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Internal Models of Target Motion: Expected Dynamics Overrides Measured Kinematics in Timing Manual Interceptions

¹ Sezione di Fisiologia umana, IRCCS Fondazione Santa Lucia, 00179 Rome; ² Dipartimento di Neuroscienze and Centro di Biomedicina Spaziale, Università di Roma Tor Vergata, 00133 Rome, Italy

Submitted 3 September 2003; accepted in final form 18 November 2003

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Prevailing views on how we time the interception of a moving object assume that the visual inputs are informationally sufficient to estimate the time-to-contact from the object's kinematics. Here we present evidence in favor of a different view: the brain makes the best estimate about target motion based on measured kinematics and an a priori guess about the causes of motion. According to this theory, a predictive model is used to extrapolate time-to-contact from expected dynamics (kinetics). We projected a virtual target moving vertically downward on a wide screen with different randomized laws of motion. In the first series of experiments, subjects were asked to intercept this target by punching a real ball that fell hidden behind the screen and arrived in synchrony with the visual target. Subjects systematically timed their motor responses consistent with the assumption of gravity effects on an object's mass, even when the visual target did not accelerate. With training, the gravity model was not switched off but adapted to nonaccelerating targets by shifting the time of motor activation. In the second series of experiments, there was no real ball falling behind the screen. Instead the subjects were required to intercept the visual target by clicking a mousebutton. In this case, subjects timed their responses consistent with the assumption of uniform motion in the absence of forces, even when the target actually accelerated. Overall, the results are in accord with the theory that motor responses evoked by visual kinematics are modulated by a prior of the target dynamics. The prior appears surprisingly resistant to modifications based on performance errors.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Interception requires precise timing, whether to catch or hit a ball, to jump on land or dive in water. How do we estimate time-to-contact (TTC) with an approaching object? Most extant theories posit that all relevant information is present in the visual stimulus without resorting to high-level computations or internal representations (Gibson 1966; Lee and Reddish 1981; Regan and Gray 2000; Tresilian 1999). However, there are limitations in the visual system that raise questions about the general validity of these theories (Lacquaniti et al. 1993a). Although several monocular and binocular cues can contribute to TTC estimates, they generally provide only first-order approximations related to the object's distance and velocity (Lee and Reddish 1981; Regan and Gray 2000; Rushton and Wann 1999; Tresilian 1995). In fact neurophysiology shows that cortical neurons specialized for visual motion processing accurately encode target velocity and direction, but contain only partial information about acceleration (Perrone and Thiele 2001). Moreover, psychophysics indicates that the human visual system poorly estimates image accelerations over short viewing periods (Bozzi 1959; Brouwer et al. 2002; Regan et al. 1986; Runeson 1975; Werkhoven et al. 1992). Accordingly, at the motor level arbitrary accelerations generally are not taken into account in timing interceptions (Bootsma et al. 1997; Engel and Soechting 2000; Gottsdanker et al. 1961; Lee et al. 1997; Port et al. 1997). Another problem is that, because of sensorimotor delays, interceptive actions must be anticipatory, based on information available 100 –200 ms before contact. Extrapolation of visual motion information into the future can lead to severe spatial and temporal misjudgments (Tresilian 1995).

To overcome these problems, it has been hypothesized that on-line visual information is combined with an a priori representation (called reference model) of target dynamics that depends on the context of the specific visuomotor task (Lacquaniti et al. 1993a). In other words, the brain could make the best guess about visible or hidden causes of object's motion at any time, and use a predictive model to extrapolate TTC from expected dynamics (kinetics) rather than from measured kinematics. In Newtonian dynamics a free particle moves with constant velocity, unless acted on by an accelerative force. If it is difficult to discriminate arbitrary accelerations visually, gravitational acceleration might be internalized based on lifelong exposure (Hubbard 1995; Lacquaniti et al. 1993a; Tresilian 1993; von Hofsten and Lee 1994; Watson et al. 1992). Visual signals about target position and velocity could be combined with an internal estimate of earth's gravity, yielding a second-order dynamic model of TTC for interception of objects whose motion is expected to be affected by gravity (Lacquaniti and Maioli 1989; Lacquaniti et al. 1993a; McIntyre et al. 2001). In a similar vein, it has been suggested that targeted movements are coupled to an intrinsic second-order guide similar to gravity (Lee 1998). The idea that an internal model of gravity can be used to disambiguate sensory information has also been developed in the field of vestibular physiology. Vestibular otoliths (as all linear accelerometers) measure gravitoinertial forces, but cannot distinguish linear accelerations from gravity. Based on eye movement responses to combinations of tilts and rotations, it has been proposed that the brain combines canal signals with otolith signals and uses an internal model of gravity to estimate linear accelerations and disambiguate tilt from translation of the head (Hess and Angelaki 1999; Merfeld et al. 1999).

Here we show that the timing of responses aimed at intercepting a visual target is very different depending on the dynamic context. Virtual targets moving vertically downward with different laws of motion were intercepted with a timing consistent with the assumption of uniform motion in the absence of forces. In contrast, when the same virtual targets were used to punch a hidden real ball arriving in synchrony, response timing was consistent with the assumption of gravity effects on an object's mass. With training, the gravity model was not switched off, but adapted to nonaccelerating targets by shifting the time for motor activation. The results indicate that responses evoked by low-level visual inputs change with the context established by dynamic predictive models from higher-level processing.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Subjects

In total 20 healthy subjects (12 women and 8 men, 29 ± 5 yr old, mean ± SD) participated in the study receiving modest monetary compensation. All but one subject participated in one experiment only. 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 (except subject G.B.). Experiments were approved by the Institutional Review Board and conformed to the Declaration of Helsinki on the use of human subjects in research.

Apparatus

First we describe the setup common to both the punching and the button-press experiments. Subjects sat on a chair placed in front of a wide vertical screen attached to the ceiling (Fig. 1). The screen (Video Spectra 1.5-gain) had a pearlescent frontal projection surface (3.94 m wide, 2.13 m high). The rear part of the screen was made of black, nontransparent material, preventing the view of any object through it. The back of the chair supported the head and torso of the subjects without preventing their motion. The vertical inclination of the back was adjusted at about 40° so as to allow a comfortable view of the screen with the subjects' eyes located at a horizontal distance of 0.5 m from the screen, 1.82 m below the top. 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 9-cm-wide box was constantly displayed at the top of the screen against a white background. In each trial a red 9-cm-diameter target sphere moved vertically downward; it was displayed initially as emerging progressively from within the start box at a predefined speed and acceleration (see Experimental procedures and protocols) and then shifted with the prescribed law of motion to finally disappear progressively at the lower border of the screen. The visual angle subtended by the target increased from 0.8°, when it was first fully visible, to 10.3° when it passed at eye level. The rest of the room was dimly illuminated, the only light sources being the BARCO and a PC monitor. Next we describe the setup specific for each set of experiments.

View larger version (46K): [in this window] [in a new window]   FIG. 1. Experimental setup. Subjects sat on a chair facing a wide vertical screen. They were asked to intercept a virtual target sphere moving vertically downward on the screen when it arrived at the bottom end of the screen. In the punching experiments, they did so by punching a real soft ball (of the same size as the virtual one), hidden behind the screen, that arrived in synchrony with the virtual target. In the button-press experiments, there was no real ball falling behind the screen. Instead subjects had to "explode" the virtual target by clicking a mouse button.

BUTTON-PRESS EXPERIMENTS. Here there was no real ball falling behind the screen. Subjects sat on the chair as before, grabbing a computer mouse in the right hand. To provide visual feedback of late interceptions, the screen surface was prolonged downward with a strip 0.9 m long and 0.33 m wide. A black start box and a blue end box were constantly displayed at the top and bottom of the regular screen, respectively, the red target-sphere moving downward from the start to the end box.

Recording system

PUNCHING EXPERIMENTS. 3D motion of selected body points was recorded at 200 Hz by means of the Optotrak 3020 system (Northern Digital, Waterloo, Ontario; ±3SD accuracy better than 0.2 mm for x, y, z coordinates). Three infrared emitting markers were attached to the skin overlying the shoulder, elbow, and wrist joint on the right arm. Three additional markers were fixed to the lower border of the screen to determine the screen plane. A miniature accelerometer (Isotron Endevco), fixed on the wrist, was sampled at 1 kHz. Electromyographic (EMG) activity of the clavicular portions of pectoralis (PE) and trapezius (TP), anterior deltoid (AD), posterior deltoid (PD), biceps (BC), triceps (TR), and flexor carpi radialis (FC) was recorded by means of surface electrodes (SensorMedics 650414; diameter of detection surface, 2 mm) in bipolar configuration (interelectrode distance, 1 cm). EMG signals were preamplified at the recording site using a pair of matched FET operational amplifiers to reduce noise, and then differentially amplified (120-dB CMRR), high-pass filtered (20 Hz), low-pass filtered (200-Hz, 4-pole Bessel), and sampled at 1 kHz. Sampling of kinematic and EMG data was synchronized with trial start.

BUTTON-PRESS EXPERIMENTS. The analog voltage signal from the mouse button was A/D converted and sampled at 1 kHz synchronized with trial start.

Experimental procedures and protocols

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 informed that in each trial a visual target sphere would move vertically downward from the start box at different randomized speeds.

PUNCHING EXPERIMENTS. Subjects were asked to intercept the target just below the lower border of the screen by punching the real soft ball that would arrive there at the same time as the visual target. They kept a free relaxed posture between trials. Before trial initiation, an alert signal instructed the subjects 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, the hand and fingers clenched in a fist. At the beginning of the trial, an audible attention signal sounded and, after a random delay ranging from 1.2 to 1.7 s, the virtual sphere exited from the black box. The movement required to punch the ball involved the forward protraction of the hand by about 20 cm, along a roughly straight-ahead, horizontal direction. Data were acquired for 5 s, starting 0.4 s before sphere exit. The intertrial interval was about 14 s. Total duration of each session was about 1 h 20 min, with 10 min of rest allowed halfway.

In any given trial the virtual sphere moved from the starting box with randomly assorted initial speeds (v₀ = 0.7, 1, 1.5, 2.5, or 4.5 m s^–1) at a constant acceleration (either A = 9.81 ms^–2 = 1g or A = 0 = 0g). The randomization procedure always avoided identical combinations of v₀ and A in consecutive trials. Subjects were assigned to one of 3 groups (with 5 subjects in each group). In each group, subjects were exposed to identical sequences of trials.

Subjects of the first 2 groups performed 2 different experiments 24 h apart. Each experiment consisted of 4 identical blocks of 55 trials. In each block, there were 50 test trials and 5 pseudorandomly interspersed "catch trials" with unexpectedly altered kinematics. For test trials, v₀ was pseudorandomly assigned to one of 5 values, and A was assigned to one fixed value for day 1 experiment (either 0 or 1g depending on the protocol) and the other fixed value on day 2. Thus A = 1g on day 1 and A = 0 on day 2 for group 1 (1g–0g protocol), whereas A = 0 on day 1 and A = 1g on day 2 for group 2 (0g–1g protocol). Each catch trial had the same v₀ as that of the previous test trial (catch 1), but different A (A = 0 if test trial was 1g and A = 1g if test trial was 0g), except for the catch trials of day 1 in group 1 that were all trials with A = 1g, v₀ = 0 m s^–1. Each experiment therefore consisted of 40 repetitions per 5 test conditions, and 4 repetitions per 5 catch conditions. The sequence of target v₀ was identical in day 1 and day 2. The first through the fifth catch trials occurred in the sequence as the 7th, 14th, 27th, 34th, and 54th trials, respectively. Catch trials were used to reveal the presence of aftereffects attributed to adaptation (Shadmehr and Mussa-Ivaldi 1994).

Subjects of group 3 (Random protocol) performed a 1-day experiment involving 200 test trials without catch trials. For each trial, both v₀ and A were randomized (20 repetitions per 10 conditions).

BUTTON-PRESS EXPERIMENTS. Subjects sat in the same posture as for punching, and kept the arm still on an armrest with the computer mouse grabbed in the right hand. Before trial initiation, an alert signal instructed the subjects to look at the box on the top of the screen. At the beginning of the trial, an audible attention signal sounded and, after a random delay ranging from 1.2 to 1.7 s, the virtual sphere exited from the black box. Subjects were asked to click the mouse button with the index finger so as to intercept the descending target just as it reached the center of the end box (interception point). If they intercepted in the allotted time (±15 ms relative to the exact interception time), the target sphere "exploded" blue with a pleasing reward sound. If they intercepted too early or too late, the target sphere was flashed red at the point of incorrect interception. Six subjects performed a 1-day experiment involving only test trials whose v₀ and A were randomized (as in Random above). One subject had participated in a punching experiment (Random protocol) about 1 yr earlier.

General data analysis

PUNCHING EXPERIMENTS. Rectified EMG and raw kinematic data were numerically low-pass filtered (bi-directional, 20-Hz cutoff, 2nd-order Butterworth filter) to eliminate impact artifacts. Instantaneous tangential velocity of the hand was computed by numerically differentiating the recorded x, y, z coordinates of the wrist marker. We measured motor timing based on the time of occurrence of the first positive peak of hand acceleration because this parameter was very reliably derived from the data (see Fig. 3 for examples of acceleration records). The time of this acceleration peak relative to the interception time will be denoted TTC. We also estimated the onset time (the time when the acceleration first reached 10% of the first positive peak), and the time of zero-crossing of hand acceleration. Movement duration was defined as the interval between onset time and zero-crossing time.

View larger version (17K): [in this window] [in a new window]   FIG. 3. Interception of 1g vs. 0g targets in a punching experiment. Time profiles of hand acceleration and rectified EMG activity of the indicated muscles of subject M. (Protocol 1g–0g) for 3 repetitions (1st, 2nd, and 11th, in red, green, and blue, respectively) of accelerated (left) and constant velocity (right) targets. Traces are aligned on interception time (dashed lines) and negative time is before that time.

An exponential function was fitted to the series of repetitions of TTC values for each condition (v₀, A) to characterize the rate at which subjects adapted. The function has 3 free parameters: offset b₀, gain b₁, and time constant b₂

(1)

where TTC_i is the TTC value on repetition i.

BUTTON-PRESS EXPERIMENTS. TTC was measured as the time of the button press relative to the interception time. To calculate interception rate we used 2 different time windows relative to the interception time: ±15 or ± 60 ms. The first window corresponds to the time interval used to provide positive or negative feedback of success to the subjects during the experiments. The second window corresponds to the time interval that was used to compute the interception rate in the punching experiments. For both time windows, the interception rate was derived as the percentage of trials whose TTC was included in the window.

STATISTICS. Differences between conditions were assessed using ANOVA followed by Bonferroni–Dunn tests, and using Wilcoxon signed ranks or t-statistics (P < 0.05, level).

Few trials (0.2% of all trials) were excluded from the analysis attributed to the presence of artifacts or lack of subject's attention during the trial as marked in the experiment notebook.

Models of interception timing

Figure 2 defines the critical timing variables for both punching and button-press experiments. The instantaneous height h(t) of the virtual target above the interception point is given by the standard equation of motion

(2)

where h₀ and v₀ are the initial height and velocity, respectively, and

View larger version (10K): [in this window] [in a new window]   FIG. 2. Timing interceptive actions. Instantaneous height h(t) of a target above the interception point is plotted as a function of time. IT denotes the interception time expected by the subject, not necessarily coinciding with the true interception time. Our models of interception assume that, when a visual or internalized variable has reached a critical threshold value, a motor response is programmed with some lead time (LT) relative to IT. LT depends on neural and mechanical delays. LT = PT + MT, where PT and MT represent the processing time and the movement time, respectively. RT denotes the response time (time of occurrence of the motor response relative to trial onset). TT = IT – LT. h0 and v0, hTT and vTT, hRT, and vRT denote h(t) and v(t) at t = 0, t = TT, and t = RT, respectively.

In the following, we provide an expression for the time of the motor response from trial onset predicted by each model (RT*, asterisk denotes prediction) as a function of the initial conditions of target motion (h₀, v₀, A) and of the model parameters (PT, and h_TT or or , depending on the model; see following text). In addition, we provide an expression for predicted by each model as a function of . For these models, is linearly related to , with an intercept (c₀) and slope (c₁) that differ according to the model and the initial conditions, as follows

(3)

All expressions are derived from Eq. 2 in a straightforward manner.

DISTANCE MODEL. It postulates that the response is programmed when the target has reached a certain threshold height h_TT at t = TT (van Donkelaar et al. 1992; Wann 1996). The distance model provides a zero-order approximation of target motion because it incorporates information about current distance of the target, but ignores velocity and acceleration.

In the case of 0g motion

(4)

and c₀ = h_TT and c₁ = –PT in Eq. 3.

In the case of 1g motion

(5)

and c₀ = h_TT + 0.5gPT² and c₁ = –PT.

-MODEL. It postulates that the response is programmed when IT – TT has decreased below a certain threshold time (Lee and Reddish 1981; Lee et al. 1983). By definition

(6)

The -model provides a first-order approximation of target motion because it incorporates information about target distance and velocity, but ignores acceleration. Thus it represents a 0g model.

In the case of 0g motion

(7)

and c₀ = 0 and c₁ = ( – PT).

In the case of 1g motion

(8)

and c₀ = (0.5PT – t)gPT and c₁ = ( – PT).

1g MODEL. It postulates that the response is programmed when IT – TT has decreased below a certain threshold time (Lacquaniti and Maioli 1989; Lacquaniti et al. 1993a; McIntyre et al. 2001). By definition

(9)

where is an internalized estimate of gravitational acceleration. In the following we assume that = 1g (the acceleration of earth's gravity at sea level). The -model provides a second-order approximation of target motion, because it incorporates information about target distance and velocity, and always assumes that the target is accelerated by gravity. Thus it represents a 1g model.

In the case of 0g motion

(10)

and c₀ = 0.5g² and c₁ = ( – PT).

In the case of 1g motion

(11)

and c₀ = 0.5g( – PT)² and c₁ = ( – PT).

The experimental values of RT (either the times of the positive peak of hand acceleration or the times of the button press relative to trial onset) were (least-squares) fitted using Eqs. 4–11 to obtain estimates of the model parameters (PT, and h_TT or or ). To assess the robustness of the estimates, we followed different fitting procedures. First, the results for 1g targets and 0g targets were fitted simultaneously, using the global model obtained by coupling Eqs. 4 and 5 for the distance model, Eqs. 7 and 8 for the -model, and Eqs. 10 and 11 for the -model. Second, the results for 1g targets were fitted separately from those of 0g targets. This procedure could be used for both 1g and 0g targets in the case of the distance model, only 1g targets for the -model, and only 0g targets for the -model.

An additional analysis was performed for the punching experiments based on the height–velocity (h–v) phase plots introduced by Port et al. (1997). Here the experimental values of h_RT were linearly regressed against the corresponding values of v_RT using Eq. 3 to obtain estimates of the intercept (c₀) and slope (c₁). The results for 1g targets were fitted separately from those for 0g targets. Several different temporal landmarks were used for RT, based on the time when hand acceleration first reached a given percentage level (10, 25, 50, 75, and 100%) of its positive peak. This analysis allows discrimination among the various models based on the quadrant where the intercept–slope values (determined from the experimental data) fall.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Intercepting 0g versus 1g targets by punching a ball

In all experiments a virtual target sphere moved vertically downward on the projection screen with a randomized law of motion. In the first series of experiments, subjects were asked to intercept this target by punching a real soft ball (of the same size as the virtual one) that fell hidden behind the screen. Because the virtual target arrived at the interception point at the same time (interception time) as the real ball, subjects had to estimate the TTC of the former to punch the latter correctly.

On the first day, one group (group 1) of subjects was presented with visual targets moving with 1g acceleration and randomly assorted initial velocities (v₀). All subjects punched the ball by propelling the hand forward with a ballistic, stereotypical movement. The responses were correctly time-locked to the interception time from the first attempt and varied little in successive trials. The 1st, 2nd, and 11th presentations of the same condition from one such experiment are superimposed in Fig. 3 (left, in red, green, and blue, respectively). Acceleration and EMG profiles are essentially indistinguishable among these repetitions.

On average, hand acceleration reached the first positive peak at –64 ± 31 ms (mean ± SD across all 1g test trials of 5 subjects, n = 997). Negative (positive) values indicate times before (after) the interception time. The zero-crossing of hand acceleration occurred at –2 ± 24 ms, indicating that subjects generated maximum momentum to punch the incoming ball at the right time. Limb propulsion was determined by a burst of muscle activity of shoulder flexors and elbow extensors, and it was subsequently braked by shoulder extensors and elbow flexors.

The same group of subjects was trained 24 h later with targets at 0g acceleration and the same randomly assorted v₀ as the previous day. Limb movement and arm muscles activation were similar to those of the previous day, but their timing was very different (Fig. 3, right). When subjects were first exposed to each 0g condition, their responses started too early and the hand passed the interception point well before the interception time. On average, hand acceleration reached the first positive peak at –158 ± 83 ms (mean ± SD across the first presentations of each 0g test trials of 5 subjects, n = 25), 94 ms earlier than for the 1g targets of the previous day. The zero-crossing of acceleration occurred at –93 ± 78 ms. As a measure of the spatial error, we computed the difference between the anteroposterior horizontal spatial coordinate of the hand at the interception time and the corresponding mean value for the 1g targets of the previous day. On average, in the first 0g trial the hand was 7.3 ± 6.6 cm more anterior than in the 1g trials. Consequently, it was hit by the ball at the metacarpus or carpus. With training, the early inappropriate responses were progressively delayed (compare the 1st, 2nd, and 11th presentations in Fig. 3, right), but they remained premature as compared with 1g responses.

The timing but not the waveform of hand acceleration depended on the law of target motion. All presentations of the same condition from one experiment are superimposed in Fig. 4. Acceleration profiles for 1g trials are essentially superimposable for most trials. The positive component of the acceleration profiles for 0g trials is very similar to that of 1g trials, staggered in time as a function of presentation order. The negative component (braking the movement) tends to be slightly greater for 0g trials than that for 1g trials. Several spatiotemporal landmarks of the motor responses systematically covaried across trials and conditions. Thus the time of onset, first positive peak, and zero-crossing of hand acceleration were always strongly correlated (P < 0.001) between each other. Moreover, movement duration varied little across all conditions (on average, its SD was 32 ± 13 ms). In the following, we mainly concentrate on the TTC of the acceleration peak, considered as the motor correlate of the time-tocontact of the target estimated by the subject.

View larger version (17K): [in this window] [in a new window]   FIG. 4. Time profiles of hand acceleration for all 40 repetitions of accelerated (top row) and constant velocity (bottom row) targets from the same subject of Fig. 3. Traces are aligned on interception time (dashed lines) and negative time is before that time. Each column corresponds to the indicated initial velocity.

View larger version (12K): [in this window] [in a new window]   FIG. 5. Effects of training in the Random protocol of punching experiments. Time to contact (TTC) of peak positive acceleration (time relative to interception time) was averaged across all subjects of group 3. Mean TTC values (±SE, n = 5) are plotted as a function of repetition number for each initial velocity (red and blue points correspond to 1g and 0g trials, respectively). Dark lines through 0g trials correspond to the exponential best fit, whereas dark lines through 1g trials represent their mean value over all repetitions.

View larger version (35K): [in this window] [in a new window]   FIG. 6. Ensemble averages and best-fitting (least-squares) surfaces. TTC values were averaged across all subjects and experiments for each repetition to a given target.

The TTC values of 0g test trials were significantly (t-test, P < 0.001) longer than those of 1g test trials: on average, by 114 ± 23 ms (means ± SE across all v₀ conditions and subjects) in the first repetition and by 56 ± 9 ms over the last 5 repetitions. Two-factor ANOVAs showed that TTC of 0g trials shortened (the time of acceleration peak shifted closer to contact) significantly with both increasing v₀ (P < 0.001) and repetition (P < 0.001) in all groups. These changes were substantial, resulting in a mean range of variation of 237 ± 44 ms. In general, TTC shortened rapidly with repetition (within the first 4–5 repetitions) in the conditions with low v₀ (0.7, 1 m s^–1). Exponential fitting (Eq. 1) of these changes yielded a time constant of 1.8 ± 0.6 (mean ± SE).

We can rule out the hypothesis that subjects synchronized their movements to the start of target motion by assuming that the real ball was dropped at the same time as the virtual one. This hypothesis would predict a constant time of the motor response (RT) relative to trial onset. This hypothesis was rejected because 3-factor ANOVAs showed a highly significant effect of acceleration (P < 0.001), velocity (P < 0.001), and repetition (P < 0.005) in all groups.

In contrast with the clear dependency of the timing (TTC) of arm movements on the law of target motion, their maximum speed showed no systematic relation with target motion. Thus peak tangential velocity of the hand (v_P, occurring close to nominal interception time; see above) did not depend significantly on target acceleration or v₀ in 2 groups, whereas it was slightly but significantly (P < 0.001) higher for 0g targets than for 1g targets in group 1 (on average, v_P = 3.54 ± 0.96 and 3.25 ± 0.61 m s^–1 for 0g and 1g targets, respectively; mean ± SD, n = 1,000). However, the overall range of variation of v_P across conditions (v₀ and acceleration changes) was always <13% of the mean v_P.

Consistent with the accurate timing, the interception rate of 1g trials was high from the outset and improved only slightly with practice. On average, it was 80 ± 16% (mean ± SD) in the first repetition and 92 ± 10% in the last 5 repetitions of each condition for all subjects and groups. Instead, the interception rate of 0g trials increased substantially with practice but was always significantly (t-test, P < 0.001) smaller than that of 1g trials. On average, it was 20 ± 10% in the first 0g repetition and 59 ± 13% in the last five 0g repetitions.

Aftereffects

Occasional (9% incidence) trials with unexpectedly altered kinematics, termed "catch trials," were interspersed among the test trials of protocols 1 and 2 to verify for the presence of aftereffects attributed to adaptation. The 1g catch trials with v₀ = 0 m s^–1 unexpectedly presented during immersive training with 1g test trials with random v₀ (day 1 in group 1) did not cause aftereffects. TTC of these catch trials did not differ significantly from that of the preceding trial.

In contrast, the unexpected occurrence of 1g catch trials during immersive 0g training did cause significant (P < 0.001) aftereffects in groups 1 and 2 (Fig. 7). The size of the aftereffects did not change significantly as a function of the time of occurrence during 0g training, probably because of the fast time constant of adaptation (the first 1g catch trial occurred in the sequence after most adaptation had occurred). Two-factor ANOVAs showed no significant effect of repetition in any group and a slight significant effect of v₀ (P = 0.003) in group 1 but not in group 2. When present, the effect of v₀ was small (on average, 6 ms TTC change per 1 m s^–1 velocity change). On average, the interception rate for these 1g catch trials was 37 ± 21%, significantly (t-test, P < 0.001) lower than the interception rate of 1g test trials.

View larger version (20K): [in this window] [in a new window]   FIG. 7. Aftereffects of 0g training. Mean (± SE, n = 5) TTC values for 1g catch trials are plotted for groups 1 and 2. Dashed line represents the average value of TTC for 1g test trials presented in the other day of training.

Modeling the data

In theory, subjects might time their responses on the basis of different kinds of TTC estimate of the visual target: exact estimates or approximations of different order. Exact estimates would lead to a correct motor timing for both 1g and 0g targets. Zero-order approximations of the distance model incorporate information about target distance, but ignore target velocity and acceleration. The application of this model would overestimate the TTC of targets with shorter flight duration and underestimate the TTC of targets with longer flight duration. First-order approximations of the -model incorporate information about target distance and velocity, but ignore target acceleration. Because the -model is equivalent to a 0g model, it would lead to correct TTC estimates for 0g targets, but would overestimate TTC of 1g targets. Finally, second-order approximations based on an internalized estimate of gravitational acceleration incorporate information about target distance and velocity, and always assume that the target is accelerated by gravity. Therefore the 1g model correctly estimates TTC of 1g targets, but underestimates TTC of 0g targets.

The finding that the motor timing was premature and the success rate was low in 0g test trial compared with that of 1g test trials indicates that the TTC of 0g targets was underestimated, whereas the TTC of 1g targets was estimated correctly. This finding apparently refutes both exact as well as zero-order and first-order mechanisms and is compatible instead with a second-order mechanism based on the 1g model. However, in line of principle a difference in timing of the motor responses to either the 0g targets or the 1g targets could result from the application of a low-order TTC estimate but with an offset of the motor responses. For instance, subjects could program the response when the target has traveled a given distance but the response would occur only after a given delay attributed to neural and mechanical transmission times. The consequences of a delay are not intuitive and need to be assessed quantitatively.

To account for transmission delays, we fitted Eqs. 4–11 to the experimental RT values and obtained the values of RT* and TTC* (= flight duration – RT*) predicted by each model. In a first analysis, the results for all 1g targets and for 0g targets were fitted simultaneously, under the assumption that the model parameters are invariant across all tested initial conditions. Moreover, a lower boundary value of 100 ms was imposed on PT, , and in the fitting procedure, to avoid unrealistic results (minimum delays for visuomanual responses are of that order of magnitude; Carlton 1981). The values of TTC* are plotted as a function of v₀, along with the mean experimental values of TTC for the first repetition of each condition in group 2, in Fig. 8. The results were essentially identical for all groups.

View larger version (9K): [in this window] [in a new window]   FIG. 8. Predictions of different models for the punching experiments. Dots: mean (n = 5) experimental TTC values for the first repetition of each target in group 2. Thick, thin, and dash-dotted curves: TTC* values predicted by the 1g model, the distance model, and the -model, respectively.

The distance model (thin curve) reproduced the experimental data for the lower speed (v₀ 1.5 m s^–1) 0g targets, but grossly overestimated (on average, by 106 ms) both the values of TTC experimentally obtained for higher speed 0g targets, as well as the values for all 1g targets. The best fit was obtained with a PT of 100 ms and a threshold height (h_TT) of 0.39 m. The linear regression of all predicted TTC* values versus the corresponding experimental TTC values was significant (mean r² = 0.86 across groups), but the slope (1.5) and the intercept (133 ms) were significantly (P < 0.05) greater than unity and zero, respectively.

The -model (dash-dotted curve) reproduced the experimental TTC values for 1g targets but did not reproduce those for 0g targets (mean r² = 0.47 across groups). The best fit was obtained with a processing time (PT) of 267 ms and a threshold time () of 476 ms.

Next, we fitted the 1g targets separately from the 0g targets. The results did not change markedly with the 1g model (mean r² = 0.76, PT = 157 ms, = 208 ms). The distance model reproduced the mean TTC of 1g data better than before, but overestimated or underestimated the TTC of targets with shorter or longer flight durations, respectively (mean r = –0.31); its fit of 0g data did not change appreciably (mean r² = 0.76, PT = 100 ms, h_TT = 0.36). The -model fitted the data separately worse than before (mean r² = 0.08).

Then we removed the boundary values in the fitting procedure. The results for both the 1g model and the -model remained identical to those obtained under constrained fitting. Instead, the unconstrained best-fitting of the distance model yielded negative values incompatible with a causal system (PT = –121 ms and h_TT = –0.40 m).

The second analysis is related to the h–v phase plots introduced by Port et al. (1997). The models we are considering predict that the height of the target above the interception point at a given time after trial onset is linearly related to the corresponding value of target velocity, for all v₀ at a fixed acceleration (either 1g or 0g). However, the intercept and slope of the h–v relationship differ according to the model and the initial conditions (see Models of interception timing in METHODS). Figure 9 shows the results of simulating the performance of the different models. Each curve in the figure corresponds to the locus of all pairs of intercept–slope values obtained by varying model parameters (see figure legend for details). Red and blue curves correspond to 1g and 0g targets, respectively. This graphic analysis allows discrimination among the various models on the basis of the quadrant where the intercept–slope values fall.

View larger version (38K): [in this window] [in a new window]   FIG. 9. Analysis of height–velocity (h–v) relationships for different responses times. Models predict a linear relationship between the height and the velocity of the target at a given time. Each curve corresponds to the locus of all pairs of intercept–slope values obtained by varying model parameters in the following ranges: 0.1 < hTT < 1.6 m for the distance model, 100 < < 300 ms for the -model, and 100 < < 300 ms for the 1g model. For all models, 0 < PT < 300 ms. Red and blue curves correspond to 1g and 0g targets, respectively. Intercept–slope values predicted by the 1g model, distance model, and -model fall in the yellow, orange, and green quadrants, respectively. Black circles in the yellow quadrant correspond to the intercept–slope values estimated from the linear regression of the experimental values of hRT vs. the corresponding values of vRT (first repetition of each target in group 2). RT was varied in steps between 100 and 10% of the time of occurrence of the positive peak of hand acceleration (arrows). The azure circle denotes the time when the pairs of intercept–slope values predicted by the 1g model for 0g targets becomes identical to that for 1g targets, and TTC = .

Predicted curves were compared with the experimental data in the following way. The experimental values of h_RT for the first repetition of each condition were linearly regressed against the corresponding values of v_RT using Eq. 2 to obtain estimates of the intercept and slope. Here the response time RT was varied in steps based on the time of occurrence of different percentage levels (10, 25, 50, 75, and 100%) of the positive peak of hand acceleration. This allowed generalization of the results to different temporal landmarks of the response. The results for 1g targets were fitted separately from those for 0g targets.

Experimental values of h_RT generally were linearly related to those of v_RT (mean r = 0.91). The estimated pairs of intercept–slope values (black circles in the yellow quadrant of Fig. 9) consistently satisfied the predictions of the 1g model. Each circle corresponds to a different RT. With decreasing values of RT, the experimental points shift progressively higher (arrows) along the curves predicted by the 1g model. Notice the good agreement between the path described by the experimental points and the theoretical curves.

The 1g model also predicts that the pairs of intercept–slope values for 0g targets becomes identical to that for 1g targets when TTC = (azure circle). Thus the time threshold can be estimated independently from 3 parameters of the model: 1) the TTC at which the intercept–slope curve for 0g targets intersects that for 1g targets, 2) the corresponding intercept value (c₀ = 0.5g²), and 3) the slope (c₁ = ). In the first repetition, the 3 estimates of were 214, 210, and 202 ms, respectively. Notice the good agreement of the different estimates between each other and with those derived above. Training with 0g targets resulted in a systematic decrement of -values. This issue is addressed in the next section.

Adaptation of 1g model to repeated presentations of 0g trials

We previously showed that the responses to 0g trials were premature and that with training the performance improved but there was a consistent residual timing error (Figs. 4 and 5). Here we show that this behavior is compatible with the persistent use of the 1g model throughout practice, with a progressive adaptation of the threshold time .

Application of the 1g model leads to predictable differences in timing between 1g and 0g targets. A 1g target arrives at the time predicted by the 1g model, but a 0g target arrives later than expected (Fig. 10 left). Thus motor activity triggered by the 1g model starts earlier than necessary at 0g. For any given value of , the timing error () between a 0g target and a 1g target decreases nonlinearly with increasing v₀ (Fig. 10 middle). We tested the hypothesis that adaptation reduces by progressively decreasing by fitting the prediction of the 1g model (* = g²/2v₀) to the set of experimental -values.

View larger version (12K): [in this window] [in a new window]   FIG. 10. Adaptation of 1g model to repeated presentations of 0g targets. Left: height of the 0g target above the interception point is plotted vs. time. The 1g model predicts that a motor response is programmed when the time remaining before the estimated interception time falls below a given time threshold . A 0g target arrives later than predicted by the 1g model, and is the timing error between 0g and 1g conditions. The shorter the -value (decreasing from red to green to blue), the later is the response and the smaller the error. Middle: each curve depicts the -values (as a function of v0) predicted by the 3 -values of left panel. Right: experimental results. Mean (n = 5) -values from the 1st, 2nd, and 6th repetition of group 2. Values of were computed as the difference between the TTC of peak acceleration for individual 0g test trials of one day and the mean values for 1g test trials of the other day.

View larger version (12K): [in this window] [in a new window]   FIG. 11. Adaptation of across all repetitions of group 2. Values of computed over all v0 (n = 5) at incremental repetitions of 0g trials. Continuous line is exponential fitting function.

Virtual interception

A striking result of the punching experiments was that, even after prolonged exposure, visual targets moving vertically downward at constant velocity were misrepresented as accelerated by gravity. To verify whether these illusions depend on the nature of the visual stimuli or they depend on the dynamic context of interception, we carried out an additional series of experiments. Here the subjects were presented with targets descending with randomly assorted initial velocities and accelerations as in the Random protocol of punching experiments, but there was no real ball falling behind the screen. Instead the subjects were required to "explode" the virtual target by clicking the mouse at the interception time.

For most conditions, the results were the reverse of those of punching, both at a population level and at an individual level (one subject participated in both sets of experiments 1 yr apart). Thus the responses to slower 0g targets (that were timed too early in punching) were correctly time-locked to interception time from the first attempt and varied little in successive trials in virtual interception (Fig. 12). On average, the TTC of the button press was 0 ± 31 ms (mean ± SD, n = 720) for all 0g targets at v₀ 1.5 m s^–1. Conversely, the responses to slower 1g targets (that were timed correctly in punching) were slightly but significantly (P < 0.001) late in virtual interception. On average, TTC was 21 ± 40 ms (mean ± SD, n = 958) for all 1g targets at v₀ 2.5 m s^–1. The timing differences between 0g and 1g targets at corresponding v₀ were highly significant (P < 0.001). The interception rate computed within ±15 ms relative to the interception time was not significantly different between these 0g targets (39 ± 10%) and 1g targets (40 ± 12%). However, the interception rate computed within the time window (±60 ms) comparable to that used for punching experiments was significantly (t-test, P < 0.001) higher for 0g targets (95 ± 4%) than for 1g targets (84 ± 15%).

View larger version (13K): [in this window] [in a new window]   FIG. 12. Training in the Random protocol of button-press experiments. Mean TTC values (±SE, n = 6) of button press are plotted as a function of repetition number for each initial velocity (red and blue points correspond to 1g and 0g trials, respectively). Black lines through the conditions A = 1g, v0 = 0.7, 4.5 m s–1, A = 0g, v0 = 0.7, 1, 4.5 m s–1 correspond to significant exponential fits, whereas the brown and dark blue lines through the remaining conditions correspond to their mean value over all repetitions for 1g and 0g trials, respectively.

Three-factor ANOVA showed a significant effect of acceleration (P < 0.001) and v₀ (P < 0.001), but no significant effect of repetition (P = 0.772) when all the trials were pooled together. However, the mean changes of TTC with repetition were significantly (P < 0.05) fitted by an exponential (Eq. 1) in a number of individual 0g and 1g conditions (see fitting curves in Fig. 12), indicating the presence of training effects. The changes were of limited amplitude and could involve either a decrement or an increment of TTC with repetition. They were rapid for slower targets (time constant of 1.6 ± 0.9, means ± SE), and more gradual for fast targets (15.3 ± 9.9).

The finding that the TTC of most 0g targets was estimated better than that of most 1g targets indicates the use of a dynamic model that assumes uniform motion in virtual interception. This hypothesis was supported quantitatively by the results of modeling (Fig. 13). The TTC* values predicted by different interception models are plotted as a function of v₀, along with the mean experimental values of TTC using the same analysis previously applied to punching (see Fig. 8). The overall best fit (mean r² = 0.61, P < 0.01) of the ensemble of all TTC data was provided by the -model (dash-dotted curve) with a processing time (PT) and a threshold time () of 161 ms. As expected, this model did not accurately reproduce TTC values corresponding to high initial velocities (v₀ = 3.5 or 2.5 m s^–1 for 1g and 0g trials, respectively). The other 2 models (1g and distance model) failed to fit the ensemble of data (yielding low, nonsignificant correlations).

View larger version (8K): [in this window] [in a new window]   FIG. 13. Predictions of different models for the button-press experiments. Dots: mean (n = 240) experimental TTC values for each target. Same format as in Fig. 8.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

Visual information

Although humans are able to analyze visual motion in general scenes, they can easily be deceived into seeing incorrect motion of simple stimuli (Weiss et al. 2002). We must then address the question of possible perceptual biases arising in the present experiments arising from the nature of the visual stimuli employed. The visual system normally integrates information from multiple sources to judge the relative position and motion of objects. Changes of retinal image size, binocular disparity, optical gap between target and interception point, and eye movements are known to contribute to TTC estimates under different conditions (Regan and Gray 2000; Tresilian 1999). These cues are probably combined to generate more robust estimates, as in the dipole