### Abstract

1. Global geometric variables represent high-order parameters in the control of cat posture. In particular, limb length and orientation are accurately controlled in response to tilts of the support platform. There is now electrophysiological evidence, obtained in anesthetized cats, that spinal sensory neurons projecting to the cerebellum are broadly tuned to limb length and orientation. Limb length and orientation specify the position of the limb end-points in body-centered polar coordinates. They define an intended posture in a global manner, leaving the detailed geometric configuration of the limbs undetermined. The planar covariation of limb joint angles described in the accompanying paper suggests the existence of an intermediate processing stage that transforms 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 freedom of angular motion in each limb exceeds that of endpoint motion in world space, several different angular configurations are compatible 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, the first pattern affecting selectively limb length and the second pattern affecting limb orientation. This decomposition, however, was found in only a few sporadic cases. 3. We also tested the possibility that the coordinate transformation involves the Moore-Penrose generalized inverse. We found that this algorithm produces a planar covariation of the joint angles, but with an orientation orthogonal to the experimental plane. By contrast, a linear transformation with constant, position-independent terms can fit the experimental plane of angular covariations but predicts large errors in endpoint position. 4. The particular orientation 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 joint angles that correspond with an invariant position of the endpoint) at an acute angle. Consequently the specification of limb orientation is little sensitive to joint configurations: relatively small changes in orientation can be produced by large changes in joint configurations. This is exactly the opposite of what is predicted by the Moore-Penrose generalized inverse, which tends to minimize the changes in joint angles. The variability of data points around the plane could result from a fuzzy implementation of inverse kinematics. This region of variation encompasses long segments of the constant position lines, suggesting that a one-to-many inverse mapping between desired endpoint position 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 global variables are mapped in the joint angle space according to different 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 orientation. We have generalized this observation to the dynamic responses evoked by ramp rotations of the support platform in the nose-down or nose-up direction. These perturbations affect both limb length and orientation, but with a different time course. The paths described in the three-dimensional space of joint angles tend to diverge in different directions 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 remains essentially the same in both cases. The bulk of the experimental evidence appears consistent with the hypothesis that the dynamics of the postural system is governed by a chaotic attractor coinciding with the plane of angular covariation.

Original language | English |
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Pages (from-to) | 1496-1515 |

Number of pages | 20 |

Journal | Journal of Neurophysiology |

Volume | 72 |

Issue number | 4 |

Publication status | Published - 1994 |

### ASJC Scopus subject areas

- Physiology
- Neuroscience(all)

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## Cite this

*Journal of Neurophysiology*,

*72*(4), 1496-1515.