BOiNT OF VIEW
Motor Control, 1998, 2, 10-12
0 1998 Human Kinetics Publishers, Inc
Rejecting the
Equilibrium-Point Hypothesis
Gerald L. Gottlieb
The lambda version of the equilibrium-point (EP) hypothesis as developed by
Feldman and colleagues has been widely used and cited with insufficient critical
understanding. This article offers a small antidote to that lack. First, the hypothesis implicitly, unrealistically assumes identical transformations of lambda
into muscle tension for antagonist muscles. Without that assumption, its definitions of command variables R, C, and lambda are incompatible and an EP is
not defined exclusively by R nor is it unaffected by C. Second, the model
assumes unrealistic and unphysiological parameters for the damping properties of the muscles and reflexes. Finally, the theory lacks rules for two of its
three command variables. A theory of movement should offer insight into
why we make movements the way we do and why we activate muscles in
particular patterns. The EP hypothesis offers no unique ideas that are helpful
in addressing either of these questions.
For over 35 years, the lambda version of the equilibrium-point hypothesis
has been a fertile intellectual model for discussing how the central nervous system
(CNS) controls voluntary movement. In progressively elaborating on this hypothesis, its originator and advocates have attempted to create a more complete description of how hypothesized command signals from the brain are transformed
into muscle contraction and movement by peripheral neuromuscular structures.
The most recent step in this process has been the publication of a detailed model
for single joint movement (St-Onge, Adamovich, & Feldman, 1997). That work
and the earlier essay by Feldman and Levin (1995) have convinced me that the
notion of the CNS controlling movement by specifying the trajectory of a moving
equilibrium point remains subtly but profoundly flawed. Because it proposes a
conceptually attractive way of thinking about movement control, and perhaps because of its longevity, it has been widely used and cited with what is in my opinion
insufficient critical understanding. This paper offers a small antidote to that lack
and describes what I consider the most egregious problems with the hypothesis.
Let us begin, however, with a principal point of agreement. The neuromuscular system creates a movable equilibrium position (EP) for a joint such that, in
the absence of external forces, the joint will approach and rest at its EP. This essen-
a-
.-
Gerald L. Gott
- Cummington St., Bos
Equilibrium-Point Hypothesis
11
tial truth has always been at the core of the equilibrium-point hypothesis but it is
also at the core of all modem theories of movement. What is required of a theory
of movement is a clear and complete definition of its independent or control variables, the rules by which those variables specify changes in movement features
such as distance, speed, and load, and any major assumptions that may underlie or
limit its range of validity. This essay presents three reasons why the equilibriumpoint hypothesis in its most widely cited formulation (Feldman & Levin, 1995; StOnge et al., 1997) fails to satisfy these requirements.
Feldman and Levin (1995) defined the control variables in two different
ways. The theory is formulated in terms of a variable called lambda, from which it
derives its name, that is defined as a threshold at which the tonic stretch reflex will
start to activate a motor neuron pool. This is a bioelectrical definition. Lambda is
also described as the angle at which the torque-angle relation of the tonic stretch
reflex crosses the abscissa and muscle torque equals zero, a biomechanical definition. Two new variables called reciprocal (R) and coactivating (C) central commands are then defined respectively as the sum and the difference of the lambdas
of the antagonist muscles. R and C are also described in terms of their mechanical
properties, R being the determinant of the equilibrium position (EP) of the joint
and C modulating its stiffness.The theory postulates that to initiate movement, the
CNS specifies the temporal patterns of the central commands R and C and of
another central command variable called y that controls feedback gain. These commands, in combination with peripheral feedback signals from muscle proprioceptors, activate the muscles. According to Feldman and Levin (1995, p. 731), "A
change in the R command gives rise to a shift in the EP of a joint" and if external
forces permit, movement follows. "The C command [influencesjoint stiffness but]
does not affect the EP of the joint."
An implicit assumption here is that the transformation of lambda into muscle
tension, a transformation that involves the full spectrum of motoneuron, muscle
fiber, and joint biomechanical properties, is identical for both flexors and extensors. The antagonist muscleslreflexes must have, in the model's terminology, identical static "invariant characteristics" and identical dynamical joint torque (e.g.,
twitch and forcelvelocity)characteristics.At few (and perhaps no) joints will these
conditions be met, the ankle dorsiflexorslplantar flexors and the jaw openers1
closers being just two examples of joints with strong asymmetries. Compounding
the effects of muscle asymmetries is the fact that the dynamical properties of shortening and lengthening muscles are very different from each other. Without those
assumptions, these definitions of R, C, and lambda are incompatible. As the control variables are presently defined, the C command does affect the EP of the joint
because it will change net joint torque. While in principle it may be possible to
address this by redefining the relationships between RIC and the lambdas, the
computational problems this would raise appear daunting and physiologically implausible, although the issue has never been addressed. Except under idealized
conditions, R does not define an EP.
A second problem is that the highly constrained patterns of R and C described by Feldman and his colleagues (Feldman & Levin, 1995; St-Onge et al.,
1997) require high levels of joint viscosity for a simulated movement to terminate
smoothly at its target. Those levels appear to be far above the values that have
been estimated by experiment (e.g., Gomi & Osu, 1996). Hence, in contradiction
of experimental data, the model assumes unrealistic and unphysiological param-
eters for neuromuscular properties. This problem might be addressed by using
different R and C patterns (e.g., Latash's N-shaped patterns; Latash, 1993), but
this idea has been rejected by Feldman (Feldman & Levin, 1995). Its discussion
must await another "Point of View."
The most attractive feature of the lambda hypothesis is that it lets us conceptualize movement control in terms of positional concepts such as a virtual trajectory. This is simple, elegant, and seductive. However, it ignores the issue that leads
to my third criticism. Control is not effected by the construction of a virtual trajectory or R command alone but only by the added construction of the equally important C command as well as others that control the strength of feedback signals
(e.g., p). Despite 35 years of development including the articles cited above, there
are still no explicit, systematic rules for any variable other than R (St-Onge et al.,
1997,Figure 3). The theory and the simulations give us little guidance as to how C
or p change to move a single joint over different distances, at different speeds, or
with different external loads. The rules for C and p are neither obvious nor simple,
and it is impossible to define their behavior without explicit consideration of the
dynamics of the moving system. Yet it is the claimed ability to describe control in
positional terms or shifting "frames of reference," rather than in dynamical terms,
that has made the equilibrium-point hypothesis so attractive.
In light of the above, I suggest that it is not useful to think of controlling
movement in terms of this proposed formulation of R, C, and p commands. As yet,
the published record does not tell us in a reasonable way how this happens. This
version of the equilibrium-point hypothesis assumes parameters with values that
are unphysiological, it inaccurately describes the effects of its control variables on
movement except under implied and unphysiological assumptions, and by defining the behavior of only one of its control variables, it has obscured how incomplete a description it offers of movement control.
Above all, a theory of movement should offer insight into why we make
movements the way we do and why we activate the muscles in particular patterns.
I do not think the equilibrium point hypothesis offers any unique ideas that are
helpful in addressing either of these questions. It is time to turn to new paradigms.
References
Feldman, A.G., & Levin, M.E (1995). The origin and use of positional frames of reference
in motor control. Behavioral Brain Science, 18,723-806.
Gomi, H., & Osu, R. (1996). Human arm stiffness and viscosity in interaction with environments on a horizontal plane (No. ISRL-96-3). Atsugi: NTT Basic Research Labs.
Information Science Research Lab.
Latash, M.L. (1993). Control of human movement. Champaign, IL, Human Kinetics.
St-Onge, N., Adamovich, S.V., & Feldman, A.G. (1997). Control processes underlying elbow flexion movements may be independent of kinematic and electromyographic
patterns: Experimental study and modeling. Neuroscience, 79,295-3 16.
Manuscript accepted: August 19, 1997