Two dimensionless parameters and a mechanical analogue for the HKB model of motor coordination

J F Cass, S J Hogan*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
82 Downloads (Pure)


The widely cited Haken-Kelso-Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue.

We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model.

The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations.
This observation has implications for the development of virtual partner interaction (VPI) and the human dynamic clamp (HDC), and potentially for the HKB model itself.
Original languageEnglish
Pages (from-to)343-364
Number of pages22
JournalBiological Cybernetics
Issue number4
Early online date5 Jun 2021
Publication statusPublished - Aug 2021

Bibliographical note

Funding Information:
(information that explains whether and by whom the research was supported): Not applicable

Publisher Copyright:
© 2021, The Author(s).

Structured keywords

  • Engineering Mathematics Research Group


  • Haken-Kelso-Bunz model
  • motor coordination
  • mechanical analogue
  • nondimensionalisation
  • bifurcation


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