Source
- Haken, Kelso and Bunz (1985), LIT-588, read in full in NOTE-1424: a theory paper with one simulation and no new data. Its figures were not viewed; what they show is taken from their captions and the text.
- Kelso, Scholz and Schöner (1986), LIT-1851, read in full in NOTE-1433, figures included.
What was proposed and what was measured
The model (LIT-588, §§2–3). Relative phase φ between the hands is the order parameter, “created by the cooperation of the individual components” and in turn governing them (pp. 348–349). It moves in a potential V = −a cos φ − b cos 2φ: in-phase and anti-phase are both stable while |b| > |a|/4, and only in-phase below it, b/a falling as movement frequency rises. The φ equation is derived from two self-sustained oscillators, one per hand, whose coupling must be nonlinear and involve delay or derivatives; the authors say the choice “is not unique” (p. 351). One simulation with small noise shows the switch. The data modelled are Kelso’s earlier experiments, summarized and not shown: two stable phase relations, an abrupt switch from anti-phase to in-phase at a critical frequency, and the system staying in-phase when the frequency is lowered (p. 348).
The fluctuations (LIT-1851). As wrist or finger movement is paced faster, the standard deviation of relative phase rises as the anti-phase pattern nears the switch, peaks in the transition and falls once in-phase is reached; started in-phase, the same movement shows neither the switch nor the rise (Figs. 1–4). The wrist data average 30 runs on four of the ten subjects run, the finger data ten runs of one representative subject of seven. No statistical test is reported. The authors read it as critical fluctuations, “one of the key features of a nonequilibrium phase transition” (p. 283).
What this does not say
- That the switch is a nonequilibrium phase transition with critical slowing and hysteresis as early warning. Critical slowing is predicted in LIT-1851, which calls its own evidence “very preliminary”; hysteresis is reported from Kelso’s experiments in LIT-588 with no data. That is THEORY-069’s question.
- That the fluctuations hold across subjects. Most subjects run are not shown, and in the transition region the measure partly reflects the switch itself (NOTE-1433, Limitations).
- That the result is replicated. Both papers are by the model’s authors.
- That HKB-like dynamics reaches beyond bimanual movement. Kelso’s later extensions are reported in THEORY-069, from LIT-601.
Connections
- CLAIM-172 draws its criterion of a combined form from this model: a pair whose relative phase has its own attractors, fluctuations and switch.
- THEORY-069 is the general account, Proposed, with the promote_when this entry does not meet.