor behavior in human and nonhuman primates, using approaches based both upon the hard-wired and dynamic models.     

An Analysis of Air-Stepping in Normal Infant Vervet Monkeys

Limb movements during air-stepping were analyzed in three neonatal vervet monkeys over a three-week period. The movements had similar temporal organization both across animals and across time. For example, the duration of both the hind and the forelimb cycle equaled about 500 ms, with the hind limb return strokes lasting much longer than the hind limb power strokes. Furthermore, there were clear indications of both intra- and interlimb coordination. Specifically, all the joints of a limb tended to flex and extend simultaneously, and contralateral and ipsilateral limb pairs had an average phase relationship of approximately 50% of cycle duration. Despite a qualitative similarity between limb movements during air-stepping in the neonates and overground locomotion in older animals, there were notable differences both in temporal relationships and joint displacement patterns. Finally, there appeared to be important similarities between air-stepping in these monkeys and stepping in newborn humans. Most notably, both tended to disappear after a limited period. The implications of these similarities, as well as the overall results, are discussed in relation to the understanding of the development of locomotor behavior in human and nonhuman primates, using approaches based both upon the hard-wired and dynamic models.     

The planning and execution of sequential eye movements: saccades do not show the one target advantage

The present experiment examined the one-target advantage (OTA) with regard to saccadic eye movements. The OTA, previously found with manual pointing responses, refers to the finding that movements are executed faster when the limb is allowed to stop on the target compared to the situation where it has to proceed and hit a second target. Using an adapted limb movement OTA task, saccades of 5 degrees and 15 degrees were made to (a) a single target (one-target), (b) one target and immediately to another target without a change in direction (two-target-extension), and (c) one target and immediately back to the start location (two-target-reversal). Unlike manual movements, the movement times for the initial saccade in the two-target-extension condition were not prolonged compared to either of the other two conditions. Moreover, this pattern of results was found for both the shorter and longer amplitude saccades. The results indicate that the OTA does not occur in the oculomotor system and therefore is not a general motor control phenomenon.     

Perception of the symmetrical patterning of human gait by infants

Two experiments tested 3- and 5-month-old infants' sensitivity to properties of point-light displays of human gait. In Experiment 1, infants were tested for discrimination of point-light displays of a walker and a runner, which, although they differed in many ways, were equivalent with regard to the phasing of limb movements. Results revealed that 3-month-old, but not 5-month-old, infants discriminated these displays. In Experiment 2, the symmetrical phase-patterning of the runner display was perturbed by advancing two of its limbs by 25% of the gait cycle. Both 3- and 5-month-old infants discriminated the walker display from this new phase-shifted runner display. These findings suggest that 3-month-old infants respond to the absolute and relative motions within a single limb, whereas 5-month-old infants respond primarily to the relations between limbs and, in particular, to the bilateral symmetry between the limbs.     

On the space-time structure of human interlimb co-ordination

In three experiments we show, using behavioural measures of movement outcome, as well as movement trajectory information and resultant kinematic profiles, that there is a strong tendency for the limbs to be co-ordinated as a unitary structure even under conditions where the movements are of disparate difficulty. Environmental constraints (an obstacle placed in the path of one limb, but not in the other) are shown to modulate the space-time behaviour of both limbs (Experiment II). Our results obtain for symmetrical (Experiment I) as well as asymmetrical movements that involve non-homologous muscle groups (Experiment III). These findings suggest that in multi-joint limb movements, the many degrees of freedom are organised to function temporarily as a single coherent unit that is uniquely specific to the task demands placed on it. For movements in general, and two-handed movements in particular, such units are revealed in a partitioning of the relevant force demands for each component (a force scaling characteristic) and a preservation of the internal “topology” of the action, as indexed by the relative timing among components. These features, as well as systematic deviations from perfect synchrony between the limbs can be rationalised by a model that assumes the limbs behave qualitatively like non-linear oscillators.