ثبات محلی در حرکات ریتمیک هماهنگ: نوسانات و زمان آرام سازی

An experiment was conducted to examine the stability of the anti-phase and in-phase modes of coordination by means of both fluctuations and relaxation times. Participants (n=6) performed a rhythmic bimanual forearm coordination task that required them to oscillate their forearms in-phase and anti-phase while grasping two manipulanda at fixed frequencies ranging from 0.6 to 1.8 Hz. Relaxation times were measured as the time taken to return to a stable mode following the application of a transient mechanical torque. It was found that relaxation times were not different statistically across participants, frequencies, and coordinative modes. However, fluctuations, as indicated by the mean S.D. of relative phase across individual frequency plateaus, were significantly greater in the anti-phase than in the in-phase mode of coordination, p<0.05. Whilst providing new empirical support for the notion that relaxation times should be of the same order of magnitude at frequencies outside transition regions, the findings suggest that the level of stochastic noise in the anti-phase mode is greater than that of the in-phase mode. Implications are made for the future assessment of local pattern stability.

Dynamical systems theory has provided opportunities to determine principles that govern the organization and development of stable coordination patterns. Central to the dynamical systems approach has been the observation of stable, rhythmic anti-phase and in-phase modes (e.g., finger or forearm oscillations) at the level of the order parameter relative phase. The identification of pattern switches or phase transitions from the anti-phase to the in-phase mode in response to increases in the control parameter oscillation frequency led to the development of the HKB model of bimanual coordination (Haken, Kelso, & Bunz, 1985). To explain changes in the variability of coordination patterns, Schöner, Haken, and Kelso (1986) provided a stochastic extension of the HKB model. Within the current paper indices of local pattern stability (i.e. fluctuations and relaxation times), stemming from the work of Schöner et al. (1986), are compared below critical regions. Whilst findings provide information on stochastic and deterministic contributions to pattern stability they will have direct implications for the future assessment of local pattern stability with relaxation times and the mean S.D. of relative phase. 1.2. Standard deviations Deviations from stable coordination patterns or attractor states are caused by stochastic noise of a supposedly constant strength that arises from the system's many interacting degrees of freedom (Kelso, Schöner, Scholz, & Haken, 1987). To account for the presence of such fluctuations, Schöner et al. (1986) extended the HKB (1985) model by incorporating a stochastic noise term ξ with strength parameter Q. In doing so, Schöner et al. (1986) were able to provide a quantitative account for the enhanced fluctuations observed by Kelso, 1981 and Kelso, 1984 in the vicinity of phase transitions from anti-phase to in-phase coordination. Fluctuations are typically assessed by measuring the standard deviation (S.D.) of relative phase across individual frequency plateaus and averaging those across multiple trials. As the size of fluctuations is determined by the stability of an attractor and the magnitude of stochastic noise in the system ( Schmidt, Treffner, Shaw, & Turvey, 1992), they provide a relative indication of the local stability of a particular mode. Theoretically, the influence of stochastic noise on the stable stationary behavior of the system may be understood in terms of the attractor landscape defined by a so-called potential. Contained within the landscape is the in-phase attractor with a narrow, steep-sided concave well. Across scaled changes in the required movement frequency the system minimally deviates from the in-phase attractor in response to perturbations from stochastic noise. In contrast, the anti-phase attractor has a shallower, less concave well and less pronounced minimum (Kelso & Ding, 1993). When the system is in the anti-phase attractor constant stochastic noise causes greater deviations from the minima, relative to those observed in the in-phase attractor. When approaching a transition region elevated fluctuations termed `critical fluctuations' may arise due to the anti-phase potential becoming shallower, less concave and therefore less able to resist perturbations from constant stochastic noise. Eventually, perturbations may push the system toward the in-phase mode (Kelso, Scholz, & Schöner, 1986). This theoretical description has been widely supported at a coordinative level in experiments that have examined fluctuations across frequencies close to participants' transition regions (e.g., Jeka, Kelso, & Kiemel, 1993; Kelso et al., 1986; Schmidt, Carrello, & Turvey, 1990; Wimmers, Beek, & van Wieringen, 1992). In the motor behavior literature, the local stability of steady state in-phase and anti-phase coordination has commonly been determined by assessing fluctuations of relative phase in participants' upper limbs (e.g., Fontaine, Lee, & Swinnen, 1997; Lee, Swinnen, & Verschueren, 1995; Swinnen, Lee, Verschueren, Serrien, & Bogaerds, 1997; Swinnen et al., 1998) and index fingers (Zanone and Kelso, 1992 and Zanone and Kelso, 1997) at low movement frequencies between 1.0 and 1.33 Hz. These studies have reliably shown that prior to learning a new movement pattern below critical regions, the anti-phase mode is locally less stable than the in-phase mode. 1.3. Relaxation times Stemming from the treatment of observed fluctuational behavior in the stochastic dynamical model of Schöner et al. (1986) are predictions relating to local relaxation times and their behavior in the approach to transitions. Local relaxation times are a measure of local pattern stability, indicative of the time taken by the system to relax to a stable stationary state following a small perturbation (Schöner et al., 1986). When the system is perturbed the relaxation time is determined by the stability of the attractor that the system resides within. The shallower the attractor, the longer the relaxation time, and therefore the less stable the coordination pattern (Schmidt, Shaw, & Turvey, 1993). Although relaxation times and fluctuations (e.g., standard deviations) are both indices of local stability, relaxation times are governed by deterministic aspects (i.e., the potential or gradient dynamics) independently of stochastic noise, thereby providing an absolute indication of coordinative pattern stability.1 Strong increases in relaxation time that are indicative of a loss of local pattern stability and an imminent phase transition are referred to as critical slowing down (Scholz & Kelso, 1989). Despite the differing aspects of local stability assessed with fluctuations and relaxation times, evidence of critical slowing down should be accompanied by a concurrent increase in critical fluctuations (Jeka & Kelso, 1989). To test predictions regarding relaxation times, Scholz and Kelso (1989) applied perturbations to participants' bilateral oscillating index fingers using mechanical torques (see also Scholz, Kelso, & Schöner, 1987). When perturbations of equivalent magnitude were applied to participants' intrinsically stable anti-phase mode at frequency plateaus approaching their pre-determined mean transition frequency toward the in-phase mode, relaxation times became significantly longer than those measured in the in-phase mode at the same frequency.2 Relaxation times in the in-phase mode either remained constant or decreased. Similar findings were reported in a study by Wimmers et al. (1992) that extended the bifurcation paradigm for the case of bilateral coupling to unilateral coupling. In a rhythmical forearm tracking task, Wimmers et al. (1992) observed significantly larger fluctuations and longer relaxation times at pre-transition frequency plateaus in the anti-phase mode compared to the in-phase mode at the same frequency. Across changes in movement frequency relaxation times in the in-phase mode remained constant. In summary, the findings of Scholz and Kelso (1989) and Wimmers et al. (1992) are consistent with the predictions of Schöner et al. (1986), that: (a) a loss of local stability in the anti-phase attractor underlies the observed transitions to the in-phase mode; and (b) the stability of the in-phase mode remains invariant across scaled changes in movement frequency. 1.4. Relaxation times: Steady state analysis Until recently the analysis of participants' relaxation times at a coordinative level has been confined to five and two frequency plateaus before participants' critical transition frequency, respectively.3 Adopting this methodological strategy has enabled predictions of the HKB model and Schöner et al.'s (1986) extension to be robustly identified across a variety of systems (Beek, Rikkert, & van Wieringen, 1996; Post et al., 2000b). In the stochastic extension to the HKB model Schöner et al. (1986) estimated the duration of relaxation times for the anti-symmetric and symmetric modes away from transition regions (Trel=0.25 s). However, these qualitative estimates based upon the time participants' oscillating index fingers adapted to systematic changes in the driving frequency prescribed by a metronome, were only to provide the adequate order of magnitude of relaxation time. Additionally, the informational nature of the perturbation that assumed a constant strong perceptual coupling between participants' and the `non-forcing' specificational information provided by the stimulus of a metronome ( Kelso, 1994) suggests the validity of these findings be viewed with caution. Post et al. (2000b) recently examined the local stability of steady state anti-phase and in-phase relations between 0.75 and 2.25 Hz with the S.D. of relative phase and an alternative measure for the swiftness of the relaxation process following a mechanical perturbation. For anti-phase coordination the S.D. of relative phase was significantly higher than for in-phase coordination, whilst for both modes larger S.D. were observed toward higher movement frequencies. The strength of attraction, of coordinative patterns was examined by fitting an exponential decay function to the continuous relative phase between participants' oscillating forearms. The resultant exponential decay parameter (β), that was inversely related to relaxation time, was equivalent for anti-phase and in-phase coordination. The interpretation of this finding was that the deterministic aspects of the dynamics governing these intrinsically stable patterns was equivalent. In contrast to predictions of Schöner et al. (1986), but consistent with previous empirical observations ( Beek et al., 1996), significant effects of frequency for fluctuations and the decay parameter β indicated that both anti-phase and in-phase became less stable toward higher movement frequencies. Based on the mean relative phase, S.D. and values of β, Post et al. (2000b) determined that the parameter Q, the strength of the intrinsic noise in the dynamical system, was significantly larger for the anti-phase than the in-phase mode. This finding which opposed the estimation of Schöner et al. (1986), that noise strength is the same for anti-phase and in-phase coordination, indicated that variability measures of relative phase may not always be suitable indices for comparing local pattern stability. For example, where the S.D. of relative phase has previously been employed to demonstrate that fluctuations in the anti-phase mode are larger than those of the in-phase mode, the findings of Post et al. (2000b) suggest that differences may have arisen due to a greater level of stochastic noise in the anti-phase mode rather than a reduction in pattern stability. In a subsequent study, Post, Peper, and Beek (2000a) examined the effects of frequency and amplitude on the stability of relative phase. From identical perturbation trials to those employed by Post et al. (2000b), results for the S.D. of relative phase and the decay parameter β, were largely equivalent The only discrepancy in the findings was that no effect of frequency was observed for the decay parameter β, thereby indicating that pattern stability remained constant. In contrast to the findings of Post et al. (2000b) but supportive of the prediction of Schöner et al. (1986), no statistical effect of coordination mode on strength parameter Q was observed. The interpretation of these findings was that the use of variability measures of relative phase was justified. The contrasting findings of Post and colleagues with regard to estimates of Q suggest that this issue remains unresolved. In consideration of the previous widespread use of S.D. of relative phase as an indicator of pattern stability in the motor learning and control literature, research is required to clarity this issue. 1.5. Testable predictions The purpose of the current study is to contribute to the emerging body of knowledge with regard to the local dynamic relations governing rhythmical forearm movements at a coordinative level in stable stationary regions. Specifically, the primary aim is to evaluate the stability of anti-phase and in-phase coordination by assessing fluctuations and relaxation times. The level of fluctuations will be assessed by determining the S.D. of relative phase. Based upon a wealth of previous research fluctuations are predicted to: (a) be greater for the anti-phase rather than the in-phase mode of coordination; and (b) increase in both modes coordination across scaled changes in movement frequency. Relaxation times will be measured as the duration of the return process following a mechanical perturbation. The criteria for determining the duration of the return process will be based on those adopted in the original work of Scholz and Kelso (1989). In consideration of the recent empirical findings of Post and colleagues relaxation times in both modes of coordination are predicted to be equivalent. If results obtained with an alternative methodology to that employed in the work of Post et al., 2000a and Post et al., 2000b were consistent with these predictions support would not only be gained for the local stability of antiphase and in-phase coordination to be equivalent, but indirect support would also be gained for the finding of Post et al. (2000b) that the level of stochastic noise in the anti-phase mode is greater than that of the in-phase mode. If results were to show that relaxation times remained constant across steady state regions in the face of scaled changes in frequency (cf. Post et al., 2000a) the interpretation would be that deterministic aspects of pattern stability were not affected by changes in movement frequency. However if relaxation times increased toward higher movement frequencies across steady state regions the interpretation would be that changes in movement frequency do elicit changes in the gradient or potential dynamics that govern pattern stability (cf. Post et al., 2000b).