Full length articleMaximum Lyapunov exponent revisited: Long-term attractor divergence of gait dynamics is highly sensitive to the noise structure of stride intervals
Introduction
Analysis of the nonlinear variability of human locomotion has attracted growing interest over the past decade [1]. This approach postulates that decoding nonlinear dependence among consecutive gait cycles (strides) can help to better understand gait control. A popular nonlinear method is the local dynamic stability (LDS) of the gait [[2], [3], [4], [5]]. LDS is derived from the maximum Lyapunov exponent, which is used to highlight the deterministic chaos in nonlinear systems. Gait LDS has been proven particularly useful for detecting patients at risk of falling [6].
The majority of LDS studies use the Rosenstein’s algorithm that computes the distance between trajectories of an attractor reflecting the gait dynamics [3,4]. A logarithmic divergence curve is then built to assess the exponential divergence rate—the divergent exponent (DE)—by means of linear fitting over a given range. Two ranges have been proposed: a short-term range over 0–1 or 0–0.5 stride (the short-term DE), and a long-term range over 4–10 strides (the long-term DE) [4]. Puzzling results have been found when these two LDS indexes are used together to assess fall risk: both indexes most often vary in opposite directions [7,8]. Further theoretical and experimental studies have shown that only the short-term DE is a valid gait stability measure [2,9,10]. However, it is not excluded that the long-term DE is associated with other gait features given its responsiveness to various conditions [[11], [12], [13]].
Another approach for studying nonlinear gait variability is the analysis of the noise structure of stride-to-stride fluctuations. In healthy individuals, basic gait parameters, such as stride interval, stride length and stride speed, fluctuate among strides within a narrow range of 2%–4% [14]. It has been shown that these fluctuations are not random, but exhibit long-range correlations and a scale-free, fractal-like pattern [[14], [15], [16]]. This particular noise structure is observed in many different physiological signals, and is considered a hallmark of the complexity of living-beings [17]. Interestingly, this fractal structure can be altered when external cues are used to intentionally drive the steps, such as synchronizing gait to a metronome, or by following marks on the floor [16].
In 2009, Jordan et al. [18] analyzed both gait stability and complexity in treadmill walking and running. They observed a strong correlation (r = 0.80) between a measure of gait complexity (the scaling exponent of stride intervals) and the long-term DE. In 2012, Sejdic et al. [12] assessed the noise structure of stride intervals as well as the LDS (short-term and long-term DEs) during normal walking with and without external cueing (metronome walking). The results showed that, with auditory cueing, the long range-correlations of stride intervals changed to anti-correlated patterns along with a substantial decrease of long-term DEs, but with no change of short-term DEs. Similarly, in 2013 [4], we analyzed the gait stability and complexity of treadmill walking, confirming that both long-term DE and the noise structure of stride intervals were similarly modified by external cueing. A significant correlation between scaling exponents and long-term DEs (r = 0.57) was also observed. In summary, the long-term DE seems more associated with the noise structure of stride intervals than with local stability and fall risk. Complex fluctuations that occur over dozens of consecutive strides seem to induce a less dampened divergence curve, resulting in a higher long-term DE.
The current study’s objective was to further explore whether the long-term DE should be interpreted as an index of gait complexity rather than an index of gait instability. To this end, stride intervals of natural gait acceleration signals were replaced with artificial time series exhibiting known noise structure. The hypothesis was that higher long-term DEs were associated with a more complex variability of stride-to-stride fluctuations. It was also assumed that short-term DEs were, in contrast, not sensitive to the noise structure of stride intervals.
Section snippets
Setting
A large, anonymized dataset of acceleration signals obtained from our previous studies was re-analyzed [19,20]. In short, 100 healthy individuals aged between 20 and 69 years walked at preferred speed on a treadmill for five minutes in two sessions, separated by one week. A 3D accelerometer, attached to the sternum, recorded the trunk acceleration.
Data pre-processing
Each of the two-hundred acceleration signals was pre-processed using Matlab (R 2017a; Mathworks, Natick, MA, USA). First, the vertical signal was
Stride-interval time series
The analyzed database contained 200 acceleration signals. Among them, 109 signals from 69 participants were judged of sufficient quality to be included in the analyses. Included subjects had the same mean age as the excluded subjects (44 yr. vs. 45 yr. t-test p = 0.58). The included time series of stride intervals exhibited a mean CV of 2.9% (SD = 1.6%) and a correlated structure with a mean scaling exponent of 0.71 (SD = 0.14) (Table 1). DFAs of the artificial times series confirmed that the
Discussion
The results supported the hypothesis that long-term DE is responsive to the noise structure of stride intervals. Indeed, hybrid signals with identical shapes but modified stride intervals had long-term DEs that varied strongly according to the type of noise applied. Alternatively, short-term DE varied within a narrower range (∼20%) and were not sensitive to noise type.
Regarding the analytical method, we used two different ranges to compute long-term DE: across the span of 2—4 strides and across
Conclusion
The present study’s findings further support the idea that the short-term DE and the long-term DE, although they are both computed from attractor divergence curves, should not be interpreted in a similar manner. Accordingly, we propose a new term to better differentiate between them. Because long-term DE is an index of complexity computed from a multidimensional attractor, the term attractor complexity index (ACI) is thought to be appropriate. We hope that the empirical clarification of the
Authors’ contributions
Funding acquisition: PT. Conceptualization: PT. Methodology: PT. Data curation: PT, FR. Investigation: FR. Formal analysis: PT. Writing—original draft: PT. Writing—review and editing: PT, FR. Visualization: PT. Supervision: PT, FR. Project administration: FR. Both authors read and approved the final version of the manuscript.
Conflict of interest statement
None declared.
Acknowledgements
The authors wish to thank Dr. Olivier Dériaz for his administrative help and useful advice. The SUVA and the clinique romande de réadaptation were the main sponsors of the study through internal funding. A gift from the Loterie Romande also supported the study. The Institute for Research in Rehabilitation is funded by the State of Valais and the City of Sion. The study’s sponsors are not implied in the study design; the collection, analyses, or interpretation of data; the writing of the
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