Stochastic Analysis of Human Postural Sway

Yaradanakul, Naci
"The decisive step in the transition from ape to man", in other words, inherently unstable dynamics of human erect posture, is investigated throughout the thesis considering the random features with a spatiotemporal perspective employing stochastic dynamical system theory and statistical mechanics. During the investigation, mechanistic reasoning and control theory is not mentioned, but the complex dynamical output of four basic sensors somatosensory, vestibular, visual and proprioceptive, physicochemical actuator striated muscles, multidegree of freedom plant having redundancies depending on the task, and the nervous system, namely postural sway, is modeled as a black box that is characterized by CoPx time series. Keeping this objective in mind, the first chapter is devoted to the explanation of the loaded philosophical content of postural sway with a phylogenetic and ontogenetic point of view. Furthermore, the contrast between Newtonian determinism in time and Boltzmanian understanding of chance in space are considered in terms of the concept of time, its direction, causality, and entropy to clarify the spatiotemporal domain, in which the dynamics of interest to be analyzed with stochastic analytical tools. In this context, in Section 1.2, philosophical ideas that have been developed were employed in an attempt to explain the particular problem by means of stochastic process and cybernetics to answer the question "What is the dice ruling the postural sway?". Besides the philosophical content, Chapter 2 continued with neurophysiological reasons behind the never-ending postural sway, possibly serving for the ecological search of the equilibrium point (conservative level) and the boundaries of stability (operative level) with trial and error. For instance, the output of this complex system, $CoP_x$, two superposed oscillations at different frequencies constituting postural sway are explained considering the works of Zatsiorsky's rambling trembling decomposition, Gurfinkel's conservative and operative levels, Feldman's $\lambda$-threshold theory, and Hogan's virtual trajectory explanation; in spirit, all reconciling similar dualities. Considering the neurophysiological aspects in Section 2.2, measurement methods of postural sway are explained for the quantification of the problem. Later, stochastic dynamics of the $CoP_x$ signal are investigated in a pathwise manner with proposed stochastic differential equations starting from the visual verification of normality of decision makings, i.e., displacements. The first step is started with the integration of the Wiener process obeying $\sigma \left(\Delta t\right)^{1/2}\mathcal{N}(0,1)$. Due to the linearly increasing variance, which would result in falling down, linear mean reversion was imposed on the dice ruling the sway to constraint the variance inside the area under the foot. While the resulting process is named as Ornstein Uhlenbeck, it is linked to viscoelasticity known as Langevin Equation. For the analytical consideration of the two-level nature of postural sway, the double Ornstein Uhlenbeck process was defined. In this way, the baseline fluctuations around zero are converted into oscillations around $\theta_t$ which describes the rambling equilibria. The covariance function of the proposed system of differential equations having a double exponential form revealed distinct time scales consistent with the posture literature. A final improvement was performed by imposing a cubic nonlinearity on the model for the assessment of the multimodality of sway.


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Citation Formats
N. Yaradanakul, “Stochastic Analysis of Human Postural Sway,” M.S. - Master of Science, Middle East Technical University, 2022.