Frequency-Specific Fractal Analysis of Postural Control Accounts for Control Strategies

Figures

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  Figure 1. (A.1) Artificial fractal signal with α = 0.8 (Generated using Davies and Harte method Davies and Harte, 1987. (A.2) Example of a detrending for a fixed value of n = 181 of the artificial signal after integration. (A.3) Fluctuation function and value of α obtained by the slope of the curve. (B.1) Typical position signal obtained for the displacement of COP along the AP axis. (B.2) Example of a detrending for a fixed value of n = 181 of the typical signal after integration. (B.3) Fluctuation function: at least two different scalings are observed here. Note that both DFA show F(n) vs. n in a log-log plot.

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  Figure 2. Illustration of the fGn/fBm continuum, with corresponding Hurst exponent H, scaling exponent α and β index. This illustration was largely inspired by Marmelat (Marmelat et al., 2012).

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  Figure 3. Example of the estimation of the cutoff box of nilow(α, fc) and nihigh(α, fc), done in the low-passed signal Xilow(α, fc) (thick black dotted line), and high passed signal Xihigh(α, fc) (thick black full line). In this example, we chose Xi=1low(α=0.8, fc=0.056). The crossover was considered to be at the intersection of the asymptotes in each signal (thin dashed straight lines). Note that the asymptote of Xilow(α, fc) for high values of n and the asymptote of Xihigh(α, fc) for low values of n (thin dashed blue straight lines) are perfectly superimposed since they correspond to the unaltered frequency ranges of the original signal Xi(α) (not plotted here for clarity), which has a unique scaling, hence a unique slope.

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  Figure 4. Estimation of the slope aα. Values are mean ± SD from n = 10 simulated signals for each value of α.

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  Figure 5. Calculations of scaling exponents α on a typical CoP displacement along the AP axis (A) and of the same signal, derived (B).

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  Figure 6. Individual changes in short term scaling exponent on CoP velocity and long-term scaling exponent on CoP position, respectively with fatigue and eye closing, for all subjects.

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  Figure 7. Typical PSD of COP velocity on the AP axis (A), and its schematic representation (B) revealing four key parameters: Pf0, β1, β2, f0.