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<TITLE>Stride Time Dynamics</TITLE>





  
  

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<BR> <P>





  
  

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<H3><A NAME="SECTION00030300000000000000">Stride Time Dynamics</A></H3>





  
  

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<P>





  
  

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To study the intrinsic stride-to-stride dynamics and its changes with age, some





  
  

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 preprocessing was performed on each time series.  The first





  
  

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sixty seconds and the last five seconds of each time





  
  

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series were not included  to eliminate any





  
  

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start-up or ending effects and 





  
  

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to allow the subject to become familiar





  
  

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with the walking track. The time series were also processed to





  
  

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remove any pauses (stride time &gt; 2 seconds and the 5 seconds before and





  
  

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after any pauses) as well as any large spikes or outliers.  These





  
  

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outliers, which occurred infrequently, were removed so that the





  
  

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intrinsic dynamics of each time series could be more readily





  
  

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analyzed. This  was accomplished using





  
  

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previously established methods  (8,10) by: i) determining the mean and standard deviation of the





  
  

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stride time while excluding the 5% of the data with the lowest and





  
  

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highest values, and then ii) removing from the original time series





  
  

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all data that fell more than 4.0 standard deviations away from this





  
  

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mean value. The number of pauses (typically 0) and the number of





  
  

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strides excluded (typically 2 %) were similar in all three 





  
  

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age groups.





  
  

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<P>





  
  

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As shown in Table 1 and 





  
  

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summarized below, we applied several measures





  
  

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to analyze the variability and temporal structure of the stride time





  
  

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dynamics.





  
  

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<P>





  
  

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<P><P>





  
  

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<B>Stride-to-Stride Variability Measures</B>





  
  

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<P>





  
  

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To estimate the overall





  
  

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stride-to-stride variability, we calculated the standard deviation





  
  

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of each time series and 





  
  

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the coefficient of variation (CV) (100<IMG WIDTH=8 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline304" SRC="img7.png">standard deviation/mean), 





  
  

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an index of





  
  

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variability normalized to each subject's mean cycle duration.





  
  

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Both the standard deviation and the CV provide a measure of overall variations





  
  

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in gait timing during the entire walk, i.e., the amplitude of the





  
  

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fluctuations in the time series with respect to the mean.  However,





  
  

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these measures may be influenced by  trends in the data (e.g., due





  
  

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to a change in speed) and cannot distinguish between a walk with large changes





  
  

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from one stride to the next and one in which stride-to-stride





  
  

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variations are small and more long-term, global changes (e.g., a change in





  
  

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average value) result in a large standard deviation.  Therefore, 





  
  

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to estimate variability independent of 





  
  

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local changes in the mean, we quantified 





  
  

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successive stride-to-stride changes (i.e., the difference between the





  
  

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stride time of one stride and the previous stride) by determining the first difference of





  
  

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each time series.  The first difference, a discrete analog





  
  

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of the first derivative, is one standard method for removing slow varying trends and is calculated by subtracting the previous value in the 





  
  

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time series from the current value. The standard deviation of the first





  
  

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difference time series provides 





  
  

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a measure of variability after detrending.





  
  

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<P>





  
  

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<P><P>





  
  

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<B>Temporal Structure Measures</B>





  
  

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<P>





  
  

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To study the temporal organization, we applied three methods to analyze





  
  

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different aspects of the dynamical structure of the 





  
  

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time series of the stride time.





  
  

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<P>





  
  

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<B>Spectral Analysis:&nbsp;&nbsp;</B> Fourier spectral analysis is a standard method for





  
  

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examining the dynamics of a time series. To





  
  

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insure that these dynamical measures were independent of the average





  
  

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stride time or the stride time variability, we studied the first 256





  
  

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points of each subject's time series (after the 60 second ``start-up'' period) by first subtracting the mean and





  
  

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dividing by the standard deviation. This produces a time series





  
  

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centered at zero with a standard deviation of 1.0. Subsequently,





  
  

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standard Fourier analysis using a rectangular window was performed on each time





  
  

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series.  To quantify any differences in the spectra, we calculated the





  
  

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percent of power in the high frequency band (0.25--0.50





  
  

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strides<IMG WIDTH=15 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline306" SRC="img8.png">) and the ratio of the low (.05 -- 0.25 strides<IMG WIDTH=15 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline306" SRC="img8.png">)





  
  

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to high frequency power.





  
  

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This ratio excludes the power in the lowest frequencies and





  
  

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thus is independent of very large scale changes in the stride time.





  
  

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By computing the ratio of the fluctuations over relatively long time scales





  
  

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(i.e., low frequencies)





  
  

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to short time scales (i.e., high frequencies), an index of the 





  
  

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frequency ``balance'' of the 





  
  

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spectra  is obtained. A large low/high 





  
  

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ratio is indicative of nonstationarity.





  
  

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Therefore, to the extent that  the gait of the younger children is more





  
  

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nonstationary, one would expect this spectral





  
  

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ratio to decrease with maturation.





  
  

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<P>





  
  

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<B>Autocorrelation Decay:&nbsp;&nbsp;</B> As a complementary method for analyzing





  
  

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the temporal structure of gait dynamics, we examined the autocorrelation properties of





  
  

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the stride time series.  The autocorrelation function estimates how a





  
  

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time series is correlated with itself over different time lags and





  
  

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provides a measure of the ``memory'' in the system, i.e., for up to





  
  

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how many strides is the present value of the stride time correlated





  
  

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with past values.  After direct calculation of the





  
  

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autocorrelation function in the time domain  (20),  





  
  

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we calculated two 





  
  

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indices of autocorrelation decay: <IMG WIDTH=31 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline310" SRC="img9.png"> and





  
  

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<IMG WIDTH=31 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline312" SRC="img10.png">, the number of strides for the autocorrelation to decay





  
  

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to 37 % (1/e) or 63 % (1-1/e) of its initial value,





  
  

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respectively.  To minimize any effects of data length, mean or





  
  

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variance, we applied this analysis to the first 256 strides and





  
  

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normalized each time series with respect to its mean and standard





  
  

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deviation. This autocorrelation measure emphasizes the correlation





  
  

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properties over a very short time scale, where the correlation decays





  
  

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most rapidly. 





  
  

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If the ``memory'' of the system increases with maturity, one would





  
  

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expect to see longer decay times in older children.





  
  

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<P>





  
  

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<B>Stride Time Correlations:&nbsp;&nbsp;</B> To further study the temporal





  
  

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structure of the stride time dynamics (independent of the overall





  
  

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variance), we also applied detrended fluctuation analysis





  
  

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(DFA)  (11,18) to each subject's stride time time series. DFA is a





  
  

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modified random walk analysis that can be used to quantify the





  
  

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long-range, fractal properties of a relatively long time series or, in





  
  

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the case of shorter time series, (i.e., the present study), it can be





  
  

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used to measure how correlation properties change over different time





  
  

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scales or observation windows  (10,18).  Methodologic details have





  
  

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been provided elsewhere  (10-12,18).  Briefly, the root-mean





  
  

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square fluctuation of the integrated and detrended time series is





  
  

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calculated at different time scales and the slope of the relationship





  
  

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between the fluctuation magnitude and the time scale determines a fractal





  
  

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scaling index, <IMG WIDTH=10 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline314" SRC="img11.png">. To determine





  
  

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the degree and nature of stride time correlations, we used previously





  
  

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validated methods  (10) and calculated <IMG WIDTH=10 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline314" SRC="img11.png"> over the region <IMG WIDTH=100 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline318" SRC="img12.png"> (where n is the number of strides in the window of





  
  

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observation). This region was chosen as it provides a statistically





  
  

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robust estimate of stride time correlation properties that are most





  
  

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independent of finite size effects (length of data)  (17) and because





  
  

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it has been shown to be sensitive to the effects of neurological





  
  

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disease and aging in older adults  (10). Like the autocorrelation method,





  
  

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the DFA method quantifies correlation properties. However, the DFA method





  
  

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assumes that within the scale of interest the correlation decays in a power-law





  
  

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manner and, therefore, a single exponent (<IMG WIDTH=10 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline314" SRC="img11.png">) can quantify the scaling. 





  
  

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Whereas the autocorrelation method was applied to examine the dynamics over very short time scales, 





  
  

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the DFA method as applied here examines scaling over relatively





  
  

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longer time periods.





  
  

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If the stride-to-stride fluctuations are more random (less correlated) in younger children,





  
  

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one would expect that <IMG WIDTH=10 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline314" SRC="img11.png"> would be closer to 0.5 (white noise)





  
  

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in this group. In contrast, 





  
  

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an <IMG WIDTH=10 HEIGHT=9 ALIGN=BOTTOM ALT="tex2html_wrap_inline314" SRC="img11.png"> value closer to 1.5 would indicate fluctutations





  
  

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with a  brown noise quality, indicatinig





  
  

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the dominance of  slow moving  (18).





  
  

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<P>





  
  

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