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Exp Bra in Res (1993) 95 :308-318

9 Springer-Verlag 1993

Open loop and c losed loop control of posture:A rand om w alk analys i s o f center of pressure trajectoriesJ . J . C o l l in s C . J . D e L u c aNeuroMuscular Research Center and Department of Biomedical Engineering, Boston U niversity, 44 C umm ington St.,Boston, MA 02215, USAReceived: 5 M ay 1992/A ccepted: 21 Decem ber 1992

Abstract . A n e w c o n c e p t u a l a n d t h e o r e t i c a l f r a m e w o r kf o r s t u d y i n g th e h u m a n p o s t u r a l c o n t r o l s y s t e m is i n tr o -d u c e d . M a t h e m a t i c a l t e c h n i q u e s f r o m s t a t i s t i c a l m e c h -a n i c s a r e d e v e l o p e d a n d a p p l i e d t o t h e a n a l y s i s a n di n t e r p r e t a t i o n o f s t a b i l o g r a m s . T h i s w o r k w a s b a se d o nt h e a s s u m p t i o n t h a t t h e a c t o f m a i n t a i n i n g a n e r ec tpos ture could be v i ewed, i n pa r t , a s a s tochas t i c p rocess .T w e n t y - f i v e h e a l t h y y o u n g su b j e c ts w e r e s t u d i e d u n d e rq u i e t - s t a n d i n g c o n d i t i o n s . C e n t e r - o f - p r e s su r e C O P ) tr a -j e c t o r i e s w e r e a n a l y z e d a s o n e - d i m e n s i o n a l a n d t w o -d i m e n s i o n a l r a n d o m w a l k s. T h i s n o v el a p p r o a c h l ed t o t h ee x t r a c t i o n o f r e p e a ta b l e , p h y s i o l o g i c a l l y m e a n i n g f u l p a r a -m e t e r s f r o m s t a b i l o g r a m s . I t i s sh o w n t h a t a l t h o u g hi n d i v i d u a l s t a b i l o g r a m s f o r a s i n g l e su b j e c t w e r e h i g h l yv a r i a b l e a n d r a n d o m i n a p p e a r a n c e , a c o n s i s t e n t, su b j e ct -sp e c if ic p a t t e r n e m e r g e d w i t h t h e g e n e r a t i o n o f a v e r a g e ds t a b i l o g r a m - d i f f u s i o n p l o t s m e a n sq u a r e C O P d i sp l a ce -m e n t v s t i m e i n t er v a l) . I n a d d i t i o n , s i g n i f ic a n t i n te r - su b j e c td i f fe rences were found in the ca l cu la t ed re su l t s . Thi ssu g g e s t s t h a t t h e s t e a d y - s t a t e b e h a v i o r o f t h e c o n t r o lm e c h a n i s m s i n v o l v ed i n m a i n t a i n i n g e r ec t p o s t u r e c a n b eq u i t e v a r i a b l e e v e n a m o n g s t a p o p u l a t i o n o f a g e - m a t c h e d ,a n t h r o p o m e t r i c a l l y s i m i l a r , h e a l t h y i n d i v i d u a l s . T h e sep o s t u r o g r a p h i c a n a l y s e s a l s o d e m o n s t r a t e d t h a t C O Pt r a j e c t o r i e s c o u l d b e m o d e l l e d a s f r a c t i o n a l B r o w n i a nm o t i o n a n d t h a t a t le a s t t w o c o n t r o l sy s t e m s - a sh o r t -t e r m m e c h a n i s m a n d a l o n g - t e r m m e c h a n i s m - w e r eo p e r a t i n g d U r in g q u i e t s t a n d i n g . M o r e sp e c if i ca l ly , t h ep r e se n t r e su l t s su g g e s t t h a t o v e r sh o r t - t e r m i n t e r v a l so p e n - l o o p c o n t r o l s c h e m e s a r e u t i l i z e d b y t h e p o s t u r a lc o n t r o l sy s t e m , w h e r e a s o v e r l o n g - t e r m i n t e r v a l s c l o se d -l o o p c o n t r o l m e c h a n i sm s a r e c a l l e d i n t o p l a y . T h i s w o r ks t r o n g l y s u p p o r t s t h e p o s i t io n t h a t m u c h c a n b e l e a rn e da b o u t t h e f u n c t i o n a l o r g a n i z a ti o n o f t h e p o s t u r a l c o n t r o ls y s t e m b y s t u d y i n g t h e s t e a d y - st a t e b e h a v io r o f t h e h u m a nb o d y d u r i n g p e r i o d s o f u n d i s t u r b e d s t an c e .Key words: P o s t u r a l c o n t r o l - O p e n - l o o p c o n t r o l -C l o s ed - lo o p c o n tr o l - R a n d o m w a l k - H u m a nCorrespondence to J.J. C ollins

In tr o duc t io nT h e t a s k o f m a i n t a i n i n g a n u p r i g h t p o s t u r e i n v o l v es ac o m p l e x s e n s o r i m o t o r c o n t r o l s y s t e m . E v e n w h e n ay o u n g , h e a l t h y i n d i v i d u a l a t t e m p t s t o s t a n d s t i l l , t h ec e n t e r o f g r a v i t y o f h i s o r h e r b o d y a n d t h e c e n t e r o fp r e s su r e C O P ) u n d e r h i s o r h e r fe e t m o v e r e la t i v e t o ag l o b a l c o o r d i n a t e sy s t e m . A p l o t o f t h e t i m e - v a r y i n gc o o r d i n a t e s o f t h e C O P i s k n o w n a s a s t a b i l o g r a m F i g . 1 ).A n u m b e r o f b io m e c h a n i c a l r e se a r ch e r s h a v e a t t e m p t e d t oe v a l u a t e p o s t u r a l sw a y b y u s i n g a f o r c e p l a t f o r m t om e a su r e t h e a n t e r o p o s t e r i o r a n d m e d i o l a t e r a l d i sp l a c e -m e n t s o f t h e C O P o v e r th e p l a n e o f s u p p o r t. H o w e v e r ,p r e v i o u s a t t e m p t s a t i n t e r p r e t i n g s t a b i l o g r a m s f r o m am o t o r c o n t r o l p e r sp e c t iv e h a v e n o t b e e n su c ce s s f u l. M a n yo f t h e e a r l i e r s t u d i e s l i m i t e d t h e a n a l y s i s o f t h e se p l o t s t osu m m a r y s t a t i s ti c s , i. e. , c a l c u l a t i o n o f t h e l e n g t h o f sw a yp a t h , a v e r a g e r a d i a l a r e a, e t c. D i e n e r e t a l. 1 9 8 4 ; K i r b ye t a l. 1987; No r r6 e t a l. 1987; H asa n e t a l . 1990). By do ingso , t h e se i n v e s t i g a t i o n s i g n o r e d t h e d y n a m i c c h a r a c t e r -i st ic s o f s t a b i lo g r a m s , i .e ., t h e m a g n i t u d e a n d d i r e c t i o n o fd i sp l a c e m e n t s b e t w e e n a d j a c e n t p o i n t s , t h e t e m p o r a lo r d e r i n g o f a s e r ie s o f C O P c o o r d i n a t e s , e t c .T h e m a j o r i t y o f c o n t e m p o r a r y s c i e n t i f i c a n d c l i n i c a li n v e s t i g a t i o n s i n p o s t u r a l c o n t r o l h a v e d i r e c t e d t h e i r a t -t e n t i o n t o a n a l y z i n g t h e r e s p o n se o f t h e h u m a n b o d y t ov a r i o u s e x t e r n a l p e r t u r b a t i o n s M o o r e e t a l. 1 9 8 8; W o o l -l aco t t e t a l . 1988; D ien e r e t a l . 19 88 , 1991; H or ak e t a l .1990; D ie t z e t a l. 1991) . A l t ho ug h th i s re f l ex ive app roa che n a b l e s o n e t o e x a m i n e t h e i n p u t / o u t p u t c h a r a c t e r i s t ic s o fd i f f e re n t c l o se d - l o o p f e e d b a c k sy s t e m s N a sh n e r 1 9 71 ,1972) , i t does n o t cons ide r expl i c i t l y the s t ab i l i z ing ro l e s o fp o s s i b l e o p e n - l o o p c o n t r o l s c h e m e s o r t h e s t e a d y - s t a t eb e h a v i o r o f th e h u m a n b o d y d u r i n g p e r i o d s o f u n d is t u r -b e d s t a n c e. F u r t h e r m o r e , t h e e x p e r i m e n t a l p r o t o c o l s a s so -c i at e d w i t h d y n a m i c p o s t u r o g r a p h y a r e c o n s i d e r a b lym o r e h a z a r d o u s a n d p h y s i c al l y t a x in g t h a n t h o s e i n v o l v edi n s t a t i c p o s t u r o g r a p h y . A l so , t h e o u t p u t o f a p e r t u r b a t i o no r i n p u t i m p u l se is m o d u l a t e d b y t h e s t a t e o f e x c i t a t i o n o ra l e r tness o f t he ind iv idua l Ho rak e t al . 1989; Beckley e t a l .1 99 1) . T h u s , f r o m a c l i n i ca l s t a n d p o i n t , s t a t ic p o s t u r o g r a -

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d i s p l a c e m e n t x m m )Fig. 1. Typical 30-s stabilogram for a healthy young individualduring quiet standing. Mediolateral and anteroposterior COP dis-placements are plotted along the x-axis and y-axis, respectively

phy is a much simpler and safer test to perform andadminister to aged an d disabled individuals. Nonetheless,to date, the utility of static postu rog rap hy in the clinic andlaboratory has been limited by the lack of a reliable,consistent ly useful approach or technique for extract ingrepeatable, physiological ly meaningful information fromstabilograms.In the present study, the problem of characterizingstabil ogra ms is approa ched from the perspective of statist-ical mechanics. In particular, it is postulated that themove ment of the center of pressure during quiet stan dingcan be modelled as a system of coupled, correlated r and omwalks, i.e., the motion is considered to be the result of acomb inati on of determinist ic and stochast ic mechanisms.In order to test this hypothesis, probabilist ic tools andtechniques from statistical mechanics will be introducedinto the experimental domain of posturography. One ofthe aims of this work is to develop a general stochasticmodell ing framework to examine and interpret center-of-pressure time series. In this way, it is hoped that thepresent approach wil l lead to a greater u nders tandi ng ofthe strategies utilized by the postural control system tomain tain the complex, mult i-degree-of-freedom structureof the musculosk eletal system in equi libr ium with externalforces during quiet standing.

ater i a l s and m ethods

Appl i ca t ion o f s ta t i s ti ca l mechanicsFundamental concepts and principles from statistical mechanicshave been applied to the study of a number of different neuro-physiological systems and phenomena (Holden 1976; Tuckwell1989). Bartol et al. (1991), for instance, utilized diffusion theory andstochastic methods for modelling miniature endplate current genera-tion in neuromuscular junctions. In a classic study, Gers tein andMandelbro t (1964) represented the statistical properties of spiketrains of single neurons as random walks towards an absorbingbarrier. Recently, this work has been extended to more detailed

analyses of how information is encoded and t ransmitted by neurons(Gorse and Taylor 1990; Longtin et al. 1991). Others have usedstatistical physics models to simulate and analyze the collectivedynamics and emergent properties of large networks of coupledneurons (Peretto 1984; Sompolinsky 1988).The general driving principle of statistical mechanics is thatalthough the outcome of an individual random event is unpredict-able, it is still possible to obtain definite expressions for the probabil-ities of various aspects of a stochastic process or mechanism. Aclassic example of a statistical mechanical phenomenon is Brownianmotion. The simplest case of Brownian motion is the randommovement of a single particle along a straight line. This construct isknown as a one-dimensional andom walk. In 1905, Einstein studiedBrownian motion and showed that the mean square displacement(Ax2) 1 of a one-dimensional random walk was related to the timeinterval At by the expression:(Ax 2) =2DAt (1)where the parameter D is the diffusion coefficient. In words, thediffusion coefficient is an average measure of the stochastic activity ofa random walker, i.e., it is directly related to its jump frequencyand/or amplitude. The above result is easily extended to higherdimensions, i.e., random walks in a plane or in three-dimensionalspace. In each case, the mean square displacement and time intervalare still linearly related.The term f i actional Brownian motion was introduced byMandelbrot and van Ness (1968) to designate a generalized family ofGaussian stochastic processes. This mathematical concept, which isan extension of classical or ordinary Brownian motion, has beenused to model a number of different natural objects and phenomena,including andscape terrains and fluid turbulence (Mandelbrot 1983).Accessible introductions to the subject are provided by Feder (1988),Saupe (1988) and Voss (1988). For the purposes of the present study,it is important to point out that for fractional Brownian motion theEinstein relation given by Eq. 1 is generalized to the following scalinglaw:(Ax 2) ~ At~" (2)where the scaling exponent H can be any real number in the range0 < H < 1. For classical Brownian motion, H =1 . As can be seen fromEq. 2, the scaling exponent H can be determined from the slope of thelog-log plot of the mean square displacement versus At curve.An impor tant feature of fractional Brownian motion is that pastincrements in a particle's displacement are correlated with futureincrements. The only exception to this rule is the case H =89 which,as stated earlier, corresponds to a classical random walk.2For fractional Brownian motion, the correlation function C, which istime-independent, is given by the expression (Feder 1988):C= 2 2 2H ~-1). 3)

Note that for H > 89 the stochastic process is positively correlated,i.e., C>0. In this case, a fractional Brownian particle moving ina par ticular direction for some t o will tend to continue in the samedirection for t > t o. In general, increas ing (decreasing) trends in thepast imply on the average increasing (decreasing) trends in the future(Feder 1988; Saupe 1988). This type of behavior is known aspersistence.The opposite situation occurs for H< 89 - past and futureincrements are negatively correlated. Thus, an increasing (decreas-ing) trend in the past implies a decreasing (increasing) trend inthe future. This type of stochastic behavior is referred to asanti-persistence.

1The angled brackets ( -) denote an average over time or anensemble average over a large number of samples.2 The increments in displacement making up ordinary Brownianmotion are statistically independent.

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310E x p e r i m e n t a l m e t h o d sTw e nty- f iv e he a l thy m a le subje c t s of s im i la r a ge (19 27 ye a rs , m e a n22.3 ye a r s ) a nd s iz e (body we ight 59 .1-85 .0 kg , m e a n 71 .8 kg; he ight165.1-186.7 c m , m e a n 174.9 c m ) we re inc lud e d in the s tudy . T hem e m b e r s o f t h e s u b je c t p o p u l a t i o n h a d n o e v i d e n c e o r k n o w n h i s t o r yo f a g ai t , p o s t u r a l o r s k e l e t a l d is o r d e r . I n f o r m e d c o n s e n t w a so b t a i n e d f r o m e a c h s u b j e ct p r i o r t o p a r t i c i p a t i o n . P o s t u r a l s t a b i l i tyw a s e v a l u a t e d b y u s i n g a K i s t l e r 9 2 8 7 m u l t i c o m p o n e n t f o r c e p l a t -f o r m t o m e a s u r e t h e t i m e - v a r y i n g d i s p la c e m e n t s o f t h e C O P u n d e r asubje c t s fe e t. Ea c h subje c t wa s ins t ruc te d to s t a nd in a n u pr igh tp o s t u r e i n a s t a n d a r d i z e d s t a n c e o n t h e p l a t fo r m . I n t h e s t a n d a r d i z e ds ta nc e , the subje c t s f e e t we re a bduc te d 10~ a nd the i r he e l s we res e p a r a t e d m e d i o l a t e r a l l y b y a d i s t a n c e o f 3 c m . D u r i n g t h e t e s ti n g ,t h e s u b j e c ts s t o o d b a r e f o o t w i t h t h e i r a r m s c o m f o r t a b l y a t t h e i r s i d esa n d t h e i r e y es o p e n a n d f i x e d o n a p o i n t i n f r o n t o f t h e m . E a c h t r i all a s t ed f o r a p e r i o d o f 30 s a n d t h e f o r c e p l a t f o r m d a t a w e r e s a m p l e d a ta f r e que nc y o f 100 Hz . A se rie s of 30 t r i a ls we re c on du c te d for e a c h ofthe f i r s t 10 subje c t s . Thi s l a rge num be r of t e s t s wa s r e qu i re d to a s se sst h e r e l i a b i l i t y o f t h e p r o p o s e d s t a t i s t i c a l m e c h a n i c a l m e t h o d o l o g y .R e s t p e r i o d s o f 6 0 s a n d 5 m i n w e r e p r o v i d e d b e t w e e n e a c h t r i a l a n dbe twe e n e a c h se t of 10 t r i a l s , r e spe c t ive ly . Te n t r i a l s we re c onduc te dfor e a c h of the r e m a in ing 15 subje c t s .

In orde r to a s se s s the noi se c ha ra c te r i s t i c s of the K is t l e r p la t form ,a l a r g e m a s s o f a p p r o x i m a t e l y 1 0 0 k g w a s p l a c e d o n t h e p l a t f o r m .T h e k i n e t i c d a t a w e r e s a m p l e d a t 1 0 0 H z f o r a p e r i o d o f 3 0 s . I nt h e o r y , t h e C O P o f a s t a ti c o b j e c t s h o u l d b e c o n s t a n t a s a f u n c t io n o ft i m e . I t w a s f o u n d , h o w e v e r , t h a t e x p e r i m e n t a l n o i s e i n t r o d u c e dv a r i a t i o n s t h a t w e r e l es s t h a n + _ 0. 5 m m i n m a g n i t u d e i n t h e m e a s -u r e d c o o r d i n a t e s o f t h e C O P o f t h e t e s t m a s s .

Data ana ly s i s and s tab i l ogram-d i f f us ion p lo t sT h e C O P t r a j e c t o r i e s w e r e s t u d i e d a s o n e - d i m e n s i o n a l a n d t w o -d i m e n s i o n a l r a n d o m w a l k s . T h e d i s p l a c e m e n t a n a l y s i s w a s c a r r i e do u t b y c o m p u t i n g t h e s q u a r e o f t h e d i s p l a c e m e n t s b e t w e e n a l l p a ir sof poin t s s e pa ra te d in t im e by a spe c i f i e d t im e in te rva l A t ( s e eF i g . 2 a ) . T h e s q u a r e d i s p l a c e m e n t s w e r e t h e n a v e r a g e d o v e r t h en u m b e r o f A t m a k i n g u p t h e C O P t i m e s e ri e s. 3 T h i s p r o c e s s w a sr e p e a t e d f o r i n c r e a s in g v a l u e s o f A t. A n i m p o r t a n t p o i n t t o n o t e i st h a t t h e n u m b e r o f c a l c u l a t e d s q u a r e d i s p l a c e m e n t s w a s i n v e r s e lyp r o p o r t i o n a l t o t h e s i ze o f t h e t i m e i n t er v a l . A p l o t o f m e a n s q u a r eC O P d i s p l a c e m e n t v e r s u s t i m e i n t e r v a l A t w i l l b e r e f e r r e d t o a s as tab i logr am-d i f fu s ion p lo t (Fig. 2b).E x p e r i m e n t a l s t u d i e s c o n c e r n e d w i t h d i f f u s i o n - l i k e p r o c e s s e st y p i c a ll y a n a l y z e e i t h e r a l o n g t i m e s e r ie s o f d a t a m e a s u r e m e n t s o r al a r g e n u m b e r o f s m a l l e r ti m e s e r ie s o f su c h m e a s u r e m e n t s ( S h l e s in g e ra n d W e s t 1 9 8 4 ; M o n t r o l l a n d L e b o w i t z 1 9 8 7 ). I n a p o s t u r o g r a p h i ci n v e s t i g a t i o n , i t w o u l d b e i m p r a c t i c a l , h o w e v e r , t o h a v e s u b j e c t ss t a n d o n a f o r c e p l a t f o r m f o r e x t e n d e d p e r i o d s o f t im e . P h y s i o l o g i c a lf a c tor s suc h a s f a t igue would t e nd to obsc ure the r e su l t s . In thep r e s e n t s t u d y , i t w a s t h e r e f o r e d e c i d e d t o c o l l e c t a l a r g e n u m b e r o f30-s t r i a l s for e a c h su bje c t a nd to a na ly z e a ve ra g e d se t s of the r e su l t sde r ive d f rom the se t e s t s . Spe c i f i c a l ly , s t a b i logra m -di f fus ion p lo t sw e r e c o m p u t e d f o r e a c h s u b j e c t t r i a l a n d t h e n 1 0 s u c h c u r v e s w e r ea v e r a g e d t o o b t a i n a r e s u l t a n t s t a b i l o g r a m - d i f f u s i o n p l o t f o r ap a r t i c u l a r s u b j e c t . T h r e e r e s u l t a n t p l o t s w e r e t h u s g e n e r a t e d f o r asubje c t who pa r t i c ipa te d in 30 t r i a l s .D i f fus ion c oe f f i c ie n t s D (Eq. 1) we re c a lc u la te d f ro m th e s lope s oft h e r e s u l t a n t l i n e a r -l i n e a r p lo t s o f m e a n s q u a r e C O P d i s p l a c e m e n tve r sus t im e in te rva l c urve s , i .e ., (Ar 2 vs A t , A x 2 ) v s A t , an d ( A y 2 )v s A t. S i m i l ar l y , s c a l in g e x p o n e n t s H ( E q . 2 ) w e r e c o m p u t e d f r o m t h ere sul t a n t lo g- log p lo t s o f suc h c urve s . In a l l c a se s , the s lope s we red e t e r m i n e d b y u t i l i z i n g th e m e t h o d o f le a s t s q u a r e s t o f i t s t r a i g h t3 I n t h e p r e s e n t s t u d y , { x , } a n d { y , } a r e t h e m e d i o l a t e r a l a n da n t e r o p o s t e r i o r C O P t i m e s e ri e s, r e sp e c t iv e l y , a n d { A r 2 ) = { A x 2 )q- Ay2) .

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6O T iCri t ic a l P o i n t = A t r c , < A r 2 >c )(ram2) 40

200 0 2 4 6 8 10

Time Interval (s)F i g . 2 . a D i a g r a m s h o w i n g t h e m e t h o d f o r c a lc u l a t i n g m e a n s q u a r ep l a n a r d i s p l a c e m e n t ( f i r a ) a s a f u n c t i o n o f ti m e i n t e r v a l A t f o r aC O P t r a je c tory m a de up of N da ta poin t s (x~ , Y l ; x2 , Y2;- . -; xN , YN}-b . A typic a l r e su l t a n t p la na r s t a b i log ra m -d i f fus ion p lo t ( (Ar 2) vs A t )g e n e r a t e d f r o m C O P t i m e s e r i e s a c c o r d i n g t o t h e m e t h o d s h o w n i n(a) . The d i f fus ion c oe f f i c ien t s D~ a nd D ,~ a re c o m p ute d f rom thes l o p e s o f t h e l i n e s f i t t e d t o t h e s h o r t - t e r m a n d l o n g - t e r mre gions , r e spe c t ive ly . The c r i t i c a l po in t , (A t rc , (ArZ)c ) , i s de f ine dby the in te r se c t io n of the l ine s f it t e d to the two re g ions of theplo t . The sc a l ing e xpo ne nts Hrs a nd Hr~ a re c a lc u la te d f ro m thes l o p e s o f t h e l o g - l o g p l o t s o f t h e s h o r t - t e r m a n d l o n g - t e r m r e g i o n s,r e spe c t ive ly

l i n es t h r o u g h d e f i n e d p o r t i o n s o f t h e a f o r e m e n t i o n e d p l o t s. A l lp a r a m e t e r s w e r e d e t e r m i n e d b y a s i n g l e in v e s t i g a t o r .In t r a c la s s c o r re la t io n c oe f f i c ie n t s ( ICCs) we re c a lc u la te d to de te r -m i n e t h e d e g r e e o f a g r e e m e n t b e t w e e n t h e r e s p e c ti v e s t a b i l o g r a m -d i f fu s i o n p a r a m e t e r s w h i c h w e r e e x t r a c t ed f r o m t h e t h r e e r e s u l t a n tp lo t s for e a c h of the f i r s t 10 subje c t s . Th e IC C e q ua t i on for a r a n d o me f fe c t s m ode l , i . e . , ICC e qua t ion (2 ,1) a s de sc r ibe d by Shrout a ndFle i s s (1979) , wa s use d in the pre se nt s tudy . Th is e qu a t ion i s g ive n by

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the expression:BMS EMSlCC 2, 1)= 4)B MS + k - 1) EMS + k J M S- EMS) /n

where n is the number of subjects, k is the number of repeatedmeasures judges) per subject, and B M S, JMS and E MS are thebetween-subjects mean square, between-judges mean square anderro r mean square, respectively. See Shrout and F leiss 1979) forfurther details.Although there are no rigorous, universally accepted standardsfor assessing reliability, Flei ss 1986) offer ed the following generalguidelines: ICC values less than 0.4 represent p oo r reliability, onesbetween 0.4 and 0.75 represent fair to good reliability, and valuesgreater than 0.75 correspond to excellent reliability. This categoriz-ation was adopted in this investigation.

R e s u l t s

R e s u l t a n t p l a n a r s t a b i l o g r a m - d i f f u s i o n p l o t s f o r f o u r r e p -r e s e n t a t i v e s u b j e c t s a r e s h o w n i n F i g. 3 . I t is i m p o r t a n t t on o t e t h a t t h e p o s t u r o g r a p h i c r e s u l t s o f F i g . 3 w e r e q u a l i -t a t iv e l y d i ff e re n t f r o m t h o s e e x p e c t e d f o r o r d i n a r y B r o w -n i a n m o t i o n . W h e r e a s t h e m e a n s q u a r e d i s p l a c e m e n t o f ar a n d o m w a l k g r o w s l i n e a r l y w i t h a c o n s t a n t s l o p e f o ri n c r e a s i n g t i m e i n t e r v a l [ s e e E q . 1 ], t h e s t a b i l o g r a m -d i f f u s i o n c u r v e s c h a n g e d s l o p e a f t e r a t r a n s i t i o n o r c r i t ic a lp o i n t a t s o m e s m a l l A t . T h i s g e n e r a l f e a t u r e w a s f o u n d i nt h e c a l c u l a t e d r e s u l t s f o r al l 2 5 s u b j e c t s w h o p a r t i c i p a t e di n t h e p r e s e n t i n v e s t i g a t i o n . A s w i l l b e d e s c r i b e d b e l o w ,C O P t r a j e c to r i e s a l s o d i ff e re d f r o m c l a ss i c al B r o w n i a nm o t i o n i n o t h e r w a y s .I n o r d e r t o p a r a m e t e r i z e t h e a b o v e s t a b i l o g r a m - d i f f u -s i o n p l o t s , t w o r e g i o n s w e r e i d e n t i fi e d - a s h o r t - t e r mr e g i o n a n d a l o n g - t e r m r e g io n . T h e s e r e g i o n s w e r e s e p a r -a t e d b y a t r a n s i t i o n p e r i o d w h e r e t h e s l o p e o f t h es t a b i l o g r a m - d i f f u s i o n p l o t c h a n g e d c o n s i d e r a b l y . D i f f u -s i o n c o e ff i c ie n t s a n d s c a l i n g e x p o n e n t s w e r e c a l c u l a t e d f o re a c h r e g i o n F i g . 2 b ). S u b s c r i p t s s a n d l w i ll b e u s e dt h r o u g h o u t t h e m a n u s c r i p t t o d e n o t e t h e s h o r t - t e r m a n dl o n g - t e r m r e g i o n s , r e s p e c t i v e l y . T h e l i n e s f i t t e d f o r c o m -pu ta t io n o f D ~s, D jz , H j~, a nd H j l w h e re j = x , y , r ) ha d r 2va lue s t h a t r a n ge d f ro m 0 .97 to 1 .00, 0 .65 to 0 .99 , 0 .90 to1 .00 , a nd 0 .68 to 1 .00 , r e spe c t ive ly . A n e s t ima te fo r e a c hc r i t i c a l p o i n t w a s d e t e r m i n e d a s t h e i n t e r s e c t i o n p o i n t o ft h e s t r a i g h t l i n e s f it t e d t o t h e t w o r e g i o n s o f t h e l i n e a r -l i n e a r v e r s i o n o f e a c h r e s u l t a n t s t a b i l o g r a m - d i f f u s i o n p l o tFig . 2b) .

The f ir s t groupDiffusion coefficients. T h e m e a n s a n d s t a n d a r d d e v i a t i o n so f t he c a l c u l a t e d d i f fu s ion c oe f f i c i e n t s fo r t he f i r s t 10s u b j e c t s a r e g i v e n in T a b l e 1. S e v e r a l g e n e r a l p o i n t s s h o u l db e n o t e d . F i r s t l y , i n a l l c a s e s , t h e s h o r t - t e r m d i f f u s i o nc o e f f i c i e n t s w e r e m u c h g r e a t e r t h a n t h e r e s p e c t i v e l o n g -t e rm c oe f f i c i e n t s , i .e ., D ~s > D ~ w h e re j= x , y , r . Se c o nd ly ,f o r t h e m a j o r i t y o f s u b je c t s, t h e a n t e r o p o s t e r i o r d i ff u s io nc o e ff ic i en t s w e r e g r e a t e r t h a n t h e i r m e d i o l a t e r a l c o u n t e r -par ts , i .e . , Dy i > Dx i w h e r e i=s I . T h i s r e s u l t w a s n o tu n e x p e c t e d g i v e n t h e f a c t t h a t a n t e r o p o s t e r i o r p o s t u r a l

l o o80

< ~ I_ 2 > 6 0(ram2 40

200

4 6Time Interval (s)

311

10

IOO80

60(rama ) 40

200

4 6Time Interval (s)

10

1(3080

60m m 2 ) 4O

2O0

0 2 4 6Time Interval (s)

8 10

100.806 0< A r 2 >

r a m 2 ) 4 0 -

20-0 . . . . . . .

0 2 4 6 8 10Time Interval (s)

F i g . 3 a - d . R e s u l t a n t p l a n a r s t a b i l o g r a m - d i f f u s i o n p l o t s . E a c h r e s -u l t a n t c u r v e i s g e n e r a t e d f r o m t e n d i f f e r e n t 3 0 -s C O P t i m e s e ri e s. T h ea p p r o x i m a t e d s h o r t - t e r m r e g i o n in e a c h g r a p h i s s h a d e d , a S u b j e c t 1 ;b subject 3; e subject 4; d subject 5. Th e calculated results for thesesubjects are given in Tables 1 3

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312a b l e 1. Diffusion coefficients (mm 2 s 1):means and standard deviations for thefirst population of subjects (n = 10)

Su bje ct Mediolateral (x) Anteroposterior (y) Planar (r)D : ~ D xz D y~ y l D ~ D~l

1 3 . 2 3_ +0 . 52 0 . 60_+0. 28 5 . 05_+0. 25 1 . 37_+0. 07 8 . 28_+0. 74 1 .97_+0.232 2 . 3 3_ +0 . 16 0 . 24_+0. 07 4 . 10_+0. 62 0 . 55 -+0 . 4 9 6 . 43_+0. 53 0 .79-+0 .453 3 . 35 -+ 0 . 22 0 . 32-+0 . 14 6 . 10_+0. 86 1 . 13_+0. 53 9 . 45_+0. 67 1 .45_+0.574 2 . 9 1_ +0 . 54 0 . 64_+0. 05 4 . 06_+0. 19 1 . 09_+0. 25 6 . 97_+0. 47 1 .73+_0 .215 3 . 4 7_ +0 . 49 0 . 31_+0. 08 3 . 16_+0. 58 1 . 02_+0. 15 6 . 62_+0. 70 1 .32_+0.136 4. 31 _+ 0. 33 0.19_+ 0.14 7.26_+ 1.84 1.00_+0 .30 11.57_+2.06 1.19_+0.337 2.89 __+ 0.44 0.96 _+0 .63 2.72_+0.29 1.11 _+0.12 5.60 _+0 .73 2.06_+0.578 1 . 2 0+ _0 . 23 0 . 21_+0. 14 2 . 46_+0.41 0 . 45_+0. 11 3 . 66_+0. 32 0 .66_+0.039 2 . 5 3_ +0 . 21 0 . 80_+0. 38 4 . 08_+0. 53 1 . 47_+0. 38 6 .61_+0. 65 2 .27_+0.2910 0. 58 _+ 0. 16 0.13_+ 0.04 1.63_+0.11 1.06_+ 0.25 2.21_+ 0.26 1.19_+0.22

GM _+SD 2.68_+1.10 0.44_+0.35 4.06_+ 1.76 1.02 +_0 .40 6.74_+2.68 1.46_+0.59Gr oup means (GM ) and standard deviations (SD) for the respective parameters arelast row given in the

s w a y i s ty p i c a l l y g r e a te r t h a n m e d i o l a t e r a l s w a y . T h i r d l y ,s ince( A r 2 ) = ( A x 2 ) q - ( A y 2 ) ( 5)i t f o l l o w s t h a t t h e r e s p e c t i v e p l a n a r d i f f u s i o n c o e f f i c i e n t sa r e l i n e a r c o m b i n a t i o n s o f t h e d i f f u s i o n c o e f fi c i e n t s c a l c u -la ted fo r the x a nd y d i rec t io ns , i .e ., Dri = D~ Dy .I t s h o u l d a l s o b e p o i n t e d o u t t h a t t w o s u b j e c t s , s u b -j e c t s 8 a n d 1 0, e x h i b i t e d r e l a t i v e l y s m a l l d i f f u s i o n c o e ff i c -i en t s Da i (Tab le 1 ) . S im i la r ly , ve ry sm a l l (c lose to ze ro)l o n g - t e r m m e d i o l a t e r a l d i f f u s i o n c o e f fi c i e n t s w e r e c a l c u -l a t e d f o r s u b j e c t 6 . I n t h e c a s e o f s u b j e c t 6, h i s C O P h a df u ll y e x p l o r e d t h e c h a r a c t e r i s t ic s p a c e f o r m e d i o l a t e r a ls w a y d u r i n g t h e e a r l y s t ag e s o f t h e l o n g - t e r m r e g i o n o f th es t a b i l o g r a m - d i f f u s i o n p l o t . I n o t h e r w o r d s , a f t e r s o m es m a l l A t, t h e C O P n o l o n g e r m o v e d f a r t h e r a w a y a l o n g t h ex - a x is , o n a v e r a g e , f r o m s o m e r e l a t i v e c e n t r a l p o i n t . U n d e rt h e s e c o n d i t i o n s , t h e C O P t r a j e c t o r y is sa i d t o h a v es a t u r a t e d t o s o m e b o u n d a r y v a l u e . S i n c e t h e C O P i sl i m i t e d t o t h e a r e a o f s u p p o r t d e f i n e d b y a s u b j e c t s f e e t, i ti s e x p e c t e d t h a t a s t a b i l o g r a m w o u l d a l s o s a t u r a t e t o as y s t e m a t i c b o u n d a r y v a l u e i n t h e a n t e r o p o s t e r i o r d i r e c -t i o n . H o w e v e r , f o r t h e m a x i m u m t i m e i n te r v a l c o n s i d e r e din the p resen t s t ab i logram -di f fus ion p lo t s (F ig . 3 ) , i . e . ,At,,~x = 1 0 s, s u c h a n e f f e c t w a s n o t f o u n d f o r a n y o f t h es u b je c ts . L o n g e r C O P t i m e s e ri e s a r e n e e d e d t o s t u d y t h i sp h e n o m e n o n . 4S c a l i n g e x p o n e n t s . T h e m e a n s a n d s t a n d a r d d e v i a t io n s o ft h e c o m p u t e d s t a b i l o g r a m - d i f f u s io n s c a li n g e x p o n e n t s f o rt h e f i r st 1 0 s u b j e c t s a r e p r e s e n t e d i n T a b l e 2 . F o r t h e s h o r t -t e r m r e g i o n , t h e s c a l i n g e x p o n e n t s H i s w e r e , i n g e n e r a l ,m u c h g r e a t e r t h a n 0 . 5 . T h u s , o v e r s h o r t - t e r m i n t e r v a l sd u r i n g q u i e t s t a n d i n g , C O P t r a j e c to r i e s e x h i b it p e r s is t e n tb e h a v i o r . O n t h e o t h e r h a n d , l o n g - t e r m s c a li n g e x p o n e n t sH j l w e r e in n i n e o u t o f t e n c a s e s m u c h l e ss t h a n 0 .5( T a b l e 2 ) . T h u s , o v e r l o n g - t e r m i n t e r v a l s , s t a b i l o g r a m se x h i b i t a n t i -p e r s i st e n c e . T h e o n l y e x c e p t i o n t o t h e a b o v e4 In a preliminary posturographic study using 60-s time series, t wasfound th at several subject~ saturated to a characteristic boundaryvalue in the anterop osterior direction after some relatively long timeinterv al, i.e., At = 30 s.

s t a t e m e n t w a s s u b j e c t 1 0 w h o d i s p l a y e d m e a n v a l u e s o f0 . 53 and 0 . 47 fo r Hyz an d Hr~ , re spec t iv e ly . S ub jec t 10 ss h o r t - t e r m b e h a v i o r w a s a l s o d i ff e r e n t f r o m t h e o t h e rs u b j e c t s ; f o r e x a m p l e , h i s m e d i o l a t e r a l s c a l i n g e x p o n e n tw a s c lo s e t o t h a t e x p e c t e d f o r a n u n c o r r e l a t e d r a n d o mw a l k , i . e . , H x s ~ 0 . 5 . F i n a l l y , i t s h o u l d b e n o t e d t h a t t h el o n g - t e r m m e d i o l a t e r a l s c a l in g e x p o n e n t H x , f o r s u b j e c t 6w a s n e a r l y z e r o ( T a b l e 2 ) . T h i s i m p l i e d t h a t s u b j e c t 6 sm e a n s q u a r e C O P d i s p l a c e m e n t i n t h e x d i r e c t i o n w a se s s e n t i a l ly c o n s t a n t f o r i n c r e a s i n g v a l u e s o f A t. T h i s r e s u l tis a d ir e c t c o n s e q u e n c e o f t h e fa c t t h a t t h e C O P t r a j e c t o r -i e s f o r s u b j e c t 6 s a t u r a t e d t o a m e d i o l a t e r a l b o u n d a r yv a l u e d u r i n g t h e e a r l y s t a g e s o f t h e l o n g - t e r m r e g i o n ( a sn o t e d a b o v e ) .

C r i t i c a l p o i n t c o o r d i n a t e s . T h e e s t i m a t e d v a l u e s o f t h et i m e i n te r v a l s a n d m e a n s q u a r e d i s p l a c e m e n t s d e f i n i n g th es t a b i l o g r a m - d i f f u s i o n c r i t i c a l p o i n t s f o r t h e f i r s t 1 0 s u b -j e c t s a r e p r e s e n t e d i n T a b l e 3 . I n g e n e r a l , t h e t r a n s i t i o npoin t s occur red a t re la t ive ly sm a l l t im e in te rva l s , i . e . , At jcr a n g e d f r o m 0 .3 3 t o 1 .6 7 s w i t h a m e a n o f a p p r o x i m a t e l y1 .0 s . O n t h e o t h e r h a n d , t h e c r i t ic a l m e a n s q u a r e d i s p l a c e -m e n t s r a n g e d f r o m v e r y s m a l l v a l u e s , i .e ., 1 . 10 m m 2 f o r( A x 2 ) c , t o r a t h e r l a r g e o n e s , i .e ., 2 9 .3 7 m m 2 f o r ( A r 2 ) c . I ti s i m p o r t a n t t o p o i n t o u t t h a t t h e c r i t i c a l p o i n t s w e r e i l l -d e f i n e d f o r m a n y o f th e s u b j e c t s . I n t h e s e c a s e s , t h e s l o p e o fa s t a b i l o g r a m - d i f f u s io n p l o t d i d n o t c h a n g e a b r u p t l y a t ad i s t in c t p o i n t , b u t r a t h e r i t c h a n g e d m o r e g r a d u a l l y o v e r ab r i e f t i m e p e r i o d . C o n s e q u e n t l y , t h e n u m b e r s g i v e n inT a b l e 3 s h o u l d o n l y b e lo o k e d u p o n a s r o u g h e s t im a t e s .R e l i a b i l i t y . T h e i n t r a c l a s s c o r r e l a t i o n c o e f f i ci e n t s f o rt h e d i f f u s i o n c o e f f i c ie n t s , s c a l i n g e x p o n e n t s a n d c r i t ic a lp o i n t c o o r d i n a t e s f o r t h e f i r s t 1 0 s u b j e c t s a r e g i v e n i nT a b l e 4 . T h e s h o r t - t e r m d i f f u s i o n c o e f f i c i e n t s D j ~ a n ds c a l i n g e x p o n e n t s H i s w e r e h i g h l y r e l i a b l e - I C C v a l u e sr a n g e d f r o m 0 . 7 6 t o 0 . 9 2 . T h e r e l i a b i l i t y m e a s u r e s f o r t h el o n g - t e r m s c a l i n g e x p o n e n t s H ~z w e r e l o w e r b u t , i n g e n e r a l ,g o o d t o e x c e l l e n t (F l e is s 1 9 8 6) : I C C v a l u e s r a n g e d f r o m0 . 5 9 t o 0 . 8 3 . T h e I C C v a l u e s f o r t h e l o n g - t e r m d i f f u s i o nc o e f f i c ie n t s D~g w e r e l o w e r t h a n t h o s e f o r t h e s h o r t - t e r mdi f fus io n coe f f i c ien t s , i .e ., 0 . 46-0 . 68 vs 0 . 83-0 . 90 . In a l lc a s e s , t h e r e l i a b i l i t y m e a s u r e s f o r t h e l o n g - t e r m s t a b i l o -

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able2. Scaling exponents: means andstandard deviations for the first popula-tion of subjects (n = i0)

313

Subject Med iolateral (x) Anteroposterior (y) Plan ar (r)H ~ Hx, Hy~ Hy H~ Hrl1 0 .7 1_ + 0 .02 0 .26_+ 0 .05 0 .73_+ 0.01 0 .39_+ 0 .05 0 .72_+ 0 .01 0 .35_+ 0 .032 0 .7 4_ + 0 .02 0 .18_+ 0 .02 0 .8 l_+ 0 .03 0 .17_+ 0 .10 0 .78+ _0 .03 0 .17_+ 0 .073 0 .7 6_ + 0 .01 0 .18_+ 0 .02 0 .78_+ 0 .02 0 .34 -+ 0 .08 0 .77_+ 0.01 0 .28_+ 0 .064 0 . 78 -+ 0 . 03 0 .31 -+ 0 .05 0 .79 -+ 0 .01 0 .30+ _0 .07 0 .79 -+ 0 .02 0 .31+_0 .065 0.78 _+0 .01 0.11 _+0.05 0.76-+0.03 0.31 -+ 0.0 1 0.77_ +0.0 1 0.24_+0.026 0 . 79 -+ 0 . 01 0 .06 -+ 0 .04 0 .85 -+ 0 .02 0 .17_+ 0 .04 0 .82+ _0 .02 0 .14 -+ 0 .037 0 .7 7_ + 0 .01 0 .27 -+ 0 .1 0 0 .79_+ 0 .01 0 .34 -+ 0 .02 0 .78_+ 0.01 0 .31_+ 0 .048 0 .6 5_ + 0 ,02 0 .21_+0 .11 0 .71_+ 0 .03 0 .24_+ 0 .05 0 .69_+ 0 .02 0 .23+0 .019 0 . 73 -+ 0 . 03 0 .29 -+ 0 .07 0 .79 -+ 0 .03 0 .37_+ 0 .08 0 .76 -+ 0 .03 0 .34_+ 0 .0310 0 . 57 _+ 0. 03 0 .26_+0.05 0 .72_+0.02 0 .53_+0.05 0 .68_+0.01 0 .47_+0.04

GM_+ SD 0.73_+0.07 0 .21_+0.10 0 .77_+0.05 0 .31 -+0.1 2 0 .76_+0.05 0 .28_+0.10Gr ou p means (GM ) and standard deviations (SD) for the respective parameters are given in thelast row

able 3. Critical point coordinates [timeintervals (s) and mean squaredisplacements (mm2)]: meansand standard deviations for the firstpopulation of subjects (n = 10)

Subject Mediolateral (x) Anteroposterior 0 ) Pla nar (r)Atxc AX2 )c Atyc Ay~ Atrc Ar2)c1 0 . 88 _+ 0 . 27 5 .38_+1 .01 0 .42_+ 0 .32 3 .67_+ 2.93 0 .61_+ 0 .28 9 .26 -+ 3 .572 1 . 20 _+ 0. 16 5 .28_+0.40 1 .67-+0.15 12.85-+1.99 1 .50_+0.10 18.15_+2.253 0 .8 7 - + 0 .1 7 5 .44_+ 1 .49 0 .33 -+ 0 .15 3 .55 -+ 2 .14 0 .55 -+ 0 .08 9 .34_+ 1.224 0 .8 1 -+ 0. 1 4 4 .15_+1.02 1 .28 9 .68_+5.02 1 .06_+0.33 13.80-+5.405 0 .8 3 - + 0 .0 8 5 .32 -+ .02 1 .57_+0 .54 9 .57_+ 4 .21 1 .15 -+ 0 .27 14 .28 -+ 3 .146 1. 30 _+ 0. 16 10.48 -+2.15 1.36_+0.19 18.86_+3.37 1.33_+0.05 29.37-+5.047 1 . 25 -+ 0 . 36 7 .15_+ 1 .62 1 .21_+ 0 .54 6 .68_+ 2 .37 1 .23 -+ 0 .40 13.90_+ 3 .458 1 . 23 _+ 0. 31 2 .84_+1.24 1 .35_+0.21 6 .51_+2.03 1 .35- +0.1 0 9 .46_+1.539 1 .1 0_ + 0 .68 5 .65 -+ 3 .24 1 .00_+ 0 .25 7 .22_+ 0 .97 0 .85 -+ 0 .26 11.03_+ 3 .7610 1 . 04 -+ 0 . 39 1 .10 -+ 0 .26 0 .56_+ 0 .18 1 .58 -+ 0 .39 0 .72_+ 0 .23 2 .80_+0 .64

GM _+SD 1.05_+0.32 5.28_ +2.73 1.07_+ 0.55 8.02 _+5 .44 1.04_+0.38 13.14_+7.34Gr ou p means (GM ) and standard deviations (SD) for the respective parameters are given in thelast row

Table 4. Intraclass correlation coefficients: ICC (2, 1) for the repeatedmeasures of the stabilogram-diffusion parameters for the first 10subjectsDiffusion coefficientsD:,~ D:,~ D~s D~,~ D~ D,.~0.90 0.46 0.83 0.46 0.90 0.68Scaling exponentsH:,s Hxl H~.s Hyl H rs Hrz0.92 0.59 0 .76 0.73 0 .86 0.83Critical point coordinatesAtxc (AX2>c Atyc

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314Table 5. Rang es, group m eans andstandard deviations (SD) for the diffusioncoefficients, scaling expo nents, a ndcritical point coordinates for thesecond population of 15 subjects

Parameter Range Group Mean+_SDDiffusion coefficients m i tt 2 s - 1 ) Dxs 1.0 5- 12 ,44 3.94+2.81Dxz 0.17-2.09 0.54 _ 0.47Dys 2.57 1 6 .0 3 7.27+_4.11Dyt 0.43-6.73 2.51 _ 1.76

Dr, 3.62 26.03 11.21 + 6.43Drt 0.60-7.51 3.05 + 2.06Scaling expon ents Hx~ 0.634).81 0.74 -- 0.05H-x~ 0.1 24 ).32 0.23 +_0.05Hy~ 0.70 4).83 0.77 _+0.04H rz 0 . 1 8 - 0 . 5 0 0 . 3 8 _ + 0 .1 0Hr~ 0.70-0 .82 0.76 +_0.04Hr~ 0.18-0.45 0.34 +- 0.08Critical point coordinates (s) At~c 0.63 1 . 8 5 1.06 _+ 0.42Aty 0.35-1.85 0.97 0.46At~ 0.44-1 .30 0.92 -+ 0.29A x 2 ) c 1 .8 8 2 0 . 7 0 7 . 7 5+ _ 5 . 74Critical point coordinates (ram2) < A y 2 > ~ 3 . 8 7 - 4 7 . 6 3 1 2 .6 9 _+ 0 .7 1A r 2 )c 5.50-64.64 19.65 -+ 15.72

t e r m a n t e r o p o s t e r i o r s c a li n g e x p o n e n t ( H y~ ) e q u a l t o 0 .5 .A s m e n t i o n e d e a r l i e r , t h i s i s t h e r e s u l t e x p e c t e d f o r ac l a s s i c a l , i . e . , u n c o r r e l a t e d , r a n d o m w a l k .Q u a n t i t a t i v e l y , t h e g r o u p m e a n s f o r t h e d i ff u s i o n c o e f -f i c i en t s D jl g i v e n i n T a b l e 5 w e r e , i n a l l ca s e s, l a r g e r t h a nt h o s e l i s t e d in T a b l e 1 . T h i s w a s d u e m a i n l y t o t h e f a c t t h a ta s m a l l n u m b e r o f th e s u b j e c ts i n th e s e c o n d p o p u l a t i o nh a d v e r y l a r g e d i f f u s i o n c o e f f i c i e n t s . O n t h e o t h e r h a n d ,t h e g r o u p m e a n s f o r t h e s c a li n g e x p o n e n t s a n d c r i ti c a l t i m ei n t e r v a ls f o r th e s e c o n d e x p e r i m e n t a l p o p u l a t i o n ( T a b l e 5 )w e r e i n c lo s e a g r e e m e n t w i t h t h o s e f o r t h e f ir s t p o p u l a t i o n( T a b l e s 2 a n d 3 , r e s p e c t i v e l y ) . H o w e v e r , t h e r e s p e c t i v eg r o u p m e a n s f o r t h e cr i ti c a l m e a n s q u a r e d i s p l a c e m e n t sg i v e n i n T a b l e 5 w e r e l a r g e r , i n a l l c a s e s , t h a n t h o s ec a l c u l a t e d f o r t h e f i r s t 1 0 s u b j e c t s ( T a b l e 3 ) . T h e s e d i s -p l a c e m e n t d i f fe r e n ce s w e r e a d i r e c t c o n s e q u e n c e o f th e f a c tt h a t t h e c r i t i c a l t i m e i n t e r v a l s f o r t h e t w o g r o u p s w e r ee q u i v a l e n t b u t t h e a v e r a g e s h o r t - t e r m d i f f u s i o n c o e f f i c -i e n t s f o r t h e s e c o n d s u b j e c t p o p u l a t i o n w e r e l a r g e r t h a nthose fo r the f i r s t .

athematical model developmentI n t h i s s tu d y , it w a s s h o w n t h a t p o s t u r o g r a p h i c t i m e s e ri e sw e r e q u a l i t at i v e l y d i f f er e n t f r o m t h o s e p r e d i c t e d f o r o r d i -n a r y B r o w n i a n m o t i o n . A s d e s c r i b e d e ar l ie r , t h e d i f f u si o nc u r v e s f o r a c l a s s i c a l r a n d o m w a l k g r o w l i n e a r l y a n du n b o u n d e d w i t h i n c r e a s in g t i m e i n t e r v a l w h e r e a s s t a b i l o -g r a m - d i f f u s i o n c u r v e s c h a n g e s l o p e a f t e r a t r a n s i t io n p o i n ta t s o m e s m a l l A t a n d t h e n s a t u r a t e t o a r e l a t i v el y c o n s t a n tm e a n s q u a r e d i s p l a c e m e n t a f t e r s o m e l a r g e t im e i n t e r v a l.F u r t h e r m o r e , w h e r e a s t h e s c a l i n g e x p o n e n t f o r a n u n -c o r r e l a t e d r a n d o m w a l k s h o u l d b e e q u a l t o 0 . 5 , i t w a sd e m o n s t r a t e d t h a t t h e s ca l in g e x p o n e n t s f o r C O P t r a je c -t o r i e s a r e g r e a t e r t h a n a n d l e s s t h a n 0 . 5 ( p e r s i s t e n c e / a n t i -p e r s i s t e n c e ) f o r s h o r t - t e r m a n d l o n g - t e r m i n t e r v a l s ,r e s p e c t i v e l y .

T o a c c o u n t f o r t h e s e e x p e r i m e n t a l o b s e r v a t i o n s a n do f f e r a p o s s i b le p h y s i o l o g i c a l e x p l a n a t i o n , t h e m o v e m e n t so f t h e C O P d u r i n g u n d i s t u r b e d s t an c e c a n b e m o d e l l e d asa s y s te m o f c o u p l e d , b o u n d e d r a n d o m w a l k s. T h e p r e -s u m e d f o r m o f th e m o d e l w a s m o t i v a t e d b y t h e c a l c u l a t e dd y n a m i c d i f f er e n c es b e t w e e n t h e s h o r t - t e r m a n d l o n g - t e r mr e g i o n s o f t h e s t a b i l o g r a m - d i f f u s i o n p l o t s a n d b y t h ef o l l o w i n g m e c h a n i c a l / p h y s i o l o g i c a l c o n s i d e r a ti o n s : ( 1) t h ee r e c t h u m a n b o d y c a n , i n t h e o r y , a s s u m e a n i n f i n i t en u m b e r o f d i f f er e n t g e o m e t r i c c o n f i g u r a t i o n s a n d r e m a i ni n e q u i l i b r i u m w i t h e x t e r n a l f o r c e s ; ( 2 ) s i n c e s k e l e t a lm u s c l e is i n c a p a b l e o f p r o d u c i n g p u r e l y c o n s t a n t f o r c es( D e L u c a e t al . 19 8 2) , b o d y s e g m e n t s a c t e d u p o n b y a c t i v em u s c l e s a re i n c a p a b l e o f m a i n t a i n i n g p u r e l y c o n s t a n tp o s i t i o n s a n d / o r o r i e n t a t i o n s . T h e p r e l i m i n a r y m a t h e m a t -i c a l m o d e l i s c o m p o s e d o f t w o s t o c h a s t i c s y s t e m s ( F i g . 4 ).T h e f i rs t sy s t e m c o n s i s ts o f a r a n d o m w a l k e r w h i c h i sa t t a c h e d t o a n o n l i n e a r s p r i n g a t t h e c e n t e r o f a b o u n d e dc i r c u l a r a r e a . If t h e r a n d o m w a l k e r m o v e s a w a y f r o m t h ec e n t e r o f t h e c i r c l e ( it s e q u i l i b r i u m p o i n t ) , i t is a c t e d u p o nb y a n o n l i n e a r , e l a s t i c r e s t o r a t i v e f o r c e . T h e s e c o n d s y s -t e m , w h i c h i s l i n e a r l y s u p e r i m p o s e d o n t o t h e f i r st , c o n s i s t so f a n o t h e r r a n d o m w a l k e r w h i ch i s b o u n d e d b y a s m a l le rc i r c u l a r a r e a . I n t h e s m a l l e r c ir c l e, t h e w a l k e r i s a c t e d u p o nb y a s y s t e m o f t h r e s h o l d - b a s e d a l t e r n a t i n g s p r i n g s, w h i c ha r e a t t a c h e d a r o u n d t h e b o u n d a r y . O n c e t h e r a n d o mw a l k e r r e a c h e s t h e p e r i m e t e r o f i ts b o u n d e d a r e a , th es p r i n g a c t in g u p o n t h e w a l k e r i s d e a c t i v a t e d a n d t h e o n el o c a t e d o n t h e o p p o s i t e s i d e o f t h e b o u n d e d a r e a i s c a l le di n t o p l a y . T h u s , f o r t h is c o n f i g u r a t i o n , t h e c e n t e r o f t h eb o u n d e d a r e a i s not a s t a b l e e q u i l i b r i u m p o i n t . I n s u m -m a r y , th e C O P r a n d o m w a l k e r f o r e a c h m o d e l le d s y st e mi s p e r t u r b e d a t a n y g i v e n t i m e b y b o t h a r a n d o m l yf l u c t u a t in g f o r c e a n d a s t a b i l iz i n g a n d / o r d e s t a b il i z in gs p r i n g f o r c e .T h e b e h a v i o r o f t h e a b o v e p o s t u r o g r a p h i c m o d e l c a n b ed e s c r i b e d b y a s i m p l e se t o f e q u a t i o n s . F o r e x a m p l e , t h ed y n a m i c s o f t h e fi rs t C O P r a n d o m w a l k e r ( w h ic h m o v e s

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r b

F i g . 4 . D i a g r a m o f t h e m o d e l s y s te m o f co u p l e d , b o u n d e d r a n d o mw a l k s . T h e f i r s t s y s t e m c o n s is t s o f a C O P r a n d o m w a l k e r w h i c h i sa c te d upo n by a spr in g (w i th s t if fne ss k~) whic h i s a t t a c he d to thec e n t e r o f th e l a r g e r c i r c u l a r a re a . L i n e a r l y s u p e r i m p o s e d o n t o t h i sr a n d o m w a l k e r i s a n o t h e r b o u n d e d s y s t e m . I n t h e s m a l l e r b o u n d e da r e a , a s e c o n d C O P r a n d o m w a l k e r i s a c t e d u p o n b y a s y s t e m o ft h r e s h o l d - b a s e d a l t e r n a t i n g s p r i n g s . I n t h e d i a g r a m , t h e r a n d o mwa lk e r i s in i t i a l ly a c te d upo n b y a spr ing (w i th s t i ffne s s k , ) a t t a c he d a tr E . A f t e r t h e r a n d o m w a l k e r c r o s se s t h e b o u n d a r y a t r ~ ,, t h e f i r s tspr ing i s tu rne d o f fa nd the spr ing (w i th s t if fne s s k , ) a t t a c he d a t - r~ ,i s a c t iva te d

100-80-

60 -(mm2) 40-

20-

2 4 6 8 ITime nterval s)F i g . 5 . R e s u l t a n t p l a n a r s t a b i l o g r a m - d i f f u s i o n p l o t s g e n e r a t e d b yt h e p o s t u r o g r a p h i c m o d e l o f F i g . 4. E a c h r e s u l t a n t c u r v e i s g e n e r a t e dfrom ten different 30-s simulated C OP time series. The approximatedshort-term region for the system is shaded

w i t h i n t h e l a r g e r b o u n d e d a r e a ) i s g o v e r n e d b y t h ee x p r e s s i o n : 5l _ l ~ l i ir , + l - r , + F ( r o , 0 - r , ) (6 )

w h e r e ~ a n dn + 1 r n a r e t h e p o s i t i o n s o f t h e r a n d o m w a l k e ra t t im es t = n + 1 and t = n , re spe c t ive ly , ~z repr esen t s thei n f l u e n c e o f a r a n d o m l y f l u c t u a t i n g f o r c e , a n d F ~( r ) r e p r e -s T h e o r i g i n o f ea c h l o c a l c o o r d i n a t e s y s t em i s l o c a t e d a t t h e c e n t e r o fi t s b o u n d e d a r e a .

315s e n t s t h e i n f l u e n c e o f a r e s t o r a t i v e s p r i n g l o c a t e d a t t h ec e n t e r (rZ o,0) o f t h e b o u n d e d a r e a . T h e b e h a v i o r o f t h es e c o n d C O P r a n d o m w a l k e r ( w h i c h m o v e s w i t h i n t h es m a l l e r b o u n d e d a r e a ) i s g o v e r n e d b y t h e f o l l o w i n ge q u a t i o n s :r~+~ = r. + 4.~ + F ~ (r ~ - r s) if [r.~[ < I gJ (7)r,~+l = r, + ~ + F S ( - r~, - r, ) if lrSl _> I g] (8)w h e r e t h e a b o v e t e r m s a r e e q u i v a l e n t t o t h o s e i n E q . 6e x c e p t t h a t F S (r ~) i n E q . 7 r e p r e s e n t s t h e i n f l u e n c e o f ap e r i m e t e r - b a s e d s p r i n g a t t a c h e d a t p o i n t r ~. I f t h e s e c o n dC O P r a n d o m w a l k e r c r o ss e s t h e b o u n d a r y a t r{ ,, t h e n t h ef i r st s p r i n g a c t i n g i n E q . 7 is t u r n e d o f f a n d a s p r i n ga t t a c h e d a t p o i n t - r ~ , i s a c t i v a t e d , a s s h o w n i n E q . 8 a n dFig. 4.T h e a b o v e e q u a t i o n s f o r m e d t h e b a s i s o f a n i n i t ia l -p h a s e c o m p u t e r p r o g r a m . I n o r d e r t o c o m p a r e t h e s im u -l a t i o n r e su l t s w i t h e x p e r i m e n t a l d a t a , t h e c o m p u t e r m o d e lw a s u s e d t o g e n e r a t e d i f f e r e n t s e t s o f 1 0 t i m e s e r ie s o f 3 0 0 0d a t a p a i r s . R e s u l t a n t s t a b i l o g r a m - d i f f u s i o n c u r v e s w e r et h e n c a l c u l a t e d a c c o r d i n g t o t h e m e t h o d s d e s c r i b e de a r li e r. R e p r e s e n t a t i v e r e s u lt s o f t h e m o d e l a r e s h o w n i nF i g. 5 . T h e s i m u l a t i o n c u r v e s o f Fi g . 5 a r e r e m a r k a b l ys i m i l a r i n s h a p e a n d f o r m t o t h e e x p e r i m e n t a l p l o t s o fF i g . 3. F o r e x a m p l e , i t i s c l e a r t h a t t h e s i m u l a t e d d i f f u s i o nc u r v e s c h a n g e d s l o p e a f t e r s o m e s m a l l t i m e i n t e r v a l , i . e . ,A t ,- ~ 1 .0 s . M o r e o v e r , o v e r s h o r t - t e r m a n d l o n g - t e r m i n t e r -v a l s , t h e s t a b i l o g r a m m o d e l e x h i b i t e d p e r s i s t e n t ( s i m u -l a t e d H j ~ > 0 . 5 ) a n d a n t i - p e r s i s t e n t ( s i m u l a t e d H i t < 0 . 5 )b e h a v i o r , r e s p e c ti v e ly . F i n a ll y , a n i m p o r t a n t f e a t u r e o f t h ep r e s e n t c o m p u t e r m o d e l w a s t h a t t h e q u a l i t a t i v e a n dq u a n t i t a t i v e c h a r a c t e r is t i c s o f t h e s i m u l a t e d s t a b i l o g r a m -d i f f u si o n c u r v e s c o u l d b e m o d i f i e d b y v a r y i n g t h e m a g n i -t u d e o f th e d i f f e r e n t p a r a m e t e r s t h a t d e f i n e d t h e m o d e l sb e h a v i o r .

i scussion

e l iab i l i t yB e f o r e a n y n e w t e c h n i q u e c a n b e u s e f u ll y e m p l o y e d a s as c i e n ti f ic a n d / o r c l i n ic a l t o o l ,, i t s r e l i a b i l i t y m u s t b e a s s e s-s e d. I n t h i s s t u d y , i n t r a c l a s s c o r r e l a t i o n c o e f f i c i e n t s w e r ec a l c u l a t e d t o d e t e r m i n e t h e d e g r e e o f a g r e e m e n t b e t w e e nr e p e a t e d m e a s u r e s o f t h e r e s p e c t iv e s t a b i l o g r a m - d i f f u s i o np a r a m e t e r s - d i f f u s i o n c o e f f i c ie n t s , s c a l i n g e x p o n e n t s , a n dc r i ti c a l p o i n t c o o r d i n a t e s . I t w a s s h o w n t h a t t h e m a j o r i t yo f t h e p r o p o s e d p o s t u r o g r a p h i c p a r a m e t e r s d e m o n s t r a t e dg o o d t o e x c e l l e n t r e l ia b i li t y . T h e k e y a n a l y t i c a l a d v a n c e -m e n t o f t h e p r e s e n t w o r k w a s t h e u t i l i z a t io n o f tr i ala v e r a g in g . D u e t o t h e s t o c h a s t i c n a t u r e o f s t a b i lo g r a m s , i ti s d i f f ic u l t , c o n c e p t u a l l y a n d p r a c t i c a l l y , t o o b t a i n r e p e a t -a b l e p a r a m e t e r s f r o m i n d i v i d u a l 3 0 - s C O P t i m e s er ie s. A se v i d e n c e d b y t h e r e p o r t e d r e s u l t s , t h i s t a s k i s g r e a t l ys i m p l i f ie d b y l o o k i n g a t e n s e m b l e a v e r a g e s o f a r e l a t i v e l ys m a l l n u m b e r o f e x p e r i m e n t a l s e q u e n c e s , i .e ., te n 3 0 - s t es ts .A s m e n t i o n e d e a r l i e r , i t i s e x p e c t e d t h a t t h e r e l i a b i l i t y o ft h e r es p e c t iv e p a r a m e t e r s w o u l d i m p r o v e a s th e n u m b e r o fC O P t i m e s e r i e s m a k i n g u p t h e r e s u l t a n t s t a b i l o g r a m -

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316d i f fu s i o n p lo t s w a s i n c r e a se d . H o w e v e r , b e c a u s e o f t h ec o m p l i c a t i o n s a s s o c i a t e d w i t h s u b j e c t f a t i g u e , a n y c l i n i c a lo r s c i e n ti fi c i n v e s t i g a t io n i n p o s t u r o g r a p h y w i ll h a v e t oa c c e p t s o m e t r a d e - o f f b e t w e e n r e l ia b i li t y a n d e x p e r i m e n t a lp r a c t i c a l i t y .

Open loop and closed loop control strategiesB y a n a l y z i n g s ta b i l o g r a m s a s f r a c t io n a l B r o w n i a n m o t i o n ,i t w a s r e v e a l e d t h a t a t l e a s t t w o d i s t i n c t l y d i f f e r e n t n e u r o -m u s c u l a r c o n t r o l m e c h a n i s m s - o n e w h i c h e x h i b i ts p e rs i s-t e n c e a n d a n o t h e r w h i c h e x h i b i t s a n t i -p e r s i s te n c e - a r ef u n c t i o n i n g d u r i n g q u i e t s t a n d i n g . M o r e s p e c i f ic a l l y , t h e s ea n a l y s e s s u g g e s t t h a t o v e r s h o r t - t e r m i n t e r v a ls o p e n - l o o pc o n t r o l s c h e m e s a r e u t i l i z e d b y t h e p o s t u r a l c o n t r o l s y s -t e m , w h e r e a s o v e r l o n g - t e r m i n t e r v a l s c l o s e d - lo o p c o n t r o lm e c h a n i s m s a r e c a l l e d i n t o p l a y . T h e s e i s s u e s w i l l b ed i s c u s s e d i n g r e a t e r d e t a i l i n t h e f o l l o w i n g s e c t i o n s .

Physiologically mean ingful posturographic param etersP o s t u r o g r a p h y h a s b e e n l i m i t e d b y t h e la c k o f a n a n a l y t -i c a l t e c h n i q u e f o r e x t r a c t i n g p h y s i o l o g i c a l l y m e a n i n g f u lp a r a m e t e r s f r o m s t a b i l o g r a m s . S i n c e t h e C O P is a m e a s -u r e o f w h o l e - b o d y d y n a m i c s , i t r e p r e s e n t s th e s u m m e de f fe c t o f a n u m b e r o f d i ff e r e n t n e u r o m u s c u l o s k e l e t a l c o m -p o n e n t s a c t i n g a t a n u m b e r o f d if f e r en t j o i n ts . T h i s i n h e r -e n t f e a tu r e h a s c o n f o u n d e d t h e m a j o r i t y o f p r e v i o u sa t t e m p t s a t i n t e r p r e t i n g s t a b i l o g r a m s f r o m a m o t o rc o n t r o l p e r sp e c t iv e . A n a d v a n t a g e o f t h e p r o p o s e d s t a b i l o -g r a m - d i f f u s i o n p a r a m e t e r s i s t h a t t h e y c a n b e d i r e c t l yr e l a t e d t o t h e r e s u l t a n t s t e a d y - s t a t e b e h a v i o r a n d f u n c -t i o n a l in t e r a c t i o n o f t h e o p e n - l o o p a n d c l o s e d - l o o p n e u r o -m u s c u l a r m e c h a n i s m s u n d e r l y i n g p o s t u r a l c o n t r o l .D i f f u s i o n c o e f f i ci e n t s, f o r e x a m p l e , r e f l e c t t h e l e v e l o fs t o c h a s t i c a c t i v i ty o f t h e C O P a l o n g t h e m e d i o l a t e r a l o ra n t e r o p o s t e r i o r a x is o r a b o u t t h e p l a n e o f s u p p o r t . T h e s em e a s u r e s c a n t h u s b e u s e d t o q u a n t i f y p o s t u r a l i n s ta b i l it y ,i . e . , l a r g e r D j i c o r r e s p o n d t o a l e s s t i g h t l y r e g u l a t e d o rm o r e r a n d o m c o n t r o l s y st e m a n d v i ce v e r sa . 6 I n l ig h t o ft h e s e c o m m e n t s , s e v e r a l a s p e c t s o f t h e c a l c u l a t e d r e s u l t ss h o u l d b e d i s c u s s e d . F i r s t l y , i t w a s f o u n d t h a t t h e s h o r t -t e r m d i f fu s i o n c o ef f ic i en t s w e r e m u c h l a r g e r t h a n t h e l o n g -t e r m c o e ff i ci e nt s . T h i s s u g g e s ts t h a t t h e o p e n - l o o p c o n t r o lm e c h a n i s m s w h i c h d o m i n a t e s h o r t - t e r m i n t e r v a l s h a v e ah i g h e r l e v e l o f s t o c h a s ti c a c t i v i ty t h a n t h e c l o s e d - l o o pc o n t r o l m e c h a n i s m s w h i c h a r e u ti l iz e d o v e r l o n g - t e r mi n t er v a l s. S e c o n d l y , i t w a s s h o w n t h a t t h e a n t e r o p o s t e r i o r6 It is important to point out, however, that an individual may haverelatively large diffusion coefficientsDj~and relatively small criticalmean square displacements and /or saturation points. In this case, thepostural control system allows a relatively high lev el of stochasticactivity within a relatively small region of the base of support. Itshould also be noted th at it is possible for two individuals to havesimilar scaling exponen ts bu t sign ificantly different diffusion coeffic-ients. Fo r th is situation, the CO P step increments for the twosubjects are sim ilarly correlated, bu t the average mean squaredisplacem ent (over a given time interval) is larger for one subject ascompared to the other.

d i f f u s i o n c o e ff i c i en t s w e r e g r e a t e r t h a n t h e r e s p e c t i v em e d i o l a t e r a l c o e ff ic i en t s. T h i s a s y m m e t r y c a n b e a t t r i b u -t e d l a r g e ly t o t h e g e o m e t r y o f t h e l o w e r l im b . T h e a n k l e o rt i b i o t a r s a l j o i n t i s, f o r e x a m p l e , m a i n l y a s i m p l e h i n g e j o i n tw h i c h a l lo w s r o t a t i o n ( p l a n t a r f le x i o n / d o r s i f le x i o n ) i n t h es a g i t t a l p l a n e . T h u s , f r o m a p a s s i v e m e c h a n i c a l s t a n d -p o i n t , u p r i g h t b i p e d a l s t a n c e i s c o n s i d e r a b l y m o r e s t a b l ei n t h e f r o n t a l p l a n e t h a n i n t h e s a g i t t a l p l a n e . T h i r d l y , i tc a n b e s e e n i n T a b l e s 1 a n d 5 t h a t t h e m a g n i t u d e s o f t h ed i f f u si o n c o e f fi c ie n t s w e r e h i g h l y v a r i a b l e a m o n g s t y o u n gh e a l t h y s u b j e c t s . F o r i n s t a n c e , t h e r e w a s a n o r d e r o fm a g n i t u d e d i f fe r e n c e i n t h e c a l c u l a t e d v a l u e o f t h e l o n g -t e r m p l a n a r d i f f u s i o n c o e f f i c ie n t D r t f o r s u b j e c t 1 0 ( T a b l e 1 )a n d o n e o f t h e s u b j ec t s o f t h e s e c o n d p o p u l a t i o n ( T a b l e 5 ):1 .1 9 v s 2 6. 0 3 m m 2 s - 1 . T h i s i m p l i e s t h a t s u b j e c t 1 0 w a sm o r e s t a b l e t h a n t h e o t h e r i n d i v i d u a l . O n a l a r g e r s c a l e ,t h e se d i f f e r en c e s s u g g e s t t h a t t h e s t e a d y - s t a t e b e h a v i o r o ft h e c o n t r o l m e c h a n i s m s i n v o l v e d i n m a i n t a i n i n g e r e c tp o s t u r e c a n b e q u i t e v a r i a b l e e v e n a m o n g s t a p o p u l a t i o no f a g e - m a t c h e d , a n t h r o p o m e t r i c a l l y s i m i la r , h e a l t h yi n d i v i d u a l s .T h e s e c o n d s e t o f s t a b i l o g r a m - d i f f u s io n p a r a m e t e r sw e r e t h e s c a li n g e x p o n e n t s , w h i c h q u a n t i f y t h e c o r r e l a t i o nb e t w e e n t h e s t e p i n c r e m e n t s m a k i n g u p a n e x p e r i m e n t a ls t a b i l o g r a m t i m e s e r i e s . F o r t h e s h o r t - t e r m r e g i o n , t h ec o m p u t e d s c a l i n g e x p o n e n t s w e r e m u c h g r e a t e r t h a n 0 . 5 .T h u s , o v e r s h o r t - t e r m i n t e r v a l s d u r i n g q u i e t s t a n d i n g , t h eC O P b e h a v e d a s a p o s i t i v e ly c o r r e l a t e d r a n d o m w a l k , i.e .,o n e w h i c h t e n d s t o m o v e a w a y f r o m s o m e r e l a t i v e e q u i li -b r i u m p o i n t f o l l o w i n g a n e x t e r n a l p e r t u r b a t i o n ( i n d i c a ti v eo f o p e n - l o o p c o n t r o l ). O n t h e o t h e r h a n d , i t w a s f o u n d t h a tt h e s c a l i n g e x p o n e n t s w e r e m u c h l e s s t h a n 0 .5 f o r t h e l o n g -t e r m r e g i o n . T h u s , o v e r l o n g - t e r m i n t e rv a l s , t h e C O Pb e h a v e d a s a n e g a t i v e l y c o r r e l a t e d r a n d o m w a l k , i.e ., o n ew h i c h t e n d s t o r e t u r n t o a r e l a t i v e e q u i l i b r i u m p o i n tf o l l o w i n g a p e r t u r b a t i o n ( i n d ic a t iv e o f c l o s e d - l o o p c o n -t r o l) . T h e s e r e s u l t s i m p l y t h a t f o r t h e m a j o r i t y o f su b j e c t st h e m o v e m e n t s o f t h e C O P d u r i n g u n d i s t u r b e d s t a n c e a r en o t p u r e l y r a n d o m , 7 i .e ., H j i r I n s t e a d , t h e s e m o t i o n sm o s t l i k e l y r e p r e s e n t t h e c o m b i n e d o u t p u t o f c o - e x i s t e n td e t e r m i n i s t i c a n d s t o c h a s t i c m e c h a n i s m s / F i n a l l y , i ts h o u l d b e n o t e d t h a t t h e s c a l i n g e x p o n e n t i n t e r - s u b j e c tv a r i a b i l i t y ( T a b l e s 2 a n d 5 ) w a s l e s s t h a n t h a t f o r t h ed i f fus ion coe f f i c ien t s (Tab les 1 and 5): the coe f f i c ien t s o fv a r i a t i o n f o r t h e s c a l i n g e x p o n e n t s w e r e , i n g e n e r a l ,s m a l l e r t h a n t h o s e f o r t h e d i f f u s i o n c o e f fi c i e n ts . T h u s , t h i sf u n c t i o n a l a s p e c t o f t h e p o s t u r a l c o n t r o l s y s t e m , i.e ., t h es t a b i l i z i n g / d e s t a b i l i z i n g e f f e c t s o f d i f f e r e n t c o n t r o l m e c h -

7 A sm all number of the subjects had scaling exponents which w ereapprox imately eq ual to 0 .5, i.e., the resu lt expected for classicalBrownian motion.s It is important to emphasize the fact that the C OP displacementsmeasured in the present study w ere principally associated withpostural swa y and not instrumentation no ise - the maximumfluctuations introduced by instrumentation noise were less than 5%of the maximum displacement of the CO P during a typical 30-s trial(see Materials and methods). The presence of low levels of elec-tronic/instrumentation noise would, how ever, decrease the correla-tion between the CO P step increments and thereby shift the values ofthe resp ective scaling exponen ts closer to 0.5.

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317a n i sm s , m a y b e m o r e c o n s i s t e n t w i t h i n a p a r t i c u l a r s u b j e c tp o p u l a t i o n .T h e t h i rd s et o f p o s t u r o g r a p h i c p a r a m e t e r s w e r e t h ec r i ti c a l p o i n t c o o r d i n a t e s t h e c r i ti c a l t i m e i n t e r v a l s a n dc r i ti c a l m e a n s q u a r e d i s p l a c e m e n t s . F r o m a n a n a l y t i c a ls t a n d p o i n t , t h e s e c o o r d i n a t e s a p p r o x i m a t e t h e t r a n s i t i o np o i n t a t w h i c h t h e s l o p e o f a r e s u l t a n t s t a b i l o g r a m -d i f f u si o n p l o t c h a n g e s c o n s i d e r a b l y . F r o m a p h y s i o l o g i c a ls t a n d p o i n t , t h e s e c o o r d i n a t e s r e p r e s e n t t h e p o i n t a t w h i c ht h e p o s t u r a l c o n t r o l s y s te m s w i t ch e s o v e r f r o m o p e n - l o o pc o n t r o l t o c l o s e d - l o o p c o n t r o l . O n a v e r a g e , t h is c r o s s o v e rp o i n t o c c u r r e d a t r e l a t i v e l y s m a l l t i m e i n t e r v a l s , i . e . ,At jc ~ 1 .0 s , an d m ean squ a re d i sp lac em ents , i .e . ~ A j 2 c w a sl es s t h a n 2 0 .0 m m 2. A r e l a t e d p a r a m e t e r w h i c h w a s n o tf u ll y e x p l o r e d i n t h e p r e s e n t i n v e s t i g a t io n w a s t h e s a t u r a -t i o n p o i n t . T h i s p o i n t , w h i c h i s a f u n c t i o n o f m e a n s q u a r ed i s p l a c e m e n t a n d t i m e i n t e r v a l , c o r r e s p o n d s t o a s y s -t e m a t ic b o u n d a r y v a l u e f o r l a rg e - sc a le C O P m o v e m e n ta n d t h e c h a r a c t e r i s t i c ti m e i t t a k e s t h e C O P t o s a t u r a t e t ot h a t v a l u e . B e y o n d t h e s a t u r a t i o n p o i n t , t h e d i f f u s i o nc o e f fi c ie n t s a n d s c a li n g e x p o n e n t s s h o u l d a p p r o x i m a t ez e r o . T h e s a t u r a t i o n p o i n t c a n t h u s b e v i e w e d a s a na v e r a g e m e a s u r e o f t h e o p e r a t i o n a l s a f e t y l i m i ts a l l o w e db y a n i n d i v i d u a l ' s p o s t u r a l c o n t r o l s y s t e m d u r i n g q u i e ts t a n d i n g . I t w a s o b s e r v e d t h a t s e v e r a l s u b j e c t s s a t u r a t e d t oa b o u n d a r y v a l u e i n th e m e d i o l a t e r a l d i r e c t i o n i n A t < l 0 s .A s i m i l a r a n t e r o p o s t e r i o r e f f ec t w a s n o t s e en i n a n y o f t h es u bj ec ts . T h i s p o s t u r o g r a p h i c p h e n o m e n o n - a n t e r o p o s -t e r i o r p o s t u r a l s w a y s a t u r a t i o n - r e q u i r e s l o n g e r C O Pt i m e s e ri es a n d c o r r e s p o n d i n g l y l a rg e r A tm , x .

P h y s i o l o g i c a l i n t e r p r e t a t i o n sD u r i n g a n y g iv e n t a s k , t h e h u m a n p o s t u r a l c o n t r o l s y s te mr e c ei v e s i n f o r m a t i o n f r o m t h e v i s ua l , v e s t i b u l a r a n d s o m a -t o s e n s o r y s y s t e m s . I n t h e p a s t , i t w a s g e n e r a l l y b e l i e v e dt h a t t h e s e a f f e r e n t s i g n a l s w e r e u t i l i z e d t o r e g u l a t e a n dc o n t i n u a l l y m o d i f y t h e a c t iv i t y o f t h e m u s c u l a t u r e d u r i n gq u i e t s t a n d i n g . T h e p r e s e n t p o s t u r o g r a p h i c r e s u l t s , h o w -e v e r , d o n o t s u p p o r t t h i s v ie w . I n s t e a d , t h e s e a n a l y s e si n d i c a t e t h a t i n a d d i t i o n t o t h e a b o v e c l o s e d - l o o p f e e d b a c km e c h a n i s m s , t h e p o s t u r a l c o n t r o l s y s t e m a l s o e m p l o y so p e n - l o o p c o n t r o l s c h em e s , t h e o u t p u t o f w h i c h m a y t a k et h e f o r m o f d e s c e n d i n g c o m m a n d s t o d i f f er e n t p o s t u r a lm u s c le s . S i n ce s k e l e ta l m u s c l e s a r e i n c a p a b l e o f p r o d u c i n gp u r e l y c o n s t a n t f o r c e s ( D e L u c a e t a l . 1 9 8 2 ) , t h e s e o p e n -l o o p a c t i v a t i o n s i g n a l s r e s u l t i n s m a l l m e c h a n i c a l f l u c t u -a t i o n s a t v a r i o u s j o i n t s o f t h e b o d y . T h e p r e s e n t w o r ks u g g e s t s t h a t t h e s e f l u c t u a t i o n s a n d t h e i r a s s o c i a t e d d r i f te f fe c ts a r e l e ft u n c h e c k e d b y t h e p o s t u r a l c o n t r o l s y s t emu n t i l t h e y e x c e e d s o m e s y s t e m a t i c t h r e s h o l d , a f t e r w h i c hc o r r e c t i v e f e e d b a c k m e c h a n i s m s a r e c a l le d i n t o p l a y . A sn o t e d a b o v e , t h e s t a b i l o g r a m - d i f f u s i o n c r i t i c a l p o i n t c o -o r d i n a t e s q u a n t i f y t h e s p a ti a l a n d t e m p o r a l c h a r a c t e r i s t ic so f t h is s w i t c h in g p h e n o m e n o n . I t is i m p o r t a n t t o p o i n t o u tt h a t w i t h i n t h i s c o n c e p t u a l m o d e l , t h e c e n t r a l n e r v o u ss y s t em s ti ll c o n t i n u a l l y r e c e i v es a f f e r e n t i n f o r m a t i o n f r o mp e r i p h e r a l s e n s o r y o r g a n s ; h o w e v e r , s u c h i n f o r m a t i o n i sn o t u s e d t o m o d u l a t e t h e e f f e r e n t s ig n a ls t r a n s m i t t e d t op o s t u r a l m u s c l e s u n l e s s a c e r t a i n t h r e s h o l d v a l u e i s e x -

c e e d e d . T h i s o p e n - l o o p / c l o s e d - l o o p c o n t r o l s t r a t e g y ,w h i c h a l l o w s a c e r t a i n a m o u n t o f s l o p p i n e s s i n b a l a n c ec o n t r o l , m a y h a v e e v o l v e d t o t a k e a c c o u n t o f t h e i n h e r e n tt i m e d e l a y s o f f e e d b a c k l o o p s a n d t o s i m p l if y th e t a s k o fi n t e g ra t i n g v a s t a m o u n t s o f s en s o r y i n f o r m a t i o n w h e n t h es y s t em is n o t i n j e o p a r d y o f i n s ta b i l it y ,T h e r e a r e a n u m b e r o f p o s s i b le s o u r c e s w h i c h m a y s e tt h e c o o r d i n a t e s o f t h e s t a b i l o g r a m - d i f f u s i o n c r i t ic a l p o i n t .F i r s tl y , fo r e x a m p l e , it s p o s i t i o n m a y b e d e t e r m i n e d b y ap r o p r i o c e p t i v e d e a d z o n e , i .e ., a r e g i o n o v e r w h i c h s l i g h tv a r i a t i o n s i n b o d y s e g m e n t p o s i t i o n a n d o r i e n t a t i o n a r el e f t u n c h a n g e d . W i t h i n t h i s s c e n a r i o , t h e c r i t i c a l m e a ns q u a r e d i s p l a c e m e n t q u a n t i f i e s a n i n d i v i d u a l ' s f i r s t - l e v e ls t a b i l i t y l i m i t , i . e . , a p r i m a r y f e e d b a c k t h r e s h o l d . A na l t e r n a t i v e w a y o n e c o u l d g e t a d e a d z o n e w o u l d b et h r o u g h t h e i n t e r a c t i o n o f p o s t u r a l r e s p o n s e s w i th t h eb o d y ' s i n e r t i a . P o s t u r a l r e s p o n s e s h a v e t i m e d e l a y s o f~ 1 0 0 m s ( N a s h n e r 1 9 77 ), w h e r e a s t h e m u s c u l o s k e l e t a ls y s t e m h as r i g i d - b o d y t i m e c o n s t a n t s w h i c h m a y b e o n t h eo r d e r o f ~ 1 s. T h e s y s t em i n e r t i a m a y t h u s c a u s e at e m p o r a l d e a d z o n e f o r r e s p o n se d u r i n g w h i c h t h ec e n t r a l n e r v o u s s y s t e m i s i n a n o p e n - l o o p c o n t r o l m o d e .A n o t h e r p o s s i b il i ty i s th a t t h e p o s i t i o n o f th e t r a n s i t i o np o i n t i s r e l a t e d t o t h e d e s t a b i l i z i n g i n f l u e n c e o f g r a v i t y . I ft h i s w e r e t h e c a s e , i t w o u l d b e e x p e c t e d t h a t t h e c r i t i c a lp o i n t c o o r d i n a t e s w o u l d c h a n g e i n a r e d u c e d - g r a v i t ye n v i r o n m e n t . F i n a l l y , t h e s t a b i l o g r a m - d i f f u s i o n c r i t i c a lp o i n t m a y b e e s t a b li s h e d b y f ix e d , p r e - p r o g r a m m e d c e n t -r a l c o m m a n d s t h a t a r e u t i li z e d in q u i e t s t an c e . C e n t r a lp r o g r a m s f o r p o s t u r a l c o n t r o l h a v e p r e v i o u s l y b e e n d is -c u s s ed i n t h e c o n t e x t o f p e r t u r b a t i o n e x p e r i m e n t s ( H o r a ka n d N a s h n e r 1 9 8 6 ; D i e t z 1 9 9 2 ) .Acknowledgements. This study was supported by the R ehabilitationand Development Service of Veterans Affairs and Liberty M utualInsurance Company. The authors w ould like to thank ProfessorGuido Sandri for many helpful discussions and M ichael Rosensteinfor his assistance with software development and the preparation offigures.

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