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Biology Division, California Institute of Technology, Pasadena, California
Submitted 20 February 2006; accepted in final form 16 July 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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). Across repetitions, for each point in their motion path there will be a coincidence in space and time. Likewise, when subjects perform reaching movements in a motor control experiment, their timing achieves very precise duration and a consistent temporal strategy. But what happens during the course of learning a new motion? Is the system aware of how long every new movement should take? Is there a predefined speed profile? And if so, where does this ideal timing come from?
For years, the field of motor control has focused on reaching movements that are highly automatic. They consistently follow a spatiotemporal profile that for the nonredundant cases can be derived by solving the equations of motion under the classical mechanics scheme. The movement trajectories minimize the energy expressed as an integral over a predefined time interval. The energy-minimizing solution simultaneously describes a motion in space and time (Alexander 1997; Flash and Hogan 1985; Harris and Wolpert 1998; Todorov and Jordan 2002; Uno et al. 1989, 1995). Therefore space and time are tightly coupled. Current computational models of motor control follow this approach when solving the motion dynamics (see reviews by Jordan and Wolpert 1999; Todorov 2004) and often argue over whether the kinematics (position, velocity, etc.) or the dynamics (forces and force-related quantities) are more important for the planning of a motion. Regardless of the side one takes in this argument, the temporal aspects of the movement have already been predefined, so they do not form part of the learning process.
The developmental literature, however, teaches us that infants master reaching and grasping motions at an early age and that the process can be clearly divided into two phases: 4 to 8 wk after reach onset (occurring at 20 wk) there is rapid improvement of motion parameters in hand and joint space, followed by a second phase of fine-tuning where more gradual changes (such as a reduction in path variability) are observed that last well into the 2 yr of age (Konczak and Dichgans 1997). Thus these studies already hint at some separable process during early learning.
We also know from studies involving adult humans that subjects can traverse the same planar reaching hand path with different speeds (and loads) (Atkeson and Hollerbach 1985), the same postural (and three-dimensional hand) paths under different speeds during reaching (Nishikawa et al. 1999), and orientation matching (Torres and Zipser 2004). However, why or how this should be possible has not been formally addressed with a computational theory.
This study explores the question of space–time decoupling during learning. To elicit the need for learning, and to monitor learning over time, we introduce a novel experimental paradigm that during each session forces the system to switch between two tasks. The paradigm interleaves straight-line reaches with novel reaches around obstacles that significantly change the curvature and length of the hand paths, thus demanding movements of longer duration (see METHODS and Fig. 1A for further details). The key element in this paradigm is that the new task is simple enough that it requires no training of the animals, yet complex enough that it changes the arm path geometry and calls for the learning of a new spatiotemporal strategy.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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The experimental paradigm consisted of a first block of straight reaches (10–15 trials), a second block of 30 trials of reaches around obstacles, and a third block of 30 trials of straight reaches after obstacle exposure. The animals saw the experimenter both positioning and removing the obstacles with the lights temporarily on. An electromagnetic tracking system (Polhemus Fastrak) recorded the arm motions in space at a sampling rate of 120 Hz for four sensors. Three sensors were mounted on small pieces of Plexiglas and attached with thick Velcro to the left sleeve of a rhesus primate jacket (Harvard Apparatus). The fourth sensor was used separately to measure shoulder position. In figures in the main text, filled cylinders mean that there was an obstacle at that location during that block of trials, whereas wire-frame cylinders mean that there was no obstacle present in that block. The wire-frame cylinder represents the locations where the obstacle was.
Obstacle properties and configurations affect reaches differently
Several obstacle shapes and lengths were explored before settling on the chosen one. One obstacle alone on location 7 or/and 9 did not elicit sufficiently curved paths to cause temporal adaptation. It should be noted that only when two obstacles were positioned at locations 7, 12, or/and 9, 14, did we observe highly curved arm paths that resulted in temporal adaptation, causing aftereffect paths; otherwise, most movements showed from the first trial a single bell-shaped temporal strategy similar to that from the straight reaches. The changes were in the magnitude of the speed (Fig. 2A). This is important to keep in mind to replicate our results. There will be temporal adaptation only if the geometric deformation of the paths is significant enough to call for a change in the temporal strategy previously used for straight reaches. In addition, the length and shape of the obstacles will affect the path-solution strategy (Fig. 2B).
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We distinguished the geometric-based temporal learning from the automatic epochs based on the speed profiles [i.e., learning (slow broken profiles) vs. automatic (fast smooth profiles)] and for data analysis we partitioned the blocks into the first 10 trials (early) versus the last 10 trials (late) in the order in which they were acquired. To extract the motion from the sensors' output, we defined the beginning and the end of the movement as 5% maximum velocity along a speed profile. For each trajectory, we determined the points where the velocity dropped to 5% of the maximum and eliminated the data beyond those points. Overall movement speed ranged from 70 to 185 cm/s (minimum and maximum, respectively).
A measure of curvature K was determined by projecting each point of the curved path onto the straight line joining the initial and final points and obtaining the normal distance from each point along the curved path to its corresponding projection on the straight line (range 10–26 cm). A perfectly straight path would have K = 0.
The Wilks's test statistic (Rencher 1995) for points of the hand paths was used in standard multivariate ANOVA on the positional hand paths of Fig. 4, A and B and in the analysis of the postural paths (Table 1).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Cognitive goals determine the spatial path before movement initiation
HAND SPACE.
During the obstacle condition both subjects resolved the spatial solution paths with no errors, i.e., without colliding against the obstacles or missing the targets. The spatial solution paths complied with distance-based constraints arising from the obstacle location, shape, size, and the fixed distances between the primate chair and the board. These visually determined task constraints influenced the choice of path to prevent not only the hand but also the rest of the arm from colliding with the obstacles in complete darkness. A geometric model of obstacle-avoidance path specification based on these kinds of goals and constraints for an arm with degrees of freedom (df) = 7 inspired the design of this experiment (see APPENDIX). However, we reserve for the discussion some key aspects of this theoretical approach. Figure 2A versus Fig. 3 contrasts the differences in solution paths resulting from obstacle positioning. On average, most paths in Fig. 2A inherited the same temporal strategy as those in the straight reaches. Temporal learning was far less evident than that observed when two obstacles were present, as in Fig. 3. In this case paths were much longer and curved, rendering the former speed profiles obsolete for the new geometry [consistent with earlier findings by Abend et al. (1982)]. Figure 2B shows the choice of spatial route as a function of obstacle size. For a longer cylinder movement to all three targets avoided the obstacle by passing below it. For a shorter cylinder movements to one of the targets avoided the obstacle by passing above it.
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POSTURE SPACE.
Across targets this consistency in the hand paths extended to the forearm and to the upper arm. It was also reflected in the seven-dimensional joint angle positional paths that best reconstructed the motion sensor paths [reconstruction method described in detail in Torres (2001)]. Figure 5A shows the seven-joint-angle paths resampled in space (100 points, with no temporal information), where no significant differences were found for all joint angles according to Wilks's lambda ratio (Table 1) between the learning and the automatic epochs across all targets. Notice, however, that for paths to the most affected ipsilateral target shown in this figure, the individual hand abduction joint angle showed differences between the first few trials in the early obstacle-avoidance epoch compared with the last trials of the late epoch. This revealed that the major transport components of the reach involving flexors and extensors remained consistent as the tempo was being learned. Figure 5B illustrates the similarity of the paths for the learning and the automatic phases for each one of the sensors at the hand, the forearm, and the upper arm. These results for one subject were the same for the second subject.
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Geometric-based (temporal) learning: when movement timing is not predefined
The transition from simple straight reaches (SRs) to early obstacle avoidance (OB-learning) demanded drastic changes in the geometry of the movement. These large geometric changes influenced the motion dynamics, thus changing the temporal course along the spatial path. Table 1 (columns 1 and 2) shows that across targets that are maximally affected by the obstacle there were significant differences in path length, curvature, movement duration, and number of velocity peaks compared with those of straight reaches.
The required changes in spatial path and movement timing followed a clear order. Although the spatial solution path for obstacle avoidance was anticipated at the postural level, succeeded from trial one and was conserved for the remaining trials, the temporal path that accommodated it had to be learned. This was evidenced in the changes observed in the time duration, the number of velocity peaks, and the evolution of the first significant velocity peak value from low to high from OB-learning to OB-automatic (one-tail critical t-test for two groups at the 0.01 alpha-level; Tables 1 and 2). Figure 4B contrasts the jerky versus the smooth speed profiles during the learning and the automatic phases of motion.
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Recent work proposed the existence of a spatial via point to aid in sequentially forming complex curved motions (Hatsopoulos et al. 2003; Morasso and Mussa-Ivaldi 1982; Viviani and Flash 1995; Viviani and Schneider 1991; Wada and Kawato 2004). Although the thought of breaking up the spatial paths according to some geometric prescription and storing these spatial segments as "motor primitives" is intuitively sound, our results suggest an additional and new idea: to use temporal rather than spatial points as reference in the search for a smooth motion and to store temporal rather than spatial information as motor primitives. This temporal point should be tied to the tangential peak velocity of the movement in relation to the goals of the task, not to an arbitrary point along the spatial path. Tangential peak velocity points are relevant because they relate to curvature and deceleration phases of the motion. Peak velocity and thus first deceleration were consistently reached near to the point of highest curvature along the path, regardless of whether the system was still in the learning or in the automatic phase. In other words, the first impulse of the motion strongly determined how the system temporally moved along the remainder of the path.
Figure 6, A and B shows this for both the learning and the automatic phase and aid in examining the validity of this new idea of forming motor primitives in the temporal domain. The figure contrasts the differences in the point along the spatial path where the first velocity peak was reached during the learning and the automatic phases. The 10 first and 10 last trials in the block where used in the analysis. Initially there was a region of interest in space, plotted here for each trial (Fig. 6A). This region eventually converged to a tight cluster, a single location in space (Fig. 6B). Each point is plotted at the end of the distance traveled before the hand started to slow down.
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Because during obstacle avoidance the total path length of the motion remained stable and the time to reach the first peak was reliable, as the distance traveled up to that point in time changed, so did the peak velocity value. Figure 6D shows significant separation of the means for the distance traveled along the path up to the temporal point when comparing the early and late epochs of the OB block [F = 16.15 > (F0.05,1,18* = 4.41, F0.01,1,18* = 8.29), P = 0.0008]. Comparison of straight reaches to the early and late OB epochs revealed significant differences as well [F = 12.06 > (F0.05,2,27* = 3.35, F0.01,2,27* = 5.48), P = 0.0002]. Figure 6E shows significant differences for the peak velocity values when comparing straight to early and late OB-avoidance reaches (F = 16.15, P = 0).
The trend observed in Fig. 6, D and E shows a cautious strategy for the peak velocity of the motion where the system initially moved significantly slower than it could afford, increasing the velocity as it covered more distance in the same amount of time. It is possible that an overall higher-level plan similar to that in straight reaches is evidenced in these data, that is, to reach the peak velocity value at midway along the spatial path. However, the hand trajectories in Fig. 6B suggest that dynamics constraints for the highly curved motion settled this point at one third of the total path length.
A longitudinal analysis for board locations 1 and 5 on the evolution of the time to reach the first peak, the total path length and the value of the first velocity peak over 15 consecutive days of experiments revealed consistency in the first two parameters and significant changes on the third. During each experimental session the time to first peak and the path length were the same during both the learning and the automatic OB-avoidance phases with a separation between ipsi- and contralateral space locations. In contrast, the value of the first velocity peak showed significant changes from slow OB-learning to fast OB-automatic during each experimental session and across time. Figure 7, A and B shows that both the time to reach the first peak and the path length were more robust to learning, with a slight (nonsignificant) trend for path length to decrease when comparing day 1 to day 15. Figure 7C shows that the value of the first velocity peak changed each day and evolved over time. The first day there was a significant separation of the means. The overlap starts at the fifth consecutive day of training. The error bars represent unit SDs. The inset in Fig. 7E shows the evolution across trials for each one of the days. Notice that on day 15 the value of the peak velocity overlapped for the earlier and later trials. There was also an overall increase in this parameter when compared with days 5 and 10. Figure 7D shows (for the most affected ipsilateral target on board location 1) the effect of learning on the distance traveled up to the time when the motion reached the first velocity peak. Initially these values were significantly different between the early and the late OB-avoidance epochs, but as time progressed their mean values became closer. A similar result was observed for the contralateral case.
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To examine this question we analyzed two cases: 1) for the same OB configuration and same starting hand location, does the temporal point shift as a function of the target location?; and 2) for the same starting location of the hand and the same final target location, does the temporal point change as a function of the task condition? (i.e., for the contexts of no obstacle, one obstacle, two obstacles).
In the first case, we found that for both ipsi- (Fig. 8A) and contralateral (Fig. 8B) target locations, the same obstacle configuration (two OB) across 20 trials shifted the time to reach the first peak as a function of space location. Hand motions to location 11 reached the first peak at 250 ms on average before starting the deceleration phase of the movement. Hand motions to board location 2 reached the first velocity peak at 300 ms on average. The first OB that the hand passed by affected the temporal point with a significant effect for different target locations {one-way balanced ANOVA for ipsilateral targets on board locations 2 and 11; [F = 72.75 > (F0.05,1,38* = 4.10, F0.01,1,38* = 7.35), P = 0, and F = 19.65, P = 0] for contralateral targets on board locations 5 and 15}. In the cases of board locations 2 and 5 the second OB that the hand passed by on the way to the final target was also taken into consideration for the subsequent temporal search.
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The experiment took place in the dark, yet the subjects saw both the positioning and the removal of the obstacle in full light. In switching from the straight to the curved reaches the goals and constraints imposed by the perceptual system dominated the choice of path strategy. A straight path to the target would have collided against the obstacle, but the system immediately solved this problem with a highly curved path. The perceptual goals were different from those of simple reaches and they determined the hand paths.
In switching from the automatic OB-avoidance reaches back to the straight reaches the perceptual goals were the same: reach for a spatial target. However, during the OB block the motor system had developed dynamics that did not comply with the straight reaches in demand. The arm system had to build the right inertial forces to enable very fast motions along many of the curved paths that did not overshoot the targets. In this case, the system in charge of the dynamics overrode the perceptual system, so residual curved paths with inadequate speed profiles for straight reaches were observed. Figure 10, A and B shows the hand positional paths for ipsilateral target on board location 1 and for contralateral target on board location 5. This kind of persistence resembles the notion of an "aftereffect" observed in learning studies involving exposure to force fields (Shadmehr and Mussa-Ivaldi 1994).
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These features extended to the joint angle domain. Figure 11A displays the joint angle paths also capturing the differences in endpoint paths despite the fact that the final spatial goal was the same. These residual paths were also observed in the forearms and the upper arms, which is depicted in Fig. 11B together with the differences in the initial and final postures.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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This division suggests differential roles for the perceptual and the motor systems during the acquisition of a new motor skill. It poses the question of whether different but necessarily interrelated brain areas are involved in the learning and the automatic phases.
The spatial solution path for the new complex motion was resolved before movement execution. The best evidence for the precomputation of a spatial solution path lies in three facts.
This result implies that the perceptual system is involved not only in the identification of goals (target, obstacle, time to first peak) but also in the computation of the ideal spatial solution path for the task. The length of this ideal path plays a role in the estimation of movement duration for the computation of the optimal dynamics by the execution system. Spatial hand path constancy versus temporal path search hints that the former can be used as a reference while the latter is being learned.
These arm motions were unconstrained and occurred in three dimensions with ample room for other choices of hand path across different trials. Not only were the hand paths conserved but also the forearm and the upper-arm paths did remain invariant while the temporal course of the motion was dramatically changing. This spatial consistency extended to the joint angle paths recovered from the arm motion sensors. This is important because with more joint angles to control than dimensions in the perceptual goals, different postural solution paths across trials could have easily emerged as the system searched for the adequate temporal strategy. The joint-angle path robustness to drastic changes in temporal profiles suggests that the set of perceptual goals had an analogous set in the joint-angle space.
In particular, the postural adjustments observed before the obstacle-avoidance reaches set the stage for the new geometry-compliant dynamics. Before the movement started both animals determined already what portion of the space to use and translated that to the postural domain by rotating the arm to a posture more compliant with the OB-target configuration and the curvature of the upcoming path, which was executed 1 s later. Previous experimental work on pointing (Soechting et al. 1995) and orientation matching (Torres and Zipser 2004) showed that the initial position of the arm determines the final posture. Our present result on the postural paths extends these notions to a more complex task. In the context of task switching the spontaneous repositioning of the initial posture in anticipation of different dynamics is a new result that also agrees with the theoretical prediction that the joint-angle paths contain high-level cognitive information about the goals (Torres and Zipser 2004) and that, in addition to the final posture, the entire path depends on the initial posture (Torres and Zipser 2002).
A priori temporal information
The data revealed the existence of a temporal value that seemed important in shaping the evolution toward the final temporal course along the path. The perceptual system had a mapped representation of the time to reach the first velocity peak that smoothly varied across the target spatial locations and, within each spatial location, smoothly changed across tasks. This new result lends further support to our proposition that the motion's time is a dependent variable of the task space.
The representation of this particular temporal parameter was not subject to learning as was the full temporal course of the motion. While the value of the first velocity peak was changing (thus causing a change in the temporal duration under a fixed total path length), the time to reach this peak remained robust across days. This kind of movement feature is a good candidate for a motor primitive in trajectory formation. Whereas spatial features of the target object, such as location, orientation, shape, volume, and luminance, are identified by the visual system, it is an open question what system would identify this kind of temporal primitive for action. It is known, however, that the posterior parietal cortex (PPC), an early area involved in reference frame determination to locate spatial targets for action (Zipser and Andersen 1988), is also involved in the mental representation of the temporal aspects of simulated movements (Sirigu et al. 1996). It is possible that a space–time map for action exists in this area and is modifiable through experience as the tasks' demands change.
Just as the spatial goals determined the consistent spatial strategy, the time to first impulse clearly set perceptual constraints for trajectory generation. This is important because path generation between two points in space is an overcomplete problem. There are infinite numbers of ways to connect these points in space and time, yet having reliable spatiotemporal goals entirely determined the solution trajectory.
Another key aspect of these data were how early in the movement the peak velocity value was being updated as the system was probing how far along the path the hand could travel before starting to slow down to stop at the target. For instance, Fig. 6, A and B shows the evolution of the distance traveled up to the first peak, where all the points at the end of this first impulse were reached between 200 and 250 ms. At 250 ms there was not enough time to receive sensory feedback from the motor component of the signal, yet at the execution of each trial the value of the velocity peak was being changed under cognitive control.
Speed is distance traveled in time. For a fixed time to first peak, this suggests that the kind of speed information under control came from a geometric distance-based estimation that could be obtained before the motion execution and bypass the time delays that sensory-motor feedback is subject to. This is in contrast to a temporal-based computation derived from the dynamics of the executed action. A good candidate for this type of fast and early, dynamics-invariant planning is the PPC because it seats between earlier perceptual and later motor areas. In particular we suggest the parietal reach region (PRR) as a region of interest for this kind of on-line learning mechanism. This processing would engage the perceptual (rather than the motor system) and rely on an abstract, geometric reference signal for the early portion of the movement. It was recently reported that rTMS causing disruption of the intraparietal sulcus in the PPC impairs correction during the first impulse of the motion (Glover et al. 2005). Lesions of this area cause severe misreaching behavior as well (Karnath and Perenin 2005).
Theoretical implications of these results
The data described here suggest that a form of action simulation takes place before actual movement execution. There is an important difference between the types of planning that our data suggest and those that have dominated the computational field for many years (for recent reviews see Jordan and Wolpert 1999; Todorov 2004). Earlier optimal control models (Alexander 1997; Flash and Hogan 1985; Harris and Wolpert 1998; Todorov and Jordan 2002; Uno et al. 1989, 1995) solved the planning problem in a coupled manner, whereby a detailed spatiotemporal trajectory was generated before movement. In such models space and time are coupled. Regardless of whether the planning was based on extrinsic or intrinsic parameters, time in these models was predefined, so they could not account for temporal learning. None of these methods captures the flavor of our data: that paths are anticipated before the movement takes place, show an immediate spatial strategy, the paths are time invariant, and that the first impulse of the motion is under cognitive control. The type of precomputation that best characterizes our data is geometric—i.e., independent of the motion dynamics—yet provides a signal ready for the execution system.
During the learning period of motor skill acquisition this signal can be characterized using Hamilton–Jacobi's principle of least action from variational mechanics (Feynman 1965; Jose and Saletan 1998; Lanczos 1970). This principle considers mechanical systems whose Lagrangian function does not contain time explicitly and brings out the relationship between conservative systems and the non-Euclidean geometry of the underlying space. In particular the path in the learning stage can be thought of as the shortest straight line (a geodesic) between two definite endpoints in a Riemannian manifold.
We previously proposed that the brain simulates its actions in this space before execution (Torres and Zipser 2002). It is a space that links perceptual goals to an abstract representation of the biomechanical system that solves the dynamics of motion. Thus the simulated action contains perceptual information about the cognitive goals of a given task already translated to the language of the actual biomechanical system. What the principle of least action establishes is that the problem of finding the solution of a given dynamical problem is mathematically equivalent to the problem of finding the geodesics of the underlying space.
This simulated signal for action was recently modeled (Torres 2001) as a geodesic direction that coexists at the visual and the proprioceptive levels (see APPENDIX). According to this model there are two possible ways to generate the motion path: 1) iterative path computation, which allows on-line error correction; and 2) recursive spatial integration of the geodesic direction before motion execution, to provide the length of the geometrically optimal path for the dynamical system. The precomputed time-independent path is not a detailed description of the actual movement. It is neither the most optimal path of the dynamics, but it brings the system to a "good enough" configuration that solves the task. This path contains key spatial information that facilitates its dynamical implementation and temporal estimates linked to distances and geometric properties of the task. When coupled to the execution system this signal will lead to the efficient time course of the geometric movement.
Motions that are still being learned versus those that have reached automaticity can be thought of as simulated actions that are separable in space and time, as opposed to physical actions that can no longer be decoupled. The former are best described by a time-independent process. The motion path exists in space independent of time and can be well characterized as the solution to a time-independent partial differential equation. It answers a simpler question of "how to get from point A to point B in space." Time at this stage depends on the task space and enters into the learning process.
In contrast, the solution path that solves a time-dependent integral describes the motion simultaneously in space and time. This characterization better suits the kinds of behaviors that are automatic. It answers a more complex question: "how to get from point A now to point B exactly X ms later?" Before this study it was never justified in the computational arena where the limits of integration for the dynamics were chosen from for the predefined time interval involving a given motion. It was simply always assumed that the system had a good estimate of what the duration of a given movement was before its execution.
In fact experimental work in motor control has reinforced this notion for many years. If one carefully reads the METHODS sections of many papers, one can see that subjects are instructed to perform motions within a predefined time window and receive some form of feedback on their success at it. These paradigms can afford such instructions because these simple motions were learned during infancy. There is no motor learning of the kind described here taking place during those experiments. A good estimate of movement duration is quite automatic.
Exquisite a priori temporal estimation of the entire motion might indeed exist for movements that have reached automaticity. However, we have empirically shown here that for a new task this notion emerged later with practice and repetition, and that early during the learning process, time was not a free variable. Its duration and course (speed profile) had to be learned.
From a computational perspective our results clearly establish a new kind of motion planning that is separable from the motion execution. The intermediate geometric planning stage proposed to exist in the PPC (Torres and Zipser 2002, 2004) acts as a translator between perceptions and actions. Its output provides an abstract representation of the motion that decouples space and time. This simulated action can be used as reference for its dynamics implementation. It was recently shown that the disruption of the parietal system impairs the learning of trajectory adjustment under new dynamics conditions (Della-Maggiore et al. 2004).
New experimental paradigm useful to study motor learning
In its current block-design form, the experimental paradigm's switching from one task to another resulted in a temporal visuomotor adaptation that over time permanently altered the relationship between the two tasks. As the obstacle-avoidance behavior reached a permanent automatic nature, task switching from straight to OB-avoidance reaches and back occurred in one step.
This new experimental paradigm will have implications for the neurophysiological aspects of this problem. Two distinct but necessarily interrelated cortical regions must be involved in geometric-based learning and the acquisition of automaticity, respectively. Neurophysiologists can use this paradigm in concurrent recordings of the PPC (which we target as the geometric transformer) and the M1 cortex (which we propose to be involved in the acquisition of motor coordination and temporal-based motor programs). Although both areas will be engaged through feedback loops, the PPC should lead initially while the motion is still geometric, i.e., separable in space and time, and its output must be used by the execution system. Yet later when the goal of space–time synchronization prevails and the system strives for a well-coordinated motion, M1 should lead and be highly and differentially engaged.
The subjects in our experiment had no visual feedback about the obstacle. It is an open question whether changing the task by lighting the peripheral obstacle while fixating straight ahead would change the trajectories' evolution. This is predicted because the underlying geometry of the perceptual space would change as well. An explicitly cued spatial via point could accelerate the temporal learning required to make the motion automatic. Simple modifications to the current paradigm will enable us to further investigate this and other important questions.
The present experimental results have implications for the understanding of the acquisition of a new motor skill in general because they formally distinguish learning from automaticity. The data show that temporal learning is separable from spatial learning in that it occurs later, it takes longer to master, and that over time it causes a permanent alteration of the visuomotor transformation relating an old well-known task to the new one. The performance of the new task also changes permanently, as it becomes "second nature," thus signaling the acquisition of a new motor program. This suggests that in a new task, the temporal aspects of the motion enter in the motor error signal for learning. It also suggests that temporal adaptation is crucial for the achievement of good motor coordination as defined by the tight synchronization of space and time. We have clearly shown that the temporal course for a new complex motion and its duration cannot be arbitrarily predefined when the system is faced with its geometry for the first time.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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; Torres and Zipser 2002, 2004). The theory provides a general equation
 | (A1) |
Path determination to achieve the set of goals in a given purposeful action comes from following a geometrically optimal direction (a geodesic direction) in posture and in hand space generated with Eq. A1. The function f maps the initial posture qinit to the starting location of the hand and xgoals defines the goals of the task. Our optimization scheme defines r, the distance in X, as the objective function. The idea is to treat r as the line element of the spaces of interest.
The line element 
denotes the infinitesimal distance between two neighboring points in space, expressed in terms of the coordinates and their differentials. In the Pythagorean case, 
= dx12 + dx22 +... . This expression is a consequence of the Euclidean postulates and the coordinates x1, x2,... . However, if the coordinate lines of the reference frame system are no longer straight lines, but arbitrary curves, then the following general form is used instead
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). They are the elements of a symmetric, positive, definite matrix, which in flat space is the identity matrix. In general, the matrix elements are constants only if rectangular, or more generally "rectilinear," coordinates (with oblique instead of rectangular axes) are used. In the case of curvilinear coordinates, values of gij change smoothly from point to point in space.
In our formulation, each task generates a different r distance in X, which is a function of the xgoals. For instance, the spatiotemporal goals in the obstacle-avoidance task relate to the obstacle location, the target, and the time to reach the first velocity peak. The composition of r with f builds a map (r°f):Q 
X R+, which is a function on Q. This construction is the pullback of r by f, denoted by f*r = (r°f), which allows the control of the rate of change of r arising from changes in the set of joint angles representing a posture in Q.
In general
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, we operate on the gradient of f*r, which we defined for simulations to gain insight into the true metric and its associated gradient from empirical data.
A geometrically optimal gradient—a geodesic direction—is obtained by pulling back into Q the geometry of X by its metric tensor and the Jacobian: Gqµ = JTGx
J, where Gx is an m x m matrix, J is m x n, and Gqµ is n x n (for simplicity we take n = 7 joint angles to represent postures). The Gqµ is the new metric arising from the change of coordinates from X to Q. It preserves the geometry of X through the Gx metric. In addition, it is also possible to modify the metric in Q to include other joint-angle constraints as in Torres and Zipser (2002). In tensorial language, the gradient in Eq. A1 is a (10)-tensor, a covariant tensor of order one. The gij metric in E is a (20)-tensor conjugate to the gij metric in the space E*, the dual of E, a (02)-tensor, i.e., [gij]nn =[gij]nn–1. In Eq. A1 applying G–1 = [gij]nn–1 to the covariant gradient raises the index to create a new contravariant (01)-tensor (Kay 1988), i.e., the differential dq that is tangent to the curve at each point describing the arm path. In what follows we denote dq = 
.
The tensorial transformation in Eq. A1 depends only on the relative orientations and scales of the coordinate axes at that point, not on the absolute values of the coordinates. It builds a local isometric linearization of the map f:Q 
En X Em, where Q and X are assumed to be open subsets of topological spaces with a norm, representing posture and cognitive spaces, respectively.
Preserving the notion of distance through the pullback operation makes the linear map df:TqQ Tf(q)X an isometry. Here the derivative map df operates on the tangent spaces to the posture and the task manifolds, respectively. Let Y denote the vector field created by the transformation in Eq. A1, where each vector in the field is given by Yq = (q, ) at each q. Let Z denote a vector field in X, where each vector in the field is given by Zx = (x, 
) at each x = f(q). The derivative map Tqf · (Y) = (f*Y)(x), is the pushforward of Y. It tells what the new vector field Z is in X under the transformation. Thus 
TqQ is pushed forward to get the geodesic direction dx = Tf(q)X at x (relative to the hand) by df() = , where = qT 
q
in the sense of vector decomposition into the tangential and normal components, so that we can identify with qT in the subspace relevant to the task and distinguish it from the redundant dimensions.
The corresponding inverse map df–1 starting at a given posture is locally injective, and the path thus obtained gives a continuous map onto its image df–1[Tf(q)X], i.e., an embedding (Do Carmo 1992). This local operation preserves in Q the notion of distance in X and makes df–1 a local isometric embedding.
To guarantee that the direction TqQ with q f–1(x) maps uniquely to the desired direction Tf(q)X, which the task demands, two conditions must be met.
) = , i.e., maps linearly through the differential df of the map f to the goal-related direction . 
, q + 
= 0 
Ker(df) = {|df() = 0}, i.e., q + for all changes in postures that do not change the configuration of the hand (self-motion subspace). In this way one can restrict the solution to be the unique local geodesic direction = 
q for posture space with a corresponding direction = x for the hand space that iteratively builds the geodesic paths.
Equation A1 builds vector flows on the tangent spaces to the posture and the cognitive manifolds. The projection of such paths in three-dimensional space has curvature that should be informative of the underlying geometric changes that arise from changes in the task goals and constraints. Equation A1 generates unit-speed paths (parameterized by arc-length) that minimize the r-distance defined by the task. Notice that Eq. A1 is recursive. It self-reevaluates the notion of distance and stops only when the value of r is 0, i.e., when all the goals are met. This generates paths autonomously. In addition, the iterative version allows coupling of the geometric signal with the dynamics. This is useful for on-line error correction (Torres and Zipser 2002) in the presence of sudden perturbations, adaptation, and so forth.
The 
refers to the step size for the unit-length gradient direction, and not to the time parameter for speed. During the first impulse of the motion speedfirst = dtraversed/timefirst and by manipulating the value of the distance traveled to the first peak and/or the time to reach the first peak the system can change the magnitude of the first velocity peak. In this task for a given target location the time to the first peak remained constant, so changing the dtraversed was equivalent to changing the magnitude of the speed. In terms of transmission delays in the motor system, this type of geometric manipulation can take place before the sensory motor feedback is available and engage the perceptual rather than the motor system.
When the reaching action involves obstacle avoidance the spatial path curves around the obstacle. To avoid overshooting the target or hitting the obstacle it makes sense to keep the timefirst
