Relative posture between head and finger determines perceived tactile direction of motion

The hand explores the environment for obtaining tactile information that can be fruitfully integrated with other functions, such as vision, audition, and movement. In theory, somatosensory signals gathered by the hand are accurately mapped in the world-centered (allocentric) reference frame such that the multi-modal information signals, whether visual-tactile or motor-tactile, are perfectly aligned. However, an accumulating body of evidence indicates that the perceived tactile orientation or direction is inaccurate; yielding a surprisingly large perceptual bias. To investigate such perceptual bias, this study presented tactile motion stimuli to healthy adult participants in a variety of finger and head postures, and requested the participants to report the perceived direction of motion mapped on a video screen placed on the frontoparallel plane in front of the eyes. Experimental results showed that the perceptual bias could be divided into systematic and nonsystematic biases. Systematic bias, defined as the mean difference between the perceived and veridical directions, correlated linearly with the relative posture between the finger and the head. By contrast, nonsystematic bias, defined as minor difference in bias for different stimulus directions, was highly individualized, phase-locked to stimulus orientation presented on the skin. Overall, the present findings on systematic bias indicate that the transformation bias among the reference frames is dominated by the finger-to-head posture. Moreover, the highly individualized nature of nonsystematic bias reflects how information is obtained by the orientation-selective units in the S1 cortex.

www.nature.com/scientificreports www.nature.com/scientificreports/ The somatotopic frame must take the body posture into account because the perception of physical environments by the hand cannot be disassociated from certain body postures 36 . In practice, body posture is an integrated result derived from the positions of multiple joints 37,38 , including the shoulder, elbow, hand, and wrist [39][40][41] . Strong evidence suggests that humans have the ability to estimate the end-point of the body extremities (i.e., the hands) with a high degree of precision through the integration of proprioceptive information [42][43][44] . In other words, the body posture is well organized into the somatotopic frame. Regarding the cutaneous senses of the hand, previous studies have shown that the perception of tactile orientation by the hand cannot be explained by any single reference frame of the body posture. When performed on the horizontal 45,46 , midsagittal 47 , or frontoparallel plane 48 , the perception of tactile orientation must include information from the angles of multiple joints to be integrated with the cutaneous senses. In other words, tactile perceptual bias may be attributed to the existence of intermediate reference frames that are used for multi-sensory integration [49][50][51] .
However, perceptual bias may also stem directly from cutaneous senses. In general, cutaneous information must be integrated with information regarding the body posture in order to achieve precise haptic processes. For example, when participants were asked to judge whether a bar presented to one palm was parallel to a second bar presented to the other palm, the participants' judgment was intermediate between the allocentric and somatotopic reference frames 45,46 . Furthermore, tactile direction and orientation presented to the left index fingerpad yielded a clockwise bias of 20°-25° when the left forearm was positioned in a forward and volar side up posture 52,53 . In other words, it appears that tactile orientation and direction share a common transformation process from an object's physical condition to an individual's perception.
The present study aimed to characterize the rules governing the perceptual bias underlying the transformation of reference frames. Tactile motions were presented to the left index fingerpad using a miniature tactile stimulator. A video screen was placed between the eyes of the participant and the stimulator, and the participants used a mouse to report the perceived direction of motion of the stimulus on the screen. This study used a design in which the stimulator, left index fingerpad, center of the video screen, and eyes were perfectly aligned along the posterior-to-anterior axis. In performing the trials, the participants' head and finger postures were manipulated in a controlled manner in order to investigate the effect of these postures on perceptual bias. Results revealed that the perceptual bias was a linear summation of two different types of bias, namely systematic bias and nonsystematic bias. The systematic bias correlated linearly with the difference between the finger and the head postures for all participants. By contrast, the nonsystematic bias is highly individualized, and has a phase determined mainly by the finger posture. The two co-existing biases may represent the underlying principles governing the transformation of reference frames in the perception of tactile motion.

Results
Study design. This study evaluated the effect of the relative head and finger posture on motion perception at the fingerpad. A rotating aluminum ball with a groove depth of 500 μm, a wavelength of 4 mm, and a 45% duty cycle was used as the tactile stimulus to deliver tactile motions (Fig. 1a). Motion stimulus was delivered to the fingerpad of the left index finger using the miniature tactile stimulator with three motors that can precisely control the speed, direction and indentation depth of the stimulus 54 (Fig. 1b). Each participant sat in front of a table with the angle of elbow joint kept constant. The participant's left upper limb was supported by the arm holder with the palm facing the participant's face such that the left index fingerpad contacted the aluminum ball during stimulation. The participant's head, eyes, video display, stimulus, and left index fingerpad were precisely aligned along the posterior-to-anterior axis (Fig. 1c, see also Experimental set-up for details).
The perceived direction of the tactile motion presented to the participant's left index fingerpad was analyzed for carefully controlled angles of the head and finger. In each trial, the rotating grating ball indented the fingerpad and was driven in a particular direction resulting in tactile stimulation. The participant visually fixated on a cross presented at the center of the video display and reported the perceived direction of motion (R i ) using a mouse to click on an appropriate point on a circle shown on the video display (Fig. 2a). The stimulation trials were performed in accordance with a 3-by-4 factorial finger-and-head posture combination design consisting of three head postures and four finger postures. In particular, the finger postures (θ F ) were set as 90°, 60°, 30° or 0°, while the head postures (θ H ) were set as 120°, 90° or 60°; yielding a total of 12 different posture combinations (Fig. 2b). The effect of the head and finger postures on the perceived direction of tactile motion was evaluated by comparing the difference between the veridical and perceived directions across all postures.

Systematic bias.
This study aimed to understand how the perceived direction of motion of the stimuli presented to the fingerpad was modulated by the relative posture of the finger and head. According to the data obtained from a single sample participant (Fig. 3a), the systematic bias was found to be modulated by both the head posture (θ H ) and the finger posture (θ F ). Specifically, the highest systematic bias occurred for θ H = 120° and θ F = 0°, while the lowest systematic bias occurred for θ H = 60° and θ F = 90°. Interestingly, the systematic bias was approximately zero for postures of θ H = 120° and θ F = 90°, θ H = 90° and θ F = 60°, and θ H = 60° and θ F = 30°, i.e., postures under which the difference between the finger and head angles was −30° (θ H − θ F = −30°). Notably, a clockwise shift of the finger induced a counterclockwise change in the systematic bias, indicating that the finger posture modulated the systematic bias in the opposite direction (Fig. 3b, middle). By contrast, a clockwise shift of the head posture resulted in a clockwise change in the systematic bias. In other words, the head posture modulated the systematic bias in the same direction (Fig. 3b, bottom). These patterns of the systematic bias were found to be similar across all six participants (Fig. 3c), with high values of pairwise correlations of systematic bias across finger-head postures (0.93 ± 0.04, Mean ± SD, across all participants, Fig. 3d).
In general, the results presented in Fig. 3 indicate that the systematic bias can be robustly predicted by the head and finger postures. A further analysis was thus performed to clarify the detailed relations of the systematic bias with finger, head and finger-head postures, respectively (Fig. 4a-c    www.nature.com/scientificreports www.nature.com/scientificreports/ six participants). Results showed that systematic bias was modulated by finger posture in the sample participant An investigation was performed to determine the ability of finger posture to predict systematic bias when the head posture was controlled. A good goodness-of-fit (as measured by the R 2 coefficient) was found for the finger posture in both the sample participant ( 14, 9.68, 6.29, 7.33], all p < 0.001, df = 16, data from biases averaged within participants). A final investigation was performed to examine the degree to which systematic bias could be predicted by the relative finger-head posture (i.e., the difference between finger posture and the head (θ F-H )). Results showed almost perfect correlation of systematic bias with finger-head posture (Fig. 4c, for sample participant #1: slope = −0.63, R 2 = 0.98, t = −22.56, p < 0.001, df = 10, data from averaged biases for each finger-head posture. Figure 4f, for all six participants, slope = −0.58, R 2 = 0.99, t = −57.18, p < 0.001, df = 4, data from biases averaged across participants); indicating that finger-head posture almost completely determined systematic bias. Finally, the systematic bias vs. finger-head posture plot had a slope of −0.63 for the sample participant and −0.58 across all the participants. In other words, a change in the finger-head posture induced a change of systematic bias in the opposite direction with a similar magnitude across all six participants.   www.nature.com/scientificreports www.nature.com/scientificreports/ Nonsystematic bias. In addition to the systematic bias described above, which correlated linearly with the relative posture between the finger and the head, a further bias (denoted as nonsystematic bias) which exhibited different values for different directions of the tactile stimulation was also observed. Nonsystematic bias varied across all stimulus directions (Fig. 5a, blue line); hence, it was fitted with a cosine function. A preliminary investigation found that the best fit was obtained using a cosine function with moment = 2 (Fig. S1a). After the cosine fit was established, two parameters were extracted, namely the amplitude (A) of the nonsystematic bias at its peak position and the corresponding phase (θ p ) (Fig. 5b) Figure 6a shows the nonsystematic bias for all the finger-head posture combinations of the sample participant. When the finger shifts from a vertical position to a horizontal position (θ F : from 90° to 0°), the phase of the nonsystematic bias shifts gradually and congruently (θ p : from 87° to −13°). However, a change in head posture has no noticeable effect on the phase of the nonsystematic bias. For example, the phase of the nonsystematic bias remains virtually unchanged when the head is moved from the leftward to rightward postures (θ H : from 120° to 60°).
An investigation was performed to examine the detailed relations of the finger, head and finger-head postures with the nonsystematic bias across all six participants and the factors affecting the amplitude of the nonsystematic bias (Fig. 6b,c for the phase of the nonsystematic bias, Fig. 6d,e for the amplitude of the nonsystematic bias). Correlation was found between the phase of the nonsystematic bias with both the finger posture (Fig. 6b left, for the sample participant #1: slope = 1.14, R 2 = 0.93, t = 11.17, p < 0.001, df = 10, data from averaged biases for each finger-head posture. Figure 6c left, for all six participants: slope = 1.08, R 2 = 0.73, t = 7.76, p < 0.001, df = 22, data from biases averaged across participants) and the finger-head posture (Fig. 6b right Systematic bias (º) www.nature.com/scientificreports www.nature.com/scientificreports/ participant: slope = 0.71, R 2 = 0.55, t = 3.52, p = 0.006, df = 10, data from averaged biases for each finger-head posture. Figure 6c right, for all six participant: slope = 0.79, R 2 = 0.72, t = 9.39, p < 0.001, df = 34, data from biases averaged across participants), but not between the phase and the head posture (Fig. 6b middle, for the sample participant: slope = 0.09, R 2 = 0.003, t = 0.18, p = 0.86, df = 10, data from averaged biases for each finger-head posture. Figure 6c, middle, for all six participants: slope = −0.17, R 2 = 0.06, t = −1.02, p = 0.32, df = 16, data from biases averaged across participants). The slope of the nonsystematic bias phase vs. the finger posture plot equaled approximately 1 (Fig. 6c, left). In other words, a change in finger posture had a positive and congruent effect on the phase of the nonsystematic bias. However, no correlation was found between any of the three postures and the amplitude of the nonsystematic bias; indicating that the amplitude of the nonsystematic bias is not posture-related (Fig. 6d for [ 16,34], data from biases averaged across participants). Pairwise correlations across all six participants equaled 0.71 ± 0.11 (R 2 , Mean ± SD) for the phase of the nonsystematic bias (Fig. S2a,b) and 0.15 ± 0.16 for the amplitude of the nonsystematic bias (Fig. S2c,d). Hence, it is inferred that the phase of the nonsystematic bias is consistent across all six participants, but the amplitude of the nonsystematic bias is highly individualized.
Given that the slope of the nonsystematic bias is close to 1 and the phase is similar across participants, this study further examined whether the phase of nonsystematic bias is anchored on the somatotopic reference frame or any other reference frame that is aligned with the skin, such as finger-centered or forearm-centered reference frames. To this end, the results shown in Fig. 6b,c when mapped on the somatotopic reference frame were analyzed (Fig. S3). It was found that the phase is consistent across finger postures when analyzed on the somatotopic reference frames, suggesting that nonsystematic bias is phase-locked to the skin or any other reference frame that is aligned with the skin.

Systematic vs nonsystematic bias. The systematic and nonsystematic bias differ in two main regards.
First, the systematic bias is dominated by the combined information of the finger and head postures (Fig. 7a, for v.s. θ F-H : Student's t test, t = 5.61, p < 0.001, df = 10, data from regression coefficients of bias v.s. postures of all participants). On the contrary, the phase of the nonsystematic bias is primarily determined by the finger posture (Fig. 7a, for v.s. θ F : Student's t test, t = −2.31, p = 0.04, df = 10. Fig. 7a, for v.s. θ H : Student's t test, t = 7.13, p < 0.001, df = 10). The head posture correlates only weakly with the systematic bias and has no effect on the phase of the nonsystematic bias. Second, the changes in direction of the bias caused by posture shifts for the systematic bias and the phase of the nonsystematic bias are different. In particular, the slope is negative for the systematic bias and positive for the phase of the nonsystematic bias given changes in the head posture and finger-head posture. Similarly, the slope is positive for the systematic bias and negative for the phase of the nonsystematic bias given changes in the finger posture (Fig. 7b). A further investigation was performed to examine the relationship between the systematic bias and the phase of the nonsystematic bias (Fig. S4). Results showed a consistent negative correlation between the systematic bias and phase of the nonsystematic bias across all six participants; indicating that both biases are modulated by the finger postures.
The sample size of six participants may raise the issue of a lack of statistical power, and it remains unclear whether the aforementioned biases have adequate test-retest reliability. To this end, we performed test-retest experiments that included an additional six participants (Supplementary Text). The results showed that the systematic (Fig. S5) and nonsystematic (Fig. S6) biases were reliable between the test (Fig. S5a,b) and retest (Fig. S6a,b) experiments. Also, the results obtained in these additional six participants were analogous to those observed in the formal experiment (Figs. S5c,d and S6c,d). www.nature.com/scientificreports www.nature.com/scientificreports/ Finally, we examined whether the posture-related changes of systematic and nonsystematic biases were simply mediated by the fact that the participants reported the finger or head postures. We showed that it was not the case as participants reported the perceived directions of motion instead of the finger or head postures (Fig. S7). www.nature.com/scientificreports www.nature.com/scientificreports/

Discussion
The present study has revealed plausible mechanisms underlying the reference frame transformation between the somatotopic and allocentric reference frames in response to tactile stimulation. In particular, a consistent deviation in transformation, referred to in this study as a systematic bias, has been found between the veridical and observed directions of the allocentric reference frame. The systematic bias is similar across different participants and is perfectly and linearly predicted by the relative finger-head posture. An additional nonsystematic bias has also been observed: the phase of the nonsystematic bias differs with different stimulus directions and is modulated primarily by the finger posture (i.e., not the head posture). The nonsystematic bias can be well-fitted by a phase-locked cosine function with moment = 2. In other words, an identical nonsystematic bias is induced by tactile stimulations applied in opposite directions. This finding suggests that the nonsystematic bias is associated with inhomogeneous cutaneous senses and is probably involved with the orientation-selective units in the S1 cortex 55 . Furthermore, given a change in the finger or head orientation induced dramatically different changes in the systematic and nonsystematic biases, it is inferred that the two biases are mediated by distinctively different mechanisms.
The present results have shown that the systematic bias is linearly predicted by the relative finger-to-head posture; indicating the importance of both the finger posture and the head posture in determining the bias. These findings are reminiscent of the observations of Carter et al. 56 , who found that the eye position affected the perceived tactile direction, and those of Volcic et al. 57 , who found that the head posture affected the perceived direction in parallelity tests. Tactile orientation was originally hypothesized to be perfectly encoded in an allocentric reference frame. However, this notion was challenged by the data collected by Hammerschmidt 18 . Moreover, in recent years, parallelity and mental rotation experiments have shown that the perceived orientation cannot be explained by any single reference frame when performed on the horizontal 45,47 , midsagittal 47 , or frontoparallel planes 48 . However, a reference frame that is an intermediate of multiple frames can account for such tactile orientation bias [49][50][51] . Many different sources may contribute to the reference frame, including the position or posture of the skin 16 , hand 19,58 , arm [59][60][61] , and body 62 . These findings are analogous to those obtained in studies on motor-sensory coordination [59][60][61] , which show that a hybrid frame of reference is constructed to combine parallel multisensory information. In general, these previous findings imply a multisensory nature of tactile perception. That is, tactile information needs to be mapped onto other reference frames such that the somatosensory, visual, auditory and motor functions can be integrated as a single holistic system 16,[56][57][58][59][60][61][62][63][64] .
The phase of the nonsystematic bias between the reported tactile direction and the veridical direction is anchored on the somatotopic reference frame (i.e., not the head posture). To the best of our knowledge, the nonsystematic bias of human touch has never been reported. However, the finding that nonsystematic bias can be fitted by a cosine function with a moment of two is reminiscent of the theory of tactile anisotropy (also known as the oblique effect), which states that tactile acuity tends to be better at certain orientations 62,65,66 . Neuronal data provide plausible support for the origin of the nonsystematic bias 55 . Specifically, some neurons in the primary somatosensory cortex are highly selective for the orientation or direction of scanning gratings 55,67 . In other words, transformation of the reference frames may be mediated by these orientation-selective units, in which a high percentage of neurons prefer the proximal-distal orientation 68 . Another possible explanation for the origin of the nonsystematic bias may be inhomogeneous finger compliance or inhomogeneous receptor properties 69 in response to motion stimulus presented to the finger. However, all these suggestions seem possible because the nonsystematic bias is constant when calculating the bias in the somatotopic reference frame 62,65,66,68,69 .
The evidence presented in this study suggests that systematic and nonsystematic biases reflect two completely different properties underlying the transformation from the somatotopic frame to the other reference frames. In particular, systematic bias is determined primarily by the relative finger-head posture; reflecting its multisensory nature. By contrast, nonsystematic bias is determined only by the finger posture, indicating most probably that www.nature.com/scientificreports www.nature.com/scientificreports/ it has a somatosensory nature. Furthermore, the slope of the phase of nonsystematic bias is close to 1 for most of the participants, suggesting that nonsystematic bias is phase-locked to the skin or any other reference frame that is aligned with the skin, such as finger-centered and forearm-centered reference frames. The present findings are inconsistent with the hypothesis that all sensory modalities are remapped to a common frame of reference [70][71][72] . In fact, neurons in the posterior parietal cortex, such as the ventral intraparietal (VIP) area, apply a variety of reference frames, including intermediate somatosensory and visual reference frames, to encode the stimulus location 73 , indicating that the integration between touch and vision is mediated by a coexistence of multiple reference frames.
The formal experiment included only six participants, a sample size that might be susceptible to type II errors. Also, this small sample size could limit our ability to analyze the variance across participants. Except for that, the experimental design adopted in this study has several important advantages. First, it enables the linearity of change in bias phase to be examined as a function of posture; a property that parallels the gradual shift of the receptive field in multisensory neurons observed in the ventral intraparietal area 73,74 , lateral intraparietal area 75 , ventral premotor area 76 , and superior colliculus 70,71 . Second, the present study on the perceived direction of stimulus motion has ecological value as haptics usually involves motion between the finger and the object 6,77 . However, tactile flow, i.e., the motion information obtained by the finger, is still required for subsequent motor planning. Finally, the tactile stimulation applied in this study is implemented with a directional precision of 1° and an indentation depth precision of 1 μm 54 . As a result, it provides the means to extract bias patterns with relatively small magnitudes, such as the nonsystematic bias reported herein. Tactile stimulator. The ball measured 20 mm in diameter and was engraved with square-wave gratings with a depth of 500 μm, a wavelength of 4.0 mm and a 45% duty cycle (Fig. 1a). A three-motor controller was employed to control the tactile stimulus (Fig. 1b) so that the direction of motion and indentation depth could be precisely controlled (see Pei et al. 2014 for details). During the experiment, white noise was played through an earphone to prevent the participant from hearing the motor noise.

Participants.
Experimental set-up. Each participant sat in front of a table with the left upper arm and forearm held by arm holders to maintain the arm position (Fig. 1c). The angle of the elbow joint was not measured but was kept constant across all the experiments. Specifically, for the left upper limb, the participant's index finger and wrist were kept at neutral position (straight), forearm supinated at 90°, and elbow flexed at 90°. The finger orientation was adjusted by changing the participant's shoulder abduction and internal rotation postures. In order to make sure that these postures were kept stationary during each session of the experiment; a forearm holder was utilized to support the wrist and an arm holder was employed to support the elbow. For the head posture, a pad was placed lateral to the head and the participant needs to tilt the head to touch the pad so that the head posture can be maintained.
The stimulus ball was placed immediately in front of the left index finger. In addition, a video display screen was placed between the participant and the tactile stimulator to provide experimental instructions and to enable the participant to report the perceived direction of motion of the ball. In setting up the experimental process, from posterior to anterior with respect to the participant's head, the eyes, video display, tactile ball, and left index fingerpad were perfectly aligned along the posterior-to-anterior axis. The allocentric reference frame (coordinate) was defined on the frontoparallel plane (Fig. 1c).

Experimental design of motion stimulation.
For each participant, the finger, forearm, and head holders were adjusted to ensure between-participant consistency of finger and head postures. For the intensity of tactile stimulation, tactile motion was presented by the miniature tactile motion stimulator 54 (Fig. 1a,b) with indentation depth of 1000 μm, and had the indentation rate of 2 mm/s which was far beyond the sensation threshold of indentation rate (>0.3 mm/s) 78 . In this setup, the discriminability of stimulus direction 54 was 11.4° ± 2.5°. Tactile motion was presented with a speed of 40 mm/s using a square wave grating ball with a wavelength of 2 mm 79 and temporal frequency of 10 Hz 80 , both of which were beyond the threshold of tactile perception 13 .
At the beginning of each trial, the participant visually fixated on a cross presented at the center of the video display. After 1 second fixation, the rotating ball was indented on the index fingertip with indentation depth of 1 mm. Tactile stimulation was then applied for 1 second in one of the 24 different directions (0° to 345° in 15° increments) at a speed of 40 mm/s. The ball was then withdrawn from the fingerpad (Fig. 2a). After the ball was moved away, the participant reported the perceived direction of motion (R i ) using a mouse to click on an appropriate point on a circle shown on the video display. After that, a gray blank appeared on the screen for 1 second (Fig. 2a left). After the participant clicked on the circle, the screen was presented in gray blank for a period of 1 second and then shown again for the following trial.
The stimulation trials were performed in accordance with a 3-by-4 factorial finger-and-head posture combination design consisting of three head postures and four finger postures. In particular, the finger postures (θ F ) were set as 90°, 60°, 30° or 0°, while the head postures (θ H ) were set as 120°, 90° or 60°; yielding a total of 12 different posture combinations (Fig. 2b). Note that parameters θ H and θ F were both defined on the allocentric reference frame. A total of 96 trials were performed for each posture combination (24 directions * 2 times * 2 blocks).
www.nature.com/scientificreports www.nature.com/scientificreports/ Definition of bias. The perceptual bias (B i ) was quantified as the difference between the reported stimulus direction (R i ) and the veridical direction (V i ) on the allocentric reference frame (B i = R i -V i ). For each posture combination, the systematic bias, S, was computed as the mean of the perceptual biases across all 24 directions, i.e., = ∑ = S B i i 1 24 1 24 . In addition, the nonsystematic bias, NS , i for the veridical direction was computed as the difference between the perceptual bias and the systematic bias, i.e., Fitting for nonsystematic bias. The nonsystematic bias was fitted as a function of the veridical direction using a cosine function with moment = 2 (Fig. S1) that yields two full oscillations in one complete direction cycle, i.e., θ = − ( ) NS A V cos 2 2 , i i p where A and θ p represent the amplitude and phase corresponding to the maximal nonsystematic bias and are free parameters determined using the least-squares fitting method.
Statistical analysis. The perceptual bias, the difference between the perceived and veridical directions on the allocentric reference frame, was first computed for each trial. For each participant, the perceptual bias for each direction of motion was the circular mean of perceptual bias across the four repetitions. The systematic bias for each posture was the mean across stimulus directions (Fig. 3). The amplitude (A) and phase (θ p ) of the nonsystematic bias were retrieved from cosine fit to the nonsystematic bias at its peak position. To evaluate the relationship between postures and biases, we applied Pearson's correlation (simple regression model) to compute the correlation between posture angles to the parameters of systematic (Fig. 4) or nonsystematic bias (Fig. 6). For each of the three posture conditions, θ F , θ H , and θ F-H , we applied Student's t test to compare the values of regression coefficients (R 2 ) between systematic and nonsystematic biases to evaluate the degree to which the two biases was modulated by the postures (Fig. 7). To examine whether there was inconsistency of goodness-of-fit (R 2 ) of cosine fit across postures, we applied repeated-measures ANOVA for each of the finger posture (θ F ), head (θ H ), and finger-head postures (θ F-H ) (Fig. S1b). We used Pearson's correlation to examine whether nonsystematic bias is a function of finger posture on the allocentric reference frame (Fig. S3), and, finally, to evaluate the relationship between systematic bias and the phase of nonsystematic bias for each participant (Fig. S4).