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1Department of Neuromotor Physiology, Scientific Institute Foundation Santa Lucia, Rome; and 2Department of Neuroscience and 3Center of Space Bio-medicine, University of Rome Tor Vergata, Rome, Italy
Submitted 28 February 2005; accepted in final form 4 April 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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; Wolpert et al. 2001). Internal models are often thought to be unique for the objects with which we have learned to interact. According to this view, a collection of specific internal models is stored in motor memory, and sensory or cognitive information provides the identifying cues to select the appropriate internal model (Flanagan and Wing 1997; Gordon et al. 1993; Wolpert and Kawato 1998). For instance, after subjects learn to manipulate two different tools (a rotated mouse and a velocity mouse), they can easily switch between the two, while distinct, spatially segregated regions are engaged in the cerebellum by each tool (Imamizu et al. 2003).
However, learning and switching between multiple models is not always easy or even possible. Thus learning of two conflicting environments, such as opposing force fields or opposing visuomotor rotations, has been reported to be difficult or very slow, probably because of long-lasting interference in memory (Caithness et al. 2004; Krakauer et al. 1999). Osu et al. (2004) reported that subjects can learn two opposing force fields when provided with contextual cues and random, frequent switching. However, concurrent learning was slower and more difficult than separate learning. Karniel and Mussa-Ivaldi (2002) taught subjects, on separate days, to move in two different velocity-dependent force fields applied at the hand by a robot. Even after this experience, subjects were unable to move accurately when the same two fields alternated after each movement. This suggests that subjects did not use two separate models and switched between them according to the sequence context. Instead, they tried to represent the alternating perturbations with a single internal model. From this and other evidence (see Martin et al. 1996), one is led to conclude that, in addition to multiple internal models specific for individual objects, there also exist single internal models that can cope with a variety of conditions.
Whether we learn a new model or tune an existing one might also depend on the ecological significance of the specific environment (Georgopoulos 2004; Gibson 1966; Shepard 1984). Natural selection favors adaptation to any biologically relevant property of the world, especially those properties that hold throughout all habitable environments. The 24-h period of the terrestrial circadian cycle (consequence of the conservation of angular momentum) and the gravitational acceleration are examples of ecological constraints that are presumably internalized. On Earth, normally we experience an invariant gravity (1g), and the maximum variation that we sporadically encounter by changing latitude or altitude is <1%. There is increasing evidence that an internal model (called "1g model") calculating the effects of Earth gravity on seen objects is stored in the brain. Indeed, the human visual system poorly estimates arbitrary image accelerations (Brouwer et al. 2002; Werkhoven et al. 1992), and these accelerations generally are not taken into account in timing manual interceptions (Port et al. 1997). Nevertheless, visually guided interceptions of objects falling under gravity are accurately timed by synchronizing the changes of anticipatory muscle activity, as well as those of reflex activity and limb impedance, to the time of collision (Lacquaniti and Maioli 1987, 1989; Lacquaniti et al. 1991, 1992, 1993a,b; Michaels et al. 2001). Evidence for internalized visual gravity was provided by the observation that, in the absence of gravity-determined sensory cues, astronauts initially expect the effects of Earth gravity on a dropped object (moving at constant speed) when they attempt to catch it in the Spacelab, and they adapt to the new environment only after few days of flight (McIntyre et al. 2001).
It should be noticed that a fixed model of free fall would not be appropriate to deal with a variety of natural conditions even on Earth. Thus vertical motion of objects is accelerated by gravity and decelerated to a variable extent by air (or other fluid) friction depending on the object's mass, size, shape, texture, and fluid viscosity. Does the brain develop an internal model for each specific environment or does it adapt a single general model by tuning its parameters?
Here we addressed this issue by investigating how interceptive responses change with changes of the visual environment produced in the laboratory. In our paradigm, a virtual target moves vertically downward on a screen with different laws of motion. Subjects are asked to punch a hidden ball that arrives in synchrony with the visual target. Using this paradigm, it was previously found (Zago et al. 2004) that subjects time their motor responses very differently when the visual target moves with an acceleration equivalent to gravity (1g) or when it moves at constant speed (0g). Motor responses generally are time-locked to the arrival of 1g targets, whereas the responses to 0g targets are premature. Here we used this paradigm to test between alternative strategies for learning to deal with the novel environment of visual targets descending at 0g. One possibility is that extensive training results in the development of a 0g model appropriate for the new situation. The alternative possibility is that subjects continue using the 1g model throughout the exposure to 0g targets, but adapt its internal parameters. These two hypotheses entail completely different schemes of neural implementation for the temporal control of interception of falling objects. The first hypothesis predicts that, with training, the brain develops and stores multiple internal models of target motion in different neural populations, equivalent to modules (see Imamizu et al. 2003). The second hypothesis, instead, predicts that adaptation to a new target motion involves a different temporal processing through cortical and subcortical regions interconnected with the internal model of gravity stored in the brain. The following section is devoted to the formal treatment of the different model schemes for dealing with 1g and 0g targets.
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MODELING |
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TOPABSTRACTINTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The instantaneous height h(t) of a descending target above the interception point is given by
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; Michaels et al. 2001; Zago et al. 2004, 2005). Moreover, target contact occurs close to the time of peak arm velocity, and the deceleration phase is simply a follow-through. For this type of interceptions, it is generally considered that subjects time the movement based on a perceptual estimate of the remaining time-to-arrival at the interception point (referred to as time-to-contact [TTC]) of the target (Fitch and Turvey 1978; Lee et al. 1983; Michaels et al. 2001; Tresilian 1999, 2004). The movement is centrally triggered when TTC reaches a given criterion value (Lee et al. 1983; Michaels et al. 2001; Tresilian 2004). The criterion value is also sometimes called time margin or time threshold relative to the expected collision, and will be denoted 
in the following. A time margin is necessary to compensate for signal transmission delays (typically, between 80 and 300 ms; see Port et al. 1997). Thus , which includes the neuromechanical delays, is equal to the sum of central processing time and peripheral movement time (Lacquaniti and Maioli 1989; Lee et al. 1983; Port et al. 1997; Tresilian 2004). Note that, for these TTC models, the control of interception is under on-line visual control up to time : the visually derived estimate of target TTC is compared continuously (or intermittently) with the preset criterion value, as in a countdown process. When the time threshold is reached, the action is launched with little (if any) correction based on feedback (Tresilian 2004).
Within the class of these TTC-models, the 1g model specifically predicts that subjects estimate TTC using on-line visual information about target distance and velocity, under the assumption that it is accelerated by gravity (Lacquaniti and Maioli 1989; Lacquaniti et al. 1993a; McIntyre et al. 2001; Zago et al. 2004). At time 
after the onset of target motion, the remaining time-to-arrival at the interception point is TTC(), target velocity is v(), and the height above the interception point is h(). Thus according to the 1g model, when the target arrives at the interception point
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) and h() as initial conditions. TTC() is then given by
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) = . Predictions of the 1g model for 1g and 0g motions
For a successful punch, the hand must reach the interception point at the same time as the target does. The hand arrives at time + after the onset of target motion, whereas the target arrives at time T. For a successful interception: + = T. The 1g model correctly estimates the time of arrival of 1g targets if h() and v() are measured reliably by vision, and the punch is timed exactly on T as a result (Fig. 1A). By contrast, the 1g model underestimates the time of arrival of 0g targets by 
, and the punch occurs at time before target arrival even with correct measurements of h() and v() = v0 (Fig. 1B). The value of the temporal error is derived as follows. For a constant speed target
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For simplicity, the preceding analysis neglects the existence of possible margins of error for the interceptive action. If present, these margins would result in a temporal tolerance window (µ) around the ideal trigger time. This window depends on the spatiotemporal configuration of the moving hand relative to the moving target. Thus for 1g targets, T – – µ– < < T – – µ+. We will consider this aspect later (see INTERCEPTION SCORE in METHODS).
Learning
Here we consider alternative strategies for learning to deal with 0g targets. Training might result in the development of a 0g model appropriate for the new situation. According to the 0g model
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Equation 8 follows from Eq. 2 by taking g = 0. TTC() is then given by
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) = . If subjects were able to use a 0g model with the 0g targets of the present experiments, they would correctly estimate the time of arrival of these targets and produce motor responses time-locked to target arrival.
Alternatively, however, subjects might continue using the 1g model throughout the exposure to 0g targets, and they would adapt its parameters. An adaptive 1g model has two internal parameters, and 
. was defined earlier as the time margin for triggering the interceptive action; is an internalized estimate of target acceleration. In Bayesian estimation theory, would represent the prior for target kinetics. In principle, the value of this prior could depend on the probability distribution of target accelerations within any one experiment. In the default case of targets accelerated by gravity, = 1g; otherwise, could take any arbitrary value, depending on the neural estimate of target acceleration. In the limit case that = 0, the adapted 1g model becomes identically equal to the 0g model considered above.
If we let the internal estimate of acceleration vary in the 1g model, Eq. 7 becomes
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with 0g targets could be reduced by decreasing either or or both. A decrease of could result from a compression of central processing times up to minimum values dictated by physiological constraints. In this case, one would expect that the error with 0g targets decreased, but did not go below the minimum signal transmission delays (Fig. 2B). Performance with 1g targets should not be affected by a physiological decrease of , as long as h() and v() continue to be correctly estimated (Fig. 2A).
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, there are no a priori physiological limits for the adaptation of the internal estimate of target acceleration (). In principle, the brain could adjust this estimate down to the true acceleration ( = 0g), and succeed in zeroing the error for 0g targets (Fig. 2D). Note, however, that progressive adaptation of to values <1g (as well as simultaneous adaptation of both and ) would affect the responses to both 0g targets and 1g targets. Responses to 0g targets would improve according to Eq. 10. At the same time, responses to 1g targets would become worse. An internal estimate of target acceleration to < 1g overestimates the time of arrival of 1g targets by 
, and the punch would occur at time after target arrival (Fig. 2C). The value of this temporal error is derived as follows
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METHODS |
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TOPABSTRACTINTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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In total twenty-nine healthy subjects (fourteen women and fifteen men, 30 ± 6 yr old, mean ± SD) participated in the study. The subjects were right-handed (as assessed by a short questionnaire based on the Edinburgh scale), had normal vision or vision that was corrected for normal, and were naïve to the purpose of the experiments. They gave informed consent to procedures approved by the Institutional Review Board of IRCCS Fondazione Santa Lucia, in conformity with the Declaration of Helsinki on the use of human subjects in research.
Experimental setup
Subjects sat on a chair placed in front of a vertical screen (3.94 m wide, 2.13 m high) attached to the ceiling of a dimly illuminated room. Subjects' eyes were located at a horizontal distance of 0.5 m from the screen, 1.82 m below the top. The back of the chair, vertically inclined by 40°, supported the head and torso of the subjects. In that position, subjects could easily reach below and beyond the lower border of the screen by protracting their arm forward. Images were generated by a PC and displayed on the screen by a Barco Graphics 808s (1,024 x 768 pixels, 85-Hz refresh frequency). A black square box (width, 9 cm) was constantly displayed at the top of the screen against a white background. In each trial, a red target sphere (diameter, 9 cm) moved vertically downward, emerging progressively from within the start box at the predefined initial speed and acceleration (see Protocols) and disappearing progressively at the lower border of the screen. Subjects did not perceive any transient acceleration related to the start of target motion because the target appeared already in motion, as it would occur when an object emerges after a period of visual occlusion. During the same time interval, a real ball was dropped by an electromagnet behind the screen, hidden to the subject until it emerged from below the lower border of the screen. The ball (diameter, 9 cm; weight, 70 g) was made of soft rubber, with a small steel washer (outer diameter, 1.4 cm) on the surface to ensure magnetic contact with a precision laminated electromagnet (G.W. Linsk, Clifton Springs, NY). The electromagnet, driven by a PC-controlled device, held the ball and released it with an accuracy better than 1 ms. Spatial trajectories of the center of the virtual sphere on the screen and of the real ball were aligned in the vertical plane orthogonal to the seat of the chair, offset by 7.5 cm in depth. The release time of the real ball and the start time of virtual motion were set by the computer to ensure the synchronous arrival at an interception point located just below the lower border of the screen.
The time of synchronous arrival will be denoted interception time in the following. Synchronous timing was determined by means of an optic calibration procedure that involved 1-kHz–shuttered video recordings. The timing error was always less than the refresh rate of image display for all tested laws of virtual motion. Because the release of the ball from the electromagnet and its subsequent fall were noiseless and invisible to the subject until the emergence from below the screen, the visual information about the virtual target provided the only perceptually available time-to-contact information before the ball contact with the arm. A miniature accelerometer (either custom-made or from Isotron Endevco, San Juan Capistrano, CA, depending on the experiment) was attached by means of a Velcro strap fastened to the wrist, roughly corresponding with the midline between processus styloideus radii and processus styloideus ulnae. In this position, the device was not bent by wrist flexion–extension. In addition, three-dimensional motion of selected body points was recorded by means of the Optotrak 3020 system (Northern Digital, Waterloo, Ontario, Canada; ±3 SD accuracy 0.2 mm for x-, y-, z-coordinates). Two infrared-emitting markers were attached to the skin overlying the shoulder and elbow joint on the right arm. Two other markers were attached to the wrist strap at a close distance from the accelerometer. Three additional markers were fixed to the lower border of the screen to determine the screen plane. Accelerometer data were sampled at 1 kHz and Optotrak data at 200 Hz for 5 s, starting 0.4 s before target appearance in synchrony with trial start. Intertrial interval was 14 s. Total duration of each session was 1 h 20 min, with 10 min of rest allowed halfway.
Task
Before the experiment, subjects received general instructions and familiarized with the setup in front of the screen (they could not see behind it). They were told that they should punch a hidden falling ball as it emerged below the lower border of the screen. They were further informed that, to be successful, they had to visually monitor the motion of the sphere on the screen because it arrived at the same time as the hidden ball at the interception point. Between trials, subjects kept a free relaxed posture until an alert auditory signal instructed them to look at the box on the top of the screen and to recoil their arm in the starting posture: with the adducted shoulder, the upper arm was roughly vertical, the forearm horizontal, the wrist midpronated, and the hand and fingers clenched in a fist. At trial onset, another auditory signal was followed by the visual target emerging from the box, after a random delay ranging from 1.2 to 1.7 s. Subjects were asked to punch with the finger knuckles so as to deviate the ball trajectory. According to the instructions, a valid interception required contacting any point of the ball with any part of the finger knuckles. Instead, contacting the ball outside the knuckles (such as with the metacarpus, carpus, or forearm) represented an invalid trial. Six practice trials were run before the experiment so that subjects could become familiar with the task. In these trials, target motion (v0 = 0, a0 = 1g) was different from any motion used during the actual experiment.
Protocols
The visual target descended from the start box with pseudorandomly assorted initial speeds (v0 = 0.7, 1, 1.5, 2.5, or 4.5 m s–1) and accelerations (a0 = 1g = 9.81 m s–2 or a0 = 0g). The randomization procedure avoided identical conditions in consecutive trials. Each experimental session consisted of blocks of 55 or 50 trials (depending on the protocol) repeated four times consecutively. The probability of a given v0 was always 20%, but the probability of 0g versus 1g acceleration could vary depending on the protocol and the day of practice. We included five protocols that differed in the relative probability of 0g versus 1g targets on any given day: P9, P50, P91, P100, and P0 (Table 1). The label denotes the probability (P) of 0g targets on day 1, P ranging between 0% (P0) and 100% (P100). There were seven subjects in group P50, seven subjects in group P100, and five subjects for each group P0, P9, and P91. All subjects of a group were exposed to identical sequences of trials. Subjects of group P50 performed a 1-day experiment with 50% of 0g targets and 50% of 1g targets. In the other groups there were two experimental sessions 24 h apart. Subjects of group P100 were exposed to an identical sequence of trials on both days, involving only 0g targets (100% probability) except for the six practice trials that preceded each session (see above). Subjects of group P9 were exposed to 9% of 0g targets and 91% of 1g targets on day 1. On day 2, they were exposed to an identical sequence of target v0 but reversed accelerations relative to day 1, resulting in 91% of 0g trials and 9% of 1g trials. Subjects of group P91 practiced on day 1 (day 2) the same sequence experienced on day 2 (day 1) by subjects of group P9. Thus they had 91% of 0g trials on day 1 and 9% of 0g trials on day 2. The sequence of target v0 was identical for trials at the same acceleration in P9, P91, and P100.
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MOTOR TIMING.
Accelerometer data were numerically low-pass filtered (bidirectional 10-Hz cutoff, 2nd-order Butterworth filter) to eliminate impact artifacts. Motor timing was quantified from the time of occurrence (denoted as cross-time) of the zero-crossing relative to the interception time, which is the time when the acceleration first crosses the zero-line after the first positive peak (see Fig. 3 for typical recordings). In correctly intercepted trials, the zero-crossing occurs very close to the interception time. Therefore the cross-time reliably reflects the subject's estimate of the arrival time of the target at the interception point. We previously showed that the timing of the zero-crossing co-varies tightly with that of other landmarks of the acceleration trace, such as the first positive peak and the first negative peak: the r2 between each pair of landmarks is always >0.90 (Zago et al. 2005). By correlating the motor timing with the spatial trajectories of the limb, we also showed that the temporal landmarks of wrist acceleration are good predictors of the wrist position measured at interception time (Zago et al. 2005). Thus when the zero-crossing occurs earlier relative to the interception time (negative cross-time) the hand has moved past the interception zone when the ball arrives. Conversely, when the zero-crossing is later than the interception time (positive cross-time), the hand has not yet reached the interception zone when the ball arrives.
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ADAPTATION RATE.
An exponential function was fitted to the series of repetitions of the chosen parameter 
(cross-time or interception score) for each condition (a0 and v0) to characterize the rate at which subjects adapted during an experiment. The function has three free parameters: offset b0, gain b1, and learning constant b2, expressed as
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i is the value estimated for repetition i.
STATISTICS.
Differences between conditions were assessed using single- or multiple-factor ANOVAs. The threshold for statistical significance was set at 
= 0.05. Statistically significant differences of interception scores were assessed using the exact test for matched pairs (McNemar 1947). Statistical determination of correlation coefficients was performed on the normally distributed, Z-transformed values (Kendall and Stuart 1969).
MODEL FITTING.
We took the cross-time (time of occurrence of the zero-crossing of hand acceleration) as the subject's estimate of the arrival time of the target at the interception point. Therefore the cross-times reflect the timing errors ( for 0g trials or for 1g trials). Estimates of model parameters ( and ) for any given repetition of a given condition were obtained by least-squares fitting the experimental values of cross-times averaged over all subjects of a group. All analyses were implemented in Matlab code (The MathWorks, Natick, MA).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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Interception of 0g versus 1g targets
In agreement with the previous results obtained with a similar paradigm (Zago et al. 2004, 2005), we found that the initial performance with 0g targets was very different from that with 1g targets. Motor responses generally were time-locked to the arrival of 1g targets, whereas the responses to 0g targets often were very premature, and the hand passed through the interception point well in advance of the target. As a result of premature timing in 0g trials, the falling ball often hit the forearm instead of the hand, generating an error feedback. Close scrutiny of the changes of the acceleration waveforms with prolonged exposure to 0g targets provides critical clues to discriminate between the two hypotheses of learning outlined above. If subjects learned a new internal model specific for the 0g targets, they should become able to estimate the target TTC correctly and, as a result, produce responses to 0g targets with the same timing with respect to impact as for 1g targets. Accordingly, one would expect that the amplitude of the early, inappropriate response gradually diminished, whereas another, distinct response developed time-locked to the nominal interception time, becoming increasingly predominant (Witney et al. 1999). The alternative hypothesis that subjects adapt the preexisting 1g model by stretching its parameters predicts that the original response does not diminish but, with learning, it progressively migrates to a later time, increasingly closer to the nominal interception time.
Figure 3 shows the time profiles of hand acceleration of three repetitions (1st, 2nd, and 12th in blue, green, and red, respectively) of target v0 = 0.7 m s–1 for a representative subject. It can be seen that the responses to 1g targets were time-locked to the nominal interception time (time 0 on the abscissa) from the first repetition and varied little in the following repetitions (Fig. 3, left). Thus the zero-crossing of hand acceleration occurred very close to the interception time, indicating that subjects generated maximum momentum to punch the incoming ball at the right time. By contrast, when subjects were first exposed to the 0g target, their responses started too early (Fig. 3, right). With training, the early, inappropriate response did not disappear; instead, it shifted closer to the nominal interception time, but remained premature as compared with 1g responses. This finding is consistent with the adaptation of a preexisting model rather than the development of a new model.
Note that the waveform of hand acceleration was fairly stereotyped, independent of target initial velocity, acceleration, and repetition. Hand movements always consisted of a brief acceleration followed by an equally brief deceleration, each phase lasting about 130 ms. To quantify the temporal relationship between different landmarks of wrist acceleration as a function of target acceleration, we computed the time interval between a given landmark in each 1g trial and the same landmark in the corresponding 0g trial. Thus 
PP was the time interval between the positive acceleration peak in each 1g trial and the same landmark in the corresponding 0g trial, whereas ZC was the time interval between the zero-crossing in the 1g trial and the same landmark in the 0g trial. When the data were pooled together (n = 3,700, all trials of all subjects in P9, P50, P91, and P0), they appeared scattered through a wide range because the timing of the responses varied across 0g trials (see Fig. 3). Importantly, however, the best-fitting regression line of ZC versus PP (r2 = 0.88, intercept –0.58 ms, slope 0.98) was similar to that one would expect if the landmarks on the acceleration waveform maintained the same temporal relationship in 0g trials and 1g trials. Moreover, the time interval between the positive peak and the zero-crossing of each acceleration trace varied only slightly across all conditions (by pooling all trials of all subjects, the mean SD was only 16 ± 2 ms).
Learning rate
Learning rate was quantified from the changes of two parameters as a function of repetition: 1) the time of occurrence (denoted cross-time) of the zero-crossing of hand acceleration relative to the interception time, and 2) the interception score (see METHODS). Mean cross-time values and interception scores are plotted as a function of repetition number in the top and bottom rows of Fig. 4, respectively. These data were obtained in two immersive protocols: 100% of targets were either 1g trials over one session (day 1 of P0, Fig. 4, left column) or they were 0g trials over two sessions performed one day apart (day 1 and day 2 of P100, Fig. 4, right column). The timing of the responses to 1g targets did not change significantly with repetition (cross-time = –4 ± 8 ms, mean ± SD, n = 40; one-factor ANOVA, P = 0.8). The interception score was high from the outset and improved only slightly with repetition. On average, it was 88 ± 12% in the first four repetitions and 98 ± 6% in the last four repetitions (the difference between these two mean scores was statistically significant: P < 0.005, McNemar's exact test for matched pairs). By contrast, cross-time and interception score changed drastically over the first few repetitions of 0g targets, reaching plateau levels that remained substantially lower than those for 1g targets. The changes of these parameters with repetition were fitted by an exponential function (Eq. 16), with learning constants of 1.3 and 2.2 repetitions for cross-time and score, respectively. Mean cross-time over the last four repetitions of day 2 was –89 ± 23 ms, and mean score was 45 ± 24%, both values being significantly (P < 0.001) lower than the corresponding values of 1g targets.
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Adaptation of the 1g model to immersive 0g exposure
As explained in MODELING, the 1g model has two internal parameters, and : is the time margin necessary to trigger the interceptive action before the expected collision, to compensate for neuromechanical delays; is the internalized estimate of target acceleration. The error that would result by applying the 1g model to 0g targets might be reduced with training by adaptively decreasing either or or both parameters (Eq. 10). A decrease of could result from a compression of signal transmission delays up to a minimum value dictated by physiological constraints. In this case, one would expect that the error with 0g targets decreased, but did not go below a minimum value. In contrast with the adaptation of , there are no a priori physiological limits for the adaptation of the internal estimate of target acceleration (). In principle, one could expect that the brain adjusts this estimate from the initial default of gravitational acceleration ( = 1g) up to the true acceleration ( = 0g). If so, subjects should ultimately succeed in zeroing the error for 0g targets (Fig. 2D).
The values of cross-time measured in protocol P100 (Fig. 4) can be fitted equally well by one or the other adaptation scheme (Fig. 5). Based on modeling, either or decrease exponentially as a function of repetition. The steady-state value of is 123 ms, a value that is compatible with the known visuomotor delays for interception (Lacquaniti and Maioli 1989; Port et al. 1997). The steady-state value of is 0.35g. Because this was an immersive protocol with prolonged exposure to 0g targets over two sessions 1 day apart, it appears strange that never approached the true 0g value of the visual targets. Nevertheless, both adaptation schemes are compatible with the data of this protocol. The data from the next protocol help to discriminate between the two schemes.
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Adaptation of both and predict a progressive decrease of the error with 0g targets. However, progressive changes of should not affect the response timing with predictable 1g targets presented concurrently with 0g targets within the same session (see MODELING). Instead, progressive changes of do affect the response timing with these 1g targets. An internal estimate of target acceleration equal to < 1g overestimates the time of arrival of 1g targets by , and the punch will occur at time after target arrival (see Fig. 2C). The smaller the value of as a result of adaptation, the more delayed will be the punch relative to the correct time and the lower will be the interception score. In other words, the performance in 1g trials should become progressively worse, in parallel with the improvement of the performance in 0g trials. Therefore to discriminate between the hypothesis of adaptation and the hypothesis of adaptation, we should scrutinize the responses to 1g targets in a protocol where target acceleration randomly switches between 1g and 0g from trial to trial.
In protocol P50, 1g and 0g trials were randomized with the same probability (50%) within the same session (Fig. 6). Thus subjects equally expected either type of target. Adaptation of with repetition (red line) fits the cross-time values of 0g trials slightly better than does adaptation of (green line, mean r2 = 0.82 and 0.79 for and adaptation, respectively; see Fig. 6B). However, the major difference between the predictions of the two adaptation schemes involves the 1g data (Fig. 6A). Adaptation of with repetition (dashed green line) predicts progressive, systematic delays of the cross-time for 1g trials that should parallel the changes of timing for 0g trials as a function of repetition. No such trend was present in the experimental data (one-factor ANOVA, P = 0.21), in contrast with the hypothesis of adaptation of and in agreement with the hypothesis of adaptation of (red line).
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. In fact, the data showed the opposite trend (Fig. 7). The interception score for 1g targets was low initially (64 ± 26% in the first repetition) and increased exponentially with repetition (the difference between the mean score in the first four repetitions and that in the last four repetitions was significant: P < 0.0001, McNemar's exact test for matched pairs). Performance with 0g targets was similar to that of the other protocols, with an exponential increase and a plateau lower than that for 1g targets.
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In Bayesian estimation theory, represents the prior for target kinetics. In principle, the value of this prior should depend on the probability distribution of target accelerations within any one experiment (Kording and Wolpert 2004). If subjects acted in a Bayesian way in the face of uncertainty, they should combine a priori information about the most plausible target acceleration with evidence about the real acceleration experienced in previous presentations. Therefore one would expect that the higher the probability of 0g trials in any one session, the greater the extent of adaptation of and the lower the residual error with 0g targets at the end of practice. A similar prediction stems from a different but related hypothesis: that subjects learn to use a combination of two distinct internal models, the original 1g model and a new 0g (or fractional g) model, the relative weight of each one changing as a function of the context (Davidson and Wolpert 2004). If this were the case, one would expect that the higher the probability of 0g trials in any one session, the greater the weight of the corresponding 0g model and the lower the residual error with 0g targets at the end of practice.
We tested these two hypotheses by comparing five different protocols with different probabilities of 0g trials in any one session. Figure 8A plots the mean values of cross-time over the last four repetitions of the same 0g target in sessions with 50% probability (day 1 P50), 91% probability (day 1 P91, day 2 P9, and day 2P0), and 100% probability (day 2 P100). Contrary to the Bayesian prediction, there was no significant correlation (r2 = 0.15, P = 0.52) between these cross-times (reflecting the error ) and the corresponding probability of 0g trials during that session. This negative finding provides further evidence against the hypothesis of adaptation of as a strategy for dealing with 0g trials.
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with 0g targets was significantly (r2 = 0.91, P < 0.02) related to the probability of 1g trials on the first experimental day (Fig. 8B): the higher the previous exposure to 1g targets, the better the steady-state performance with 0g targets (the lower the residual error ). Obviously, the ranking co-varied in the opposite direction with the probability of 0g trials on the first experimental day, with the apparent paradox that the steady-state performance with 0g targets was worse in the case of a higher previous exposure to 0g targets.
Adaptation of in the 1g model
The bulk of the evidence presented above is against the adaptation of and in favor of the adaptation of . Figure 9 shows the changes of the estimated values as a function of repetition, plotted separately for all protocols. As predicted by the hypothesis of 1g model adaptation, values tended to decrease monotonically with 0g repetitions. Exponential fitting of the changes of with repetition showed a rapid decrement from initial values of around 200 ms to steady-state values that varied as a function of protocol, according to the same ranking noted in the previous section. Thus the higher the previous exposure to 1g targets, the greater the extent of adaptation (corresponding to lower steady-state values). Steady-state values were 143, 137, 96, 104, and 78 ms for P100 (0% of 1g targets on day 1), P91 (9%), P50 (50%), P9 (91%), and P0 (100%), respectively.
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TOPABSTRACTINTRODUCTIONMODELINGMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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The 1g model belongs to the class of threshold-based TTC models (Lee et al. 1983; Port et al. 1997; Tresilian 1999, 2004). It predicts that the interceptive movement is centrally triggered when TTC reaches a given threshold value , equivalent to a time margin before the expected collision time. The control of interception is under on-line visual control up to , by comparing the visually derived estimate of target TTC with the preset threshold value. The threshold value is specified by information about target distance and velocity, under the assumption that the target is accelerated by gravity. At time before the expected collision, the action is launched with little (if any) correction based on feedback. We showed that the punching movements of the present experiments were compatible with the 1g model. Indeed, all movements started from a fixed initial position and involved brief and stereotyped acceleration waveforms. It should be noted, however, that although we found no evidence for modifications of the interceptive action once initiated, we cannot exclude subtle on-line changes that may have gone undetected in our analysis. Thus the arguments we develop about adaptation of the 1g model strictly apply to the trigger of the movement up to the time of maximum momentum (corresponding to the measured cross-time of hand acceleration), and we make no claim about the control of other aspects of the action. In principle, a modified version of the model should be applicable to other types of interceptive actions, such as those that involve slower responses developing under more variable conditions than those of the present experiments (such as variable starting position of the limb, movement duration, or final position of the target). These latter types of interceptive actions presumably rely on visual feedback until the end of the movement (Bootsma et al. 1997; Brenner et al. 1998; Dessing et al. 2002). In particular, the RRVITE (relative and required velocity integration to endpoint) model predicts that TTC information is combined with gaze-centered velocity and position information about both the target and the hand to provide continuous control of interception of slow targets (Dessing et al. 2005). As hypothesized by these authors, prior knowledge about gravity might be a direct input to the TTC stream. Thus the 1g model could be coupled with the RRVITE model for the control of slower, more variable interceptions than the present ones.
A different model has been proposed that may also account for gravity effects: the 
-guide model (Georgopoulos 2002; Grealy et al. 2004; Lee 1998). This model assumes that an action is coupled to an intrinsic 2nd-order guide akin to gravity. It departs from the original -model that ignores accelerations (Lee et al. 1983) and formally resembles the 1g model. However, the current version of the -guide model does not take into account signal transmission delays (see Brouwer et al. 2003) and thus it should be modified to account for the present results showing adaptable delays. Still another model has been proposed by Tresilian (1993, 1999): this model assumes that, for short heights of fall, TTC can be estimated by using internalized gravity and a perceptual estimate of drop height (H) by solving the equation TTC = (2H/g)0.5. Also this model fails to account for adaptable delays. Moreover, it requires that subjects are able to see the target starting from rest. It cannot predict TTC when the target first appears in view with an unpredictable initial velocity, as was the case for the targets used in the present experiments.
Adaptation of the 1g model
The hypothesis that subjects apply the 1g model predicts that they should correctly time the interception of 1g targets, but that they should make systematic timing errors with 0g targets. In other contexts, it has been shown that practice of arm movements in a novel environment leads to formation of the appropriate internal model (Flanagan and Wing 1997; Gandolfo et al. 1996; Gordon et al. 1993; Imamizu et al. 2003; Wolpert and Kawato 1998). Here, we considered the possibility that training with 0g targets in our experimental context might result in the development of a 0g model appropriate for these targets. No evidence for this 0g model was found, even in an immersive protocol with prolonged exposure to 0g targets over two sessions performed one-day apart (day 1 and day 2 of P100). This is striking given that: 1) human vision is much more sensitive to motions at constant speed than to accelerated motions (Brouwer et al. 2002; Werkhoven et al. 1992); 2) a 0g model is equivalent to a 1st-order estimate of target TTC based on visual information about the ratio between target distance and velocity, an estimate that has been shown to be used in several other tasks that do not involve a falling mass (Merchant et al. 2003; Port et al. 1997; Regan and Hamstra 1993; Rushton and Wann 1999; Tresilian 1999); and 3) it was previously shown that, when 0g visual targets projected in front of the subjects under conditions identical to those used here are intercepted virtually by clicking a mouse button in the absence of the real ball, the responses are correctly timed in accordance with a 0g model (see Fig. 12 in Zago et al. 2004).
In the present experiments, the failure to develop a 0g model for 0g targets occurs in the face of potentially conflicting information about the physical laws of motion. Vision should have signaled to the subjects that the visible target descended at constant speed. However, previous experience with other falling objects should have informed the subjects that the hidden real ball to be punched was acted on by gravity. Moreover, body graviceptors attested to the presence of the Earth's gravitational field. Overall, subjects appeared to rely on an internal model of the physical world in which a downward moving target should accelerate. Remarkably, the same behavior occurs even in the microgravity conditions of orbital flight, when all body sensors (including the graviceptors) and cognitive information concur in signaling the lack of measurable gravity. Thus astronauts trigger muscle responses earlier with respect to impact when catching a ball launched "downward" at constant speed, as if they continued to anticipate the effects of gravity (McIntyre et al. 2001). The responses were well accounted for by the 1g model over three different sessions, performed on days 3, 9, and 15 of flight, with limited adaptation over time.
Therefore both the present experiments as well as the previous microgravity experiments indicate that the a priori of Earth gravity for objects moving along the vertical is highly resistant to changes, despite strong, compelling evidence that a specific visual target is not affected by gravity, and despite the feedback about performance errors repeated trial after trial. In the present experiments, visual and somatosensory error feedback was generated when the falling ball hit the forearm instead of the hand, as a result of premature timing of the movement toward 0g targets. Here we were able to show that, not only was there no evidence for the development of a 0g model, but also there was no evidence for adaptation of the internalized estimate of target acceleration () to values intermediate between 1g and 0g when subjects were concurrently exposed to both sets of targets. When these targets are randomly intermixed with equal probability (protocol P50), adaptation of to values intermediate between 1g and 0g would guarantee a globally adequate performance because the error with 0g targets would be drastically reduced at the expense of a modest deterioration of the responses to 1g targets. However, in this protocol we failed to observe the predicted progressive deterioration of the responses to 1g targets in parallel with the improvement of the responses to 0g targets (Fig. 6). In fact, the interception score for 1g targets showed the opposite trend, tending to improve with repetition (Fig. 7).
The present results also rule out the possibility that subjects learned to use a combination of two distinct internal models, the original 1g model and a new 0g (or fractional g) model, the relative weight of each one changing as a function of the context (Davidson and Wolpert 2004). If this were the case, one would expect that the higher the probability of
50%) trials are presented, except for the 0g trials of day 1 of P9, which are plotted adjoined to those of day 2. Thick lines correspond to exponential fits (P < 0.05). Learning constants are 1.4, 1.0, 0.9, 1.4, and 2.1 repetitions for P100, P91, P50, P9, and P0, respectively.