muscle activity and balance – how do we keep standing? ms emg cop.pdf · 2005-02-25 · muscle...

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Muscle activity and balance – how do we keep standing?

Frank Borg (1), Maria Finell (2), Mika Herrala (1) & Ismo Hakala (1)

(1) Chydenius Institute, Jyväskylä University, 67100 Kokkola-Karleby, Finland(2) Physio Medical Co, Kokkola-Karleby, Finland

From a biomechanical point of view quiet standing can be described as the motion of a humaninverted pendulum pivoted at the ankle joints. According to this model the motion isdetermined by gravity, the physical properties of the pendulum (moment of inertia, mass,center of mass) and the active and passive force-contributions from muscle-tendon systemsacting over the ankle joint. The muscle-tendon system of central importance for mantainingerect posture is the “shaft” muscle gastrocnemius (GA) and its tendon. This is emphasized bymeasurements which show a close relationship between the GA muscle activity as measuredby electromyography (EMG) and the forward-backward motion location of the center-of-pressure (COP). While GA seems to be in charge of the phasic (modulating) control mode, thesoleus (SL) sets the basic tonus. Our hypothesis is that the GA-activity is linked to “dynamicstiffness control” which is necessary in order to achieve postural stability given the complianttendon. Thus, the characteristic swayings observed in quiet standing constitute an integraland essential part of the stabilization system and are not just some “disturbance” or “noise”.

Key words: quiet standing, stiffness, center-of-pressure (COP), electromyography (EMG),gastrocnemius, inverted pendulum model.

I. Introduction

Quiet standing is an interesting example of the action of a neuro-muscular control-system. It can becompared to the classical problem of controlling an inverted pendulum in engineering. The“simplest” solution would be to increase the stiffness of the muscle-tendon system such that itwould act as powerful spring, bringing automatically back the body into a vertically erect positionwhenever a deviation occurs. Winter et al. (1998, 2001, 2003) have emphasized the role of the“passive” muscle stiffness as an explanation for the restoring torque. A second alternative (e.g.Peterka 2000, 2002) emphasizes feedback control while a third group (e.g. Morasso & Schieppati1999, Morasso et al. 1999, Loram & Lakie 2002) claims that passive stiffness and active feedbackare not enough to maintain posture, but that a feedforward control is also needed and indicated byexperimental data. Indeed, measurements indicate that the stiffness of the Achilles tendon is too lowto support normal standing posture. Everyone though seems to agree that quiet standing in thetypical case (“ankle strategy”) is well described by the model of an inverted pendulum pivoted at theankles. There is less agreement about the details of the neuro-muscular control mechanism of thehuman inverted pendulum. As Morasso et alii (1999) emphasize, the problem of quiet standingseems from a number of perspectives to be more enigmatic than that of walking or running.

A traditional method of studying both quiet and perturbed standing is by using a force plate.Typically a force plate is a rectangular plate with force transducers attached to each corner by whichthe force plate rests on the floor. From the load distribution between the force transducers one cancalculate the center-of-pressure (COP) of the person standing on the force plate. The COP-point canbe defined as the point in the plane of the force plate where the total force (due to the weight and

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possible movements) of the person is acting on the force plate. Figure 1 shows a “typical” trace(posturogram) of the COP during quiet standing. The apparent similarity of the posturogram withthe trace of a Brownian particle model (Einstein 1905) in two dimensions has inspired theapplications of statistical physical methods (Collins & De Luca 1994; Eurich & Milton 1996; Lauket al. 1998; Bosek, Grzegorzewski & Kowalczyk 2004). From the posturogram it is thus obviousthat the swayings in quiet standing cannot be described as oscillations about some single fixedpoint. The COP-point seem more like wandering around within the area of stable support.

The present study aims at clarifying and reviewing some of the above issues. It is inspired by ourrecent measurements of the muscular activity in the quiet and perturbed standing states and also bythe findings by other groups about the tendon properties (Maganaris & Paul 2002; Loram et al.2004). In section II we describe the generally accepted human inverted pendulum model for bipedalquiet standing together with evidence for its validity. This model defines the physical constraints ofupright balancing when the person adheres to the “ankle strategy”. In section III we review some ofthe models that have been proposed for balance control. A major issue is to what extent the controlinvolves a “passive” element, active feedback or a feed-forward control loop. In section IV wediscuss the importance the tendon stiffnes properties for the stability of the postural control whichhas been overlooked in most of the previous models. Our force plate and EMG-measurement resultsindicate that the gastrocnemius (GA) muscle plays a major phasic role in this control loop. TheGA-EMG data also provide new checks for theories about the EMG-force relation. The experimen-tal procedures are described in a final section while some physiological basics, the EMG-forcerelation, phase-shift and the experimental methods are covered in the Appendices.

II. The human inverted pendulum

The centre of gravity (COG) cannot in general be measured in such a simple manner as the COP; inorder to obtain it one has to track the position of the body segments which typically requires video-based measurement systems. Simultaneous measurements (e.g. Winter et al. 1998; Gatev et al.1999) of COP and COG have revealed that they are very close to each other such that COG appearsto be a smoothed version of COP. (We have verified these findings using a simpler system formeasuring COG, applicapble in the quiet standing case, which is described further below – see alsofig. 14.) The COP-COG relation can be highlighted by recapitulating the analysis of the invertedpendulum model. The considerations next will be restricted to motions in the sagittal plane(forward-backward sways), see figure 2. For this we will take the forward-backward (anterior-posterior, A/P) direction to be along the y-axis, the vertical direction is taken as the z-axis, and wetake y = 0, z = 0, to be the position of the ankle joint in the sagittal plane. Denote further by F thetotal force acting on the feet by the force plate at the point (u, −ζ,) which is the COP in the sagittalplane. Finally we denote the COG-coordinates in the sagittal plane by (y, z).

From Newton's equations for an inverted pendulum we obtain then in term of the angle ofinclintaion θ, assuming that the feet do not move or deform, and that the ankle joint can beapproximated as a hinge (L is the the distance between COG and the ankle joint; I is the moment ofinertia; m is the body mass minus feet mass; g is the gravitational acceleration ≈ 9.81 m/s2),

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(1)m y=F y

m z=F z−m gI =−u F z− F ym g Lsin

.

The component Fz is equal to the total vertical force (directed upwards) measured by the force trans-ducers. For small deviations around the z-axis we may replace θ by y/L and as a first approximationwe may also set Fz = mg (though e.g. the beating of the heart causes a fluctuation in the verticalforce of the order of 5 - 8 N detectable in the measured total vertical force Fz) then we obtain from(1)

(2) y−u≈g Im g L⋅y

Thus, the difference between the projection of the COG and the COP on the y-axis is proportional tothe second derivative of the COG-coordinate y. We can estimate the difference of y – u by approxi-

mating the RHS of (2) asLg⋅y and using L ≈ 1 m. If we assume a sway of 5 mm of the COG in 1

second of time we can estimate y to be around 1/100 m/s2 (correspodning to a horizontal force ofthe order of 1 N for an adult) and the corresponding typical difference y – u to be of an magnitudearound 1 mm. If we take formally the Fourier transform of (2) we obtain the equation

(3)y f =u f ⋅ 1

1 f 2

f c2

where fc is a characteristic frequency of the body given by

(4) f c=1

2 ⋅ g

Im L

which can be estimated using antropometrical data to be typically around 1/3 Hz for adults. From(3) it is apparent that the COG-coordinate y is a low-pass filtered version of the COP-coordinate uand that there is no phase shift between y and u no matter what the feedback-law for u as long as yoscillates around an equilibrium position in a (quasi-)stationary manner. Indeed, we have substanti-ated these relations, and thus the validity of the pendulum model, by measuring A/P COG simulta-neously with COP using a thin wire attached to the standing person at the waist level and runningover a pulley the other end being finally connected to a rotational optical encoder via a small leverarm. The estimated A/P COG was obtained by applying the filter (3) to the A/P COP data (usingcut-off fc = 1/3 Hz) and then comparing with the A/P COG obtained via the optical encoder data(fig. 14). The COG-curve y(t) is thus obtained from the COP-curve u(t) as its "average" over a time

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interval whose characteristic size is typically of the order 1/ωc ∼ 0.5 seconds (ωc = 2πfc). This timeinterval defines a characteristic time-unit in term of which equ (2) becomes

(2*) y s– u s= y s s= tc

One might solve the linear equation (2*) directly,

(5a) y s=∫0

s

sinh v−su v dv y 0⋅cosh s

y 0⋅sinh s

(by assumption this is restricted to the case | y/L | << 1). Equation (5a) emphasizes that we get run-away solutions if u stays nearly constant (and u ≠ 0). The characteristic time around 0.5 secondsimplies that any feedback-control in general must have a delay less than this in order to be able tostabilize the system. The formulations (2) and (5a) suggest that the COP-coordinate u is to beconsidered as a control variable while the COG-coordinate y is a “state variable”. This also empha-sized by the results (fig. 4) showing that the GA-muscle activation is connected with the COP-uvariable. The equ (2) can also be read as follows: given a desired motion y(t) of the A/P COG, thenequ (2) can be solved for u(t) which in turn prescribes the torque the neuro-muscular system mustgenerate.

Note on the computation of A/P COG from A/P COP. −When calculating COG from COP applyingthe Fourier-method (numerically implemented by using the FFT-transform) based on the relation (3)this will correspond in time-space to using the non-causal solution

(5b) y s=1 2 ∫

−∞

∞

exp –∣v−s∣u v dv .

One may wonder how this can be reconciled with the causal version (5a) which depends on theinitial values at a given time s = 0, while (5b) apparently depends on no initial values at all.Inserting u using the equation (2*) into (5*) and doing the partial integrations we see that we indeedrecover y if we may assume the following boundary condition

(5c) ∣y s∣∣y s∣⋅exp −∣s∣0 when s±∞ .

The condition (5c) is satisfied e.g. if y and dy/dt are bounded. Examples of “forbidden” solutions arey(s) = exp(±s) corresponding to u = 0. Thus, if we know that the control-signal u varies so as to keep

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y bounded, then (5b) and the Fourier-method are justified1.–

As a control problem it remains to determine how u varies with time in order to mantain erectposture, and it is here that further physiological considerations must enter into the picture. Theplantar and dorsal flexors generate a torque T with respect to the ankle joint, represented in (1) by

(6) T =u F z F y≈u m gm y

A consequence of (6) is that the torque T is to a good approximation proportional to the COP-u,since the second term m y is usually much smaller in comparison, its magnitude typicallybeing less than a promille of m g u. Indeed, a basic observation is that people normally in quietstanding lean a bit forward with the vertical line through the COG lying about 2 – 5 cm ahead of theankle joints (talocrural joints) which means that m g u is a positive quantity bounded from below2.Especially it follows that there should be no systematic time-lag between muscle activation(generating the torque T) and COP-u, save for the viscoelastic properties of the muscle-tendonsystems. This means that the plantar flexors such as the gastrocnemius (GA) and soleus (SL)3 mustbe constantly active in order to counteract the gravitational pull. The simplest model to account forthese facts is depicted in figure 1. One muscle group (GA) – or two if we think of both legs – hasbeen added which generates a rectifying torque with regards to the ankle joints in a forward leaningposition. In our measurements using electromyography (EMG) we found strong evidence that GA isresponsible for the modulating force, see fig. 3. Other muscles such as SL are also active andessential and their role during quiet standing seem to be to set the the basic tonus. The idea is thatthis simplified model is still able to reveal the essential mechanisms of balance in quiet standing.More detailed models can be developed to address specific issues regarding e.g. the influences ofcertain sensory inputs, to accommodate other sorts of experimental conditions which may involvemulti-segment models, etc (for a 3-segment model see e.g. van Soest, Haenen & Rozendaal 2003).The physiological basis for the human inverted pendulum model of balance subsumes: (a) the“ankle strategy” for postural control, which is typical in quiet standing; (b) the co-activation ofmuscles around the hip and knee joints as well as the postural muscles in the neck and the trunkwhich freezes the degrees of freedom (all in all there are no less than 639 muscles in the humanbody) thus justifying the “solid body” model. A quite different physical continuous model, the socalled pinned polymer model, has been suggested by Chow and Collins (1995), which basically hasan infinite number of degrees of freedom. Such a model is perhaps too ambitious in face of theapparently good validity of the much simpler human inverted pendulum model for quiet standing4.However, an interesting suggestion coming from the pinned polymer model is to apply thefluctuation dissipation theorem (FDT) to balance data (Lauk et al. 1998), a method which can aswell be used for a “noise”-driven human inverted pendulum model (such as e.g. described by equ(7) below). Lauk et alii proposed that such a method could be employed in order to predict, using

1 Prof. Olof Staffans (Åbo Akademi University, Finland) pointed out to us that the condition (5c) for the solution (5b)is related to the concept of “exponential dichotomy” in functional analysis.

2 In equ (6) it is assumed that we are using the coordinate system with u = 0 at the ankle joint. The actual A/P COPdata are though often presented with a different reference point depending on the measurement setup and dataanalysis. Typically one centers the data around the average point.

3 GA and SL are collectively referred to as the triceps surae and are all inserted to the heel bone – calcaneous – viathe Achilles tendon

4 It is possible that an elastic model could be used to describe small deviations (segmental oscillations) from thehuman inverted pendulum model. In this context one could also mention a recent contribution (Koshenev 2004)which analyzes human walking while explicitely avoiding the inverted pendulum models.

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only quiet standing data, the response to small disturbances.

Summary of main points. − We have discussed the evidence that: (a) quiet standing can bephysically modelled by the human inverted pendulum model; (b) the swayings exhibit an apparentrandom fluctuation around an average forward leaning position between ca 2 to 3 degrees; (c) theGA-muscles are mainly responsible for modulating the balance while SL sets the basic tonus; (d)the inverted pendulum model predicts that the muscle force (or rather the produced torque) and themeasured COP should be in phase.

III. Postural control - review

Postural control is active on several levels5. In order of increasing complexity, processing andlatency, we have the fast myotatic stretch reflexes (ca 40 ms) of which the patellar reflex is a typicalexample and which are mediated by the spinal pathway; the learned (“programmed”) automaticpostural responses (ca 100 ms) needed e.g. when we ride a bike and which are mediated by thebrainstem and subcortical pathway; the voluntary postural movements (more than 150 ms) whichare mediated by the cortical pathway. In quiet standing we expect the two first levels to be in focussuggesting some sort of feedback model for postural control (the physiological basis of feedback isoutlined in Appendix B). The following sections present brief reviews of some of the central modelsthat have been proposed for the postural control.

A. PD-Feedback control

Peterka (2000, 2002, 2003) has, among others, employed a PID-control model6 borrowed fromengineering in order to attempt to explain a number of features of the balance data for quiet andperturbed standing. While Peterka is confident that the feedback model can explain the essentialfeatures of postural control, others like Morasso et alii (1999), who have also studied the invertedpendulum model as a PD-control problem, have concluded that an anticipatory mechanism (internalmodel) is required in order to explain the stability of the system. However, the working of such ahypothetical anticipatory mechanism has remained a mystery. The model by Peterka (2000) isapplied to the anterior-posterior sway. Masani et alii (2003) adopted the same model but theydropped the integral-term obtaining a reduced PD-model whose equation of motion can be written(which is similar to a form considered earlier by Morasso & Schieppati 1999)

(7) I t K D t−K Pt−−m g L⋅sin t =⋅∫−∞

t

eu−t f n udu

The ansatz (7) is thus based on the assumption that the gravity resisting torque T(t) at time tgenerated by the muscle-tendon is given by

5 A convenient summary is provided by Kejonen (2002). Aspects of the stretch reflex (for the wrist) has been probedby one of us (MH) in an earlier M. Sc. thesis work (Herrala 2002).

6 For an introduction to control theory and proportional-integrative-derivative models in a physiological context seeKhoo (2000).

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(8) T t =K D t−K Pt−T noise

The Tnoise term in (8) has been put in a parenthesis because in the litterature it has not been madeclear whether the stochastic component of the r.h.s. of (7), represented in the model by a filteredGaussian noise n(u), should be counted as part of the muscular torque, and thus be included in (8),or as an “external” fluctuation. In fact, it is this fluctuating part in the model which produces theswayings, yet the origin of these fluctuations has not been accounted for.

From a physicist's point of view equ (7) represents a model for the Brownian motion of a particle ina “retarded” force field. Observed swayings seem however to differ from the simple Brownianmodels in one important respect; namely, the COP-motion is often characterized by intermittency.The COP-point seems to “oscillate” now around one “center point” and then (typically within 0.5 –2 s) moves to another center point (Zatsiorsky & Duarte 2000). This is a bit reminiscent of thesaccadic movementes of the eye. Apparently we have at least two different mechanisms at workhere. {statistical measure for deviation from Brownian motion?} The “saccadic” shifts could bepart of a strategy to avoid fatigue and sensory adaption and thus work as a constant wake-up call(Borg 2003).

B. λ-Theory

One set of control models which involve thresholding on the macroscopic level is related to theequilibrium point hypothesis (“λ-model”) advanced by A G Feldman in the mid sixthies whichattempts to describe the CNS-control of movements7 Micheau, Kron & Bourassa (2003) havepresented λ-model for postural control which postulates that the muscular torque is represented by(here given in a linearized8 version)

(12) T t =[ K D t−K Pt−−t−e]+

where [x]+ is defined by [x]+ = x if x > 0 and otherwise 0. The efferent delay τe is the time it wouldtake for the central “command” to reach and activate the muscle. An alternative delimiter issuggested by Staude, Dengler & Wolf (2000) defined by [x]γ

0 = 0 if x < 0, [x]γ0 = x if 0 < x < γ, and

[x]γ0 = γ otherwise; thus, γ is sort of a saturation level. For small excursions around the operating

point these two delimiters agree and they take care of eliminating the physiologically unrealisticnegative muscle forces – muscles cannot push (TA, acting in the opposite direction, would bepresented by a similar term but with a negative sign). The parameter λ represents the way CNS isthought to influence the equilibrium point of the equation of motion, which in the MKB-model iswritten as

(13) I T t – m g Lsint =0

7 Feldman's λ-model is extensively discussed in Cordo & Harnards (eds.), 1994; for recent contributions to the controldebate see Feldman & Latash (2004) and Ostry & Feldman (2003).

8 Strictly speaking (12) is not a linear model due to discontinuous delimeter function. MKB employ the exponentialsuggested by Feldman – see equ (16).

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In contrast to the PD-model (7) there is no explicit stochastic term in (13), instead it is replaced bythe lambda CNS-term λ(t). Therefore the PD-model (7) is basically equivalent (save for the lack ofa delimiter) to the MKB-model (12-13) to the extent that CNS-commands look random and can bemodelled as a filtered Gaussian noise. The MKB-group did not attempt to model the evolution ofthe “hidden variable” λ but extracted it instead from the measurement data using the relation (12).(As an aside they report in their paper of having measured EMG for quiet standing but that theyfound that “its signal-to-noise ratio is too small” to be useful.) The significance of (12) is that theparameter λ functions as a threshold such that the muscle will be activated (EMG increased) if theexpression K D t−K Pt− exceeds λ,

(14) K D t−K Pt−t−e⇒ activation

This is similar to the form presented by Feldman & Latash (2004) who write the activationcondition as (in terms of the total muscle-tendon length x, which for the human inverted pendulumand the GA is coded by the inclination angle θ)

(15)x*⇒ activation where *=F− x f t

.

The λF in (15) is a rescaled version of the λ in (14), λF = λ/KP. The version (15) also includes the“effect of inter-muscular interaction” ρ, and a “history-dependent change of the threshold” f(t). Theformulations (14, 15) do not say what the activation level will be; equ (12) implies a linearrelationship between the resulting torque and [x – λ*]+. Feldman has termed the force, as a functionof [x – λ*]+ , as an “invariant characteristic” (IC) and proposed (Feldman 1966) the exponential IC-form

(16) F m=k exp[ x−]+−1 .

while e.g. Shadmehr & Arbib (1992) have used the form

(17) F m=1

ln {exp k x−1 }

(which apparently is always positive even without the delimiter if α, k > 0, but has the odd propertyof giving a nonzero force ln(2)/α for zero elongation case, x = λ).

Many authors (e.g. Gatev et al. 1999; Loram et al. 2004; Casadio et al. 2004) have suggested that

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quiet standing involves a feedforward control mechanism. The model (7) may in contrast seem likea pure feedback model plus a stochastic input; however, as pointed out above, the “stochastic” partmay be exactly where the “open loop” control is hiding. Masani et al. (2003) have, usingsimulations based on (7), showed that the model can produce the observed EMG phase lead. Thetheoretical reason for this effect (in the model) is described in Appendix B. The maximum timeadvancement according to (B.4) (when f approaches zero) is given by KD/KP. Figure 8 shows howthe time advancement varies with the frequency in the range 0.01 – 5 Hz in the cases KD/KP = 1/2and KD/KP = 1/5 considered by Masani et al. (2003). As we can see, the time advancement exceeds200 ms and 140 ms respective case when f < 1 Hz.

In conclusion, the salient points of the λ-models are that: (a) the neural control is directed atcontrolling the muscle length; (b) the control is of a threshold type and therefore basically non-linear. The models however, as such, do not explain the origin of the swaying.

C. Stiffness hypothesis

Winter at alii (1998, 2001, 2003) have emphasized the “passive” control via muscle stiffness. Oneof the main the objections (Morasso et al. 1999; Loram et al. 2001, 2002, 2004; Hof 2001) againstthe stiffness control is that the measured stiffness KP seems to be inconsistent with the requirementof stability (that is, it was found that sometimes KP < m g L which in equ (7) leads to instability).The charge is that models like (7) seem to work only because one uses physiologically unrealistichigh stiffness parameters KP. (Masani et al. (2003) e.g. use the same “big” stiffness values asemployed by Peterka (2000).) This controversy arises perhaps due different things people read intothe concept of stiffness, whether one includes the active stiffness component present in reflexivemuscles or not. Evidently, if one uses a model like equ (7) the “stiffness” KP must be hypercriticalon order to be in line with the fact the we can keep standing. Also a source of confusion is that theterm “passive” is sometimes used to indicate that no forward control is involved (no anticipatorymechanism), and sometimes to emphasize the properties of an areflexive muscle (which does notreceive feedback from the spindles). The areflexive criterion is more or less equivalent to the usageby Peterka (2002) who calls “feedback” passive if it does not involve time-delay. This is in accordwith the stiffness control hypothesis by Winter et alii (1998) in which the control of quiet standingis modelled by the equation

(7a) I t B t K – m g Lt =F t

with no time-delay. Here F stands for some fluctuation thought to be explained by a process where“energy is continuously generated into this mass, spring, and damper system”, a system which theauthors compare with a “tuned mechanical circuit”9. A further assumption is that K > m g L and thatthe “resonance frequency” is given by

9 This formulation is a rather compelling one and all the models which use a noise-source such as (7) and the PMD-control model to be described below are models where energy is “dissipated” in a mass-sprong system.

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(7b)res=n≡ K e

I

K e≡K – m g L

.

They propose to measure Ke (and thus the stiffness K) by finding the resonance peak of thespectrum of the difference A/P COP – COG (y – u in equ (2)). Since, according to the basic relation(2), this difference is proportional to the second derivative of θ (or y) the resonance frequency willcorrespond to the peak of the spectrum for θ multiplied with ω2, which yields

(7c)res=

n

1 – B2

2 I 2 n2

instead of (7b) used by Winter et alii who neglect the influence of the damping constant B. In caseof a small damping constant we do get (7b). For one set of measurements Winter et alii (1998)quote, however, the following average values

fn (Hz) = 0.620 ± 0.110Ke (N m/rad) = 901 ± 338 ⇒ K = Ke + mgL ≈ 1600 B (N m s/rad) = 346 ± 93

which are close to the critically damped case B

2 I n=1 (actual value 0.76). For these values

the theoretical spectrum would not even have a peak, which requires that B

I n2 . The peak

values observed for the COP – COG spectrum is likely to be a consequence of the fact that thedifference is a band-passed signal whose spectrum thus must have a maximum somewhere in thisband. The large value for damping parameter B given by the authors is also inconsistent with apassive control because it would mean that the ankle joint normally dissipates a significant part ofthe kinetic energy, which biologically makes little sense. Still it might be true that an active B-component can be sizeable; indeed, it plays a major role in some of the models referred to above.Eurich and Milton (2000) even assumes that in quiet standing we are dealing with an over-dampedsystem that can be described by a Langevin-type equation. The above method of determining the stiffness constant by “resonance” assumes implicitly that thespectrum of the fluctuation F in (7a) is more or less flat (“white noise”) over the relevant band-width (0 – 10 Hz). This assumption does not seem to be valid. For a mechanical system one mightindeed probe the resonance frequencies by using an artificial white noise input, but in the case ofquiet standing any perturbation is likely to trigger e.g. the muscle spindles which via feedback mayalter the “constants” K, B, etc. A typical perturbation scheme is to tilt the supporting surface. (Adifferent perturbation system used by us is shown i fig. 10.) If the forward tilt angle is denoted θs

then Peterka (2002) proposed that the feedback torque is given by (we have dropped the integral I-

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term for simplicity)

(19) T =−K t −B t – K P t−−st−−K D t−−st−

which includes a passive component (no time-delay) and an active feedback component (with delayτ). Inserting this into the equation of the inverted pendulum and taking the Fourier-transform leadsto the transfer function

(20) H =s

=−K – i B – K Pi K De−i

−I 2 K – m g Li BK Pi K De−i

Estimating the transfer function from perturbation data via standard cross-correlation analysis andfitting it to the model (“system identification”), Peterka (2002) obtained the following estimates forthe parameters10

K (N m/rad) 91.7 ± 28.4KP 968.3 B (N m s/rad) 24.6 ± 10.9KD 349.5

The estimated “passive” parameters K and B are about ten times smaller than the values quotedfrom Winter et alii above, while the active parameters KP and KD are somewhat closer. This maysuggest that Winter et alii have in fact measured the “active” parameters instead of the assumed“passive” ones. However, one can also question the reliability of the system identification methodbased on (20) or similar equations since they take for granted that we are dealing with a simplelinear feedback system. Some measure of arbitrariness is also involved in the system identificationprocess since, if we suppose we have a feedback-term of the form

(19*) T =K t B t K Dt− ,

then to the first order we have t−≈t −⋅t and the expression (19*) can be approxi-mated by KK D⋅t B− K D⋅t with no explicit time-delay term. Another point of viewis that if we assume an active feedback term of the form

K⋅t ,

where t is an estimation of the angle at time t given data at time t - τ, then if we may take for

10 Peterka (2002) finds also in his data, by the way, that there is a negative “linear” correlation between the time-delayτ and the parameter KP. Indeed, if the algorithm for extracting the parameters tries to offset the phase-shift due to thetorque term (see Appendix B) then we would expect an inverse relationship between the estimated delay τ and KP.

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the estimation the expresson t =t−⋅t− which would thus predict that

(19**) BK

≈

Even if one were to adopt some of the above stiffness values, and methods of their measurements,as about correct little has been said about the cause of the swayings. While Winter (2005: 108)recognizes, on the basis of the fundamental relation (2), that, in order to maintain stability “the COPmust move ahead of and behind the COM”, the role of the stiffness and the principal mechanism ofthe control system responsible for this movement remain unclear. In the next sction we will try toaddress these issues.

IV. Dynamic stiffness and PMD-control

As pointed out earlier, the measurements of muscle stiffness thus seem as yet to be riddled withsome principal uncertainties and definite conclusions may therefore have been premature. However,for the tendon stiffness there are already some conclusive evidence which will have importantimplications for our understanding of the postural control, as we will try to show in this section.There is also another point which has often been overlooked by modelers. Since quiet standing isassociated with a forward leaning posture the models should predict, or be consistent with a positivetime-average ⟨⟩ (around 2° - 3°) for the inclination. One obvious reason for this forward leaningposture is that the average COP/COM-point will then be situated around the middle of the feetalong the A/P-direction which is the safest position considering the inevitable swayings back andforth. However, since the average velcocity and acceleration must be zero, and if we may assumethe same for the “fluctuation” force, then the model (7a) is consistent with a nonzero averageinclination only if we have exactly K = m g L; i.e. Ke = 0. Such a fine tuning would cry out for anexplanation. The same sort of problems also afflicts the PID-models. Mathematically theseproblems could of course be partially fixed by shifting the origin of the θ-coordinate. Yet what isstill missing is a proper account of the muscle-tendon mechanics in which the relation betweenmuscle and tendon lengths and the inclination is of central importance, as has been emphasized e.g.by Loram and his group.

Maganaris and Paul (2002) have studied the tensile properties of the human GA tendon in vivo andconfirmed that it behaves as a nonlinear spring with hysteresis. For their data of force F vs elonga-tion ∆l (measured by ultrasound), when fitted with a second order polynomial, they reported an R2

value of 0.98 ± 0.01. Extracting data from their figure 3 of mean F (N) versus elongation ∆x2 (mm)for increasing loading condition one finds as a second order best fit (with the condition that we havezero force for zero elongation)

(21) F=39.1⋅ x2 3.4⋅ x22 .

The quoted mean data cover a force range about 0 – 870 N, and an elongation range 0 – 11 mm. In

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fact the maximum tendon force for their group (n = 6, young male adults) was reported to be 875 ±85 N. During unloading the corresponding elongation for a given force was larger than in theloading case (hysteresis). Differentiating F in (21) visasvis ∆l we thus get an estimate the stiffnessof the series elastic component (SEC),11

(22) K s=∂ F

∂ x2 .

To present some numbers we assume as an example that m = 76 kg, L = 0.9 m, and an averageforward leaning of 0.05 m (3.2°). The weight will then be m g = 746 N corresponding to (in case ofsymmetrical loading) of 373 N per leg and tendon. According to equ (21) this will furthercorrespond to a 6.2 mm elongation of the tendons. By elementary geometrical reasoning one infersthat the total muscle-tendon length x changes only by ca 1 mm per degree of leaning; thus, since intypical cases the swaying amplitude is around 1 degree the muscle-tendon length change duringquiet standing will be around 1 mm. If we assume that the GA muscle-tendon is in a neutral positionat θ = 0, then in order to reach the forward leaning “equilibrium” position the tendon has to extendto a length x2 = x - x1 (x is total muscle-tendon length, x1 is the muscle length – see fig. 12) for whichthe corresponding tendon force (both legs combined) generates a torque Ttend equal to the gravita-tional torque. The equilibrium condition

(23) T tend x−x1=m g L x

defines in fact an “equilibrium curve” xequ = x(x1). Along the curve (23) we have naturally

(24)d T tend

d =m g L .

This may explain measurements of ankle “stiffness” - which yield values close to critical stiffness -if the “fluctuations” or perturbations somehow adhere close to the the equilibrium curve. E.g. Loramand Lakie (2002) report measurements that indicate the average “intrinsic stiffness” to be 0.91 ±0.23 of the critical value (m g L). The “stiffness hypothesis” assumes though that the stiffness ishypercritical but it is not explained how this can be achieved. Loram et al. (2004) have also usedultrasound in order to measure the GA muscle length during voluntary plantar/dorsal flexiondemonstrating the compliance of the tendon. The muscle length was found to shorten by ca 4 – 5mm when the person leaned forward by about 5 degrees.

In order to calculate the stiffness we would have to know the angle θ as a funcion of tendon lengthx2, θ = θ(x2), when the system is perturbed from the equilibrium position. This in turn requires a

11 Comparing with the mean stiffness values given by Maganaris and Paul (2002: fig. 4) their values seem to be about30% larger than the ones computed via (22) and (21). They report mean maximum stiffness ranging from 49 ± 12N/mm to 150 ± 28 N/mm for load levels of 0 – 10 % MVC and 90 – 100 % MVC respectively (MVC = maximumvoluntary contraction).

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muscle-activation model to begin with. For simplicity, suppose that the muscle keeps its length (x1)constant at the forward leaning “equilibrium” position (the example discussed above), then any totallengthening would be due to the elongation of the tendon length x2 alone while the tendon at thispoint would have the stiffness of ca 81 N/mm. If 1 mm total lengthening (∆x) corresponds to a 1degree change in the forward leaning, then the effective stiffness will be (taking both legs intoaccount and assuming a moment arm length of 0.05 m) only ca 464 N m/rad to be compared withthe critical value of m g L = 671 N m/rad. The equilibrium point is thus inherently unstable andeven the smallest perturbations or mismatches are likely to send the system on a run-away courseunless the muscle responds properly. Erect posture depends therefore on the dynamic activitypattern of the muscle. Unlike the case of a very stiff tendon the muscle cannot lock the system into astatic postion. Therefore swaying is not merely a consequence of “disturbances” and “noise” but anintegral and essential part of the dynamical stabilization process. This suggests that the conclusionby Collins and de Luca (1994) that there is no sign of deterministic chaos in balance data could bepremature.

For a given movement and time “dynamical stiffness” can be defined by

(25) K =∂T tend

∂ =∂T tend

∂ x2 ⋅∂ x2

∂ x ⋅∂ x∂ .

Given that, for the above mentioned case

(26) ∂T tend

∂ x2 ≈8.1 N m/mm and ∂ x∂≈56 mm/rad ,

in the neighbourhood of the “equilibrium” point, we find that (25) exceeds critical stiffness if(tendon length is x2 = x – x1 )

(27) ∂ x2

∂ x = 1

1 ∂ x1

∂ x2≥ 1.5 ⇒ k min≤−∂ x1

∂ x2≤k max .

with kmin ≈ 0.7 and kmax = 1 in the above example. The question arises whether stability - in thesense of the swayings being bounded - can be ensured in this case if the neuro-muscular controlimplements the following rule:

the tendon should lengthen faster than the muscle contracts but

(27*) no faster than about one and a half as fast as the muscle contracts; or, equivalently, the tendon should lengthen one and a half as fast as the total muscle-tendon length.

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Could there be some form of a “dynamic stiffness control” in contrast to the passive stiffnesscontrol (where the tendon stiffness is assumed to be hypercritical) in which a rule like (27*) willplay a fundamental role? Before we proceed with this question let us abstract the essential elementsof the posture control problem and pose the following set of conditions and (rescaled) equationscorresponding to a springy control of the inverted pendulum:

(29)

a x t =x t – u t b x=x1 x2

c u t = f x1 t− , x1 t− , u t−≥0 d u t =h x2 t , x2 t , t e ∂x2

h x2, x2, t 0 f equilibrium point x1 , x2 =x1

0 , x20 with u0≡h x2

0= f x10 , 0 , u0=x1

0x20≡x0

g ≡∂x2hx2

0 , 0 , t 1

Here (29.c) f denotes the feedback force which is assumed to depend only on the time delayedvalue of the muscle length x1(t-τ) and the force level u(t-τ). Next (29.d) expresses the condition thatthe tendon force h(x2) must be equal to the feedback force, furthermore the tendon force is assumed(in (29.e)) to be an increasing function of the “tendon” length x2. Finally the condition (29.g)implies that the equilibrium point is an unstable in the sense that the stiffness is “subcritical” at thispoint; that is, a stable solution cannot be obtained by trying to keep x1 constant and equal to theequilibrium value x1

0. In (29.d) we have indicated that the tendon force may depend also on the rateof change of the tendon length x2 and on the time (hysteresis effects). In the simplest version thoughone may consider only dependence on x2, typically in form of a quadratic equation as in (21). Ageneral question is then what sort of non-trivial systems and bounded solutions (29) are allowedaround the the equilibrium point. We would like the solutions to be bounded also for small random“perturbations” added to the variables or parameters. The set (29) may also be looked on as asomewhat odd example of a coupled oscillator problem with x1 and x2 designating the twooscillators (muscle and tendon).

Now, if according to a rule of the form (27, 27*), x (or x1) and x2 were forced to depend on eachother as

(30) x2 =a⋅xb

we could easily, by using the right choice of the parameter a in (30), obtain bounded solutions. If wecan neglect the hysteresis then, since we have a relation between force and the tendon length x2, thefeedback law (29.c) can indeed be expressed in term of a sum x = x2 + x2, or in term of an estimate

x of the sum (neglecting the delimiter for a moment),

(31) u t =k x t l .

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Equ (31) together with u(t) = h(x2) does at least imply a relation between x2 and an estimate of x.One simple estimate, suggested earlier above, is given by

(32) x t =x t−⋅x t− .

If we replace the derivative in (32) with a difference (or a time-average of the derivative) we obtaina form of feedback that can be compared to the proportional minus delay controller (PMD) intro-duced by Suh and Bien (1979). Atay (1999) has applied the PMD-control model to the invertedpendulum described by the delay equation

(31) x t k x t =a⋅x t – 1b x t−2

obtaining the following proposition:

Proposition [Atay 1999]. Suppose k < 0. Then the zero solution of (30) is asymptoticallystable iff

(a) k > -1,

(b) −kb2

2

– k

and

(c) −2 b⋅cos bk ak – b .

The proposition can be proved by investigating the characteristic equation of (31),

(32) 2 k – a e−−b e−2 =0 ,

obtained by inserting x(t) = exp(λt) into (30). The stable (a, b)-regime corresponds to solutions of(32) with Re(λ) < 0. We can apply this result to the human inverted pendulum (2) and write12,

(33) x t =2 x t x t− x t – 2

12 In our model the geometrical relation between A/P COG y and muscle-tendon length x is about ∆y/∆x = 15.6 forsmall inclinations.

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If we rescale the time with the delay τ, t → t/τ, then (33) can be rewritten on the form

(34) x t =2 2 x t a x t−1b x t – 2

where −2 2 can be identified with the parameter k in the proposition. Here x corresponds tothe total muscle-tendon length x1 + x2, and by shifting the origin we may assume that the equilibriumpoint corresponds to x = 0. The first stability condition (a) translates into

(35) 1

.

A typical value of ω = 2 Hz implies thus an upper bound of 500 ms for the delay τ. Using τ = 0.2 swe obtain k = -0.16. The conditions (a-c) define, given the k-value, a stability area in the (a, b)parameter space (see fig. 15). An example of parameter choice in the stable region is a = -0.6, b =0.5. A direct numerical solution of the characteristic equation (32) for this choice gives the root λ =-0.982 demonstrating the stability. In order to get physiologically interesting results we have tostudy modifications of (34) by adding e.g. noise or “perturbation” terms. It is of course hardly likelythat the neuro-muscular system is able to exactly assess the total muscle-tendon length x or generatean exactly determined response (feedback). The muscle spindles do provide information about themuscle length x1, while the tendon elongation x2 can be estimated from the force (gauged by theGolgi tendon organ, GTO) by using the force-elongation relation for the tendon (function h in (29)).These informations are though to be exptected to be somewhat “fuzzy”. One modification of (34)which introduces “perturbations” is of the form

(36) u t =[−a x t−1−b x t – 2c x2t−1−x20t e0 ]

+−e0

Here ε denotes a noise contribution (in simulations we have used filtered Gaussian noise) and e0 is anon-negative threshold parameter. Thus, (36) allows “negative” feedback force (down to -e0) butthis is due to the shift of the origin only. The new x2-term in (36) is added in order to reflect the factthat neuro-muscular system may not be able to assess the sum x1 + x2 exactly. In this way it is quiteeasy to generate non-trivial models which exhibit irregular swayings. These models demonstratethat anticipatory control is not necessary in order to stabilize the human inverted pendulum.

V. Summary and Conclusions

We have presented evidence that quiet standing can be analyzed in term of the human invertedpendulum model which involves a phasic postural control mode, mainly mediated by the GA-muscles. Investigations of the tendon properties reported by other groups imply that the quiet stanceis an inherently unstable system which requires the muscle to continuously adapt in order to createan effective stiffness off-setting the effect of gravity. This activity gives rise to the characteristic

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sway patterns observed for quiet standing. These findings will have implications for interpretationsof the parameters obtained in posturological measurements.

The undulatory character of the postural control mechanism becomes also evident using a feedbacksystem that simultaneously shows A/P COG and A/P COP on a computer screen, and by which theuser may voluntarily control the A/P COG position e.g producing a smooth and slow swaying backand forth. The “organic control system” translates the voluntary commands into a proper A/P COPcontrol signal resulting in the desired COG-output. Though it is possible to quite smoothly controlthe COG via the visual feedback system it appears, according to our experience, to be impossible to“control” COP smoothly. The reason for this behaviour is the inherent unstability of the systemwhich forces the muscle to continually respond to the changes in the tendon length, which “it” canassess from the load. The instability means that any fluctuation in the neuro-muscular system maybecome amplified into observable swayings. Fluctuations in biological control systems are indeed tobe expected since they have to do “calculations” in a noisy environment (Koch 1999). Thegeneration of an action potential is a threshold on-off phenomenon and is thus suspectible torandom influences. Furthermore, the process of the release of the transmitter susbstances at thesynapses is a stochastic phenomenon itself. A threshold-type activation means that small naturalfluctuations in the action potentials can be amplified since their contributions can make thedifference whether the threshold is crossed or not13. To this we have to add the continuouslychanging properties of the muscles due e.g. fatigue; the internal environment is changing all thetime.

The “chaotic” nature of the human inverted pendulum has also been tested with a simplemechanical model. A spring (corresponding to the Achilles tendon) is connected to an invertedpendulum (steel bar) and further connected to steel wire which runs over a pulley (corresponding tothe heel bone, calcaneous). By pulling from the wire (corresponds to the GA-activity) one tries tokeep the pendulum from falling. The spring constant is such that the equilibrium positions areunstable similarily to the case of the real muscle-tendon system. In the absence of significantfriction it is impossible to keep the inverted pendulum stationary, one is forced to constantly varythe force and thus create a swaying motion in order to keep it from toppling over. This action webelieve basically mimics the activity of the triceps surae at quiet standing and shows that theswaying is an integral part of the stabilization mechanism.

In a number of sets of measurements one of us (FB) has attached EMG-electrodes to GA, TA andSL while performing swaying movements back and forth (see fig. 13). When displaying the EMG inreal time on the computer screen one feature becomes evident - a sudden increase in GA activity isseen as soon as one makes the smallest movement forward, and a similar sudden increase in TAactivity is seen for backward movement14. In this sense the favoured “operating point” in quiet

13 This may remind some readers of the concept of “stochastic resonance” (SR), and indeed, there are someexperimental evidence of SR for quiet standing (Priplata et al. 2002). Influence of haemodyanmics and breathinghave also been suggested as cause of the swayings, but they seem too regular (though the amplitudes vary) in orderto explain as such the random character of the swayings. The size of the force of the heart oscillation is of the order 5N. Supposing its center of mass is shifted by 5 cm with respect to the ankle joints in the horizontal plane, this gives aperturbation torque of the order of 0.25 N m to be compared with the typical total gravitational torque (m g L θ) ofthe order of 30 – 40 N m. However, these perturbations may contribute to the threshold-crossing phenomenon. Afurther observation is that when we subjected the participants to a tug from behind corresponding to about 14 N m,the GA modulation in some cases all but ceased while the TA on the opposite side started to pull although the 14 Nm did not offset the gravitational torque (though this torque will subsequently be reduced when the body is pulledbackward shifting the AP COP by 2 – 3 cm). Conceivably, by adjusting the perturbing torque one could tune theproportion of TA and GA modulating activity.

14 If one lets oneself fall forwards freely (“ballistic motion”) then the GA activity ceases till one starts to brake the fall.

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standing seems to correspond to the minimum GA and TA activity. Furthermore, the SL activityexhibits smoother transients than GA. These observations suggest that, around the “equilibriumpoint”, SL has a dominantly tonic role setting the basic tonus, while GA has a phasic rolemodulating the force around the “equilibrium point”. This phasic modulation by the GA can belinked to the dynamic stiffness control. However, a detailed comparison of the roles of SL and GAwill remain the subject of future studies. Here our central aim has been to highlight the generalcontrol mechanisms involved in making it possible for us to stand upright.

Acknowledgments

This work was made possible by a project sponsored in large part by the National TechnologyAgency of Finland, TEKES ( www.tekes.fi). We are indebted to Noraxon Co (www.noraxon.com,AZ, USA) for an extended loan of their Myosystem 1400 EMG-device. Dr Peter Konrad (Noraxonhas provided useful inputs about EMG-measurement issues. Hur Co (www.hur.fi, Finland) andCEO Mats Manderbacka have been one of the main supporters of the project. We thank Ian D.Loram (Birmingham, UK), At L. Hof (Groningen, NL), Gerhard Staude (Neubiberg, Germany) andMarco Schieppati (Pavia) for offprints and correspondence. Special thanks to the participants of thetests, and Ms Linnea Luokkala, secretary of the regional (Central Ostrobothnia) MS-society.

Abbreviations

DSC dynamic stiffness controlGA (medial) gastrocnemius (“calf” muscle)TA tibialis anterior (“shin” muscle)SL soleusGTO Golgi-tendon organEMD electromechanical delayEMG electromyography/gramCOP center of pressureCOG center of gravity (here equal to the center of mass, COM)A/P anterior/posteriorM/L medial/lateralCNS central nervous system (brain + spinal cord)LP low-pass (filter)MU motor unitMUAP motor unit action potentialMVC maximum voluntary contractionPID proportional-integrative-derivative controlPMD proportional-minus-delay control

Appendix A. EMG-force relationship

In order to get some information about the activity of the postural muscles one can useelectromyography (EMG) (for an overview see e.g. Winter 2005, Cram, Kasman & Holtz 1998) or

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Basmaijan & De Luca 1985). Surface EMG is a non-invasive method where surface electrodesattached (usually) to the belly of a (surface) muscle picks up a small current induced by the weakelectric field generated by the motor unit action potentials (MUAP). The EMG-signals range froma few microvolts (mV) up to the millivolt (mV) size. The MUAPs arise when action potentialsalong motor nerves reach the synaptic contacts with muscle fibers (muscle cells) causing localdepolarisations (when the rest membrane potential around -60 mV reverses momentarily to a ca 30mV peak value) over the fibre membranes which are then propagating as muscle action potentialsalong the fibres (with a velocity around 4 m/s) which are associated with transmembrane currents.Depending on the muscle type a single motor neuron may innervate from 10 to 2000 muscle fibers(constituting a motor unit MU) which thus amplifies the original action potential and makes itdetectable as EMG. Enoka (2002: 281) quotes a study where they for the medial GA of one subjectestimated the number of a-motor axions to be 579, and the total number of muscle fibers as1120000, the innervation ratio thus being 1934. The depolarisation of the muscle fibers triggers thecontractile process of the fibre in which the release and reuptake of the Ca++ ions plays a significantrole. Due to such electrochemical processes it may take 20-50 ms to activate the filaments in themuscle fibre while the reuptake of Ca++ may last some 80-200 ms during which the force decreases.The force generated by a single active muscle fiber is of the order 30 – 90 mN. The time from theonset of the MUAPs till an induced force is acting on the bones is named the electromechanicaldelay (EMD). For the stretched medial GA muscle an EMD around 15 ms has been reported(Muraoka et al. 2003). Such a small EMD is indeed consistent with the data reported here (seefigure 6). There is also a delay in the cessation of the force after the EMG-activity has ended whichis probably linked to the finite time of Ca++ reuptake, and this EMD is around 100 ms. Mathemati-cally the EMG-force relationship can be compared to a low-pass filter, as is also demonstrated byour data. Denoting by emg(t) the raw EMG-signal (the sampled signal is in reality a a time-discretesignal), we calculated an envelope of the EMG by first rectifying, then subtracting the mean andthen low-pass filtering the signal resulting in an output signal s(t),

(A.1)

d 1 t =∣emg t ∣ ... rectifyingd 2 t =d 1t – d 1 ... subtracting the mean d 1 of d 1t d 2 f =∫

−∞

∞

exp −i 2 f t d 2 t dt ... Fourier transformation

d 3 f = 1

1 ∣ ff c∣

n f⋅d 2 f ... LP-filtering

s t =∫−∞

∞

expi 2 f t d 3 f df ... inverse Fourier transformation

The force F generated by the muscle is assumed to be proportional to the low-pass filtered EMG,

(A.2) F t ≈k⋅s t−

for some constant k and a delay τ (which according to our data is around 200 – 300 ms). This

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“constant” k obviously must depend to some extent on the location of the electrodes, the impedanceconditions, etc. For the “order” nf of the filter we have found that for nf ≥ 2 we get best agreement,whereas for nf = 1 the envelope becomes too “hairy”. The theoretical rationale for the linearrelationship is that the EMG may be expected to be proportional to the number of MUs activated,which in turn determines the force generated by the muscle. One objection against the abovefiltering procedure (A.1-2) is that it seems to imply a non-causal transfer function h(t),

(A.3)s t =∫

−∞

∞

h t – u∣emg u∣du

ht ≠0 for t0

Indeed, the low-pass (LP) filter corresponds to a transfer function which is a sum of terms of theform (modulo multiplicative constants)

(A.4) ei 2 zk f c∣t∣

where zk are the complex roots of the equation

(A.5) 1zn f=0

These terms (A.4) are obiously non-zero for t < 0. This non-causal property is partially offset, byusing the time delay τ in (A.2), if h(t) is very small for t < -τ. If we want to link a LP-filter to aphysical model for the EMG-force transduction then we need a causal filter. Basmaijan & De Luca(1985) mentions one proposed transfer function (in frequency and time space)

(A.6)h f = 1

1 i ff c

2 ⇔ht = t2⋅e

− t t≥0, 2 f c=

1

which is a LP-filter with the cut-off frequency fc. For our data we got best fit with fc = 1 Hz (insteadof the value 2.5 Hz quoted by Basmaijan & De Luca) although the fit using (A.6) is not in general asgood as with (A.1) in our case; indeed, the filter (A.6) seems to give a result in phase with COP, butits amplitude is not correlating as well with the COP. From the filter (A.6) we can estimate the timeshift ∆t from (see also Appendix B)

(A.7) t f =− 1 2 f

arctan 2 f / f c

1 – f / f c2

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which for small f becomes t≈– 1/ f c and for fc ≈ 1 Hz this is around -300 ms. When fapproaches fc from below the time shift approaches t≈– 1/4 f c or a delay about 250 ms. (Atf = fc there is a singularity in (A.7) and the time delay changes sign. However, the part f > fc will besuppressed due to the LP-filtering.) Equ (A.7) indicates a second order system and one can see howit might arise from the “standard” visco-elastic muscle-tendon model (see figure 12 and e.g. Winter2005, ch. 8; Hof (1984, 1997, 2003) has especially emphasized the role of the muscle-tendonproperties for the interpretation of the EMG-signal). If Fm denotes the force generated by the muscleand x1 its length (minus tendon length) then the model prescribes an equation for the form

(A.8) F m=M x1 d x1 k x1

which in the frequency space implies (with ω = 2 π f, γ = d/M, and ω02 = k/M ) that

(A.9) x1 f = 1 M

⋅F m f

−2 i 02

For nearly isometric contractions (total muscle+tendon length constant) x will be proportional to theforce F transmitted via the tendon to the bone. The muscle force Fm may be assumed to beproportional to the EMG-activity. For instance, one could propose a linear relation of the form (aweighted summation of the EMG-amplitude)

(A.10) F mt =C ∫−∞

t

eu−t ∣emg u∣ du

which would contribute with a factor of the form

(A.11)1

1 i

to the filter (A.9) resulting in a third order EMG-to-force filter (transfer function) of the form

(A.12) D⋅ 1 1 i

⋅ 1 −2 i 0

2

in the frequency space. The Fourier inverse of (A.12) is (ℑ(...) stands for imaginary part, K is a

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constant)

(A.13) ht =−K e− t

⋅ℑ{ 1 i a−1}K ℑ{ e−i a t

i a−1} for t0 otherwise =0

where

(A.14) a=02 −2/4 −i /2 .

The impulse response (A.13) predicts an exponential decay of the force after an EMG-pulse, whichis observed as a muscle “twitch”. It is commonly assumed that the muscle-tendon system is acritically damped one (Winter 2005: 251) whose impulse response function will then be of the form(A.6). Winter quotes (2005: 209) an average value of 79 ms for the medial gastrocnemius twitchtime τ (f0 =1.5 Hz). The critical damped state may though seem as a fortitutious tuning of themuscle-tendon parameters. The transfer function corresponding to the second order factor in (A.12)is given by (corresponding to letting τ approach 0+ in (A.13))

(A.15) h2t =e−

t2 ⋅

sin 0 t 0

with 0 =02 −

2

4.

which for small ν0 may be indistinguishable from the form (A.6) for times t < 1/ν0.

Staude, Dengler and Wolf (2000) have on the other hand formulated the activation level in terms ofa generated EMG-signal E(t),

(18) E t =b[ x−*]0⋅w t

where w(t) represents a white Gaussian noise. The reason for the proposed square root in (18) is notquite clear, but it might perhaps be argued that for a non-coherent summation of N pulses theamplitude grows like N and, since the contractile force may be assumed to be proportional tothe number N of MUAPs, it follows that the EMG-amplitude would be proportional to the squareroot of the force thus substantiating the ansatz (18). However, evidence seems to point to a (quasi)linear relationship as a rule. This is the case too for the relationship between the envelope of theGA-EMG and the A/P COP obtained in our measurements.

Appendix B. Phase shift and complex stiffness

One direct consequence of relation of the form

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(B.1) T t =K D t−K P t−−0

is that it leads to a phase advancement off-setting the delay τ. Inserting the form t =exp i t into (B.1) we obtain

(B.2) T t =ei t−

with the phase advancement or lead δ given by

(B.3) tan = K D

K P.

which in term of time corresponds to a time advancement of

(B.4) t=1

arctan K D

K P

From (B.4) we see that the time advancement is almost independent of the frequency when K D≪K P and is then given by KD/KP. The phase shift concept can be conveniently handled if

we for linear feedback systems, as exemplified by (B.1), adopt the complex stiffness parameters

(B.5) =T

defined in term of the Fourier transforms of angular displacement and the torque. (B.5) is wellknown from the mechanics of vibrations.

Appendix C. Physiological basis of feedback

A more detailed physiological motivation for the feedback term (8) is that skeletal muscles areequipped with sensing organs (stretch receptors) called muscle spindles which basically registermuscle length and its rate of change. One of the early investigations emphasizing the role of themuscle spindles in standing was Burke & Eklund (1977). The muscle spindle is sort of a smallmuscle (consisting of intrafusal fibers) within the bulk muscle (consisting of extrafusal fibers).According to figures quoted by Enoka (2002: 233) there are about 27500 muscle spindles in thewhole human body, of which ca 7000 are in each leg. The intrafusal fibers are attached in parallel

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with the extrafusal fibers. Whereas the extrafusal fibers receive their synaptic input from the α-motor neurons, the spindle receives independent synaptic input from γ-motor neurons called γ-efferents. There are two varieties of the γ-efferents. The static γ-efferents (γs) control the lengthsensistivity, while the dynamic γ-efferents (γd) control the length-change sensitivity. The sensoryoutputs from the muscle spindle, to the central nervous system (CNS), are in turn conveyed by theprimary (Ia) and secondary (II) axons. The Ia-afferents respond mainly to length changes (velocity)whereas the II-afferents respond mainly to length itself (position). Roughly, the Ia-output can becompared to the output of a high-pass filter while the II-output to that of a low-pass filter of musclelength data. The amplitude of the outputs are controlled by the γ-efferents. Thus, if e.g. γs is in an“off-state” and γd is in an “on-state”, then the II-output will be reduced while the Ia-output - andtherefore the sensitivity to length changes - is enhanced. In general the (non-negative) “gain” factorsKD and KP in (8) should not be considered as constants but as functions depending on the efferentinput – this can be described using the λ-model, see below. KP can be regarded as a measure of the“effective” stiffness of the muscle while KD-term incorporates the velocity dependent Ia-afferentoutput of the spindle as well as effects of (mechanical) viscosity (damping). People affected withthe Charcot-Marie-Tooth syndrome (CMT) basically lacks the Ia-output because of a selectivedamage (demyelation) to the largest motor and sensory fibers, including the Ia-afferents, yet theirbalance at quiet standing is not severly affected (Nadone et al. 2000). The smaller II-afferentshowever are more or less intact. Schieppati et alii (2001) presents further (indirect) evidence thatduring quiet stance the CNS “preferably uses the input from group II than from group Ia fibres forbalance control”. Although the affected people would lack the Ia-contribution to KD there would bestill be a II-contribition and maybe a “passive” velocity-dependent contribution from viscosity.

Besides the muscle spindles there are also the Golgi-tendon organs (GTO) which aremechanoreceptors attached to the muscle tendons in series. They register the tension of the muscle-tendon system which is conveyed by the Ib-afferents to the CNS. Their role in reflex loops is usuallyto keep the force about constant. For large loads they have an inhibitory effect reducing the afferentinput to the muscle (negative feedback). Other important mechanoreceptors are the joint capsulemechano-receptors such as the Ruffini endings which respond to changes of joint angles, andPacinian corpuscles which respond to vibrations (pressure changes). We can thus infer that thereare several information channels available for any hypothetical balance feedback controlmechanism15. Khoo (2000: § 4.7.1) presents a simple integrated mathematical model of the muscleand the muscle spindle for describing the (arm) stretch reflex (see fig. 12), which as such may bedirectly applied to the human inverted pendulum. Vestibular reflexes are also important for posturebut are not normally triggered during quiet standing. Since the neuro-muscular system is providedwith a very wide set of sensory-information channels, relying to a larger or lesser degree on this orthat subset of the sensory channels, depending on the external and internal circumstances, aparticular feedback model and its parameter values may apply only for very specific conditions.

Appendix D. Materials and experimental methods

COP-data has been obtained using a standard force plate (Hur Co, www.hur.fi) with 4 x 50 kguniaxial force transducers. The accuracy of the force plate is mostly affected by noise. Measuringwith a constant weight results typically in a (Gaussian) COP-fluctuation with a standard deviationabout 0.3 mm. Surface EMG-measurements were made with Noraxon Myosystem 140015 In a recent study Backlund (2004) emphasizes the role of skin receptors as mediators of postural information. In her

investigations she applied, among other things, an air stream stimulus to finger tips which was found to improve thepostural control.

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(www.noraxon.com). The amplified outputs of the force plate and the Myosystem (signal from itsD25-connector whose gain is preset to 1000) were connected in parallel to a National Instruments(www.ni.com) 16 bit AD-card model PCI-6036E interfacing an ordinary PC. The measurementsoftware was programmed using the National Instruments Labwindows CVI tool. EMG and forcedata were sampled with the rate of 1000 samples per second (S/s). Force data was smoothed takingconsecutive 10-point averages resulting in an effective sampling rate of 100 S/s. The force data wasfinally converted to COP-x (medial-lateral) and COP-y (anterior-posterior) coordinates and the totalvertical force Fz, which were saved along with the raw EMG-data (1000 S/s). Surface Ag/AgClelectrodes (“Blue Sensor M”, www.ambu.com ) were arranged using the standard bipolar configura-tion, with placements and skin preparations made in accordance with the SENIAMrecommendations (Hermens 1999). The skin-electrode impedance was checked in each case with amultimeter. For COG-measurements (see the caption of fig. 14) a rotational optical encoder(Hengstler, Germany) was used, with the resolution of 5000 pulses per revolution, which wasconnected to the counter pin of the NI-Daq card using up/down-counting configuration and asampling rate of 100 S/s. The collected data was finally analyzed using our own custom C-programs(Labwindows CVI, National Instruments) and Mathcad-programs (Mathsoft).

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Illustrations

Fig. 1. Typical COP-trace for quiet standing (posturogram). COP-coordinates in mm.

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Fig. 2. The human inverted pendulum model. Gravity acting through the center-of-gravity (COG) and center-of-pressure (COP)balance each other on the average. The gastrocnemius (GA) muscles modulate the balance force. The average position is slightlyforwad leaning with COG being on the average ca 5 cm ahead of the ankle joints. The moment arm of the GA visavis de ankle jointis also ca 5 cm.

Fig. 3. Schematic drawing of the efferent and afferent innervations of the muscle (after Winters & Crago, eds. 2000: 25). Besidesthe α-efferents and the γ-efferents there is third category of efferent neurons called β-efferents that innervate both the extra- andintrafusal fibers. Their role is not yet fully clarfied.

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Fig. 4. Quiet standing test, JFR003 GEN_0_1. Upper curve summed rectified and LP-filtered ( fc = 1 Hz) EMG/µV fromgastrocnemius on left and right leg (summed). Lower curve anterior-posterior COP/mm. For clarity the upper curve has beenshifted vertically.

Fig. 5. Magnification of fig. 4. 300 ms separation between vertical grid lines.

Fig. 6. Perturbation test, JFR003_07_09. Anterior/posterior COP/mm and raw left gastrocnemius EMG/0.2µV.

Fig. 7. Magnification of fig. 6. 50 ms separation between the vertical grid lines.

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Fig. 8. Time/s advancement as a funcion of frequency/Hz according to equ (11). Upper curve the case KP/KD = 1/2, lower curve thecase KP/KD = 1/5.

Fig. 9. Data JFR003_0_1, quiet standing. Anterior/posterior COP/mm and raw right gastrocnemius EMG/µV. Illustrates how EMG"drives" the COP.

Fig. 10. Perturbation test, schematic setup. The torque produced by the weight visavis the ankle joints is around 14 Nm, dependingon the length of the person. This simplified perturbation device is patterned on the one described by Lauk et al. (1998).

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Fig. 11. The picture shows the sagittal section and horizontal sections of the lower limb with gastrocnemii muscles (GA) and tendon(gray area) indicated. As seen GA is biarticulate; i.e. it extends over two joints (ankle and knee). The moment-arm length at theankle joint is about 5 cm and at the knee about 3.5 cm. Average GA mass is about 160 g, fiber length 48 mm, physiological cross-sectional area (PCA) 30 cm2. Of the muscles crossing the ankle GA represents 22 % and soleus 41% in term of PCA. The dottedhorizontal line indicates the site of the cross section and the EMG-location. The hatched areas represent the bones fibula andtibialis. From the picture we can estimate the GA thickness to be about 1/6th of the leg diameter for this cross-section. For a leg witha diameter about 10-12 cm this means a thickness around 1.7 – 2 cm. For a typical bipolar EMG-electrode arrangement theelectrodes will pick up signals from a volume within ca a 2 cm radius of the electrodes which thus seems quite optimal for the GA.(Electrode, El, and the pick-up area indicated in the picture.) Legends: GAm = medial gastrocnemius, Gal = lateral gastrocnemius,SL = soleus, TP = tibialis posterior, TA = tibialis anterior, PRb = peroneus brevis, PRl = peroneus longus. The picture also showsthe two heads (medialis and lateralis) of GA from behind. GA is a so called pennate muscle where the fibers do not go from one endof the muscle to the other but are oriented sideways. The pennation angle may vry from ca 15 to 60 degrees depending on the load.Drawings partially based on Winter (2005: fig. 3.10 p. 83) and Virtual Hospital picture, plate 7.20, available online athttp://www.vh.org/adult/provider/anatomy/HumanAnatomy/7Section/20.html. Physiological parameters quoted from Winter (2005)are only representative.

Fig. 12. Basic muscle and spindle models (drawn after Khoo 2000 p. 93). The viscous properties are modelled by dashpots (B) andthe elastic elements by springs are modelled by parallel and serial springs (K). The muscle series eleastic element corresponds tothe tendon part (x1, x) while the parallel elastic element corresponds to the (passive) connective tissue of the muscle. The parallelcomponent comes into play when the muscle is lengthened beyond its opitmal length. CE denotes the contractile element of themuscle, whereas Γ0 is a contractile element of the spindle, controlled by a gamma afferent. These models apply to single fibers, themacroscopic muscle is an aggregate of such fibers which may vary in length and other properties; thus, it is with some caution onemay apply the model as such to macroscopic muscles. In the case treated by Khoo it is assumed that the γ-efferent is constant andthat the spindle length is the same as extrafusal fiber length, z = x. The spindle feedback is assumed to generate a conatractile

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muscle force Fm proportional to the spindle elongation z – z1; Fm = β (z – z1), where β is a “gain” factor responsible for the activestiffness of the reflexive muscle.

Fig. 13. Blow-up of a part of data from a test where one of the authors (FB) was voluntarily swaying back and forth. The bottomcurve shows the variation of the A/P COP/5 mm, the next curves from bottom up are raw EMG from GA, SL and TA (50 µV units).At the time about 15 sec one can see that the motion reverts to forward direction. One observes also the characteristic sharpthreshold for GA activity while SL exhibits a smoother onset. Although TA is most active when leaning backwards it overlaps withGA activiity during the forward phase around 16-17 sec.

Fig. 14. The figure is based on data from a 20 sec quiet standing (eyes open) on the force plate. The wildly fluctuating curverepresents the A/P COP/mm while the fat curve represents A/P COG as measured with a rotational optical encoder. Finally, thedotted curve is obtained by applying a second order low-pass filter (see equ (3)) to the A/P COP-data with cut-off freqeuncy fc set to1/3 Hz. The good agreement with the A/P COP data is a demonstration of the applicability of the inverted pendulum model. It alsodemonstrates that the A/P COG can be estimated from the A/P COP using a low-pass filter as predicted by theory. This method hasbeen employed i.a. by Caron, Faure and Brenière (1997) and Rougier and Caron (2000). For the measurement of the A/P COG weused the same setup as in the perturbation test (fig. 10) but now a thin wire was instead connected to a levearm attached to the shaftof a rotational optical encoder having a resolution of 5000 pulses per revolution. The other end of the wire was attached at thewaist level (L4-L5) to the person. The resolution of the system was 0.07 mm (70 µm) and can easily be improved by an other orderof magnitude.

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Fig. 15. The area of parameters for stable PMD-control for equ (34) is given by the region bounded by the two curves. The b-paramter along the horizontal axies, and the a-paramter along the vertical axis. As can be seen, the allowed range for the aparameter is raher narrow given a fixed b-value.

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