COORDINATE TRANSFORMATIONS IN THE CONTROL OF CAT POSTURE

1. Global geometric variables represent high-order parameters in the c ontrol of cat posture. In particular, limb length and orientation are accurately controlled in response to tilts of the support platform. Th ere is now electrophysiological evidence, obtained in anesthetized cat s, that spinal sensory neurons projecting to the cerebellum are broadl y tuned to limb length and orientation. Limb length and orientation sp ecify the position of the limb endpoints in body-centered polar coordi nates. They define an intended posture in a global manner, leaving the detailed geometric configuration of the limbs undetermined. The plana r covariation of limb joint angles described in the accompanying paper suggests the existence of an intermediate processing stage that trans forms endpoint coordinates into the angular coordinates of the joints (inverse mapping). In this paper we address the question of the nature of this coordinate transformation. Because the number of degrees of f reedom of angular motion in each limb exceeds that of endpoint motion in world space, several different angular configurations are compatibl e with any given endpoint position in world space. Thus the problem of coordinate transformation is a priori indeterminate. We have tested a number of different hypotheses. 2. Coordinate transformation could be accomplished implicitly by means of discrete kinematic synergies. Any given geometric configuration of the limb would result from a weighed combination of only two distinct patterns of angular covariations, th e first pattern affecting selectively limb length and the second patte rn affecting limb orientation. This decomposition, however, was found in only a few sporadic cases. 3. We also tested the possibility that t he coordinate transformation involves the Moore-Penrose generalized in verse. We found that this algorithm produces a planar covariation of t he joint angles, but with an orientation orthogonal to the experimenta l plane. By contrast, a linear transformation with constant, position- independent terms can fit the experimental plane of angular covariatio ns but predicts large errors in endpoint position. 4. The particular o rientation in joint space of the experimental plane, coupled with the scatter of data points around the plane, bears a specific implication for the problem of inverse mapping. The experimental plane crosses the constant position lines (the loci of all possible changes of the join t angles that correspond with an invariant position of the endpoint) a t an acute angle. Consequently the specification of limb orientation i s little sensitive to joint configurations: relatively small changes i n orientation can be produced by large changes in joint configurations . This is exactly the opposite of what is predicted by the Moore-Penro se generalized inverse, which tends to minimize the changes in joint a ngles. The variability of data points around the plane could result fr om a fuzzy implementation of inverse kinematics. This region of variat ion encompasses long segments of the constant position lines, suggesti ng that a one-to-many inverse mapping between desired endpoint positio n and joint angular configurations is available to the control system. 5. By considering clusters of data grouped according to the values of either limb orientation or limb length, one finds that these two glob al variables are mapped in the joint angle space according to differen t rules. Limb length maps linearly in discrete subsets of joint angles , whereas limb orientation maps in broad regions of joint angle space. 6. The analysis of static posture has demonstrated the existence of a planar constraint on the admissible covariations of limb joint angles . This constraint allows a high degree of flexibility and adaptability of the specific geometric configurations (i.e., joint angles) of the limb that are used to implement the desired values of length and orien tation. We have generalized this observation to the dynamic responses evoked by ramp rotations of the support platform in the nose-down or n ose-up direction. These perturbations affect both limb length and orie ntation, but with a different time course. The paths described in the three-dimensional space of joint angles tend to diverge in different d irections depending on stimulus direction and initial table position. All paths, however, are confined within a small volume surrounding the plane of static angular covariations. We have been able to prove also that dynamic perturbations may evoke postural responses that involve a given trajectory in joint angle space in one trial but a completely different trajectory in another trial, even though endpoint position r emains essentially the same in both cases. The bulk of the experimenta l evidence appears consistent with the hypothesis that the dynamics of the postural system is governed by a chaotic attractor coinciding wit h the plane of angular covariation.