Tuesday 13 September 2011

Coordination dynamics and relative speed

The Bingham model of coordinated rhythmic movement makes three predictions. First, it predicts that movement stability is a function of perceptual ability, and we confirmed this in two ways (by showing how people can move stably at non-0° with transformed visual feedback (Wilson et al, 2005) and by showing that perceptual learning of 90° led to improved movement stability without practice at the movement task; Wilson et al, 2010). This prediction is also supported by recent work by Kovacs and Shea, who are busy demonstrating that transformed, Lissajous feedback breaks the classic pattern of movement stability in coordination tasks. The second prediction is that relative phase is specified by the relative direction of motion; we confirmed this by selectively perturbing various components of motion and showing selective effects on performance (Wilson & Bingham, 2008). 

The third prediction was that the detection of relative direction was conditioned on the relative speed; the latter was simply a noise term. de Rugy, Oullier & Temprado (2008) tested this prediction by using an amplitude manipulation to alter the relative speeds. Their data did not support the model predictions, and they concluded that the approach taken by the Bingham model was flawed. We recently replicated their experiment (Snapp-Childs, Wilson & Bingham, in press as of Friday; download) and identified numerous critical flaws in their design and analysis which invalidated their criticism.

A problem for the model: de Rugy et al (2008)
de Rugy et al ran a simple study. They had participants control a single dot on a screen using a joystick (similar to our unimanual experimental setup), and track a computer-controlled target at either 0° or 180°. They manipulated the frequency of the required movement (0.75Hz, 1.25Hz, and 1.75Hz) to alter the overall speeds involved. They then altered the amplitude of movement the person had to track so that it was either the same as or three times as large as the person's amplitude, which remained constant. There were then 4 key conditions: moving at 0° with the same or different amplitude, and moving at 180° with the same of different amplitude. Critically, the relative speed in the 0°/different amplitude condition is the same as the 180°/same amplitude condition, and thus these conditions should show similar phase variability. The 180°/different amplitude should then be the most variable condition.

The critical result, however, was that the amplitude manipulation had no effect on movement variability; only the typical effects of frequency and required relative phase had any effect (Figure 1). They measured movement variability using a variable called SDΨ. Relative phase is a circular variable (the distribution of possible values lies on a circle) and there are special statistical methods for handling these kinds of data. One of the variables you can compute is the uniformity, which is a number that ranges from 0-1 and is 0 if the data are evenly distributed and 1 if all the data are the same; the higher the number, the more 'bunched up' the data are. High uniformity therefore corresponds to stable movement, in this case; de Rugy et al applied a standard transformation to this variable to convert it to a number equivalent to a standard deviation (ranging from 0 to infinity with a linear distribution); SDΨ = (-2 logeU)1/2.
Figure 1. Data from de Rugy et al (2008)

The rebuttal: Snapp-Childs et al (in press)
This paper poses a challenge to the model. However, that challenge was not actually as great as the authors made out. There were four critical concerns which we spotted immediately:
  1. de Rugy et al used the model to make predictions about the pattern of behaviour in a unimanual version of the task. The model was designed to capture the dynamics of the bimanual version; it includes a bi-directional coupling term which makes this version of the task more stable than the unimanual version. While we have used the unimanual task as a method which allowed us to investigate qualitative predictions of the model, we have very purposefully not fit any of these data using the model.
  2. Performance at 1.75Hz was spectacularly variable; participants could clearly not do the task. This supported our hunch that the unimanual version of the task is less stable, making those data uninterpretable (see the next point).
  3. SDΨ is a common but problematic measure. It only measures movement variability, with no regard to what the person is attempting to do; if a participant is trying to move at 90° and failing, slipping into 0°, then they will typically produce stable but incorrect behaviour. Researchers therefore usually also report the absolute error (which de Rugy et al did not do); all this does, though, is tell you whether SDΨ for that trial is a valid measure of performance. More recently (Wilson et al, 2010a, b) we have used a better solution - we report 'time on task', the proportion of time spent at the target phase, +/- an error bandwidth. This measure validly captures both accuracy and stability, and we have used it with great success. 
  4. The amplitude manipulation is only interesting if participants were, in fact, moving at the correct amplitudes. It's well known that this is difficult; people typically assimilate their amplitudes to the one they are tracking (e.g. Kovacs and Shea, 2010). de Rugy et al did not report or analyse the actually produced amplitudes. Maintaining the correct amplitude makes the unequal amplitude conditions a dual task, and thus harder.
We therefore 1) tested the stability of unimanual performance at 180° to confirm that it is less stable, and then 2) replicated de Rugy et al but then analysed the amplitudes produced and added 'time on task' to the dependent variables.

Unimanual movements are less stable than bimanual
Figure 2 shows 'time on task' when attempting to move at 180° with increasing frequency; performance past 1.25 Hz drops and performance past here remains almost entirely uncoordinated. This supported our first concerns.
Figure 2. Increasing frequency disrupts unimanual performance at much lower frequencies than the bimanual version
Is relative speed a noise term?
We then replicated and correctly analysed their experiment. We tested the three key conditions (0°/equal amplitudes, 0°/unequal amplitudes, 180°/equal amplitudes). de Rugy et al's result was that the two 0° conditions were equally stable. However, the model they claimed to be testing does not include amplitude control (part of this design), and if participants are unable to maintain the unequal amplitudes then that condition will stabilise. We asked the following questions:

Question 1: Did the participants maintain the required amplitudes? Only when the amplitudes were equal; in the 0°/unequal amplitude case, they increased their amplitude to be closer to that of the target, thus reducing the relative speed difference.

Question 2: Did the participants produce the correct relative phase? Only in the 0°/equal amplitude condition. In the other two conditions there were significant phase errors, which makes SDΨ an inappropriate measure of performance.

Question 3: Did movement stability decrease as relative speed increased when measured with time on task? Yes. The most stable condition was 0°/equal amplitude, then 0°/unequal amplitude, then 180°/equal amplitude. This ordering makes sense, given that participants reduced the relative speed in the middle condition by moving at the incorrect amplitude.

To summarise: we replicated their results when using SDΨ but when we actually measured amplitude control we found that the 0°/unequal amplitude and 180°/equal amplitude conditions did not produce the same relative speed, and that performance measured correctly reflected this fact. Relative speed is, indeed, the relevant noise term.

Modelling unimanual performance and amplitude control
The de Rugy experiment tested a model of bimanual coordination under equal amplitudes with a unimanual task that entailed amplitude control to maintain different amplitudes. It's therefore no surprise that they failed to find evidence in favour of the model, and the phase and amplitude errors also meant that their analysis using only SDΨ was not up to the task. One of their conclusions, however, was that the basic Bingham model was unlikely to be modifiable to cope with their data; we therefore addressed this as well in the final section. Recall that the reason this model is important is that it is built out of the empirically identified components of the actual system; additions to the model can only use actually perceivable properties, for instance, and not simply add terms to stand in for those properties.

Unimanual performance
We altered the basic model to be unimanual by simply removing the coupling; one oscillator (the person controlled dot) was driven by the perceived phase of the second, computer controlled dot, which was now not driven by phase (the driver was set to 0). We simulated performance at 180° and for various frequencies; like our participants, the model could not maintain coordination past ~1.5Hz.

Amplitude control
The original model does not control amplitude, but does include a parameter, c, where amplitude can be set. We therefore had to find a function that controlled this parameter, using only perceivable properties that did not violate the autonomy of the overall dynamic. The available variables come from the phase plane; we used the radius of the phase plot which uniquely specifies the amplitude throughout the trajectory. This radius is the 'energy' of the system, and there is evidence that people are sensitive to this variable (Goldfield et al, 1993). We modelled the perception of amplitude errors as perceiving the difference between the required amplitude and the current energy of the system.

Relative phase error correction
The model was designed to simulate the task under a 'non-interference' instruction: it's often the case that participants are told not to resist any transitions from, say, 180° to 0°. In this task, of course, people were told to maintain the target relative phase, and the unimanual task with amplitude variation meant people accumulated plenty of errors that needed to be corrected. We modelled phase corrections by using the model's method for judging the current relative phase (integrating Ρ over a 2s window); when this perceived relative phase was sufficiently different from the target, the model simply reset the initial conditions. This matches one correction strategy, where people move to one end and wait for the dot to catch up. We set the 'sufficiently different' threshold to be small for 0° and large for 180°.

The unimanual model which included both amplitude control and phase correction replicated the qualitative pattern of results. While there is actually a lot of work to be done to fine tune these implementations, this served as a proof-of-concept that a model following the perception-action modelling strategy could handle this task.

Why this matters
No other model is built under the rules which demand the entire model be built from actually perceived information coupling correctly formulated action components. This model therefore matters because it represents the future of perception-action dynamical systems modelling. When tested appropriately, the model passes with flying colours; but testing this correctly is hard and requires careful attention to all aspects of the task. de Rugy et al (2008) used the wrong task, the wrong measure, and failed to check the effects of their manipulations. My next project will build on the developments we developed to reveal these facts, and will extend the model to handle frequency and amplitude control, learning and error correction according to the perception-action rules.

de Rugy, A., Oullier, O., & Temprado, J. (2008). Stability of rhythmic visuo-motor tracking does not depend on relative velocity Experimental Brain Research, 184 (2), 269-273 DOI: 10.1007/s00221-007-1180-0 Download

Goldfield, E.C., Kay, B.A., & Warren, W.H. (1993). Infant bouncing: The assembly and tuning of action systems. Child Development, 64, 1128-1142. Download

Snapp-Childs, W., Wilson, A. D., & Bingham, G. P. (2011). The stability of rhythmic movement coordination depends on relative speed: the Bingham model supported Experimental Brain Research DOI: 10.1007/s00221-011-2874-x. Download

Wilson, A. D., & Bingham, G. P. (2008). Identifying the information for the visual perception of relative phase. Perception & Psychophysics, 70(3), 465–476. Download

Wilson, A. D., Collins, D. R., & Bingham, G. P. (2005). Perceptual coupling in rhythmic movement coordination – stable perception leads to stable action. Experimental Brain Research, 164(4), 517–528.

Wilson, A. D., Snapp-Childs, W., & Bingham, G. P. (2010). Perceptual learning immediately yields new stable motor coordination. Journal of Experimental Psychology: Human Perception & Performance,36(6), 1508-1514.

Wilson, A. D., Snapp-Childs, W., Coates, R., & Bingham, G. P. (2010). Learning a coordinated rhythmic movement with task-appropriate coordination feedback. Experimental Brain Research, 205(4), 513-520.

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