Entropy of balance  some recent results
 Frank G Borg^{1}Email author and
 Gerd Laxåback^{1}
DOI: 10.1186/17430003738
© Borg and Laxåback; licensee BioMed Central Ltd. 2010
Received: 19 February 2010
Accepted: 30 July 2010
Published: 30 July 2010
Abstract
Background
Entropy when applied to biological signals is expected to reflect the state of the biological system. However the physiological interpretation of the entropy is not always straightforward. When should high entropy be interpreted as a healthy sign, and when as marker of deteriorating health? We address this question for the particular case of human standing balance and the Center of Pressure data.
Methods
We have measured and analyzed balance data of 136 participants (young, n = 45; elderly, n = 91) comprising in all 1085 trials, and calculated the Sample Entropy (SampEn) for mediolateral (M/L) and anteriorposterior (A/P) Center of Pressure (COP) together with the Hurst selfsimilariy (ss) exponent α using Detrended Fluctuation Analysis (DFA). The COP was measured with a force plate in eight 30 seconds trials with eyes closed, eyes open, foam, selfperturbation and nudge conditions.
Results
1) There is a significant difference in SampEn for the A/Pdirection between the elderly and the younger groups Old > young. 2) For the elderly we have in general A/P > M/L. 3) For the younger group there was no significant A/PM/L difference with the exception for the nudge trials where we had the reverse situation, A/P < M/L. 4) For the elderly we have, Eyes Closed > Eyes Open. 5) In case of the Hurst ssexponent we have for the elderly, M/L > A/P.
Conclusions
These results seem to be require some modifications of the more or less established attentionconstraint interpretation of entropy. This holds that higher entropy correlates with a more automatic and a less constrained mode of balance control, and that a higher entropy reflects, in this sense, a more efficient balancing.
Background
The attentionconstraint interpretation (ACI)
There is a longstanding interest to analyze biological signals in terms of complexity, regularity and chaos. Measures such as entropy, the Hurst ssexponent and fractal dimensions have become popular. In physiology one can perceive two general lines of interpretations for such measures: (A) One may interpret irregularity and high entropy as signs of a healthy vigilant system; indeed, at the other extreme end we have death which is characterized by a "flat line". Irregularity may thus been seen as a mark of alertness. The system explores the "phase space" and is ready for the unexpected. An impaired system in contrast may become rigid and trapped in repeating patterns unable to successfully cope with new challenges. (B) On the other hand, irregularity and high entropy may be taken as signs that the system is loosing its structure and becoming less sustainable. This is close to the traditional interpretation of entropy as a measure of disorder and noise.
Standing posture is a case in point with regards to these dualistic interpretations. When measuring the excursions during quiet standing in terms of the center of pressure (COP) one may interpret "chaotic" excursions as a sign of poor balance and deficient postural control. On the other hand, chaotic excursions may be also interpreted as a characteristic of a successful vigilant strategy to keep balance. Obviously both interpretations can be correct, but the question is then how to decide which one is the most appropriate one in a case at hand. Or more generally, when is a high entropy, fractal dimension, etc, to be interpreted as a sign of a pathological condition and when as a sign of health [1–4]? This is also intertwined with the issue of complexity vs regularity, and what metric measures which [4]. Roughly speaking entropy is thought to be associated with regularity while various fractal measures are related to complexity, but there is no agreement on this issue. Since there is no unambiguous definition of complexity, theres is no single complexity measure. This motivates the inclusion of a fractal variable in our investigation as a complementary measure, although the interpretation of entropy visavis balance is the main focus. In the present case we use Sample Entropy [5] as the entropy measure, and the Hurst exponent α, based on the detrended fluctuation analysis (DFA) [6], as our fractal measure. The use of DFA in posturographic analysis goes at least as far back as [7] with some more recent investigations such as [8–10].
A summary of some studies of entropy in balance
Publication  Study details  Results 

[11]  Case study of a 73 y woman with a labyrinthine deficit. Balance training. Dynamic and static tests. Entropy variable: ApEn [28].  Higher entropy after training interpreted as "improved stability", "increased complexity", and as a sign of "a more selforganized system". 
[12]  30 young adults. Modified SOT test. Dual task DT (digit recall) vs single task ST. Entropy variable: ApEn.  DT > ST (APdirection, quiet standing). "Potential of ApEn to detect subtle changes in postural control." Higher ApEn interpreted as a mark of "less system constraint", and a decrease in ApEn as a "change in the allocation of attention." 
[13]  30 young adults. QS, EO, EC, DT, ST. DT = uttering words backwards. Entropy variable: SampEn [5] ("regularity") plus scaling exponent, correlation dimension and Ljapunov exponent.  ST: EC < EO; EC: DT > ST. "Regularity of COP trajectories positively related to the amount of attention invested in postural control." Increasing entropy during DT/EC interpreted as an increase in "automaticity" or "efficiency" of postural control. 
[14]  10 ballet dancers and 10 track athletes. Foam vs rigid support. Shoulder width stance. Entropy from RQA analysis [35].  Dancers < athletes; EC > EO; foam > rigid. Increasing entropy interpreted as sign of "greater flexibility". Note: the entropy here is calculated differently than SampEn or ApEn. 
[10]  14 young and 14 elderly. QS 60 sec and prolonged 30 min. Shoulder width stance (60 sec). Entropy variable: multiscale entropy MSEN [36] plus scaling exponent (DFA [6]).  Old > young (APdirection); DFA: old < young. Higher entropy for elderly found to be "inconsistent with the hypothesis that complexity in the human physiological system decreases with aging." 
[15]  11 low and 11 highly hypnotizable students. 30 sec QS with EC, plus mental computation. "Easy" = stable support; "difficult" = unstable support (foam). Feet position: 2 cm heeltoheel, 35° splay. Entropy variable: SampEn.  Difficult > easy. "No significant hypnotizabilityrelated modulation was observed." 
[16]  10 diabetics II with symptomatic neuropathy, 10 asymptomatic diabetics, and 10 nondiabetics. QS, EO, EC, COP measured in APdirection. Entropy variable: ApEn.  EC > EO stat. significant only for symptomatic diabetics. 
[17]  19 preadolscent dancers and 16 agematched nondancers. 20 sec QS with EO, EC, DT. DT = memorize words from audiotape. Entropy variable: SampEn.  Dancers > nondancers; EC < EO; DT > ST. Higher entropy interpreted as increased "automaticity of postural control." 
[18]  19 infants with typical development and 22 infants with delayed development. Sitting postural sway. Entropy variables: symbolic entropy and ApEn.  Delayed < typical in MLdirection. "Healthy postural control is seen to be more complex." 
[1]  Case study no. 2, 18 y old collegiate soccer player with cerebral concussion. Entropy variable: ApEn.  Entropy decreased during recovery from concussion. Entropy "can be considered as a measure of system complexity". "Lesser amounts of complexity are associated with both periodic and random states where the system is either too rigid or too unstable." 
COP and the feedback loop
That is, the basic assumption is that the automatic responses/control increases entropy while the volitional control decreases it. The later effect may be understood as a consequence of the longer volitional response time and consequent more sluggish behaviour. One natural hypothesis then is that volitional control determines the setpoints on a longer time scale, while the automatic control handles the fine tuning toward the setpoints on a shorter time scale. From this interpretation it does not necessarily follow that larger entropy implies smaller COP amplitude. Large entropy may either be associated with a complex fine tuned control (resulting in small COP amplitude) or a an inefficient chaotic control (resulting in a large COP amplitude).
Methods
Participants
Participant characteristics
Group  Number (♂ + ♀)  Age ± SD  BMI ± SD (kg m^{2}) 

Elderly Fallers (F)  34 (6 + 28)  81.5 ± 5.7 (68  94)  27.3 ± 4.8 (17.7  37.6) 
Elderly NonFallers (NF)  57 (14 + 43)  79.8 ± 6.2 (64  91)  29.6 ± 5.3 (20.8  46.1) 
"Young" (Y)  45 (16 + 29)  38.9 ± 11.6 (17  61)  24.3 ± 3.4 (19.5  33.8) 
Measurements
The balance measurement was performed using a standard strain gauge force plate (model B4, http://www.hurlabs.com) connected to the PC via USB. The protocol, designed at our lab for fall risk assessment, consisted of the following trials (EO = eyes open; EC = eyes closed):
EO1 First EO trial
EC1 First EC trial
EO2 Second EO trial
EC2 Second EC trial
FOAM Standing on foam EO (2 cm PEfoam)
HEAD R Autohead rotation EO (neutral → left → right → neutral)
HEAD E Autohead extension EO (neutral → up → down → neutral)
NUDGE Perturbation EO (one forward nudge at the waist level at the beginning of the trial)
Each trial lasted 30 seconds. The foot position (shoes off) was standardized [22]: clearance (heeltoheel distance) of 2 cm; 30° splay (angle between medial sides of the feet). Arms were held at the sides. A mark on the wall (3 m distance, height 1.5 m) was used for fixing the gaze. The instruction to the participant was to be relaxed (breath normally, etc) and to stand as quiet as possible.
Analysis
For calculating the Sample Entropy (SampEn) and Detrended Fluctuation Analysis (DFA) we used the computer codes obtained from Physionet [23]http://www.physionet.org/physiotools/. For SampEn we used the "default" parameter values m = 2 and r = 0.2. Before calculation the COPdata was down sampled from 200 Hz to 10 Hz since: (a) there is little of physiological significance above 10 Hz in the COP signal; (b) it lessens the computational burden of analyzing about 8 hours of data; (c) this down sampling corresponds to a lag value also used e.g. by [12]. 10 Hz corresponds to 100 ms which is of the order of the automatic responses and hence also makes physiological sense as a lag time. The sampen function was used with the n option meaning that the data was normalized before the entropy calculation (mean value is subtracted and the result is then divided by the standard variation). As a measure of the amplitude of COP we have computed its standard deviation denoted σX and σY for mediallateral and anteriorposterior direction respectively. For statistical significance level we use p < 0.05. For statistical calculations and data visualizations we have used MATHCADhttp://www.ptc.com/products/mathcad/ and the Rpackage [24]. The twosample Welch ttest for comparing the means of two sets A and B with unequal variances was calculated by the Rcommand t.test(A,B). When checking the entropy difference between the EO and EC conditions we have applied the paired ttest to S(EO 1) + S(EO 2) and S(EC 1) + S(EC 2). Statistical tests with respects to all trials have been calculated using the averages over the trials for each person. (In R one can use the aggregate command with FUN = mean to obtain the means.)
Results and Discussion
Results
Sample entropy
For mediallateral (X) vs anteriorposterior (Y) a prominent feature is that the groups of elderly have higher entropy for the Y direction: S(Y ) > S(X) (p < 0.0001). A general pattern is the higher entropy in the Xdirection for eyes closed condition (EC) compared to the eyes open (EO) condition, S(EC, X) > S(EO, X) (p < 0.0005). For Y direction the elderly fallers have a pronounced increase in the eyes closed case compared to the eyes open case (p < 0.0001). A final interesting feature is the decrease of Y entropy for the nudge trial for all groups (p < 0.0001).
COP amplitude
An expected feature is that the "young" in general have a smaller COP amplitude (p < 0.0001). One exception is the Y amplitude for the nudge trial. Since the COP Y is proportional to the righting torque the relative large COP Y for the "young" group in the nudge case reflects the ability to counteract the nudge. The elderly tend to have larger X  and Y amplitude with eyes closed compared to eyes open (p < 0.0001). The larger lateral COP X amplitude is a distinguishing feature between the elderly fallers and nonfallers for the foam (p = 0.009) and head extension (p = 0.04) conditions.
Hurst ssexponent α
We note that mean values α for the groups stay well within the range 1  1.5 characterizing antipersistence. For the elderly we have a higher αvalue in the Xdirection, α(X) > α(Y ) (p < 0.0001). Another pattern is that α(X) is lower for the "young" compared with the elderly (p < 0.0002). A conspicuous feature for the elderly is that α goes up and down from trial to trial. This is true also for the "young" in the Xdirection but not so in the Y direction.
Relations
Discussion
The attentionconstraint interpretation (ACI) seems to be in accord with lowering of entropy S(Y ) in the nudge trial (Fig. 2). However, the higher entropy in the eyes closed case, S(EC) > S(EO), seems, prima facie, to be at variance with the ACI and some results in the literature, see e.g. [13, 17] or Table 1. We may though understand the higher entropy in EC case, despite an "increasing cognitive involvement in postural control" [[13], p. 1], if the lack of visual cues cannot be compensated for by other proprioceptive cues. That is, lack of sensory information through sensory deprivation, or impairment, may imply that an increase of cognitive involvement does not translate into a corresponding constrained mode of balance. The pilot is so to speak flying blinded. Suppose the attentive control works by increasing the deterministic component in relation to the noise and that it may in this way lead to decreased entropy. However, if the sensory input is affected by noise then the output of the deterministic control will also be accordingly affected by noise, and we may see an increase in entropy instead of a reduction. The higher entropy S(Y ) for the elderly may be interpreted along these lines as an effect of a more impaired (noisy) sensory system which provides less precise input for the balance control. This is supported also by Fig. 6 where the data for elderly show an increase in the scatter of COP Y when entropy is above about 1 unit. For the young, however, an increased entropy S(Y ) is associated with a smaller COP Y . In this case increased entropy apparently signifies a more fine tuned control and not so much the contribution from noise.
One finding related to fallers vs nonfallers was the greater mediallateral (M/L) sway for fallers during the foam and head rotation conditions. M/Lsway (foam) σX ≥ 10 mm indicates for the elderly roughly an odds ratio of 4.5 for belonging to the fallers group. Several other studies have also implicated increased lateral sway as a marker for fall risk, see e.g. [25, 26]. A novel feature here may be the increased SampEn for the anteriorposterior COP Y during eyes closed condition (EC) for the elderly fallers relative to the nonfallers. This suggests that one should make further studies of the usefulness of this entropy variable as a fall risk indicator. The reason why a similar entropy increase does not show up for the M/Lsway for the EC condition is a bit of a mystery, but maybe is related to the somewhat different control mode (shifting the weight between the legs) of the M/Lsway for bipedal quiet standing, compared to the control of the A/Psway.
If we wish to establish a canon of entropy interpretation, we could proceed by measuring entropy vs COP for various groups and conditions, as exemplified by Fig. 6. Those groups which are known to have excellent balance would then define the optimal entropy relation. Hopefully this could then be followed up by a convincing theoretical framework. With an appropriate test protocol one could draw an entropyCOP diagram for an individual that could yield further clinically useful information on the weak/strong points of the balance control. A complementary approach would be to use brain imaging techniques during balancing tasks [27] to reveal whether some specific functional areas, if such areas can be identified, are correlated with the entropic measures.
Conclusions
The data presented here provide further evidence that entropy is a variable that may complement the traditional posturographical variables. Comparison of results from young and elderly reveals though that more work is needed to identify the correct physiological interpretation of entropy in a given situation. One way to proceed is to measure the entropyCOP relation for various groups of people and conditions. Those known to have excellent balance control would define the optimal entropy relation. Of clinical importance is to find those conditions (test protocols) that yield a maximum of information about deficiencies of the balance control, yet are safe and simple to administer.
Appendix
Entropy
are close to each other, z_{ i } z_{ j }<r · SD, then so will the next following points be too, x_{i+md} x_{j+md} Therefore ApEn and SampEn can be seen to estimate the degree of "surprise" in the data. Here the distance z_{ i } z_{ j } between two sequences is defined as the largest absolute difference between any two pairs of data points from the sequences. The distance is measured in terms of the fraction r of the standard deviation SD of the time series. Typical choices for the parameters are m = 2 which is the so called embedding dimension, and r = 0.2 for the so called tolerance; for more elaborate methods of selections of these parameters see [30, 31]. In our case m is restricted by the size of the downsampled time series (300 points). As a rule thumb one needs about 10^{ m } 20^{ m }data points [32].
The Hurst selfsimilarity (ss) exponent α
For 0.5 <H < 1 (1.5 <α < 2) we have the opposite property called persistence. For a pendulum, as an example, we may expect persistence for small time intervals since it tends to continue its motion in the same direction. For longer time intervals we expect antipersistence since the pendulum swings back. A smaller αvalue for quiet standing COP can thus be interpreted as a higher degree of antipersistence; that is, a higher proportion of rapid corrective impulses.
This relation suggests that with increasing α (or H) the curve becomes increasingly smooth since the higher frequency components are suppressed. Finally, in the case of selfsimilar time series x(t), the Hurst ssexponent can be related to the fractal dimension D of the graph (t, x(t)) as D = 3  α = 2  H [[34], p. 60].
List of abbreviations
 α :

Hurst selfsimilarity (ss) exponent
 ACI:

attentionconstraint interpretation
 ApEn:

approximate entropy
 A/P:

anteriorposterior
 BMI:

body mass index
 COP X:

medio/lateral center of pressure
 COP Y:

anterior/posterior center of pressure
 DFA:

detrended fluctuation analysis
 EC:

eyes closed
 EO:

eyes open
 H:

Hurst parameter
 S:

(sample) entropy
 M/L:

mediallateral
 σ :

standard deviation.
Declarations
Acknowledgements
Data gathering and analysis have been parts of projects supported by the XBranches Programme (an Innovative Action Programme supported by the ERDF in EU). We thank Magnus Björkgren, the head of the Health Science Unit (Kokkola University Consortium Chydenius) for making this study possible. We also thank the referees for pointing out errors and suggesting additional references.
Authors’ Affiliations
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