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INTRODUCTION |
Idiothetic signals (Mittelstaedt and Mittelstaedt 1973
) are the sensory signals generated by the displacement of a subject: optic flow, proprioception, efference copies, and inertial signals. Mittelstaedt and Mittelstaedt (1980
, 1982)
hypothesized the use of these idiothetic signals for spatial orientation in the "path integration" process. Through path integration the position of a moving subject is continuously updated with respect to the starting point, and therefore the subject is able to compute the homing direction at any time. Recent studies in humans (Bloomberg et al. 1991
; Glasauer et al. 1994
; Israël and Berthoz 1989
; Klatzky et al. 1990
; Loomis et al. 1993
; Mittelstaedt and Glasauer 1991
; Thomson 1983
) have shown that subjects can indeed estimate the traveled path solely from self-generated information, i.e., without external signals (visual or acoustic landmarks).
Mittelstaedt and Mittelstaedt (1980)
and Etienne et al. (1988)
found that rodents could correctly home after passive rotations in darkness, but not after passive linear displacements. Although in a more recent study, Mittelstaedt and Glasauer (1991)
showed that linear inertial forces imposed to active homing trajectories are taken into account by rodents, the former results did cast some doubt on the use of linear inertial information in the path integration process. However, passive linear displacement estimation in humans has been studied with a number of different paradigms
through verbal estimates (Guedry and Harris 1963
), saccadic eye movements (Israël and Berthoz 1989
), or button-pushing responses (Israël et al. 1993
; Mittelstaedt and Glasauer 1991
)
and all of these studies showed that the amplitude of passive linear motion can be correctly estimated.
Although formal models of path integration have been proposed (see Benhamou and Séguinot 1995
; Maurer and Séguinot 1995
for critical reviews), with or without distance estimation, the neural mechanisms involved in the process are still to be clarified. Spatial memory plays a key role here inasmuch as an internal coding of the distance and direction of the perceived motion has to be built and stored by the brain.
In a preliminary report (Berthoz et al. 1995
) we provided qualitative evidence about the type of memory encoding of simple whole body passive linear displacements in darkness. Subjects required to reproduce the distance of an imposed passive motion also reproduced its velocity profile. This implies that all the spatiotemporal properties of movement are stored and that reproduction is based on the dynamic comparison of the incoming sensory input with the stored one. However, such a process does not exclude the possibility that static parameters of motion may also be either independently stored or retrieved from spatial memory.
In the present paper, we examine in more detail, with quantitative methods, the performance of the subjects in this task and suggest an additional hypothesis that extends previous theories of path integration.
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METHODS |
Experimental setup
A mobile robot, the Robuter (Robosoft SA, Bayonne, France), with a race car seat fixed on it was used for this experiment (Fig. 1A). In this device, two motor wheels driven by two 300-W DC permanent magnet independent motors ensure propulsion of a 120-kg maximal mass at a maximal linear velocity of 1.2 m/s, with a maximal acceleration of 1 m/s2. Steering is obtained by controlling the relative speed of the two driving wheels. The robot can be controlled by a remote microcomputer (PC) through wireless modems, or by a joystick connected to the robot itself. The joystick controls the robot's linear velocity in steps of 0.05 m/s (robot velocity directly proportional to joystick angle) with a delay of 0.2 s. Such a delay originates from the hardware and software implementation of the joystick mode control of robot motion and is not due to the mechanical inertia of the robot mass. Positioning accuracy and linearity of trajectory is ensured by proportional integral derivative control loops employing optical encoding of position (resolution of 1 mm and 0.01 s) and a trajectory generation and control system operating at 250 Hz. Odometry (position on the X-axis and timing) was recorded by the robot during motion at a 50-Hz sampling rate (Fig. 1B).

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| FIG. 1.
A: subject is seated on robot, with black goggles, headphones, and bite bar in place, and is using joystick. Two modems (1 at top of robot seat and the other close to microcomputer) can also be seen. B: stimulus and response position for trial of 10-m distance and 1-m/s peak velocity in 1st (triangular velocity) condition for 15 subjects. C: stimulus and response velocity for trials in B (derivation after 5-Hz low-pass filtering of position traces). D-F: some examples of stimulus and response velocity in a previous test in which subjects had to reproduce velocity of stimulus.
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The subject was secured with three safety belts onto the seat of the robot (Fig. 1A). The subject's head was restrained by a cushioned support mounted onto the seat to impede head translations and yaw rotations; a bite bar prevented pitch movements. The subjects wore headphones relaying a wideband noise ("pink" noise) to prevent perception of external acoustic cues, and a pair of goggles with blacked-out lenses to suppress visual information.
The joystick, which subjects held in the hands, was set (software configuration) for the whole experiment so as to allow only linear movements of the robot, forward or backward along the X-axis. All stimuli delivered by the PC were linear displacements forward along the X-axis (as in natural locomotion). The experiment was performed within a corridor 1.9 m wide and 50 m long.
In a preliminary experiment (Georges-François and Israël, unpublished observations) we checked whether the subjects could satisfactorily control the robot by manipulating the joystick. We asked five naive subjects to reproduce, while blindfolded, the velocity profile of a linear passive transport to which subjects had been submitted. Some recordings from this experiment are shown in Fig. 1, D-F. By the use of different velocity profiles it was observed qualitatively that the task was apparently rather simple for the subjects. In some cases the reproduction was strikingly accurate (Fig. 1, D-F). We emphasize that this control test was performed with subjects other than those included in the present study.
Experimental procedure
Fifteen healthy volunteers, with ages ranging from 20 to 50 yr and with no history of vestibular disorder, gave their informed consent to take part in the experiment, which was approved by the local ethical committee.
The subject first learned to manipulate the joystick by driving the robot freely in the corridor, with visual and auditory cues available. After ~5 min of training, which was sufficient for the subject to feel confident with the apparatus, headphones and black goggles were put on.
CALIBRATION.
The preliminary phase of the experiment, called "calibration," was performed for several reasons: we wished 1) to force subjects to pay attention to the amplitude of displacement rather than to other contingent factors such as duration or peak velocity; 2) to avoid effects due to uncertainty in controlling a nonfamiliar, although relatively slow, transport vehicle in darkness; and 3) to obtain information about subjects' ability to estimate distance of passive transport. The subjects were requested to drive the robot a distance of exactly 2 m in complete darkness. The experimenter then told the subject the exact distance just traveled, and another attempt was made. This exercise was repeated until the response was stabilized at ~2 m, but a minimum of 10 trials was imposed even when apparently not necessary.
CONDITION 1: TRIANGULAR VELOCITY PROFILE.
The subject was passively randomly displaced along 2, 4, 6, 8, or 10 m with the headphones and the black goggles on. Velocity profiles of most stimuli (13 of 16) were triangular, i.e., with equal values of accelerations and decelerations in the range of 0.06-0.5 m/s2. Peak velocity ranged from 0.6 to 1 m/s (Table 1). This profile was chosen to produce a continuous stimulation of the otoliths by linear acceleration. Three stimuli with constant velocity profiles (0.4, 0.6, and 0.8 m/s) were also applied over distances of 4, 6, and 8 m, with acceleration and deceleration at 0.8 m/s2. In contrast with the triangular profile, this provided only brief stimulation of the otoliths at the beginning and at the end of the trajectories.
About 10 s after the end of the imposed displacement (stimulus), the subject was required to reproduce as accurately as possible the distance traveled, controlling the robot with the joystick (response). The paradigm is illustrated in Fig. 1, B and C.
The whole test included 13 trials with triangular velocity profiles and 3 trials with constant velocity profiles. The order of these 16 trials was randomly changed for the different subjects.
CONDITION 2: CONSTANT DURATION PROFILES.
In the first condition, the total distance and duration of the passive transport were not independent. Therefore, to prevent the subjects from using the duration of transport as a cue to reproduce distance, we devised a second condition in which different velocity profiles of same duration (16 s) for all distances were used.
To travel distances from 2 to 10 m in the same amount of time, different velocity profiles could be used. We chose a rectangular velocity profile (constant velocity), a trapezoid profile, and a triangular profile (Table 2). The theoretical output signal generated by the otoliths (Ormsby and Young 1977
) corresponding to each profile is shown in Fig. 2. With the triangular profile it was not possible to travel the 2- and 10-m stimuli in 16 s, given the limited velocity range of the robot: the durations were 13.33 and 20 s for the 2- and 10-m stimuli, respectively.

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| FIG. 2.
Otolith response to acceleration profiles of 2nd condition, simulated with the use of the transfer function developed by Ormsby and Young (1977) . Dashed lines: result of simulation (MOD). Solid lines: input signal. Thin lines: data from "artificial" (ideal) profiles (ART). Heavier lines: data recorded and derivated from robot odometry (REC) after 3-Hz low-pass filtering. All 3 profiles generated a 6-m distance.
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We used the same instructions as in the first condition, and the subjects (7 who also participated in the triangular velocity condition, and 2 additional subjects) reproduced the five distances with the three velocity profiles (i.e., 15 trials) presented in completely random order.
Data analysis
REPRODUCTION OF VELOCITY PROFILES.
Inspired by a two-dimensional cross-correlation analysis, a method was developed to provide an index of resemblance between the stimulus and response velocity profiles. Our aim was to obtain a comparison of shapes regardless of errors in distance, duration, or peak velocity reproduction. The first step in the procedure was a normalization of the time scales of both stimulus and response profiles, separately, from 0 to 100%. The number of samples of the profiles was reduced to 100 by averaging adjacent points. Because the number of points constituting a profile initially ranged from 1,000 to 2,000, it follows that from 10 to 20 adjacent points were averaged; this corresponds to a cutoff frequency of 5-10 Hz for a moving average filter. Because the profiles were previously filtered at 5 Hz to eliminate high-frequency, low-amplitude noise, this additional filtering did not markedly alter the relevant characteristics of the signals. Then, all of the normalized trials from the same subject or from the same stimulus were averaged, so as to reduce noise/random variability stemming from the manipulation of the joystick. Finally, the root mean square (RMS) of the differences between the averaged stimulus and response velocity profiles was taken as a quantitative index of mutual resemblance.
STATISTICAL ANALYSIS.
Appropriate analysis of variance (ANOVA) designs were used to compare the data from individual trials between different conditions, with either response total magnitude (distance or duration) or algebraic error (response
stimulus) as dependent variables. The relative error (response
stimulus)/stimulus was used to quantify overall accuracy. Linear regressions of individual trials were performed to quantify the stimulus-response relationship from the subjects. A probability level of 0.05 was considered significant. Multiple regression analysis was performed to estimate the relative importance of the respective stimulus parameters in determining the response.
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RESULTS |
Calibration
The average traveled distance of the very first trial of the calibration test was 1.55 m, i.e., this first trial induced on average a 22.5% undershoot error with respect to the required 2-m distance, and interindividual variability was ±0.52 (SD) m. The distance was much closer to 2 m (1.90 m, i.e., 5% undershoot error) and the interindividual SD was lower (±0.31 m) at the end (the 10th trial) of this preliminary exercise. It can be seen (Fig. 3) that a plateau in performance was already reached by the fifth trial (1.95 ± 0.27 m, mean ± SE), and successive trials exhibited about the same error. The difference between the 1st and the 10th trial distance was significant [F(1,14) = 5.44, P = 0.035], as was the difference between the 1st and the 5th trials [F(1,14) = 6.63, P = 0.022]. The difference was also significant between the 1st trial and each of the trials after the 5th, whereas there was no significant difference between the 5th and the 10th trials.

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| FIG. 3.
Calibration. Performance of subjects during calibration test: average and SE for all 15 subjects shown with respect to trial number.
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Triangular velocity condition
DISTANCE REPRODUCTION.
Subjects were able to reproduce the distances (Fig. 4) with an average overall accuracy of 25% (SD of the pooled relative errors, n = 232). The linear regression between stimulus and response distance was calculated for each subject with all the 16 trials (an example is shown in Fig. 4B). Regressions from the individual subjects are presented in Fig. 4A, inset. The average of the individual regression lines (Table 3) is shown in Fig. 4A, together with the means and SEs of the reproduced distances. The correlation coefficient r was highly significant for all subjects (P < 0.0001).

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| FIG. 4.
Distance reproduction (1st condition). A: average (- - -) of all individual lines of 15 subjects, and average ± SE of each reproduced distance. Average determination coefficient r2 was 0.85. Inset: individual regression lines between stimulus and response distance for 15 subjects. B: responses of subject BJ, with corresponding regression line (- - -).
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The reproduction of the shortest distance (2 m) led to a slight overshoot (2.31 ± 0.12 m), whereas that of the longer distances exhibited an undershoot (9.21 ± 0.33 m for the 10-m stimulus).
Finally, there was no significant difference in the reproduction of the distance between the trials with triangular velocity profiles and those at constant velocity.
DURATION REPRODUCTION.
As mentioned in METHODS, with the triangular velocity profile, stimulus duration and stimulus distance were interdependent. Therefore the duration of the stimulus could provide some information assisting its reproduction. Indeed, the subjects also reproduced the duration of the stimulus (Fig. 5, Table 3), although the instruction was to reproduce the distance. The value of r was highly significant for each subject (P < 0.0001, except for 1: subject EC obtained P < 0.0013). Only subject EC displayed a "step strategy," i.e., with joystick manipulations of short duration and displacement at high velocity; this subject was nevertheless as accurate as the others in reproducing distance.

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| FIG. 5.
Duration reproduction. A: individual regression lines between stimulus and response duration for 15 subjects. B: responses of subject BJ with corresponding regression line.
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The duration of the shortest stimulus (4 s) was reproduced with an overshoot (4.60 ± 0.32 s), and the longest (25 s) with an undershoot (21.27 ± 1.0 s).
VELOCITY REPRODUCTION.
There was no correlation between stimulus peak velocity and distance, and therefore stimulus peak velocity could not be of any help for the subjects to reproduce the distance. However, most subjects did reproduce stimulus peak velocity. The value of r was significant for all subjects (P < 0.01) but four (subjects LC, EC, II, and MB). The average determination coefficient (Table 3) was therefore lower than that of distance and duration.
In the constant velocity trials the subjects exceeded the plateau velocity significantly more than the peak velocity in the triangular profile trials [F(1,14) = 9.66, P < 0.008], by 0.12 ± 0.03 m/s compared with
0.04 ± 0.04 m/s in the triangular trials.
INTERDEPENDENCE AMONG DISTANCE, DURATION, AND VELOCITY.
To examine whether stimulus duration or peak velocity had been used by the subjects to reproduce distance, we applied a multiple regression analysis to our data, with reproduced distance as the dependent variable and stimulus distance, duration, and peak velocity as independent variables. The results indicate that the response distance can only be attributed to the stimulus distance (Table 4A), and neither to stimulus duration nor peak velocity. The response is correlated with stimulus duration (Table 4A), but this correlation results from the existing correlation between stimulus distance and duration.
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TABLE 4.
Multiple regression on distance reproduction and duration reproduction in the triangular velocity condition
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The same question can be asked about the reproduction of duration: was duration reproduced as a consequence of distance reproduction, or was it reproduced per se-, by the subjects ? The results of the multiple regression, with reproduced duration as dependent variable, indicate that duration was reproduced mainly because of stimulus duration, and there was a significant contribution of stimulus distance (Table 4B).
VELOCITY PROFILE.
Whereas the peak velocity was not a relevant parameter for judging and reproducing distance, subjects frequently reproduced the stimulus velocity profile (Fig. 1C). Only subject LC produced very asymmetric triangular profiles, whereas subject EC systematically used a rectangular velocity profile, reaching the maximal velocity of the robot on short duration. Subjects II and MB exhibited variable combinations of velocity profiles. Nevertheless, distance reproduction was not markedly different from that of the other subjects, who produced mostly triangular responses.
Constant duration condition
The multiple regression analysis applied on the triangular velocity profile trials allowed us to establish that duration was not the main cue used by the subjects. Furthermore, there was some indication that subjects also reproduced the velocity profile of the passive transport. We then designed a second condition with another set of stimuli, characterized by three velocity profiles of identical duration (Table 2). We wanted to check whether duration was indeed not necessary to reproduce the corresponding distance, and whether subjects would also reproduce velocity profiles when they are not triangular.
DISTANCE REPRODUCTION.
As expected, although stimulus duration was not a valid predictor of its distance, subjects did reproduce the imposed distance (Fig. 6) as well as in the first condition: overall accuracy (as measured by the SD of the relative error) was 35% (n = 132), with 25% (n = 44) for the triangular profile, 28% for the trapezoid profile, and 47% for the rectangular profile. The average of the individual regression lines (each line computed on the basis of 5 measures) for each velocity profile as well as for all three pooled profiles (15 measures; Fig. 6A) is given in Table 5. For the rectangular profile, r was significant for all subjects but subject EC. The determination coefficient (r2) obtained from the rectangular profile was lower than that resulting from the other profiles, but a two-factor repeated-measures ANOVA (profile type × distance) on the reproduced distance showed no effect [F(2,16) = 0.67, P = 0.52] or interaction [F(8,64) = 0.62, P = 0.75] due to the velocity profile.

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| FIG. 6.
Distance reproduction (2nd condition). A: average (- - -) of all individual lines of top left inset, and average and SE of each reproduced distance. Top left inset: individual regression lines between stimulus and response distance of 3 profiles pooled (n = 15) for 9 subjects. Bottom right inset: average regression lines between stimulus and response distance for 3 profiles. B: responses of subject BJ with corresponding regression line.
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The mean regression for the seven subjects who had also participated in the first condition was basically not different from that obtained in the triangular velocity condition. A two-factor within-subjects ANOVA (condition × distance) on the reproduced distance showed no effect of the condition [F(1,6) = 2.62, P = 0.16] or interaction [F(4,24) = 1.79, P = 0.16].
DURATION REPRODUCTION.
In this condition stimulus duration did not vary and the average response duration was very close to that of the stimulus: 14.14 ± 1.21 s (n = 132). There was no significant difference between the duration errors for the three profiles. The average error was
1.41 ± 0.93 s (n = 9) with the rectangular velocity profile,
0.65 ± 0.84 s with the trapezoid profile, and
2.17 ±0.63 s with the triangular profile: response duration was shorter than stimulus duration in all cases.
Subject EC again exhibited a shorter response than the other subjects: mean duration error for subject EC was
7.40 s, whereas the error of the six remaining subjects was
0.52 ± 0.36 s; the difference was larger than in the first condition (
4.91 s for subject EC vs.
0.59 ± 0.76 s for the 6 remaining subjects). Subject EC apparently applied the step strategy with still more conviction when deprived of temporal information correlated to distance, without losing accuracy in fulfilling the task.
VELOCITY REPRODUCTION.
In this condition, in which stimulus duration and distance were not correlated, stimulus maximal velocity could have been used to provide some information about distance. Stimulus peak velocity and distance of the five trials were indeed strongly interdependent for all velocity profiles.
This may have been a reason why there was a significant correlation between stimulus and response peak velocity, in all subjects but two, in all profiles (P < 0.03): subject EC had a nonsignificant correlation of stimulus-response peak velocity in all three profiles, and another subject (PG) showed a nonsignificant correlation for both the rectangular and trapezoidal profiles. Table 5 indicates the average regression line between stimulus and response peak velocity for all subjects.
Subject EC displayed the greatest peak velocity error (0.73 m/s, vs. 0.02 ± 0.03 m/s for the 6 other subjects).
With a multiple regression analysis, following the same procedure as in the triangular velocity condition, it was found that distance was again the most important predictor of distance reproduction [F(3,129) = 149.93, P < 0.0001]. The value of r was significant for stimulus peak velocity (0.74) and for stimulus distance (0.88), but not for duration, as expected.
VELOCITY PROFILE.
Subjects reproduced the velocity profile of the stimuli more closely for longer distances (Fig. 7A). A global overshoot at the onset of the reproduction can be seen, principally in the triangular profiles. This was probably due to the relatively long delay of the joystick control(0.2 s).

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| FIG. 7.
Velocity reproduction (2nd condition). A: all velocity responses to all trials of all subjects (except subject EC) are shown for each profile and each distance. Heavy line: stimulus. B: normalized response (velocity and duration) averaged over all trials of same velocity profile for 3 subjects (RG, YT, and BJ), mean ± SD, with cumulated velocity error. Dotted line: stimulus. Heavy line: mean velocity. Thin lines: mean ± SD.
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We then applied the method called "normalization" (Fig. 7B) to quantify our visual inspection. The RMS errors of the differences reproduction-stimulus appear in Table 6. A one-factor repeated-measures ANOVA revealed a significant difference among the three values [F(2,8) = 5.61, P = 0.014], and a Student-Newman-Keuls post hoc comparison confirmed that the RMS error of the triangular profile was greater than that of the trapezoid and rectangular profiles, but the errors of the latter two were not significantly different from one another. Therefore in general the subjects reproduced the rectangular and trapezoid profiles more accurately than the triangular one. However, two subjects (EC and PG) exhibited a larger error with the rectangular than with the triangular profile, and two other subjects (EM and RG) exhibited a larger error with the rectangular than with the trapezoid profile.
Finally, to test whether an accurate reproduction of velocity profiles was sufficient for an equivalently accurate reproduction of distance, we compared the velocity profile error (RMS) with the reproduced distance error, and found no significant correlation [F(1,13) = 0.93, P > 0.1]. This suggests that the velocity profile and the distance were independently reproduced.
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DISCUSSION |
A passive displacement can be represented in static as well as in dynamic terms. That is to say that we could refer to an experienced travel either as a 10-m, 10-s linear displacement or as a motion at a gradually increasing speed followed by a deceleration up to a stop. These two representations might coexist in short-term memory or, alternatively, only one might be stored. In this latter case, if the dynamic representation is the one available, static estimates could be derived by reprocessing it. On the other hand, motion dynamics would be lost if only static parameters are stored.
In this paper we applied our recently developed method for studying the memory of traveled linear distance, which is based on the overt reconstruction of a passive transport (Berthoz et al. 1995
), to address the relationship between motion perception and path integration. Blindfolded subjects were asked to reproduce the perceived distance of a linear passive displacement by directing another passive motion.
The results show that 1) the distance of the passive travel was reproduced; 2) this was accompanied by the reproduction of the stimulus duration, peak velocity, and velocity profile; and 3) distance was reproduced even when stimulus duration was kept constant. This latter finding and the statistical analysis of the results suggested that distance was reproduced per se, and not as a consequence of duration, peak velocity, or velocity profile reproduction. Therefore the present study provides evidence that both a static and a dynamic representation of passive motion are stored in memory.
The reproduction experiment was preceded by a calibration task that is discussed before the main findings are addressed.
Calibration task
In this initial experiment, subjects were asked to drive the robot in darkness over a 2-m linear path with the joystick. Klatzky et al. (1990)
used the same distance to "train" their subjects in an experiment on locomotion. It is indeed reasonable to assume that humans should have some meaningful representation of such a short length. The results showed that all subjects but two undershot this distance (i.e., subjects overestimated their own self-traveled distance) by 22% in the first trial with an interindividual variability >25%. Such a considerable undershoot and variability suggest that either subjects have a very variable representation of the required length, or the vestibular and somatosensory inputs are not calibrated to correspond to a metrical representation of distance.
A similar undershoot was observed by Israël et al. (1993)
in another linear displacement task. Subjects, blindfolded and with ears plugged, sitting on a sled that moved along the X-axis, had to push a button when they thought that the sled passed a previously seen target (the distance here was 2.4 m): the button was pushed at 1.57 ± 0.37 m during the displacement. Despite methodological differences (the displacement was passive and not self controlled; the task was goal directed and not amplitude coded; and the expected response was at 2.4 m, not 2.0 m), the error is quantitatively similar. Because the target was seen by the subjects in this former experiment, the previous hypothesis of a variable metrical representation of the 2-m length is weakened. Another explanation can be suggested for both cases: in the former experiment the undershoot was explained as a consequence of a double integration over time of otolith discharge, including the initial overshoot that is induced by an acceleration step (see Berthoz and Droulez 1982
for a review). It is possible that such a process occurred also in the present experiment.
The calibration experiment, theoretically, has no consequence on the performance in the reproduction task other than the expected one, i.e., that of polarizing subjects' attention on the distance of the displacement on the robot.
Reproduction task
REPRODUCTION OF DISTANCE.
The present work brought contrasting evidence about the mechanisms of distance perception because 1) the strategy selected by most subjects to reproduce distance (the assigned task) was to reproduce the velocity profile, i.e., the spatiotemporal dynamics of the passive transport, but 2) regression analyses indicated that distance reproduction was not correlated with the accuracy of duration, peak velocity, or velocity profile reproduction. Thus the results suggest that whereas the dynamics of passive motion are stored and available to further use, total distance is probably independently estimated.
Possibly, subjects reproduced the dynamic characteristics of the passive transport because the self-controlled transport was not goal directed but amplitude coded, and the task itself was ambiguous because "distance" is a static parameter, whereas the word "reproduction" can implicitly denote a dynamic task. Also, the joystick did not control the displacement magnitude but the robot speed, which might have induced subjects to work with dynamics. Moreover, the subjects might have felt that retrieving the dynamic properties of the displacement could help fulfill the requested task. It must be underscored that this was not the only possible strategy, because one subject (EC) systematically used a rectangular velocity profile, of short duration and high peak velocity, which resulted nevertheless in an accurate reproduction of the distance.
It might also be argued that the observed independence in distance reproduction does not necessarily imply that an accurate, independent, static internal estimate of total path length was produced and memorized: distance is processed twice in each trial, and thus even in the case in which processing was highly nonlinear or biased or totally uncalibrated, the result for reproduction would still be correct should the same processing apply to the passive and the active phase of displacement. An alternative explanation of the results of the statistical analyses could be that distance was independently reproduced as a consequence of some peculiar feature of the actively controlled transport phase, dissociating path length from motion dynamics (e.g., noise). The results from subject EC, the evidence from previous experiments (Glasauer et al. 1994
; Israël et al. 1993
; Mittelstaedt and Glasauer 1991
), and the relative accuracy in total distance reproduction do not favor, in our opinion, this last hypothesis. Subject EC must have retained or reconstructed some internal static estimate of the imposed path and disregarded the dynamic information in the reproduction task.
REPRODUCTION OF DISTANCE WITH DURATION AND PEAK VELOCITY.
The different statistical analyses indicated that no motion parameter (including the velocity profile) significantly contributed to the accuracy of reproduction of distance. On the other hand, reproduction of duration appeared secondary to distance. The indication given by this method was confirmed by the constant duration experiment, in which the duration of the stimulus was kept constant. Indeed, the accuracy in reproducing distance was the same as when duration varied proportionally to distance. Therefore we conclude that distance was not estimated from related magnitudes. This is not surprising, because we have previously shown (Israël and Berthoz 1989
) that the otoliths are necessary to estimate a passive linear whole body displacement with respect to an earth-fixed memorized visual target, and to acquire this target with eye saccades.
RANGE EFFECT.
Whereas during calibration an undershoot of the requested 2 m was observed, in the reproduction task the subjects overshot 2-m trials but undershot larger distances. These distortions in the reproduction task can be interpreted as a manifestation of the "range effect," also known as "regression to the mean effect" or "central tendency effect" (overshoot of small distances and undershoot of large distances of a given set) (Poulton 1979
; Stevens and Greenbaum 1966
). The effect was observed both for the reproduced distance and for the duration in the triangular velocity condition, whereas it almost disappeared in the constant duration condition. Because this effect is characteristic of judgments of sensory magnitude (Poulton 1979
), the subjects might have implicitly estimated both distance and duration, although the instruction was about distance only. The absence of range effect in the constant duration condition suggests that this effect can be mainly ascribed to the estimation of duration.
COMPARISON WITH LOCOMOTION EXPERIMENTS.
There is a close resemblance between our results and those obtained by Loomis et al. (1993)
. In the experiment of Loomis et al., blindfolded subjects were first led by the experimenter while walking along a path 2-10 m long (the same distances as in the present experiment), and the subjects then had to reproduce the same distance while blindfolded and without aid. The reported results are strikingly similar to the present ones: the 2-m distance was overshot by 0.26 m (0.31 ± 0.12 m in the present test) and the 10-m distance was undershot by 1.02 m (0.79 ± 0.33 m here). This resemblance suggests that there are important parallels between active locomotion and the self-controlled passive displacement we have used in the present study. It should be noted that the motion parameters (speed and acceleration) selected for passive motion were much in the physiological range of normal locomotion.
It may thus be suggested that the inertial and proprioceptive signals generated in the present task are processed in a very similar way as during locomotion when motion-related information is considered and when motion is self-driven. The double time integration of the acceleration forces on the otoliths (Israël et al. 1993
) might participate in the updating of position during motion.
The absence of bias in the stimulus-response relationship on distance suggests that the same processing of inertial signals occurred during the passive and the active transport phase. This is at variance with respect to the results obtained by Mittelstaedt and Glasauer (1991)
: those authors found that subjects passively transported in darkness toward a previously seen target tended to underestimate (in the range of linear velocity and distance used in the present experiment) the traveled distance, whereas the opposite happened during active locomotion. Mittelstaedt and Glasauer proposed the idea that a leaky path integrator, loaded with the visually estimated distance, processes incoming inertial signals: the reference distance is differently included in the processing during passive transport and active locomotion, respectively. If we applied the same rule to the present experiment, we would obtain a large undershoot of reproduced versus imposed distances, which was not the case. The similarity between the passive and active transport phases in our experiment might have caused the same processing of inertial signals to occur in both phases. Because of the important methodological differences between both experiments (i.e., the visually acquired reference distance and the proprioceptive locomotion-related signals of the above quoted experiment), further comparison would be pointless.
Somatosensory signals could also have contributed to the estimation process: arthrokinetic information is known to affect linear self-motion perception (Bles et al. 1995
; de Graaf et al. 1994
; Hlavacka et al. 1992
). Tactile cues may complement vestibular information, providing 1) a signal related to the body linear acceleration (pressure on the back, visceral shifts, etc.) and 2) a signal generated by robot vibrations. The subjects could have correlated all of these signals with visually perceived velocity during the very preliminary training (before the calibration task). However, the training phase was unlikely to influence the strategy of reproduction, because the performance of the subjects who participated in the constant duration condition was not different from that displayed in the first condition, although no training was performed before the second condition. Therefore it is highly improbable that the somatosensory input generated by vibrations (which are peculiarly linked to the robot and ground characteristics) was calibrated.
The case for the propulsion forces exerted by the motor during travel is different. It is impossible to dissociate vestibular from somatosensory contribution to motion perception because the two sensory systems are simultaneously stimulated during the transport. Pressure on the back during acceleration is unavoidably felt in sitting subjects and vibrations always occur during transport. However, a paraplegic subject who underwent the triangular velocity test was as good as normal subjects in reproducing distance (Berthoz et al. 1995
). This result does not rule out the participation of tactile cues to the estimation process, but it confirms the importance of the otolith signals.
How are the motion dynamics matched during the reproduction?
An important finding of this (and the previous) work is that motion dynamics are stored during the passive displacement and played back during the reproduction phase. There are many ways such a behavior can be modeled. Figure 8 shows a simple schema that summarizes our concept. The schema is made of two parts: 1) the passive transport phase and 2) the active transport phase. The actual acceleration drives the otoliths (and somatosensory system), the output of which is stored in a dynamic short-term memory. Then memory feeds a comparator of a negative feedback controller. The block labeled "robot control" represents the feedback gain. The robot acceleration profile provides the input to the otoliths, whose output is in turn compared with the memorized input profile. This generates an error signal driving the active reproduction of passive motion dynamics.

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| FIG. 8.
Simple schema of distance reproduction paradigm. Graph labeled "stimulus" includes all dynamic properties of passive motion. OTOLITHS actually includes all idiothetic signals available in present experiment (i.e., those due to vestibular and somatosensory systems), and box labeled "robot control" covers all robot and joystick parameters.
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The critical point in the schema is the MEMORY box, because we cannot directly access the internal representation. The otoliths (and the somatosensory system) provide signals directly related to linear acceleration, which might allow an efficient acceleration feedback control to take place. On the other hand, data from motion perception and eye movement studies in humans (Lichtenberg et al. 1982
; Shelhamer and Young 1994
; Young and Meiry 1968
) revealed that integration over time of the otolith-induced neural discharge occurs over the whole frequency range of the stimuli employed in the present study, leading to a close relationship between the perceived and the actual linear velocity (Young 1984
). On the grounds of these latter findings it can be proposed that feedback control is carried out on linear speed rather than acceleration. As mentioned above, a second time integration of sensory input may occur (Israël et al. 1993
) and feedback control could be theoretically performed on instantaneous position. This last possibility corresponds to the true path integration hypothesis (Mittelstaedt and Mittelstaedt 1982
), which is a very attractive one because it would attribute to one and the same process both the static and the dynamic characteristics of the subjects' responses. In such a case, the static estimate of distance would correspond moment by moment to the current stored value of the dynamic profile and no reprocessing of recorded signals should be needed.
We must admit that there is no evidence from the results obtained in the present experiment to favor one formulation or another. From a mathematical point of view all of them can solve the experimental task. However, they will imply substantially different consequences if the first and the second phase of the experiment are dynamically uncoupled. For example, if control is carried out on instantaneous position, by constraining the robot speed in the active phase, the reproduction of the displacement distance will not be impaired. In contrast, gross distance errors will be expected if feedback is performed on acceleration or speed. Such predictions will allow testing the above hypothesis.
It is worth noting that the greatest mismatch between the stimulus and response velocity profile was found for the triangular profile when compared with the trapezoidal and rectangular profiles. Because the triangular profile provided stimulation to the otoliths continuously throughout the displacement, whereas the trapezoidal and rectangular profiles included segments at constant velocity, this result is surprising. However, it should be noted that reproducing a triangular profile requires more complex motor skill. Subjects had to continuously increase the tilt of the joystick to achieve a velocity ramp, despite a critical delay in response time of the control (0.2 s) and low elasticity in the joystick itself. But the poor accuracy in triangular velocity reproduction could also be due to the normalization analysis, which is particularly severe with this profile because it takes into account both slope and symmetry. In contrast, the variability in distance reproduction (SD of relative error) was much larger when rectangular rather than triangular profiles were applied. Trapezoidal profiles induced intermediate accuracy in distance reproduction. Because in the triangular type of profile, otoliths are continuously stimulated (Fig. 2), it is suggested again that otolith signals play a major role in the perception of distance but a dissociation between dynamic and static components emerges. This dissociation suggests an independence between the use of sensory signals for path length estimation and for the monitoring of active reproduction, as already suggested by regression analysis results.
General remarks
A careful quantitative analysis of the results of the present experiment allows us to conclude that vestibular and somatosensory signals generated by passive transport can be used to build a dynamic as well as a static representation of the traveled path. Recent studies have already provided some neurophysiological background to understand the present findings: by transporting monkeys on a robot-mounted platform in complete darkness, O'Mara et al. (1994)
found hippocampal neurons responding to linear motion and others responding to axial rotation; cells belonging to the rat hippocampal formation that code head direction in space have now long been known (McNaughton et al. 1983
); changes in rat hippocampal theta activity correlated to the velocity of angular rotation in the yaw plane have been demonstrated (Gavrilov et al. 1996
). The improving description of neural ascending pathways (Grüsser et al. 1992
; Muller et al. 1996
) bringing multisensory motion-related information to the cortex led to the identification of computational (Wan et al. 1994
) and biological (McNaughton et al. 1996
) models of a corticothalamohippocampal navigation system that work by updating position and direction in space in real time.
Passive transport is a special case of navigation in which no active control is performed. The qualitative and quantitative similarity between our experimental results and those obtained in analogous experiments on locomotion (Loomis et al. 1993
) suggests that these two types of navigation tasks draw on common physiological processes and extend the relevance of our results to more ecological behaviors of path integration.