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Equilibrium Point Control Cannot be Refuted by Experimental Reconstruction of Equilibrium Point Trajectories

Institute for Fundamental and Clinical Human Movement Sciences, IFKB, Vrije Universiteit, Amsterdam, The Netherlands

Submitted 14 March 2007; accepted in final form 6 June 2007

In the literature, it has been hotly debated whether the brain uses internal models or equilibrium point (EP) control to generate arm movements. EP control involves specification of EP trajectories, time series of arm configurations in which internal forces and external forces are in equilibrium; if the arm is not in a specified EP, it is driven toward this EP by muscle forces arising due to central drive, reflexes, and muscle mechanics. EP control has been refuted by researchers claiming that EP trajectories underlying movements of subjects were complex. These researchers used an approach that involves applying force perturbations during movements of subjects and fitting a mass-spring-damper model to the kinematic responses, and then reconstructing the EP trajectory using the estimated stiffness, damping, and measured kinematics. In this study, we examined the validity of this approach using an EP-controlled musculoskeletal model of the arm. We used the latter model to simulate unperturbed and perturbed maximally fast movements and optimized the parameter values of a mass-spring-damper model to make it reproduce as best as possible the kinematic responses. It was shown that estimated stiffness not only depended on the "true" stiffness of the musculoskeletal model but on all of its dynamical parameters. Furthermore it was shown that reconstructed EP trajectories were in agreement with those presented in the literature but did not resemble the simple EP trajectories that had been used to generate the movement of the model. It was concluded that the refutation of EP control on the basis of results obtained with mass-spring-damper models was unjust.

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