Nonlinear Phenomena in Complex Systems, 2002, Vol.5, No.4, pp.418-427

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2002, Vol.5, No.4, pp.418-427

Stochastic Iterative Maps with Multiple Time-Scales for Modelling Human Motor Behavior.
G. Mayer-Kress and K.M. Newell

One of the most successful applications of synergetics is the analysis and modelling of human movement coordination. The celebrated Haken-Kelso-Bunz [1] model has had a profound impact in the field of kinesiology. Here we want to discuss a generalization of a modelling concept that was first introduced in [2],[3] to the study of non-linear, chaotic behavior under the influence of stochastic perturbations. Instead of ordinary differential equations (ODEs) the use of iterative maps has several advantages in a number of applications. The particular class of models that we want to focus on here, are piece-wise linear maps that capture important features of smooth non-linear maps and are especially appropriate for the study of stochastic systems. Finally we will generalize the concept of iterative maps of low-dimensional systems to a general class of models of systems of arbitrary dimension with multiple time-scales. We apply these models to the situation of human isometric force production. For most types of coordinated movement or force production various sensory-motor control loops are involved (see e.g. [4]. Our main result is that the model was shown to simulate the basic findings of the structure of human force variability that decreasing variability is correlated with an increase in dynamical complexity as measured with the "Approximate Entropy (ApEn)" statistics [5].
Furthermore we could demonstrate how stochastic perturbations can actually increase movement accuracy.
Key words: discrete stochastic nonlinear dynamics, iterative maps, kinesiology, human movement

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[2] G. Mayer-Kress, H. Haken. The influence of noise on the logistic model. Journal of Statistical Physics. 26, 149-171 (1981).
[3] H. Haken, G. Mayer-Kress. Chapman-Kolmogorov Equation and Path Integrals for Discrete Chaos in the Presence of Noise. Zeitschrift fur Physik B-Condensed Matter. 43, 18 (1981).
[4] J. J. Collins and C. J. De Luca. Open-loop and closed-loop control of posture: A random-walk analysis of center-of-pressure trajectories. Experimental Brain Research. 95, 308-318 (1993).
[5] G. Mayer-Kress, K.M. Deutsch, K.M. Newell. Modeling The control of isometric force production with piece-wise linear, stochastic maps of multiple time-scales, to app.

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