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1 Gerontology Division, ABSTRACT Hausdorff, Jeffrey M., Susan L. Mitchell, Renée
Firtion, C. K. Peng, Merit E. Cudkowicz, Jeanne Y. Wei, and Ary L. Goldberger. Altered fractal dynamics of gait: reduced
stride-interval correlations with aging and Huntington's disease.
J. Appl. Physiol. 82(1): 262-269, 1997.
nonlinear dynamics; human locomotion; long-range correlations; power-law scaling
INTRODUCTION THE DURATION OF THE GAIT CYCLE fluctuates from one
stride to the next in a complex fashion (14, 30, 43). Recently, these apparently "noisy" variations have been shown to display a hidden and unexpected fractal property (14, 15). In healthy young subjects,
this variability is not simply attributable to random fluctuations
(white noise) or solely to short-range influences. Instead, gait cycle
duration, i.e., the stride interval, exhibits long-range power-law
correlations. Fluctuations in the stride interval are statistically
correlated with variations in the stride interval hundreds of strides
earlier, and this influence decays in a scale-invariant, fractal
manner. This behavior appears to be intrinsic to the healthy locomotor
system; it persists regardless of walking speed and disappears in
healthy subjects only during metronomically paced walking (14, 15).
From a neurophysiological control viewpoint, this behavior is of
interest because it signifies the presence of long-term dependence. The
mechanism(s) responsible for these stride-interval correlations are
largely unknown. They may be a consequence of peripheral input or lower
motorneuron control, or they may be related to higher nervous system
centers that control walking rhythm. Although the breakdown of
long-range correlations during metronomic walking suggests that
supraspinal influences (e.g., a metronome) can override the normally
present long-range correlations, their origin and function remain to be
determined. To gain insight into the basis for this long-term
dependence, we investigated the effects of advanced age and of a
neurodegenerative condition, Huntington's disease, on stride-interval
correlations.
Aging is associated with a number of neurophysiological changes that
may alter the locomotor system's ability to generate stride-interval
correlations. These include diminished nerve conduction velocity,
deafferentation, loss of motorneurons, decreased reflexes, reduced
proprioception, and decreased muscle strength, as well as decreased
central processing capabilities (9, 18, 20, 21, 23, 26, 28, 29, 35, 36,
41). The magnitude of these age-related changes depends to a great
extent on an individual's comorbid medical conditions. Nevertheless,
to some degree, these changes appear to be a part of even
"normal" aging. Thus healthy elderly subjects may serve to model
subtle changes in neuromuscular control.
Huntington's disease is an autosomal-dominant neurodegenerative
disease of the central nervous system. Primary clinical features are
chorea and cognitive and personality changes (33). Most of the
pathological changes are seen in the basal ganglia, with a loss of
neural projection in the striatum (caudate nucleus and putamen) (33).
Multiple neurochemical markers (e.g., dynorphin, enkephalin, substance
P, and We hypothesized that the locomotor system's ability to produce
stride-interval correlations would be diminished in elderly subjects
and in subjects with Huntington's disease. To test this hypothesis, we
compared the stride-interval dynamics of subjects with advanced age or
Huntington's disease with healthy control subjects.
METHODS This study took place in two parts at two locations. In part 1, the effect of aging was examined, and in part 2, the effect of
Huntington's disease was investigated.
Fluctuations
in the duration of the gait cycle (the stride interval) display fractal
dynamics and long-range correlations in healthy young adults. We
hypothesized that these stride-interval correlations would be altered
by changes in neurological function associated with aging and certain
disease states. To test this hypothesis, we compared the
stride-interval time series of 1) healthy elderly subjects and
young controls and of 2) subjects with Huntington's disease
and healthy controls. Using detrended fluctuation analysis, we computed
, a measure of the degree to which one stride interval is correlated
with previous and subsequent intervals over different time
scales. The scaling exponent was significantly lower in elderly
subjects compared with young subjects (elderly: 0.68 ± 0.14; young:
0.87 ± 0.15; P < 0.003). The scaling exponent was
also smaller in the subjects with Huntington's disease compared with
disease-free controls (Huntington's disease: 0.60 ± 0.24;
controls: 0.88 ± 0.17; P < 0.005). Moreover, was linearly related to degree of functional impairment in subjects with
Huntington's disease (r = 0.78, P < 0.0005).
These findings demonstrate that stride-interval fluctuations are more
random (i.e., less correlated) in elderly subjects and in subjects with Huntington's disease. Abnormal alterations in the fractal properties of gait dynamics are apparently associated with changes in central nervous system control.
-aminoleutynic acid) are depleted in striatum of patients
with Huntington's disease (6). However, the mechanisms through which
these changes affect the ability of the basal ganglia to regulate motor
control are still being elucidated (1, 6). The net result of these
changes is that patients with Huntington's disease often display
uncontrolled "dancing" (choreiform) movements and gait ataxia,
although these features are not necessarily always linked (24).
Huntington's disease generally affects people in their 30s and 40s who
are typically free from concomitant disease and age-related
physiological changes. With impairment limited primarily to the central
nervous system, Huntington's disease offers a contrast to aging for
the study of the conditions necessary for stride-interval correlations.
n
, where the scaling index (also called the self-similarity parameter) is determined by
calculating the slope of the line relating log F(n) to log
n. For a process where the value at one step is completely uncorrelated with any previous values, i.e., white noise, = 0.5. In
contrast, long-range, persistent correlations are present if 0.5 < 
1.0. An < 0.5 signifies antipersistent correlations (a
large stride interval is more likely to be followed by a small one and
vice versa over different time scales).
In general, is equivalent to Hurst's exponent (3, 40). The
Hurst's exponent is a measure that has been widely used to evaluate
the self-similarity and correlation properties of fractional Brownian
noise, the time series produced by a fractional (fractal) Gaussian
process. Both Hurst's exponent and are used to evaluate the
presence or absence of long-range dependence and its degree in a time
series. However, local trends (nonstationarities) are often present in
physiological data and may compromise the ability of some methods to
measure self-similarity. The DFA method () was used here because it
was designed to be insensitive to these trends by removing local
nonstationarities from the analysis (32). It is important to note that,
like other measures of self-similarity, depends on the sequential
ordering of the fluctuations in the time series but not on the overall
magnitude of the fluctuations (i.e., the variance of the time series).
(Indeed, for all the stride-interval time series in this study, we
confirmed that was unchanged even if the time series was normalized
to its SD value.) Theoretically, a time series can display
self-similarity and fractal scaling with relatively small overall
variance or large variance, and, conversely, a time series can have no
correlations while having either small or large overall variance.
Because it is difficult to obtain walking data for an extended period
of time in clinical patients, we asked subjects to walk for 5 (or 6)
min. This does not allow for testing of true long-range correlations
(thousands of strides) but still enables evaluation of stride-interval
correlations. To determine the degree and nature of stride-interval
correlations, we calculated over the region 10 n 20. This region was chosen as it provides a statistically robust estimate
of stride-interval correlations1
that are most independent of finite size effects (length of data) (31).
DFA has typically been used on relatively long data sets (thousands of
points). For long data sets, it has been shown that DFA provides a
fairly accurate measure of the true scaling exponent (13, 40). To be
able to better assess the present results, we generated artificial
fractional Brownian noise time series of known and applied DFA over
the region 10 n 20. Software obtained from the National
Simulation Resource of the University of Washington's Center for
Bioengineering2 was used to
generate the time series of known by using a previously validated
method (5, 7). For 0.5 1.0, the mean of 10 simulated
realizations was within 6% of the true value for time series of
lengths 128, 256, and 512. The accuracy fell off somewhat for < 0.5.
Other measures of gait and balance.
For each subject's stride-interval time series, we also calculated the
average stride interval and the stride-interval coefficient of
variation (100 × SD/mean), a measure of the magnitude of
stride-to-stride variability and gait unsteadiness (11). In addition,
the mean gait speed of each walking trial was determined by counting
the number of laps around the track (of known length) and dividing by
the time of the walk. The "up-and-go test" was also performed as
a gross measure of functional balance and mobility (34). This test
measures how long a subject needs to rise from a chair, walk 3 m, turn
around, and return to a seated position. "Normal" elderly
subjects require between 7 and 10 s to perform this task (34). Test
time has been correlated with the Berg balance scale and the Barthel
index of activities of daily living (34).
Statistical analysis.
For continuous variables, the Wilcoxon rank sum test was performed to
compare each study group and its control group. This nonparametric test
makes no assumptions about the underlying distribution of the data
being compared. For categorical data, Fisher's exact test was used to
test for group differences. Group differences were considered
statistically different if P 0.05. Linear regression was
performed to determine whether group differences in persisted after
adjusting for any potentially confounding clinical variables (e.g.,
height) that differed significantly between the control and study
groups. Correlations between two continuous variables (e.g., and
Huntington's disease impairment) were measured by using the Spearman
correlation coefficient (r). Statistical analysis was performed
by using SAS software, release 6.04 (Cary, NC). Group results are
reported as means ± SD.
RESULTS
was 0.68 ± 0.14 for the elderly
subjects vs. 0.87 ± 0.15 for the young subjects
(P < 0.003).
) is 0.56 for this elderly
subject and 1.04 for this young subject.
Table 1 summarizes the measures of gait in these two groups. Elderly and young subjects had similar average stride intervals, and, hence, the time series length was also similar in the two groups (young: 315 ± 22 strides; elderly 316 ± 28 strides). Both groups required comparable amounts of time to perform the up-and-go test. The magnitude of stride-to-stride variability (i.e., stride-interval coefficient of variation) was also very similar in the two groups (and the percentage of points deleted by the median filter was not different in the two groups; young: 2.5 ± 1.0%; elderly: 2.6 ± 1.3%). These results show that, although
was different
in the two age groups, the gross measures of gait and mobility function
of these elderly subjects were not significantly affected by age.
Average gait speed of elderly subjects was similar to that observed in other studies of "healthy" elderly adults (2) and was slightly less than that of the young subjects. Elderly subjects were also slightly shorter (1.64 vs. 1.70 m; P = 0.06). However,
univariate analysis showed that was not associated with gait speed
(r = 
0.07; P > 0.7) or height
(r = 0.04; P > 0.8). Multivariate analysis techniques were employed to confirm whether speed and height were confounding the association between age and . After adjustment for
these potential confounders (speed and height), age still remained
independently associated with (P < 0.0005).
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= 0.40). This
indicates increased randomness and reduced stride-interval correlations
as compared with the control subject ( = 0.92). Similar results
were obtained for other subjects in these groups as
well.3 The scaling component was 0.60 ± 0.24 for the subjects with Huntington's disease and 0.88 ± 0.17 in the control subjects (P < 0.005).
is 0.40 for this subject with Huntington's
disease and 0.92 for this 23-yr-old control subject.
Among the subjects with Huntington's disease,
was inversely
correlated with disease severity as assessed by the TFC score (r = 0.78, P < 0.0005) (see Fig.
3). The scaling component was significantly lower (P < 0.005), indicative of more random
stride-interval fluctuations, in subjects with the most advanced stages
of Huntington's disease (TFC 5:
= 0.44 ± 0.18), as compared with subjects in the
early stages of the disease (TFC 
9: = 0.83 ± 0.12). In a
few subjects with the most severe impairment, was <0.5, suggesting the presence of anticorrelations.
) among subjects with Huntington's disease. Disease
severity is measured by using total functional capacity (TFC) score of
Unified Huntington's Disease Rating Scale (0 = most impairment;
13 = no impairment).
With regard to other clinical features of the disease (measured by using the Unified Huntington's Disease Rating Scale),
was not
associated with the amount of chorea (of any body segment or total),
whole body bradykinesia, dystonia, degree of abnormalities of eye
movements, the speed of alternating arm and hand movement, or gait and
balance scores. However, was significantly associated with
dysarthria score (r = 0.66, P < 0.001). When
subjects were stratified based on medication use and cognitive and
behavioral functions, there were no differences in ,
except that was slightly lower in the four subjects with signs of
dementia (P < 0.04). (The average of of the subjects
without signs of dementia was still significantly less than that of
control subjects.) Nevertheless, several subjects without signs of
dementia also had small values (near 0.4). Finally, among the
subjects with Huntington's disease, was not significantly
associated with age (P > 0.15).
Table 2 summarizes other measures of gait
in these two groups. Mean stride interval was not statistically
different in subjects with Huntington's disease and in controls.
Subjects with Huntington's disease walked more slowly, took more time
to complete the up-and-go test, and had increased stride-to-stride
variability. Consistent with these results, the number of strides
taken during the walk was slightly larger in control subjects (subjects
with Huntington's disease: 234 ± 32 strides; controls:
252 ± 12 strides; P > 0.05). The percentage of points
deleted by the median filter was similar in the two groups (subjects
with Huntington's disease: 4.8 ± 1.4%; controls: 5.6 ± 1.3%;
P > 0.05). The scaling exponent was not significantly
correlated with gait speed, up-and-go time, or gender, whereas it was
correlated with age (r = 0.39; P < 0.05) and
stride-interval variability (r = 0.62;
P < 0.001). Multivariate analysis showed that even after
adjustment for these potential confounders remained independently associated with the presence of Huntington's disease (P < 0.005). In contrast, age and stride-interval
variability were not independently associated with .
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.
Deleting points from a time series of fractional Brownian noise may
alter the correlation properties of the time series. To test the extent
to which the deletion of turning points affected the estimate of in
stride-interval time series, we deleted corresponding points on
simulated data of known and measured the effects of the deletions.
Specifically, for known ranging from 0.50 to 0.95, we generated
x simulated data time series (each with a different seed
number) using the Davies and Harte algorithm (7), where x is
the number of subjects in a given group (e.g., 10 elderly subjects or
17 subjects with Huntington's disease). From each of these simulated
time series, we then deleted the points that corresponded (in time) to
the deleted points in each of the stride-interval time series. The
length of the original (and "filtered") simulated time series was
also matched to the corresponding experimental data set. After deleting
these points, we then calculated . The average was within 4% of
the true value, no matter which group's data provided the template for
deletions (for the young and old groups who walked for an extra
minute and had less deletions, the difference was <2%).
It is, therefore, unlikely that application of the median filter
significantly altered the estimation of the "intrinsic"
self-similarity in each group's walking pattern. In fact, it is
interesting to note that after application of the median filter both
groups of control subjects had almost identical gait measures, despite
minor differences in the walking protocol (6 vs. 5 min; circular path vs. back-and-forth hallway) and the 10-yr difference in the average age
of the two groups (no subjects participated in both studies). The
median filter applied to each subject's time series was apparently effective in minimizing any effects due to the walking protocol (turning around). Before the filter, was 0.83 ± 0.15 and 0.95 ± 0.14 in the aging and Huntington's disease control subjects, respectively. After the filter, the group values were essentially identical to each other, even if we calculated an average in two
different ways.4 This indicates
the robustness of estimates of mean and may also suggest that the
median filter was effective in preserving the underlying correlation
properties while removing extraneous data points associated with
changes in gait direction.
DISCUSSION
This study demonstrates that a seemingly intrinsic mathematical property of one output of the healthy locomotor system changes both with aging and with Huntington's disease. In healthy control subjects, stride-interval fluctuations were consistent with our previous findings of fractal scaling (14, 15). In contrast, in the elderly and in subjects with Huntington's disease, fluctuations tended to be more random, and the correlations of one stride with nearby strides were reduced. Moreover, the degree of stride-interval correlations was inversely associated with the degree of functional impairment in subjects with Huntington's disease. In fact, in the most impaired subjects with Huntington's disease, the stride-to-stride fluctuations became either completely uncorrelated, like white noise, or anticorrelated.
The finding of reduced stride-interval correlations with aging and with Huntington's disease parallels other findings of changes in fractal scaling and long-range correlations with age and disease (25). For example, alterations in the fractal properties have been observed in the fluctuations in breathing during hyperoxia and hypoxia (16) and in the variations of the electroencephalographic-evoked potentials with aging and disease (4, 27, 39). One of the most widely studied examples of fractal physiology is the beat-to-beat fluctuations in cardiac dynamics. The correlations properties of heart rate are diminished in the elderly (19) and in certain subjects with cardiovascular disease (e.g., heart failure) (22, 32). Intervention studies have demonstrated that vagal blockade alters this fractal scaling of the heart beat and its dependence on intact autonomic neural function (42).
The alterations in the fractal dynamics of the stride interval are not
simply attributable to reduced gait speed or increased stride-to-stride
variability (unsteadiness) with aging or disease. Previous findings
(15) showed the presence of stride-interval correlations ( close to
1.0) even when healthy subjects walked slowly (i.e., 1.0 ± 0.2
m/s). In the present study, stride-to-stride variability was virtually
unchanged in the elderly subjects compared with young controls, yet the
stride-interval correlations were significantly reduced. Furthermore,
the magnitude of the stride-interval correlations was independent of
gait speed and stride-to-stride variability in both studies.
Apparently, stride-interval correlations depend on some aspect of the
neuromuscular control system that is not directly related to walking
velocity or gait unsteadiness.
Given the reduced stride-interval correlations in subjects with Huntington's disease, especially in the most impaired subjects, it is likely that the areas of the cerebrum that are affected by Huntington's disease play an important role in generating stride-interval correlations. While some pathological changes have been found in the cortex (6), the primary pathology is in the basal ganglia. Perhaps, the striatal pathology that leads to a decrease in fine motor control in Huntington's disease also impairs the "long-term dependence" and fine control necessary for stride-interval correlations.
Other factors that may have contributed to the differences between the
stride-interval correlations of control and Huntington's disease
subjects include age (control subjects were slightly younger), medication use, and cognitive and personality changes. Multivariate analysis suggests that the small age effect is not significant. This is
supported by 1) findings of identical degrees of
stride-interval correlations in the two (young adults vs. middle-aged
subjects) control groups, despite an almost 10-yr difference in mean
age and by 2) the fact that older control subjects showed no
reduction in (e.g., was 1.11 and 0.89 in a 57- and 52-yr-old
subjects, respectively), whereas some young Huntington's disease
subjects showed large reductions in (e.g., was 0.31, 0.40, and
0.40 in Huntington's disease subjects who were 34, 41, and 42 yr old, respectively). Interestingly, was not significantly altered in the
subjects who showed signs of depression or confusion or those who were
using neuroleptics. The independence of and neuroleptic use is
consistent with findings of Koller and Trimble (24). They found that
haloperidol therapy decreased chorea but did not change gait impairment
of subjects with Huntington's disease. The scaling exponent tended
to be lower in patients with signs of dementia. However, compared with
control subjects, was still significantly lower in those subjects
with no signs of dementia, and several subjects without signs of
dementia also had small values. Subjects with Huntington's disease
were free of other comorbidities and peripheral disease that would be
likely to affect gait. Thus the decrease in stride-interval
correlations with Huntington's disease is probably not simply
due to secondary factors associated with this condition but is most
likely related to the underlying central neuropathology.
In our group of healthy elderly subjects, the reduction in stride-interval correlations is less than that seen in the subjects with Huntington's disease. The reduced correlations in the elderly gait may be due in part to comorbidities that we did not detect. Alternatively, the alterations in gait dynamics in the elderly may be due to subtle changes in neural control that were not revealed by our clinical evaluation. These elderly subjects were free from overt neurological disease. However, even in "normal" elderly adults, there is an age-related decline in dopamine content in the basal ganglia, and it has been suggested that age-related changes in gait may result from subtle changes in striatal dopamine mechanisms (8). The reduction in stride-interval correlations may also be a manifestation of these changes in central processing. However, although the primary pathology of Huntington's disease is also in the basal ganglia, it is unlikely that the same mechanisms are affecting the elderly and subjects with Huntington's disease. Huntington's disease is primarily hyperkinetic, whereas these putative age-related changes are hypokinetic. Furthermore, the affected regions within the basal ganglia are also different. Nonetheless, it is interesting to speculate why measures of stride-to-stride variability and gross measures of mobility (up-and-go time) are unchanged in these elderly subjects, whereas stride-interval correlations are diminished. Perhaps, the ability to produce a correlated gait is a more demanding locomotor challenge. The diversity of the age-associated neuromuscular changes (e.g., pyramidal, extrapyramidal, and peripheral) and the absence of overt neurological disease in our subjects make it difficult to precisely determine the cause(s) of the changes in stride-interval correlations in these elderly subjects.
Precise elucidation of the factors affecting the fractal dynamics of gait remain to be determined. Stride-interval correlations are decreased with advanced age and with Huntington's disease and are probably dependent on intact central nervous system (basal ganglia) processing but are independent of walking speed and variability. Future studies that examine subjects with pathologies involving different portions of the extrapyramidal and pyramidal systems may provide additional insight into this unexpected property of normal walking. Finally, we note that the analysis of stride-interval dynamics may have practical utility in neurological diagnosis and in the quantitative assessment of therapeutic interventions.
ACKNOWLEDGEMENTS
The authors thank H. Edelberg, D. Kaliton, and A. Pedroza for their assistance; Dr. J. Collins for valuable discussions; and the National Simulation Resource at the University of Washington's Center for Bioengineering for the software used in generating simulated time series.
FOOTNOTES
Beth Israel Hospital, in collaboration with Pfizer
Inc., and the Pharmaceutical Research and Manufacturers of America
Foundation Award. We are also grateful for partial support from the G. Harold and Leila Y. Mathers Charitable Foundation and from the National
Aeronautics and Space Administration.
1
Although one cannot establish long-range
correlations and fractal fluctuations of the stride interval with only
5 min of walking data, it is still possible to perform the fluctuation analysis over more limited time scales. This analysis is, in turn, sufficient to indicate changes in fractal-related scaling exponents.
2
The software was obtained from
http://nsr.bioeng.washington.edu/NSR/NSR.html.
3
Because our analysis focused on the scaling region
10 n 20, where n is the number of strides in the
given window of observation, we would not expect to see much difference
if steps were analyzed rather than strides. (Local compensations from
one step or stride to the next are outside the region of our analysis.) In some subjects, we had data from the left and right foot and we were
able to confirm this.
4
In the first, we simply calculated for each
individual subject and took the average. In the second, we normalized
each subject's time series to its SD (after subtracting the mean) so that individual records could be analyzed as if they originated from
the same underlying dynamic system. We then averaged
F(n) for each box size across all subjects in the
group and extracted .
Address for reprint requests: J. Hausdorff, Beth Israel Hospital, 330 Brookline Ave., Room KB-26, Boston, MA 02215.
Received 10 June 1996; accepted in final form 10 September 1996.
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ABSTRACT
