Publicidad e inderogabilidad singular

control of motion.

For example, observing that the hand paths of reaching movements are quasi- straight, [Morasso, 1981] suggested that movement were planned in Cartesian space. In the following years, analogous arguments were used to hypothesize a control in joint angle space, as movements were found to also be quasi-straight in that space [Soechting and Lacquaniti, 1981, Lacquaniti et al., 1986]. This hy- pothesis was supported by other evidence including the predictivity of final pos- ture [Desmurget et al., 1995], variability of unconstrained reaching trajectories performed in the dark [Magescas and Prablanc, 2006], Further studies argued that both joint angle and Cartesian space frames of reference were used, either concurrently [Cruse and Br¨uwer, 1987, Carrozzo and Lacquaniti, 1994], or de- pending on the task setting [Desmurget et al., 1997]. According to the latter, free movements are controlled in joint space, while constrained movements (such as 2D movements) were controlled in Cartesian space.

To further inquire about the frames of reference used for planning reach- ing movements, experiments involving reaching to remembered targets were performed. By varying the initial conditions (arm, head or hand position), it was investigated whether the target position was stored in a FoR centered on the eyes, the head, the torso, the hand, or the environment. Results in- dicated that targets were memorized in FoR centered on the arm and the eye [Lemay and Stelmach, 2005, Beurze et al., 2006]. Other results indicated that an eye-centered FoR is used [Batista et al., 1999, Vetter et al., 1999] More gen- erally, it was argued that the nervous system could adapt the FoR depending on the task demand [Adamovich et al., 1998, Ghafouri et al., 2002].

Other kinds of invariants where investigated, such as optimality criteria gov- erning the control of reaching movements. The underlying hypothesis is that tra- jectory planning is performed by optimizing some given functional of the move- ment. Various criteria were suggested [Engelbrecht, 2001], such as minimizing the total jerk (i.e. the path integral of the third time derivative of the hand tra- jectory) [Hogan, 1984, Flash and Hogan, 1985, Hogan and Flash, 1987], mini- mum torque change [Nakano et al., 1999], minimum energy [Alexander, 1997].

It was also observed that subjects could adapt the control of their move- ments when the dynamics of the environment was modified, for example by an external force field [Shadmehr and Mussa-Ivaldi, 1994]. When subject’s arm is submitted to such an artificial force field, the reaching trajectories are first importantly disturbed. After some practice, however, the subject can again reach a target with quasi-straight trajectories, as if there was no force field. When the force field is removed, one can observe so called after-effects, i.e. the reaching trajectories are perturbed again, but this time in the other di- rection. Those after effects disappear after some practice in the natural set- ting. To explain those observations, it was suggested that the nervous sys- tem learns a forward model of the dynamics the the arm in its environment [Wolpert et al., 1995, Gandolfo et al., 1996]. According to the latter, the first a trajectory is planned and the appropriate commands for realizing this trajectory are computed using the forward model and sent to the muscles.

However, the distinction between trajectory planning and execution advo- cated by engineers [Hollerbach, 1982], never gained a full consensus among schol- ars. Many of them, in particular those coming from a biological background, objected it. According to them, movements are control by a dynamical system acting on a small number of variables. According to this dynamical system approach [Kelso, 1995], trajectory are not planned. Rather, they are implic- itly specified by a dynamical system, and unfold as time goes by. The exact nature of this dynamical system and of the variables involved is still unclear. One of the earliest and most influential model is the equilibrium point hypoth- esis, dating back to the mid-fifties [Bizzi et al., 1984, Feldman and Levin, 1995, Feldman and Latash, 2005, Feldman, 2006]. This model is based on a study of muscle properties and suggests that the dynamics are provided by the bio- mechanical properties of the arm. According to this theory, the signals sent to the muscles are thresholds for muscle activation, which indirectly specify an arm configuration were agonist, antagonist and external forces are balanced, the equilibrium point. This point can be seen as an attractor for the arm, due to the spring properties of the muscles.

Another hypothesis fitting into the dynamical system approach is the stochastic optimal feedback control theory [Todorov and Jordan, 2002]. According to this view, the control is operative only along the dimensions relevant to the particu- lar task at hand. Perturbations that do not interfere with the goal, will thus not be corrected. This is the minimal intervention principle [Todorov, 2004], also related to the concept of an uncontrolled manifold [Scholz and Sch¨oner, 1999]. The controller is thus optimal in some (still to be defined) sense, with respect to the task. This theory aims not at reproducing precise trajectories, but rather the variability among trajectories for a particular task. Any performed trajec- tory is dependent, among others, on noise at the sensory level and can hence not be precisely modeled. Due to the random aspect of the noise, trajectories are best seen as realizations of a random variable. This view is reminiscent of the interpretation of quantum mechanics given by [Prigogine, 1996]. Other dynam- ical systems were also suggested, such as the Vector Integration To Endpoint (VITE) [Bullock and Grossberg, 1988] described below.

To sum up, the main question concerning the human control of reaching movements is what is being actually controlled and in what frame of reference. This problem is still unsolved [Admiraal et al., 2004]. The evidence accumu- lated during several decades of research seems to indicate that those questions may be somewhat too simplistic and that many frames of reference can be (pos- sibly simultaneously) used to control the arm. It all depends on the task at hand. It seems reasonable to hypothesize that humans can adapt the control of their movements and choose the most appropriate frame of reference. Similarly, they can adapt to different dynamics of the environment, and they can adapt the movement trajectories, if it does not interfere with the task. This is com- patible with the dynamical system approach to motor control. Below, I present the VITE dynamical system, as it will be used in the following of this chapter.

4.4. A MODEL OF REACHING MOVEMENT CONTROL 63