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Detection of co-regulation of local structure and magnitude of stride time variability using a new local detrended fluctuation analysis
Gait & Posture 39 (2014) 466–471
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Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Detection of co-regulation of local structure and magnitude of stride time variability using a new local detrended fluctuation analysis Espen A.F. Ihlen a,*, Beatrix Vereijken b a b
Department of Neuroscience, Norwegian University of Science and Technology, Trondheim, Norway Department of Human Movement Science, Norwegian University of Science and Technology, Trondheim, Norway
A R T I C L E I N F O
A B S T R A C T
Article history: Received 27 January 2013 Received in revised form 15 August 2013 Accepted 25 August 2013
Detrended fluctuation analysis (DFA) is a popular method to numerically define the persistent structure of stride time variability. The conventional DFA assumes that the persistent structure in stride time variability is consistent in time and can be numerically defined by a single DFA scaling exponent. However, stride time regulation has to be adaptive to both environmental and internal perturbations and consequently, the persistent structure of stride time variability will have to be modulated in time. The present article introduces a new local detrended fluctuation analysis (DFAloc) that is able to detect modulation in the structure of stride time variability generated by phase-couplings between temporal scales. DFAloc was used in a reanalysis of the data set of stride time variability of Hausdorff et al. and a data set available at www.physionet.org. The results showed that there were significant phase couplings between temporal scales that generate an inverse correlation (r = 0.54 to 0.83) between the local structure and local magnitude of the stride time variability. Furthermore, the modulation of the local structure was significantly influenced by gait speed, external pace making, and age (all p’s < 0.05). These results suggest several specific modifications of contemporary theories that have been suggested for the persistent structure of stride time variability found by the conventional DFA. ß 2013 Elsevier B.V. All rights reserved.
Keywords: Walking Stride time Variability 1/f noise Detrended fluctuation analysis
1. Introduction The variability of spatiotemporal gait variables, like stride time, length, and width, can be numerically defined by both its magnitude and structure. Parameters of the magnitude, such as standard deviation and coefficient of variation, are the most common way to numerically define variability whereas parameters of structure, such as scaling exponents, have gained increased attention in several reports during the last decades [1–9]. Detrended fluctuation analysis (DFA) is one of the most common methods to numerically define the structure of variability in stride time, length, and speed [1–8]. The results from DFA indicate that stride time variability has a persistent structure but that the degree of persistency is dependent on multiple factors that are both internal and external to the nervous system. Several studies have shown that the persistency in stride time variability declines with increased age and with the presence of neurodegenerative diseases such as Huntington and Parkinson’s disease [2,5,6]. They further suggest that structural and functional
* Corresponding author at: Department of Neuroscience, Norwegian University of Science and Technology, N-7489 Trondheim, Norway. Tel.: +47 47354674. E-mail address: [email protected] (Espen A.F. Ihlen). 0966-6362/$ – see front matter ß 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.gaitpost.2013.08.024
degenerations in the central nervous system with age and diseases affect an internal pace-making mechanism of human gait [10,11]. In addition, the persistent structure of stride time variability has been shown to vanish when the cadence is set by an external metronome [1,12]. These findings further support that the persistent structure is generated by an internal pace-making mechanism within the central nervous system and that the decline in persistency reflects degeneration of the pace-making mechanism. According to this perspective, the decline in persistency of stride time variability has thus been interpreted as reflecting ‘‘unhealthy’’ gait function. However, not all reports support the suggestion that the decline in persistency of stride time variability reflects an ‘‘unhealthy’’ gait function by degeneration of an internal pace-making mechanism. A U-shaped relationship between stride time persistency and gait speed has been found in several studies on both over-ground and treadmill walking [1,13]. The persistent structure of both stride time and length variability in treadmill walking reflected the stride-to-stride regulation of the gait speed to the treadmill, not an internal pace making mechanism [8]. Furthermore, peripheral neuropathy in diabetic patients did not change the persistent structure even though this pathological condition would have affected the sensor-motor integration within the central nervous system [7]. In addition, sensor-motor noise as external input to the
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central nervous system has been found to reproduce the persistent structure of stride time variability without the influence of an internal pace-making mechanism [14]. Thus, the persistent structure of stride time variability could reflect ‘‘healthy’’ stride time regulation to the walking condition rather than an ‘‘unhealthy’’ gait function. Consequently, it is still an unresolved issue how to interpret the results from DFA. Common for all factors that affect the outcomes of DFA is their appearance in time. Motor errors and speed regulation appear as modulations of stride time variability over short time scales of milliseconds whereas aging and degeneration in the central nervous system appear as modulation of stride time variability over long time scales of years. Both kinds of modulations will yield temporal changes in the persistent structure of stride time variability. However, the conventional DFA assumes that the persistent structure is constant across strides and that the pacemaking mechanism or sensor-motor noise are stationary mechanisms of stride time variability [11,14]. Results from analyses of both heart rate and stride time variability suggest that their persistent structures are temporally modulated. The temporal modulations of the structure in heart rate variability can distinguish between pathological conditions such as ventricular fibrillation and ventricular tachycardia, which cannot be distinguished in the outcomes of the conventional DFA [15]. Furthermore, the fluctuation in the structure of stride time variability is reported to be both age and speed dependent similar to the average structure defined by DFA [12,16]. Thus, the variation in the persistent structure of stride time variability, not only its average persistent structure, might be important to identify temporal changes in the stride time regulation. The temporal modulations of the structure of variability are referred to as multifractal variations [17–20]. Multifractal variations have intermittent periods with a local decrease in the persistent structure of variability and intermittent periods with a
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local increase in the magnitudes of the variability. The intermittent coupling between the local structure and local magnitude of variability is generated by phase couplings between temporal scales. The main aim of the present study is to introduce a new local DFA (DFAloc) of stride time variability that can identify variation in the persistent structure of stride time variability generated by phase couplings across temporal scales. 2. Methods DFAloc was used to reanalyze two data sets of [1] and [21]. Data set 1 contains stride time series of 60 min over-ground walking of slow (1.0 0.1 m/s), preferred (1.4 0.1 m/s) and fast (1.7 0.1 m/ s) gait speed for ten younger adults (age: 18–29 years, body mass: 71.8 10.7 kg, height: 1.77 0.08 m). The same trials were also conducted with an external cadence-setting metronome for 30 min over-ground walking. Data set 2 contains 15 min over-ground walking for five younger adults (age: 23–29 years) and five healthy older adults (age: 71–77 years). Further information for data sets 1 and 2 can be found in [1] and [21], respectively. DFAloc was conducted in the same way as the conventional DFA with three important differences. First, the root mean square of the detrended residuals was computed in a floating time interval across the integrated stride time series instead of in nonoverlapping time intervals as in conventional DFA. The floating time intervals had a length n, referred to as the scale, and were centered at time t. Secondly, the obtained root-mean-square measure F(n,t) was dependent on both time and scale in contrast to the root-mean-square measure F(n) of the conventional DFA that is only dependent on scale (i.e., average of F(n,t) across time t). Thirdly, a local DFA scaling exponent a(t) is numerically defined by the linear slope of log[F(n,t)] versus log(n) for each time instant t instead of by the time-independent log[F(n)] versus log(n) for the conventional DFA. A modified procedure developed by [22] was
Fig. 1. Illustration of DFAloc and the corresponding Monte Carlo surrogate test for stride time variability. Panel A: A stride time series (black trace) and a surrogate time series (blue trace) where the phase couplings between temporal scales are removed. Panel B: The definition of DFAloc scaling exponents a(n, t) of stride time series and surrogate series. The root-mean-square variation F(n, t) is plotted for each scale n (in log–log coordinates) and stride number t (cluster of blue dots). The DFAloc scaling exponent a(n, t) is defined as the difference between log[F(n, t)] and the regression line alog(n) + C (vertical black arrows) of the conventional DFA (red line) for each stride t divided by the difference between the scale n = 4 to 40 of F(n, t) and the maximum scale nmax (horizontal black arrows). The slope of the green lines are max[a(n, t)] and min[a(n, t)]. Panel C: Contour plot of a(n, t) resolved in stride number t and scale n resolution where the red contours indicate intermittent periods of more persistent variation (i.e., large a(n, t)) and blue contours indicate intermittent periods of less persistent variation (i.e., small a(n, t)). The red diamonds and blue circles indicate the stride number and scale where the large and small a(n, t), respectively, are significantly influenced by phase couplings (p < 0.025). Panel D: The probability distribution p(a) of all DFAloc scaling exponents a(n, t) (green trace), and large a(n, t) (red trace) and small a(n, t) (blue trace) influenced by phase couplings. The green, red and blue intervals indicate the width of distributions p(a) numerically defined as interquartile ranges (IQRa). (For interpretation of the references to color in the text, the reader is referred to the web version of the article.)
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used to make the estimated a(t) numerically stable. In this procedure, a(t) is defined by the following equation (see Fig. 1B):
aðn; tÞ ¼
logðFðn; tÞÞ ðalogðnÞ þ CÞ þa logðnÞ logðnmax Þ
(1)
where alog(n) + C is the regression line for log[F(n)] versus log(n) computed by the conventional DFA and a is the conventional DFA scaling exponent. A limited scaling range of 4 < n < 40 was chosen for Eq. (1) to make the DFAloc localized in time t and scale n (see Fig. 1C). The local structure of the stride time variability is persistent when a(t) > 0.5, anti-persistent when a(t) < 0.5, and
uncorrelated white noise when a(t) = 0.5. Technical details and tutorial instruction on how to conduct DFAloc are represented elsewhere [18,20]. All data were processed in Matlab 2009a (Mathworks, Natick, MA). A Monte Carlo surrogate test was conducted in correspondence with DFAloc to identify phase couplings between temporal scales. 2500 surrogates were generated by iterated amplitude adjusted Fourier transform for each stride time series. These surrogates replicate the average a estimated by the conventional DFA and the stride time distribution, but eliminate all phase couplings between temporal scales. Significant phase coupling is present when less
Fig. 2. Example of the influence of phase couplings between temporal scales on the inverse correlation between the local structure a(t) and magnitude SD(t) of stride time variability. Panel A: Stride time series and the corresponding contour plot of a(n, t). The red diamonds and blue circles indicate large and small a(n, t), respectively, significantly influenced by phase couplings between temporal scales. Panel B: The local structure a(n, t) (blue trace) and magnitude SD(t) (red trace) for n = 20. The intermittent periods with large and small a(t) influenced by phase couplings generate an inverse relationship with the local magnitude SD(t) of stride time variability. Panel C: The plot of local structure a(t) versus local magnitude SD(t) for the stride time series (left panel) and the surrogate series in Fig. 1. (For interpretation of the references to color in the text, the reader is referred to the web version of the article.)
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than 2.5% (two-sided test) of the surrogate series had a(n,t) larger or smaller than the a(n,t) of the original stride time series (see red diamonds and blue circles in Fig. 1C). Technical details on the conduction of Monte Carlo surrogate test and the generation of surrogates by iterated amplitude adjusted Fourier transform can be found in [20,23]. The temporal variation in a(t) was summarized in a probability distribution P(a) (see Fig. 1D). P(a) were defined as histograms for all a(t) and those a(t) that were significantly influenced by phase couplings between temporal scales (see red and blue traces in Fig. 1D). By definition, the mode (i.e., central tendency) of P(a) is a as estimated by the conventional DFA. The width of P(a) was defined as the interquartile range (IQRa) of a(t) (see Fig. 1D). IQRa was also computed for a(t) and for the portion of a(t) significantly influenced by phase couplings between temporal scales. A Pearson correlation analysis between a(n,t) and SD(t) within a floating window of sample size n of each stride time series was conducted to test for a significant relationship between the temporal modulation of structure and magnitude of stride time variability. In data set 1, the effect of gait speed on IQRa and Pearson’s correlation coefficient was tested by repeated-measures ANOVA with post hoc pair-wise comparisons (paired samples t-tests) with Bonferroni correction for multiple comparisons. The effect of external pace making metronome on IQRa was tested by a paired samples t-test for each of the speed conditions. In data set 2, an independent samples t-test was used to identify significant differences between IQRa and Pearson’s correlation coefficient between healthy young and older adults. 3. Results Fig. 2A illustrates a representative example of a(n,t) and the influence of phase couplings between temporal scales. Intermittent periods with large stride time variability had small a(n,t) significantly influenced by phase couplings (see blue circles in the contour plot of Fig. 2A), whereas intermittent periods with small stride time variability had large a(n,t) also significantly influenced by couplings (see red diamonds in the contour plot of Fig. 2A). Consequently, all stride time series had a(n,t) in the tails of P(a)
A
All α(t) 0.2
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that were significantly influenced by phase couplings (see blue and red sub-distributions in Fig. 1D). Furthermore, the inverse relationship between the temporal modulation of the local magnitude and structure of stride time variability was confirmed by a highly significant correlation between SD(t) and a(t) (r = 0.54 to 0.87, p < 0.00001; see example in Fig. 2B and C). Thus, phase couplings between temporal scales generated an increase in the inverse correlation between local structure and magnitude of stride time variability. In data set 1, a significant decrease in the width of distribution P(a) was found with increase in gait speed (repeated-measures ANOVA: F(3,9) = 3.58, p < 0.05; see black bars in Fig. 3A). The decrease in width of P(a) was related to a decrease in the subdistribution of small a(t) significantly influenced by phase couplings (repeated-measures ANOVA: F(3,9) = 4.00, p < 0.05; see black bars in Fig. 3B). However, the post hoc paired comparisons only yielded a significant difference between IQRa for the sub-distribution between slow and fast gait speed before Bonferroni correction (t-test: p = 0.03). There was no significant influence of gait speed when the cadence was set by a metronome (see gray bars in Fig. 3A and C). Furthermore, no speed-dependent differences were found in the inverse correlation between the local structure a(t) and magnitude SD(t) for either the unconstrained or the metronome-constrained condition (see Fig. 3D). In data set 1, the width of the distribution P(a) significantly increased in the metronome constrained condition for the preferred and the fast gait speeds, but not for the slow gait speed (see stars in Fig. 3A). The increase in width of P(a) was related to an increase in the width of the sub-distribution of small a(t) influenced by phase couplings (see stars in Fig. 3B). A significantly higher inverse correlation was found between structure a(t) and magnitude SD(t) of stride time variability in the metronome-constrained condition for all gait speeds (see stars in Fig. 3D). The coupling between structure and magnitude of stride time variability reflects the increased width of P(a) and, consequently, the increased width of the sub-distribution of small a(t) significantly influenced by phase couplings between temporal scales. In data set 2, the width of the distribution P(a) significantly decreased for healthy older adults compared with healthy younger
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Fig. 3. Results of DFAloc employed to data set 1 of [1]. Panel A: The interquartile range (IQRa) of all a(t) for unconstrained (black bars) and metronome constrained gait (gray bars) and for slow, preferred and fast gait speed. Panel B: IQRa of small a(t) significantly influenced by phase couplings between temporal scales. Panel C: IQRa of large a(t) significantly influenced by phase couplings between temporal scales. Panel D: Cross-correlation coefficient between local structure a(t) and magnitude SD(t) for unconstrained (black bars) and metronome constrained (gray bars) slow, preferred and fast gait speed. Stars indicate significant differences between unconstrained and metronome constrained gait; *p < 0.05, **p < 0.0005.
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Fig. 4. Results of DFAloc employed to data set 2 [21]. Panel A: The interquartile range of all a(t), and small and large a(t) significantly influenced by phase couplings for healthy young (black bars) and healthy older adults (gray bars). Panel B: Crosscorrelation coefficient between local structure a(t) and magnitude SD(t) for healthy young and older adults. Stars indicate significant differences between healthy young and older adults; *p < 0.05, **p < 0.005.
adults (see left set of bars in Fig. 4A). In addition, the decreased width of P(a) was related to a significant decrease in the width of the sub-distribution of large a(t) significantly influenced by phase couplings (see right set of bars in Fig. 4A). However, no significant differences were found in the inverse cross-correlations between the healthy younger and older adults (see Fig. 4B). 4. Discussion The present article introduced DFAloc that can numerically define temporal changes in the structure of stride time variability that are generated by phase couplings between temporal scales. Furthermore, DFAloc can also numerically define the relationship between temporal change in local structure and magnitude of the stride time variability. These numerical features of phase couplings between temporal scales cannot be detected by the conventional DFA. Previous studies that used the conventional DFA found that both metronome-constrained gait in healthy younger adults and impaired gait in older persons and patients with neurodegenerative diseases have a less persistent stride time variability (i.e., a 0.5) compared to normal gait [1,2,5,6,12]. DFAloc indicates that metronome-constrained gait has a more active stride time regulation that is reflected by more phase couplings between temporal scales compared to unconstrained stride-time variability. The stronger influence of phase couplings reflects ‘‘healthy’’ counter-adjustments (i.e., anti-persistent variations) of stride time in intermittent periods of large stride time variability. This active regulation of stride time variability also strengthens the correlation between the local structure a(t) and the local magnitude SD(t) of stride time variability. In contrast, the gait of healthy older adults had fewer phase couplings between temporal scales in the periods with small and persistent variation (i.e., large a(t)), which resulted in reduced active regulation of the local structure of stride time variability. The latter might reflect impaired stride time regulation. Impaired stride time regulation might also be indicated by the inability to adjust the inverse correlation between the local structure a(t) and magnitude SD(t) of stride time variability. The ability to adjust the covariation between the local structure and magnitude of stride time variability might reflect a ‘‘healthy’’ stride time regulation in more challenging walking tasks and walking environments. In summary, DFAloc has the ability to differentiate between different stride time variabilities that have a similar
scaling exponent a as defined by conventional DFA and consequently, provide further insight into the stride time regulation of human walking. The new numerical features obtained by DFAloc suggest modification of previous interpretations of the conventional DFA scaling exponent a. A previous study showed that a of stride time variability can be generated by external white noise as input to the sensory-motor system [14]. This model can reproduce declines in a by a simple magnitude change in neuronal input noise. This reproduced decline in a stands in contrast to more complex reproduction of the same decline by the internal modulation of the network of central pattern generators. However, the results from DFAloc indicate, under the assumption of a stationary environment in the experimental condition, that both the variation in a(t) and its covariation with the local magnitude is generated by significant phase couplings (i.e., multiplicative interactions) between temporal scales that cannot originate from additive white noise. By definition, models with additive white noise will reproduce the mean a(t) of the stride time series and assume that a(t) is constant. However, the variation in a(t) might be a more central feature of stride time regulation, reflecting the neural modulation of white input noise by mechanisms within the central nervous system. Thus, DFAloc of stride time variability might give further insight into stride time regulation as a relationship between mechanisms in the central nervous system and the external walking conditions. Further research should therefore use DFAloc in combination with the conventional DFA to evaluate the output of model simulation of stride time variability. Several issues of DFAloc need to be further investigated. First, DFAloc has the same documented limitations as the conventional DFA [24–26]. DFAloc assumes scale invariance and self-affinity within a predefined scaling range of the stride time series. In the absence of a linear relationship between log[F(n)] and log(n) (see red line in Fig. 1B), this would result in false positive results for DFAloc. Further studies should extend DFAloc according to methods in [27] and [28] to investigate scale-dependent changes in a(t) that indicate the absence of local scale invariance. Second, like the conventional DFA, the performance of DFAloc is dependent on the number of strides contained in the stride time series [29]. The stride time series reanalyzed in the present study contained between 709 and 3822 strides and the large differences in the number of strides might have influenced the results. Future studies should therefore investigate the influence of total number of strides on the computational errors of IQRa. Furthermore, average metrics of multiple trials should be considered for stride time series with short sample size (<500 strides). Third, the conventional DFA is shown to have high test– retest reliability [30]. The test–retest reliability of the new metrics of DFAloc was not tested in the present study, but should be assessed in the future to establish the relevance of DFAloc for clinical gait analyses. Fourth, the interpretation of DFAloc when employed to healthy older adults is based on a small sample size (i.e., N = 5) with limited background information of the participants. Thus, further studies should replicate the DFAloc results of stride time variability in a larger group of older persons and, in extension, in patient groups with neurodegenerative diseases. 5. Conclusion The present article introduced a new local detrended fluctuation analysis (DFAloc) of stride time variability. The metrics introduced by DFAloc were able to identify temporal variations in the local structure of stride time variability that were generated by phase couplings between temporal scales. Furthermore, DFAloc identified an inverse correlation between the temporal modulation in the local magnitude and the structure of the stride time variability. The metrics introduced by DFAloc were dependent on
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gait speed, metronome-constrained cadence, and age. The present article suggests that DFAloc can provide further insight into the stride time regulation of human walking and might identify precursors of ongoing decline in gait function that might proceed to increased risk of falling in older adults and patients with neurodegenerative diseases. Acknowledgments PhD grants from the Norwegian University of Science and Technology. Data set 1 and 2 is available at www.physionet.org/ physiobank/database/#gait. Matlab code for DFAloc is available at www.ntnu.edu/inm/geri/software. Conflict of interest statement The authors declare that no conflict of interests is associated with the present study. References [1] Hausdorff JM, Purdon PL, Peng CK, Ladin Z, Wei JY, Goldberger AL. Fractal dynamics of gait: stability of long-range correlations in stride interval fluctuations. J Appl Physiol 1996;80(5):1448–57. [2] Hausdorff JM. Gait dynamics, fractals and falls: finding meaning in the strideto-stride fluctuations of human walking. Hum Mov Sci 2007;26(4):555–89. [3] Jordan K, Challis JH, Cusumano JP, Newell KM. Stability and the time-dependent structure of gait variability in walking and running. Hum Mov Sci 2009;28(1):113–28. [4] Terrier P, Turner V, Schutz Y. GPS analysis of human locomotion: further evidence for long-range correlations in stride-to-stride fluctuations of gait parameters. Hum Mov Sci 2005;24(1):97–115. [5] Hausdorff JM, Mitchell SL, Firtion R, Peng CK, Cudkowicz ME, Wei JY, et al. Altered fractal dynamics of gait: reduced stride interval correlations with aging and Huntington’s disease. J Appl Physiol 1997;82(1):262–9. [6] Hausdorff JM. Gait dynamics in Parkinson’s disease: common and distinct behavior among stride length, gait variability, and fractal-like scaling. Chaos 2009;19(2):026113–26114. [7] Gates DH, Dingwell JB. Peripheral neuropathy does not alter the fractal dynamics of gait stride intervals. J Appl Physiol 2007;102(3):965–71. [8] Dingwell JB, Cusumano JP. Re-interpreting detrended fluctuation analyses of stride-to-stride variability in human walking. Gait Posture 2010;32:348–53. [9] Delignie´res D, Torre K. Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdorff et al.. J Appl Physiol 2009;106(4):1272–9.
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Gait & Posture journal homepage: www.elsevier.com/locate/gaitpost
Detection of co-regulation of local structure and magnitude of stride time variability using a new local detrended fluctuation analysis Espen A.F. Ihlen a,*, Beatrix Vereijken b a b
Department of Neuroscience, Norwegian University of Science and Technology, Trondheim, Norway Department of Human Movement Science, Norwegian University of Science and Technology, Trondheim, Norway
A R T I C L E I N F O
A B S T R A C T
Article history: Received 27 January 2013 Received in revised form 15 August 2013 Accepted 25 August 2013
Detrended fluctuation analysis (DFA) is a popular method to numerically define the persistent structure of stride time variability. The conventional DFA assumes that the persistent structure in stride time variability is consistent in time and can be numerically defined by a single DFA scaling exponent. However, stride time regulation has to be adaptive to both environmental and internal perturbations and consequently, the persistent structure of stride time variability will have to be modulated in time. The present article introduces a new local detrended fluctuation analysis (DFAloc) that is able to detect modulation in the structure of stride time variability generated by phase-couplings between temporal scales. DFAloc was used in a reanalysis of the data set of stride time variability of Hausdorff et al. and a data set available at www.physionet.org. The results showed that there were significant phase couplings between temporal scales that generate an inverse correlation (r = 0.54 to 0.83) between the local structure and local magnitude of the stride time variability. Furthermore, the modulation of the local structure was significantly influenced by gait speed, external pace making, and age (all p’s < 0.05). These results suggest several specific modifications of contemporary theories that have been suggested for the persistent structure of stride time variability found by the conventional DFA. ß 2013 Elsevier B.V. All rights reserved.
Keywords: Walking Stride time Variability 1/f noise Detrended fluctuation analysis
1. Introduction The variability of spatiotemporal gait variables, like stride time, length, and width, can be numerically defined by both its magnitude and structure. Parameters of the magnitude, such as standard deviation and coefficient of variation, are the most common way to numerically define variability whereas parameters of structure, such as scaling exponents, have gained increased attention in several reports during the last decades [1–9]. Detrended fluctuation analysis (DFA) is one of the most common methods to numerically define the structure of variability in stride time, length, and speed [1–8]. The results from DFA indicate that stride time variability has a persistent structure but that the degree of persistency is dependent on multiple factors that are both internal and external to the nervous system. Several studies have shown that the persistency in stride time variability declines with increased age and with the presence of neurodegenerative diseases such as Huntington and Parkinson’s disease [2,5,6]. They further suggest that structural and functional
* Corresponding author at: Department of Neuroscience, Norwegian University of Science and Technology, N-7489 Trondheim, Norway. Tel.: +47 47354674. E-mail address: [email protected] (Espen A.F. Ihlen). 0966-6362/$ – see front matter ß 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.gaitpost.2013.08.024
degenerations in the central nervous system with age and diseases affect an internal pace-making mechanism of human gait [10,11]. In addition, the persistent structure of stride time variability has been shown to vanish when the cadence is set by an external metronome [1,12]. These findings further support that the persistent structure is generated by an internal pace-making mechanism within the central nervous system and that the decline in persistency reflects degeneration of the pace-making mechanism. According to this perspective, the decline in persistency of stride time variability has thus been interpreted as reflecting ‘‘unhealthy’’ gait function. However, not all reports support the suggestion that the decline in persistency of stride time variability reflects an ‘‘unhealthy’’ gait function by degeneration of an internal pace-making mechanism. A U-shaped relationship between stride time persistency and gait speed has been found in several studies on both over-ground and treadmill walking [1,13]. The persistent structure of both stride time and length variability in treadmill walking reflected the stride-to-stride regulation of the gait speed to the treadmill, not an internal pace making mechanism [8]. Furthermore, peripheral neuropathy in diabetic patients did not change the persistent structure even though this pathological condition would have affected the sensor-motor integration within the central nervous system [7]. In addition, sensor-motor noise as external input to the
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central nervous system has been found to reproduce the persistent structure of stride time variability without the influence of an internal pace-making mechanism [14]. Thus, the persistent structure of stride time variability could reflect ‘‘healthy’’ stride time regulation to the walking condition rather than an ‘‘unhealthy’’ gait function. Consequently, it is still an unresolved issue how to interpret the results from DFA. Common for all factors that affect the outcomes of DFA is their appearance in time. Motor errors and speed regulation appear as modulations of stride time variability over short time scales of milliseconds whereas aging and degeneration in the central nervous system appear as modulation of stride time variability over long time scales of years. Both kinds of modulations will yield temporal changes in the persistent structure of stride time variability. However, the conventional DFA assumes that the persistent structure is constant across strides and that the pacemaking mechanism or sensor-motor noise are stationary mechanisms of stride time variability [11,14]. Results from analyses of both heart rate and stride time variability suggest that their persistent structures are temporally modulated. The temporal modulations of the structure in heart rate variability can distinguish between pathological conditions such as ventricular fibrillation and ventricular tachycardia, which cannot be distinguished in the outcomes of the conventional DFA [15]. Furthermore, the fluctuation in the structure of stride time variability is reported to be both age and speed dependent similar to the average structure defined by DFA [12,16]. Thus, the variation in the persistent structure of stride time variability, not only its average persistent structure, might be important to identify temporal changes in the stride time regulation. The temporal modulations of the structure of variability are referred to as multifractal variations [17–20]. Multifractal variations have intermittent periods with a local decrease in the persistent structure of variability and intermittent periods with a
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local increase in the magnitudes of the variability. The intermittent coupling between the local structure and local magnitude of variability is generated by phase couplings between temporal scales. The main aim of the present study is to introduce a new local DFA (DFAloc) of stride time variability that can identify variation in the persistent structure of stride time variability generated by phase couplings across temporal scales. 2. Methods DFAloc was used to reanalyze two data sets of [1] and [21]. Data set 1 contains stride time series of 60 min over-ground walking of slow (1.0 0.1 m/s), preferred (1.4 0.1 m/s) and fast (1.7 0.1 m/ s) gait speed for ten younger adults (age: 18–29 years, body mass: 71.8 10.7 kg, height: 1.77 0.08 m). The same trials were also conducted with an external cadence-setting metronome for 30 min over-ground walking. Data set 2 contains 15 min over-ground walking for five younger adults (age: 23–29 years) and five healthy older adults (age: 71–77 years). Further information for data sets 1 and 2 can be found in [1] and [21], respectively. DFAloc was conducted in the same way as the conventional DFA with three important differences. First, the root mean square of the detrended residuals was computed in a floating time interval across the integrated stride time series instead of in nonoverlapping time intervals as in conventional DFA. The floating time intervals had a length n, referred to as the scale, and were centered at time t. Secondly, the obtained root-mean-square measure F(n,t) was dependent on both time and scale in contrast to the root-mean-square measure F(n) of the conventional DFA that is only dependent on scale (i.e., average of F(n,t) across time t). Thirdly, a local DFA scaling exponent a(t) is numerically defined by the linear slope of log[F(n,t)] versus log(n) for each time instant t instead of by the time-independent log[F(n)] versus log(n) for the conventional DFA. A modified procedure developed by [22] was
Fig. 1. Illustration of DFAloc and the corresponding Monte Carlo surrogate test for stride time variability. Panel A: A stride time series (black trace) and a surrogate time series (blue trace) where the phase couplings between temporal scales are removed. Panel B: The definition of DFAloc scaling exponents a(n, t) of stride time series and surrogate series. The root-mean-square variation F(n, t) is plotted for each scale n (in log–log coordinates) and stride number t (cluster of blue dots). The DFAloc scaling exponent a(n, t) is defined as the difference between log[F(n, t)] and the regression line alog(n) + C (vertical black arrows) of the conventional DFA (red line) for each stride t divided by the difference between the scale n = 4 to 40 of F(n, t) and the maximum scale nmax (horizontal black arrows). The slope of the green lines are max[a(n, t)] and min[a(n, t)]. Panel C: Contour plot of a(n, t) resolved in stride number t and scale n resolution where the red contours indicate intermittent periods of more persistent variation (i.e., large a(n, t)) and blue contours indicate intermittent periods of less persistent variation (i.e., small a(n, t)). The red diamonds and blue circles indicate the stride number and scale where the large and small a(n, t), respectively, are significantly influenced by phase couplings (p < 0.025). Panel D: The probability distribution p(a) of all DFAloc scaling exponents a(n, t) (green trace), and large a(n, t) (red trace) and small a(n, t) (blue trace) influenced by phase couplings. The green, red and blue intervals indicate the width of distributions p(a) numerically defined as interquartile ranges (IQRa). (For interpretation of the references to color in the text, the reader is referred to the web version of the article.)
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used to make the estimated a(t) numerically stable. In this procedure, a(t) is defined by the following equation (see Fig. 1B):
aðn; tÞ ¼
logðFðn; tÞÞ ðalogðnÞ þ CÞ þa logðnÞ logðnmax Þ
(1)
where alog(n) + C is the regression line for log[F(n)] versus log(n) computed by the conventional DFA and a is the conventional DFA scaling exponent. A limited scaling range of 4 < n < 40 was chosen for Eq. (1) to make the DFAloc localized in time t and scale n (see Fig. 1C). The local structure of the stride time variability is persistent when a(t) > 0.5, anti-persistent when a(t) < 0.5, and
uncorrelated white noise when a(t) = 0.5. Technical details and tutorial instruction on how to conduct DFAloc are represented elsewhere [18,20]. All data were processed in Matlab 2009a (Mathworks, Natick, MA). A Monte Carlo surrogate test was conducted in correspondence with DFAloc to identify phase couplings between temporal scales. 2500 surrogates were generated by iterated amplitude adjusted Fourier transform for each stride time series. These surrogates replicate the average a estimated by the conventional DFA and the stride time distribution, but eliminate all phase couplings between temporal scales. Significant phase coupling is present when less
Fig. 2. Example of the influence of phase couplings between temporal scales on the inverse correlation between the local structure a(t) and magnitude SD(t) of stride time variability. Panel A: Stride time series and the corresponding contour plot of a(n, t). The red diamonds and blue circles indicate large and small a(n, t), respectively, significantly influenced by phase couplings between temporal scales. Panel B: The local structure a(n, t) (blue trace) and magnitude SD(t) (red trace) for n = 20. The intermittent periods with large and small a(t) influenced by phase couplings generate an inverse relationship with the local magnitude SD(t) of stride time variability. Panel C: The plot of local structure a(t) versus local magnitude SD(t) for the stride time series (left panel) and the surrogate series in Fig. 1. (For interpretation of the references to color in the text, the reader is referred to the web version of the article.)
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than 2.5% (two-sided test) of the surrogate series had a(n,t) larger or smaller than the a(n,t) of the original stride time series (see red diamonds and blue circles in Fig. 1C). Technical details on the conduction of Monte Carlo surrogate test and the generation of surrogates by iterated amplitude adjusted Fourier transform can be found in [20,23]. The temporal variation in a(t) was summarized in a probability distribution P(a) (see Fig. 1D). P(a) were defined as histograms for all a(t) and those a(t) that were significantly influenced by phase couplings between temporal scales (see red and blue traces in Fig. 1D). By definition, the mode (i.e., central tendency) of P(a) is a as estimated by the conventional DFA. The width of P(a) was defined as the interquartile range (IQRa) of a(t) (see Fig. 1D). IQRa was also computed for a(t) and for the portion of a(t) significantly influenced by phase couplings between temporal scales. A Pearson correlation analysis between a(n,t) and SD(t) within a floating window of sample size n of each stride time series was conducted to test for a significant relationship between the temporal modulation of structure and magnitude of stride time variability. In data set 1, the effect of gait speed on IQRa and Pearson’s correlation coefficient was tested by repeated-measures ANOVA with post hoc pair-wise comparisons (paired samples t-tests) with Bonferroni correction for multiple comparisons. The effect of external pace making metronome on IQRa was tested by a paired samples t-test for each of the speed conditions. In data set 2, an independent samples t-test was used to identify significant differences between IQRa and Pearson’s correlation coefficient between healthy young and older adults. 3. Results Fig. 2A illustrates a representative example of a(n,t) and the influence of phase couplings between temporal scales. Intermittent periods with large stride time variability had small a(n,t) significantly influenced by phase couplings (see blue circles in the contour plot of Fig. 2A), whereas intermittent periods with small stride time variability had large a(n,t) also significantly influenced by couplings (see red diamonds in the contour plot of Fig. 2A). Consequently, all stride time series had a(n,t) in the tails of P(a)
A
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that were significantly influenced by phase couplings (see blue and red sub-distributions in Fig. 1D). Furthermore, the inverse relationship between the temporal modulation of the local magnitude and structure of stride time variability was confirmed by a highly significant correlation between SD(t) and a(t) (r = 0.54 to 0.87, p < 0.00001; see example in Fig. 2B and C). Thus, phase couplings between temporal scales generated an increase in the inverse correlation between local structure and magnitude of stride time variability. In data set 1, a significant decrease in the width of distribution P(a) was found with increase in gait speed (repeated-measures ANOVA: F(3,9) = 3.58, p < 0.05; see black bars in Fig. 3A). The decrease in width of P(a) was related to a decrease in the subdistribution of small a(t) significantly influenced by phase couplings (repeated-measures ANOVA: F(3,9) = 4.00, p < 0.05; see black bars in Fig. 3B). However, the post hoc paired comparisons only yielded a significant difference between IQRa for the sub-distribution between slow and fast gait speed before Bonferroni correction (t-test: p = 0.03). There was no significant influence of gait speed when the cadence was set by a metronome (see gray bars in Fig. 3A and C). Furthermore, no speed-dependent differences were found in the inverse correlation between the local structure a(t) and magnitude SD(t) for either the unconstrained or the metronome-constrained condition (see Fig. 3D). In data set 1, the width of the distribution P(a) significantly increased in the metronome constrained condition for the preferred and the fast gait speeds, but not for the slow gait speed (see stars in Fig. 3A). The increase in width of P(a) was related to an increase in the width of the sub-distribution of small a(t) influenced by phase couplings (see stars in Fig. 3B). A significantly higher inverse correlation was found between structure a(t) and magnitude SD(t) of stride time variability in the metronome-constrained condition for all gait speeds (see stars in Fig. 3D). The coupling between structure and magnitude of stride time variability reflects the increased width of P(a) and, consequently, the increased width of the sub-distribution of small a(t) significantly influenced by phase couplings between temporal scales. In data set 2, the width of the distribution P(a) significantly decreased for healthy older adults compared with healthy younger
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Fig. 3. Results of DFAloc employed to data set 1 of [1]. Panel A: The interquartile range (IQRa) of all a(t) for unconstrained (black bars) and metronome constrained gait (gray bars) and for slow, preferred and fast gait speed. Panel B: IQRa of small a(t) significantly influenced by phase couplings between temporal scales. Panel C: IQRa of large a(t) significantly influenced by phase couplings between temporal scales. Panel D: Cross-correlation coefficient between local structure a(t) and magnitude SD(t) for unconstrained (black bars) and metronome constrained (gray bars) slow, preferred and fast gait speed. Stars indicate significant differences between unconstrained and metronome constrained gait; *p < 0.05, **p < 0.0005.
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Fig. 4. Results of DFAloc employed to data set 2 [21]. Panel A: The interquartile range of all a(t), and small and large a(t) significantly influenced by phase couplings for healthy young (black bars) and healthy older adults (gray bars). Panel B: Crosscorrelation coefficient between local structure a(t) and magnitude SD(t) for healthy young and older adults. Stars indicate significant differences between healthy young and older adults; *p < 0.05, **p < 0.005.
adults (see left set of bars in Fig. 4A). In addition, the decreased width of P(a) was related to a significant decrease in the width of the sub-distribution of large a(t) significantly influenced by phase couplings (see right set of bars in Fig. 4A). However, no significant differences were found in the inverse cross-correlations between the healthy younger and older adults (see Fig. 4B). 4. Discussion The present article introduced DFAloc that can numerically define temporal changes in the structure of stride time variability that are generated by phase couplings between temporal scales. Furthermore, DFAloc can also numerically define the relationship between temporal change in local structure and magnitude of the stride time variability. These numerical features of phase couplings between temporal scales cannot be detected by the conventional DFA. Previous studies that used the conventional DFA found that both metronome-constrained gait in healthy younger adults and impaired gait in older persons and patients with neurodegenerative diseases have a less persistent stride time variability (i.e., a 0.5) compared to normal gait [1,2,5,6,12]. DFAloc indicates that metronome-constrained gait has a more active stride time regulation that is reflected by more phase couplings between temporal scales compared to unconstrained stride-time variability. The stronger influence of phase couplings reflects ‘‘healthy’’ counter-adjustments (i.e., anti-persistent variations) of stride time in intermittent periods of large stride time variability. This active regulation of stride time variability also strengthens the correlation between the local structure a(t) and the local magnitude SD(t) of stride time variability. In contrast, the gait of healthy older adults had fewer phase couplings between temporal scales in the periods with small and persistent variation (i.e., large a(t)), which resulted in reduced active regulation of the local structure of stride time variability. The latter might reflect impaired stride time regulation. Impaired stride time regulation might also be indicated by the inability to adjust the inverse correlation between the local structure a(t) and magnitude SD(t) of stride time variability. The ability to adjust the covariation between the local structure and magnitude of stride time variability might reflect a ‘‘healthy’’ stride time regulation in more challenging walking tasks and walking environments. In summary, DFAloc has the ability to differentiate between different stride time variabilities that have a similar
scaling exponent a as defined by conventional DFA and consequently, provide further insight into the stride time regulation of human walking. The new numerical features obtained by DFAloc suggest modification of previous interpretations of the conventional DFA scaling exponent a. A previous study showed that a of stride time variability can be generated by external white noise as input to the sensory-motor system [14]. This model can reproduce declines in a by a simple magnitude change in neuronal input noise. This reproduced decline in a stands in contrast to more complex reproduction of the same decline by the internal modulation of the network of central pattern generators. However, the results from DFAloc indicate, under the assumption of a stationary environment in the experimental condition, that both the variation in a(t) and its covariation with the local magnitude is generated by significant phase couplings (i.e., multiplicative interactions) between temporal scales that cannot originate from additive white noise. By definition, models with additive white noise will reproduce the mean a(t) of the stride time series and assume that a(t) is constant. However, the variation in a(t) might be a more central feature of stride time regulation, reflecting the neural modulation of white input noise by mechanisms within the central nervous system. Thus, DFAloc of stride time variability might give further insight into stride time regulation as a relationship between mechanisms in the central nervous system and the external walking conditions. Further research should therefore use DFAloc in combination with the conventional DFA to evaluate the output of model simulation of stride time variability. Several issues of DFAloc need to be further investigated. First, DFAloc has the same documented limitations as the conventional DFA [24–26]. DFAloc assumes scale invariance and self-affinity within a predefined scaling range of the stride time series. In the absence of a linear relationship between log[F(n)] and log(n) (see red line in Fig. 1B), this would result in false positive results for DFAloc. Further studies should extend DFAloc according to methods in [27] and [28] to investigate scale-dependent changes in a(t) that indicate the absence of local scale invariance. Second, like the conventional DFA, the performance of DFAloc is dependent on the number of strides contained in the stride time series [29]. The stride time series reanalyzed in the present study contained between 709 and 3822 strides and the large differences in the number of strides might have influenced the results. Future studies should therefore investigate the influence of total number of strides on the computational errors of IQRa. Furthermore, average metrics of multiple trials should be considered for stride time series with short sample size (<500 strides). Third, the conventional DFA is shown to have high test– retest reliability [30]. The test–retest reliability of the new metrics of DFAloc was not tested in the present study, but should be assessed in the future to establish the relevance of DFAloc for clinical gait analyses. Fourth, the interpretation of DFAloc when employed to healthy older adults is based on a small sample size (i.e., N = 5) with limited background information of the participants. Thus, further studies should replicate the DFAloc results of stride time variability in a larger group of older persons and, in extension, in patient groups with neurodegenerative diseases. 5. Conclusion The present article introduced a new local detrended fluctuation analysis (DFAloc) of stride time variability. The metrics introduced by DFAloc were able to identify temporal variations in the local structure of stride time variability that were generated by phase couplings between temporal scales. Furthermore, DFAloc identified an inverse correlation between the temporal modulation in the local magnitude and the structure of the stride time variability. The metrics introduced by DFAloc were dependent on
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gait speed, metronome-constrained cadence, and age. The present article suggests that DFAloc can provide further insight into the stride time regulation of human walking and might identify precursors of ongoing decline in gait function that might proceed to increased risk of falling in older adults and patients with neurodegenerative diseases. Acknowledgments PhD grants from the Norwegian University of Science and Technology. Data set 1 and 2 is available at www.physionet.org/ physiobank/database/#gait. Matlab code for DFAloc is available at www.ntnu.edu/inm/geri/software. Conflict of interest statement The authors declare that no conflict of interests is associated with the present study. References [1] Hausdorff JM, Purdon PL, Peng CK, Ladin Z, Wei JY, Goldberger AL. Fractal dynamics of gait: stability of long-range correlations in stride interval fluctuations. J Appl Physiol 1996;80(5):1448–57. [2] Hausdorff JM. Gait dynamics, fractals and falls: finding meaning in the strideto-stride fluctuations of human walking. Hum Mov Sci 2007;26(4):555–89. [3] Jordan K, Challis JH, Cusumano JP, Newell KM. Stability and the time-dependent structure of gait variability in walking and running. Hum Mov Sci 2009;28(1):113–28. [4] Terrier P, Turner V, Schutz Y. GPS analysis of human locomotion: further evidence for long-range correlations in stride-to-stride fluctuations of gait parameters. Hum Mov Sci 2005;24(1):97–115. [5] Hausdorff JM, Mitchell SL, Firtion R, Peng CK, Cudkowicz ME, Wei JY, et al. Altered fractal dynamics of gait: reduced stride interval correlations with aging and Huntington’s disease. J Appl Physiol 1997;82(1):262–9. [6] Hausdorff JM. Gait dynamics in Parkinson’s disease: common and distinct behavior among stride length, gait variability, and fractal-like scaling. Chaos 2009;19(2):026113–26114. [7] Gates DH, Dingwell JB. Peripheral neuropathy does not alter the fractal dynamics of gait stride intervals. J Appl Physiol 2007;102(3):965–71. [8] Dingwell JB, Cusumano JP. Re-interpreting detrended fluctuation analyses of stride-to-stride variability in human walking. Gait Posture 2010;32:348–53. [9] Delignie´res D, Torre K. Fractal dynamics of human gait: a reassessment of the 1996 data of Hausdorff et al.. J Appl Physiol 2009;106(4):1272–9.
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