Contribution
It extends coordination dynamics from switches that happen to people to switches people make. Before it, the HKB dynamics described spontaneous transitions as a control parameter crosses a critical point. Here an instructed, intentional switch between the same two patterns, at frequencies where both are stable, is shown to take time in proportion to how stable the starting pattern is, as a model in which intention is one more term in the same dynamics predicts.
Key insight
An intention does not override the pattern dynamics; it tilts them. Leaving the deep in-phase well takes longer than leaving the shallow anti-phase one, even when the person wants to leave, so the landscape that governs involuntary switching also shapes voluntary switching.
Assumptions
- The intrinsic dynamics stay valid under intention. Relative phase obeys φ̇ = −a sin φ − 2b sin 2φ + √Q ξ_t (Eq. 1) while an intention is in force (p. 11).
- Intention is a force toward the intended phase ψ. f_int(φ, t) = −θ(t − t₀) c_int sin(φ − ψ) (Eq. 2), switched on instantaneously at the transient’s onset t₀, “neglecting hypothesized physiological processes associated with intentional acts”.
- Parameters from the spontaneous-transition experiments. a = 1 Hz, b = 0.5 Hz, Q = 0.25 Hz, “for which the two states … differ in their stability, and the relative phase fluctuations and relaxation times are of the correct order of magnitude” (p. 11).
- Switching time is measured by eye. From where φ “first diverged from its previous mean state” to where it stabilized within ±15° for two consecutive cycles, using interactive graphic displays of the trajectories and relative phase (p. 10).
Key results
- Task (p. 9). Index fingers moved in the transverse plane, in-phase or anti-phase; a metronome set the frequency for 10 cycles and stopped; after 10 more cycles a tone cued an immediate switch to the other pattern, held for 20 cycles. Fingers tracked by WATSMART at 200 Hz, with EMG of the first dorsal interosseous.
- Asymmetry (p. 10, Fig. 2). Mean switching time, over six trials and four subjects, at 1.75–2.75 Hz: about 300–400 ms from anti-phase to in-phase and about 550–650 ms from in-phase to anti-phase, read from the figure. “Clearly, switching from the in-phase to the anti-phase mode is slower by approximately a factor of two.” Per-subject ANOVA: “Fs(1,60) = 52.5 (JB), 20.7 (JD), 27.3 (JJ) and 100.4 (JS), Ps < 0.0001.”
- Distributions (Fig. 3). In-phase to anti-phase has a larger mean and a longer tail.
- Model (pp. 11–12, Fig. 4). Solving the Fokker–Planck equation for Eq. (3) with c_int ≈ 10 Hz, “chosen to account roughly for the observed order of magnitudes”, gives densities with the same asymmetry and tail, and mean switching times of 0.41 s (anti- to in-phase) and 0.85 s (in- to anti-phase), described as “in remarkable agreement with the experimental result”.
- Holding an unstable pattern. The theory “can account for the ability of the system to sustain a pattern that is intrinsically unstable, because an intention (as part of the dynamics) can change the stabilities of the intrinsic dynamics” (p. 12).
- Cited, not shown (p. 8). “As theoretically predicted [11,12] enhanced fluctuations of relative phase [13,14] and growth in relaxation time … [15,16] — both indicative of loss of stability — accompanied the transition from one (anti-phase) to the other (in-phase) pattern”, and switching times of spontaneous transitions “were predicted [12] and found to be in striking agreement with theory [16]”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Intentional switching from in-phase to anti-phase takes about twice as long as the reverse | strong for these subjects | four subjects, six trials each per condition, F(1,60) ≥ 20.7, p < 0.0001 each; consistent across 1.75–2.75 Hz |
| C2 | The asymmetry follows from the patterns’ different stability | moderate | stochastic model with intrinsic parameters from earlier experiments; one adjusted parameter |
| C3 | Intention acts as a perturbation within the order-parameter dynamics | moderate as a model | C2’s fit; the form of f_int is the simplest chosen, not derived |
| C4 | Spontaneous transitions show enhanced fluctuations and growing relaxation time | cited | refs. 13–16; not shown here |
| C5 | The tools of stochastic dynamics are adequate for “inherently biological and even psychological questions” | asserted | the summary’s generalization |
Concepts
- Intrinsic dynamics — the relative-phase dynamics without any intention, Eq. (1), identified from spontaneous transitions.
- Intentional information — a contribution to the dynamics that attracts relative phase to an intended value ψ, Eq. (2).
- Switching time — the length of the transient from one pattern to the other, a random variable whose distribution follows from the time-dependent probability density of relative phase.
Connections
It applies the stochastic HKB theory (Schöner, Haken and Kelso 1986, not filed; the model of LIT-588) and the intentional-dynamics formulation of Schöner and Kelso (refs. 20–21, not filed) to new data, and relies on Kelso, Scholz and Schöner 1986 (LIT-1851, NOTE-1433) and Scholz, Kelso and Schöner 1987 (not filed) for the spontaneous transition’s loss of stability. Kelso 2021 (LIT-601, NOTE-467) summarizes it.
Bearing on the record
- THEORY-069: partly, and indirectly. It does not test the
spontaneous transition, so it meets none of the
promote_when’s three findings at first hand. What it adds is independent of them: the two patterns differ in stability as the model says, measured by switching times rather than fluctuations, and in-phase is the deeper well. Its introduction cites growth in relaxation time before the spontaneous switch (refs. 15–16), which is the evidence THEORY-069 still lacks; the citation points to Scholz, Kelso and Schöner 1987 (Physics Letters A 123:390, doi:10.1016/0375-9601(87)90038-7), the paper to file next. - THEORY-069’s report of a fit “with a single free parameter”. This paper’s fit has one new parameter, c_int, set to the order of magnitude of the switching times; the theoretical means (0.41 s and 0.85 s) are each above the observed (about 0.35 s and 0.6 s, read from Fig. 2). The asymmetry and the tail are reproduced; the values only roughly.
- behavioral-integration. A case where intention and pattern dynamics are not separate controllers: the intended act runs through the same landscape as the involuntary one. It bears on THEORY-069’s line that no controller chooses the pattern: here a choice is made, and the dynamics still sets its cost.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Four subjects, all named by initials; whether they were the authors or naive is not said.
- Switching time was judged interactively from displays, with a ±15° criterion; no reliability check is reported.
- The model’s fit is to order of magnitude; c_int is free, the intention is assumed to act instantly at the observed onset, and the delay between cue and onset is not analysed.
- Spontaneous-transition evidence is cited, not shown.
- A slip. The text says the theoretical means “should be compared to the data in fig. 1”; the switching-time data are in Fig. 2.
Open questions
- Does the intentional term’s strength vary with the person or the frequency, and can it be measured independently of the switching times?
- Is the cue-to-onset delay also asymmetric?
- Do switches in other effectors, or between people, show the same dependence on pattern stability?