Author's personal copy Human Movement Science 32 (2013) 91–105

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Human Movement Science journal homepage: www.elsevier.com/locate/humov

Temporal correlations in postural sway moderate effects of stochastic resonance on postural stability Damian G. Kelty-Stephen a,⇑, James A. Dixon b,c,d a

Wyss Institute for Biologically Inspired Engineering, Harvard University, 3 Blackfan Circle, 2nd Floor, Boston, MA 02115, United States Department of Psychology, University of Connecticut, 406 Babbidge Road, Unit 1020, Storrs, CT 06269, United States c Center for the Ecological Study of Perception and Action, University of Connecticut, 406 Babbidge Road, Unit 1020, Storrs, CT 06269, United States d Haskins Laboratories, 300 George St., New Haven, CT 06511, United States b

a r t i c l e

i n f o

Article history: Available online 12 January 2013 PsycINFO classification: 2221 2320 2330 2860 4010 Keywords: Stochastic resonance Postural sway Temporal correlations Fluctuations

a b s t r a c t The present work documents reanalysis of previous research by Priplata and colleagues (Priplata et al., 2002) into the effects of subthreshold vibratory stimulation to the plantar surface of the foot on postural stability during quiet standing. In stochastic resonance, stimulating a nonlinear system with noise can promote system stability. Stochastic resonance has been proposed to have clinical applications as an intervention that might help to stabilize posture. Insoles designed to stimulate the plantar surface of the foot with uncorrelated white-noise fluctuations have been shown to reduce a number of standard measures of postural variability. An important remaining concern is that the efficacy of stochastic-resonance applications is subject to strong individual differences. Our reanalysis of data from Priplata et al.’s original study provides evidence that effects of uncorrelated fluctuations in subthreshold vibratory stimulation are moderated by temporally correlated fluctuations in postural sway. We suggest how future development might capitalize on this finding to fine-tune existing stochastic-resonance applications to posture. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The role of noise in the regulation of posture constitutes a major challenge for biological and behavioral sciences. It is sometimes customary to think of ‘‘noise’’ as a complicating factor, disturbing or ⇑ Corresponding author. Tel.: +1 (300) 300 3375; fax: +1 617 432 7828. E-mail address: [email protected] (D.G. Kelty-Stephen). 0167-9457/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.humov.2012.08.006

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interfering with the more stable, more sensible ‘‘deterministic’’ aspect of structure or function. On first glance, quiet standing appears to be a trivially stable behavior, and it seems to depend on setting to zero any noise terms that might induce variability. On the other hand, quiet standing is not without its own variability, and this variability reflects noise processes exhibiting a complex variety of temporal patternings (Collins & De Luca, 1993, 1994; Duarte & Zatsiorsky, 2000, 2001; Delignières, Torre, & Bernard, 2011). Postural control depends on a smooth coordination between noise and determinism. Complicating matters further is the observation that noise can actually stabilize posture. It has been widely observed that applying noise to a complex system can in fact strengthen or stabilize the system (Jung & Mayer-Kress, 1995; Shinbrot & Muzzio, 2001; Wiesenfeld & Moss, 1995). This counterintuitive phenomenon, called ‘‘stochastic resonance’’, depends on a subtle application of noise that should be neither so strong as to interfere with stability nor so weak as to be damped out. As it turns out, the noise that can stabilize posture may not even need to be felt. Priplata et al. (2002), Priplata, Niemi, Harry, Lipsitz and Collins (2003) and Priplata et al. (2006) have pioneered a vibrating-insole technology that delivers a subthreshold (specifically, 90% of threshold) white-noise vibration to three different points on the plantar surface of the foot. That is, a temporally uncorrelated noisy signal that cannot be felt has the effect of minimizing postural variability. Quite apart from any conceptual lessons we may take from this relationship between noise and order, the finding of stochastic resonance in postural control and its efficacy for minimizing variability may be one of the more important clinical breakthroughs. Falling is a leading cause of injury and subsequent death in the elderly (CDC, 2006; Runyan & Casteel, 2004). Certainly, Priplata et al. (2002, 2003, 2006) have presented stochastic resonance and vibrating insoles as a practical, non-invasive treatment that may minimize falls and improve postural control. If the evidence continues to confirm the stabilizing effect of subthreshold vibratory stimulation, then stochastic resonance may indeed serve as a powerful clinical treatment for patients at risk for falling. Translating the clinical evidence for stochastic resonance into an effective clinical intervention will require understanding how best to tailor the vibrating-insole technology to individual patients. Essentially, the effects of stochastic resonance are contingent on a potentially wide variety of individual differences. This point has already been made explicit in Priplata et al.’s (2006) evidence that the stabilizing effects of subthreshold vibratory stimulation on posture may depend on baseline variability, that is, the amount of postural variability a participant exhibits on trials without stimulation. Galica et al. (2009) demonstrated a similar baseline-dependence in the effects of the same stimulation on stability in gait, but individual differences also moderate the effects of stochastic resonance on other aspects of human behavior not involving the vibrating-insole technology (e.g., effects of auditory white noise on memory performance; Söderlund, Sikström, Loftesnes, & Sonuga-Barke, 2010). A first step towards understanding these individual differences is to recognize that stochastic resonance relies on at least two classes of fluctuations: exogenous fluctuations and endogenous fluctuations. Exogenous fluctuations encompass the field of all noise sources external to the participant. Exogenous fluctuations might include the subthreshold vibratory stimulation from a vibrating insole, but they might also include a variety of other environmental constraints on the participant’s behavior that we have yet to identify, such as changes in rigidity of the supporting surface. Real postural-control systems will also have endogenous fluctuations. It is possible to apply noise to a perfectly deterministic system (e.g., a computer-simulated sine wave), but in practice, the systems we have available for experimentation (e.g., human bodies) are full of fluctuations themselves. These endogenous fluctuations may even do much to support the typical, healthy deterministic function of system function (e.g., Ivanov et al., 2001; West, 2006). There is nonzero postural variability with or without the stochastic vibrations from an insole. The division between endogenous and exogenous fluctuations may not always be straightforward (Johnston & Edwards, 2002), but it is nonetheless important to recognize that we are not applying noise to a dormant, passive system. A crucial point is that endogenous fluctuations may actually be meaningfully related to the effects of exogenous fluctuations. The baseline-dependence of the stochastic-resonance effect on posture may be one expression of the influence of endogenous fluctuations. However, there may be a deeper, more subtle relationship. For example, postural sway (i.e., endogenous fluctuations) is temporally correlated (Duarte & Zatsiorsky, 2000, 2001; Delignières et al., 2011). Recent research has found that changes in temporal correlations of fluctuations in perceptual-motor behavior (i.e., of endogenous fluctuations)

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help to predict individual differences in perceptual and cognitive responses to environmental stimuli (Stephen & Anastas, 2011; Stephen, Arzamarski, & Michaels, 2010; Stephen, Boncoddo, Magnuson, & Dixon, 2009; Stephen & Dixon, 2009; Stephen & Hajnal, 2011). Indeed, Thurner, Mittermaier, and Ehrenberg (2002) suggested that stochastic-resonance control systems in posture may operate on the interaction of temporal correlation in stochastic stimulus with temporal correlation in the stimulated postural system. That is, individual differences in how the human body responds to exogenous fluctuations can be predicted from the temporal correlations of endogenous fluctuations. These considerations may have strong implications for bolstering the clinical value of stochastic resonance. That is, if the effect of stochastic resonance is moderated by the temporal correlations of postural sway, one potential way to tailor the vibrating-insole technology to individual patients might involve tuning the (exogenous) fluctuations from the insole to the temporal correlations of (endogenous) postural fluctuations. The superposition (i.e., summing together) of two or more fluctuations with different degrees of temporal correlations will exhibit a compromise among all superposed fluctuations; the temporal correlation of the superposition is often an amplitude weighted average of component temporal correlations (Chen, Ivanov, Hu, & Stanley, 2002; Hu, Ivanov, Chen, Carpena, & Stanley, 2001). Similarly, fitting a temporally correlated fluctuating postural system with a temporally uncorrelated foundation (i.e., the vibrating insole) may effect a similar compromise. That is, one aspect of the vibrating insole’s efficacy for minimizing postural variability may be the de-correlation of postural fluctuations through the superposition of temporally uncorrelated noise from the insole. So, not only might temporal correlations in postural fluctuations serve as a kind of filter on the effects of vibrating insoles on postural variability, but we also expect that the uncorrelated noise from vibrating insoles serves to diminish temporal correlations in postural fluctuations. Further motivating this expectation is the fact that temporal correlation is closely associated with fast growth of variance (e.g., Eke, Herman, Kocsis, & Kozak, 2002;Bassler, Gunaratne, & McCauley, 2006). So, not only would a superposition of temporal correlations suggest that uncorrelated-fluctuation stimulus should rein in temporal correlations of postural sway, but the converse should follow as well: any reduction in descriptive statistics of postural variability following from the application of the vibrating insoles might entail that fluctuations were less temporally correlated. Note that this reasoning does not suggest any interpretation of the effect of vibrating insoles specific to the phenomenon of stochastic resonance. Previous work has already promoted the potential clinical efficacy of vibrating insoles and described the phenomenon as one of stochastic resonance (Priplata et al., 2002, 2003, 2006; Galica et al., 2009). The present work makes no comment on the appropriate physical interpretation of stochastic resonance in posture but rather seeks to understand how this phenomenon is moderated by fluctuations endogenous to the postural system. A number of current concerns in the literature on stochastic resonance are relevant here: first, a stochastic-resonance formalism for one system may not be apt for another; second, the term ‘‘stochastic resonance’’ in the present application to posture refers more accurately to observed phenomenon and not a mechanism; third, it may be more profitable to focus on better understanding how fluctuations in general, endogenous and exogenous, contribute to system function (McDonnell & Abbott, 2009). In this vein, we approach the data from Priplata et al. (2003) as a test case, and we do not intend the hypotheses to be specific to subthreshold vibratory stimulation. The present work seeks to further the understanding of how endogenous temporal correlations influences the effects of fluctuation-based experimental or clinical intervention. 1.1. Hypotheses We propose that changes in temporal correlations of postural sway help to predict changes in the amount of postural variability and, more crucially, to predict changes in the effect of subthreshold vibratory stimulation on postural variability. Using the data from Priplata et al. (2003), we expect not only to confirm the negative effect of subthreshold vibratory stimulation on measures of postural variability (see Section 2.2.1), we also intend to test two specific hypotheses. First, we expect to find significant interactions of postural temporal correlation with subthreshold vibratory stimulation on measures of postural variability (Hypothesis 1). Second, we expect diminished postural temporal correlations on trials with application of uncorrelated subthreshold vibratory stimulation (Hypothesis 2).

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2. Methods 2.1. Procedure We present a re-analysis of data originally collected and reported by Priplata et al. (2003). Priplata et al. collected postural-sway data from 15 healthy young participants (average age 23 years, SD = 2) and 12 healthy elderly participants (age M = 73 years, SD = 3) for 30-s trials of quiet standing. Young and elderly participants contributed 20 and 10 trials, respectively. During all trials, participants wore gel-based insoles in each shoe that contained vibrating components (‘‘tactors’’), two beneath the forefoot and one beneath the heel. For half of the trials, the tactors provided vibratory stimulation to the feet; for the rest of the trials (i.e., control trials), the tactors provided no stimulation to the feet. For each trial with stimulation to the feet, a white noise signal was fed through the tactors with amplitudes adjusted to 90% of the sensory threshold for each foot; this white-noise stimulation was provided for the entirety of the 30-s trial. Stimulation trials and control trials were randomized for each participant. Priplata et al. (2003) collected postural data by placing a near-infrared-reflective marker on the right shoulder of the participant and recording its positions with a Vicon motion-capture system (Oxford Metrics, Oxford, UK). The Vicon system sampled this postural data at 60 Hz. Further details of data collection may be found in Priplata et al. (2003). The original data are available on the Physionet database at http://www.physionet.org/physiobank/database/nesfdb/ (Goldberger et al., 2000).

2.2. Data analysis 2.2.1. Definition of postural-variability measures The proposed effect of white-noise vibratory stimulation is to reduce measures of postural variability. Previous research has investigated this effect on the following dependent measures of postural variability: mean radius (RMean), maximum radius (RMax), elliptical area (AEllipse), mediolateral (ML) range (RangeML), anterior-posterior (AP) range (RangeAP), ML root mean square (RMSML), AP root mean square (RMSAP), ML path length (PathML), AP path length (PathAP), sway speed, and swept area (ASwept). These measures each describe a particular aspect of variability over the course of a single trial. The notion of ‘‘radius’’ in postural sway refers to the absolute distance from the average position during standing. RMean and RMax refer to the average absolute distance from the average position during standing and maximum absolute distance from the average position during standing, respectively. AEllipse is the area of the ellipse analytically derived from the major and minor axes of postural sway. RangeML is the difference between the left- and right-most excursions of position along the coronal plane. RangeAP is the difference between forward-most and rear-most excursion of position along the sagittal plane. RMSML and RMSAP are the standard deviations of position along the coronal and sagittal planes, respectively. PathML, PathAP, sway speed, and ASwept summarize the movement of posture from sample to sample of the recording device. PathML and PathAP are the sum of total absolute Euclidean displacements of postural position along the ML axis and AP axis, respectively. Sway speed is the average planar Euclidean displacement (i.e., the average square root of the sum of squared displacements along the ML and AP axes) between each consecutive pair of samples. ASwept is the sum of the areas between each consecutive pair of radii (calculated as half the cross product of each consecutive pair of radii). The unit for all measures is millimeters, except for ASwept and AEllipse whose units are squared millimeters and for sway speed whose units are millimeters per second. The greater the foregoing postural-variability measures, the more likely a fall (Fernie, Gryfe, Holliday, & Llewellyn, 1982; Liu-Ambrose et al., 2004; Lord et al., 1994; Maki, Holliday, & Fernie, 1990; Overstall, Exton-Smith, Imms, & Johnson, 1977). Further information on the specificity and sensitivity of postural variability parameters for predicting fall risk can be found in literature on the Physiological Profile Assessment (Lord, Menz, & Tiedeman, 2003; Lord, Ward, Williams, & Anstey, 1994). The proposal that subthreshold vibratory stimulation stabilizes posture goes hand in hand with an expectation that subthreshold vibratory stimulation should reduce the postural-variability measures.

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Indeed, for a number of these measures, this expectation has already been confirmed (Gravelle et al., 2002; Priplata et al., 2003, 2006). 2.2.2. Postural fluctuation time series Our reanalysis involved time series analysis of fine-grain postural fluctuations. Specifically, for each trial, we analyzed the time series of AP position, ML position, and the planar Euclidean displacement. These time series were derived from the recorded Vicon data as the excursion from average position along the sagittal plane, excursion from the average position along the coronal plane, and the square root of the sum of squared excursions, respectively. The choice of the three time series is intended to exhaust the position and velocity information in postural sway. AP and ML position time series are standard in posture research. The planar Euclidean displacement time series is essentially the first-order difference of COP path length. Though not standard to posture research, it is a postural analog to the Euclidean displacement time series that, in previous research, provided information regarding the role of fluctuations in exploratory behaviors. Further, as will be seen, it carries independent information (i.e., pertaining to velocity) not found in either AP or ML position time series alone. Whereas AP and ML each describe only one dimension of sway, planar Euclidean displacements provide complementary information regarding the fluctuations in sway across the two-dimensional support surface. Fig. 1 depicts each of these time series from the same example trial. 2.2.3. Detrended fluctuation analysis We analyzed each of the three postural fluctuation time series using detrended fluctuation analysis (DFA). DFA is an adaptation of traditional random-walk analysis that examines the growth of root mean square (RMS) fluctuations over the course of a time series while also controlling for nonstationarities due to drift (Peng et al., 1994). It should be noted that, although DFA bears conceptual relation to Collins and De Luca’s (1993, 1994) stabilogram diffusion analysis (Duarte & Zatsiorsky, 2000, 2001), there are important algorithmic differences (Delignières et al., 2011). The intent of DFA is to estimate a scaling exponent a indexing temporal correlations (Scafetta & Grigolini, 2002). The analysis begins with the integration of a time series x(t) into a random-walk trajectory y(t), as follows:

yðtÞ ¼

XN i¼1

xðiÞ  xðtÞ;

ð1Þ

where xðtÞ is the mean of x(t). DFA calculates RMS after removing local trends. Linear regressions yn(t) detrend non-overlapping n-length bins of y(t). Fluctuation F(n) is calculated as average root meansquare (RMS) error of these regressions for each n:

FðnÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ð1=NÞ ½yðtÞ  yn ðtÞ2

ð2Þ

typically for n < N=4. However, because there are relatively fewer bins for relatively larger bin sizes, F(n) can be unstable for larger n. So, DFA can be run more conservatively by limiting F(n) to n < N=10 (Damouras, Chang, Sejdic, & Chau, 2010; Hu et al., 2001). Although the error due to scaling region may be negligible for individual estimates of a on a given trial (e.g., Stephen & Hajnal, 2011), our strategy of generating multiple estimates of a (i.e., for multiple fluctuation time series) for the same trial would risk compounding this error. Hence, for present purposes, we limited our estimates of a to the more conservative scaling region, n < N=10. Over the scaling region, F(n) increases as

FðnÞ  na :

ð3Þ

Logarithmic scaling of Eq. (3) yields

log FðnÞ  a log n:

ð4Þ

The slope of F(n) in double-log plots is taken to estimate a. Scaling exponents a greater than .5 generally suggest the presence of temporal correlations. Temporally uncorrelated empirical time series generally have scaling exponents a equal to .5. However, it

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Fig. 1. Example postural-fluctuation time series from a single trial of an example participant. The top and middle panels depict postural sway in the mediolateral and anterior-posterior dimensions, respectively. The bottom panel depicts the corresponding time series of planar Euclidean displacements.

is possible for distributional anomalies to produce scaling exponents spuriously greater than .5 in the absence of temporal correlations (Bassler, Gunaratne, & McCauley, 2006). In order to guarantee that a sufficiently large a reflects temporal correlations, it is customary to run DFA on a copy of the original time series that contains the same values in randomized order. In theory, shuffled time series should have a = .5, but in practice, sampling error and/or departures from pure normality will lead to scaling exponents that only approximates .5. In any event, scaling exponents for temporally correlated time series should exceed scaling exponents for the shuffled copies. As noted above, DFA provided an estimate of a for each of the three postural-fluctuation time series on each trial. We will denote the estimates of a for AP position time series, ML position time series, and planar Euclidean displacement time series as aAP, aML, and aPED, respectively. 2.2.4. Growth curve modeling Our primary goal was to test whether the negative effects of subthreshold stimulation on postural-variability measures were moderated by the temporal correlation of the endogenous postural fluctuations (i.e., the estimated scaling exponents a). To accomplish this goal, we analyzed the postural-variability measures using growth curve models (GCMs), a longitudinal multiple regression

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technique designed to test the effect of time-varying predictors (Singer & Willett, 2003). Relevant to present purposes, GCM has proven to be well-suited for testing the time-varying effects of endogenous physiological fluctuations on a biological organism’s response to stimulation (Stephen & Anastas, 2011; Stephen & Hajnal, 2011; Stephen et al., 2009). The output of a GCM is largely similar to that of any OLS multiple regression techniques. That is, GCMs decompose a dependent measure in terms of a weighted sum of linearly separable predictors, and they return estimates of coefficients for each predictor. The crucial difference between a GCM and OLS regression techniques (e.g., RM ANOVA) lies in the different assumptions about the distribution of error across participants and over time. Whereas OLS estimation assumes equal variance over time and across participants, a GCM uses maximum likelihood (ML) estimation to fit random effects for individual differences across participants and over time. Whereas the effect of added predictors in OLS estimation is evaluated in terms of a change in ‘‘proportion of explained variance’’ (i.e., r-squared), ML estimation for a continuous dependent measure allows no absolute goodness-of-fit statistic and hence no reliable description of proportion of explained variance (Lyons, 2008). Instead, nested models can be evaluated based on the reduction of a 2 log likelihood (2 LL) deviance statistic. Improvement in model fit following the addition of m new parameters is evaluated in terms of 2 LL deviance, where change in 2 LL is tested as a chi-square statistic with m degrees of freedom. An important detail to note in the present work is that we will be using GCM to model the effect not just of experimental manipulations (i.e., stimulation from the insole) but also statistics on endogenous fluctuations (i.e., the scaling exponents a) to predict dependent measures of postural variability. That is, GCM will encompass not only exogenous predictors but also endogenous predictors. GCMs will be described in the rest of the report in terms of the highest-order interactions, with the understanding that all lower-order interactions and main effects are necessarily included concurrently to permit their standard interpretation. In order to test whether endogenous postural fluctuations moderated the negative effect of subthreshold vibratory fluctuations on postural variability, we modeled each postural-variability measure with the same set of predictors. The predictor representing insole stimulation (Stim) was coded as 1 for trials with subthreshold vibratory stimulation and 0 for trials with no subthreshold vibratory stimulation. Similarly, the predictor representing age group (Age) was coded 0 for young adult participants and 1 for elderly participants. The predictors representing endogenous postural fluctuations were the trial-by-trial values of aAP, aML, and aPED as estimated by DFA. Specifically, the highest-order term in Model 1 was StimAgeaAPaMLaPED, all constituent lower-order interactions and main effects thereof, and a main effect of Trial. The interaction StimAgeaAPaMLaPED and all constituent terms model the interaction of stimulation, age group, and temporal correlation of postural fluctuations on postural-variability. The main effect of Trial simply controls for effects of time spent in the task (e.g., fatigue).1 3. Results 3.1. Results from DFA of postural-fluctuation time series Before describing the GCMs of postural-variability measures, we devote this section to reviewing the scaling exponents estimated from DFA of the postural-fluctuation time series. We submitted three postural-fluctuation time series (see Section 2.2.2) from each of 20 and 10 trials from 15 young participants and 12 elderly participants to DFA, for a total of 1260 (i.e., 15  20  3 þ 12  10  3) scaling exponents. We present the means for each estimated scaling exponent aAP, aML, and aPED for the original postural-fluctuations time series and for shuffled copies of the original time series (Table 1). As noted above, we estimated scaling exponents using DFA only for the scaling region within a conservative bound, that is for bin sizes n < N=10 and we also estimated scaling exponents for the shuffled copies in order to determine that the scaling exponents were due to distributional anomalies. For all 1

Additional modeling suggested that interactions of Trial with the StimAgeaAPaMLaPED term did not significantly improve model fit. Interactions of Trial with other predictors may support improvements in model fit in tasks with closer to 100 trials (e.g., Blau, Stephen, Carello, & Turvey, 2009), but tasks with approximately 20 trials need only incorporate Trial in terms of a standalone covariate for time spent in the task (e.g., Stephen & Anastas, 2011).

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Table 1 Scaling exponents for original and shuffled time series.

aAP Original Shuffled

aML

aPED

M

SE

M

SE

M

SE

1.81 .50

.01 .01

1.78 .50

.01 .01

.85 .53

.01 .01

three postural-fluctuation time series, estimated scaling exponents a exceeded those estimated for the shuffled copies. Fig. 2 depicts fluctuation functions for each of the postural-fluctuation time series on logarithmically scaled axes.2 We wish to emphasize that the following analyses do not hinge on the observation of fractal scaling. Scaling exponents a for AP and ML position time series were greater than 1.5, which is consistent with some previous findings (Blázquez, Anguiano, de Saavedra, Lallena, & Carpena, 2009) but by no means all previous findings (Duarte & Sternad, 2008). Values of a for planar Euclidean displacements fell into the range traditionally associated with fractal scaling (i.e., :5 < a 6 1) suggesting that planar Euclidean displacement series are fractional Gaussian noise (fGn) and so that values of a for these series approximate the Hurst exponent (Eke et al., 2002). Although position series with a greater than 1.5 can sometimes be fractional Brownian motions, i.e., integrations of fGn whose scaling exponents correspond to H + 1. Consistent with observations by Delignières et al. (2011), however, DFA of the differenced AP and ML position series did not show straightforward fractal scaling, suggesting that there is no simple relation of the a for the position series and the Hurst exponent. In any event, the subsequent use of principal component analysis involves centering the distribution of all three scaling exponents. Subsequent growth curve modeling hinges neither upon any single actual value of the scaling exponents nor upon the categorically fractal or non-fractal nature of these fluctuations. What is at issue is how changes in estimates of temporal correlations predict changes in postural variability with and without stimulation. 3.1.1. Correcting for collinearity among predictors One more statistical step was needed to deal with collinearity of effects in the predictors and thus provide interpretable GCMs. In one sense, as suggested above in Section 2.2.2, the postural-fluctuation time series each carried substantial independent information about postural sway. In support of this suggestion, the simple pairwise correlations among the scaling exponents aAP, aML, and aPED were relatively weak (see Table 2). However, the introduction of these terms as main effects and as part of variously higher-order interactions into the same model produces strong collinearities. That is, even when main effects (e.g., scaling exponents aAP, aML, and aPED) are not themselves strongly correlated, predictors representing higher-order interactions may be correlated (Pinheiro & Bates, 2009). As with any other multiple regression technique, GCM estimates may be misleading when predictors are correlated. Efforts must be taken to orthogonalize those main effects participating in higher-order interactions. Model 1 (i.e., StimAgeaAPaMLaPED + Trial) tested the effects of interactions among subthreshold vibratory stimulation, age, and temporal correlation in postural fluctuation as well as an effect of trial on the magnitude of postural-variability measures. It included 31 predictors, but these predictors exhibited a preponderance of very high correlation with one another (i.e., r > .9). To overcome this collinearity among the predictors, we replaced the three scaling exponents aAP, aML, and aPED with the corresponding three principal components. This step dramatically attenuated the correlations among fixed effects, with correlations remaining relatively weak (i.e., approximately 90% of correlations with absolute-value of r < .25 and no correlations with absolute-value of r > .9). 2

Use of the conservative n < N=10 scaling region provided more reliable estimates of the scaling exponents. Linear fits of logarithmically-scaled fluctuation functions exhibited significantly higher R-squared statistics over the n < N=10 scaling region (for AP, ML, and PED time series, Ms. = .99, .99, and .99, SEs = .01, .01, and .01, respectively) than over the n < N=4 scaling region (Ms. = .98, .98, and .97, SEs = .01, .01, and .01, respectively), paired-samples t(419)s = 33.13, 33.34, and 14.23, ps < .0001.

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Fig. 2. Fluctuation functions from detrended fluctuation analysis of an example time series of mediolateral sway, anteriorposterior sway, and planar Euclidean displacements, from top to bottom. Fluctuation functions have been offset from one another by constants to facilitate visualization. Table 2 Correlation matrix for estimates of scaling exponents aAP, aML, and aPED.

aAP aML aPED

aAP

aML

aPED

1.00

.20 1.00

.30 .18 1.00

Table 3 shows the loadings for the principal components. These loadings indicated that aPED, aML, and aAP contributed most to PC1, PC2, and PC3 respectively. Reinforcing this point, we found strong correlations both between aPED and PC1, r(418) = .94, p < .0001; between aML and PC2, r(418) = .88, p < .0001; and between aAP and PC3, r(418) = .87, p < .0001.3 In replacing scaling exponents with principal components, we abandoned Model 1 in favor of Model 2, including PC1PC2PC3StimAge, all lower-order interactions and main effects thereof, and the main effect of Trial. Having addressed the correlations among predictors, we proceeded to analyze each postural-variability measure listed in Section 2.2.1. 3.2. Testing Hypothesis 1: effects of temporal correlations in postural fluctuations on postural-variability summary statistics 3.2.1. GCMs of postural-variability measures We analyzed each of the 11 postural-variability measures using GCMs containing the predictors outlined in Model 2 (see Section 3.1). Because Model 2 involves 31 predictors, the present work reports only the coefficients for Stim, Age, PC1 and PC1PC2Stim as these predictors were either most

Table 3 Loadings of scaling exponents aAP, aML, and aPED on PC1, PC2, and PC3.

aAP aML aPED

PC1

PC2

PC3

.28 .35 .89

.00 .92 .39

.96 .20 .23

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Table 4 Coefficients from Model 2 of 11 postural-variability measures for effects of Stim, Age, PC1, PC1PC2Stim. Stim RMean RMax AEllipse RangeML RangeAP RMSML RMSAP PathML PathAP Sway speed ASwept

Age **

.63 1.14* 52.50* .19 2.61** .19 .67** 2.05 3.02 .14 70.87*

.06 .68 72.70 1.38 1.35 .06 .15 38.28 42.42** 2.14* 182.67

PC1

PC1PC2Stim ***

9.19 17.66** 981.10**** 20.32** 29.56** 6.42*** 7.45** 90.48** 108.70*** 5.27*** 1139.54***

149.81* 351.97* 11348.40* 134.85 661.65** 39.20 189.42** 1668.97* 1487.72* 79.50* 18512.85*

*

Denote ps < .05. Denote ps < .01. *** Denote ps < .001. **** Denote ps < .0001. **

theoretically salient (i.e., Stim and Age) or exhibited significant effects most consistently across postural-variability measures (i.e., Stim, PC1, PC1PC2Stim).4 Significantly large coefficients for each of these predictors indicated significant effects of subthreshold vibratory stimulation, age group, the first principal component, and the interaction of the first two principal components of the scaling exponents and subthreshold vibratory stimulation. Table 4 reports the coefficients for these predictors for each of the 11 postural-variability measures. The first noteworthy aspect of the coefficients in Table 4 is the consistency of direction. For each postural-variability measure, stimulation (Stim) had a negative effect, age group (Age) had a positive effect, PC1 had a positive effect, and PC1PC2Stim had a negative effect. First, these consistencies confirmed earlier findings that subthreshold vibratory stimulation reduces postural variability, and second, they provided support for the hypothesis that changes in temporal correlation of postural fluctuations (represented in the model by the three principal components of the three scaling exponents aAP, aML, and aPED) moderate the effect of subthreshold vibratory stimulation. Specifically, the first principal component (PC1) generally contributed to postural variability, and the interaction of the first two principal components with subthreshold vibratory stimulation (PC1PC2Stim) generally reduced postural variability. However, effects supporting previous findings were less consistently significant: effects of Stim and Age were significant on only 6 (i.e., RMean, RMax, AEllipse, RangeAP, RMSAP, and ASwept) and 2 (i.e., PathAP and sway speed), respectively, of the 11 postural-variability measures. Meanwhile, effects implicating the role of temporal correlations in postural fluctuations were more consistently significant: effects of PC1 and PC1PC2Stim were significant on all 11 and 9 (i.e., RMean, RMax, AEllipse, RangeAP, RMSAP, PathML, PathAP, sway speed, and ASwept), respectively, of the 11 postural-variability measures.5 3.2.2. Interpretation of temporal-correlation effects from GCMs of postural-variability measures Essentially, regardless of the reduction in postural variability independently attributable to stimulation, the GCMs demonstrated two points regarding temporal correlations. First, increases in the first principal component (PC1) of the three scaling exponents predicts significantly greater postural 4

The authors are willing to share the complete 31-predictor models for any or all of the 11 postural-variability measures with anyone who is interested in these details. However, as they include a number of effects that were not intuitively interesting or consistently significant, we omitted them here. It should be noted that, for any predictors with significant effects on more than one postural-variability measure, the direction of these effects was consistent across models. 5 The size of coefficients should be understood both in terms of the scale of the dependent measures and also of the values taken by the predictor. For instance, the scale of ASwept was sufficiently large as to leave the larger Age coefficient (182.67) nonsignificant. The relatively large magnitude of the coefficients for PC1 and PC1PC2Stim reflected the fact that the values of the principal components were centered on their mean and thus had relatively small absolute values (i.e., <.28). Hence, the interaction (i.e., multiplication) of PC1 and PC2 has even smaller absolute values (i.e., <.03). So, the miniscule-valued principal-component terms require larger coefficients in order to constitute significantly large contributions to postural variability.

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variability. That is, temporal correlations in postural sway help to predict the magnitude of postural sway. Because all scaling exponents aAP, aML, and aPED had positive loadings on PC1, increased temporal correlations of all aspects of postural sway predict increased postural sway. However, the dominant loading on PC1 was aPED, and so the temporal correlations of planar Euclidean displacement may have played a slightly larger role than those of AP or ML position. Second, the significant PC1PC2Stim effect entailed that the interaction of the first two principal components (i.e., PC1PC2) moderated the negative effect of stimulation (Stim). When PC1 and PC2 were changed in same direction (i.e., with the same sign), the original effect of stimulation on postural variability is accentuated (i.e., made more negative). Conversely, when PC1 and PC2 were changed in opposite directions, the product of multiplying PC1 and PC2 is negative, and the original effect of stimulation on postural variability is diminished (i.e., made more positive). Because the strongest loadings on PC1 and PC2 came from aPED and aML, this result indicates that temporal correlations of planar Euclidean displacements and of mediolateral sway moderate the effect of stimulation.6

3.3. Testing Hypothesis 2: effects of age group and subthreshold vibratory stimulation on temporal correlations of postural fluctuations The effects of PC1 and PC1PC2Stim appeared to serve as more reliable predictors than either Stim or Age alone. Thus, despite the original findings of differences due to Stim or Age (e.g., Priplata et al., 2003), postural-variability measures may have depended more on the temporal correlations in postural fluctuations. To determine whether the traditional predictors Stim and Age bore any relationship to the more strongly predictive effects of PC1 and PC1PC2Stim, we ran a new GCM to test for effects of Stim and Age on PC1, PC2, and PC1PC2. Given the finding that temporal correlations appeared to influence postural variability directly and to moderate the effect of subthreshold vibratory stimulation, it was important to determine whether the temporal correlations themselves responded to any differences by Stim or Age. This GCM tested for differences in PC1, PC2, and PC1PC2 concurrently so as to control for the relationship among them, using a class variable to distinguish significant differences specific to each three dependent variables. The only significant effects were a positive effect of Age on PC1 (B = .06, SE = .02, p < .01) and a negative effect of Stim on PC1 (B = .014, SE = .007, p < .05). Age thus increased values of PC1, and Stim decreased values of PC1. The correlations between scaling exponents to principal components suggest that, in terms of the original scaling exponents, Age and Stim should exert these effects on aPED (Table 3). Alternate modeling confirms this point,7 demonstrating that Age serves to increase aPED and that Stim serves to decrease aPED.8 So, Age and Stim 6

Because PC1 and PC2 were strongly correlated with aPED and aML, respectively (see Section 3.1.1), it is possible to rephrase the foregoing principal-component effects in terms of the scaling exponents aPED and aML. Specifically, the positive main effect of PC1 on postural-variability measures suggested that postural variability increases with aPED. That is, sway increases with stronger temporal correlation in the planar Euclidean displacements of posture. Also, the negative interaction effect of PC1PC2Stim on postural-variability measures suggested that the effect of Stim on postural variability can be positive or negative depending on whether aPED and aML change in similar or different directions, respectively. That is, when temporal correlations in planar Euclidean displacement and in mediolateral position both increase or decrease, the effect of subthreshold vibratory stimulation is to reduce sway. On the other hand, when temporal correlations both change in opposite directions, the effect of subthreshold vibratory stimulation is to increase sway. In order to ensure that these significant effects were not spuriously due to the transformation of scaling exponents into principal components, we ran simpler alternate models involving only those scaling exponents loading most strongly on PC1 and PC2 (i.e., aPED and aML) and confirmed positive effects for aPED and negative effects for Stim aPEDaML throughout, ps < .05. 7 We ran a GCM to test for differences in the scaling exponent most strongly correlated with PC1, that is, aPED. There was only a significant positive effect of Age on aPED (B = .05, SE = .017, p < .01) and a moderately significant negative effect of Stim on aPED (B = .01, SE = .006, p = .07). This finding provided partial confirmation of the relationship between the PC1 and aPED, but the slightly weaker significance for the negative effect of Stim on aPED than for the negative effect of Stim on aPED suggested that the PC1 may not be entirely equated with aPED. Subthreshold vibratory stimulation serves to reduce temporal correlations in postural fluctuations. However, it reduces a weighted sum of the three scaling exponents (i.e. PC1), with aPED weighted more heavily than the other two. 8 Alternate modeling testing for simple Age effects in individual scaling exponents aAP and aML demonstrated that both of these scaling exponents decrease in healthy elderly participants as previously found (Duarte & Sternad, 2008; Lin, Deol, Nussbaum, & Madigan, 2008), but these differences were not significant.

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may strengthen and weaken, respectively, the temporal correlations in the planar Euclidean displacements of posture.

4. Discussion We had posed two hypotheses. First, we hypothesized that the effects of subthreshold vibratory stimulation on measures of postural-variability would significantly interact with temporal correlations of postural fluctuations. That is, we expected that the effect of fluctuations exogenous to posture (i.e., stimulation) would be moderated by fluctuations endogenous to posture. Second, we hypothesized that subthreshold vibratory stimulation would serve to diminish temporal correlations in postural fluctuations. The results were consistent with these hypotheses. Stochastic resonance may be an important conceptual framework for the development of clinical interventions. The original finding by Priplata et al. (2003) that subthreshold vibratory stimulation on the plantar surface of the foot stabilized posture was an important existence proof. The present reanalysis may point to a twofold elaboration of the original finding that, in the first place, offers a deeper understanding of the clinical efficacy of this plantar-surface stimulation using vibrating insoles and, in the second place, suggests a possible means of tailoring vibrating insoles to individual patients. The clinical efficacy of plantar-surface stimulation with vibrating insoles may depend on the interaction of fluctuations. Though stochastic resonance, that is, the application of fluctuations to stabilize a signal, seems counterintuitive, it goes hand in hand with the notion that fluctuations organized at multiple scales are crucial for the subtle coordination of various physiological processes (Ivanov et al., 2001; Johnston & Edwards, 2002; Stephen & Hajnal, 2011; West, 2006). For instance, fluctuations in heartbeat dynamics interact with fluctuations in respiratory dynamics so as to coordinate the oxygenation of blood flowing through the human body. The effect of an intervention on a biological system must similarly be the product of the interactions between the intervention and fluctuations endogenous to the biological system (e.g., Stephen & Hajnal, 2011). We thus propose that the effects of fluctuations from vibrating insoles are moderated by the fluctuations in posture. Thus, a nonlinear relationship unfolds between postural system and vibrating insole in which the effect of the intervention on the physiological system is contingent on the activity of the very same physiological system itself. We propose that the efficacy of vibrating insoles for stabilizing posture lies specifically in a compromise between the temporal correlations of intervention on one hand and of physiological fluctuations on the other. Unlike many other clinical interventions, the vibrating insoles were explicitly designed to generate fluctuations with a specific degree of temporal correlation (specifically, zero correlation). Because physiological fluctuations are known to be temporally correlated (Ivanov et al., 2001; Stephen et al., 2010; West, 2006), temporal correlations may serve as a common currency in which to understand the relationship between the vibrating insole and postural system. Interestingly for present considerations, it is known that temporal correlations in biological systems can vary widely across the lifespan. Biological systems usually fare best when fluctuations are temporally correlated but not excessively so. Some evidence suggests that fractal (‘‘1/f’’) fluctuations (sometimes called ‘‘pink noise’’) are ideal in that they reflect a power-law balance between more random, uncorrelated fluctuations (i.e., ‘‘white noise’’) and more correlated fluctuations (‘‘Brownian noise’’; Van Orden, 2010; Van Orden, Kloos, & Wallot, 2011). The present results suggest that postural sway and vibrating insoles may reflect a similar kind of balance. The present results suggest, first, that increases in postural-variability measures (e.g., root mean square in anterior-posterior or in mediolateral directions; see Section 2.2.1) are associated with increases in temporal correlation of planar Euclidean displacements in posture fluctuations (i.e., the strongest loading on PC1; Table 3). Second, these results suggest that an interaction between temporal correlations of planar Euclidean displacements and temporal correlations of mediolateral position moderates the effect of the white-noise (i.e., temporally uncorrelated) insole vibrations. Third, these results suggest that the elderly exhibit an increase in temporal correlations. This finding of heightened temporal correlations in the elderly is consistent with observations by Thurner et al. (2002). Last, these results suggest stimulation by vibrating insoles may serve to diminish the same temporal correlations.

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Generally speaking, this result may reflect a tendency for coupling to bring about comparable complexity between coupled systems (West, Geneston, & Grigolini, 2008), as has been previously found in perceptual-motor coordination (Marmelat & Delignières, 2012; Stephen & Dixon, 2011; Stephen, Stepp, Dixon, & Turvey, 2008). More specifically, this effect is strikingly similar to what Thurner et al. proposed in their stochastic-resonance description of postural control. Important for optimizing the clinical value of vibrating insoles may be the recognition that stimulation with different degrees of temporal correlation may be suitable for different systems on different time scales. That is, one kind of noise may not suit all patients. The vibrating insoles used by Priplata et al. (2003) produce a white noise (i.e., uncorrelated) signal, but other physiological systems may benefit more from pink noise than from either white or Brownian noise (e.g., Linkenkaer-Hansen, Nikulin, Palva, Kaila, & Ilmoniemi, 2004; Nozaki, Collins, & Yamamoto, 1999). It is even the case that anterior-posterior postural sway in the healthy elderly can sometimes, in more prolonged standing than reported here, exhibit weaker temporal correlations than that in healthy young adults (Duarte & Sternad, 2008; Lin, Deol, Nussbaum, & Madigan, 2008). If the efficacy of stochastic-resonance interventions depends on balancing postural fluctuations out to match the temporal correlation of healthy young adults, then there will be no single effective kind of noise. The efficacy of applying noise may depend both on the desired time scale of the intervention and the specific excursion from 1/f noise exhibited by less stable, more variable posture. That is, the identification of what may be the ‘‘healthy’’ steady-state may depend on whether intervention should unfold slowly over long time scales or quickly over short time scales. The results of the present reanalysis speak only to the case of relatively short-term standing: applying white-noise stimulation through vibrating insoles may be most effective for stabilizing posture whose fluctuations are significantly more temporally correlated than those for young adults. Further, this conclusion implicates the temporal correlations for planar Euclidean displacements more centrally than those for anterior-posterior sway. These results motivate future directions in both the empirical investigation of stochastic resonance in postural control and the development of vibrating insoles for stabilizing posture. Further research might investigate these effects of vibrating insoles in prolonged standing to test whether they still provide a stabilizing effect and whether this effect is moderated by the same temporal correlations (i.e., those in planar Euclidean displacements). Further developments of the vibrating insole technology might explore the capability for tailoring the temporal correlations in the vibratory stimulation to ongoing changes in the temporal correlations in postural fluctuations. Perhaps sensors and processors embedded in the insole could measure postural fluctuations and calculate temporal correlations, respectively, during a daily period of quiet stance. If the insoles had direct access to temporal correlations in postural fluctuations, then it might be possible to automatically adjust the insole’s vibratory signal to a noise with compensatory temporal correlations. For instance, as postural fluctuations become more or less temporally correlated than young adults under similar conditions, the insole’s actuators might be automatically adjusted to generate less or more temporally correlated noise. That is, periodic assessment of temporal correlations in posture could help tailor the stochastic-resonance intervention to current changes in the postural system. Such tailoring could make vibrating insoles a more effective intervention for a wider range of patients, and more broadly, such work might further the understanding of how fluctuations support biological function.

Appendix A Usually, for PCA of p-dimensional datasets, it is customary to exclude all principal components that capture less than 1/p of the variability. However, alternative modeling of the posture-variabilty measures highlighted limitations of this rule of thumb. There were no clear indications that exclusion of principal components was either appropriate or helpful for demonstrating an effect of postural fluctuations moderating the effect of subthreshold vibratory stimulation. This section will discuss the entailments of removing the second and third principal components. Because it explains only 15.95%, PC3 would normally be on the chopping block. However, GCMs of the postural-variability measures outlined in Section 2.2.1 demonstrated that inclusion of PC3 and its

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interactions with PC1PC2StimAge and all lower-order interactions and main effects thereof (i.e., 16 terms) significantly improved model fit for ML range, RMSML, AE, sway speed, PathAP, and PathML indicated by significant changes in 2 LL, v2s(16) = 29.96, 30.71, 36.68, 39.28, 41.12, and 36.54, ps < .05, .05, .01, .01, .001, and .01 respectively. Excluding PC3 from the GCM for AE left the significant effect of PC1, Stim, and PC1PC2 on AE unchanged but left the interaction of PC1PC2Stim only marginally significant (p = .06). Similarly, significant effects on ML range and RMSML (not involving PC3) shown in Section 3.2.1 were unchanged when PC3 was excluded. Excluding PC3 from GCMs of the other postural-varaibility measures left all effects of PC1, Stim, PC1PC2, and PC1PC2Stim unchanged from those shown in Section 3.2.1. Capturing only 29.41% of the variability, PC2 is on the cusp of warranting exclusion but still manages to carry important predictive weight in the GCMs. Including PC2 and its interactions with PC1StimAge and all lower-order interactions and main effects thereof (i.e., 8 terms) significantly improved fit for models of all postural-variability measures except AP range and RMSAP, for which the reduction of 2 LL was relatively small, v2(8) = 9.88 and 7.79, ps = .27 and .45, respectively. For models of both AP range and RMSAP, excluding PC2 (i.e., while also excluding PC3) left effects of PC1 and Stim unchanged from those shown in Section 3.2.1 but eliminated significant effects of PC1PC2 and, more crucially, PC1PC2Stim. In general, these remarks illustrate the point that changes in model fit can provide different information from the coefficients for individual terms. 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