Handbook of Psychology, Volume 4: Experimental Psychology

Motor Coordination 337

Figure 12.14 PotentialsV() as specified by Haken et al. (1985) for dif-

ferent ratios of baof the parameters.

The nature of structural constraints on coordination and
their combination with task constraints or task-related inten-
tions is nicely captured by perhaps the most influential model
of motor coordination and its developments (Haken, Kelso, &
Bunz, 1985). This model applies to tasks that require concur-
rent oscillations of at least two effectors, for example, the two
hands. When these oscillatory movements are produced sym-
metrically, there is essentially no difficulty in speeding them
up as far as this is possible. However, when they are produced
asymmetrically, the phase relation between the oscillations is
less stable, and occasionally symmetric movement cycles in-
trude (Cohen, 1971). When the asymmetric movements are
speeded up, stability is reduced even more, provided that sub-
jects are instructed to maintain the asymmetric phase relation
(Lee, Blandin, & Proteau, 1996). However, with a “let it go”
instruction, subjects tend to switch to symmetric movements
at a certain critical frequency (Kelso, 1984).
Haken et al. (1985) modeled these phenomena in terms of
what came later to be called an intrinsic coordination dynam-
ics. The model was formulated at two levels, the level of actual
movements and the level of an order parameter that captures
the relation between the periodic movements. Basically, at the
kinematic level two nonlinear oscillators were posited, one for
each effector, with a nonlinear coupling in addition. Relative
phasewas chosen as the order parameter (or collective vari-
able); this is the phase difference between the two oscillatory
movements.Forthisvariablethedynamicswerespecified
based mainly on formal considerations: =–asin–
2 bsin 2. Better known is the formulation in terms of a po-
tential function V with  =dVd,V=–acos–b
cos 2. This potential, which is illustrated in Figure 12.14, has
stableequilibriaat=n ,n=0,1,2,...,providedthe
parametersaandbare within certain ranges. Stable equilibria
are characterized by’s being positive for smaller values of
and negative for larger values, so that relative phase will drift
back to the equilibrium angle whenever it deviates as a conse-
quence of some perturbation; in the potential function, stable
equilibria are characterized by minima.
The ratio of the parameters aandbis hypothesized to de-
pend on movement frequency, bbecoming relatively smaller
as frequency increases. When it becomes sufficiently small,
the stable equilibria at =m ,m=1,3,5,...disap-
pear (cf. Figure 12.14). This corresponds to the observation
that, as the frequency increases, only symmetric oscillations
(in-phase oscillations in formal terms) are maintained while
asymmetric oscillations (anti-phase oscillations) tend to
switch to symmetric ones.
This account of the switch is based pretty much on formal
considerations, and other models are available with stronger
reference to physiological or psychological considerations or

both (Grossberg et al., 1997; Heuer, 1993b). In addition, the

prediction that the switch should be associated with reduced

movement amplitudes (Haken et al., 1985) is not necessarily

correct (Peper & Beek, 1998). Nevertheless, the model cap-

tures nicely the soft nature of structural constraints, and an

extension of it illustrates how structural constraints bias per-

formance when they deviate from task constraints or task-

related intentions.

Yamanishi, Kawato, and Suzuki (1980) asked their sub-

jects to produce bimanual sequences of finger taps at various

phase relations. The stability of phasing was highest with

synchronous taps (relative phase of 0°), second highest with

alternating taps (relative phase of 180°), and lower at all other

relative phases. In addition the mean relative phases were bi-

ased toward the stable relative phases at 0° and 180°. Schöner

and Kelso (1988) modeled these effects by way of adding a

term to the intrinsic dynamics that reflects the “intention” to

reach a target relative phase, so that the potential becomes

V=–acos–bcos2–ccos((–)2).Whenthe

intendedrelativephasediffersfrom=0°and=180°,

the minima of this potential are broader, corresponding to

an increased variability, and shifted away from the intended

relative phase, corresponding to the observed systematic

biases.

Basic Structural Constraints on Coordination

Structural constraints on coordination are indicated by sys-

tematic errors. They have been studied mainly by means of