Fractional Stability of Trunk Acceleration Dynamics of Daily-Life Walking: Toward a Unified Concept of Gait Stability

Over the last decades, various measures have been introduced to assess stability during walking. All of these measures assume that gait stability may be equated with exponential stability, where dynamic stability is quantified by a Floquet multiplier or Lyapunov exponent. These specific constructs of dynamic stability assume that the gait dynamics are time independent and without phase transitions. In this case the temporal change in distance, d(t), between neighboring trajectories in state space is assumed to be an exponential function of time. However, results from walking models and empirical studies show that the assumptions of exponential stability break down in the vicinity of phase transitions that are present in each step cycle. Here we apply a general non-exponential construct of gait stability, called fractional stability, which can define dynamic stability in the presence of phase transitions. Fractional stability employs the fractional indices, α and β, of differential operator which allow modeling of singularities in d(t) that cannot be captured by exponential stability. The fractional stability provided an improved fit of d(t) compared to exponential stability when applied to trunk accelerations during daily-life walking in community-dwelling older adults. Moreover, using multivariate empirical mode decomposition surrogates, we found that the singularities in d(t), which were well modeled by fractional stability, are created by phase-dependent modulation of gait. The new construct of fractional stability may represent a physiologically more valid concept of stability in vicinity of phase transitions and may thus pave the way for a more unified concept of gait stability.


Technical details of the surrogate generation based on the multivariate empirical mode decompositions
The MEMD surrogate generation method has the following four steps (see Fig. A1): Step 1: Decompose the reconstructed gait dynamics, defined by Eq. 7 of the main text, into intrinsic mode functions by MEMD Step 2: Consider the sum of high frequency components, d 1 (t) to d 4 (t), as the intra-step details y high (t) and the sum of low frequency components, d 5 (t) to d N (t) together with the final residual r N (t), as the inter-step periodicity y low (t) of the gait dynamics ( Fig. A1): where the original gait dynamics x(t) = y high (t) + y low (t).
Step 3: IAAFT surrogates were generated for the intra-step details, y high (t).
Step 4: MEMD surrogate dynamics was obtained by adding the IAAFT surrogates and the inter-step periodicity, y low (t).

Multivariate empirical mode decomposition (MEMD)
The following MEMD algorithm introduced by Rehman and Mandic (2010) was used in the present study (see acknowledgment): Step 1: Generate a Hammersley sequence-based point set on a 3m -1 dimensional sphere where m is the number of lags in the state space reconstruction method (m = 1, 2, and 3 in the present study) .
Step 2: Compute the projection () k pt  of the gait dynamics x(t) = (or residual r(t) or d(t) for iterative steps) along the unit direction vectors θ k of the 3m -1 dimensional sphere.
Step 3: Find the time instant k t  that corresponding to the maxima max () k pt  of () k pt  along all k = 1, 2,…,3m -1 dimensions.
Step 4: Obtain the envelope curves, () Step 5 Step 6: The first series of details d 1 (t) around the mean m 1 (t) is defined as d 1 (t) = x(t)m 1 (t).
If d 1 (t) satisfies the selected stopping criteria, then d 1 (t) is defined as an intrinsic mode function (IMF) and Step 2 to 5 is performed on first residual, r 1 (t) = x(t)d 1 (t). The second IMF is defined as d 2 (t) = r 1 (t)m 2 (t) with residual r 2 (t) = r 1 (t)d 2 (t). Consequently, the nth IMF is defined as d n (t) = r n-1 (t)m n (t) with residual r n (t) = r n-1 (t)d n (t). This iterative shifting procedure (i.e., Step 2 to 5) is continued until two maxima max () Step 3 can no longer be found. If d n (t) do not satisfy the stopping criteria, then step 2 to 5 is performed as a iterative procedure on d n (t) until the stopping criteria is met and a IMF is defined. The stopping criteria used in the present study is similar to the stopping criteria

Iterated amplitude adjusted fourier transform (IAAFT)
IAAFT surrogate was generated for the intra-step details of the gait dynamics, y high (t), defined by Eq. A1 by following five steps (Schreiber and Schmitz, 1996): Step 1: Store the rank ordering of the amplitudes of y high (t).
Step 2: Fast fourier transform y high (t) and store the obtained spectral amplitudes.
Step 4: Initiate an iterative procedure where: 1) The spectral amplitudes of shuffled time series are obtained by a fast Fourier transformation.

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2) The obtained spectral amplitudes of the shuffled time series are substituted with the stored spectral amplitudes of y high (t).
3) The surrogate series are obtained by an inverse fast Fourier transformation.
4) The amplitudes of obtained surrogate series are ranked as in y high (t).
In the present study, 1 to 4 is iterated a maximum of 500 times to obtain a saturation effect as described by Schreiber and Schmitz (1996) where the surrogate series had equal power spectral density and probability density function as y high (t), but where the phase-dependent changes in y high (t) are removed.