静立姿态控制中开关间歇性模型切换型Fokker-Planck方程的马尔可夫链近似
A Markov chain approximation of switched Fokker-Planck equations for a model of on-off intermittency in the postural control during quiet standing
AI总结：
针对静立姿态控制中的开关间歇性，提出用有限元法将切换型Fokker-Planck方程近似为马尔可夫链模型，并验证其动力学，为后续马尔可夫决策过程研究奠定基础。
摘要：
反馈控制的间歇性开关切换被认为是人体静立姿态稳定的主要机制，该机制可通过具有不稳定子系统的切换型混合随机延迟微分方程来建模。该模型的动力学也可由相应的切换型Fokker-Planck（FP）方程描述。在此，我们开发了一种全面的数值方法，用于模拟切换型FP方程，其前提是概率流在切换边界处守恒，正如无噪声模型在边界处解具有C^0连续性那样，但该方法可扩展到具有不连续跳跃的情形。具体而言，FP方程通过有限元方法近似为有限状态马尔可夫链模型。然后，分析了马尔可夫链模型的动力学，包括概率密度函数（PDF）的时间演化、稳态PDF以及姿态晃动的功率谱。我们进一步研究了当模型重要参数变化时稳态PDF如何改变。将马尔可夫链模型的动力学与基于蒙特卡洛的模型动力学进行比较，从而验证了所开发的数值方法。所获得的马尔可夫链模型为我们未来将间歇性姿态控制作为马尔可夫决策过程进行研究奠定了基础。
英文摘要：
The intermittent on-off switching of feedback control is considered as a major mechanism of postural stabilization during human quiet standing, which can be modeled by switched-type hybrid stochastic delay differential equations with unstable subsystems. Dynamics of the model can also be described by the corresponding switched-type Fokker-Planck (FP) equations. Here, we develop a comprehensive numerical recipe to simulate switched-type FP equations in the case that the probability current is conserved at the switching boundary, as is the case for noise-free models exhibiting C^0-continuity for solutions at the boundary, but in a way extendable to cases with discontinuous jump. Specifically, the FP equations are approximated by a finite state Markov chain model using the finite element method. Then, dynamics of the Markov chain model, including time evolution of probability density function (PDF), stationary PDF, and power spectrum of postural sway are analyzed. We further investigate how the stationary PDF alters as values of important parameters of the model change. Dynamics of the Markov chain model are compared with Monte Carlo-based dynamics of the model, by which the developed numerical recipe is validated. The obtained Markov chain model forms a basis of our future investigations of the intermittent postural control as a Markov decision process.
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