# Dynamical structure of center-of-pressure trajectories in patients recovering from stroke.

**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 center-of-pressure (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 phase-randomized surrogate data. The presence of long-range 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 stroke-induced 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 non-essential 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 task-related 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.

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- Melvyn Roerdink, Andrea G Cutti, Aurora Summa, Davide Monari, Davide Veronesi, Mariëlle W van Ooijen, Peter J Beek[Show abstract] [Hide abstract]

**ABSTRACT:**During walking on an instrumented treadmill with an embedded force platform or grid of pressure sensors, center-of-pressure (COP) trajectories exhibit a characteristic butterfly-like shape, reflecting the medio-lateral and anterior-posterior weight shifts associated with alternating steps. We define "gaitography" as the analysis of such COP trajectories during walking (the "gaitograms"). It is currently unknown, however, if gaitography can be employed to characterize pathological gait, such as lateralized gait impairments. We therefore registered gaitograms for a heterogeneous sample of persons with a trans-femoral and trans-tibial amputation during treadmill walking at a self-selected comfortable speed. We found that gaitograms directly visualize between-person differences in prosthetic gait in terms of step width and the relative duration of prosthetic and non-prosthetic single-support stance phases. We further demonstrated that one should not only focus on the gaitogram's shape but also on the time evolution along that shape, given that the COP evolves much slower in the single-support phase than in the double-support phase. Finally, commonly used temporal and spatial prosthetic gait characteristics were derived, revealing both individual and systematic differences in prosthetic and non-prosthetic step lengths, step times, swing times, and double-support durations. Because gaitograms can be rapidly collected in an unobtrusive and markerless manner over multiple gait cycles without constraining foot placement, clinical application of gaitography seems both expedient and appealing. Studies examining the repeatability of gaitograms and evaluating gaitography-based gait characteristics against a gold standard with known validity and reliability are required before gaitography can be clinically applied.Medical & Biological Engineering & Computing 09/2014; · 1.50 Impact Factor - SourceAvailable from: PubMed Central[Show abstract] [Hide abstract]

**ABSTRACT:**Falls among the older population can severely restrict their functional mobility and even cause death. Therefore, it is crucial to understand the mechanisms and conditions that cause falls, for which it is important to develop a predictive model of falls. One critical quantity for postural instability detection and prediction is the instantaneous stability of quiet upright stance based on motion data. However, well-established measures in the field of motor control that quantify overall postural stability using center-of-pressure (COP) or center-of-mass (COM) fluctuations are inadequate predictors of instantaneous stability. For this reason, 2D COP/COM virtual-time-to-contact (VTC) is investigated to detect the postural stability deficits of healthy older people compared to young adults. VTC predicts the temporal safety margin to the functional stability boundary ( = limits of the region of feasible COP or COM displacement) and, therefore, provides an index of the risk of losing postural stability. The spatial directions with increased instability were also determined using quantities of VTC that have not previously been considered. Further, Lempel-Ziv-Complexity (LZC), a measure suitable for on-line monitoring of stability/instability, was applied to explore the temporal structure or complexity of VTC and the predictability of future postural instability based on previous behavior. These features were examined as a function of age, vision and different load weighting on the legs. The primary findings showed that for old adults the stability boundary was contracted and VTC reduced. Furthermore, the complexity decreased with aging and the direction with highest postural instability also changed in aging compared to the young adults. The findings reveal the sensitivity of the time dependent properties of 2D VTC to the detection of postural instability in aging, availability of visual information and postural stance and potential applicability as a predictive model of postural instability during upright stance.PLoS ONE 10/2014; 9(10):e108905. · 3.53 Impact Factor - SourceAvailable from: Jaap Van Dieen[Show abstract] [Hide abstract]

**ABSTRACT:**Patients with non-specific low back pain (LBP) may use postural control strategies that differ from healthy subjects. To study these possible differences, we measured the amount and structure of postural sway, and the response to muscle vibration in a working cohort of 215 subjects. Subjects were standing on a force plate in bipedal stance. In the first trial the eyes were open, no perturbation applied. In the following 6 trials, vision was occluded and subjects stood under various conditions of vibration/no vibration of the lumbar spine or m. Triceps Surae (TSM) on firm surface and on foam surface. We performed a factor analysis to reduce the large amount of variables that are available to quantify all effects. Subjects with LBP showed the same amount of sway as subjects without LBP, but the structure of their sway pattern was less regular with higher frequency content. Subjects with LBP also showed a smaller response to TSM vibration, and a slower balance recovery after cessation of vibration when standing on a solid surface. There was a weak but significant association between smaller responses to TSM vibration and an irregular, high frequency sway pattern, independent from LBP. A model for control of postural sway is proposed. This model suggests that subjects with LBP use more co-contraction and less cognitive control, to maintain a standing balance when compared to subjects without LBP. In addition, a reduced weighting of proprioceptive signals in subjects with LBP is suggested as an explanation for the findings in this study. Copyright © 2014. Published by Elsevier B.V.Human Movement Science 02/2015; 39:109-20. · 2.03 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 center-of-pressure trajectories

in patients recovering from stroke

Received: 31 August 2005/ Accepted: 12 March 2006/Published online: 10 May 2006

? Springer-Verlag 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

center-of-pressure (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 phase-randomized surrogate

data. The presence of long-range 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 stroke-induced 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 non-essential

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 task-related 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 Non-linear

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 time-evolution 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

E-mail: m.roerdink@fbw.vu.nl

Tel.: +31-20-5988516

Fax: +31-20-5988529

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/s00221-006-0441-7

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 time-evolutions 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

non-linear 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 non-linear character of the COP dynam-

ics. Hence, in order to identify the stochastic dynamics

of postural sway, COP analyses that account for the

non-linear 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 (eyes-open versus

eyes-closed) 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 gender-matched

healthy elderly controls, stroke patients exhibited a se-

vere weight-bearing asymmetry that was accompanied by

increased sway amplitude and sway velocity, especially in

the frontal plane. With follow-up assessments, the

weight-bearing 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 long-range 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), sway-density 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,

rambling-trembling component analysis, sway-density

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 so-called 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 post-stroke 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) lower-limb

motor selectivity, as scored according to the six motor

stages defined by Brunnstrom (cf. Brunnstrom 1966;

Fugl-Meyer et al. 1975).

During each assessment, two posturographic trials

for three quiet-standing 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., medio-lateral)=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 post-stroke (weeks)

Type of stroke (infarction/haematoma)

Hemisphere of stroke (left/right)

Functional Ambulation Categoriesa

Lower-limb 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, Non-functional (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; Fugl-Meyer et al. 1975)

1In the original study of De Haart et al. (2004), each balance

assessment consisted of two consecutive test series, incorporating

fourquiet-standingtasksandoneweight-shiftingtask,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.Theweight-shiftingtaskwasperformed

twice after (and preceding) the first (second) EC condition. A 1-min

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

30-s 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 high-dimensional phase

space by time-delaying 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

cut-off parameter3W was defined as twice the first

minimum in the mutual information function. Npis the

number of pairs i, j such that |i-j|‡W, where Np=(N-

W+1)(N-W)/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 low-dimensional structure. With the intro-

duction of the cut-off 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 cut-off 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 state-space 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

information-related 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 M-1 points (within a tolerance range r and without

allowing self-matches): the higher the entropy, the

lower the time series’ regularity.

In view of the limited length of the to-be-analyzed

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 gray-scaled 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 so-called DFA (Peng et al. 1994, 1995), in

which, subsequently, this integrated time series is divided

into non-overlapping 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

double-logarithmic 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 non-linear

analyses applied here, we exploited surrogate data by

means of both time- and phase-randomized 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 auto-correla-

tion function (cf. Kantz and Schreiber 2004). Hence,

Hurst exponents of phase-randomized and original data

should match, while, as for the shuffled surrogates,

phase-randomization 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 within-subject 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 gray-scaled 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 gray-scaled 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 least-squares fit. Lower panel:

double-logarithmic 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 long-range 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

follow-up 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,

post-hoc analysis was performed by means of t-tests

using the standard Bonferroni correction to maintain the

family-wise 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 follow-up assessments) and healthy controls)

as between-subject factor and condition as within-subject

factor. Post-hoc t-testing with Bonferroni correction was

performed for significant effects of condition, whereas for

significant interaction effects, separate repeated mea-

sures ANOVAs with condition as within-subject 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 within-subject factor (e.g., three levels:

original COP data, shuffled surrogates, and phase-ran-

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 follow-up 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

follow-up [weeks]

812

0

2.0

4.0

6.0

8.0

ns

ns

σML [mm]

0 2 4

follow-up [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

follow-up [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 follow-up 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 pair-wise two-tailed 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 post-stroke 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 post-hoc 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 phase-randomized 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

phase-randomized 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 (follow-up assess-

ments), and (c) task conditions (EO, DT, EC). In the

following, we systematically discuss to what extent new

findings and insights were obtained vis-a ` -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 long-range 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

anti-persistent 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

phase-randomization 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 non-integer 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

(non-essential) 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 stride-to-stride 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 non-essen-

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 medio-lateral 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 phase-randomized

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 follow-up 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 long-range

correlations (i.e., scaling exponents) in the COP trajec-

tories, however, did not change over the five follow-up

assessments.

Table 3 Main and interaction effects of health status (between-subject factor: two levels) and condition (within-subject 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 follow-up

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 multi-sensory 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, co-contraction), while

in the course of rehabilitation they learn to more auto-

matically exploit the subspaces of controlled and

uncontrolled variables. This possible trade-off 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 follow-up, 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

center-of-mass 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

sensori-motor 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

well-known 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 vis-a ` -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

phase-randomized

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, phase-randomized

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 dynamics-related 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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