A stochastic theory of phase transitions in human hand movement

Abstract

The order parameter equation for the relative phase of correlated hand movements, derived in a previous paper by Haken et al. (1985), is extended to a time-dependent stochastic differential equation. Its solutions are determined close to stationary points and for the transition region. Remarkably good agreement between this theory and recent experiments done by Kelso and Scholz (1985) is found, and new predictions are offered.

Deterministic and stochastic features of rhythmic human movement

“…Recently, there has been an upsurge of studies focussing on stochasticity as a hallmark property of human movement that not only needs to be addressed, but also possesses functional qualities (e.g., Harris and Wolpert 1998;Körding and Wolpert 2004;Riley and Turvey, 2002;Schöner et al 1986;Schöner 2002). For instance, variability in endpoint trajectory has been associated with task difficulty (Todorov and Jordan 2002): variability is reduced in more difficult tasks to comply with accuracy constraints, whereas in easier tasks the variability is allowed to increase to enhance system flexibility.…”

“…Another example of the usefulness of motor variability can be found in studies of interlimb coordination conducted from a dynamical systems perspective. In this context, variability has been incorporated as random fluctuations to account for phenomena like critical fluctuations and critical slowing down in the vicinity of phase transitions, that is, situations in which a system switches between stable states or attractors, e.g., switches from antiphase to in-phase coordination (Haken et al 1985;Kelso 1984;Post et al 2000;Schöner et al 1986). In relation to the attractor strength, the amount of random fluctuations competes with stability and, thus, determines the flexibility of the system.…”

“…Formally, variability may be accounted for by incorporating either additive or multiplicative random fluctuations yielding stochastic differential equations. Schöner et al (1986) and Schöner (1990) formulated stochastic models for single-and multi-limb movements similar to the abovementioned 'deterministic' limit cycle models (comprising van der Pol, Duffing and quintic stiffness terms) under the impact of stochastic forces (Gaussian white noise). Schöner (1990) also suggested similar accounts of discrete movement and postural sway, albeit without providing any empirical support.…”

Challenges for the understanding of the dynamics of social coordination

“…They predicted that the change of pattern corresponded to a phase transition encountered in physics, and thus should operate by a loss of stability of the intended anti-phase pattern. This prediction has been verified experimentally, and further developments taking into account biological noise led to stochastic predictions (e.g., critical fluctuations, first passage time, correlations), and again to converging evidence (Schöner et al, 1986). This initial round of theoretical predictions and crucial experiments, exotic as it was at the time in this field, shake the theory of biological control of movement inspired by cybernetic and computer program metaphors.…”

On a Nonlinear Master Equation and the Haken-Kelso-Bunz Model

“…The transition from bistability to monostability occurs at a = 4b. Several stochastic versions of the HKB model have been proposed, see, for example, [12,21,22,23,41]. So far, however, the HKB model has not been discussed in the context of master equations and general information measures.…”