A time–frequency approach to estimate critical time intervals in postural control

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A time–frequency approach to estimate critical time intervals in postural control a

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Hongbo Zhang , Maury A. Nussbaum & Michael J. Agnew a

Department of Industrial and Systems Engineering, Virginia Tech, 250 Durham Hall (0118), Blacksburg, VA 24061, USA Published online: 08 Aug 2014.

To cite this article: Hongbo Zhang, Maury A. Nussbaum & Michael J. Agnew (2014): A time–frequency approach to estimate critical time intervals in postural control, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2014.946915 To link to this article: http://dx.doi.org/10.1080/10255842.2014.946915

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Computer Methods in Biomechanics and Biomedical Engineering, 2014 http://dx.doi.org/10.1080/10255842.2014.946915

A time – frequency approach to estimate critical time intervals in postural control Hongbo Zhang, Maury A. Nussbaum* and Michael J. Agnew Department of Industrial and Systems Engineering, Virginia Tech, 250 Durham Hall (0118), Blacksburg, VA 24061, USA

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(Received 4 October 2013; accepted 17 July 2014) The critical time interval (CTI) is a parameter that has been used to distinguish open-loop from closed-loop control during upright stance. The aim of this study was to develop a new method to determine CTIs. The new approach, termed the intermittent critical time interval (ICTI) method, was motivated from evidence that upright standing is an intermittent rather than an asymptotic stability control process. For this ICTI method, center-of-pressure time series are first transformed to the time – frequency domain with a wavelet method. Subsequently, the CTI is assumed equal to the time span between two local maxima in the time – frequency domain within a distinct frequency band (i.e., 0.5 –1.1 Hz). This new method may help facilitate better estimates of the transition time interval between open and closed-loop control during upright stance and can also be applied in future work such as in simulating postural control. In addition, this method can be used in future work to assess temporal changes in CTIs. Keywords: intermittent postural control; critical time interval; wavelet transform

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Introduction

Maintenance of upright stance is often considered as an asymptotically stable control process (Newman et al. 1996; Morasso and Schieppati 1999; Peterka 2000, 2002; Masani et al. 2003; Peterka and Loughlin 2004), for which optimization and PID feedback control have been used as the basis of simulation models. This traditional view has been challenged by some recent studies (Jacono et al. 2004; Bottaro et al. 2005; Milton et al. 2009; Loram et al. 2011), which have suggested that an intermittent control process is involved. Intermittent stabilization control differs from asymptotic control by recognizing that the equilibrium position of upright standing is unstable, and also by suggesting that there are numerous (rather than one) equilibrium positions. Evidence such as the biphasic ‘throw and catch’ pattern of ankle torque (Loram and Lakie 2002) and the ‘rambling and trembling’ of sway trajectories (Zatsiorsky and Duarte 2000) support the assumption that body movement is intermittently controlled (King and Zatsiorski 1997). These findings imply that both open-loop and closed-loop control co-exist during intermittent motion (Loram and Lakie 2002; Loram et al. 2006). A relevant issue then is identifying the transition time interval between open and closed-loop control, which is an important control parameter for understanding and/or modeling human balance (Collins and De Luca 1993, 1994; Jacono et al. 2004; Bottaro et al. 2005; Qu et al. 2007; Milton et al. 2009; Loram et al. 2011). The stabilogram diffusion function (SDF), proposed by Collins and De Luca (1993, 1994) and adopted in

*Corresponding author. Email: [email protected] q 2014 Taylor & Francis

numerous subsequent studies (Mitchell et al. 1995; Newman et al. 1996; Newell et al. 1997), is a method suited for differentiating between open and closed-loop control of upright stance. It is based on the relationship between mean square displacements kDr 2 l and time intervals Dt, denoted by kDr 2 l ¼ 2DDt, where D is the diffusion coefficient. Both short-term and long-term diffusion coefficients (DS and DL, respectively) are determined as half of the slope of linear fits to respective regions. Between these regions, a ‘critical point”’ is defined by the time interval and the SDF coefficient at the intersection of the shortterm and long-term fits. Associated with this critical point is the critical time interval (CTI), which is used to distinguish between a short-term diffusion region and a long-term diffusion region. Collins and De Luca (1993, 1994) suggested that open and closed-loop control strategies are associated with the short-term and longterm diffusion regions, respectively, and such an interpretation has been made by others (Priplata et al. 2002). As such, the CTI is the transition time interval from open-loop to closed-loop control. While a CTI identification algorithm was not given explicitly by Collins and De Luca (1993, 1994), it can be found elsewhere (Stamp 1997). This method for CTI identification specifies that the CTI is the minimum of the second derivative of diffusion data over time intervals up to 2.5 s. Further, it was based on the assumption that the second derivative of diffusion data is proportional to the acceleration of sway. As such, the local minima of sway acceleration possibly correlate with equilibrium positions

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at which there is a transition from open-loop to closedloop control. The validity of the CTI identification method was questioned by Newell et al. (1997), who noted that the CTI among older adults should be shorter than among young adults, yet the traditional CTI method yielded the opposite. Several studies have sought to enhance SDF by more accurately identifying the CTI (Chiari et al. 2000; Norris et al. 2005). Yet, these appear to lack a strong physiological basis. All SDF-based methods, however, remain based on an assumption of continuous postural control, and as such may not capture the time interval occurring intermittently. It is also unclear whether SDFbased methods, which were optimized for specific experimental conditions, are generalizable. It is reasonable to assume that certain times during sway (or certain postures) correspond to the termination of closed-loop control and the beginning of an open-loop control process. When postural sway moves toward an equilibrium (upright) position, there is diminished muscle stretch, such as the ankle (Loram et al. 2007; Di Giulio et al. 2009), which in turn can lead to weaker feedback signals (e.g., angular position and/or velocity) and less likelihood of use of feedback signals. Previous work, using both postural control modeling (Asai et al. 2009) and measures of muscle activity (Asai et al. 2013), has suggested that within the stable manifold (a space associated with stable posture), it is likely that the postural controller adopts an open-loop control mechanism (i.e., without use of feedback). Switching from open-loop to closed-loop control is dependent on the availability of feedback signals. In the case of upright stance, feedback signals are often initiated when postural sway is near a sway singularity, or a local maximum (Loram and Lakie 2002; Bottaro et al. 2008). The occurrence of local maxima (especially spikes, for example the overshoot in control theory) often indicates that the performance of the underlying control mechanism is inefficient, and that a more effective (e.g., feedback-based) control mechanism is needed. It can thus be postulated that local maxima in postural sway represent a switching point between openloop and closed-loop control. Furthermore, the time intervals between such local maxima are an alternative approach to characterizing CTIs in postural control. The purpose of this study was to develop a new CTI algorithm to quantify the transition time between openand closed-loop control of upright stance. Our approach is based on the assumption that the time interval between two local postural maxima presents the CTI, and that this CTI derivation can be used to identify transitory periods between open and closed-loop control. A wavelet modulus maximum (MM) method was used to locate the local maxima in center-of-pressure time series during periods of quiet upright stance (Mallat and Hwang 1992), and a new method was developed to quantify the time intervals between these extracted local maxima.

2. Methods 2.1. Participants and data Data from an earlier experiment (Lin et al. 2008) were used here, specifically that from 16 young adults (gender balanced). A summary of the methods is given below, with additional details available in the noted publication. All participants gave informed consent as approved by the Virginia Tech Institutional Review Board and had no selfreported injuries, illness, musculoskeletal disorders, or falls in the year prior to the experiment. Participants completed repeated trials of quiet upright stance. During all trials, participants were instructed to stand (without shoes) as still as possible, with their feet together, arms by their sides, head upright, and eyes closed. Each trial lasted 75 s, with at least one minute between each. Participants stood on a force platform (AMTI OR6-7-1000, MA, USA), from which tri-axial ground reaction forces and moments were sampled at 100 Hz. Raw signals were low-pass filtered (Butterworth, 5 Hz cut-off frequency, fourth order, zero lag), transformed to obtain center-of-pressure (COP) time series in the anteroposterior and mediolateral directions. These times series were demeaned, and the first 10 s and the last 5 s were removed. For use in the current analysis, a subset of data from the noted experiment was extracted from 16 young adults (gender balanced). For each participant, three trials were used and that were obtained consecutively on a single day.

2.2. Intermittent critical time interval 2.2.1. Wavelet transform Diverse postural control mechanisms exist, including vision, vestibular, proprioceptive, and CNS function. However, these mechanisms appear to act on different time scales, as reflected in distinct frequencies of COP time series (Thurner et al. 2000; Singh et al. 2012). As such, a frequency analysis should be able to capture changes in postural control. Here, a wavelet approach was used for identifying CTI, as this method can capture temporal as well as frequency aspects. For brevity, the methods and results obtained using the ICTI methods are presented only for anteroposterior COP data. The general wavelet transform is: 21=2

WTða; bÞ ¼ jaj

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where w is a wavelet function, b is the translation parameter, a is the scale (frequency), f ðtÞ is a COP time series, and WTða; bÞ is the wavelet coefficient. To obtain an integral multiple of two samples, as required by the wavelet transform, only the first 40.96 s (of 60 s) data were used. The Coiflets wavelet function was used, based on preliminary analysis that indicated it was the most

Computer Methods in Biomechanics and Biomedical Engineering effective at reducing low-frequency distortion in COP data. Wavelet coefficients in the 0.5– 1.1 Hz frequency band were identified and used in subsequent analyses (see Figure 1 for results for a representative trial). This

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particular frequency range was selected based on earlier work indicating that somatosensory information is critical for postural control (Mitchell et al. 1995; Masani et al. 2003) and that somatosensory activity is associated with COP content in this frequency band (Thurner et al. 2000).

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Mathematically, a local maximum of the wavelet coefficients is also denoted as a local singularity. Features of complex data such as the contour of objects in medical images and audio signal noise can be detected through the local singularity (Mallat 1999). Mallat and Hwang (1992) indicated that the MM wavelet transform is particularly well suited to estimate the local maxima of an underlying signal. Furthermore, the original signal can be reconstructed based on the identified singularities, indicating the completeness of the singularities that can be extracted from the original signal and that the singularities are thus representative of the original signal (Mallat and Hwang 1992; Liew and Law 2000; Ayat et al. 2009). Within the MM wavelet transform, the wavelet power spectrum was adopted to detect the MM of the power spectra, where the power spectrum PSða; bÞ is equal to the square of wavelet coefficients:

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Across all trials, and similar to the pattern in Figure 1, wavelet coefficient magnitudes varied over time for a given frequency, though the magnitudes of the coefficients were similar over the 0.5 – 1.1 Hz range at specific times. Overall, the wavelet coefficients suggest a curtain like pattern (Figure 1(B)), consisting of numerous local maxima of the wavelet coefficients at specific times. These local maximum were further identified using a MM algorithm, described below, which yielded discrete points in the time – frequency domain that corresponded to local maxima in both the COP signal and the wavelet coefficients (Figure 1(A),(B)).

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Figure 1. Summary of the process used to identify wavelet MMs. COP time series (A, shown for the AP direction) are used to calculate the absolute values of wavelet coefficients of the COP signal and then calculate the wavelet power spectrum (B) based on the wavelet coefficients. Finally, the MMs (C) are identified based on the wavelet power spectra. Here, a single local minimum (LM) in the COP time series is highlighted at time t, along with the associated wavelet coefficients (WC) and modulus maxima (MM).

Wavelet MM: method and sensitivity analysis

To identify the MMs, for each distinct time (t), the wavelet power spectrum was scanned over 0.5– 1.1 Hz. At a given time t, and for a certain frequency, the local wavelet power spectrum was compared to neighboring regions, specifically over ½t 2 DtMM ; t þ DtMM
, where DtMM describes the time interval used (Figure 2). This comparison process was used to identify the local maxima, or the MM of the wavelet power spectrum. After identifying the MMs, the MMs were also compared to the global maximum (GM) of the wavelet power spectrum, to reduce potential effects of

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H. Zhang et al. for all 48 trails using a range of searching time intervals. The ratio declined slightly with the size of the interval (Figure 4). As only small changes of the ratio were evident with increases from 30 to 40 ms, the interval of 30 ms was chosen for subsequent analyses.

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Figure 2. Summary of the two criteria used for identifying MMs. The first is the MM searching region (DtMM) and the second is global maximum used to eliminate the too small MM. Power spectrum wavelet coefficients (in AP direction) are shown as a function of time, at a frequency of 0.8 Hz. Green dots indicate MMs that satisfy both criteria. Red dots indicate MMs eliminated through the MM identification process, due to an insufficient magnitude relative to the global maximum. (Color online)

noise or very small magnitudes of the wavelet power spectrum (Figure 2). Specifically, MMs were identified as . GM/constant, where different values of the constant can affect the number of identified MMs (e.g., a larger constant results in a smaller number of identified MMs). To assess the sensitivity of the MM to the constant specified, a range of values were tested (Figure 3). For a sample trial, it is evident that the number of identified MMs increases with the value of the constant (Figure 3 (A)). This increase, however, appeared nonlinear (i.e., larger effects of increasing constant from 2 to 10 vs. from 10 to 21). To assess this effect more generally, the total numbers of identified MMs were obtained in each trial (across the 0.5 –1.1 Hz band) for a range of constant values, and a ratio was then determined by dividing the number of maxima by trial time (40.96 s). Across all 48 trials, this ratio increased nonlinearly with constant (Figure 4). As increases in the ratio beyond , 21 were quite small, a constant ¼ 21 was maintained for subsequent analyses. With respect to the influence of the local searching region, DtMM , the number of identified MMs were only minimally affected by changes of DtMM from 10 to 30 ms (Figure 3(B)). This suggests that the method is relatively insensitive to the size of local searching region. To again assess this more generally, the MM ratio was determined

2.2.4. Wavelet MML The MM of the wavelet power spectrum appeared to be aligned vertically (across frequencies) and to appear intermittently over time (Figure 3). This suggests intermittent behaviors underlying postural control, and as such, there was particular interest in quantifying the time intervals between these vertical lines (thereby yielding the desired CTIs). However, the MMs identified from the earlier-noted procedure are discrete points, and lines need to be formally constructed before such time intervals can be determined. For this, a MM line (MML) construction method was developed. A MML at time t (MMLt) was defined as a line consisting of the MM wavelet power spectrum points at time t across the frequency band from f 1 to f 2 . This can be described by d

MMLðf ; tÞ¼fMMðf ;tÞ; t ¼ t min ; .. . ;t max ; f ¼ f 1 ; .. .; f 2 }: ð3Þ s.t. jt max 2 t min j # 2DtMML ; where f is 0.5– 1.1 Hz and t is between 0 and 40.96 s; and MMLðf ; tÞ is a MML consisting of MM MMðf ; tÞ. Specifically, for time t, the MML is restricted within a line searching time interval of 2DtMML . The steps involved in MML construction and ICTI identification are shown in Figure 5. Initially, the algorithm searched for MMs and then attempted to identify the MMLs. For a specific time t, the search scanned each frequency in the 0.5 – 1.1 Hz band (starting at the higher end) and marked the positions of the MMs in the region of ½t 2 DtMML ; t þ DtMML
. Within this region, the mean across identified MMs at a given frequency was treated as one data point that belonged to the MML at time t. After scanning all frequencies in the noted band, an array of data points were identified, and then the mean value of these points was regarded as the position (time) of the MML at time t. This scanning and identification process was repeated for all times t in a trial 0 to 40.96 s and yielded a set of MMLs. Subsequently, the time intervals between two consecutive MMLs were determined (as CTIs). As the MMLs appeared intermittently (Figure 5), these time intervals are termed ICTIs. Of note, some MMLs did not cover a larger proportion of the 0.5 – 1.1 Hz frequency band (Figure 5). To account for these, a length ratio was determined for

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Figure 3. Illustration of MM sensitivity to the specifications of the constant ratio (see text) and the local searching region (DtMML), in the left and right columns respectively. These results are for the same representative trial as shown earlier (Figure 1), in the AP direction.

each MML, as the proportion of the entire frequency band that was spanned by a given line. This length ratio was then compared to length threshold, and a given MML was excluded if this threshold was not exceeded. For a given trial, a final ICTI was obtained at the mean across all identified time intervals between non-excluded MMLs.

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Sensitivity of wavelet MML

To identify the influences of length ratio and local searching region (DtMML), MMLs and ICTIs were obtained using different values of each. With increasing length ratio, changes in resulting MMLs were clearly evident, with fewer MMLs identified and those MMLs

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spanned more of the frequency band (Figure 6). Larger length ratios also led to increased values of ICTI. Using a length ratio of 60%, changes in identified MMLs with DtMML are shown in Figure 7. With DtMML between 110 and 310 ms, MMLs were not uniform in terms of the time intervals between consecutive MMLs. When DtMML increased to 390 ms, the ICTIs became more uniformly distributed. However, with DtMML increased further to 430 ms, the ICTI slightly decreased, and some overlaps between MMLs began to appear.

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Figure 4. Sensitivity of MM identification to the constant ratio (left) and the local searching region (DtMML, right). Results are means of the MM ratio for all COP time series (n ¼ 48), with error bars indicating standard deviations.

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The influences of length ratio and DtMML on ICTI were tested across all 48 trials, by separately assessing the effects of a range of respective values on ICTIs. ICTIs increased nonlinearly with length ratio (Figure 8). The distribution of MMLs, however, becomes more nonuniform when length ratio increased, especially with the length ratio . 60% (as in Figure 6). As such, a length ratio of 60% was chosen as the final recommended value. ICTI decreased nonlinear with DtMML (Figure 8), particularly when less than 300 ms. There was a relatively smaller effect of DtMML on ICTI when the former was between 350 and 450 ms. Given this, and as larger values of DtMML, will lead to overlap between local searching regions and MMLs (Figure 7), 390 ms was chosen as the final recommended value.

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Figure 5. Summary of the MML construction process. As described in the text, MMLs are constructed using two criteria – the local searching region (DtMML) and the length ratio – and ICTIs are obtained between adjacent MMLs. Here, a subset of a trial (as in Figure 1) is illustrated, in which three ICTIs are derived. The symbol £ denotes MMLs eliminated as p not meeting the length ratio threshold, whereas the symbol indicates those MMLs both within the local searching region and exceeding the length ratio threshold. Red vertical lines indicate the identified MMLs, and red dots indicate the positions of the identified MMLs. (Color online)

Discussion

A new intermittent CTI identification method (ICTI) was created, based on recent experimental findings of intermittent muscular control of quiet upright stance (Loram et al. 2005, 2006), and which does not assume one global equilibrium position. In this method, a COP time series is split into multiple equilibrium positions, thereby inherently reflecting the intermittent nature of the control mechanism(s) involved during quiet upright stance. The ICTI method is driven by the underlying hypothesis that local maxima of wavelet power spectra of the COP data distinguish the transitory nature of intermittent control (Loram and Lakie 2002; Gawthrop et al. 2009; Asai et al. 2013). In the original work of Collins and De Luca (1993), respective mean (SD) CTIs of 1.07 (0.55) and 1.05 (0.32) s in the anterioposterior (AP) and mediolateral (ML) directions were reported. Subsequent work, using a similar identification method, indicated CTIs of 1.08 (0.44) s (Chiari et al. 2000), while Norris et al. (2005) reported AP

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and ML values of 1.12 (0.49) and 1.34 (0.51) s, respectively. In this study, overall AP and ML ICTIs m were 1.13 (0.19) s and 1.09 (0.16) s (using the final values of length ratio ¼ 60% and DtMML ¼ 390 ms). ICTIs in the

current work are thus comparable in both directions, not surprising given that all studies used similar experimental protocols (e.g. quiet upright standing) and participant samples (e.g. healthy individuals).

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Figure 7. Illustration of the influence of the local searching region (DtMML) on MML and ICTI for a COP trial (same trial shown in Figure 1). Note that small red dots indicate identified MMLs. (Color online)

Computer Methods in Biomechanics and Biomedical Engineering 2.5 Intermittent Critical Time Interval (Sec)

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Although all these CTI results have similar magnitude, CTIs derived from the current approach have smaller standard deviations, which may suggest improved reliability and/or precision (though the difference may instead be related to differences in the variability of

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Figure 8. Sensitivity of ICTIs to the length ratio (left) and local searching region (DtMML, right). Results are means of ICTIs for all COP time series (n ¼ 48), with error bars indicating standard deviations.

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Figure 9. Illustration of differences in derived ICTI using COP position (A), COP velocity (B), and COP acceleration (C) for a COP trial (same trial shown in Figure 1). (Color online)

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In contrast, the new CTI method scans and searches the signal in a temporal-space domain and implicitly recognizes that posture is intermittently controlled. It is thus possible for the new CTI method to be used in more general postural conditions, for example to identify changes in the CTIs such as due to prolonged or irregular patterns of stimulus and perturbations. Whether this new CTI identification can detect such time-dependent changes, however, still requires formal investigation. A potential limitation of the current approach is that equilibrium positions are assumed to exist between two local maxima in the wavelet power spectra of COP data, which means its treats sway displacement as the feedback signal of interest. However, sway velocity may also serve as an important feedback signal (Jeka et al. 2004; Qu et al. 2007). This raises the question whether the identification of equilibrium positions should be based on raw COP (positional) data, or a derivative of a given order (i.e., COP velocity). To assess this, we explored using the current approach with COP derivatives (representative example shown in Figure 9 for COP position and velocity). ICTIs derived using COP velocity were somewhat shorter (by 100 –300 ms) than those using COP position. When using COP acceleration, in contrast, ICTIs were substantially shorter, and on the order of 500 –800 ms. This latter result is of note, given recent evidence (Loram et al. 2005) that muscular impulses during quiet stance are generated by the gastrocnemius at a frequency of 2– 3 Hz, comparable to ICTIs obtained from COP acceleration. The use of COP velocity or acceleration to calculate ICTIs may thus be most appropriate for simulating or modeling upright stance with an intermittent control method (Bottaro et al. 2008), where the time interval between neural control signal ‘bursts’ is an important parameter. Another potential limitation of the new CTI method stems from delays due to mechanical and kinematics factors (such as gravity and body segment inertia). Such delays could lead to CTIs derived from COP data being longer than the actual transitions between muscle contractions. The current CTI method shares similarities with the conventional CTI method, in that both methods target distinguishing transitions of the postural control mechanisms involved during quiet upright stance. Yet, there are distinctions between these approaches. The conventional CTI method was proposed to separate the transition between open and closed-loop control, whereas the new CTI method, as noted above, identifies the transitory time intervals involved in intermittent control. As such, the new CTI method may have benefit in future research, yielding a quantitative measure to evaluate the intermittent postural control mechanism(s) involved in quiet upright stance. There are some specific potential benefits of the new method, but admittedly these are mainly speculative at this point. First, the new CTI method we describe derives from evidence of intermittent postural control. With this

method, it seems possible, as noted earlier, to identify the transitions or intervals between intermittent (or corrective) control signals. Second, the new CTI method is able to target specific frequency bands within the COP signal in the process of identifying the CTI. In our current work, we have emphasized the 0.5– 1.1 Hz frequency range as this band is associated with specific postural control mechanisms (Collins and De Luca 1994). Using this approach, and perhaps with other/additional frequency bands, the new CTI method may be a more sensitive and/ or reliable method for assessing several intra and interindividual differences, such as the effects of age, fatigue, proprioception, vision, etc. on postural control. However, future work is clearly needed to assess such potential benefits. In summary, this work provides a new method that can be used to calculate CTIs, and which can be used to discriminate between open and closed-loop control of posture. The presented method may also provide information regarding how to determine multiple equilibrium positions when investigating sway with intermittent control theory. Future work is needed, though, to improve the ability to identify local maximum lines. In future work, other approaches such as Bayesianbased statistical or nonlinear dynamics models might also be applicable.

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