Gates Deanna H, Dingwell Jonathan B
Department of Biomedical Engineering, University of Texas, Austin, TX 78712, USA.
J Biomech. 2009 Jun 19;42(9):1345-9. doi: 10.1016/j.jbiomech.2009.03.015. Epub 2009 Apr 19.
Measures of local dynamic stability, such as the local divergence exponent (lambda*(s)) quantify how quickly small perturbations deviate from an attractor that defines the motion. When the governing equations of motion are unknown, an attractor can be reconstructed by defining an appropriate state space. However, state space definitions are not unique and accepted methods for defining state spaces have not been established for biomechanical studies. This study first determined how different state space definitions affected lambda*(s) for the Lorenz attractor, since exact theoretical values were known a priori. Values of lambda*(s) exhibited errors <10% for 7 of the 9 state spaces tested. State spaces containing redundant information performed the poorest. To examine these effects in a biomechanical context, 20 healthy subjects performed a repetitive sawing-like task for 5 min before and after fatigue. Local stability of pre- and post-fatigue shoulder movements was compared for 6 different state space definitions. Here, lambda*(s)decreased post-fatigue for all 6 state spaces. Differences were statistically significant for 3 of these state spaces. For state spaces defined using delay embedding, increasing the embedding dimension decreased lambda*(s) in both the Lorenz and experimental data. Overall, our findings suggest that direct numerical comparisons between studies that use different state space definitions should be made with caution. However, trends across experimental comparisons appear to persist. Biomechanical state spaces constructed using positions and velocities, or delay reconstruction of individual states, are likely to provide consistent results.
局部动态稳定性的度量，如局部分散指数（lambda*(s)），量化了小扰动从定义运动的吸引子偏离的速度。当运动的控制方程未知时，可以通过定义适当的状态空间来重建吸引子。然而，状态空间的定义并非唯一，并且尚未为生物力学研究建立公认的定义状态空间的方法。本研究首先确定了不同的状态空间定义如何影响洛伦兹吸引子的lambda*(s)，因为其精确的理论值是先验已知的。在所测试的9个状态空间中的7个中，lambda*(s)的值表现出小于10%的误差。包含冗余信息的状态空间表现最差。为了在生物力学背景下研究这些影响，20名健康受试者在疲劳前后进行了5分钟的重复锯切样任务。针对6种不同的状态空间定义，比较了疲劳前后肩部运动的局部稳定性。在此，所有6种状态空间的lambda*(s)在疲劳后均降低。其中3种状态空间的差异具有统计学意义。对于使用延迟嵌入定义的状态空间，增加嵌入维数会使洛伦兹数据和实验数据中的lambda*(s)均降低。总体而言，我们的研究结果表明，对于使用不同状态空间定义的研究之间的直接数值比较应谨慎进行。然而，实验比较中的趋势似乎持续存在。使用位置和速度构建的生物力学状态空间，或单个状态的延迟重建，可能会提供一致的结果。
