^{*}

Conceived and designed the experiments: DD KT PLB. Performed the experiments: PLB. Analyzed the data: DD KT. Contributed reagents/materials/analysis tools: DD KT. Wrote the paper: DD KT.

The authors have declared that no competing interests exist.

The displacement of the center-of-pressure (COP) during quiet stance has often been accounted for by the control of COP position dynamics. In this paper, we discuss the conclusions drawn from previous analyses of COP dynamics using fractal-related methods. On the basis of some methodological clarification and the analysis of experimental data using stabilogram diffusion analysis, detrended fluctuation analysis, and an improved version of spectral analysis, we show that COP velocity is typically bounded between upper and lower limits. We argue that the hypothesis of an intermittent velocity-based control of posture is more relevant than position-based control. A simple model for COP velocity dynamics, based on a bounded correlated random walk, reproduces the main statistical signatures evidenced in the experimental series. The implications of these results are discussed.

Postural control during quiet standing is usually conceived of as the control of position: when position goes beyond a given threshold, corrective mechanisms are engaged to restore equilibrium. In this paper, we question this conception and show that postural control is based on an intermittent control of velocity, with a reversal in its dynamics when the absolute value of velocity reaches a given threshold. This hypothesis presents some counterintuitive implications. Notably, it means that the active control or correction processes do not intervene at the periphery of postural sways, as generally assumed. According to our findings, control occurs in the central region of the posturogram, where velocity reaches its maximal absolute values. The present study suggests new variables of interest in the study of postural control, especially the maximal absolute velocity of the center-of-pressure, which could describe and predict postural disorders.

Postural control during quiet stance has mainly been studied at the macroscopic behavioral level by assessing the displacement of the center-of-pressure (COP). The highly complex dynamics of COP has often been reduced to the magnitude of its variability and examined comparatively between different conditions of stance (

In the past few decades, researchers have studied COP trajectory using a variety of nonlinear time series analyses. They have assessed, for instance, the predictability of the COP trajectory using recurrence quantification analysis

Such time series analyses are based on the idea that the temporal structure of COP fluctuations captures the organization of the complex, nonlinear, and dynamical “control” processes of the postural system. While these approaches have provided original and interesting insights into the processes underlying COP dynamics, one cannot help but note that the results have sometimes been contradictory and often not directly comparable. The main issue with such methods is that while they may require specific preconditions to be properly applied to the time series, they always give

Based on the general assumption that COP dynamics can be represented by the family of stochastic processes, Collins and De Luca

The results of Collins and De Luca

In this present paper, we clarify some of the methodological issues related to the use of fractal analysis in the studies by Collins and De Luca

Two common methods for characterizing the serial correlation properties of postural data are stabilogram diffusion analysis (SDA)

Fractal processes can be categorized in two families:

Representation of the continuum of fractal processes, with: the two families of fractional Gaussian noise and fractional Brownian motion, the typical correlation and diffusion properties characterizing the two types of processes, and the associated

fBm and fGn processes are related by integration and differentiation, and they are characterized by the same

With respect to our present methodological issue, the correspondence between fGn and fBm implies that one can assess the diffusion properties of a fBm in order to infer the correlation structure of the corresponding fGn. In other words, if one wants to assess the correlation properties of a fGn (or stationary series) using methods for working on diffusion properties, the series under study needs to be integrated prior to analysis.

Both SDA and DFA work on the diffusion properties of series and are based on the scaling law of Equation 1. Basically, SDA computes the mean variance of COP displacement for a given time interval length Δ

DFA is also based on the assessment of variability within intervals of varying lengths. However, the DFA algorithm differs slightly from the SDA algorithm and especially in a first step the series is integrated. The mean standard deviation of this integrated series is then determined as a function of the interval lengths (see

Schematic representation of the typical log-log diffusion plots resulting from SDA and DFA. This figure illustrates how the cross-over phenomenon can be detected using diffusion analysis.

Now, the crucial difference between DFA and SDA is that the DFA algorithm includes the integration of the analyzed series whereas the SDA does not. In a previous paper

Twenty-six participants were asked to maintain quiet stance on a force platform. The position of the COP was recorded as time series, with a sampling frequency of 40 Hz (see

We first applied SDA on position series, following the procedure proposed by Collins and De Luca

Average log-log diffusion plots obtained from SDA and DFA, and log-log power spectra on the COP position and velocity data (ML axis) collected during quiet standing. The dashed lines in the upper (SDA) and middle graphs (DFA) represent the boundary slopes between persistent and anti-persistent correlation (slope = 1.0 for SDA, and 0.5 for DFA, see text for details). The SDA shows a cross-over phenomenon when applied to position series while both the DFA and PSD analyses show a cross-over in velocity series.

We then applied DFA to COP position and velocity series. For position series, the DFA diffusion plot revealed persistent correlations over both short and long terms. In other words, DFA did not evidence any cross-over when applied to COP position series. When applied to velocity series, however, the DFA diffusion plot showed a cross-over, with positive correlations over the short term and negative correlations over the long term (

We then applied spectral analysis as a complement to the above methods. This method is likely to provide an immediate and visually salient representation of the cross-over phenomenon, with positive slopes in the log-log power spectrum indicating anti-persistence and negative slopes indicating persistence. Thus, the cross-over is expected to be revealed by a positive slope in low frequencies and a clear inflection toward a negative slope in high frequencies. Spectral analysis confirmed the results evidenced by DFA: the cross-over phenomenon was obtained only with velocity series, but not with position series (

These results clearly showed that bounding essentially affects COP velocity and not COP position, and one can thus assume that the COP trajectory is the consequence of velocity-based control. In concrete terms, the evidenced bounding means that COP velocity evolves between two (upper and lower) limit values. Its evolution from one limit to the other looks similar to a fractional Brownian motion, yielding the persistent correlations evidenced in the short term. The long-term evolution of COP velocity is characterized by a quite systematic to and fro motion within the range defined by the upper and lower limits; these systematic reversals yield the anti-persistent correlations observed in the long term.

Series from participant #10, ML axis.

On the basis of these results, we propose a very simple model for COP velocity dynamics in order to determine whether a simple bounding control of velocity would generate the complex trajectories observed in COP. This model accounts for velocity dynamics using a first-order autoregressive process (see _{t}_{t}

Average log-log diffusion plots and power spectra obtained from DFA and PSD with simulated position and velocity series. These graphs are based on point-by-point averaging of the results obtained from 26 randomly selected simulated series. The dashed line in the upper plots (DFA) represents the slope of 0.5, corresponding to the boundary between persistent and anti-persistent correlation.

Let us briefly summarize the rationale for the present study and its main results. Collins and De Luca

In order to test this hypothesis, we analyzed experimental COP position and velocity data using three different methods: SDA, DFA, and spectral analysis. The DFA algorithm includes a preliminary integration process and thus allows detection of a cross-over in the analyzed series. Spectral analysis also reveals cross-over in a straightforward way. Our results showed that SDA replicated (qualitatively) the earlier results of Collins and De Luca: the diffusion plot of position series showed a cross-over. On the other hand, both DFA and spectral analysis evidenced a cross-over in velocity but not in position series. These results clearly support the hypothesis that bounding affects primarily COP velocity.

This two-scale dynamics suggests that an intermittent control of velocity underlies the COP trajectory, reversing its dynamics when the absolute value of velocity reaches a given threshold. Note that this hypothesis could be conceived as a velocity-based analog of the two-regime model proposed by Collins and De Luca

This hypothesis presents some counterintuitive implications. Notably, it means that the active control or correction processes do not intervene at the periphery of the COP trajectory,

Finally, while the dynamics of the COP trajectory has usually been described as very complex, our results suggested a quite simple model of velocity dynamics. This model reproduced the main statistical properties evidenced experimentally. A number of models have been proposed to account for COP dynamics during quiet stance

The present study suggests new variables of interest in the study of postural control. Beyond the signatures of serial correlations addressed here, the determinants of the threshold that bounds the dynamics of velocity may be of particular interest. The value of this threshold can be empirically estimated by computing the

Results are given for the AP axis.

This study shows how sophisticated methods for the assessment of the complex properties of experimental time series must be used with much caution and regard to their theoretical and methodological foundations. Perhaps more than for other more “classical” analysis, the conclusions drawn from such methods are directly dependent on the specific properties of the algorithms and procedures implemented.

This study was conducted according to the principles expressed in the Declaration of Helsinki. The study was approved by the Institutional Review Board of the University of Montpellier 1. All patients provided written informed consent for the collection of samples and subsequent analysis.

Twenty-six male volunteers (19.3 yrs±2.1) took part in the experiment. The participants were asked to maintain quiet stance on a force platform (Medicapteurs “40 Hz/16b”) of 530 mm×460 mm×35 mm, equipped with three pressure gauges. Participants held their arms alongside their body and focused on a visual reference mark fixed 90 cm in front of them. The feet were oriented with an angle of 15° from the sagittal midline, and the heels were positioned 4 cm apart. The participants had a 30-s familiarization period before testing began.

The vertical ground reaction forces were recorded using a 12-bit A/D converter, with a sampling frequency of 40 Hz. The system was linked to Medicapteurs Winposture2000 software, providing COP series on the anteroposterior (AP) and mediolateral (ML) axes. The duration of each recording was 25.6 s, in order to obtain time series with 1024 points. The collected series were filtered by a low-pass filter, with a cut-off frequency of 8 Hz.

We analyzed COP position and COP velocity data in the ML and AP axes. The velocity series were obtained by differentiating the position series. Note that the velocity series were not further filtered after differentiation. First, we applied SDA on the position series, following the procedure proposed by Collins and De Luca ^{2} within all pairs of points separated by a time interval Δ^{2} to be based on three non-overlapping intervals.

Second, we applied DFA to the COP position and velocity series. DFA includes a series of operations: First the analyzed series

The integrated series is then divided into non-overlapping intervals of length _{n}(k)

Because DFA needs a minimal number of points to compute the standard deviation within each interval, we considered intervals from

In complement to the above methods in the temporal domain, we applied power spectral density (PSD) analysis, which was likely to provide an immediate and visually salient representation of the cross-over phenomenon. PSD allows assessing serial correlation in a signal because the scaling law of Equation 2 can be expressed as follows in the frequency domain:

Preprocessing operations were used before the application of the fast Fourier transform algorithm: First the mean of the series was subtracted from each value, and then a parabolic window was applied: each value in the series was multiplied by the following function:

Third, bridge detrending was performed by subtracting from the data the line connecting the first and last points of the series. These preprocessing operations have been recommended by Eke et al.

In order to avoid any bias due to the logarithmic distributions of the points in the diffusion plots and power spectra, we divided the abscissa into intervals of equal lengths (24 intervals of 0.1(log_{10}Δ_{10}_{10}Δ

These analyses were performed separately on the ML and AP series. To test statistically for the persistence/anti-persistence of serial correlations, we used one-sample

The results obtained with the three methods on the COP position and velocity series are summarized in

Position | Velocity | ||||

Method | Slope | AP | ML | AP | ML |

SDA | Short-term slope | 1.60 (0.19) | 1.75 (0.09) | - | - |

Long-term slope | 0.48 (0.40) | 0.36 (0.33) | - | - | |

DFA | Short-term slope | 1.65 (0.08) | 1.70 (0.07) | 1.00 (0.17) | 1.17 (0.12) |

Long-term slope | 1.22 (0.22) | 1.00 (0.29) | 0.43 (0.12) | 0.23 (0.12) | |

PSD | High-frequency slope | −3.24 (0.39) | −3.32 (0.33) | −1.44 (0.39) | −1.52 (0.33) |

Low-frequency slope | −1.80 (0.50) | −1.60 (0.58) | 0.71 (0.54) | 1.20 (0.61) |

The results of SDA on the COP position series showed a typical two-regime diffusion plot (

For the position series, the DFA diffusion plot showed a globally positive trend over the short and long terms (

For the velocity series, the DFA diffusion plot showed a much more pronounced inflection between the short- and the long-range regions (see

For the position series, the log-log power spectrum exhibited a globally negative trend (

For the velocity series, the spectral analysis showed two qualitatively different scaling behaviors, with a positive mean slope in the low-frequency region (AP:

Given these results, we proposed to model the velocity dynamics, considering the COP trajectory as the consequence of velocity-based control. Our results indicated slightly diffusive velocity dynamics, close to Brownian motion, over the short term. Basically, this dynamics can be modeled by a first-order autoregressive process including a constant that induces a linear trend in the series:_{t}_{t}

As shown experimentally, the long-term dynamics of COP velocity is anti-persistent: the evolution in velocity reverses its direction when it reaches an upper or lower limit,

This equation yields series that reproduce the expected to and fro of velocity between the two boundary values, but in an excessively systematic manner. More realistic dynamics can be obtained by making the linear trend of Equation 8 dependent on the current absolute velocity (

According to this equation, the higher the absolute velocity, the higher the contribution of the linear trend to its dynamics. Note that the goal of Equation 9 was just to mimic the short-term behavior of velocity as closely as possible. We had no specific assumptions about possible correspondences between the terms included in the model and the neurophysiological processes involved in postural control. Our aim was simply to check whether the bounding of velocity, imposed on this short-term dynamics, would allow us to simulate the empirically observed dynamics, for the velocity and position series.

We simulated 100 series of 1024 points using Equation 9, with

We thank Cathy Stott-Carmeni for proofreading the manuscript.