Abstract
Based on an assumption of movement control optimality in reach-to-grasp movements, we have recently developed a mathematical model of transport–aperture coordination (TAC), according to which the hand–target distance is a function of hand velocity and acceleration, aperture magnitude, and aperture velocity and acceleration (Rand et al. in Exp Brain Res 188:263–274, 2008). Reach-to-grasp movements were performed by young adults under four different reaching speeds and two different transport distances. The residual error magnitude of fitting the above model to data across different trials and subjects was minimal for the aperture-closure phase, but relatively much greater for the aperture-opening phase, indicating considerable difference in TAC variability between those phases. This study’s goal is to identify the main reasons for that difference and obtain insights into the control strategy of reach-to-grasp movements. TAC variability within the aperture-opening phase of a single trial was found minimal, indicating that TAC variability between trials was not due to execution noise, but rather a result of inter-trial and inter-subject variability of motor plan. At the same time, the dependence of the extent of trial-to-trial variability of TAC in that phase on the speed of hand transport was sharply inconsistent with the concept of speed–accuracy trade-off: the lower the speed, the larger the variability. Conversely, the dependence of the extent of TAC variability in the aperture-closure phase on hand transport speed was consistent with that concept. Taking into account recent evidence that the cost of neural information processing is substantial for movement planning, the dependence of TAC variability in the aperture-opening phase on task performance conditions suggests that it is not the movement time that the CNS saves in that phase, but the cost of neuro-computational resources and metabolic energy required for TAC regulation in that phase. Thus, the CNS performs a trade-off between that cost and TAC regulation accuracy. It is further discussed that such trade-off is possible because, due to a special control law that governs optimal switching from aperture opening to aperture closure, the inter-trial variability of the end of aperture opening does not affect the high accuracy of TAC regulation in the subsequent aperture-closure phase.
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Notes
The difference in residual error magnitude (the black columns in Fig. 1a) between the low-speed condition and the normal-speed condition was significant both for the short-transport distance (t test: t(24196) = 29.8, P < 0.001) and the long-transport distance (t(31898) = 40.3, P < 0.001).
To someone who is used to thinking about motor control in terms of kinematic parameters as continuous sequences of values within a specific time interval, it might seem that, since, for instance, acceleration as a function of time can be computed as a time derivative of velocity, it must be sufficient to include only one such parameter in equations. In the case of the equation describing transport-aperture coordination; however, instantaneous values of such parameters are involved and, therefore, a different logic applies. Knowledge of hand velocity at a certain time point t in general does not allow one to calculate hand acceleration and vice versa. For this reason, these kinematic variables are viewed in theoretical mechanics as state coordinates independent of each other.
From an optimality approach perspective, this means that expenditures related to the utilization of neuro-computational resources must be included in the criterion of task performance optimality as a component with a significant weight, while the weight of performance time in that criterion is much less significant. Under different experimental conditions, in which time expenditures are critical, for example, when reach-to-grasp is performed in a context of a game in which the overall speed improves the score, the criterion of reach-to-grasp optimality should include the time duration of hand transport optimality.
According to the traditional two-component model for aiming movements, the initial component of hand transport is controlled in a ballistic manner for saving movement time, and during the later component, the arm is guided to the target in a (visual) feedback control mode (Elliott et al. 2001). However, since the TAC model for the aperture-closure phase is highly accurate even without vision (Rand et al. 2007), the TAC model’s precision does not critically depend on processing visual feedback.
It is assumed here that the “turning on” does not take significant time because the CNS is a highly parallel system.
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Acknowledgments
This research was supported by grants from NINDS NS 39352 and 40266.
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Appendix
Appendix
The material presented below is important for understanding how TAC variability is related to the general idea of motor variability. It also provides a more detailed description of the control mechanism of switching from aperture opening to aperture closure. It is assumed that the reader is well familiar with the concepts of dynamical systems and state-space geometry, including the notion of uncontrolled manifold.
Variability of TAC from the general perspective of motor variability
The phenomenon of motor variability has been studied extensively since the early observations of Bernstein that even well-optimized, stereotypic movements vary from trial to trial, they are “repeated without repetition” (Bernstein 1967). Perhaps the most important feature of motor variability is that it is not isotropic with respect to its effect on task performance accuracy: Some variability types significantly affect task performance, while some others do not (Darling and Stephenson 1993). Furthermore, The CNS decreases variability in the directions critical for task performance, while letting it to be relatively large in other directions (Todorov and Jordan 2002).
The variability of reach-to-grasp movements can be constructively described and analyzed in terms of a state-space coordinates of which are the movement parameters included in the TAC model. The relationship between those parameters described by a linear approximation of the TAC model (Eqs. 2 or 3) can be geometrically interpreted as a state-space hyperplane. One TAC hyperplane corresponds to the aperture-opening phase, in which TAC regulation is inaccurate. The other TAC hyperplane corresponds to the aperture-closure phase, in which TAC regulation is highly accurate (Rand et al. 2008). Despite the fact that the variability of the relationship between movement parameters described by the TAC model is minimal within the aperture-closure phase, the temporal profiles of those parameters significantly vary between trials (Rand et al. 2008). This means that the aperture-closure part of the trajectory in the above state space significantly varies between trials. However, its variation is confined to a very thin state-space manifold that can be approximated with a single hyperplane. That hyperplane is a geometrical interpretation of the TAC model (Eq. 2), which suggests that the state-space direction perpendicular to the TAC hyperplane can be viewed as a direction of least motor variability. This is so because any component of motor variability that has a significant projection on that direction would violate the TAC model and thus affect the reach-to-grasp task performance. Since inter-trial variations of the state-space trajectory during aperture closure are essentially confined to the aperture-closure TAC hyperplane, that hyperplane can be viewed as an uncontrolled manifold (Scholz and Schoner 1999).
In contrast to the aperture-closure phase, the part of the state space containing state-space trajectories that correspond to the aperture-opening phase from different movement trials cannot be approximated by a single TAC hyperplane. This is so because of significant inter-trial variability of TAC in the aperture-opening phase. The above part of the state space can be viewed as a manifold consisting of a set of TAC hyperplanes, where each hyperplane corresponds to an individual trial, within which TAC varies very little (as shown in this study). The “thickness” of the aperture-opening TAC manifold can be measured as the magnitude of the residual errors of fitting the TAC model. Since TAC hyperplane orientation (represented by the direction perpendicular to it) varies between different trials, the state-space direction of least motor variability in the aperture-opening phase can be obtained by averaging across directions perpendicular to each individual hyperplane from the above-mentioned set.
Geometrical interpretation of switching from aperture opening to aperture closure
To explain how the CNS can afford the inaccuracy of TAC regulation in the aperture-opening phase, it is mentioned in “Discussion” that there exists a special control mechanism that identifies an optimal state for switching from aperture opening to the aperture closure and performs the switching. That procedure can be easily understood in terms of the above state space, coordinates of which are the movement parameters included in the TAC model. The aperture-opening phase corresponds to a part of the state-space trajectory that is oriented at an angle with respect to the TAC hyperplane corresponding to the aperture-closure phase. At the state where the state-space trajectory crosses the aperture-closure TAC hyperplane, the CNS switches arm control to aperture closure. This simple rule is a geometrical interpretation of the control law for aperture-closure initiation. Since the state-space trajectory significantly varies between trials, the location of the end-point of the aperture-opening part of the state-space trajectory also varies from trial to trial considerably. That variability, however, must be constrained to the aperture-closure TAC hyperplane due to the existence of the above mechanism for switching from aperture opening to aperture closure. This constraining is indeed observed experimentally, which is evident from the fact that the set of state-space points at which aperture initiation occurs in different trials fits with high precision the TAC model for aperture closure (Rand et al. 2008). Also note that, since TAC variability is minimal during the aperture-closure phase, one can expect that TAC variability (but not necessarily the variability of the state-space trajectory) measured within a sliding time window (see “Materials and methods”) should decrease as the state of arm movement during aperture opening approaches the aperture-closure hyperplane. That decrease toward the end of the aperture-opening phase is in fact clearly visible (Fig. 5). It is obvious from the above geometrical interpretation that, due to the mechanism for switching from aperture opening to aperture closure, the variability of TAC in the aperture-opening phase does not affect the precision of TAC regulation during aperture closure, which means that the precision of grasping is not significantly affected either.
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Rand, M.K., Shimansky, Y.P., Hossain, A.B.M.I. et al. Phase dependence of transport–aperture coordination variability reveals control strategy of reach-to-grasp movements. Exp Brain Res 207, 49–63 (2010). https://doi.org/10.1007/s00221-010-2428-7
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DOI: https://doi.org/10.1007/s00221-010-2428-7