Biol. Cybern. 87, 262–277 (2002) DOI 10.1007/s00422-002-0333-2 Ó Springer-Verlag 2002
Multisensory fusion and the stochastic structure of postural sway Tim Kiemel1 , Kelvin S. Oie2;3 , John J. Jeka2;3 1
Department of Biology, University of Maryland, College Park, MD 20742, USA Department of Kinesiology, University of Maryland, College Park, MD 20742, USA 3 Program in Neuroscience and Cognitive Science, University of Maryland, College Park, MD 20742, USA 2
Received: 20 March 2001 / Accepted: 17 April 2002
Abstract. We analyze the stochastic structure of postural sway and demonstrate that this structure imposes important constraints on models of postural control. Linear stochastic models of various orders were fit to the center-of-mass trajectories of subjects during quiet stance in four sensory conditions: (i) light touch and vision, (ii) light touch, (iii) vision, and (iv) neither touch nor vision. For each subject and condition, the model of appropriate order was determined, and this model was characterized by the eigenvalues and coefficients of its autocovariance function. In most cases, postural-sway trajectories were similar to those produced by a third-order model with eigenvalues corresponding to a slow first-order decay plus a fasterdecaying damped oscillation. The slow-decay fraction, which we define as the slow-decay autocovariance coefficient divided by the total variance, was usually near 1. We compare the stochastic structure of our data to two linear control-theory models: (i) a proportional–integral–derivative control model in which the postural system’s state is assumed to be known, and (ii) an optimal-control model in which the system’s state is estimated based on noisy multisensory information using a Kalman filter. Under certain assumptions, both models have eigenvalues consistent with our results. However, the slow-decay fraction predicted by both models is less than we observe. We show that our results are more consistent with a modification of the optimal-control model in which noise is added to the computations performed by the state estimator. This modified model has a slow-decay fraction near 1 in a parameter regime in which sensory information related to the body’s velocity is more accurate than sensory information related to position and acceleration. These findings suggest that: (i) computation noise is responsible for much of the variance observed in postural sway, and (ii) the postural control system under the
Correspondence to: T. Kiemel (Tel.: +1-301-4056176, Fax: +1-301-3149358 e-mail:
[email protected])
conditions tested resides in the regime of accurate velocity information.
1 Introduction The control of human upright stance requires sensory input from multiple sources to detect center-of-mass excursions and to generate appropriate muscle responses for upright stance control. Without appropriate knowledge of self-orientation, equilibrium control is severely compromised. An intensive research effort over the last two decades has contributed considerable knowledge about how sensory information is used for upright stance control (for reviews see Nashner 1981; Horak and MacPherson 1996). In parallel, quantitative models have been developed that characterize the integration of sensory inputs in postural control and spatial orientation. In this paper, we restrict our attention to models that describe posture at the level of variables such as the center of mass. It can be argued that the behavior of the postural system at this level of description might be approximately linear (e.g., Johansson et al. 1988; Kuo 1995). Subsequent experimental and modeling efforts have shown that multisensory fusion involves substantial nonlinearities under conditions such as imposed movements of the body (Mergner and Rosemeier 1998) and movements of the sensory environment (Jeka et al. 2000) in which there are conflicts or ambiguities present in the sensory information. Although these results point out the limits of linear models of postural control, they do not necessarily exclude their use to model conditions such as quiet stance in a stationary environment. A direct evaluation of the suitability of linear models to model postural-sway trajectories was performed by Johansson et al. (1988). In their study, posture was perturbed by vibrating the calf muscles of subjects. Using the anterior–posterior sway trajectories to fit parameters in linear stochastic models of various orders, they concluded that human postural dynamics can be described by a third-order linear stochastic model. More recently,
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Collins and De Luca (1993) have argued that diffusion plots generated from postural-sway trajectories are indications of a nonlinear stochastic process. In response, Newell et al. (1997) and Peterka (2000) have argued that the diffusion plots do not rule out a linear model. In this paper, our overall goal is to illustrate how the stochastic structure of postural sway imposes important constraints on models of postural control. We accomplish this by first fitting linear stochastic models of various orders to postural-sway trajectories from an experiment in which different combinations of sensory information were available for quiet stance. For each subject and sensory condition, an appropriate model order is determined and the stochastic structure of the model is characterized by its eigenvalues and autocovariance-function coefficients. We then compare our experimental results to the predicted stochastic structure of two types of well-known linear control-theory models: (i) a third-order proportional–integral–derivative (PID) control model in which the system’s state is assumed to be known, and (ii) a fourth-order optimalcontrol model in which the postural system’s state is estimated based on noisy multisensory information using a Kalman filter. Based upon the failure of these models to account for important aspects of the data, we then consider a modification of the optimal-control model in which noise is added to the computations performed during state estimation. We identify a parameter regime for this model in which the model’s behavior is qualitatively similar to the data. In this parameter regime, sensory information provides more accurate information about the body’s velocity than about its position or acceleration.
1.1 Optimal-control model Optimal-control theory (Bryson and Ho 1975) has often been used to model multisensory fusion and postural control (e.g., Gusev and Semenov 1992; Kuo 1995; van der Kooij et al. 1999). In the simplest case, one considers a single-segment inverted pendulum whose movements are restricted to a plane and whose dynamics are linearized around vertical. Then, the optimal-control model is x_ 1 ¼ x2 ;
ð1aÞ
x_ 2 ¼ cx1 c1 x^1 c2 x^2 þ rnðtÞ ;
ð1bÞ
^x_ 1 ¼ x^2 þ K11 ðz1 x^1 Þ þ K12 ðz2 x^2 Þ þ K13 ðz3 c^ x1 Þ;
ð1cÞ
^x_ 2 ¼ c^ x1 c1 x^1 c2 x^2 þ K21 ðz1 x^1 Þ þ K22 ðz2 x^2 Þ x1 Þ ; þ K23 ðz3 c^
ð1dÞ
where z1 ¼ x1 þ r1 n1 ðtÞ ;
ð1eÞ
z2 ¼ x_ 1 þ r2 n2 ðtÞ ¼ x2 þ r2 n2 ðtÞ ; z3 ¼ €x1 þ r3 n3 ðtÞ þ c1 x^1 þ c2 x^2
ð1fÞ
¼ cx1 þ rnðtÞ þ r3 n3 ðtÞ ;
ð1gÞ
and the coefficients c1 and c2 and the matrix K are chosen to minimize 1 J ¼ E a1 x21 þ a2 x22 þ bu2 ; 2
ð1hÞ
where u ¼ c1 x^1 c2 x^2 , and nðtÞ, n1 ðtÞ, n2 ðtÞ, and n3 ðtÞ are independent white-noise processes. The four variables in the model are: x1 ðtÞ, the position of some point on the body in terms of lateral displacement from vertical (one can equivalently use angle rather than displacement); x2 ðtÞ, the velocity of the chosen point; x^1 ðtÞ, the estimated position; and x^2 ðtÞ, the estimated velocity. The behavior of the optimal-control model is determined by seven parameters: c, r, r1 , r2 , r3 , a1 =b, and a2 =b. Equations (1a) and (1b) describe the dynamics of the inverted pendulum. The term cx1 ðtÞ is the acceleration produced by gravity, which acts to destabilize vertical posture. The remaining terms on the right-hand side of (1b) constitute the acceleration due to applied muscular forces that act to stabilize vertical posture: the control signal u ¼ c1 x^1 ðtÞ c2 x^2 ðtÞ is the applied acceleration specified by the controller, and the process noise rnðtÞ is the difference between the actual and specified applied accelerations. Equations (1e)–(1g) describe three types of noisy sensory measurements: z1 ðtÞ is the position measurement, z2 ðtÞ is the velocity measurement, and z3 ðtÞ is the transformed acceleration measurement, which is obtained by subtracting the control signal u from the original acceleration measurement €x1 þ r3 n3 ðtÞ. If two or more sensory measurements of the same type are available from different sensory modalities, the corresponding zk ðtÞ is taken to be a weighted average of the different measurements (Gusev and Semenov 1992). Thus, for example, adding a sensory modality that provides velocity information decreases the velocity-measurement noise level r2 . Equations (1c) and (1d) are a steady-state Kalman filter, which uses the noisy sensory measurements to generate the state estimates x^1 ðtÞ and x^2 ðtÞ. The terms x^2 and c^ x1 c1 x^1 c2 x^2 on the right-hand side constitute the internal-model or forward-model component of the Kalman filter, which describes the predicted evolution of the system over time. The remaining terms on the righthand side describe how sensory measurements are used to change the current state estimates x^1 and x^2 . The coefficients Kjk make up the Kalman gain matrix K, which specifies how the different types of sensory measurements are weighted. Equation (1h) is the performance index. The parameters a1 and a2 describe how heavily deviations from zero of position and velocity are penalized, and the parameter b describes how heavily nonzero control signals u are penalized. 1.2 PID-control model In the optimal-control model above, the specified control depends only on the estimated position and velocity. This is optimal in cases in which the parameters
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specifying the dynamics of the system are known exactly. However, if there are unknown biases in the system’s dynamics, then it is often advantageous to have the control also depend on the integral of position (Stengel 1994), leading to a PID-control model for posture. Assuming that there are no errors in the estimated position x^1 and estimated velocity x^2 in (1b) and adding an integral term to the control signal, (1a) and (1b) become €x1 ðtÞ ¼ q1 x1 ðtÞ q2 x_ 1 ðtÞ Z t x1 ðsÞds þ rnðtÞ ; q0
ð2Þ
t0
where q1 ¼ c1 c and q2 ¼ c2 . Model (9) has four parameters: q1 , q2 , q3 , and r. Johansson et al. (1988) present a PID-control model for posture that is equivalent to model (2), except that their model does not explicitly contain any noise terms. They argue for the existence of the integral term in the model, which leads to third-order dynamics rather than second-order dynamics, based on their analysis of postural-sway trajectories described above. Fig. 1. Experimental setup
2 Methods 2.1 Experimental methods Subjects. Nine subjects from the University of Maryland participated in the study. Subjects ranged in age from 20 to 34 years and had no known musculoskeletal injuries or neurological disorders that might have affected their ability to maintain balance. All subjects were given both oral and written task instructions and gave written consent according to guidelines implemented by the Institutional Review Board at the University of Maryland before undergoing the experimental protocol. Experimental setup. The experimental setup illustrated in Fig. 1 was used to present static visual and somatosensory information. The visual display consisted of a computer-generated whole-field pattern of random dots and was positioned approximately 40 cm in front of the subject. A horizontal touch plate was positioned at approximately hip level at a comfortable distance directly to the right of the subject. A tracking sensor meant to approximate the vertical and medial–lateral position of the center of mass was affixed to the subject’s lower back at the level of the fourth or fifth lumbar vertebra. The location of this sensor was measured using an ultrasound position tracking system (Logitech) at a sampling rate of 50 Hz. Further details of this experimental paradigm are available in Jeka et al. (1998, 2000) and Jeka and Lackner (1994).
open and lightly contacted the touch plate with their right index fingertip; (ii) light touch, in which subjects stood with eyes closed and lightly contacted the touch plate; (iii) vision, in which subjects stood with eyes open and hands hanging loosely by their side; and (iv) neither touch nor vision, in which subjects stood with eyes closed and hands hanging loosely by their side. Procedure. Each subject stood on the force platform in a modified tandem Romberg stance with the back of the right foot and front of the left foot aligned on the same frontal plane, and with the inside edges of the feet aligned on the same sagittal plane. Foot position was marked in order to maintain the same foot position for each trial. The subject was instructed to maintain the specified stance with their head level. Subjects were told that an auditory alarm would sound if the applied fingertip forces exceeded 1 N. They were instructed to reduce the amount of force to stop the alarm without losing contact with the touch plate. Analysis of contact forces indicate that all subjects maintained mean applied forces below 1 N on all trials. Mean contact forces averaged across trials were 0.4400.148 N (meanSD) for the light-touch condition and 0.3590.152 N for the light-touch and vision condition. Individual trials were 4 min in duration and completely randomized. Subjects were allowed at least 90 s of seated rest between trials. Data collection lasted 2 h. 2.2 Data analysis methods
Sensory conditions. Three trials were performed in each of four separate sensory conditions: (i) both light touch and vision, in which subjects stood with eyes
Every fifth point of the medial–lateral center-of-mass postural-sway trajectories were used for analysis, corre-
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sponding to a time step h of 0.1 s. (By comparison, Johansson et al. (1988) used at time step of 0.2 s.) The trials for each condition were analyzed together. Thus, ðjÞ the data had the form fx1;k : j ¼ 1; . . . ; m; k ¼ 1; . . . ; nj g, where j is the trial index and k is the time step index. In most cases, m was 3 and nj was 2400. Three trials were discarded and one trial was shortened due to technical difficulties during data collection. The data were used to fit parameters in the ðp; qÞ autoregressive moving-average (ARMA) process (Wei 1990) ðjÞ
ðjÞ
ðjÞ
ðjÞ
X1;k ¼ x1 þ /1 ðX1;k1 x1 Þ þ ðjÞ
ðjÞ
ðjÞ
ðjÞ
;
ðjÞ
ðjÞ
ðjÞ
P ðX1;9 ¼ x1;9 ; . . . ; X1;nj ¼ x1;nj jX1;9p
j¼1 ðjÞ
ðjÞ
ðjÞ
¼ x1;9p ; . . . ; X1;8 ¼ x1;8 Þ :
ðjÞ
d0 ;
of the ARMA process, where d0 ¼ 1 and dd ¼ 0 for d 6¼ 0. When jpþ1 is positive, it can be interpreted as the variance of output noise. The eigenvalues l1 ; . . . ; lp and coefficients j1 ; . . . ; jp were arranged in order of decreasing jjl ll j. The variance of the process is jtot ¼ j1 þ þ jpþ1 . To convert from time step indices k to time t in units ðjÞ ðjÞ of seconds, we let X1 ðkhÞ ¼ X1;k and computed the eigenvalues kl ¼ logðll Þ=h. Then ðjÞ
ðjÞ
ðjÞ
¼ j1 ek1 s þ þ jp ekp s þ jpþ1 ds ;
where is the random variable corresponding to the ðjÞ measured value x1;k (see Eq. 3 below), p is the autoregressive order, /1 ; . . . ; /p are the autoregressive coefficients, q is the moving-average order, h1 ; . . . ; hq are the ðjÞ moving-average coefficients, and the ak are independent normally distributed random variables with standard ðjÞ deviation ra . The parameters x1 are the asymptotic ðjÞ means limk!1 E½X1;k for each trial. Sampling one variable from a pth-dimensional linear (in the narrow sense) stochastic differential equation (Arnold 1974) at fixed time intervals produces a ðp; p 1Þ ARMA process. Adding output noise (independent normally distributed random variables) to a ðp; p 1Þ ARMA process produces a ðp; pÞ ARMA process. With this in mind, we let q ¼ p; the total number of parameters is then 2p þ m þ 1. Parameters were fitted for models of order p ¼ 1; . . . ; 8 using the method of maximum likelihood. The likelihood function was the product of conditional probability densities: ðjÞ
ðjÞ
¼ j1 ld1 þ þ jp ldp þ jpþ1 dd ;
ðjÞ
ðjÞ X1;k
m Y
ðjÞ
E½ðX1 ðtÞ x1 ÞðX1 ðt þ sÞ x1 Þ
ðjÞ
þ /p ðX1;kp x1 Þ þ ak h1 ak1 ðjÞ hq akq
ðjÞ
E½ðX1;k x1 ÞðX1;kþd x1 Þ
ð3Þ
No assumption was made about the initial states ðjÞ ðjÞ X1;1 ; . . . ; X1;8 . Thus, our fitting procedure allowed for the possible existence of transients. In particular, a slow trend in the data could be represented in the model as a slowly decaying transient. The logarithm of the likelihood function was computed using an exact method (cf. Ansley 1979) and maximized using a quasi-Newton method (Dennis and Schnabel 1983). Pairwise comparisons of models of different orders were made using a likelihood-ratio test at a significance level of 0.05. We selected the model of maximum order that fit the data significantly better than all models of lesser orders. For the selected order p, we computed the eigenvalues ll and coefficients jl defining the autocovariance function
s0 ;
ð4Þ
where t and s are multiples of the time step h. For each postural-sway measure of interest, we tested for a significant dependence on sensory condition using a one-way repeated-measures ANOVA with a Greenhouse–Geisser correction factor. Post-hoc pairwise comparisons were made using the Ryan–Einot–Gabriel– Welsch multiple-range test. 3 Results 3.1 Experimental results For each of the nine subjects and for each of the four sensory conditions, we used the subject’s postural-sway trajectories to fit parameters in linear stochastic models of orders 1 to 8 (see Sect. 2.2). We then selected the appropriate order based on likelihood-ratio tests. The goal was to find a model whose postural-sway trajectories were statistically similar to the experimental trajectories. Figure 2 illustrates this fitting procedure. The top graph shows one of the three experimental postural sway trajectories of subject R.H. in the touch condition. The remaining four graphs show simulated trajectories from linear stochastic models of orders 1 through 4. For each order, parameters in the model were estimated based on the three experimental trajectories. For this subject and condition, the fourth-order model was selected. For each subject and sensory condition, the selected model can be characterized by the eigenvalues kl and coefficients jl of its autocovariance function (4). For the fourth-order model in Fig. 2, the term j4 ek4 s in the autocovariance function is quite small when s is equal to or greater than the time step h = 0.1 s; this was typical. In 34 of the 36 cases, jjl ekl h j was smaller than 1% of the total variance jtot for some l. We refer to the corresponding kl and jl as ‘‘insubstantial.’’ The insubstantial kl and jl may correspond to biological processes or to small sources of error introduced during data collection. We will not report their values in what follows, although they remain part of the fitted model. In our experience, the use of higher-order models including both substantial and insubstantial eigenvalues and coefficients tends to produce substantial eigenvalues and coefficients that are more consistent across subject.
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Fig. 2. Experimental and simulated postural-sway trajectories. The experimental trajectory is of subject R.H. in the touch condition. We are modeling sway trajectories as the outputs of a stochastic process. (Even under the same conditions, experimental trajectories vary from trial to trial.) Therefore, we do not expect the model trajectories to match the exact time course of experimental trajectories, only their stochastic properties. For the first-order model, k1 ¼ 0:225 s1 , j1 ¼ 0:081 cm2 , and j2 ¼ 0:001 cm2 . For the second-order model, k1;2 ¼ 1:311 1:155i s1 , j1;2 ¼ 0:040 0:046i cm2 , and j3 ¼ 0:000 cm2 . For the third-order model, k1 ¼ 0:189 s1 , k2;3 ¼ 0:856 2:850i s1 , j1 ¼ 0:058 cm2 , j2;3 ¼ 0:011 0:005i cm2 , and j4 ¼ 0:000 cm2 . For the fourth-order model, k1 ¼ 0:125 s1 , k2;3 ¼ 1:055 2:666i s1 , k4 ¼ 30:328 s1 , j1 ¼ 0:052 cm2 , j2;3 ¼ 0:013 0:007i cm2 , j4 ¼ 0:002 cm2 , and j5 ¼ 0:002 cm2
Table 1 presents information describing the model of selected order for each subject and sensory condition. The second-to-last column shows the postural-sway variance jtot . The average sway variance across subjects was 0:19 0:12 cm2 in condition B (touch and vision), 0:33 0:25 cm2 in condition T (touch), 0:48 0:28 cm2 in condition V (vision), and 0:97 0:58 cm2 in condition N (neither touch nor vision). All differences between conditions were statistically significant (P < 0:05) except the difference between conditions B and T and the difference between conditions T and V. Table 1 also shows the substantial eigenvalues kl and coefficients jl , arranged in order of decreasing jjl ekl h j. The first eigenvalue k1 was real-valued in all cases, corresponding to a first-order decay component of the autocovariance function. The eigenvalue k1 had a mean of 0:23 0:27 s1 and did not vary significantly across sensory conditions (P > 0:05). In 33 of the 36 cases, there was a pair of complex conjugates among the substantial eigenvalues, corresponding to a dampedoscillatory component of the autocovariance function. In 25 of the 36 cases, the real-valued k1 and one pair of complex-conjugate eigenvalues were the only substantial eigenvalues present. In seven cases there were two real-valued eigenvalues and a pair of
complex-conjugate eigenvalues. In two cases (A.B.:V and J.O.:V) there were three real-valued eigenvalues. In one case (J.J.:V) there were two real-valued eigenvalues. Finally, in one case (R.C.:B) there was one realvalued eigenvalue and two pairs of complex-conjugate eigenvalues. In the remainder of this section, we limit our attention to the 33 cases in which there was a pair of substantial complex-conjugate eigenvalues. In the case R.C.:B, we consider only the first of the two pairs present. For these 33 cases, the average of k1 was 0:19 0:25 s1 . The real part of the complex-conjugate eigenvalues had an average of 1:43 0:59 s1 and was always more negative than k1 , indicating that the damped-oscillatory component of the autocovariance function always decayed more quickly with time than the first-order component. Therefore, we will refer to k1 as the slowdecay eigenvalue. The absolute value of the imaginary part of the complex-conjugate eigenvalues had a mean of 1:95 0:70 s1 , corresponding to a frequency of 0.310.11 Hz for the damped-oscillatory component of the autocovariance function. The slow-decay eigenvalue and the real and the imaginary parts of the complexconjugate eigenvalues did not show a significant dependence on sensory condition (P > 0:05).
267 Table 1. Model fits based on experimental postural-sway trajectories. The first column lists the subject’s initials and sensory condition: B, both; T, touch; V, vision, and N, neither. The integer p is the order of the model. Only substantial eigenvalues and
coefficients are shown. Case R.C.:B had a second pair of substantial complex-conjugate eigenvalues k4;5 ¼ 3:382 2:328i with coefficients j4;5 ¼ 0:000 0:053i (asterisks)
Case
p
Eigenvalues kl (s1 )
Coefficients jl (cm2 )
jtot (cm2 )
j1 =jtot
A.B. B T V N
7 6 6 6
0:465 2:027 2:318i 0:013 0:655 1:813 3:430i 0:614 0:015 5:418 0:595 0:034 2:151 3:064i
0:104 0:001 0:009i 0:003 0.200 0:004 0:021i 0.337 0:094 0:048 0.820 0:113 0:026 0:058i
0:099 0.209 0.386 0.879
1:047 0.957 0.874 0.933
A.P. B T V N
4 3 4 7
0:037 0:987 1:586i 0:026 0:891 1:913i 0:066 1:176 0:913i 0:055 0:502 1:768i
0.077 0.128 0.092 0.649
0:004 0:004i 0:005 0:003i 0:024 0:035i 0:024 0:010i
0.086 0.138 0.139 0.692
0.898 0.926 0.658 0.938
E.B. B T V N
8 7 7 7
0:032 1:502 1:206i 0:047 1:894 3:278 3:025i 1:113 2:031 2:145i 0:022 0:274 1:782 2:339i
0.167 0.140 0.370 0.167
0:012 0:019i 0:081 0:014 0:010i 0:039 0:059i 0:007 0:001 0:009i
0.191 0.192 0.284 0.164
0.871 0.729 1.300 1.017
J.D. B T V N
7 3 4 5
0:079 1:584 1:007i 0:255 1:679 2:373i 0:079 0:985 1:327i 0:021 0:935 1:088i
0.063 0.353 0.439 0.778
0:007 0:017i 0:002 0:017i 0:065 0:063i 0:092 0:092i
0.078 0.348 0.569 0.963
0.805 1.014 0.771 0.808
J.J. B T V N
8 7 7 7
0:026 1:582 1:627i 5:053 0:791 0:030 1:399 2:581i 0:887 1:714 0:012 1:249 1:543i
0.071 0:015 0:030i 0.011 0.146 0.036 0:001 0:022i 0:520 0:274 0.389 0:191 0:147i
0.113 0.184 0.248 0.768
0.632 0.795 2.095 0.506
J.L. B T V N
5 8 6 6
0:126 1:275 1:093i 0:079 2:345 0:715i 0:330 1:015 2:019i 0:155 1:300 2:184i
0.235 0.213 0.470 0.456
0:055 0:071i 0:040 0:160i 0:013 0:034i 0:057 0:049i
0.345 0.294 0.490 0.572
0.681 0.724 0.958 0.797
J.O. B T V N
5 7 8 7
0:127 1:574 2:386i 0:230 1:159 2:624i 0:393 0:034 3:491 0:058 0:857 1:659i
0.426 0.721 0.722 1.796
0:007 0:016i 0:017 0:036i 0:454 0:108 0:160 0:113i
0.441 0.755 1.077 2.117
0.966 0.955 0.670 0.849
R.C. B T V N
7 8 8 5
0:027 2:814 1:201i 0:028 1:081 1:934i 2:115* 0:020 1:250 1:238i 0:082 1:114 2:030i
0.163 0.658 0.678 1.576
0:023 0:161i 0:035 0:059i 0.065* 0:063 0:074i 0:131 0:108i
0.210 0.792 0.805 1.840
0.777 0.831 0.842 0.857
R.H. B T V N
7 4 8 6
0:123 0:860 2:872i 0:125 1:055 2:666i 0:113 0:912 2:875i 0:091 0:902 1:682i
0.105 0.052 0.319 0.484
0:004 0:003i 0:013 0:007i 0:007 0:007i 0:146 0:091i
0.114 0.079 0.334 0.777
0.924 0.658 0.956 0.623
It is sometimes convenient to describe the slow-decay eigenvalue k1 and the complex-conjugate eigenvalues kk and kl in terms of the parameters b ¼ k1 , the rate constant of the slow decay; a ¼ ðkk þ kl Þ, the damping pffiffiffiffiffiffiffiffiffi parameter of the damped oscillation; and x0 ¼ kk kl , the eigenfrequency of the damped oscillation. The eigenfrequency x0 is the angular frequency of the damped oscillation in the limit as a ! 0. The result above that jk1 j < jReðkk;l Þj translates to b < a=2. The fact that kk and kl are complex conjugates translates to a < 2x0 . Figure 3A–C shows the mean a, x0 , and b across subject for the four sensory conditions. None of these measures showed a significant dependence on sensory condition (P > 0:05). The mean across all 33 cases was
2.851.18 s1 for a, 2.490.71 s1 for x0 , and 0.190.25 s1 for b. The fraction j1 =jtot comparing the slow-decay coefficient j1 to the total variance jtot is helpful in distinguishing among different models of postural control (see Sect. 3.2–3.4). We will refer to j1 =jtot as the slowdecay fraction. The slow-decay fraction was positive in all cases and had a mean of 0.850.15. It did not vary significantly across sensory condition (P > 0:05, Fig. 3E). 3.2 PID-control model We now compare the stochastic structure of our experimental postural-sway trajectories to the experi-
268
that the experimental slow-decay fraction j1 =jtot was typically in the general vicinity of 1. The eigenvalues of the PID-control model (2) are determined by the parameters q0 , q1 , and q2 . Computing eigenvalues from the fitted parameter values of Johansson et al. (1988) from the eyes-closed condition, for each of their six subjects we obtain a real eigenvalue k1 (0:49 0:15 s1 across subjects) and a pair of complex-valued eigenvalues (real part: 2:17 0:66 s1 ; absolute value of imaginary part: 5.471.25 s1 ). Thus, although their eigenvalues tend to be larger in absolute values than ours, the same structure is present: a slow first-order decay and a faster-decaying damped oscillation. This similarity exists despite differences in the experimental protocols. In terms of the parameters b, a, and x0 , the Johansson et al. (1988) parameters translate to b ¼ 0:49 0:15 s1 , a ¼ 4:33 1:32 s1 , and x0 ¼ 5:90 1:35 s1 . The PID-control model is a third-order model, so in this respect it is consistent with our experimental results. The model makes no prediction about the structure of the eigenvalues. Although the model they used to analyze their data included noise terms, Johansson et al. (1988) do not present any of the fitted noise parameters. Therefore, it is not possible to compare the autocovariance coefficients jl that we have computed to their results. The slow-decay fraction j1 =jtot of the PID-control model is determined by its eigenvalues (see Appendix): j1 k1 ðk2 þ k3 Þ ab ¼ 2 ¼ : ðk2 k1 Þðk3 k1 Þ jtot x0 ab þ b2
ð5Þ
Since we are assuming that k1 is real-valued and that k2 and k3 are complex conjugates, the slow-decay fraction (5) is negative. However, our experimental j1 =jtot was usually near 1 and was always positive, in contradiction to the PID-control model.
3.3 Optimal-control model Fig. 3A–C. Mean postural-sway measures across subjects. Means are based on the values in Table 1. The three cases in the vision condition that did not have a pair of substantial damped-oscillatory eigenvalues are excluded. Error bars denote standard errors. See text for further explanation
mental results of Johansson et al. (1988) and to the PIDcontrol model (2), which is a stochastic version of the model proposed by Johansson et al. In this and subsequent sections, we will consider only our typical experiment result and ignore exceptional cases. In most cases, the linear stochastic model fitting our experimental sway trajectories contained three substantial eigenvalues. These eigenvalues consisted of a real-valued k1 and a pair of complex conjugates k2 and k3 with jk1 j < jReðk2;3 Þj. In terms of the autocovariance function (11), k1 corresponds to a slow exponential decay and k2 and k3 correspond to a faster-decaying damped oscillation. It will be critical in what follows
Having shown that the PID-control model (2) is inconsistent with the stochastic structure of our experimental postural-sway trajectories, we now turn our attention of the optimal-control model (1). Our goal is to understand the stochastic structure of the sway trajectories produced by the model. Since the model is fourth-order, this stochastic structure can be characterized by the four eigenvalues and corresponding four coefficients of the autocovariance function (4). ^ ¼ ð^ x1 ; x^2 Þ0 , C ¼ ðc1 ; c2 Þ, and We first let x ¼ ðx1 ; x2 Þ0 , x 0 z ¼ ðz1 ; z2 ; z3 Þ (the prime symbol denotes transpose), so that model (1) can be written in the standard form x_ ¼ Fx GCx^ þ wðtÞ; x^_ ¼ ðF GCÞ^ x þ Kðz H^ xÞ ; where the sensory measurements z, performance index J , and control signal u are given by
269
z ¼ Hx þ vðtÞ; 1 J ¼ E½x0 Ax þ u0 Bu þ 2x0 Nu ; 2 u ¼ C^ x ; and the process noise wðtÞ and measurement noise vðtÞ obey E½wðt1 Þwðt2 Þ0 ¼ dðt1 t2 ÞQ; E½vðt1 Þvðt2 Þ0 ¼ dðt1 t2 ÞR; E½wðt1 Þvðt2 Þ0 ¼ dðt1 t2 ÞT ; and d is the Dirac delta function. The matrices specifying the model are 0 1 1 0 0 1 0 F¼ ; G¼ ; H ¼ @0 1A ; ð6aÞ c 0 1 c 0 A¼ Q¼ T¼
a1 0
0 ; a2
0 0
0 ; r2
0
0
0
0
0
r2
B ¼ ðbÞ; 0
N¼
r21
B R¼@ 0
0 r22
0
0 ; 0
0
ð6bÞ
0
1
0
C A;
3.3.1 Control-function eigenvalues. Here it is shown that the control-function eigenvalues kc1 and kc2 show the greatest resemblance to the experimental complex-conk3 . Therefore, we let a ¼ jugate eigenvalues k2 and pffiffiffiffiffiffiffiffiffiffiffiffi . A simple calculation kc2ffiffiffiffiffiffiffiffiffiffiffiffi ðkc1 þ kc2 Þ and x0 ¼ kc1p shows that a ¼ c2 and x0 ¼ c1 c. The matrix C ¼ ðc1 c2 Þ that specifies the optimal controller is found by a two-step procedure (Bryson and Ho 1975). First, SF F0 S A þ ðSG þ NÞB1 ðN0 þ G0 SÞ ¼ 0
2
r þ r23
ð6cÞ
:
To compute the eigenvalues of the model, it is convenient to express the model in terms of the vector y_ ¼ ðx1 x^1 ; x2 x^2 ; x^1 ; x^2 Þ0 : y_ ¼ Uy þ zðtÞ;
is solved for S. Then, C ¼ B1 ðN0 þ G0 SÞ. This procedure yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 c1 ¼ c þ c 2 þ ; b ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a 1 2 2 c2 ¼ 2 c þ c þ þ : b b We consider the nature of the control-function eigenvalues kc1 and kc2 for three types of performance index J :
0
E½zðt1 Þzðt2 Þ ¼ dðt1 t2 ÞZ ; where F KH 0 U¼ ; KH F GC Q þ KRK0 TK0 KT0 Z¼ KT0 KRK0
The separation theorem or certainty-equivalence principle of optimal-control theory (Bryson and Ho 1975) states that minimizing the performance index (1h) can be achieved by combining an optimal deterministic controller with an optimal state estimator. In terms of the postural model, this means that the choice of the controlfunction coefficients c1 and c2 depends only on c and the parameters a1 , a2 , and b specifying the performance index, whereas the choice of the Kalman gain matrix K depends only on c and the noise levels r, r1 , r2 , and r3 . Therefore, the control-function eigenvalues kc1 and kc2 depend only on c, a1 , a2 , and b, whereas the estimation eigenvalues ke1 and ke2 depend only on c, r, r1 , r2 , and r3 .
TK0 KRK0 KRK0
ð7Þ
:
Since U is a lower-triangular block matrix, two of its eigenvalues are the eigenvalues of F GC and the other two eigenvalues are the eigenvalues of F KH. The eigenvalues of F GC depend only on the parameter c, which specifies the dynamics of the uncontrolled inverted pendulum, and the coefficients c1 and c2 , which specify the control signal u as a function of the state estimates x^1 and x^2 . We will denote these controlfunction eigenvalues as kc1 and kc2 , and their corresponding coefficients as jc1 and jc2 . The eigenvalues of F KH depend only on c and the Kalman gain matrix K. They describe the dynamics of the estimation error x x^. We will denote these estimation eigenvalues as ke1 and ke2 , and their corresponding coefficients as je1 and je2 .
1. If only nonzero control signals pffiffiffi u are penalized (a1 ¼ a2 ¼ 0), then kc1 ¼ kc2 ¼ c (a ¼ 2x0 ), corresponding to a critically damped system. 2. If both nonzero control signals and deviations in velocity from zero are penalized (a1 ¼ 0), then kc1 and kc2 are real-valued (a > 2x0 ), corresponding to an overdamped system. 3. If both nonzero control signals and deviations in position from vertical are penalized (a2 ¼ 0), then kc1 and kc2 are complex conjugates (a < 2x0 ), corresponding to an underdamped system. Therefore, the experimental complex-conjugate eigenvalues k2 and k3 are qualitatively similar to the control-function eigenvalues kc1 and kc2 of the model if the performance index penalizes both nonzero control signals and deviations in position from vertical. From of a and x0 , we have that ffiffiffi pffiffiffi (8) and ourpdefinitions a 2 c and x0 c. The parameter c characterizes the effect of gravity on the body, considered as an inverted pendulum that pivots about the ankles. It is given by c ¼ mgl=I, where g ¼ 9:8 m=s2 is the acceleration due to gravity, m is the mass of the body segment that consists of everything above the ankles, l is the distance between this
270
segment’s center of mass and the ankles, and I is the moment of inertia about the ankles of this segment (Johansson et al. 1988). Under the simplifying assumption that all subjects have the same body proportions and pattern of mass distribution, c ¼ g=h for some constant g, where h is the height of the subject. Using anthropometric parameters (Winter 1990) and typical body proportions, we have estimated that g is roughly 13 m=s2 (data not shown). This gives a c of 7.6 s2 for a typical height of 1.7 m. Therefore, the model predicts that a 5:5 s1 and x0 2:8 s1 for a typical subject. Our experimental means of a ¼ 2:85 s1 and x0 ¼ 2:49 s1 are of the correct order of magnitude, but fall below the predicted ranges, especially in the case of a. 3.3.2 Autocovariance-function coefficients and estimation eigenvalues. We now consider how the model’s behavior depends on the measurement noise levels r1 , r2 , and r3 . In terms of the model, adding a sensory modality causes one or more of these noise levels to decrease. As discussed at the beginning of Sect. 3.3, the controlfunction eigenvalues kc1 and kc2 do not depend on the measurement noise levels. In what follows, we will assume that these eigenvalues are complex conjugates: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a2 : ð9Þ kc1 ; kc2 ¼ i x20 2 4 However, the expressions below also hold if a > 2x0 . Recall that our analysis of experimental sway trajectories typically showed the presence of only three substantial autocovariance-function coefficients. Therefore, to be consistent with this experimental result, one of the coefficients je1 , je2 , jc1 , and jc2 of the model must be small. This will be true only in limiting cases of the model parameters. In the Appendix we outline a perturbation analysis of the model. In this analysis, the estimation eigenvalues ke1 and ke2 and coefficients je1 , je2 , jc1 , and jc2 are approximated under the assumption that exactly one of the measurement noise levels is small. The results of the analysis are presented here. There are three cases to consider. Low position-measurement noise If the noise level for the position measurement is low (r1 small), then the estimation eigenvalues are complex conjugates and are strongly stable, that is, they have large negative real parts: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffiffiffi rr3 ke1 ; ke2 ¼ ð1 iÞu t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oð r1 Þ ; 2r1 r2 þ r23 where the notation Oðk Þ denotes a term of order k for small. The estimation autocovariance coefficients are of order r21 : je1 ; je2 ¼
ar21 5=2 ð1 iÞ þ Oðr1 Þ : 2
The control-function autocovariance coefficients are 0 1 jc1 ; jc2 ¼
r2 B ia C @1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 4ax20 2 4x a2 0
0
1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 8r1 r3 u B C @1 þ at qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ Oðr1 Þ 3 2 2 r ðr þ r3 Þ and the total variance is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r2 1 u 2r3 r1 r33 jtot ¼ þ tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr1 Þ : 2ax20 x20 ðr2 þ r2 Þ3 3
Note that the lowest-order pffiffiffiffiffi dependence of the variance jtot on r1 is of order r1 , and that this dependence is due solely to jc1 and jc2 . Therefore, for r1 sufficiently small, the original fourth-order model is well approximated by the second-order model vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u u 8r1 r33 u 2 u €x1 þ a_x1 þ x0 x1 ¼ rt1 þ at qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðtÞ ; r ðr2 þ r23 Þ3 where nðtÞ is white noise. Thus, in the case of low position-measurement noise, the only substantial eigenvalues are the two control eigenvalues kc1 and kc2 . This is in contrast to our experimental sway trajectories, which typically showed the presence of three substantial eigenvalues.
Low velocity-measurement noise If the noise level for the velocity measurement is low (r2 small), then both estimation eigenvalues are real-valued with ke1 of order r2 (weakly stable) and ke2 of order 1=r2 (strongly stable): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 r2 þ c2 r21 ke1 ¼ þ Oðr32 Þ; rr1 rr3 ke2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr2 Þ : r2 r2 þ r23 The estimation autocovariance coefficients je1 and je2 are of order r2 and r42 , respectively: je1 ¼
ðc þ x20 Þrr1 r2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr22 Þ; 2 x0 r2 þ c2 r21
je2 ¼
ðr2 þ r23 Þar42 þ Oðr52 Þ : r2 r23
The control-function autocovariance coefficients are
271
0
1
2
r B ia C @1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 4ax0 4x2 a2
jc1 ; jc2 ¼
0
0
1
2ar2 r3 C B @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ Oðr22 Þ r r2 þ r23 and the total variance is 0 2
jtot ¼
1
x20 Þr1
r r3 B ðc þ C rr2 þ @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2ax0 r2 þ c2 r2 r2 þ r2 x 0 1
þ
Oðr22 Þ
3
:
The lowest-order dependence of the variance jtot on r2 is of order r2 and this dependence is due to je1 , jc1 , and jc2 . Therefore, for r2 sufficiently small, the fourth-order model has only three substantial eigenvalues: the realvalued ke1 and the pair of complex conjugates kc1 and kc2 , with jke1 j jReðkc1 Þj ¼ jReðkc2 Þj. These eigenvalues are qualitatively similar to the experimental eigenvalues. The estimation eigenvalue ke1 corresponds to the experimental slow-decay eigenvalue k1 , and the control-function eigenvalues kc1 and kc2 correspond to the experimental complex-conjugate eigenvalues k2 and k3 . However, the slow-decay fraction of the model je1 2ðc þ x20 Þar1 r2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr22 Þ jtot r r2 þ c2 r21 is of order r2 , whereas the experimental slow-decay fractions were typically near 1. Low acceleration-measurement noise If the noise level for the acceleration measurement is low estimation eigenvalues are complex (r3 small), then the p ffiffiffiffiffi conjugates of order r3 (weakly stable): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ur r2 þ c2 r2 t 3 pffiffiffiffiffi3 1 þ Oð r3 Þ : ke1 ; ke2 ¼ ð1 iÞ 2rr1 The pffiffiffiffiffiestimation autocovariance coefficients are of order r3 : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r3 r31 r3 c þ x20 u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q t þ Oðr3 Þ : je1 ; je2 ¼ x20 2 ðr2 þ c2 r2 Þ3
Fig. 4A–D. The stochastic structure of the optimal-control model (solid lines) and noisy-computation model (dashed lines) for varying levels of velocity-measurement noise. The optimal-control model performs state estimation without computation noise. The noisycomputation model includes computation noise in its state estimator and optimizes estimation rather than control. For both models, c ¼ 4 s2 , r ¼ 1 cm s3=2 , r1 ¼ 10 cm s1=2 and r3 ¼ 10 cm s3=2 . For the optimal-control model, a1 ¼ 4, a2 ¼ 0 s2 , and b ¼ 1 s4 . The resulting control-function coefficients are c1 ¼ 8:47 s2 and c2 ¼ 4:12 s1 . For the noisy-computation model, c1 ¼ 8:47 s2 , c2 ¼ 4:12 s1 , rc1 ¼ 0:1 cm s1=2 , rc2 ¼ 5 cm s3=2 , d1 ¼ 1, and d2 ¼ 1 s2 . The results shown were obtained by numerically minimizing the appropriate performance index
1
The control-function autocovariance coefficients are 0 1 r2 B ia C jc1 ; jc2 ¼ @1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA þ Oðr3 Þ 2 4ax0 2 2 4x a 0
and the total variance is
jtot
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r2 c þ x20 u 2r3 r31 r3 ¼ þ tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ Oðr3 Þ : 2ax20 x20 ðr2 þ c2 r2 Þ3 1
In the expression for jtot , the zeroth-order term is due to the control-function pffiffiffiffiffiautocovariance coefficients jc1 and jc2 , and the Oð r3 Þ term is due to the estimation autocovariance coefficients je1 and je2 . Therefore, it is
272
not possible to approximate the original fourth-order model by a lower-order model without losing the dependence of the variance jtot on the measurement noise levels. In summary, of the three parameter regimes studied, only in the regime of low velocity-measurement noise (r2 small) does the optimal-control model have eigenvalues consistent with our experimental eigenvalues. In this regime, the fourth-order model has three substantial eigenvalues, consistent with our experimental results. The estimation eigenvalue ke1 is small, consistent with our experimental slow-decay eigenvalue k1 . However, there is a striking inconsistency between the model and experimental autocovariance coefficients: the slow-decay fraction je1 =jtot of the model is small, whereas our experimental slow-decay fractions were typically near 1. The solid lines in Fig. 4 illustrate certain key measures of the model’s behavior as r2 is varied. Note that the slow-decay fraction je1 =jtot only begins to approach 1 for relatively large values of the slow-decay rate constant b ¼ ke1 . Also, note the ratio je2 =jtot . When this ratio is small, je2 is insubstantial and the model has only three substantial coefficients. 3.4 Noisy-computation model Given the failure of the optimal-control model (1) to account for slow-decay fractions near 1, we are led to consider a modification of the model. Specifically, we add noise terms rc1 nc1 ðtÞ and rc2 nc2 ðtÞ, where nc1 ðtÞ and nc2 ðtÞ are independent white-noise processes, to the Kalman filter equations (1c, d): ^x_ 1 ¼ x^2 þ K11 ðz1 x^1 Þ þ K12 ðz2 x^2 Þ þ K13 ðz3 c^ x1 Þ þ rc1 nc1 ðtÞ ;
ð10aÞ
^x_ 2 ¼ c^ x1 c1 x^1 c2 x^2 þ K21 ðz1 x^1 Þ þ K22 ðz2 x^2 Þ þ K23 ðz3 c^ x1 Þ þ rc2 nc2 ðtÞ :
ð10bÞ
We will refer to the processes rc1 nc1 ðtÞ and rc2 nc2 ðtÞ as computation noise. Computation noise affects the dynamics of x^1 and x^2 even in the absence of sensory information (Kjk 0), unlike measurement noise, whose effects scale with the measurement weights Kjk . Computation noise is biologically plausible, given that neural computations cannot be performed with perfect accuracy. The separation theorem of optimal-control theory does not hold when computation noise is present. Therefore, the link between optimal estimation and optimal control is broken and we must decide which to pursue. We choose to optimize estimation for two reasons. First, it is reasonable to expect that estimates of the body’s position and velocity are used for purposes other than postural control. If we were to optimize control, there would be no constraint preventing the estimation error x x^ from becoming large. Thus, x^ would no longer serve as a general-purpose state
estimate. The second reason is technical and is related to the first. Given the way we have represented computation noise, its effect on the performance index (1h) can be made arbitrarily small by rescaling certain parameters and the estimates x^1 and x^2 . This solution is not biologically meaningful, because it leads to arbitrarily large values of x^1 and x^2 . Since we are optimizing estimation rather than control, we take the control-function coefficients c1 and c2 to be fixed. We consider the problem of choosing the measurement weights Kjk to minimize the performance index h i J ¼ E d1 ðx1 x^1 Þ2 þ d2 ðx2 x^2 Þ2 ; ð10cÞ where d1 and d2 are positive. We will refer to model (1) with modifications (10) as the noisy-computation model. The model has 11 parameters: c, r, c1 , c2 , r1 , r2 , r3 , rc1 , rc2 , d1 , and d2 . In the Appendix we outline a perturbation analysis of the model under the assumption that the noise levels r, r2 , and rc1 are small compared to the noise levels r1 , r3 , and rc2 . Specifically, we assume that r, r2 , and rc1 are of order , where is a small parameter. We choose this parameter regime because, as we will see, it leads to model behavior similar to our experimental results. Note that our assumption that r2 r1 ; r3 corresponds to the case of low velocity-measurement noise in Sect. 3.3. In addition, we are assuming that r, the level of process noise, and rc1 , the level of computation noise for the position estimate x^1 , are low. The results of our perturbation analysis are as follows. As before, the control-function eigenvalues are given by (9). Both estimation eigenvalues are real-valued with ke1 of order (weakly stable) and ke2 of order 1= (strongly stable): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½r23 r2c2 þ c2 r21 ðr23 þ r2c2 Þ ðr22 þ r2c1 Þ þ Oð2 Þ; ke1 ¼ r1 r3 rc2 rc2 þ Oð1Þ : ke2 ¼ r2 Therefore, ke1 corresponds to a slow-decay component of the autocovariance function, as in the case of low velocity-measurement noise for the optimal-control model. However, now ke1 does not approach 0 as r2 ! 0, if the computation-noise level rc1 is nonzero. The estimation autocovariance coefficient je1 is of order : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r22 þ r2c1 ðc þ x20 Þ2 r1 r3 rc2 je1 ¼ x40 r23 r2c2 þ c2 r21 ðr23 þ r2c2 Þ þ Oð2 Þ :
ð11Þ
The other autocovariance coefficients are smaller than je1 : je2 ¼ Oð5 Þ; Therefore,
jc1 ; jc2 ¼ Oð2 Þ :
273
jtot ¼ je1 þ Oð2 Þ;
je1 ¼ 1 þ OðÞ : jtot
Thus, the sway trajectories x1 ðtÞ of the model in this parameter regime are qualitatively similar to our experimental postural-sway trajectories. The coefficient je2 is small, implying that the fourth-order model has only three substantial eigenvalues, consistent with our experimental results. The estimation eigenvalue ke1 corresponds to the experimental slow-decay eigenvalue k1 , and the control eigenvalues kc1 and kc2 correspond to the experimental complex-conjugate eigenvalues k2 and k3 . The slow-decay fraction je1 =jtot is close to 1, consistent with our experimental slow-decay fraction j1 =jtot . The dashed lines in Fig. 4 illustrate the behavior of the model in our chosen parameter regime. Comparing Fig. 3C–E with Fig. 4A–C, we see that the effect of changing sensory condition in the experiment is qualitatively similar to the effect of changing the velocitymeasurement noise level in the noisy-computation model. For the experimental data, the variance (Fig. 3D) shows a significant dependence on sensory condition, whereas the slow-decay rate (Fig. 3C) and slow-decay fraction (Fig. 3E) do not. For the noisycomputation model, the variance (Fig. 4B) shows a large increase as the velocity-measurement noise level is increased, whereas the slow-decay rate (Fig. 4A) and slowdecay fraction (Fig. 4C) change relatively little. The lowest-order term for the variance jtot , which is given by je1 in (11), depends on all three measurement noise levels. Therefore, changing the position- and acceleration-measurement noise levels can, in some cases, have effects similar to those produced by changing the velocity-measurement noise level. Consequently, when comparing our experimental results with the noisycomputation model, it is not possible to uniquely identify changes in sensory condition with changes in one particular type of measurement noise level. However, the noisy-computation model does offer the following interpretation of our experimental results: changes in sensory condition correspond to changes in one or more measurement noise levels such that the system remains in the regime of low velocity-measurement noise across all the sensory conditions tested. Note that the analytical results presented above for the noisy-computation model do not depend on the parameters d1 and d2 in (10c). Numerical calculations of postural-sway measures for the model (for example, those in Fig. 4) also do not show any dependency on d1 and d2 , assuming both are positive. This implies that there is no trade-off between reducing position estimate errors and reducing velocity estimate errors; both are minimized for the same measurement weights Kjk . 4 Discussion 4.1 Descriptive and mechanistic models of postural control In this paper, we have considered both descriptive and mechanistic models of the postural control system. We
first used descriptive models to characterize the stochastic structure of experimental postural sway. We then compared that structure to the structure predicted by several simple mechanistic models. Our major findings are: 1. Simple classical control-theory models are inconsistent with postural-sway data, even in the case of quiet unperturbed stance. 2. A modified optimal-control model with computation noise is qualitatively consistent with the data. 3. In terms of this modified model, the postural control system under our experimental conditions operates in a parameter regime in which sensory input provides more accurate information about the body’s velocity than its position or acceleration. The descriptive models we used were ARMA models, which are a standard type of discrete-time linear stochastic model. For each subject and condition, we used the experimental sway trajectories to select a model of appropriate order. The selected model was characterized in terms of the eigenvalues and coefficients of its autocovariance function (4). In most cases, the selected model had three substantial eigenvalues. (Eigenvalues whose contributions to the autocovariance function were small were deemed insubstantial.) The eigenvalue k1 was real-valued and small, corresponding to a slow first-order decay component of the autocovariance function. The eigenvalues k2 and k3 were usually complex conjugates, corresponding to a faster-decaying damped oscillation. The slow-decay fraction j1 =jtot , which compares the coefficient of the slow-decay j1 to the total variance jtot , was typically in the general vicinity of 1. We analyzed the behavior of three linear mechanistic models of postural control: (i) the third-order PIDcontrol model (2); (ii) the fourth-order optimal-control model (1); and (iii) the fourth-order noisy-computation model, which is the optimal-control model with modifications (10). The noisy-computation model includes noise in the computation performed by its state estimator and optimizes estimation rather than control. Of these three models, the behavior of the noisy-computation model is most consistent with our experimental results. We were able to identify a parameter regime for this model in which (i) the model has three substantial eigenvalues, ke1 , kc1 , and kc2 ; (ii) the estimation eigenvalue ke1 corresponds to a slow decay; (iii) the control-function eigenvalues kc1 and kc2 correspond to a faster-decaying damped oscillation; and (iv) the slowdecay fraction je1 =jtot is near 1. The qualitative difference in behavior between the optimal-control and noisy-computation models (Fig. 4) is due to the different constraints under which the two models optimize estimation. The noisy-computation model optimizes estimation under the constraint that the computations performed by the estimator are subject to some unavoidable level of errors. The optimal-control model optimizes estimation without this constraint. Thus, viewing postural control in terms of optimization
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requires a careful consideration of the constraints under which optimization takes place. The comparison between the two models also illustrates the importance of using as many experimental sway measures as possible when developing and testing models of postural control. Both models predict that removing sensory information will increase sway variance (Fig. 4B). Therefore, variance alone is not sufficient to distinguish between these two models. Only by considering additional measures – such as the slow-decay rate (Fig. 4A) and slow-decay fraction (Fig. 4C) – is it possible to observe qualitative inconsistencies between the optimal-control model and the experimental data. These inconsistencies led us to modify the optimal-control model by adding computation noise. In this paper we have considered only relatively simple models of postural control. These simple models have the advantage that their behavior can be mathematically analyzed, thereby yielding insights into the problem of postural control. Ultimately, one wants to apply the knowledge gained from simple models to more detailed models of postural control that contain features such as multiple body segments, sensory time delays, sensory dynamics, and adaptive sensory weights (e.g., Kuo 1995; van der Kooij et al. 1999, 2001). The quietstance behavior of such models has been primarily described in terms of sway variance. Therefore, it remains to be seen whether these models are consistent with the additional experimental sway measures described here. In particular, any model with greater than three variables must be structured so that all but three terms of its autocovariance function (4) are small. 4.2 Estimation and control The noisy-computation model provides the following interpretation of our experimental results. Postural control consists of two separate processes: (i) the fusion of multisensory information to produce state estimates, and (ii) the use of the state estimates to specify appropriate muscular responses via a control function. Each of these processes is associated with its own set of eigenvalues. The experimental slow-decay eigenvalue k1 corresponds to the model estimation eigenvalue ke1 . This eigenvalue reflects the dynamics of state estimation and depends only on the noise levels in the system and on the parameter c, which describes the effect of gravity on the body. The model has an additional estimation eigenvalue ke2 , which is insubstantial; that is, its coefficient je2 is small (Fig. 4D). The experimental complex-conjugate eigenvalues k2 and k3 correspond to the model controlfunction eigenvalues kc1 and kc2 . These eigenvalues depend on c and the control-function coefficients c1 and c2 . The view of postural control in terms of estimation and control comes from optimal-control theory. Thus, although the optimal-control model failed to account for important aspects of our experimental data, it served as a useful starting point for considering the problems associated with postural control. In an alternative view of
postural control (Scho¨ner 1991; Jeka et al. 1998, 2000), the individual sources of sensory information are used to directly specify the control signal without any initial fusion into state estimates. This leads to a second-order model of the system in which additional sensory information reduces the postural-sway variance by increasing the effective damping ~a. Our experimental results are inconsistent with such a postural model in two respects. First, we found that postural sway typically shows the presence of three substantial eigenvalues, thus requiring a model of order 3 or higher. Second, although we observed significant changes in sway variance with sensory condition, none of the eigenvalues varied significantly with condition. In particular, our damping parameter a, which most resembles ~a, did not vary significantly with condition. 4.3 Sources of variability in postural sway In the noisy-computation model, the postural-sway variance that is present, even in quiet stance, is attributed to: 1. Process noise associated with the execution of movement. Process noise is the difference between the actual acceleration produced by the muscles and the acceleration specified by the controller. Multifunction muscles are involved in much more than just than the control of upright stance. Therefore, the flexibility of function necessitates a trade-off with precise control. 2. Measurement noise associated with sensory information. Measurement noise includes both noise related to the transduction of stimuli by sensory receptors and noise in the initial processing of sensory information by the nervous system to extract the relevant information (position, velocity, or acceleration) for the given modality. 3. Computation noise associated with central processing of sensory information to produce state estimates. In the parameter regime of the model most consistent with our experimental data, process noise makes only a small contribution to the variance. Most of the variance in postural sway is due to measurement noise and computation noise, which can be seen by examining the lowest-order term for the variance jtot , which is given by the lowest-order term for je1 in (11). This term does not depend on r, the level of process noise, but does depend on all the other noise levels. Thus, most postural sway variance is due to errors in the control signal u specifying the intended acceleration to be produced by the muscles, not errors in translating this control signal into actual muscular activity. The control signal u ¼ c1 x^1 c2 x^2 depends on the state estimates x^1 and x^2 . Thus, errors in the control signal u are due to errors in the state estimates. Errors in the state estimates, in turn, are mainly due to measurement noise and computation noise. Computation noise does not appear in classical control-theory models. From an engineering perspective, it may be reasonable that control algorithms assume that there are no errors in the computations performed by the
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controller. Control-theory algorithms are often developed for machines (e.g., robots) in which the major sources of noise are in the sensors (measurement noise) and the plant (process noise), and not in the onboard computer that determines the control signal. From a biological perspective, however, the presence of significant levels of noise due to errors in the computations performed by a neural (central) estimator is entirely plausible. The effect of computation noise may depend on factors such as age. There is evidence, for example, that poor balance control in elderly individuals arises from central rather than peripheral processes (Horak et al. 1989; Teasdale et al. 1991).
4.4 Sensory information for posture There are numerous descriptions as to how sensory information contributes to the control of upright posture. Typically, sensory feedback is described by the unique contribution of each modality. For example, somatosensation is thought to provide information concerning (i) contact surface forces and properties such as texture and friction, and (ii) the relative configuration of body segments such as ankle angle (Dietz 1992; Horak and Macpherson 1996). Alternative descriptions focus on information derived from sinusoidal sensory stimuli. The sinusoidal vertical-axis rotation technique rotates seated subjects at a range of frequencies to measure the gain and phase of eye movements in the dark as a measure of vestibular function (Krebs et al. 1993; Howard 1982). Likewise, oscillating stimuli have been used to demonstrate the coupling of visual and somatosensory information with whole-body posture (Lee and Lishman 1975; Berthoz et al. 1979; Soechting and Berthoz 1979; van Asten et al. 1988; Dijkstra et al. 1994; Peterka and Benolken 1995; Jeka et al. 1997, 1998). These techniques have determined that rate information is derived from sensory stimuli – for example, the vestibular system provides information about angular and linear acceleration of the head (Wilson and Melvill Jones 1979; Benson 1982) – while the visual and somatosensory system are sensitive to the velocity of a stimulus (Scho¨ner 1991; Dijkstra et al. 1994; Jeka et al. 1997, 1998). The simple models considered in this paper do not identify different forms of sensory information (position, velocity, and acceleration) with specific sensory modalities. Instead, we have emphasized that the behavior of a postural control model depends on which form of sensory information is assumed to be most accurate, regardless of the sensory modalities involved. Specifically, our model analyses suggest that the postural control system, under the sensory conditions tested, operates in a regime in which sensory input provides more accurate information about the body’s velocity than its position or acceleration. Operating in the accurate velocity regime implies that fundamental changes in postural behavior would be expected only if velocity information was eliminated or severely reduced, forcing reliance upon position or
acceleration for estimation of body dynamics. This helps to explain why little change was observed in the stochastic structure of sway (e.g., eigenvalues and slowdecay fraction) in the experimental results presented here. Velocity information could be derived from either vision, touch at the fingertip, and/or somatosensation at the feet/ankles, enabling the postural control system to remain in the accurate velocity regime. One prediction of this view is that the properties of sway should change if all accurate sensory information related to velocity is removed. For example, when a subject stands on a solid surface with eyes closed, the most accurate source of velocity information is thought to come from proprioceptive input from the ankles (e.g., muscle spindles). Vestibular input is thought to provide angular velocity information over some range of frequencies, but only indirectly by transforming the head’s angular acceleration (Wilson and Melvill Jones 1979). Therefore, we predicted that removing modalities that provide velocity as a source of information should change the properties of postural sway. In an experiment designed to test this prediction, our preliminary unpublished data show that the slow-decay rate and the damped-oscillatory fraction (the absolute value of a damped-oscillator coefficient divided by the total variance) change when subjects stand on a foam or swayreferenced support surface with eyes closed. These changes are consistent with predictions from the noisycomputation model. The increase in the oscillatory nature of sway with the removal of ankle proprioception through sway referencing or foam is particularly intriguing in comparison to previous work on ischemic blocking of leg afferents (Mauritz and Dietz 1980; Diener et al. 1984). The ischemic studies found increased sway at frequencies higher than those typically observed during quiet stance, resembling the sway observed in patients with tabes dorsalis. Acknowledgements. We thank Tjeerd Dijkstra for bringing the slow-decay component of postural sway to our attention. Funding for this research was provided by National Institutes of Health grant R29 N35070–01A2, John J. Jeka, PI.
Appendix Here we outline the analysis of the three models considered in the paper. The results of this analysis are presented in Sects. 3.2–3.4. Throughout, our reference for linear stochastic processes is Arnold (1974), and our reference for optimal-estimation theory and optimalcontrol theory is Bryson and Ho (1975). Symbolic computations were performed using Mathematica. Each of the three models is a multidimensional linear stochastic differential equation. Thus, we can use the following general procedure to compute the eigenvalues and coefficients of the autocovariance function (4). First, the model is written in the form y_ ðtÞ ¼ U yðtÞ þ zðtÞ E½zðt1 Þzðt2 Þ0 ¼ dðt1 t2 ÞZ ;
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where yðtÞ 2 Rp , U; Z 2 Rpp , and the medial–lateral postural sway x1 ðtÞ is given by x1 ðtÞ ¼ r0 yðtÞ for some r 2 Rp . The eigenvalues k1 ; . . . ; kp are the eigenvalues of the matrix U. To compute the autocovariance coefficients, we first solve UY þ YU0 þ Z ¼ 0 for the covariance matrix Y ¼ cov½yðtÞ ¼ E½yðtÞyðtÞ0 . Then, the coefficients j1 ; . . . ; jp are given by j ¼ S1 s, where S 2 Rpp , s 2 Rp , and sk ¼ r0 Uk1 Yr :
Skl ¼ kk1 l ;
ðA1Þ
A.1 PID-control model For the PID-control model (2), p ¼ 3, r ¼ ð0; 1; 0Þ0 , 0 1 0 1 0 B C U¼@ 0 0 1 A; q0 q1 q2 0 1 ðA2Þ 0 0 0 B C Z ¼ @0 0 0 A ; 0
0
r2
where q0 ¼ k1 k2 k3 , q1 ¼ k1 k2 þ k1 k3 þ k2 k3 , and q2 ¼ ðk1 þ k2 þ k3 Þ. Applying the procedure for computing j described above, we obtain the slow-decay fraction (5). A.2 Optimal-control model For the optimal-control model (1), p ¼ 4, r ¼ ð1; 0; 1; 0Þ0 , and (7) give the matrices U and Z in terms of the Kalman gain matrix K and the matrices F, G, H, Q, R, T from (6). By optimal-estimation theory, K can be computed by first solving FP þ PF0 þ Q ðPH0 þ TÞR1 ðT0 þ HPÞ ¼ 0
ðA3Þ
for P ¼ cov½xðtÞ x^ðtÞ . The result is 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 þ ~c2 r ~1 r ~21 r22 þ r 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2/ / 2r ~ ~ ~ r r r 1 B 1 2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; @ 2 2 2 2 2 ~1 Þð~ ~1 þ 2/Þ / r2 ð~ r þ ~c r rr
P¼
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~1 r2 ~cr ~2 þ ~c2 r ~ 21 ; ~1 þ r /¼r cr23 ; þ r23 rr3 ~ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; r r2 þ r23 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ r23 ~ 1 ¼ r1 : r r2 þ c2 r21 þ r23
~c ¼
r2
Then, K is given by K ¼ ðPH0 þ TÞR1 . Using this explicit expression for K, we can compute the matrix U and, in particular, the submatrix F KH defining the estimation eigenvalues ke1 and ke2 . Thus, we obtain explicit expressions for ke1 and ke2 . The other two eigenvalues – the control-function eigenvalues kc1 and kc2 – we express in terms of a and x0 using (9). To compute the autocovariance coefficients, we need the covariance matrix Y. By optimal-estimation theory, this matrix is block diagonal: P 0 Y¼ ; ^ 0 X ^ ¼ cov½^ where X xðtÞ is given by ^ þX ^ ðF GCÞ0 þ KRK0 ¼ 0 : ðF GCÞX Letting M ¼ KRK0 , we have that 0 2 2 1 ða þx0 ÞM11 þ2aM12 þM22 M11 2 2ax20 ^ ¼@ A : X x20 M11 þM22 M11 2 2a From (A3) and the expression for K; M ¼ FPþ PF0 þ Q. ^ can be expressed in terms of P. Therefore, X Now that we have explicit expressions for the eigenvalues ke1 , ke2 , kc1 , and kc2 and the matrices U and Y, we can proceed to compute the autocovariance coefficients je1 , je2 , jc1 , and jc2 . To simplify the computation, we assume that exactly one of the measurement noise levels – r1 , r2 , or r3 – is small. We compute the vector s using (A1) and approximate s and the estimation eigenvalues ke1 and ke2 by finite power series in the small parameter. We then obtain the matrix S using (A1). Finally, we compute j ¼ S1 s.
A.3 Noisy-computation model The noisy-computation model consists of model (1) with modifications (10). The matrix Z in (7) is replaced by Z¼
Q þ KRK0 TK0 KT0 þ Rc
TK0 KRK0 Rc
KT0 KRK0 Rc
KRK0 þ Rc
where 2 rc1 Rc ¼ 0
0 r2c2
:
The matrix K is chosen to minimize the performance index J in (10c). We assume that r, r2 , and rc1 are of order , where r2 , is a small parameter, and write r ¼ r, r2 ¼ rc1 , rc1 ¼ K ¼ 1 Kð1Þ þ Kð0Þ þ Kð1Þ þ Oð2 Þ; Y ¼ Yð1Þ þ 2 Yð2Þ þ 3 Yð3Þ þ Oð4 Þ; J ¼ Jð1Þ þ Oð2 Þ :
:
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We let D ¼ UY þ YU0 þ Z. Then, 0 ¼ D ¼ 2 Dð2Þ þ 1 Dð1Þ þ Dð0Þ þ Dð1Þ þ 2 Dð2Þ þ Oð3 Þ : ð1Þ
The condition Dð2Þ ¼ Dð1Þ ¼ 0 implies that K11 ¼ ð1Þ ð1Þ ð1Þ K13 ¼ K21 ¼ K23 ¼ 0. The condition Dð0Þ ¼ 0 ð0Þ ð1Þ ð0Þ implies that K11 ¼ K12 ¼ K13 ¼ 0 and allows us to ð1Þ ð1Þ ð1Þ ð1Þ ð0Þ ð1Þ write Y12 , Y22 , Y23 , and Y24 in terms of K21 , K22 , and ð0Þ ð2Þ K23 . The conditions Dð1Þ ¼ 0 and D11 ¼ 0 allow us to ð1Þ ð2Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð1Þ ð1Þ write Y11 , Y12 , Y13 , Y14 , Y22 , Y23 , Y24 , Y33 , Y34 , and ð1Þ ð1Þ ð0Þ ð1Þ ð0Þ ð1Þ ð0Þ Y44 in terms of K11 , K12 , K13 , K21 , K22 , and K23 . ð1Þ ð0Þ Setting the derivatives of Jð1Þ with respect to K11 , K12 , ð1Þ ð0Þ ð1Þ ð0Þ K13 , K21 , K22 , and K23 equal to 0, we are able to solve for these measurement weights. In particular, ð0Þ ð0Þ K21 ¼ K23 ¼ 0. We now compute the matrix U and the eigenvalues ke1 , ke2 , kc1 , and kc2 . Next, we compute the coefficients je1 , je2 , jc1 , and jc2 from j ¼ S1 s, ¼ r=, where S and s are given by (A1). Finally, we let r 2 ¼ r2 =, and r c1 ¼ rc1 = to obtain the results shown in r Sect. 3.4. References Ansley CF (1979) An algorithm for the exact likelihood of a mixed autoregressive-moving average process. Biometrika 66: 59–65 Arnold L (1974) Stochastic differential equations: theory and applications. Wiley, New York Asten NJC van, Gielen CCAM, Denier van der Gon JJ (1988) Postural adjustments induced by simulated motion of differently structured environments. Exp Brain Res 73: 371–383 Benson AJ (1982) The vestibular sensory system. In: Barlow HB, Mollon JD (eds) The senses. Cambridge University Press, New York, pp 333–368 Berthoz A, Lacour M, Soechting JF, Vidal PP (1979) The role of vision in the control of posture during linear motion. Prog Brain Res 50: 197–209 Bryson AE, Ho YC (1975) Applied optimal control: optimization, estimation, and control. Wiley, New York Collins JJ, De Luca CJ (1993) Open-loop and closed-loop control of posture: a random-walk analysis of center-of-pressure trajectories. Exp Brain Res 95: 308–318 Dennis JE, Schnabel RB (1983) Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Englewood Cliffs, N.J. Diener HC, Dichgans J, Guschlbauer B, Mau H (1984) The significance of proprioception on postural stabilization as assessed by ischemia. Brain Res 296: 103–109 Dietz V (1992) Human neuronal control of automatic functional movements: interaction between central programs and afferent input. Physiol Rev 72: 33–69 Dijkstra TMH, Scho¨ner G, Giese MA, Gielen CCAM (1994) Frequency dependence of the action-perception cycle for postural control in a moving visual environment: relative phase dynamics. Biol Cybern 71: 489–501 Gusev V, Semenov L (1992) A model for optimal processing of multisensory information in the system for maintaining body orientation in the human. Biol Cybern 66: 407–411 Horak FB, Macpherson JM (1996) Postural orientation and equilibrium. In: Shepard J, Rowell L (eds) Handbook of physiology. Oxford University Press, New York, pp 255–292
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