Dynamical structure of centerofpressure trajectories in patients recovering from stroke.
ABSTRACT In a recent study, De Haart et al. (Arch Phys Med Rehabil 85:886895, 2004) investigated the recovery of balance in stroke patients using traditional analyses of centerofpressure (COP) trajectories to assess the effects of health status, rehabilitation, and task conditions like standing with eyes open or closed and standing while performing a cognitive dual task. To unravel the underlying control processes, we reanalyzed these data in terms of stochastic dynamics using more advanced analyses. Dimensionality, local stability, regularity, and scaling behavior of COP trajectories were determined and compared with shuffled and phaserandomized surrogate data. The presence of longrange correlations discarded the possibility that the COP trajectories were purely random. Compared to the healthy controls, the COP trajectories of the stroke patients were characterized by increased dimensionality and instability, but greater regularity in the frontal plane. These findings were taken to imply that the stroke patients actively (i.e., cognitively) coped with the strokeinduced impairment of posture, as reflected in the increased regularity and decreased local stability, by recruiting additional control processes (i.e., more degrees of freedom) and/or by tightening the present control structure while releasing nonessential degrees of freedom from postural control. In the course of rehabilitation, dimensionality stayed fairly constant, whereas local stability increased and regularity decreased. The progressively less regular COP trajectories were interpreted to indicate a reduction of cognitive involvement in postural control as recovery from stroke progressed. Consistent with this interpretation, the dual task condition resulted in less regular COP trajectories of greater dimensionality, reflecting a taskrelated decrease of active, cognitive contributions to postural control. In comparison with conventional posturography, our results show a clear surplus value of dynamical measures in studying postural control.

Article: Alterations in Postural Control during the World's Most Challenging Mountain UltraMarathon.
[Show abstract] [Hide abstract]
ABSTRACT: We investigated postural control (PC) effects of a mountain ultramarathon (MUM): a 330km trail run with 24000 m of positive and negative change in elevation. PC was assessed prior to (PRE), during (MID) and after (POST) the MUM in experienced ultramarathon runners (n = 18; finish time = 126±16 h) and in a control group (n = 8) with a similar level of sleep deprivation. Subjects were instructed to stand upright on a posturographic platform over a period of 51.2 seconds using a doubleleg stance under two test conditions: eyes open (EO) and eyes closed (EC). Traditional measures of postural stability (center of pressure trajectory analysis) and stabilogramdiffusion analysis (SDA) parameters were analysed. For the SDA, a significantly greater shortterm effective diffusion was found at POST compared with PRE in the mediolateral (ML; Dxs) and anteroposterior (AP) directions (Dys) in runners (p<0.05) The critical time interval (Ctx) in the ML direction was significantly higher at MID (p<0.001) and POST (p<0.05) than at PRE in runners. At MID (p<0.001) and POST (p<0.05), there was a significant difference between the two groups. The critical displacement (Cdx) in the ML was significantly higher at MID and at POST (p<0.001) compared with PRE for runners. A significant difference in Cdx was observed between groups in EO at MID (p<0.05) and POST (p<0.005) in the ML direction and in EC at POST in the ML and AP directions (p<0.05). Our findings revealed significant effects of fatigue on PC in runners, including, a significant increase in Ctx (critical time in ML plan) in EO and EC conditions. Thus, runners take longer to stabilise their body at POST than at MID. It is likely that the mountainous characteristics of MUM (unstable ground, primarily uphill/downhill running, and altitude) increase this fatigue, leading to difficulty in maintaining balance.PLoS ONE 01/2014; 9(1):e84554. · 3.73 Impact Factor  SourceAvailable from: Gregoire P Millet
Article: Alterations in Postural Control during the World's Most Challenging Mountain UltraMarathon
[Show abstract] [Hide abstract]
ABSTRACT: We investigated postural control (PC) effects of a mountain ultramarathon (MUM): a 330km trail run with 24000 m of positive and negative change in elevation. PC was assessed prior to (PRE), during (MID) and after (POST) the MUM in experienced ultramarathon runners (n = 18; finish time = 126±16 h) and in a control group (n = 8) with a similar level of sleep deprivation. Subjects were instructed to stand upright on a posturographic platform over a period of 51.2 seconds using a doubleleg stance under two test conditions: eyes open (EO) and eyes closed (EC). Traditional measures of postural stability (center of pressure trajectory analysis) and stabilogramdiffusion analysis (SDA) parameters were analysed. For the SDA, a significantly greater shortterm effective diffusion was found at POST compared with PRE in the mediolateral (ML; Dxs) and anteroposterior (AP) directions (Dys) in runners (p<0.05) The critical time interval (Ctx) in the ML direction was significantly higher at MID (p<0.001) and POST (p<0.05) than at PRE in runners. At MID (p<0.001) and POST (p<0.05), there was a significant difference between the two groups. The critical displacement (Cdx) in the ML was significantly higher at MID and at POST (p<0.001) compared with PRE for runners. A significant difference in Cdx was observed between groups in EO at MID (p<0.05) and POST (p<0.005) in the ML direction and in EC at POST in the ML and AP directions (p<0.05). Our findings revealed significant effects of fatigue on PC in runners, including, a significant increase in Ctx (critical time in ML plan) in EO and EC conditions. Thus, runners take longer to stabilise their body at POST than at MID. It is likely that the mountainous characteristics of MUM (unstable ground, primarily uphill/downhill running, and altitude) increase this fatigue, leading to difficulty in maintaining balance.PLoS ONE 01/2014; · 3.73 Impact Factor  SourceAvailable from: Alexis LionHadrien Ceyte, Alexis Lion, Sébastien Caudron, Badreddine Kriem, Philippe P Perrin, Gérome C Gauchard[Show abstract] [Hide abstract]
ABSTRACT: In many daily situations, balance control is associated with a cognitive activity such as reading or a simple calculation. The objective of this study was to investigate the relationship between these two specific human activities, especially the influence of visual cues and support surface stability on body sway during a calculation task. A Sensory Organization Test, which can disrupt or suppress sensory inputs, was performed on 71 healthy young adults. The evaluations were performed both with and without mental arithmetic tasks which consisted of backward counting by three or thirteen. Our results showed that the addition of a calculation task induced an increase in body sway only when visual cues were available. They also showed the same instability effect of the support surface on the amount of body sway no matter what the associated cognitive task was. Moreover, no difference in body sway was observed between the two calculation tasks no matter what the visual context and/or the stability of the support surface were. We suggest that focusing on fulfilling the requirements of the mental calculation challenge may be responsible for the increase in body sway. This increase may be related to the use of oculomotor activity as unintentional attempts to increase arousal by selfgenerated body movement. Thus, this activity facilitates information processing rather than minimizing unbalance by a visual anchor point.Experimental Brain Research 03/2014; 232:2221–2228. · 2.22 Impact Factor
Page 1
RESEARCH ARTICLE
M. Roerdink Æ Æ M. De Haart Æ Æ A. Daffertshofer
S. F. Donker Æ Æ A. C. H. Geurts Æ Æ P. J. Beek
Dynamical structure of centerofpressure trajectories
in patients recovering from stroke
Received: 31 August 2005/ Accepted: 12 March 2006/Published online: 10 May 2006
? SpringerVerlag 2006
Abstract In a recent study, De Haart et al. (Arch Phys
Med Rehabil 85:886–895, 2004) investigated the recovery
of balance in stroke patients using traditional analyses of
centerofpressure (COP) trajectories to assess the effects
of health status, rehabilitation, and task conditions like
standing with eyes open or closed and standing while
performing a cognitive dual task. To unravel the under
lying control processes, we reanalyzed these data in terms
of stochastic dynamics using more advanced analyses.
Dimensionality, local stability, regularity, and scaling
behavior of COP trajectories were determined and
compared with shuffled and phaserandomized surrogate
data. The presence of longrange correlations discarded
the possibility that the COP trajectories were purely
random. Compared to the healthy controls, the COP
trajectories of the stroke patients were characterized by
increased dimensionality and instability, but greater
regularity in the frontal plane. These findings were taken
to implythat the stroke patients actively (i.e., cognitively)
coped with the strokeinduced impairment of posture, as
reflected in the increased regularity and decreased local
stability, by recruiting additional control processes (i.e.,
more degrees of freedom) and/or by tightening the
present control structure while releasing nonessential
degrees of freedom from postural control. In the course
of rehabilitation, dimensionality stayed fairly constant,
whereas local stability increased and regularity de
creased. The progressively less regular COP trajectories
were interpreted to indicate a reduction of cognitive
involvement in postural control as recovery from stroke
progressed. Consistent with this interpretation, the dual
task condition resulted in less regular COP trajectories of
greater dimensionality, reflecting a taskrelated decrease
of active, cognitive contributions to postural control. In
comparison with conventional posturography, our re
sults show a clear surplus value of dynamical measures in
studying postural control.
Keywords Motor control Æ Posture Æ Nonlinear
dynamics Æ Stroke Æ Rehabilitation
Introduction
In quiet standing, the position of the center of mass
varies continuously, resulting in changes in the forces
exerted by the human body on the support surface and in
the corresponding ground reaction forces. This postural
sway can be studied by using force platforms that mea
sure the displacement of the application point of the
ground reaction force, that is, the center of pressure
(COP). The timeevolution of the resulting COP trajec
tories is often viewed, usually implicitly, as a manifesta
tion of random fluctuations in the postural control
system. This view underlies the application of conven
tional averaging techniques in posturography that aim to
M. Roerdink (&) Æ A. Daffertshofer Æ S. F. Donker
A. C. H. Geurts Æ P. J. Beek
Faculty of Human Movement Sciences,
Institute for Fundamental and Clinical
Human Movement Sciences,
Vrije Universiteit, Van der Boechorststraat 9,
1081 BT Amsterdam, The Netherlands
Email: m.roerdink@fbw.vu.nl
Tel.: +31205988516
Fax: +31205988529
M. De Haart
Department of Rehabilitation, Amsterdam Medical Centre,
University of Amsterdam,
Amsterdam, The Netherlands
M. De Haart Æ A. C. H. Geurts
Department of Research, Development, and Education,
St. Maartenskliniek,
Nijmegen, The Netherlands
S. F. Donker
Department of Otorhinolaryngology,
Vrije Universiteit Medical Centre,
Amsterdam, The Netherlands
A. C. H. Geurts
Department of Rehabilitation Medicine,
University Medical Centre, St. Radboud,
Nijmegen, The Netherlands
Exp Brain Res (2006) 174: 256–269
DOI 10.1007/s0022100604417
Page 2
identify scalar values, like the mean COP velocity, by
averaging out the assumed noisy or random character of
postural sway. Such descriptive statistical values have
been shown to change with pathologies and aging and to
vary over a range of sensory and cognitive conditions.
There are ample indications in the literature, however,
that more detailed analyses of COP timeevolutions may
provide further insight into postural control.
In this context, the nature of postural sway has been
characterized both as deterministic chaotic (e.g., Newell
et al. 1993; Yamada 1995; Pascolo et al. 2005) and sto
chastic (e.g., Collins and De Luca 1993; Newell et al.
1997; Delignie ` res et al. 2003). Chaotic time series appear
random and unpredictable but arise from deterministic
nonlinear processes, whereas stochastic time series are
governed by chance alone (e.g., Brownian motion, see
below) or by a combination of deterministic and random
processes (e.g., biased random walk). Although con
ceptually different, these approaches both focus on the
dynamical structure of COP trajectories, which may
contain information about the postural control exerted.
The ‘smoothness’ of the COP trajectories hints at strong
deterministic components in the stochastic postural
sway dynamics and comparatively weak influences of
noise (see also Collins and De Luca 1993, 1995; Riley
et al. 1999; Riley and Turvey 2002). Using linear systems
theory and corresponding identification techniques,
Kiemel et al. (2002) similarly concluded that COP tra
jectories reflect a mixture of deterministic and stochastic
components. However, this more traditional approach
tends to downplay the importance of the repeatedly
demonstrated nonlinear character of the COP dynam
ics. Hence, in order to identify the stochastic dynamics
of postural sway, COP analyses that account for the
nonlinear and stochastic temporal evolution of postural
sway appear appropriate. While it has already been
demonstrated that analyses borrowed from the field of
dynamical systems can be meaningfully applied to
measured COP trajectories (e.g., Newell et al. 1993;
Yamada 1995; Pascolo et al. 2005), we expected that
they would also be valuable in evaluating the effects of
health status, rehabilitation, and task manipulations in
clinical studies (cf. Raymakers et al. 2005).
From this expectation, we reanalyzed the COP data
that were collected and analyzed by De Haart et al.
(2004) in an encompassing longitudinal study of the
recovery of standing balance in stroke patients. Those
data are particularly interesting from a clinical point of
view because they cover over 30 patients whose postural
control was assessed at five stages, during a 3 month
recovery period under both sensory (eyesopen versus
eyesclosed) and cognitive (standing with eyes open while
performing an arithmetic dual task) manipulations. With
conventional posturographic measures (e.g., mean posi
tion, sway amplitude, and sway velocity), De Haart et al.
(2004) found that, compared to age and gendermatched
healthy elderly controls, stroke patients exhibited a se
vere weightbearing asymmetry that was accompanied by
increased sway amplitude and sway velocity, especially in
the frontal plane. With followup assessments, the
weightbearing asymmetry reduced, as did the sway
amplitude and velocity, suggesting that posture stabilized
in the course of rehabilitation. Under the arithmetic dual
task, COP velocity in the sagittal plane was greater,
which was interpreted as an indication that patients
‘stiffened up’ to some degree, perhaps as a result of an
increased alertness. Finally, without vision, a larger COP
amplitude and velocity were observed.
We reanalyzed the data of De Haart et al. (2004) using
a set of complementary measures borrowed from
dynamical systems theory to evaluate our expectation
that they would reveal new aspects of the data that were
not disclosed by means of the conventional measures
mentioned above. Since a plethora of measures may be
used to analyze the temporal structure underlying COP
trajectories, we restricted the analysis to four: the cor
relation dimension, the largest Lyapunov exponent, the
sample entropy, and the Hurst exponent. This particular
choice was motivated from the consideration that these
measures are all defined operationally in terms of readily
interpretable and complementary (i.e., deterministic and
stochastic) properties of motor control, namely, number
of degrees of freedom, local stability, regularity, and
scaling behavior. More specifically, the correlation
dimension (e.g., Grassberger and Procaccia 1983) mea
sures the dimensionality of the COP time series (e.g.,
Newell 1998; Yamada 1995) and may provide an esti
mate of the number of (active) dynamical degrees of
freedom involved in postural control (e.g., Kay 1988).
Lyapunov exponents quantify the convergence or
divergence of nearby points in the postural control state
space, with the largest Lyapunov exponent characteriz
ing a system’s local stability (Wolf et al. 1985; Rosenstein
et al. 1993; Yamada 1995), that is, the sensitivity of the
postural control system to local perturbations. Sample
entropy (Richman and Moorman 2000) quantifies the
regularity (or predictability) of a time series, while scaling
factors like the Hurst exponent [e.g., determined via
detrended fluctuation analysis (DFA), Peng et al. 1994,
1995] quantify the extent to which a recorded COP time
series exhibits longrange correlations. To date, these
dynamical measures have rarely been used in clinical
studies of upright standing (see, e.g., Newell et al. 1993;
Pascolo et al. 2005, for exceptions), in spite of their
capacity to provide new insights, as widely demonstrated
in, for instance, physiology and the study of ‘‘dynamical
diseases’’ (e.g., Goldberger et al. 1996, 1997, 2002; Glass
2001; Lipsitz 2002; Kyriazis 2003).
With our choice of measures, we abstained from
using other techniques that have recently been applied in
the study of postural control, such as recurrence quan
tification analysis (RQA, Riley et al. 1999), rambling–
trembling component analysis (Zatsiorsky and Duarte
1999), swaydensity analysis (Baratto et al. 2002), and
stabilogram diffusion analysis (Collins and De Luca
1993, 1995). Although the various output measures of
RQA are conceptually related to some of the measures
used in the present study, especially those indexing
257
Page 3
stability and regularity, they are not identical and
cannot be simply mapped onto each other. Unlike RQA,
ramblingtrembling component analysis, swaydensity
analysis, and stabilogram diffusion analysis have been
constructed specifically to examine postural sway
dynamics, but the aim of our study was to evaluate
whether generally motivated measures derived from
dynamical systems theory would have surplus value in
the analysis of COP trajectories compared to standard
posturographic measures.
Methods
Participants and procedure
We reanalyzed the COP data from 33 stroke patients
(mean age 61.2 years, SD 13.0 years) out of an inception
cohort of 37 stroke patients (viz. data from four patients
were excluded due to missing values) and 22 healthy
elderly (mean age 63.9 years, SD 9.3 years) that was
used by De Haart et al. (2004).
In this study, stroke patients receiving standard
rehabilitation training were evaluated five times over a
12 week period from the moment that each patient was
able to stand without assistance for at least 30 s [i.e.,
grade 4 according to the standing balance scale de
scribed by Bohannon (1995)], the socalled baseline
assessment, as well as 2, 4, 8, and 12 weeks after that
moment. Stroke type (infarction/haematoma), location
(left/right hemisphere), and the time poststroke of the
33 included patients are provided in Table 1, together
with an indication of the level of recovery at the baseline
assessment, as quantified by the clinical evaluation of the
patients’ (1) walking skills, as rated according to the six
point (range 0–5) Functional Ambulation Categories (cf.
Collen et al. 1990; Wade 1992), and (2) lowerlimb
motor selectivity, as scored according to the six motor
stages defined by Brunnstrom (cf. Brunnstrom 1966;
FuglMeyer et al. 1975).
During each assessment, two posturographic trials
for three quietstanding conditions [eyes open (EO), eyes
open while performing an arithmetic dual task (DT),
and eyes closed (EC)] were performed1. In each condi
tion, the participants were asked to stand as still and
symmetrically as possible (i.e., barefoot, arms alongside
the trunk (if possible), feet placed with the heels 8.4 cm
apart, with the toes pointing outward at a 9? angle from
the sagittal midline), while their COP was recorded for
30 s at a sample frequency of 60 Hz. The healthy elderly
underwent the same procedure (see Nienhuis et al. 2001).
We refer to the original study by De Haart et al. (2004)
for more detailed information about the participants,
equipment, and the procedures.
Posturographic data analysis
Prior to all the analyses, each signal’s mean was sub
tracted.Timeseriesweresampledbymeansoft fi ti,with
i=1, 2, 3, ..., N and N indicating the total number of
samplesintheCOPtimeseries.Thesignalsaredenotedas
x(ti), with x=ML (i.e., mediolateral)=AP (i.e., anterio
posterior) COP displacements. We used the conventional
standard deviation rxto quantify the variability of the
postural sway dynamics x(ti). Besides the mean and var
iance of the time series, which, by definition, ignore the
temporal structure of the COP trajectories, we assessed
the COP dynamics by calculating its correlation dimen
sion, largest Lyapunov exponent, sample entropy, and
scaling behavior. In the following, we explain how those
Table 1 Characteristics of the 33 stroke patients at the baseline assessment
Age (years)
Time poststroke (weeks)
Type of stroke (infarction/haematoma)
Hemisphere of stroke (left/right)
Functional Ambulation Categoriesa
Lowerlimb motor selectivity (Brunnstrom stage)b
61.2 (SD 13.0; range 27–82)
9.8 (SD 5.2; range 3.3–24.1)
26/7
10/23
2 (range 1–4)
IV (range II–VI)
aValues are median scores (range). Functional Ambulation Categories are defined as follows: 0, Nonfunctional (unable): patient cannot
walk or requires help of two or more people; 1, Dependent (level 2): patient requires firm continuous support from one person who helps
carrying weight and with balance; 2, Dependent (level 1): patient needs continuous or intermittent support of one person to help with
balance and coordination; 3, Dependent (supervision): patient requires verbal supervision or standby help from one person without
physical contact; 4, Independent (on level ground): patient can walk independently on level ground, but requires help on stairs, slopes, or
uneven surfaces; 5, Independent: patient can walk independently anywhere (cf. Collen et al. 1990; Wade 1992)
bValues are median scores (range). Brunnstrom stages are defined as follows: I, flaccid paralysis; II, increased muscle tone without active
movement; III, increased muscle tone with active movements mainly in rigid extension synergy; IV, increased muscle tone with alternating
gross movements in extension and flexion synergies; V, muscle tone normalization with some degree of selective muscle control (i.e.,
combined active knee extension and foot dorsiflexion against some resistance); and VI, normal muscle tone and control (cf. Brunnstrom
1966; FuglMeyer et al. 1975)
1In the original study of De Haart et al. (2004), each balance
assessment consisted of two consecutive test series, incorporating
fourquietstandingtasksandoneweightshiftingtask,presentedina
fixed sequence. This sequence was repeated in reverse order to con
trol for time effects. Between the DT and EC condition, participants
conducted a trial while looking at a vertical black bar, which served
asavisualmidlinereference.Theweightshiftingtaskwasperformed
twice after (and preceding) the first (second) EC condition. A 1min
rest was given after each balance test, whereas a longer pause was
allowed between the two test series. The arithmetic task in the DT
condition consisted of a (varying) verbal sequence of eight single
digit additions (e.g., 7+4=11 or 3+5=7) equally timed over the
30s period. The participants were instructed to verbally indicate
the correctness of each summation by good or fault response.
258
Page 4
calculations were performed in such a manner that the
analyses can be replicated if desired.
Correlation dimension
To calculate the correlation dimension of the time series,
we followed Grassberger and Procaccia (1983) by con
ducting a phase space reconstruction of the COP
dynamics (Takens 1981). To this end, the measured COP
time series was embedded in a highdimensional phase
space by timedelaying the original time series as x(ti),
x(ti+s), x(ti+2s), ... (see, for instance, Abarbanel 1996;
Packard et al. 1980). The embedding delay s corre
sponded to the first minimum of the mutual information
function2(Fraser and Swinney 1986; see also Abarbanel
1996), that is, the time at which the original and delayed
signals were maximally independent. As shown in Fig. 1,
the modified correlation sum Cmof the COP time series
x(ti) was finally computed for each embedding dimension
CmðrÞ ¼1
Np
X
N?W
i¼1
X
N
j¼iþW
H r ? Xi? Xj
????
??
ð1Þ
in which m refers to the embedding dimension and HðzÞ
denotes the unit step function: HðzÞ =1 if z‡0, HðzÞ =0
otherwise.Xi
isgiven
x(ti+(m?1)s)], Xi?Xj is the Euclidean distance be
tween Xiand Xj, and r is a distance on a log scale. The
cutoff parameter3W was defined as twice the first
minimum in the mutual information function. Npis the
number of pairs i, j such that ij‡W, where Np=(N
W+1)(NW)/2 (Theiler and Rapp 1996). Importantly,
for small distances r, the function Cm(r) behaves as a
power law, i.e.,
as[x(ti),x(ti+s), ...,
CmðrÞ / rD2
where D2is the correlation dimension that is approxi
mated via the linear region of the slope of the log–log
display of Cm(r) as a function of r. Notice that for very
small r, estimating D2becomes difficult because usually
the number of data points (samples) becomes very small,
yielding inaccuracies similar to the case in which r be
comes comparable to the attractor size. Hence, the slope
was determined for an interval between ra, i.e., the dis
tancer capturing0.5%
(Cm(ra)=0.005) and rb, i.e., the distance r capturing 75%
of the pairs of points (Cm(rb)=0.75) (see also Fig. 1),
upper panel4. The slopes were determined per embed
ding dimension m according to:
ð2Þ
ofthepairsofpoints
dm¼log10CmðrbÞ ? log10CmðraÞ
log10rb? log10ra
ð3Þ
10
–1
10
0
10
r
1
10
2
10
3
10
–3
10
–2
10
–1
10
0
original
shuffled
Dimension Analysis
Cm
Cm (ra )
Cm (rb )
012345678910 11 12
0
5
10
original
shuffled
2dm+1=m
m
2dm + 1
Fig. 1 Dimensionality analysis. Top panel: correlation sum Cm
plotted for each embedding dimension m for original (thick black
lines) and shuffled surrogate (thinner gray lines) COP time series
(ML, EO, stroke patient). The dotted horizontal lines indicate the
boundaries between which the slopes dmwere estimated (Kaplan
et al. 1991; see text for further details). Lower panel: a plot of the
slopes dmfor the original (black solid circles) and shuffled surrogate
(gray open circles) correlation sums for increasing embedding
dimension m. Note that at the sixth embedding dimension, the
slope of the original COP time series satisfies the condition
m>2dm+1 (i.e., the corresponding dimension estimate D2is 2.26),
whereas for the shuffled surrogate data, Cm(r) scales with rm
2Forourdata,thefirstminimumofthemutualinformationoccurred
at11(mean11.14,SE0.15)and10(mean9.91,SE0.20)datasamples
forMLandAPswaycomponentsofthestrokepatients,respectively.
These minima did not change with rehabilitation or condition
(P>0.05). For the healthy controls, the first minimum of the mutual
information occurred at 11 data samples for both ML (mean 11.10,
SE 0.27) and AP (mean 10.54, SE 0.21) sway components. Again, no
change with condition was found (P>0.05). Note that the choice of
the time delay s was not based on these group averages but was
determined independently for each trial.
3Correlations between consecutively sampled points can produce
spurious indications of lowdimensional structure. With the intro
duction of the cutoff parameter W>1, it is possible to minimize
these correlations (Grassberger 1986; Theiler 1986). Therefore, all
pairs of points that are closer together in time than some cutoff W
were excluded. W=1 returns the standard Grassberger and Pro
caccia (1983) formula.
4The dimension is often calculated by looking at the slope of the
most linear segments of Cm(r), requiring a means of evaluating a
score for each plausible linear segment (i.e., based on the length of
the segment or the goodness of fit to a line). The ‘optimal’ linear
segment is chosen. In this way, these techniques emphasize the
possible existence of strange attractors. A drawback of such
methods is that the length scale chosen can depend discontinuously
on the underlying signal, because a small change in the signal can
change the relative ranks of the candidate linear segments and
thereby change the calculated dimension substantially (Kaplan
et al. 1991). Because the applied dimension analysis in this study
did not involve examination of the linear scaling of Cm(r), it would
be incorrect to interpret the estimated dimension D2 as the
dimension of the attractor. Similarly, it would be incorrect to infer
from this analysis that an attractor must exist.
259
Page 5
We estimated the dimension D2by looking for cases in
which the slopes dmsaturated with increasing embedding
dimension m, with m>2dm+1 (Fig. 1, lower panel).
Finally, whenever m>2dm+1 was fulfilled, the estimate
D2could be considered reliable.
Largest Lyapunov exponent
We quantified the mean exponential divergence d(t) at
time t of initially close statespace trajectories by means
of dðtÞ / Cekmaxt(e.g., Rosenstein et al. 1993). In this
form, the exponent kmaxreflects a system’s local stabil
ity, that is, its sensitivity to small and local perturbations
and is referred to as the largest Lyapunov exponent.
Briefly, if kmaxis negative, then any perturbation will
exponentially damp out and initially close trajectories
will stay close. In contrast, if kmaxis positive, the d(t)’s
will diverge, i.e., the distance between trajectories will
increase exponentially. Positive kmaxvalues indicate the
presence of chaos, provided that the system stays in fi
nite vicinity, implying that some attractor exists. In the
latter case, nearby points diverge exponentially (local
instability), causing a lack of predictability.
To calculate kmax, for each Xiof embedding dimen
sion m (m>2dm+1), the nearest neighbor was identified
as the point closest to Xiwith a temporal separation
larger than twice the first minimum in the mutual
information function (see above). Next, distances be
tween neighboring trajectories in state space were cal
culated as a function of time (i.e., jDt=3 s) and averaged
over all original pairs of nearest neighbors i. The kmax
were finally estimated from the slopes of
yðjÞ ¼1
Dt
1
N
X
N
i¼1
ln diðjÞð4Þ
after fitting a range from jDt=0 to 0.75 s (Rosenstein
et al. 1993).
Sample entropy
In addition to the aforementioned measures, we used
more statistical descriptions to characterize not only
the deterministic structure of the COP dynamics, but
also its stochastic features. In general, entropy or
informationrelated measures are first candidates when
analyzing a system’s regularity or ordering. Since we
investigated time series rather than arbitrary statistical
ensembles, we used the sample entropy (Richman and
Moorman 2000), which builds on the conditional
probability that a signal of length N will repeat itself
for M points, provided that it already repeated itself
for M1 points (within a tolerance range r and without
allowing selfmatches): the higher the entropy, the
lower the time series’ regularity.
In view of the limited length of the tobeanalyzed
time series, we chose the window parameter as
M=3—note that the choice of r is limited since too
small a tolerance yields low confidence, while the dis
criminative capacity drops when increasing the tolerance
range (Pincus 1991; Pincus and Goldberger 1994). Re
cently, Lake et al. (2002) proposed to use the maximal
value of the relative errors of sample entropy and con
ditional probability for a range of values of r and M to
optimize the choice of r and M while enhancing the
efficiency of the entropy estimate by penalizing condi
tional probabilities near 0 and near 1. Here, the sample
entropy analysis parameters r and M were selected based
on minima in the grayscaled relative error map
(Fig. 2)—cf. Lake et al. (2002) for further details.5
Scaling: detrended fluctuation analysis
The Hurst exponent H quantifies the leading order of
the temporal change of a time series’ correlation (or least
squared displacements). Formally one may write
r2
DxðDtÞ / Dt2Hþ ???
i.e., the variance r2of the displacement Dx changes in
time Dt according to a power law (with exponent 2H).
For the simplest diffusion process, the free Brownian
motion or ordinary random walk, the Hurst exponent is
H=0.5. Other values in the range 0<H<1 are typically
referred to as fractional Brownian motion (Mandelbrot
and van Ness 1968): H>0.5 implies persistence, i.e., the
trajectory tends to continue in its current direction and
thus produces enhanced diffusion; H<0.5 implies anti
persistence, i.e., the trajectory tends to return from
where it came and thus suppresses diffusion.
Importantly, COP trajectories are always bounded
withintheareaofsupport,
Brownian motion is, in principle, unbounded. Indeed,
Eq. (5) does not hold for diffusion processes that remain
limited in that the corresponding variance saturates be
yond a critical time interval (see Delignie ` res et al. 2003).
Thus, in order to estimate the Hurst exponent using Eq.
(5), some preprocessing needs to be applied. The basic
idea is that if the original signal is bounded, then the
integrated time series is unbounded and exhibits quan
tifiable scaling properties (Delignie ` res et al. 2003; see
also Eke et al. 2000, 2002). This integration is the first
step in the socalled DFA (Peng et al. 1994, 1995), in
which, subsequently, this integrated time series is divided
into nonoverlapping intervals. Within each interval, the
time series is linearly detrended to remove trivial corre
lations (Fig. 3, center panel) and, finally, the root mean
square fluctuations Fnof the residual are displayed as
function of the interval size n (Fig. 3, lower panel). In
the presence of a power law, the according log–log
representation yields a scaling exponent a that relates to
the Hurst exponent as H=(2a–1)/4.
ð5Þ
whereas(fractional)
5In agreement with, e.g., Lake et al. (2002) and Richman and
Moorman (2000), time series were normalized to unit variance.
Sample entropy software was obtained from PhysioNet (Goldber
ger et al. 2000).
260
Page 6
Detrended fluctuation analysis was applied to all x(ti)
time series using logarithmically spaced interval lengths
n from 10 to N/2 samples6(N=1,800). The slope of the
doublelogarithmic plot of Fnversus n was estimated
between n=0.26 s and n=8.66 s. The largest values of n
were disregarded, since these points were based on only
a few data segments, rendering the estimate unreliable.
Surrogate data
In order to guarantee the validity of the nonlinear
analyses applied here, we exploited surrogate data by
means of both time and phaserandomized COP tra
jectories (Theiler et al. 1992). We first generated surro
gate data by randomly selecting samples of x(ti), i.e., we
preserved the data’s statistics (mean, variance, etc.), but
‘destroyed’ the data’s temporal correlation by shuffling
its temporal ordering (cf. Fig. 3, upper panel). Notice
that, due to the absence of temporal correlations, shuf
fled surrogate data have a huge dimension (converging
to infinity), a high sample entropy, and a Hurst expo
nent close to zero. In addition, we randomized the data’s
phaseafterFouriertransformation.In general,
randomizing the phase does not alter the spectral power
distribution and thus preserves the data’s autocorrela
tion function (cf. Kantz and Schreiber 2004). Hence,
Hurst exponents of phaserandomized and original data
should match, while, as for the shuffled surrogates,
phaserandomization is expected to result in high cor
relation dimension and sample entropy.
Statistics
All the dependent variables for the stroke patients were
analyzed using a repeated measures analysis of variance
(ANOVA) with withinsubject factors rehabilitation (five
0.10.20.30.4
r
0.5 0.60.7
1
2
3
4
5
6
7
M
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Fig. 2 Visual selection of the optimal tolerance r for estimating the
sample entropy of 66 randomly chosen, normalized ML COP time
series (i.e., rML=1) of healthy adults, based on the median value of
the relative error. As can be seen from this grayscaled map, a
window length of M=1 allows a confident estimation of entropy
for a wide range of r. For M‡2, however, optimum values of r are
clearly evident and lie generally between 0.02 and 0.1. The relative
error rises very steeply for r<0.02 and for larger r values, stressing
the fact that many combinations of M and r are intolerable. Based
on inspection of the grayscaled visual relative error plot at the
M=3 trace, r=0.035 was the optimal choice for the sample
entropy parameter r (lowest relative error). Similarly, r=0.035 and
r=0.045 were selected as optimal parameter values for ML and AP
sample entropy estimates, respectively. These r values were the
same for the stroke patients and the healthy controls
0102030
–10
0
10
xML
time
Detrended Fluctuation Analysis
0102030
0
1000
time
XML
original
shuffled
−10
log10 n
1
0
1
2
log10 Fn
α = 1.36
α = 0.50
Fig. 3 Detrended fluctuation analysis. Upper panel: ML COP time
series xMLof a stroke patient with eyes open (black line) and its
shuffled surrogate counterpart (gray line). Middle panel: integrated
time series XML. The vertical dashed lines indicate an interval of
length n=5 s, the dotted straight line segments represent the trend
estimated in each interval by a leastsquares fit. Lower panel:
doublelogarithmic plot of fluctuation Fnversus interval length n.
The slope a relates to the scaling exponent H according to
H=(2a?1)/4. Note that for the shuffled data H=0, indicating an
uncorrelated random process, whereas for the COP time series
H=0.43, indicating longrange correlations
6Notice that a multivariate extension of the detrended fluctuation
analysis algorithm yields identical results when applied to the
embedded time series (see above) since we assumed stationarity.
261
Page 7
followup levels: baseline (week 0) and 2, 4, 8, and
12 weeks later) and condition (three levels: EO, DT, and
EC). In the case of significant main or interaction effects,
posthoc analysis was performed by means of ttests
using the standard Bonferroni correction to maintain the
familywise error rate at 5%. The effect of health status
was examined by means of a repeated measures ANO
VA with health status (two levels: stroke patients (aver
aged over followup assessments) and healthy controls)
as betweensubject factor and condition as withinsubject
factor. Posthoc ttesting with Bonferroni correction was
performed for significant effects of condition, whereas for
significant interaction effects, separate repeated mea
sures ANOVAs with condition as withinsubject factor
were performed for the two groups. For main and
interaction effects of rehabilitation, condition, and health
status, Cohen’s f was used as a measure of effect size;
large effect sizes were operationalized by convention as
f>0.4 (Cohen 1988). To quantify the effects of ran
domization, we used a repeated measures ANOVA with
randomization as withinsubject factor (e.g., three levels:
original COP data, shuffled surrogates, and phaseran
domized surrogates), for stroke patients and healthy
elderly, separately.
Results
The results (F, P, and f values) of the rehabilitation by
condition ANOVA and the health status by condition
ANOVA for the dependent COP variables are presented
in Tables 2 and 3, respectively.
Rehabilitation
With followup assessments, COP variability (rAPand
rML) decreased significantly and local stability im
proved as evidenced by a significantly reduced kmaxin
both ML and AP directions. In addition, COP regu
larity decreased in the ML direction as indexed by a
significantly increased sample entropy (Fig. 4; Table 2).
Scaling exponents7(H), dimension estimates (D2), and
sample entropy in the AP direction were not affected
significantly by rehabilitation (Table 2).
Condition
Figure 5 displays condition effects for stroke patients (see
also Table 2). The COP variability was significantly
greater for EC as compared to both EO and DT. The
COP regularity was clearly lower (significantly higher
sample entropy) for DT than for EO and EC. The ML
scaling exponents (H) were significantly lower for EC
than for EO and DT. In contrast, AP scaling exponents
were significantly higher for EO than for the two other
conditions. The estimated dimension was significantly
larger for DT than for EO. The largest Lyapunov
exponent kmaxwas significantly larger for EC (i.e., lo
cally less stable) than for EO and DT. No significant
rehabilitation · condition interaction was observed for
either of the COP measures. For the healthy elderly,
qualitatively similar condition effects were observed.
Health status
Stroke patients differed significantly from healthy elderly
in all COP measures except for the ML scaling expo
nents (see also Table 3 and Fig. 6)8. COP variability
0 2 4
followup [weeks]
812
0
2.0
4.0
6.0
8.0
ns
ns
σML [mm]
0 2 4
followup [weeks]
8 12
0
0.2
0.4
0.6
0.8
1.0
ML sample entropy
P < 0.05
P < 0.05
0 2 4
followup [weeks]
8 12
1.0
2.0
ML λmax
ns
ns
Fig. 4 Rehabilitation effects for rML, ML sample entropy and ML
kmaxvalues. Error bars denote the 95% confidence intervals for
difference after Bonferroni adjustment for multiple comparisons.
rMLand kmaxdecreased significantly with followup assessments
[except when explicitly stated so (ns)]. Sample entropy was
significantly higher after 8 and 12 weeks when compared to
baseline (0 weeks)
7To avoid false or spurious conclusions, Hurst exponents were also
determined by means of a rescaled range analysis (Hurst 1965;
Rangarajan and Ding 2000; Delignie ` res et al. 2003; cf. Wing et al.
2004 for a related power spectral approach), yielding slightly higher
estimates of the diffusion process than the DFA. To compare these
two methods, the pairwise twotailed Pearson correlation coeffi
cient between the scaling exponent based on the rescaled range
analysis (H fi HR/S) and the detrended fluctuation analysis
(H fi HDFA) was determined for all the trials of the stroke patients
(N=990). For both the AP and ML scaling estimates, the corre
lation analysis showed a good agreement between HR/Sand HDFA
(r=0.918, P<0.01 and r=0.895, P<0.01, respectively).
8The significant results reported in Table 3 were all preserved when
the averaged poststroke values were replaced by the earliest post
stroke values.
262
Page 8
was significantly larger for stroke patients than for the
healthy controls. COP regularity in the ML direction was
higher (significantly lower sample entropy) for stroke
patients, whereas COP regularity in the AP direction was
significantly higher for the healthy controls. For the
stroke patients, the scaling exponents for AP were sig
nificantly lower, while the estimated dimension and kmax
were significantly higher in both directions when com
pared to the healthy controls. There were two significant
health status · condition interactions (Table 3), but these
could not be interpreted as none of the posthoc com
parisons was significant.
Randomization
As expected, all Hurst exponents vanished for the
shuffled surrogates (see above and Fig. 3, lower panel)
and H did not differ from the original value after phase
randomization (P>0.05; see Fig. 7) for both the stroke
patients and the healthy elderly. This implies that strong
correlations were present in the time evolution of the
COP trajectories, suggesting a deterministic dynamical
structure. Furthermore, ML and AP sample entropy
estimates for the shuffled and phaserandomized surro
gate time series were both significantly higher than their
original counterparts for the stroke patients and the
healthy elderly alike (Fig. 7). Higher entropy values
imply that more information is required to describe
the surrogate data due to the applied time and phase
randomization, which again suggests a deterministic
component in the original COP trajectories.
The dimension analysis revealed that Cm(r) of shuf
fled surrogates scaled with rmrather than rD2(see Fig. 1,
lower panel), reflecting the presence of very high (or
infinite) dimensional noise in the surrogate data that was
absent in the original data. Irrespective of the health
status and movement direction, the D2 and kmax of
phaserandomized surrogates were significantly higher
than their original counterparts (Fig. 7). In line with the
previous results, this again indicates the presence of a
pronounced deterministic component.
Discussion
In the present study, we reanalyzed the COP data of De
Haart et al. (2004) using dynamical rather than con
ventional measures to examine whether this would lead
to novel insights into the changes in postural control as a
function of (a) health status (stroke patients versus
healthy elderly), (b) rehabilitation (followup assess
ments), and (c) task conditions (EO, DT, EC). In the
following, we systematically discuss to what extent new
findings and insights were obtained visa ` vis those re
ported by De Haart et al. (2004), using the three afore
mentioned independent variables as entry points. We
then conclude with a general evaluation of the signifi
cance of studying postural sway dynamics.
Health status
As expected, we replicated the finding of De Haart et al.
(2004) that stroke induced substantial differences in the
global characteristics of the COP trajectories in com
parison with the healthy controls (see also De Haart
et al. 2004). Postural sway variability (i.e., deviation
from the mean) was larger in patients than in healthy
Table 2 Main and interaction effects of rehabilitation (five levels) and condition (three levels) on standard deviation r, Hurst exponent H,
dimension estimate D2, sample entropy, and largest Lyapunov exponent kmaxfor ML and AP COP components of 33 stroke patients
Rehabilitation ConditionInteraction
F(4, 128)
a
PfF(2, 64)
a
PfF(8, 256)
a
Pf
Standard deviation r
ML
AP
Hurst exponent H
ML
AP
Dimension estimate D2
ML
AP
Sample entropy
ML
AP
Lyapunov exponent kmax
ML
AP
24.29
11.97
<0.001
<0.001
0.87
0.61
5.80
15.93
<0.01
<0.001
0.43
0.71
1.67
1.42
ns
ns
0.23
0.21
0.65
0.89
ns
ns
0.14
0.17
8.88
8.94
<0.001
<0.001
0.68
0.53
1.57
0.55
ns
ns
0.22
0.13
0.32
0.36
ns
ns
0.10
0.21
5.35
3.96
<0.01
<0.05
0.41
0.39
0.73
1.17
ns
ns
0.15
0.06
5.70
0.66
<0.005
ns
0.42
0.14
4.76
17.75
<0.05
<0.001
0.38
0.75
1.14
1.31
ns
ns
0.19
0.20
24.35
26.95
<0.001
<0.001
0.87
1.02
8.30
5.94
<0.005
<0.01
0.51
0.48
0.95
0.66
ns
ns
0.17
0.16
ns not significant
aIn case the sphericity assumption was violated, the number of degrees of freedom was adjusted using the Huynh–Feldt method. Missing
values arose for the dimension estimate D2and the Lyapunov exponent kmaxdue to the m>2dm+1 criterion for six stroke patients for the
AP COP component only (nine missing values in total). No missing value analysis was performed, resulting in a loss of six patients in the
rehabilitation · condition repeated measures ANOVA
263
Page 9
elderly. Analyses of surrogate data revealed, however,
that the observed increase in variability was not due to
an increase in noise. Unlike their shuffled and phase
randomized counterparts, the original data exhibited
dynamical features other than pure randomness (see
Fig. 7), suggestingthepresence
dynamical structure in the recorded COP trajectories.
Importantly, the structure (i.e., scaling behavior) of the
correlations in the COP trajectories was preserved
qualitativelyinthestroke
impairment. That is, stroke did not lead to a breakdown
of longrange correlations, as has been found for other
ofacorrelated
patients despitetheir
pathologies (e.g., Goldberger et al. 1996, 1997, 2002).
Our scaling analysis revealed Hurst exponents that
varied between white noise and Brownian motion (i.e.,
0<H<0.5), indicating that the COP trajectories of both
the healthy elderly and the stroke patients exhibited an
antipersistent behavior: a decreasing trend is followed
by an increasing future trend (cf. Delignie ` res et al. 2003,
see also Collins and De Luca 1993; Frank et al. 2001).
By definition, those results regarding the correlated
dynamical structure in the recorded COP trajectories
could only be obtained by applying dynamical methods
of data analysis.
Marked differences between patients and healthy
controls were found in the three other dynamical
measures (Fig. 6). Dimensionality was increased in AP
direction due to stroke (Fig. 6). This increased dimen
sionality may simply be interpreted as a result of in
creased noise, which would be consistent with the
increased dimensionality and sample entropy after
phaserandomization and shuffling (Fig. 7). However,
the increased dimensionality may also be interpreted as
a change in postural control. One option is to interpret
the D2values literally in terms of chaos theory, which
would imply that the data were chaotic because D2
always saturated at a noninteger value, and that the
dimension of the chaotic COP attractor was greater for
the patients than for the healthy controls. However,
claims about the presence of deterministic chaos have
to be made with great caution because distinguishing
between chaos and stochasticity on the basis of finite
datasets might be difficult. Hence, we focused on the
differences between groups and across conditions rather
than the possibly chaotic nature of the data. Accord
ingly, the dimensionality findings may be interpreted to
imply that the stroke patients recruited additional
control processes (degrees of freedom), for instance to
compensate for the reduced efficacy of ankle mecha
nisms on their paretic side (De Haart et al. 2004; Ge
urts et al. 2005). Alternatively, it could also be the case
that the already present control structure (defined over
essential degrees of freedom) was tightened so that
(nonessential) degrees of freedom were released from
control, resulting in greater dimensionality due to
greater expression of noise along uncontrolled dimen
sions (i.e., the notion of uncontrolled manifold; cf.
Scho ¨ ner 1995; Scholz and Scho ¨ ner 1999). Unfortu
nately, at the level of the COP variable, i.e., an output
variable integrating many subsystems, it seems hard, if
not impossible, to unambiguously relate the observed
increased dimensionality to an increased noise level (be
it directly or indirectly via an increasing number of
released (or uncontrolled) degrees of freedom) versus
an increased number of recruited control processes. As
will become apparent in the following, however, the
stability analysis and the regularity analysis provided
clues to tentatively resolve this impasse.
The largest Lyapunov exponents (kmax) of the COP
trajectories were significantly greater for the stroke
patients than the healthy elderly, demonstrating a
2.0
4.0
6.0
8.0
*
*
Condition
σ
ML AP
*
EO
DT
EC
0
0.5
1.0
*
sample entropy
MLAP
0
0.2
0.4
*
*
H
ML AP
0.5
1.5
2.5
3.5
D2
*
*
ML AP
1.0
2.0
*
*
λmax
ML AP
Fig. 5 Condition effects of ML and AP COP variability r, sample
entropy, Hurst exponent H, dimension estimate D2, and largest
Lyapunov exponent kmax. Eyes open (EO: white bars), dual task
(DT: gray bars), and eyes closed (EC: black bars) conditions for
stroke patients (healthy adults showed qualitatively similar results)
are depicted. Error bars denote the 95% confidence intervals. Stars
placed at an error bar of a specific condition denote a significant
difference from the other two conditions, whereas stars placed
above connecting lines represent significant differences between the
two connected conditions (see Table 2 for F, P, and f values for
condition effects)
264
Page 10
decreased local stability or a deteriorated neuromuscular
control. Interestingly, Buzzi et al. (2003) found similar
dynamical signatures (i.e., larger dimensionality and
decreased local stability) in the variability of joint
kinematics of gait with aging, which they attributed to
deficiencies in the ability to actively control joint mo
tion. Elderly walkers were unable to compensate for the
natural stridetostride variations, which could increase
the risk of falling (Buzzi et al. 2003). Similarly, stroke
patients may have a reduced ability to compensate for
small (internal and/or external) perturbations, forcing
them to actively adjust postural control either by
recruiting more degrees of freedom or by tightening the
control of essential variables while releasing nonessen
tial degrees of freedom. Thus, in all likelihood, the ob
servedincrease in dimensionality
significance and is not simply a reflection of an increased
noise level.
Further evidence for this interpretation was found in
the analysis of the regularity of the COP trajectories. In
the stroke patients, the postural sway was more regular
in the frontal plane (i.e., lower sample entropy), whereas
it was more regular in the sagittal plane in the healthy
controls (Fig. 6). Furthermore, the stroke patients
showed greater sway variability in the frontal plane,
whereas the healthy controls showed greater sway vari
ability in the sagittal plane (Fig. 6). Hence, the direction
with the largest sway variability also showed the greatest
regularity in the COP movements, suggesting that pos
tural sway was more tightly controlled along this
direction. Notice that the studied rehabilitation cohort
was characterized by severe impairments in frontal plane
balance (De Haart et al. 2004; see also Paillex and So
2005). Furthermore, Brown et al. (2002) provided
hasfunctional
evidence for increased attention demands for quiet
standing in stroke patients. Possibly, therefore, stroke
patients actively (i.e., by means of increased cognitive
control) compensated for the loss of accurate sensory
information from the paretic leg and for other stroke
mediated impairments hampering balance control. The
observed increased regularity of the mediolateral COP
trajectories in the stroke patients could therefore reflect
an elevated cognitive contribution to postural control
(see also below). Moreover, this finding excludes the
possibility that the accompanying increased dimension
ality is an expression of increased noise, since that would
have produced higher rather than lower sample entropy,
as was the case for the shuffled and phaserandomized
surrogates (Fig. 7). All in all, it is fair to conclude at this
stage of the discussion that using dynamical measures
had surplus value in assessing the effects of health status
on postural control, although not all of those effects
could be interpreted yet in a conclusive manner.
Rehabilitation
With followup assessments the COP variability de
creased significantly (Fig. 4), as did the corresponding
regularity in the frontal plane (i.e., ML sample entropy
increased, Fig. 4). Initially, the postural sway was very
large and fairly regular, whereas 3 months later it was
smaller and markedly less regular, which is in partial
agreement with the findings and theoretical perspective
of Goldberger et al. (1996, 1997, 2002). The longrange
correlations (i.e., scaling exponents) in the COP trajec
tories, however, did not change over the five followup
assessments.
Table 3 Main and interaction effects of health status (betweensubject factor: two levels) and condition (withinsubject factor: three levels)
on standard deviation r, Hurst exponent H, dimension estimate D2, sample entropy, and largest Lyapunov exponent kmaxfor ML and AP
COP components of 33 stroke patients and 22 healthy controls
Health status ConditionInteraction
F(1, 53)
a
PfF(2, 106)
a
PfF(2, 106)
a
Pf
Standard deviation r
ML
AP
Hurst exponent H
ML
AP
Dimension estimate D2
ML
AP
Sample entropy
ML
AP
Lyapunov exponent kmax
ML
AP
63.35
20.62
<0.001
<0.001
1.09
0.62
4.40
23.98
<0.05
<0.001
0.29
0.67
3.34
0.44
<0.05
ns
0.25
0.09
0.02
15.11
ns 0.02
0.53
8.27
12.63
<0.001
<0.001
0.40
0.49
0.03
1.04
ns
ns
0.02
0.14<0.001
3.67
24.69
=0.061
<0.001
0.26
0.68
5.90
8.68
<0.005
<0.001
0.33
0.41
1.13
0.51
ns
ns
0.15
0.10
7.38
9.60
<0.01
<0.005
0.37
0.43
0.15
7.88
ns0.05
0.38
1.37
0.40
ns
ns
0.16
0.08<0.005
126.34
37.23
<0.001
<0.001
2.77
0.83
14.36
30.53
<0.001
<0.001
0.52
0.76
0.29
5.89
ns
<0.005
0.07
0.33
ns not significant
aIn case the sphericity assumption was violated, the number of degrees of freedom was adjusted using the Huynh–Feldt method. Due to
the averaging of conditions with rehabilitation, no missing values for the factor health status were present (i.e., 33 stroke patients versus 22
healthy adults)
265
Page 11
The fact that the dimensionality of the COP trajec
tories was consistently higher for the stroke patients
than for the healthy controls suggests that the reported
differences in postural control persisted in time. How
ever, in the course of rehabilitation, postural control
improved in the stroke patients, as evidenced by in
creased local stability and decreased regularity. Both
these novel findings are theoretically significant.
The observed increase in local stability is important
in view of the suggestion of De Haart et al. (2004) that
posture stabilized in the course of rehabilitation. By
assessing postural stability directly by means of the
largest Lyapunov exponent rather than by assuming that
postural stability is inversely related to postural sway
variability (which is not necessarily valid, cf. Newell
et al. 1993), we demonstrated that postural stability
increased in the course of rehabilitation. Thus, by
reanalyzing the data using dynamical measures the
proposition of De Haart et al. (2004) was confirmed
empirically.
The observed decreased regularity with followup
assessments is important in that it may reflect a reduc
tion in the cognitive component directed to postural
control in the course of rehabilitation, possibly due to
improved multisensory integration and progressive
internalization of altered body dynamics (see also Ge
urts et al. 2005). It could be speculated that, at the
beginning of independent standing, stroke patients ac
tively (i.e., cognitively) recruit additional control pro
cesses (e.g., additional strategies, cocontraction), while
in the course of rehabilitation they learn to more auto
matically exploit the subspaces of controlled and
uncontrolled variables. This possible tradeoff between
increased number of control processes and increased
number of uncontrolled degrees of freedom can leave the
observed dimensionality unchanged, despite marked
changes in postural control. It should be noted in this
context that after 3 months of followup, the frontal
plane balance was still more regular and less stable in the
stroke patients than in the healthy controls (Fig. 6),
indicating that, congruent with the proposed relation
between COP regularity and the cognitive contribution
to postural control, the cognitive involvement in posture
was still slightly elevated in the stroke patients.
Thus, in assessing the effects of rehabilitation, using
dynamical measures was clearly beneficial in that they
allowed for a confirmation of a tentative interpretation
regarding the effect of recovery on postural stability, as
well as a novel, albeit admittedly speculative, interpre
tation of the relationship between COP regularity and
the cognitive regulation of posture. The cognitive dual
task manipulation may provide a means to further assess
this novel interpretation.
Task conditions
Without visual information, COP variability and kmax
increased significantly, while no difference in dimen
sionality was observed with or without vision. These
findings agree with the study of Me ´ grot et al. (2002) on
centerofmass trajectories when standing on an unstable
platform with eyes open and closed. In addition, sample
entropy values did not differ with and without vision
(Fig. 5), indicating that there was no change in postural
sway regularity as a function of the availability of visual
information. The scaling behavior indicated that suc
cessive data points were more negatively correlated with
eyes closed than with eyes open. The corresponding in
crease in COP variability with eyes closed might have
brought the postural control system close to its stability
limits, which might have amplified the negative serial
correlation between points in the COP time series in
order to stay upright. It seems likely that other sensory
systems (such as the vestibulum and muscle and joint
receptors) may be facilitated to a greater extent in order
2.0
4.0
6.0
8.0
*
*
Health Status
MLAP
σ
patients
controls
0
0.5
1.0
*
*
MLAP
sample entropy
0
0.2
0.4
*
ns
H
MLAP
0.5
1.5
2.5
3.5
*
P = 0.061
D2
ML AP
1.0
2.0
*
*
λmax
MLAP
Fig. 6 Health status effects of ML and AP COP variability r,
sample entropy, Hurst exponent H, dimension estimate D2, and
largest Lyapunov exponent kmax. The effect of health status is
visualized by comparing stroke patients (black bars) and healthy
controls (white bars). Error bars denote the 95% confidence
intervals. Stars placed at the error bars denote significant
differences between patients and controls (see Table 3 for F, P,
and f values for health status effects)
266
Page 12
to compensate for the visual deprivation. The fact that
with eyes closed local stability strongly decreased indi
cates that vision plays a crucial role in modulating
postural dynamics (e.g., Nienhuis et al. 2001; Me ´ grot
et al. 2002; Geurts et al. 2005).
When the cognitive involvement in postural control
was diminished by introducing an arithmetic dual task,
the COP trajectories became less regular (sample en
tropy increased, see Fig. 5), which supports the rela
tionship between postural sway regularity and cognitive
contributions to postural control as proposed in the
preceding: larger (smaller) cognitive involvement yields
more (less) regular COP trajectories. De Haart et al.
(2004) reported larger sway velocity related to a shift in
COP median frequency in the DT condition without
accompanying changes in the COP’s spectral power (or
variability). We observed increased dimensionality in the
COP trajectories accompanied by less regularity when
performing a dual task as opposed to standing with eyes
open. It could be that the dual task led to a greater
contribution or expression of noise to the COP dynamics
(i.e., either direct or indirect). An increased noise level is
traditionally viewed as detrimental for control, but the
generality of this interpretation has recently been refuted
by demonstrations of the beneficial effects of noise in
sensorimotor control (e.g., Collins 1999; Cabrera et al.
2004; see also Wiesenfeld and Moss 1995; Thurner et al.
2002; Collins et al. 2003). Interestingly, under the DT
condition, local stability did not differ from that in the
EO condition, which suggests that the postural control
system could have exploited noise in a like fashion.
In assessing the effects of sensory manipulations (i.e.,
EO versus EC conditions), the use of dynamical mea
sures only yielded marginally novel findings: with
increasing sensory difficulty stability diminished, as is
wellknown from previous studies using linear systems
measures (e.g., Peterka 2002). In contrast, in assessing
the effects of a cognitive dual task on postural control,
the use of dynamical measures had surplus value as it led
to the discovery of effects that were absent when using
conventional posturographic measures. Moreover, those
effects were fully consistent with the interpretation of the
effects of health status and rehabilitation on postural
control provided in the preceding: decreased cognitive
involvement in postural control results in less regular
COP trajectories.
Significance of studying postural sway dynamics
The present study was motivated from the expectation
that, compared to the measures traditionally used in the
study of COP trajectories, dynamical measures would
have surplus value in studying postural control. We
examined this expectation by reanalyzing an encom
passing data set on the recovery of postural control
following stroke, which not only included COP mea
surements while standing upright with eyes open, but
also during sensory and cognitive manipulations. Not
withstanding the fact that the strength of statistical ef
fects found for the dynamical measures were quite
similar to those found for the standard deviation (see
Tables 2, 3), this reanalysis led to several new important
results and discoveries visa ` vis the original analysis of
the data using conventional posturographic measures. In
particular, it was established that the data’s variability
was temporally structured, that postural stability in
creased during rehabilitation, and that postural sway
regularity was positively related to the degree of cogni
tive involvement in postural control.
By combining the observed effects of health status,
rehabilitation, and task conditions on the dynamical
measures of interest, we arrived at a coherent theoretical
interpretation that may be summarized in less scientific
terms as follows. In stroke patients, maintaining balance
is more difficult due to neuromuscular impairments,
resulting in reduced postural stability. To cope with this
reduced stability, postural control is actively (i.e., cog
nitively) increased, resulting in more regular yet higher
dimensional COP trajectories. In the course of rehabil
itation, postural stability improves, allowing the patients
to relax their cognitive involvement in postural control,
which leads to less regular COP trajectories. This pro
cess is reminiscent of that of automatization in skill
0
1
2
3
4
patients controls
ML AP ML AP
sample entropy
original
phaserandomized
shuffled
0
0.1
0.2
0.3
0.4
0.5
patients controls
ML AP ML AP
H
0
1
2
3
patients controls
ML AP ML AP
D2
0.5
1.5
2.5
patients controls
ML AP ML AP
λmax
Fig. 7 Randomization effects for sample entropy, Hurst exponent
H, dimension estimate D2, and largest Lyapunov exponent kmax
depicted for both the stroke patients and the controls for ML and
AP COP directions. Original COP data (black bars), phase
randomized (gray bars) and shuffled (white bars) surrogate data
are depicted. Note that, apart from only H for original and phase
randomized surrogate data (P>0.05), original, phaserandomized
and shuffled surrogates all differed significantly from each other
(P<0.005). Note that all Hurst exponents vanished for shuffled
surrogates, i.e., H=0, whereas for D2 and kmax no shuffled
surrogates values could be determined due to the m>2dm+1
criterion (cf. Fig. 1, Cm(r) scales with rm)
267
Page 13
acquisition (cf. Huys and Beek 2002; Harbourne and
Stergiou 2003; Huys et al. 2003, 2004; Milton et al.
2004). In line with this interpretation, the introduction
of a cognitive dual task reduced the cognitive contri
bution to postural control, resulting in less regular COP
trajectories of larger dimension but similar stability.
Thus, by combining findings gathered from a comple
mentary set of dynamicsrelated analyses under various
task conditions and its recovery during rehabilitation
after stroke, we could make readily interpretable infer
ences about (changes in) the underlying postural control.
All in all, it is fair to conclude that the results of the
present study supported our expectation that the use of
dynamical measures would have surplus value in the
analysis of COP trajectories. The implication of this
overall conclusion is that, in future studies of postural
control, it should be deemed worthwhile to incorporate
both dynamical and conventional measures in the
analysis of COP trajectories.
Acknowledgments This research was conducted while the first au
thor was working on a grant of the Netherlands Organization for
Health Research and Development (ZonMw grant 1435.0004).
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