Todorov E, Jordan M I
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
J Neurophysiol. 1998 Aug;80(2):696-714. doi: 10.1152/jn.1998.80.2.696.
The speed profiles of arm movements display a number of regularities, including bell-shaped speed profiles in straight reaching movements and an inverse relationship between speed and curvature in extemporaneous drawing movements (described as a 2/3 power law). Here we propose a new model that simultaneously accounts for both regularities by replacing the 2/3 power law with a smoothness constraint. For a given path of the hand in space, our model assumes that the speed profile will be the one that minimizes the third derivative of position (or "jerk"). Analysis of the mathematical relationship between this smoothness constraint and the 2/3 power law revealed that in both two and three dimensions, the power law is equivalent to setting the jerk along the normal to the path to zero; it generates speed predictions that are similar, but clearly distinguishable from the predictions of our model. We have assessed the accuracy of the model on a number of motor tasks in two and three dimensions, involving discrete movements along arbitrary paths, traced with different limb segments. The new model provides a very close fit to the observed speed profiles in all cases. Its performance is uniformly better compared with all existing versions of the 2/3 power law, suggesting that the correlation between speed and curvature may be a consequence of an underlying motor strategy to produce smooth movements. Our results indicate that the relationship between the path and the speed profile of a complex arm movement is stronger than previously thought, especially within a single trial. The accuracy of the model was quite uniform over movements of different shape, size, and average speed. We did not find evidence for segmentation, yet prediction error increased with movement duration, suggesting a continuous fluctuation of the "tempo" of discrete movements. The implications of these findings for motor planning and on-line control are discussed.
手臂运动的速度曲线呈现出多种规律，包括直线伸手动作中呈钟形的速度曲线，以及即兴绘图动作中速度与曲率之间的反比关系（被描述为2/3幂律）。在此，我们提出一种新模型，通过用平滑度约束取代2/3幂律，同时解释这两种规律。对于手部在空间中的给定路径，我们的模型假设速度曲线将是使位置的三阶导数（或“急动度”）最小化的曲线。对这种平滑度约束与2/3幂律之间数学关系的分析表明，在二维和三维空间中，幂律等同于将沿路径法线方向的急动度设为零；它生成的速度预测与我们模型的预测相似，但明显不同。我们在二维和三维的多项运动任务中评估了该模型的准确性，这些任务涉及沿任意路径的离散运动，由不同肢体节段描绘。新模型在所有情况下都能非常紧密地拟合观察到的速度曲线。与2/3幂律的所有现有版本相比，其性能始终更好，这表明速度与曲率之间的相关性可能是产生平滑运动的潜在运动策略的结果。我们的结果表明，复杂手臂运动的路径与速度曲线之间的关系比之前认为的更强，尤其是在单次试验中。该模型的准确性在不同形状、大小和平均速度的运动中相当一致。我们没有发现分段的证据，但预测误差随运动持续时间增加，这表明离散运动的“节奏”存在持续波动。讨论了这些发现对运动规划和在线控制的影响。
