Mark R. Hinder & Theodore E. Milner
School of Kinesiology, Simon Fraser University, Burnaby, Canada

https://doi.org/10.36866/pn.53.20

Intrinsic muscle properties have for a long time been known to stabilize limb posture and movement. Specifically, muscles act like springs, producing restoring forces when the position or trajectory of a limb is perturbed. Limb posture represents an equilibrium between the muscle forces and external forces acting on the limb. It was proposed that limb movement could be controlled by shifting this equilibrium point from one position to another. Two versions of what is commonly referred to as the equilibrium point hypothesis (EPH) were formulated – the λ-model and the α-model. The λ-model was based on the hypothesis that a fixed voluntary motor command represented a set of equilibrium positions called an invariant characteristic. Changing external torque would cause the joint angle to shift along the invariant characteristic until equilibrium was once again established (Feldman, 1966a, b; Feldman & Astrayan, 1965). A different invariant characteristic could be selected by changing the voluntary motor command, initiating movement to some new desired position. The voluntary motor command which selected the equilibrium position was referred to as the reciprocal command while the command which determined the stability of the equilibrium position, by changing the joint stiffness, was referred to as the co-contraction command. Over time, other features were added to the λ­model, such as velocity sensitivity. However, none of these features affect the essential tenet that the same final position should be achieved if the steady state forces are not altered, a property which we will refer to as equifinality. The α-model arose from observations of deafferented monkeys trained to move their heads or arms to specific target positions (Bizzi and Polit, 1978). Transient perturbations to the position of the head or arm were shown to have no effect on the ability to achieve the intended target position, leading to the hypothesis that the final equilibrium position was completely specified by the level of activation in agonist and antagonist muscles, another form of equifinality.

Although the simplicity of the EPH is very appealing, given that no inverse dynamics calculations are necessary, it seems unlikely that it can explain the large repertoire of human motor behaviour, particularly rapid adaptation to novel environmental dynamics. An alternative view that has gained much popularity is the concept of an internal model, which transforms desired motor behaviour into specific motor commands based on knowledge of mechanical properties of the limbs and environmental dynamics acquired through experience (Kawato, 1999). Support for this idea has been acquired by examining the after effects of adaptation to novel dynamic environments (Shadmehr & Mussa-Ivaldi, 1994; Lackner & Dizio, 1994; Conditt et al. 1997). After subjects have become proficient in compensating for novel dynamics, unexpected alteration of the dynamics results in perturbed trajectories that reflect the forces learned during adaptation. However, these studies have not been specifically designed to pit the hypothesis of internal model formation against EP control. Consequently, proponents of the EPH have argued that the results are not, in fact, incompatible with the EPH (Feldman et al. 1998).

We recently reported new evidence that supports the formation of an internal model of the required task dynamics but not EP control (Hinder & Milner 2003). We designed an experiment where equifinality would not be predicted by internal model formation. Briefly, we considered a single degree-of-freedom wrist flexion movement in which subjects moved to a target while assisted by a velocity dependent torque. Following extensive learning, the strength of the assisting torque was unexpectedly reduced by 25, 50, 75 or 100% on randomly selected trials, prior to movement onset. Subjects consistently stopped short of the target on these perturbed trials (Fig. 1). The undershoot was directly proportional to the reduction in the assisting torque. EMG analysis suggested that subjects generated the same feedforward motor commands whether or not the movement was perturbed. Consequently, the muscle torque was insufficient to drive the limb to the intended final position on the perturbed trials without the full assisting torque.

This result is entirely consistent with formation of an internal model, but would not be predicted under the EPH. Under the EPH, the central nervous system specifies an equilibrium trajectory, which is effected by a continuous shift in the instantaneous equilibrium position. Ghafouri and Feldman (2001) recently claimed that this trajectory reaches its final position at approximately the time of the peak movement velocity. This would imply that the unexpected increase in load should have produced an increase in agonist muscle force sufficient to drive the wrist to the intended final position. Although we have only shown that the EPH does not hold under one specific condition, we plan to further investigate the generality of our finding. In particular, we believe that equifinality is more likely to be achieved if unexpected changes in dynamics assist movement or if stretch reflex gain is high.

The results of a similar experiment, examining reaching movements in the horizontal plane, also suggest that equifinality may not hold when an internal model has been formed (unpublished observations). In this experiment, subjects adapted to a force which assisted motion regardless of movement direction (negative damping). For most directions subjects adapted by increasing the stiffness of the arm by means of co-contraction. In these cases, equifinality was achieved when the assisting force was unexpectedly eliminated. However, for forward reaches, which involved the least co­contraction, several subjects actually reversed direction and stopped closer to the start than the target position (Fig. 2). This suggests that not only was the agonist muscle force no longer adequate to fully extend the arm, but the antagonist muscle force that had been used to oppose the assisting force and decelerate the arm, was now sufficient to reverse the movement direction.

We expect proponents of the EPH to continue to raise objections to such counter examples, although formulations of the EPH, which can account for adaptation to novel dynamics, require that increasingly complex control signals be learned (Gribble & Ostry, 2000). Furthermore, a recent theory of internal model formation, which includes impedance control (co­-contraction commands), is able to circumvent the need for complex inverse dynamics calculations (Franklin et al. 2003). It may only be a matter of time before the EPH gives way to the rapidly evolving concepts of internal model formation.

References

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