## Abstract

This study shows that center-of-pressure (COP) traces that closely resemble physiologically measured COP functions can be produced by an appropriate selection of model parameters in a simple feedback model of the human postural control system. Variations in the values of stiffness, damping, time delay, and noise level determine the values of 15 sway measures commonly used to characterize spontaneous sway. Results from model simulations indicate that there is a high degree of correlation among these sway measures, and the measures cluster into three different groups. Only two principal components accounted for about 92% of the variation among the different sway measures analyzed. This model can be used to formulate hypotheses regarding the cause of postural control deficits reported in the literature. This is accomplished using a multidimensional optimization procedure to estimate model parameters from a diverse set of spontaneous sway measures. These model parameters describe physiologically meaningful features of the postural control system as opposed to conventional sway measures that provide only a parametric description of sway. To show the application of this method, we applied it to published data of spontaneous sway from elderly subjects and contrasted it to the data of young healthy subjects. We found that modest increases in stiffness and damping and a fairly large increase in noise level with aging could account for the variety of sway measures reported in the literature for elderly subjects.

## INTRODUCTION

Upright stance is inherently unstable. Small deviations from an upright body position result in a gravity-induced torque acting on the body, causing it to accelerate further away from the upright position. Corrective torque must be generated to counter the destabilizing torque due to gravity. This process of continuous small body deviations countered by corrective torques creates a pattern known as spontaneous sway. The mechanisms underlying spontaneous sway are not fully understood, and controversy remains regarding the organization of sensory and motor systems contributing to spontaneous sway.

Numerous authors have suggested that active feedback control mechanisms contribute to the maintenance of upright stance (e.g., Fitzpatrick et al. 1996; Johansson and Magnusson 1991; Peterka and Benolken 1995; Peterka and Loughlin 2004; van der Kooij et al. 1999). However, the relevance of feedback mechanisms for postural control is still debated. Some authors concluded from their experiments that corrective torque originating from feedback control is insufficient for stabilizing the body (Fitzpatrick et al. 1996). Others suggested additional sources for corrective torque, like prediction (Morasso et al. 1999; van der Kooij et al. 1999), or have proposed more complex concepts like throw and catch movements (Loram and Lakie 2002a), anticipatory “saccadic” control (Baratto et al. 2002), or open-loop and closed-loop control (Collins and De Luca 1993).

However, recent studies have shown that a model based primarily on a feedback mechanism with a 150- to 200-ms time delay can account for postural control during a broad variety of perturbations (Peterka 1995, 2002; Peterka and Benolken 1995; Peterka and Loughlin 2004) and can yield a spontaneous sway pattern that resembles normal (Peterka 2000) or pathological spontaneous sway (Parkinson’s disease; Maurer et al. 2003). The simple feedback model we aim to investigate here is consistent with models previously used to simulate upright stance behavior in humans (Ishida and Imai 1980; Ishida and Miyazaki 1987; Johansson et al. 1988; Peterka 1995, 2000, 2002; Peterka and Benolken 1995; Peterka and Loughlin 2004). In this model, the body is treated as a single-link inverted pendulum. The angular deviation from upright stance is detected by sensory systems, and this sensory information is used to generate corrective torque that compensates for disturbances. Johansson et al. (1988) showed that stabilization of an inverted pendulum requires two components of corrective torque: one component proportional to the angular deviation and the other proportional to the time derivative of the angular deviation. When adding a third component proportional to the mathematical integral of the angular deviation signal, Johansson’s experimental data were even better explained. This three-component controller is commonly referred to as a PID controller (for proportional, integral, and derivative controller). By including an internal random disturbance torque, this simple model generates sway patterns that resembles spontaneous sway. Sway measures calculated from this model-generated spontaneous sway can be compared with experimental sway measures to determine the extent to which the model accounts for experimental data.

Spontaneous body sway is most often described with measures based on movements of the center-of-pressure (COP) measured with a force platform. In general, we can distinguish time domain from frequency domain measures (Prieto et al. 1996). Time domain measures estimate a parameter associated with either the displacement or the velocity of the COP trace. Frequency domain measures characterize the area or shape of the power spectral density of the COP trace. In the past, many studies characterized postural stability based on a single displacement- or velocity-related measure (Baloh et al. 1998; Brocklehurst et al. 1982; Era and Heikkinen 1985; Kolleger et al. 1992; Overstall et al. 1977). Some studies included multiple measures (Chiari et al. 2002; Laughton et al. 2003; Maki et al. 1990; Murray et al. 1975). Depending on the cause of the postural instability, velocity-related sway measures were often reported to separate stable postural control from reduced stability better than displacement-related sway measures. This was shown for Parkinson’s disease (Burleigh et al. 1995; Maurer et al. 2003; Rocchi et al. 2002), peripheral neuropathy (Dickstein et al. 2001; Horak et al. 2002; Uccioli et al. 1995), and postural instability in elderly adults (Maki et al. 1990; Prieto et al. 1996).

One of the more recent methods for analyzing COP time series is the determination of a stabilogram diffusion function (SDF). The SDF summarizes the mean square COP displacement as a function of the time interval between COP comparisons (Collins and De Luca 1993). A typical SDF shows that spontaneous sway is characterized by a two-part behavior, which led the authors to assume that, over short time intervals, the postural system is not controlled (i.e., it operates open-loop), while at longer time intervals, there is active feedback control (i.e., closed-loop control). However, Peterka (2000) showed that a simple feedback mechanism combined with a feedback time delay could generate realistic SDFs.

Only a few studies have compared a large number of COP-based measures with the explicit aim of finding the most significant measure to distinguish between different groups and conditions (Baratto et al. 2002; Hufschmidt et al. 1980; Prieto et al. 1996; Rocchi et al. 2004). Using measures of sway path length, mean sway amplitude, mean sway frequency, and sway area, Hufschmidt et al. (1980) reported differences of equally high significance for both sway path and mean sway amplitude when examining patients with cerebellar lesions and patients with labyrinthine lesions. From a large variety of time and frequency domain measures, Prieto et al. (1996) isolated mean sway velocity as the one measure with the highest significance for separating age groups as well as visual conditions (eyes-closed/eyes-open). Based on principal component analysis, Rocchi et al. (2004) provided a recommended set of sway measures for characterizing anterior-posterior and medio-lateral sway.

It is currently unknown why one sway measure or another should be superior in distinguishing between normal and pathological conditions. The statistical relationships between alternative sway measures and between these measures and the underlying dynamic properties of the postural control system are also unknown. This study was undertaken to gain insight into these relationships. Spontaneous body sway traces were created using a simple postural control model that was previously shown to generate realistic SDFs (Peterka 2000) and was shown to account for responses to various external stimuli (Peterka 2002). In this study, we *1*) show that numerous other COP-related measures of spontaneous sway reported in the literature for normal subjects are also compatible with this model, *2*) characterize the relationship between the sway measures and the sensitivity of the sway measures to the model parameters, *3*) show that model parameters representing fundamental characteristics of the postural control system can be derived from a set of spontaneous sway measures using an optimization procedure, *4*) apply this optimization procedure to existing sets of spontaneous sway measures in elderly and young subjects to determine how fundamental postural control parameters differ between young and elderly subjects, and *5*) investigate how variations in the model architecture (active vs. passive control models) influence the quality of the model parameter estimates.

## METHODS

### Model description

We used a simple model of human postural control (Fig. 1) with all inputs and outputs restricted to the sagittal plane (anterior-posterior direction). We assumed that a neural control system senses a deviation of the body away from an upright reference position and sends commands to generate a corrective torque to resist the deviation of body position away from upright. An inverted pendulum represents the human body (Johansson et al. 1988; Peterka 2000) with parameters set to values typical for an average adult male subject. Specifically, *J*_{B} = 66 kg/m^{2} is the moment of inertia of the body about the ankle joint axis, *m*_{B} = 76 kg is body mass excluding the feet, *m*_{F} = 2.01 kg is the mass of the feet, *d*_{B} = 0.87 m is the height of the body center of mass (COM) above the ankle joint axis, *h*_{F} = 0.085 m is the height of the ankle joint axis above the ground, and *d*_{F} = 0.052 m is the horizontal distance between the ankle joint axis and the COM of the feet.

The input to the body is the torque exerted about an axis through both ankle joints. The torque consists of two components. One (*T*_{d}, see Fig. 1) is a random disturbance torque that contributes to the generation of spontaneous body sway patterns. The other is a control torque (*T*_{c}) that corrects for the disturbance torque and the torque due to gravity. For most simulations, *T*_{c} was generated entirely by the action of a “neural controller,” i.e., *T*_{c} = *T*_{a} in Fig. 1. The input to the neural controller is the sensed body sway with a time delay. We assume that the sensory systems provide an accurate measure of body sway (*sensory systems* = 1 in Fig. 1). The neural controller properties are specified by the three constants *K*_{P}, *K*_{D}, and *K*_{I}, which scale the components of the control torque that are proportional to the deviation from zero of the angular position, angular velocity, and time integral of the angular deviation, respectively. For a subset of simulations, we assumed a more complex model structure that included passive stiffness (*K*_{pas}) and passive damping (*B*_{pas}) components. In that case, *T*_{c} is a sum of *T*_{a} and the passive torque (*T*_{p}) determined by the intrinsic ankle stiffness and damping. The outputs of the system are body sway angle (θ) and COP displacement (*x*_{COP}). The computation of *x*_{COP} is based on a derivation by Barin (1989), which relates the sagittal body displacement (*x*_{B}) and the sagittal, vertical, and angular body accelerations (ẍ_{B}, ÿ_{B}, θ̈_{B}, respectively) to *x*_{COP} for an inverted pendulum (1) where *g* = 9.81 m/s^{2} is the acceleration due to gravity.

To simulate the continuous, small deviations of body sway around an equilibrium position (Prieto et al. 1996), a disturbance torque (*T*_{d}) is added to the corrective torque (*T*_{c}). We are aware of the fact that a major contribution to the overall noise in human postural control might stem from the sensory organs (van der Kooij et al. 2001). Thus it may be plausible to add noise to the sensory signal, too. However, we tested the model with different sites of noise injection, and the simulation results were very similar. For simplicity, we chose to use only one noise source, which we inject as a random disturbance torque. The disturbance torque is a low-pass filtered Gaussian noise source with a first-order low-pass filter time constant of τ_{N} = 100 s for all of the simulations presented in this paper. The use of this low-pass filter led to a noise source with characteristics of a random walk process in velocity whose spectral density declines with increasing frequency for frequencies above the filter cutoff frequency (Milotti 2002). The resulting waveforms created sway characteristics similar to those seen physiologically. In addition, the resulting sway measures were largely insensitive to the exact value of the filter time constant. A higher value for the time-constant could be compensated for by a higher noise gain.

### Model simulations

Simulations were performed using Simulink version 5.0 of Matlab 6.5 (The MathWorks, Natick, MA). As a noise source, we used the standard Matlab block “band-limited white noise” with zero mean, unity variance, and with a correlation time of 0.01 s. The simulation duration was 800 s, and the period between model outputs was 0.01 s. The Dormand-Prince algorithm (ode5) with a fixed step size of 0.01 s was used for all simulations to solve the differential equations associated with the model.

Based on published sway data for young adults and using an optimization procedure to be detailed later, a center-normal set of model parameters was defined that provided a reference for investigating how variations in model parameters influenced sway measures. Coefficients of variation (CVs) for sway measures were obtained from the results of 100 repeated simulations, with each simulation using the same center-normal model parameters but a different random seed for the band-limited noise time series. CVs of the model parameters were obtained by performing optimization procedures using the same 100 simulations with randomly varied noise seeds.

The resulting center-normal model parameters were similar to the ones used by Peterka (2000). The gain factors for the PID controller were *K*_{P} = 16.7 Nm · deg^{−1} for the proportional part (stiffness), *K*_{D} = 4.83 Nm · s · deg^{−1} for the derivative part (damping), and *K*_{I} = 0.60 Nm · s^{−1} · deg^{−1} for the integrative part. The feedback time delay (τ_{d}) was 0.171 s. The noise level gain (*K*_{N}) was 462 Nm, which corresponded to a standard deviation of the disturbance torque of 10.7 Nm. A 30-s sample of an 800-s COP time series generated from the Fig. 1 model with the center-normal model parameters is shown in Fig. 2*A*. The sample in Fig. 2*A* includes the body COP displacement, *x*_{COP}, and COM displacement, *x*_{COM} = *d*_{B}·sin(θ).

Each of the five model parameters (*K*_{P}, *K*_{D}, *K*_{I}, τ_{d}, *K*_{N}) were varied in nine equal increments between 90 and 110% of their center-normal values, and a simulation was performed for each possible combination of the nine values of the five parameters, leading to 9^{5} = 59,049 simulations.

### Sway measure calculations

Fifteen sway measures (Table 1) were calculated from each of the 800-s COP traces resulting from the 59,049 simulations. In the following equations for calculation of the sway measures, *n* = 80,000 is the number of data points included in the analysis, and *T* = 800 s is the period of time selected for analysis. To simplify the following definitions, we referenced the COP time series to the mean, i.e., we subtracted the mean COP position from the COP time series.

##### TIME-DOMAIN MEASURES.

The measures described in this section are the most commonly used measures of postural stability. We adapted these measures from Prieto et al. (1996). Time domain measures provide measures associated with either the displacement or the velocity of the COP. The mean distance (*MD*) is the mean of the absolute value of the COP time series and represents the average distance of the COP from its mean (2) The root mean square distance (*RMS*) from the mean COP is equivalent to the SD (*SD*) of the zero-meaned COP time series (3) The maximum distance (*MAXD*) is the peak-to-peak range of COP values (4) The velocity of the COP time series was calculated by subtracting consecutive positions of the COP path and multiplying by the sampling rate (*n/T*) (5) The mean velocity (*MV*) is the average of the absolute value of the COP velocity (6) The root mean square velocity (*RMSV*) is the standard deviation of the COP velocity time series (7) The mean frequency (*MFREQ*) is a frequency, in Hertz, of a sinusoidal oscillation derived from the mean distance *MD* and the mean velocity *MV*^{1} (8)

##### FREQUENCY-DOMAIN MEASURES.

The frequency domain measures described here are adapted from Prieto et al. (1996). They were originally selected for their ability to characterize the area or shape of the power spectral density of the COP, *G*(*f*). We calculated the one-sided spectral density function *G*(*f*) via discrete Fourier transforms following the methods given by Bendat and Piersol (2000). The spectral moments, μ_{k}, which are needed to compute the different frequency domain measures, were calculated for *k* = 0, 1, and 2 as (9) where Δ*f* = 0.00125 Hz is the frequency increment and *m* = 8,000 is the total number of calculated discrete power spectral density estimates, *G*(*i*·Δ*f*), with frequencies ranging from *f* = 1 · Δ*f* = 0.00125 Hz to *f* = 8,000 · Δ*f* = 10 Hz. This frequency range covers virtually the entire bandwidth of the COP signal. Specifically, Dichgans et al. (1976) showed that <1% of the total power is located above 10 Hz.

The total power (*POWER*) is the integrated area of the power spectrum (10) Theoretically, if all power is accounted for, this is the mean square value of the time series. This makes *POWER* redundant to *RMS,* which is the square root of the mean square value. Therefore to avoid the redundancy, we did not use *POWER* in some of the analysis procedures (correlation study and optimization procedures). However, we show simulated values for *POWER* because experimental measures of *POWER* were available for young and elderly adults (Prieto et al. 1996).

The 50% power frequency (*P50*), the median power frequency or the frequency below which 50% of the total power is found, is *u*Δ*f,* where *u* is the smallest integer for which (11) The 95% power frequency (*P95*), the frequency below which 95% of the total power is found, is *v*Δ*f,* where *v* is the smallest integer for which (12) The centroidal frequency (*CFREQ*), the frequency at which the spectral mass is concentrated, is the square root of the ratio of the second to the zeroth spectral moment (13) The frequency dispersion (*FREQD*), a unitless measure of the variability in the frequency content of the power spectral density, is given by (14) The frequency dispersion is zero for a pure sinusoid and increases with spectral bandwidth to a maximum of one.

For most simulation results reported in our study, the calculations of the frequency domain measures were as described above. However, for the purpose of comparing our simulated sway measures to experimental values given by Prieto et al. (1996), we replicated the calculation methods used by those authors. Specifically, the total simulation time of 800 s was divided into 40 periods of 20 s, which were comparable to the time periods analyzed by Prieto et al. (1996). Each 20-s COP trace was filtered with a first-order low-pass filter with a cut-off frequency of 5 Hz. The spectral density functions were calculated from 0.05 to 5 Hz, but the lowest two frequency components at 0.05 and 0.1 Hz were not included in the calculation of the frequency domain measures. Then, the 40 spectral density functions were averaged, and the sway measures were calculated. The affected sway measures were *POWER*, *P50*, *P95*, *CFREQ*, and *FREQD* in Table 2 and Table 6.

##### STABILOGRAM DIFFUSION FUNCTION MEASURES.

SDFs were calculated following the methodology of Collins and De Luca (1993) (15) where 〈·〉 symbolizes a calculation of the mean value, and *m* ranged from 0 to 1,000 corresponding to a time shift of 0 to 10 s between the COP traces. At *m* = 0, the SDF value is always zero. The short-term diffusion coefficient, *D*_{S}, the long-term diffusion coefficient *D*_{L}, and the critical point coordinates, Δ*T*_{c} and 〈Δ*X*^{2}〉_{c}, were computed from the results of two linear regressions to the short- and long-term regions of the SDFs (see Fig. 2*B*). In most cases, a linear fit to the short-term region of the SDF included data ranging from 0 to 0.5 s, and the linear fit to the long-term region included data ranging from 2 to 10 s. *D*_{S} and *D*_{L} correspond to one-half times the slope of the respective linear fits to the SDFs. The critical point values were determined from the point of intersection of the linear regression fits to the short-term and long-term regions. In some cases, as the SDFs changed as a function of simulation parameters, the time periods included in the short-term and long-term regions were altered slightly to maintain good quality linear fits to the data. An SDF calculated using the full 800-s COP time series from a simulation with center-normal model parameters is shown in Fig. 2*B*. The SDF shows a two-part functional form typical of experimental SDFs (Collins and De Luca 1993).

### Statistical procedures

The statistical procedures were conducted with JMP 5.0.1a (SAS Institute, Cary, NC). Using results from the 59,049 simulations, we calculated a standard correlation matrix, which gave a matrix of correlation coefficients that summarizes the strength of the linear relationships between each pair of sway measures. Principal component analysis was applied to the correlation matrix. Principal component analysis decomposes the correlation matrix by calculating a linear combination of the original sway measures such that the first principal component accounts for the maximum normalized variation among the sway measures, the second principal component accounts for the maximum variation not accounted for by the first component, and so on. The principal components are given by the eigenvectors of the correlation matrix, and the eigenvalue of each principal component indicates the amount of variation accounted for by that principal component (Jolliffe 1986). All principal components with eigenvalues > 1 are reported. A factor rotation process (varimax method) was applied to the principal components to identify factors that provide insight into the relationships among the sway measures (Jackson 1991). The factor rotation process attempts to associate each sway measure with a minimal number of factors.

### Relationships between model parameters and sway measures

To understand why certain sway measures seem to be better than others in distinguishing between normal and altered conditions, we determined how the different sway measures changed as a function of the model parameters *K*_{I}, *K*_{P}, *K*_{D}, τ_{d}, and *K*_{N}. That is, we computed the correlations between the changes in the parameter values and the changes in the sway measures using all 59,049 simulations. In addition, we performed a sensitivity analysis by varying one parameter at a time and keeping all other parameters at their center normal values. Nine simulations were performed for each of the five parameters, with the parameter varying from 90 to 110% in 2.5% increments. Sway measures were calculated from each simulation. The normalized change in the sway measure was compared with the normalized change in parameter value.

### Estimation of model parameters from sway measures

Another set of simulations was performed to investigate whether the information contained in the different sway measures is sufficient to determine the values of different model parameters. Preliminary results showed that variations in gain of the integrative part of the neural controller did not influence the sway measures to a large degree. Therefore we kept the gain of the integrative part at *K*_{I} = 0.6 Nm · s^{−1} · deg^{−1} and varied the remaining parameters (*K*_{P}, *K*_{D}, τ_{d}, and *K*_{N}) simultaneously and independently such that each parameter assumed one of three possible values: 90, 100, and 110% of its center-normal value. This procedure led to a total of 3^{4} = 81 different simulations, each with a different set of model parameter values. Sway measures were calculated from the COP trace from each of these simulations. The sway measures from each of these simulations were used in an optimization procedure to obtain estimates of the model parameters that could optimally account for each set of sway measures. A comparison of the actual model parameters with the estimated model parameters provided information about the ability of this optimization procedure to accurately identify fundamental postural control parameters from a set of COP sway measures.

We used the Matlab Optimization toolbox function “fminsearch” to find the minimum of a scalar error function of the sway measures, starting at an initial estimate. The algorithm “fminsearch” uses the simplex search method of Nelder-Mead (Lagarias et al. 1998). We used the center-normal model parameters as the initial estimate. With each iteration of the optimization procedure, the current model parameters were used in a simulation to generate a COP time series. Sway measures were calculated from the COP time series, and a scalar error function was evaluated. Then the optimization procedure made changes to the model parameters, a new COP time series was simulated, sway measures were calculated, and the error function was re-evaluated. This sequence was repeated until the error function was minimized. To normalize the error contribution from each sway measure and to achieve a symmetrical error distribution around the reference value, we used the error function (16) where *n* = 14 is the number of sway measures, *M*_{i} are the reference sway measures obtained using 1 of the 81 sets of model parameters, and v*M _{i}* is the estimated sway measure obtained with each iteration of the optimization procedure.

We also applied this optimization procedure to the means of experimental sway measures of young and elderly adults from previous studies (Collins and De Luca 1993; Collins et al. 1995; Prieto et al. 1996). All experimental sway measures we used were originally based on COP traces derived from force platform recordings during 20–30 s of quiet standing. Prieto et al. (1996) compared a variety of time and frequency domain measures between a group of 20 healthy young adults (21–35 yr) and a group of 20 healthy elderly adults (66–70 yr). Collins and De Luca (1993) described SDF-derived sway measures from 25 healthy male subjects (19–27 yr). Collins et al. (1995) presented SDF measures from 25 healthy elderly males (71–80 yr).

In a final set of optimizations, we considered alternative model structures that included passive elements that contributed to the corrective torque without time delay (*T*_{p} in Fig. 1). We tested three different model versions that included passive dynamic contributions. In all versions, the model parameters *K*_{P}, *K*_{D}, τ_{d}, and *K*_{N} were always allowed to vary during the optimization. The passive dynamic elements *K*_{pas} and *B*_{pas} were either fixed or were allowed to vary. In one model version, the passive stiffness *K*_{pas} was set to 10.2 Nm · deg^{−1} (i.e., 90% of the stiffness necessary to provide minimal stabilization, following the experimental results of Loram and Lakie 2002b), and the passive damping *B*_{pas} was set to 0. In the second version, *K*_{pas} was set to 10.2 Nm · deg^{−1}, and the passive damping *B*_{pas} was set to 3.0 Nm · s · deg^{−1} (i.e., values that were in equal proportion to the active stiffness and damping, respectively, implemented in the center normal model). The optimization procedure was used to identify the other model parameters (*K*_{P}, *K*_{D}, τ_{d}, and *K*_{N}) that minimize the *Eq. 16* error criterion. In the third version, *K*_{pas} and *B*_{pas} were included as free parameters in the optimization.

## RESULTS

We first show that the Fig. 1 model predicts sway measures that agree with experimental data of young adults. Then we present the relationships among sway measures predicted by the model and the sensitivity of sway measures to variations of model parameters. After verifying that our optimization procedure is able to reliably identify model parameters, we apply the optimization procedure to experimental data from elderly adults and investigate other model structures that include passive stiffness and/or damping.

### Center-normal sway measures

Table 2 (columns “active control” and “experimental data”) shows that all sway measures derived from simulations with the center-normal model parameters were well within the 1 SD range of the mean experimental data taken from studies of (Collins and De Luca 1993; their Table 5) and the results of young healthy adults from Prieto et al. (1996). The CVs of the sway measures were all <0.05 for all sway measures, indicating that the simulations yielded reliable estimates of the sway measures. The CVs of the model parameters were <0.06, indicating that model parameters obtained from the optimization procedure were minimally affected by the particular noise time series used in the optimization.

### Correlations among sway measures

All sway measures aggregated into primarily three different measurement groups, with high correlations within each group, and in most cases, lower correlations between groups (Table 3; Fig. 3*A*). The first group of the aggregated sway measures contained the displacement-related measures (*MAXD*, *RMS*, *MD*) and two measures from the SDF analysis (*D*_{L} and 〈*ΔX*^{2}〉_{c}). The second group contained the velocity-related measures (*MV*, *RMSV*), as well as some frequency-related measures (*FREQD*, *P50*) and *D*_{S} from the SDF. The third group contained measures related to the frequency distribution of sway (*P95*, *CFREQ*, *MFREQ*) and the SDF measure Δ*T*_{c}. Sway measure *D*_{L} was an exception, in that *D*_{L} showed a high correlation with nearly all sway measures in groups 1 and 3. Although the selection of correlation thresholds of 0.9 and 0.8 for display in Fig. 3*A* is arbitrary, the results shown in Fig. 3*A* provide a basis for understanding the inter-relationships among the sway measures predicted by the model shown in Fig. 1.

### Principal component and factor analysis

About 92% of the variability among all the different sway measures was explained by the first two principal components (Table 4, columns 1 and 2). The third principal component accounted for only 6% additional variability among the sway measures (eigenvalue = 0.86).

Factor rotation led to one factor (rotated factor 1 in Table 4) that contributed strongly to the displacement-related sway measures (*MAXD*, *RMS*, *MD*) and by the SDF measures *D*_{L} and 〈Δ*X*^{2}〉_{c} (group 1 in Fig. 3 and Table 4). The second factor (rotated factor 2 in Table 4) contributed strongly to the velocity-related sway measures (*MV*, *RMSV*) and by *D*_{S}, *FREQD*, and *P50* (group 2 in Fig. 3 and Table 4). Both factors contributed to the frequency-related sway measures *P95*, *CFREQ*, and *MFREQ,* and the SDF measure Δ*T*_{c} (group 3 in Fig. 3 and Table 4). The absolute values of factor loadings corresponding to the 14 sway measures are shown in Fig. 3*B*. Comparison of Fig. 3*B* with the correlation results in Fig. 3*A* shows a similar grouping of the sway measures.

### Relationship between sway measures and model parameters

Sway measures typically increased with increasing τ_{d}, decreased with increasing *K*_{D}, and were unaffected by *K*_{I} (Table 5). Effects of *K*_{P} and *K*_{N} varied. Group 1 sway measures (*D*_{L}, 〈Δ*X*^{2}〉_{c}, *MD*, *RMS*, *MAXD*) were negatively correlated with *K*_{P} and positively correlated with *K*_{N}. In addition, except for *D*_{L}, all group 1 measures were negatively correlated with *K*_{D} and positively correlated with τ_{d}. These findings are in line with the intuitive expectation that an increase in stiffness (*K*_{P}) and/or a decrease in noise level (*K*_{N}) decrease the sway amplitude. With two exceptions (*P50*, *FREQD*), all group 2 sway measures (*D*_{S}, *MV*, *RMSV*) were positively correlated with *K*_{P}, *K*_{N}, and τ_{d}, and negatively correlated with *K*_{D}. This again is in line with the intuition that an increase in stiffness and/or a decrease in damping increases the mean sway frequency and thereby increases sway velocity. While an increase in noise level increases the overall sway velocity, it does not affect the frequency range of sway (symbolized by *P50*). In fact, the first exception, *P50*, did not correlate with *K*_{N}; the second exception, *FREQD*, again did not correlate with *K*_{N} and, in addition, showed in all other relationships a reversed sign compared with the rest of the group 2 measures. Group 3 sway measures were almost exclusively correlated with *K*_{P}. They did not correlate with *K*_{N}, and the correlations with *K*_{D} and τ_{d} were low. The model parameter *K*_{I} showed only weak correlations with the sway measures. Therefore as mentioned in methods, we consider it unlikely that information from sway measures could be used to characterize changes in the value of *K*_{I}.

### Sensitivity analysis

The sway measures differed considerably in their sensitivity to changes in model parameters. Figure 4 shows that measures derived from the stabilogram diffusion function (*D*_{S}, *D*_{L}, 〈Δ*X*^{2}〉_{c}, and Δ*T*_{c}) were very sensitive to changes in one or more of the model parameters. In addition, *P50* showed high sensitivity to all parameter changes except *K*_{N}. Other measures showed lower sensitivity to variations of all of the model parameters. However, the lower sensitivity to parameter changes did not necessarily lead to poorer correlations between these sway measures and some of the model parameters (Table 5). One reason is that the correlation coefficient characterizes the clustering of data points around a line representing the relationship between two variables, irrespective of the scales the variables are measured on.

Figure 4 also shows that several sets of sway measures vary in similar ways with changes in model parameters. Sway measure groups with similar relationships are (*MD*, *RMS*, and *MAXD*), (*MV*, *RMSV*), and (*MFREQ*, *CFREQ*).

### Determining model parameters from sway measures

Given the variety of inter-relationships among the model parameters and sway measures (Fig. 4; Table 5), we predicted that model parameters could be estimated from a particular set of sway measures using an optimization procedure. To test this prediction, a reference set of sway measures was calculated for a particular set of model parameters in which one or more of the parameters deviate by ±10% from the center-normal parameters. An optimization procedure was performed, starting from the center-normal values of the model parameters, to estimate the postural control parameters *K*_{P}, *K*_{D}, τ_{d}, and *K*_{N} that best explained the reference set of sway measures. Figure 5 shows the mean (with 95% CIs) results of 3^{4} = 81 optimizations, where the estimated parameter is shown as a function of the actual parameter used to generate the target sway measures. The graphs show that the optimization procedure was able to identify the actual postural control parameters within a small error tolerance.

### Applying the optimization procedure to real datasets

We applied the optimization procedure for parameter identification to real experimental data sets for elderly adults to determine whether model parameters could be identified that accounted for all of the experimental data. Specifically, we used a combination of sway measures given by Prieto et al. (1996) and by Collins et al. (1995) as the target set of sway measures. The optimization procedure identified postural control system parameters of *K*_{P} = 19.7 Nm · deg^{−1}, *K*_{D} = 6.18 Nm · s · deg^{−1}, τ_{d} = 0.165 s, and *K*_{N} = 693 Nm. This translates to an 18% increase of *K*_{P}, 28% increase of *K*_{D}, 2.9% decrease of τ_{d}, and a 50% increase of *K*_{N} with respect to the center-normal parameter values that characterize the sway measures of young adults (Table 2). Table 6 shows a comparison of the mean ± SD range of experimental measures collected from Prieto et al. (1996) and Collins et al. (1995) and the simulated sway measures achieved by setting the model parameters to the values determined by the optimization procedure. We found that all simulated sway measures were within 1 SD of the experimental sway measures.

### Determining the influence of passive stiffness and/or damping

We found that introducing passive dynamics slightly worsened the quality of the data fit. Specifically, the scalar error function increased from 0.466 with no passive dynamics to 0.505 with passive stiffness (*K*_{pas} = 10.2 Nm · deg^{−1}; Table 2, column “passive stiffness”) and to 0.553 with passive stiffness and passive damping (*K*_{pas} = 10.2 Nm · deg^{−1}, *B*_{pas} = 3.0 Nm · s · deg^{−1}; Table 2, column “passive stiff. + damp.”). This increase in the scalar error function was mainly caused by a slightly reduced ability of the models with passive stiffness and/or passive damping to account for some of the experimental sway measures (*RMS*, *FREQD*) compared with a model with only active stiffness and damping. Finally, we applied an optimization procedure that allowed all six parameters (*K*_{pas}, *B*_{pas}, *K*_{P}, *K*_{D}, τ_{d}, *K*_{N}) to vary, to determine if the optimization procedure could identify any combination of *K*_{pas} and *B*_{pas} that provided a lower error than the active feedback model. We found that the values for *K*_{pas} and *B*_{pas} both converged to zero.

## DISCUSSION

Our results showed that a very simple feedback model of human postural control was able to reproduce realistic sway behaviors. The sway measures obtained from model simulations were well within the range of 1 SD of experimental data published by Collins and De Luca (1993) and by Prieto et al. (1996) for young healthy adults. Our results extend the previous finding of Peterka (2000) that showed that simulated spontaneous body sway traces yield realistic SDFs. We show here for the first time that numerous other COP-related sway measures reported in the literature for normal subjects are also compatible with this feedback control model.

Next, we showed that the different sway measures obtained from the simulated COP traces aggregated into three groups with high correlations within each group, while correlations between groups were considerably lower. The first group contained displacement-related measures, the second group mainly velocity-related measures, and the third group mainly frequency-related measures of sway. By deriving principal components from the correlation matrix, we showed that about 92% of the variability among all different sway measures was explained by only two principal components. Factor rotation of the principal components led to factors that were closely related to the sway measure aggregates (groups 1–3) derived from the correlation matrix. The first factor mainly determined the first group (displacement-related measures). The second factor determined the second group (mostly velocity-related measures). Both factors contributed to the frequency-related sway measures (group 3).

By analyzing the relationships between sway measures and the underlying model parameters, we showed that most of the group 1–related sway measures (specifically 〈Δ*X*^{2}〉_{c}, *MAXD*, *RMS*, *MD*) were positively correlated with time delay (τ_{d}) and noise level (*K*_{N}), but were negatively correlated with stiffness (*K*_{P}). Group 2–related measures (specifically *RMSV*, *MV*, *D*_{S}) were positively correlated with *K*_{P}, τ_{d}, and *K*_{N}. Together with the sensitivity relationships displayed in Fig. 4, this correlation matrix provides insight into why one sway measure might be more or less reliable than another in distinguishing between different categories of subjects. For example, an individual sway measure would be unreliable if *1*) the particular change in a postural control parameter between subject categories had little effect on that sway measure (i.e., low sensitivity, e.g., *FREQD*) or *2*) if two or more postural control parameters changed between subject categories and the effects on that sway measure cancelled. Alternatively, an individual sway measure would reliably distinguish between subject categories if the particular change in the postural control parameter between subject categories had a large effect on the sway measures and/or there was an additive effect or at least no cancellation of effects among the different changes of postural control parameters.

We found that information contained in the relationships among the different sway measures is sufficient to identify the values of different postural control model parameters. By using an optimization procedure, we demonstrated that it is possible, in principle, to identify model parameters by analyzing the set of 14 sway measures we investigated. This optimization procedure is able to accurately identify model parameters because the set of sway measures includes a wide range of functional relationships between model parameters and the individual sway measures (Fig. 4).

It is likely that it is not necessary to include all 14 of the sway measures we used in the optimization procedure. For example, the sway measure *FREQD* showed very low sensitivity to parameter changes. One would also anticipate that sway measures with high sensitivity to model parameter changes would be particularly useful in identifying model parameters. Interestingly, the parameters with some of the highest sensitivities were parameters derived from SDFs (*D*_{S}, *D*_{L}, 〈Δ*X*^{2}〉_{c}, Δ*T*_{c}), suggesting that parameters derived from stabilogram diffusion analysis should be more sensitive at detecting changes in postural control dynamics than most other sway measures. However, in identifying model parameters, it is not only important that the sway measures are sensitive to parameter changes, but also that the sway measures are not highly correlated to one another.

We then showed that the optimization procedure of parameter identification is applicable to real datasets. We used a combination of sway measures given by Prieto et al. (1996) and by Collins et al. (1995) for elderly subjects, accepting the fact that there were different individuals in these two subject groups. Another limitation arose from the fact that we did not know subjects’ body masses and moments of inertia. However, the optimization procedure converged to a solution that found that modest increases in stiffness (*K*_{P}) and damping (*K*_{D}) and a fairly large increase in noise level (*K*_{N}) compared with the center-normal parameter set that corresponds to young adults could account for the variety of sway measures reported in the literature. Specifically, our model parameters predicted sway measures that were well within 1 SD of the experimental data collected by Prieto et al. (1996) and Collins et al. (1995). One could speculate that the increased noise level in elderly subjects is due to degradation of sensory function with age such that actual body sway is less precisely encoded by sensory systems. This increase in noise level is partially compensated for by increases in both stiffness and damping. That is, increases in stiffness and damping tend to reduce body sway displacement that is caused by an increased noise level (*RMS*, *MD*, *MAXD* measures in Fig. 4 all decrease with increasing *K*_{P} and *K*_{D}). However, the model predicts that an increase in stiffness and noise level both contribute to an increase in velocity-related sway measures, consistent with the experimental finding of Prieto et al. (1996) that *MV* was the one measure with the highest significance for separating age groups.

Finally, we determined whether the inclusion of passive stiffness and/or damping components in the model could provide a better explanation for the experimental sway measures of young adults than a model without passive components. There is disagreement about the relative importance of active versus passive contributions to corrective torque generation, with some studies showing a dominant role of active mechanisms (Peterka 2002; Peterka and Loughlin 2004) and others a dominant contribution from passive mechanisms (Loram and Lakie 2002a, b). We showed that the optimization procedure yielded a slightly better fit to experimental sway measures for pure active control compared with control provided by a combination of a smaller active and a larger passive component. The decrease in fitting quality was shown by an increase in the scalar error function and by a reduced ability of the model with a passive component to predict the experimental sway measures (i.e., *RMS* and *FREQD* were more than 1 SD away from the experimental sway measures given by Prieto et al. 1996 and Collins and De Luca 1993). When we included passive stiffness and damping as freely varying model parameters in the optimization procedure, both passive stiffness and passive damping converged to zero. However, the scalar error function values of these different model structures had similar values for all models tested. Therefore it is unlikely that results from the optimization procedure can be used to determine the relative contributions of active and passive components to torque generation. Moreover, the poorer fit with the inclusion of the passive components could be the result of using experimental data from multiple sources.

In conclusion, this study shows that COP traces that closely resemble physiologically measured COP functions can be produced by an appropriate selection of model parameters in a simple feedback model of the human postural control system. Variations in the values of stiffness, damping, time delay, and noise level control the values of different sway measures, leading to a characteristic pattern of sway measures. Selection of appropriate model parameters can generate COP traces whose sway measures match the measures reported for COP traces recorded from young and elderly adults. Conversely, a given set of sway measures can be used to estimate a set of model parameters. These model parameters describe physiologically meaningful features of the postural control system as opposed to conventional sway measures that provide only a parametric description of sway. Although our modeling results are consistent with a continuous feedback-regulation strategy for upright stance control and are consistent with other recent results (Peterka 2002; Peterka and Loughlin 2004), these results do not prove that this is the actual control strategy used by the nervous system. That is, other control strategies might exist that are also able to explain the statistical properties of spontaneous body sway. For example, the apparent simplicity of the model could mask the potential complexity of sensory information processing required for postural control during quiet stance (Mergner et al. 1997, 2002; Peterka 2002). Nevertheless, it may be instructive to consider the implications for differences in postural stability in terms of our model’s structure. The proposed model is quite simple, and yet is able to account for the statistical properties of body sway as revealed by the different sway measures.

## GRANTS

This work was supported by National Institute on Aging Grant AG-17960 to R. J. Peterka.

## Footnotes

↵1 Note that this formula, used by Prieto et al. (1996) for calculating the one-dimensional

*MFREQ*(i.e., the frequency of a sinusoid that has values of*MD*and*MV*for its mean distance and mean velocity measures), was obtained from Hufschmidt et al. (1980). The correct formula should read*MFREQ*=*MV*/(2·π·*MD*) which means that*MFREQ*used here is smaller than the true*MFREQ*by a factor of 4·/(2·π) ≈ 0.90032. Because we compared our simulated sway measures to the values given by Prieto et al. (1996), we used*Eq. 8*for the calculation of*MFREQ*. However, the use of the incorrect formula did not affect the relationships found involving*MFREQ*, since correlations are unaffected by a constant multiplier.The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2005 by the American Physiological Society