J. A. Scott Kelso (2021), Biological Cybernetics 115(4):305–322, “60th Anniversary Retrospective” (published online 18 August 2021).
Key takeaways
- The model. Relative phase φ between the two hands is the order parameter; dφ/dt = −a sin φ − 2b sin 2φ (Eq. 1), with potential V(φ) = −a cos φ − b cos 2φ (Eq. 2). For b/a > 1/4 (slow movement) both φ = 0 and φ = π are stable; for b/a < 1/4 (fast movement) only in-phase remains. The switch is a pitchfork bifurcation, and with noise (Eq. 3) a nonequilibrium phase transition, with critical fluctuations and critical slowing before it.
- From components to collective variable. The φ dynamics is derived from two coupled “hybrid” (Van der Pol–Rayleigh) oscillators (Eqs. 4–6), and later from excitatory–inhibitory neural populations, which the author takes to put the mechanism-versus-description objection “to bed”.
- Symmetry breaking and metastability. Adding the eigenfrequency difference Δω (Eq. 7) tilts the potential; past a saddle-node bifurcation no fixed points remain but “coordination tendencies” do. This metastable regime is offered as the theory of von Holst’s relative coordination.
- Generality. The same coordination dynamics is reported between limbs, between a limb and a stimulus, between people, between humans and other species, and between humans and virtual partners, which the author reads as evidence that coupling is informational, not only mechanical. Combined with Kuramoto’s model it handles groups (generalized HKB, Eq. 10).
- Brain. Phase transitions and stability-dependent recruitment were found in MEG/EEG and fMRI; TMS over premotor and supplementary motor cortex switched anti-phase to in-phase but not the reverse, as the model predicts.
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 substitute for two works.
Why this paper. The brief named Kelso’s Dynamic Patterns (1995) and Haken, Kelso & Bunz (1985). The book has no lawful full text (LIT-578), and the 1985 paper is behind Springer’s paywall with no repository copy (LIT-588). This paper is by Kelso, is free to read at the publisher, and gives in eighteen pages the HKB equations with their parameter regimes, the stochastic and symmetry-breaking extensions, the metastable regime it credits the book with naming as relative coordination (“Kelso 1995”), and the link to Kuramoto that postdates the book. It reproduces the book’s coordination-potential figure as its Box 1 (“adapted from Kelso 1995”) and points to the book’s chapter 3 for the history of HKB and chapter 8 for the brain’s rhythms. The book’s contents were not seen, so what it carries beyond this paper is not known; it stays filed as a Deferred seed because it is the citation others reach for. Kelso’s encyclopedia article “Coordination Dynamics” (2009/2013), which this paper cites, would have been the other candidate; no lawful copy was found to compare.
Relation. It extends HKB 1985 (LIT-588): the paper is a statement
and extension of that model, with symmetry breaking, noise, metastability and
the many-oscillator generalization, and it could not stand without it.
Caveats for a reader. It is a retrospective by one of the model’s authors. It says it relies on “the work and opinions of others” where it can, but its §5 list of consensus impacts and its §6 dismissal of the mechanism question are the author’s view. The data it cites (Schöner et al. 1986; Meyer-Lindenberg et al. 2002; Jantzen et al. 2009) are reported, not shown.
Where it bears on what the record holds:
- Participatory sense-making (LIT-571) takes absolute versus relative coordination from Kelso 1995; this paper’s metastable regime is the formal account of the latter.
- Raja et al. (LIT-598) use coupled pendulums measured by relative phase, citing Haken et al. 1985 and Kelso 1995, as a system already explained by its own collective variable, and ask what a Markov-blanket partition would add. This paper is the read statement of that collective-variable explanation.
- Behavioral integration. HKB is a worked case of many degrees of
freedom acting as one, Bernstein’s problem, solved by a low-dimensional
collective variable rather than a controller; that is why the record tags
it
behavioral-integration. Kelso also lists the “excitator” and structured-flows models as descendants. - Philosophy of science. §6 takes up whether dynamical models of this kind explain or merely describe (Kaplan & Craver 2011; Chemero), and warns that “Dynamical Systems Theory” is often invoked without identified collective variables or control parameters.