Dynamics of Stride Interval Characteristics during Continuous Stairmill Climbing

It has been shown that statistical persistence in stride intervals characteristics exist during walking, running and cycling and were speed-dependent among healthy young adults. The purpose of this study was to determine if such statistical persistence in stride time interval, stride length and stride speed also exists during self-paced continuous stairmill climbing and if the strength is dependent on stepping rate. Stride time, stride length, and stride speed were collected from nine healthy participants during 3 min of stairmill climbing at 100, 110, and 120% of their preferred stepping rate (PSR) and 5 min of treadmill walking at preferred walking speed (PWS). The amount of variability (assessed by standard deviation and coefficient of variation) and dynamics (assessed by detrended fluctuation analysis and sample entropy) of the stride time, stride length, and stride speed time series were investigated. The amounts of variability were significantly higher during stairmill climbing for the stride time, stride length, and stride speed and did only change with increased stepping rate for stride speed. In addition to a more irregular pattern during stairmill climbing, the detrended fluctuation analysis (DFA) revealed that the stride length fluctuations were statistical anti-persistent for all subjects. On a group level both stride time and stride speed fluctuations were characterized by an uncorrelated pattern which was more irregular compared to that during treadmill walking. However, large inter-participant differences were observed for these two variables. In addition, the dynamics did not change with increase in stepping rate.

The purpose of this supplementary material 1 was to investigate the effect of different box sizes on the scaling exponent obtained with detrended fluctuation analysis (DFA).
Twenty examples of three different mathematical signals of 2500 data points with different characteristics (periodic, chaotic, and random) were created using the Colored noise generator function in Matlab (MathWorks R2011b). These signals included a brown noise signal (power spectrum of 1/f 2 ) which was considered to be periodic, a pink noise signal (power spectrum of 1/f) which was considered to be chaotic, and a white Gaussian noise signal (constant power spectral) which was considered to be random. These time series were cut to include either 65 data points (very short), 165 data points (short), or 2000 data points (long). The scaling exponent of white noise signal of long data series lengths (>2000 data points) can be expected to be close to 0.5, of pink noise close to 1.0, and of brown noise close to 1.5. In this case the use of different box sizes should have very limited effect on the precision of the DFA. To verify this claim, the present investigation included data series of 2000 data points. Furthermore, to investigate the effect of box size of relative short data series for the DFA, the present investigation included very short and short data lengths. DFA was performed with four different box sizes of [2,N], [4,N], [8,N] and [16,N] for each time series. In order to best fit a curve to the linear part of the log N-log F(N) plots, two different scaling regions were chosen, 1:14 for the very short and short time series and 20:50 for the long time series (figure S1.1-S1.3). One-sample t-tests were applied to investigate if the alpha values from each of the four box sizes significantly differed from 0.5, 1.0, and 1.5, respectively. To account for multiple statistical comparisons, level of significance was set to 0.001.
Across the three short signals, box size of [2,N] provided a scaling exponent closest to the expected (figure S1.1-S1.3) for the very short and short time series. For the long time series, choice of box size had less effect on the precision of the scaling exponent. However, large box sizes appeared to provide scaling exponent with greater difference to the expected value (figure S1.1-S1.3). When using the longest data series there was a general tendency to a smaller deviation from the theoretical value and a smaller standard deviation compared to the shorter data series. Thus, the present supplementary material supports the used box size in the main study. Figure S1.1: Representative log N-log F(N) plots for white noise signals with different time series length (left graphs: N=65, middle graphs: N=165, right graphs N=20000) different box sizes (first row: box size=2:N, second row: box size=4:N, third row: box size=8:N, fourth row: box=16:N) , and different scaling regions (left and middle colon: scaling region=1:14, right colon: scaling region 20:50). Bottom graph: mean and standard deviation of scaling exponents for twenty iterations of white noise when using different time series length, box sizes, and scaling region. Significant difference (p<0.001) between the average scaling exponent and 0.5 is indicated by asterisks. Figure S1.2: Representative log N-log F(N) plots for pink noise signals with different time series length (left graphs: N=65, middle graphs: N=165, right graphs N=20000) different box sizes (first row: box size=2:N, second row: box size=4:N, third row: box size=8:N, fourth row: box=16:N) , and different scaling regions (left and middle colon: scaling region=1:14, right colon: scaling region 20:50). Bottom graph: mean and standard deviation of scaling exponents for twenty iterations of pink noise when using different time series length, box sizes, and scaling region. Significant difference (p<0.001) between the average scaling exponent and 1.0 is indicated by asterisks. Figure S1.3: Representative log N-log F(N) plots for pink noise signals with different time series length (left graphs: N=65, middle graphs: N=165, right graphs N=20000) different box sizes (first row: box size=2:N, second row: box size=4:N, third row: box size=8:N, fourth row: box=16:N), and different scaling regions (left and middle colon: scaling region=1:14, right colon: scaling region 20:50). Bottom graph: mean and standard deviation of scaling exponents for twenty iterations of brown noise when using different time series length, box sizes, and scaling region. Significant difference (p<0.001) between the average scaling exponent and 1.5 is indicated by asterisks.
The purpose of this supplementary material 2 was to validate that the number of included strides did not affect the reliability of the DFA and SaEn. The twenty time series of three different mathematical signals described in supplementary material 1 were also used in this investigation. Additionally, the stride time, stride length, and stride speed time series during the walking trial from each of the nine subjects were included. Each time series (both mathematical and biological) were resized into nine different time series with a length ranging from 50 to 165 strides (figure S2.1). DFA and SaEn were calculated from each of the 540 theoretical time series (3 mathematical signals x 20 iteration x 9 different time series lengths) and each of the 81 biological time series (9 subject x 9 different time series lengths). A one-way ANOVA for repeated measures with a level of significance at 0.05 was applied to test the effect of the length of time series. In case of a significant effect, a Holm Sidak post hoc test was applied. In addition, the standard deviation across all iterations of the mathematical signals (figure S2.1) and across all subjects for each of the 9 different time series length was plotted against the number of data points (figure S2.2). The standard deviation of the scaling exponent and SaEn calculated on different noise signals (figure S2.2) did change with increase the number of data points. While the standard deviation of the scaling exponent of all three signals and the SaEn of the white noise signal decreased gradually with increase in included data points, the SaEn of both the pink and brown noise signal appeared to level off and remain relative constant with data lengths above 80 and 95 data points, respectively. The standard deviation of the different stride variables (figure S2.2) did not appear to change substantially when 95 or more data points were used. When comparing the average scaling exponent and SaEn for the twenty iterations of the three signals at different data series length ( figure S2.3), no significant effect was observed for the white and pink noise signals. A significant effect of data series length was observed for brown noise (p=0.012). The post hoc test revealed that the average scaling exponent at a data length of 125 and 165 data points was significantly higher compared to the scaling exponent at data length 50 (p<0.039 and p<0.030, respectively). For the average SaEn, there was significant effect of data length for all three noise signals (p≤0.002 in all cases). The post hoc test showed that the average SaEn for the white noise signal at data length 65 was significantly higher compared to that of data length 165. For the pink noise signal, the SaEn for data lengths of 50 to 95 data points was in all cases significantly higher than the corresponding values for data lengths of 110 to 165 data points. For the brown noise signal, SaEn was significantly higher for data length 50 compared to the six longest data lengths, SaEn for data length 65 was significantly higher compared to the four longest data lengths, and SaEn for data length 80 was significantly higher compared to the two longest data lengths ( figure S2.3). There was no significant effect of the data length of the stride variable time series on DFA and SaEn ( figure S2.4).
While the presented results on mathematical signals indicate a significant effect of data length on both reliability and precision of the variable outcome, this does not seem to be the case with biological data such as the stride time, stride length, and stride speed used in the present study. While figure S2.2 does indicate that the between subject variance increased when few data points were used, the results presented in figure S2.4 indicate that the groups average was less affected by changes within the range of data point used in the present study.  Based on the presented results, it can be concluded that the reliability and precision of DFA and SaEn was only limited affected by the data series length when biological data was used. This was not the case for mathematically generate signals.
This supplementary material presents the surrogate analysis results. In each figure the dependent variable (alpha-values or sample entropy on stride time interval, stride length or stride speed) is plotted as a star for each subject. In addition, the dependent variable is calculated on 39 surrogate time series and plotted as box plots with 95% confidence intervals. The results of the statistical analysis are presented in the main study. Figure S3.1: Alpha-value (left graphs) and sample entropy (right graphs) calculated on surrogate time series (boxplot) and original time series (star) for stride time intervals (top graphs), stride length (middle graphs) and stride speed (bottom graphs) for each subject during treadmill walking at preferred walking speed. Figure S3.2: Alpha-value calculated on surrogate time series (boxplot) and original time series (star) for stride time intervals (top graphs), stride length (middle graphs) and stride speed (bottom graphs) for each subject during stairmill walking at 100%, 110% and 120% of preferred stepping rate. Figure S3.3: Sample entropy calculated on surrogate time series (boxplot) and original time series (star) for stride time intervals (top graphs), stride length (middle graphs) and stride speed (bottom graphs) for each subject during stairmill walking at 100%, 110% and 120% of preferred stepping rate.