1Department of Informatics,
Morasso, Pietro G. and
Marco Schieppati.
Can Muscle Stiffness Alone Stabilize Upright Standing?.
J. Neurophysiol. 82: 1622-1626, 1999.
A
stiffness control model for the stabilization of sway has been proposed
recently. This paper discusses two inadequacies of the model: modeling
and empiric consistency. First, we show that the in-phase relation
between the trajectories of the center of pressure and the center of
mass is determined by physics, not by control patterns. Second, we show
that physiological values of stiffness of the ankle muscles are
insufficient to stabilize the body "inverted pendulum." The
evidence of active mechanisms of sway stabilization is reviewed,
pointing out the potentially crucial role of foot skin and muscle receptors.
Despite its apparent simplicity, the nature of the
control mechanisms that allow humans to stand up is still an object of controversy. Visual, vestibular, proprioceptive, tactile, and muscular
factors all contribute to the stabilization process. A model proposed
by Winter et al. (1998) Misconceptions in the stiffness control model
The first misconception involves the relationships between the
center of mass (COM) and the center of pressure (COP). The authors
found that the oscillations of the two signals are in-phase and that
there is a strong correlation between the acceleration of the COM and
the COM-COP difference. However, as we show in the following section,
such relationships are consequences of physical laws and cannot be used
to prove one control theory over another. Moreover, the authors argue
that the COM/COP correlation rules out the active/reactive control of
balance because the delays in the sensory feedback would cause the COP
to lag behind the COM, which is only partly correct. The main effect of
the delay in the feedback loop is not the phase shift between the
control and the controlled variable but the global destabilization of the controlled system, which is intrinsically unstable. Therefore if we
attribute functional importance to active mechanisms, because humans do
succeed in stabilizing their balance, there must be something in the
control circuitry that compensates for the original delays. Admittedly,
this argument per se does not rule out the stiffness control model,
simply cancels out one possible motivation for its "biological
inevitability." The phase-lock between COP and COM is a necessary
consequence of physical laws, after the system has been stabilized.
The possibility of active/reactive control of balance is ruled out also
from the sensory point of view, because vision is not functionally
relevant and proprioceptive feedback signals are below physiological
thresholds. However, the former observation is definitely contradicted
by a large body of clinical evidence in adult and elderly subjects
(e.g., Maki and McIlroy 1996 Theoretical framework
For an inverted pendulum (see Fig.
1) the system equation is
ABSTRACT
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ABSTRACT
INTRODUCTION
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INTRODUCTION
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ABSTRACT
INTRODUCTION
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attributes muscle stiffness as
the single factor involved in solving the problem. According to this
theory, the intervention of the CNS is limited to the selection of an
appropriate tonus for the muscles of the ankle joint, in order to
establish an ankle stiffness that stabilizes an otherwise unstable
mechanical system. Thus in this view the stabilization of quiet
standing is a fundamentally passive process without any significant
active or reactive component, except for the background setting of the
stiffness parameters. In this paper we describe the flaws of the
stiffness control model, review some of the relevant evidence in favor
of an active intervention of the CNS in the stabilization process, and
outline an alternative modeling framework.
; Paulus et al.
1984
) and the latter fails to take into account the potential role of foot skin receptors (Kavounoudias et al. 1998
;
Wu and Chiang 1997
). Thus the crucial question for
assessing the feasibility of the stiffness control model is the
following: are the muscles stiff enough to carry out the job of
stabilizing the human inverted pendulum? To determine this we first
outlined a theoretical framework of the system for expressing the
conditions of stability and then evaluated the theoretical
effectiveness of empiric stiffness levels.
where
(1)
is the sway angle, m and
Ip are the mass and moment of inertia of the
body (minus the feet), h is the distance of the COM from the
ankle, g is the acceleration of gravity,
ankle is the total ankle-torque, and
z stands for the set of external or internal disturbances
(such as respiration) that perturb the standing
posture.1 The ankle-torque
must also satisfy an equilibrium equation for the foot:
ankle + fvu
0, where fv is the vertical component of the ground reaction force and u is the COP position. If
we take into account that fv
mg in quiet standing, then this equation tells us that
variations of muscle torque are immediately and linearly translated
into variations of the COP position. The two equations can then be
combined into a single dynamic sway equation that relates the
controlled variable y and the control variable u.2
In other words, the COM-COP difference is bound to be
approximately proportional to the acceleration of the COM for purely mechanical reasons, because the postural noise z' is small
in quiet standing. Moreover, the same equation tells us that for the
horizontal component of the ground reaction force it must be
fH
(2)
(y
u)
since fH = m
according to Newton's law. Therefore although no specific receptors
exist that detect the COM, its position y can be
indirectly estimated through measurements of u and
fH and some computational process that
"fuses" them.
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Fig. 1.
Scheme of the human "inverted pendulum" for sway in the sagittal
plane. Center of mass (COM), position is denoted by the variable
y; center of pressure (COP), position is denoted by the
variable u; ground reaction force, (f);
sway angle, ( ); acceleration of gravity, (g); torque
of the muscles around the ankle, (
ankle);
and distance of the COM from the ankle, (h).
Let us now consider the phase relation implied by Eq. 2
during normal sway, using a simple method that can be applied to the different harmonic components of both oscillations. Let
Ay, y and
Au,
u be the
amplitude and phase parameters of y
and u, respectively. Then
=
y
u is always 180° out of phase relative to
y which implies that
u =
y (and Au > Ay). Again, this phase relationship is a
physical necessity, not something that must be proved experimentally to
discriminate the specific control action. Therefore the observation of
null phase delay neither proves nor disproves the theory of stiffness control. Equation 2 also suggests an alternative way of
calculating the COM from the COP with respect to the brute-force
approach used by the authors that is based on a complex whole body
model and the 3-D measurements of 21 markers: it is sufficient indeed to integrate Eq. 2 considering u(t) as
the forcing input. Morasso and Spada (unpublished observations)
developed an algorithm based on a variational approach and spline
functions3 (Fig.
2).
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Stability
For analyzing conditions of stability of the postural control
system, we consider the open loop transfer function (Fig.
3A) that is derived from the
Laplace transform of Eq. 2:
s2Y(s) =
g/he[Y(s) U(s)] + Z'(s). From this
we get
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(3) |
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(4) |
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(5) |
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Stiffness control: stabilization by passive muscle properties
In the stiffness control paradigm, the two parameters of the
control law can be associated with the elastic and damping coefficients of the ankle joint impedance, respectively. In particular, it is easy
to show, by means of a change of coordinates from y to , that the elastic parameter
Kp is linked to the measurable ankle stiffness Ka by the following relation
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(6) |
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(7) |
Active mechanisms of stabilization
The supporters of the passive stabilization of sway point out that
the limited range of sway movements may not stimulate the different
kinds of sway receptors beyond physiological thresholds. The
experimental evidence, however, is somehow contradictory but certainly
indicates that physiological levels of sway are very close to such
thresholds in relation to vestibular, joint, and muscle receptors (see
e.g., Fitzpatrick and McCloskey 1993; Konradson et al. 1993
). Regarding muscle receptors, functional loss of
group Ia spindle afferent fibers has been considered responsible for postural disequilibrium (Griffin et al. 1990
;
Weiss and White 1986
). Spindle group II fibers, of
smaller diameter but at least as numerous as the group Ia fibers, may
be perhaps even more relevant as the origin of information utilized by
postural control circuits. Both leg and foot muscles are the site of
postural segmental reflexes (Schieppati et al. 1995
),
mainly because of spindle group II fibers (Schieppati and
Nardone 1997
). Where very slow movements of the body occur,
such as during maintenance of quiet stance, it is conceivable that
length signals coming from the less adaptable spindle secondaries
provide an appropriate input to the CNS for detecting low-frequency
displacements occurring mainly about the ankle (Gurfinkel et al.
1995
) and for assisting foot and calf muscle reflex responses
(Schieppati et al. 1995
). The contribution provided by
the plantar cutaneous receptors is frequently overlooked, with the
exception of a few studies (Kavounoudias et al. 1998
; Magnusson et al. 1990
). These receptors do not measure
sway but are related to different parameters of the ground reaction
force f that are affected indirectly by the sway movements:
1) the vertical component fV,
2) the horizontal component or shear force
fH, and 3) the point of application
or COP position u. In principle,
fV can be derived by adding up the output of the
receptors specifically affected by the vertical component of the
contact forces. This parameter is not relevant for sway control because
it is approximately constant during quiet sway. However, the
COP position u can be computed by using the same
receptor information but with a different computational process, which
takes into account the position of the receptors as well as the
intensity of the detected signals. This is a complex computational task
that integrates information of a number of sensory channels with
appropriate transduction characteristics, like the Ruffini and Meissner
terminations that are slowly or moderately fast adapting and have small
receptive fields.
Shear force fH is frequently ignored because it is small when compared with the weight force; however, it is in the Newton-range according to Eq. 2 and so is detectable by the plantar receptors as well as the foot muscle spindles. Moreover, because fH is proportional to the COM-COP difference, it carries clear sway-relevant information; in the A/P direction, for example, if the force is directed forward it means that the COP is behind the COM and conversely if the force is directed backward. Therefore such shear contact forces have a dominant phasic component in contrast with the basically tonic nature of the vertical forces. This means that the appropriate receptors must be very fast adapting; moreover, there is no need of small receptive fields because localization in this case is not required. Pacinian corpuscles fit this description, although other receptors in the foot muscles might play a role as well.
Regardless of how the CNS processes this information, the indirect
evidence that active mechanisms of stabilization underlie sway control
is ample and multifaceted. Certainly information about mutual positions
of body links, muscular torques, and interaction with the support has
access to the CNS, and subjects can consciously evaluate the gross
amplitude of their own sway during stance (Schieppati et al.
1999), possibly with respect to a reference position
(Gurfinkel et al. 1995
). Furthermore, the available data
do not favor a constant level of antigravity leg and foot muscle
activity during stance. Instead a relationship between the
anteroposterior oscillations of the center of pressure and the profile
of the rectified and integrated EMG of those muscles (Scheppati
et al. 1994
) indicates moment-to-moment action of a system of
stance control based on timely produced muscle impulses.
In conclusion, having excluded a dominant effect of muscle stiffness, a
plausible alternative computational scheme can be outlined that is
based on the indirect estimate of a postural state vector
x = [y,
]Tobtained from the
complex combination of a variety of sway-related sensory signals. Use
of sensory signals in a feedback control mechanism that modulates the
activity of the calf muscles is shown in Fig. 3B. The
critical element of the scheme is the potentially catastrophic
influence of even small delays in the feedback loop. (In the simulation
model shown in Fig. 2 a delay of about 50 ms is sufficient to
destabilize the control system.) Restabilization rules out the
reflexive nature of the control mechanism and strongly suggests a
central computational process that carries out two main functions:
1) integrating the multisensory information into a unifying
estimate of the state vector and 2) compensating the transmission delays with an anticipatory action, i.e., a short-time prediction of the postural time series. From this point of view, the
absence of short-latency reflexes of the muscles around the ankle,
linked to the stimulation of cutaneous fibers of the foot (Abbruzzese et al. 1996
), is paradoxically in favor of a
supra-segmental role of such afferences in a more complex computational
mechanism such as the one outlined previously. Detailed analysis of
this point is beyond the scope of this paper, however, information on
research in this area of motor control and sensory adaptation is
available and is focused on the acquisition of "internal models" (Miall et al. 1993
; Morasso and Sanguineti
1997
; Morasso et al. 1999
; Wolpert and
Kawato 1998
).
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ACKNOWLEDGMENTS |
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This paper was partially supported by Consiglio Nazionale delle Ricerche, ISS, and the Ministero dell' Universitàe della Ricerca Scientifica e Tecnologica.
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FOOTNOTES |
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Address for reprint requests: P. Morasso, Dept. of Informatics, Systems, Telecommunication, University of Genova, Via Opera Pia 13, I-16145 Genoa, Italy.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
1
The assumption of additive noise in the posture
control model might seem artificial to the readers. However, there is
experimental support for the role of noise in upright standing
(Dijkstra et al. 1994; Paulus et al.
1984
). Dijkstra et al. (1994)
also provide arguments in favor of a contribution of visual input to stance.
2
Because the body sway angle is small, we can use
the following approximations: sin ()
, cos (
)
1,
/h. The moment of inertia can be
expressed in general as Ip = mh2ks where
ks is a shape factor:
ks = 1 if we assume that the body mass is concentrated in the COM and
ks = 1.33 if it is uniformly distributed along a rod-like shape; for the human body the value of the
coefficient is closer to the latter than the former estimate. We define
an "effective" value of h:
he = hks.
Eq. 2 also includes the "noise" term z' = z/mhe.
3
The solution y(t) of the
equation, over a given observation time, is approximated by means of a
B-spline function B(t), which depends
linearly on a set of parameters p as well as its second time derivative (t). Substituting
the corresponding expressions into Eq. 2 we get a LSE
(least square estimate) problem in p that can be solved
with standard methods.
4 The two elements of the "apparent" ankle stiffness Ka in Eq. 6 vary with the fourth power of the body size h since m goes with h3 and Ip goes with mh2. Muscle strength and stiffness go with the cross-sectional area of the muscles and thus vary with h2 but the ankle stiffness due to the muscles goes with h4 since Ka=Kmr2, as noted above. Thus as body size varies both the apparent ankle stiffness and the stiffness due to the muscles change in the same way and we can conclude that the calculation scheme is roughly independent of the body size.
5 Please note that with this term we denote the intrinsic mechanical stiffness as well as the tonic part of the reflex stiffness because the mentioned experiments of stiffness estimate are in fact sensitive to the cumulative effect of the two elements.
Received 19 January 1999; accepted in final form 4 June 1999.
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