Contribution
It is the first quantified report that variability of relative phase grows before the switch from anti-phase to in-phase coordination. Before it, the HKB model (LIT-588) and its stochastic extension predicted such critical fluctuations, and earlier work only “hinted” at them, for example in the widening of Lissajous trajectories (p. 280). After it, the switch had one of the signatures that separate a loss of stability from a programme changing on a cue.
Key insight
If the switch is a loss of stability, the anti-phase pattern should wobble more as its attractor flattens, before it is abandoned, and the in-phase pattern, whose attractor does not flatten, should not. That is what was found: fluctuation grows only in the pattern that is about to be lost.
Assumptions
- Relative phase is the order parameter and movement frequency the control parameter, following HKB (pp. 279–280).
- The SD of relative phase measures order-parameter fluctuation. In the wrist experiments, from a point estimate (one wrist’s peak relative to the other’s) averaged over four-cycle segments; in the finger experiments, from a continuous estimate every 5 ms, φ_R = tan⁻¹(Ẋ_R/X_R) with normalized position and velocity, and φ = φ_R − φ_L (p. 282).
- Stationarity within each frequency plateau in the finger experiments, checked with a 0.5 s window moved through each 4 s plateau; “Stationarity was guaranteed less than 1 s after the parameter change”, so means and SDs use the last 3 s (600 samples) (pp. 282–283).
- Restriction to the relevant degrees of freedom. Forearms fixed so that only wrists or fingers move: “in order to reproduce the phenomenon precisely, movements must be restricted” (p. 280).
Key results
- Subjects (p. 280). “A total of 10 using movements around the wrist joints, and 7 using movements of the index fingers.” The figures show subsets.
- Wrist (pp. 280–282, Figs. 1–2). Frequency raised every 3–4 s on the experimenter’s instruction, not precisely controlled. Mean phase modulus stays near 180° for segments S1–S3, changes from S4, and settles at about 25° by S6. The SD is “relatively low and nearly constant” for S1–S3; “A significant increase in the magnitude of relative phase variation occurs at S4, immediately prior to the transition region”; it peaks in the transition and then falls to a constant level in-phase. Fig. 2 averages over 30 runs on four subjects. Pre-transition frequency (S4) ranged from 2.41 to 2.88 Hz across subjects; by S9, 2.99 to 4.04 Hz. (“Significant” is not tied to any test.)
- Two doubts the authors raise about the wrist data (p. 282). Few data points per segment from the point estimate, and possible non-stationarity from the uncontrolled rate of frequency change.
- Fingers (pp. 282–283, Figs. 3–4). Metronome-paced, 0.25 Hz steps at 4 s intervals. Fig. 4, ten runs of “a representative subject”, shows: (1) an abrupt change in mean relative phase from the asymmetrical to the symmetrical pattern at a critical frequency, and “No such shift … in the symmetrical pattern”; (2) a clearer enhancement of SD in the anti-phase pattern as the transition nears, and “No such fluctuational growth occurs in the symmetrically prepared pattern”; (3) below the transition the in-phase pattern is less variable at every frequency, down to 1.25 Hz; (4) above it the two preparations’ data coincide; (5) in the transition region the SD peaks while mean |φ| ≈ 85°, near the 90° of an equidistribution.
- Conclusion (p. 283). “We believe this observation provides strong evidence for the presence of critical fluctuations”; the experiments have “documented, for the first time in the context of coordinated biological motion, one of the key features of a nonequilibrium phase transition”.
- Critical slowing: predicted, not measured (p. 283). Relaxation time of the anti-phase mode after a perturbation should increase toward the transition, and not for the in-phase mode. “Recently, very preliminary evidence has been obtained that line narrowing does in fact occur [13]. However, longer frequency plateaus are required … and these — as well as direct measures of relaxation times using perturbation methods — are incorporated into new experiments that we hope to report soon.”
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Relative-phase variability grows as anti-phase approaches the switch and peaks in the transition | moderate | wrist: 30 runs, four subjects (Fig. 2); fingers: ten runs, one representative subject (Fig. 4); no statistical test reported |
| C2 | No such growth, and no switch, occurs when the movement starts in-phase | moderate | Fig. 4, one subject |
| C3 | In-phase is more stable than anti-phase at all pre-transition frequencies | moderate | lower SD at every frequency from 1.25 Hz, Fig. 4 |
| C4 | The growth is critical fluctuation, a mark of a nonequilibrium phase transition | moderate as interpretation | C1–C2 with the HKB theory; the authors ask for “further exploration” of the identity with other such transitions |
| C5 | Critical slowing down precedes the switch | weak; prediction | ”very preliminary” spectral evidence (ref. 13), not shown |
Concepts
- Critical fluctuations — enhanced fluctuation of the order parameter near a critical value of the control parameter, as a state loses stability (abstract; Haken’s Synergetics §6.7).
- Critical slowing down — growth of the time to return to a stable state after a small perturbation as the transition nears; visible as narrowing of the spectral line of relative phase (p. 283).
- Point and continuous estimates of relative phase — the first from peak times, once per cycle; the second from each finger’s phase-plane angle at every sample.
Connections
It tests a prediction of the stochastic generalization of HKB (Schöner, Haken and Kelso 1986, ref. 11, not filed) of the model in LIT-588 (NOTE-1424). The direct measures of relaxation time it promises appeared as Scholz, Kelso and Schöner 1987, “Critical slowing down and switching time” (Physics Letters A 123:390–394, doi:10.1016/0375-9601(87)90038-7, not filed). Kelso, Scholz and Schöner 1988 (LIT-1873, NOTE-1449) cites this paper and that one for the loss of stability, and moves from spontaneous to intended switching. Kelso 2021 (LIT-601, NOTE-467) summarizes all three.
Bearing on the record
- THEORY-069: partly. Its
promote_whenasks for three findings at first hand. This paper gives one in full and part of another. It shows the switch running from anti-phase to in-phase as frequency rises, with no switch from in-phase (Figs. 1–4), and it shows the variance of relative phase rising before the switch, in the pattern that switches and not in the other: the “critical fluctuations” THEORY-069 had only from Kelso’s retrospective. It does not show the other half of the second finding, relaxation time after a perturbation; it predicts it and calls its own evidence “very preliminary”. It does not test hysteresis. And it is by the model’s authors, not an independent replication. The paper that would complete the second finding is Scholz, Kelso and Schöner 1987, not filed. - THEORY-069’s refutation clause (“refuted if the fluctuations and relaxation time do not grow before the switch”): the fluctuation half is met against refutation here, in a few subjects and without a reported test.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Thin reporting. Ten wrist and seven finger subjects were run, but the figures show four subjects (wrist) and one representative subject (fingers). No statistical test is reported, and “significant” is used descriptively. The other subjects’ finger data are not shown.
- The wrist protocol had uncontrolled frequency changes and few points per segment, as the authors say.
- A ceiling on the measure. In the transition the phase spreads toward uniform, so the SD’s peak there partly reflects the switch itself; the pre-transition rise (S4, and the finger frequencies before the jump) is the part that bears on early warning.
- Not shown: critical slowing, hysteresis, and whether the rise in SD depends on how fast frequency is raised.
Open questions
- Does relaxation time after a perturbation grow before the switch in the anti-phase mode only? (Scholz, Kelso and Schöner 1987.)
- Does the pre-transition rise hold across the other subjects, with a test?
- How far does the identity with physical nonequilibrium phase transitions go, as the authors themselves ask (p. 283)?