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LIT-588: A theoretical model of phase transitions in human hand movements

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StatusActive — read 2026-10-30 (NOTE-1424) in full, from the publisher's scan posted on Kelso's lab page at Florida Atlantic University (https://ccs.fau.edu/hbblab/pdfs/1985_Haken_Kelso_Bunz_Biol_Cyb.pdf); worth reading as the paper that states the HKB model: relative phase as order parameter, the potential V = −a cos φ − b cos 2φ with anti-phase lost when b/a falls below 1/4, and its derivation from two nonlinearly coupled limit-cycle oscillators. Read it knowing that it is a theory paper with one numerical simulation and no new data: the switch and the hysteresis it models are summarized from Kelso's earlier experiments, and critical fluctuations and critical slowing are named as future work, not reported.
Version3
Tagscomplex-systems, behavioral-integration, natural-sciences, embodied-cognition
Extended byLIT-601 — The Haken–Kelso–Bunz (HKB) model: from matter to movement to mind
Cited as “extends” byLIT-1851 — Nonequilibrium phase transitions in coordinated biological motion: critical fluctuations, LIT-1873 — Dynamics governs switching among patterns of coordination in biological movement
Cited as “paper” byNOTE-1424 — Haken, Kelso & Bunz — A theoretical model of phase transitions in human hand movements
Cited as “source” byTHEORY-069 — Rhythmic bimanual coordination switches from anti-phase to in-phase through a nonequilibrium phase transition in relative phase, captured by the HKB equation, with critical fluctuations and critical slowing as early warning, THEORY-1227 — Haken, Kelso and Bunz model the switch of rhythmic bimanual coordination from anti-phase to in-phase at a critical frequency as the loss of one stable state of an order parameter, relative phase, derived from two coupled oscillators; Kelso, Scholz and Schöner report the variability of relative phase rising before the switch in the anti-phase pattern only, in a few subjects and with no statistical test, THEORY-1301 — In the Haken–Kelso–Bunz potential V(φ) = −a cos φ − b cos 2φ, switching the parameters in time from no landscape (a = b = 0) to two wells (a = b = 1) to one well (a = 1, b = 0) makes the spread of the relative phase across trials rise while both wells are open and fall after one is removed, the pattern Sánchez-Cañizares compares qualitatively with post-stimulus quenching of trial-to-trial variability in decision tasks; under Gaussian white noise of the stated strength q = 1 the spread of cos φ settles near 0.40, not at 0

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Hermann Haken, J. A. Scott Kelso and Heinz Bunz (1985), Biological Cybernetics 51(5):347–356.

Read in full on 2026-10-30 (NOTE-1424), from the publisher’s scan on Kelso’s lab page.

Key takeaways

  • The phenomena modelled (§2, p. 348). Four features of Kelso’s experiments (1981a, 1984), summarized and not shown: only two stable phase relations; an abrupt switch from anti-phase to in-phase at a critical cycling frequency; only in-phase beyond it; and, when frequency is lowered again, no return to anti-phase. The transition frequency divided by each subject’s preferred frequency was constant across subjects (p. 347).
  • The order-parameter equation (§2, pp. 349–351). φ̇ = −∂V/∂φ with V = −a cos φ − b cos 2φ, chosen as the simplest periodic, symmetric potential. Anti-phase is lost when |b| < |a|/4; b is tied to movement amplitude, which falls as frequency rises.
  • The derivation (§3, pp. 351–353). Two self-sustained oscillators (a Van der Pol variant whose amplitude falls as 1/ω) with a coupling that must be nonlinear and must involve a time delay or time derivatives; Eq. (3.26) recovers the φ equation, with a critical amplitude. Noise is needed for a transition to start (Eqs. 3.28–3.29).
  • What it does not report. Critical slowing and critical fluctuations are named as phenomena “that will also be analyzed” in future studies (p. 355). They were reported the next year, in Kelso, Scholz and Schöner (LIT-1851).

Standing in the record

Filed on 2026-10-03 at the owner’s request, in the coordination-dynamics part of the agency batch, as the canonical citation for the HKB model: Raja et al. (LIT-598) cite it for coupled oscillators measured by relative phase. The record first read the model through Kelso 2021 (LIT-601), which extends it; the original was reached on 2026-10-30. Kelso’s book Dynamic Patterns (LIT-578) is the book-length treatment. The first-hand data on fluctuations and on intentional switching are in two Physics Letters A papers filed the same day (LIT-1851 and LIT-1873). It bears on THEORY-069. No instruction for machine-learning practice.

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