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

Espen A F Ihlen,Kimberley S Van Schooten,Jaap H Van Dieën,Sjoerd M Bruijn,Mirjam Pijnappels

doi:10.3389/fphys.2017.00516

Abstract

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.

Highlights

The number of studies on gait stability has rapidly increased during the last two decades
Current definitions of gait stability refer to the resistance of the gait kinematics to disturbances or the ability to recover gait kinematics after perturbations, e.g., “the ability to Fractional Stability of Human Gait maintain functional locomotion despite the presence of small kinematic disturbances or control errors” (p. 172, England and Granata, 2007) and “gait that does not lead to falls in spite of perturbations” (p. 2, Bruijn et al, 2013)
We introduced a novel approach toward gait stability, based on fractional stability, because gait dynamics with its phase transitions violates the assumptions for exponential stability

Summary

Introduction

The number of studies on gait stability has rapidly increased during the last two decades (as evidenced by two review papers Hamacher et al, 2011; Bruijn et al, 2013). Despite this increase in interest, a lack of consensus remains regarding the definition of gait stability and the numerical operationalization. The concept of exponential stability has been used to assess gait stability in several patient groups, experimental perturbation studies, walking models, and daily-life walking (Dingwell et al, 2000; Buzzi et al, 2003; Su and Dingwell, 2007; Kurz et al, 2010; McAndrew et al, 2011; Roos and Dingwell, 2011; van Schooten et al, 2011; Bruijn et al, 2012; Hamacher et al, 2016; van Schooten et al, 2016; de Melker Worms et al, 2017)

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