Coordination and the Haken-Kelso-Bunz Model

Other coordination posts are here.

Now we have the basics out of the way, we can explore some of the key characteristics of coordinated rhythmic movements.

Kelso's early experiments established a set of basic phenomena. Recall that, at the moment, the task is to wiggle your fingers in and out (towards the midline) at some mean relative phase and at some frequency. Moving your fingers in-phase (0° mean relative phase) has the fingers moving in the same direction at the same time, and entails the co-activation of homologous muscle groups in each hand.

Key Phenomenon 1: Only 0° and 180° are stable; attempts to move at other mean relative phases are highly unstable, exhibiting high variability about the target mean.

Key Phenomenon 2: 0° is more stable than 180°. If a participant begins moving at 180° and steadily increases the frequency of their movements, 180° gets progressively more unstable and (under a 'non-interference' instruction) the person will spontaneously transition from 180° to 0° around 3-4Hz. The reverse is not true: if you begin at 0°, you will never spontaneously switch into 180° even though your movement variability will increase.

Kelso made contact with some of the first researchers applying the tools of non-linear dynamical systems to human behaviour and other complex systems. In 1985, Haken, Kelso and Bunz published a simple model that captured the phenomena using tools and concepts from the study of complex systems. This model, the HKB model, has structured the empirical and theoretical investigations of this task, and contains numerous hypotheses about the observed structure of the two key phenomena (hence the 'HKB pattern').

The model is simply the superposition of two cosine functions; the frequency of the movement is specified by the ratio of the two parameters, b and a:

 V = -a cosφ –b cos 2φ 

This is a potential function: the y-axis reflects the energy of the system at each mean relative phase. Low energy means stable. Local minima in the function represent attractors, or states to which the system behaviour can be drawn. Local maxima in the function represent repellors, or states from which the system behaviour will run away from. The net result is this famous (if you're in the field) picture:

The HKB Model, ~1Hz

The balls are there to draw the intuition of this as a landscape; the minima are attractors in the sense that behaviour close to those minima is drawn to them, in the same way a ball placed partway up a slope would roll down and settle at the bottom. 

All the key phenomena are present in this simple model: While there are attractors at both 0° and 180°, 0° has lower potential energy than 180° and is thus more stable. There are no other attractors in the space, although there is a repellor at 90° - behaviour here is indeed maximally unstable (without training - more on this later).

The changes with frequency are modelled by toggling the two parameters:

The HKB Model, ~2Hz

The HKB Model, ~3-4Hz

While all states become less stable (higher energy), the attractor at 0° remains, while the attractor at 180° becomes shallower (and thus easier to 'pop' out of) and then finally disappears altogether. This simple model exhibits a phase transition from a bi-stable to a mono-stable regime.

The HKB model successfully describes the observed behaviour of a person producing bimanual coordinated rhythmic movements with a simple potential function. There are two things worth noting:

1. The model is entirely abstract, with no reference to anything about the kind of system implementing this dynamic pattern (no muscles, no perception, not even a neural network!).This is useful, because it allows the model to be applied to finger wiggling, wrist rotation, arm vs. leg systems, etc - all examples of coordination tasks in which the HKB pattern can  be found in humans.
2. The model is entirely descriptive: there is no proposed mechanism for why this particular pattern obtains. This phenomenological modelling approach aims solely to capture behaviour in terms of simple order parameters. These reflect the macroscopic behaviour of a system, and in the study of complex systems often provide a sensible level of analysis that allows one to ignore the tremendous complexity at the microscopic level. For instance, while it is true that the macroscopic properties of matter arise from the microscopic properties of atoms, analysing the former in terms of the latter is often unnecessarily complicated; with the correct choice of order parameter you can describe the macroscopic behaviour of interest without getting bogged down. The HKB model assumes that the order parameter for a coordination task is relative phase.

References

Haken, H., Kelso, J.A.S., Bunz, H. (1985). A theoretical model of phase transitions in human hand movements Biological Cybernetics, 51 (5), 347-356 DOI: 10.1007/BF00336922 Download