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Gaze, pointing, and reaching movements are thought to provide a window to internal cognitive states. In the case of numerical cognition, it has been found that the left-right deviation of a reaching movement is modulated by the relative magnitude of values in a number comparison task. Some have argued that these patterns directly reflect the representation of a logarithmically compressed mental number line (direct mapping view). However, other studies suggest that the modulation of motor outputs by numerical value could be a more general decision-making phenomenon (response competition view). Here we test the generality of interactions between the motor system and numerical processing by comparing subjects' reach trajectories during two different nonverbal tasks: numerosity comparison and facial expression comparison. We found that reaching patterns were practically identical in both tasks – reach trajectories were equally sensitive to stimulus similarity in the numerical and face comparisons. The data provide strong support for the response competition view that motor outputs are modulated by domain-general decision processes, and reflect generic decision confidence or accumulation of evidence related to mental comparison.

When asked to point to the larger of two numbers, one presented to the left and another to the right, subjects trajectories take a medial route when the numerical distance is close, and a more direct one when it is far (

Although prior research suggests that reach trajectories are affected by cognitive processes across a variety of domains, a strong test of the domain-generality of these effects would be to compare performance with stimuli from different domains within a common task. Such a test is important for determining whether number-based modulations of motor responses are domain-specific (

There are two hypotheses about how number representations relate to reach trajectories: 1) direct mapping, and 2) response competition (

Previous studies using number stimuli in reach tasks have shown that reach follows a numerically-modulated spatial path resembling a logarithmically compressed mental number line (

In the response competition view, what influences reach is not the values of the numerical representations but how the response options compete during decision-making (

In brief, what distinguishes the direct mapping hypothesis from the response competition hypothesis is that 'direct mapping' is a domain-specific hypothesis about the underlying nature of number representations whereas 'response competition' is a domain-general hypothesis about decision-making.

We investigated reaching dynamics across domains by comparing behavior on a reach task with stimuli from two different categories: numbers and faces. The prediction under a 'direct mapping' of number to reach is that number stimuli will logarithmically modulate reach trajectories and facial stimuli will not. Faces do not have an inherently uni-dimensional spatial representation or logarithmic spatial scaling (

Alternatively, under 'response competition' the similarity of the stimuli should modulate reach trajectories in comparable ways within both the face and number tasks (including the possibility of logarithmically scaled reach). Response competition affects decision-making similarly across all comparison tasks (

Here we examined subjects' reach trajectories during mental comparisons of numerical values and facial expressions. Numerical and facial judgments provide a strong test of domain-general effects because they are known to engage distinct perceptual, cognitive, and neural circuits (

22 participants were recruited for the number task (13 female; Age:

Participants saw two arrays of dots, presented on the left and right side of a computer screen. After 200 ms, the arrays disappeared and subjects had to report with their index finger the stimulus that had more dots (

Number and face comparison task. Participants were asked either to report the larger number or the happier face.

Subjects sat approximately 55 cm from the screen (monitor information: 22” diagonal size, 1920x1080 resolution, 120 Hz). To initiate each trial, they had to hold a button with their right index finger, located 29 cm from the screen, centered along its midline (VPixx VP-BB-3 button box, dimensions of box: = 23 x 13 x 6 cm). Once the button was held for 1 s, two squares appeared—these two squares were the possible target landing spots for the reaching response. After 700 ms, the non-overlapping arrays of dots appeared on the screen for 200 ms, one to the left and the other to the right side of the screen’s vertical midline. Participants were instructed to use their index finger to point to the target that had the greater number of dots. Movement sampling stopped when the index finger was 4 cm from the screen (this was a convenient distance used to avoid technical problems related to marker occlusion and signal fading). Feedback was not provided. Subjects were allowed to release the button and initiate the pointing response to one of the response squares at their own pace. The task consisted of 24 training trials and 420 test trials. Test trials were separated into 4 blocks, with 105 trials in each block. We presented numerical values in 5 numerical ratios (0.1, 0.25, 0.5, 0.75 and 0.9), with a maximum of 25 dots per array. Each ratio was shown 84 times. There were two trial types: 1) equal dot size for both sets (diameter: 40 pixels) and 2) equal cumulative area for both sets. Both types of trials appeared equally often. The order of the ratios and type of trial were randomized, as well as the side that had more dots (i.e., the response side). Dots were randomly configured within each array.

The experimental procedure was identical to the number experiment but instead of arrays of dots, subjects saw two pictures of the same person with different expressions, ranging from neutral to happy (

Example of face stimuli. A: An example of all the ratios presented for one of the faces. B: 3 samples of the 42 pairs used in the experiment.

To create a continuum of neutral to happy, we morphed the neutral expression with the happy expression in 100 steps. On any given trial, subjects saw a picture with the original happy expression and a morphed picture. Ratios were defined by the amount of morph-steps between the stimuli. For example, a 0.1 ratio trial was defined as a comparison between a neutral expression that was 10 steps into the morph versus the standard happy expression (

Response trajectories in which the subject pulled back or the marker was occluded were discarded (5.8% of trajectories). Of the remaining trajectories, only correct test trials were considered for all data analyses of reach and response times. Incorrect trials were few and mostly present in the hardest ratios (0.75 and 0.9). There was no additional information in these trials as response times and reach were slow and medial, as for correct trials. A correct trial was defined as a trajectory with 50% or more of points, including the end point, on the appropriate response side as defined by the midline of the screen (90% of trajectories). Considering only correct trials with this strict criterion is a conservative approach because any modulation is related to motor plans that were heading toward the target rather than the incorrect choice.

Trajectory coordinates were standardized to have 101 equally time-spaced points by means of linear interpolation (following standard practices in the literature e.g.

Stimulus similarity effects (based on numerical and face ratios) were tested with

All statistical tests and analysis were carried out with R (

Overall, accuracy was equivalent in the number and face task (Average accuracy number = 90%; faces = 91%). An ANOVA with Task (number and faces) and Ratio (0.1 to 0.9) showed no main effect of Task (^{2}_{g} = .018^{i}), a main effect of Ratio (^{2}_{g} = .884) and an interaction between Task and Ratio (^{2}_{g} = .056).

Performance and kinematics. Shading represents 2 s.e.m.

For response time (RT), there was a significant effect of Task (^{2}_{g} = .289) and Ratio (^{2}_{g} = .065). No effect of Side of Response (^{2}_{g} < .001) nor any interaction (all

Since RT is defined as the time between stimulus onset and liftoff, the slower RT for faces can be at least partly explained by differences in how the stimuli are encoded, taking less time for arrays of dots than facial expressions (

Discrete motor output was similar between the facial expression and numerosity tasks (^{2}_{g} < .001; Movement Time: ^{2}_{g} = .033) and Side (Velocity: ^{2}_{g} = .002; Movement Time: ^{2}_{g} = .029). There was no effect of Task (Velocity: ^{2}_{g} < .001; Movement Time: ^{2}_{g} = .030) nor any interaction (all

We estimated Bayes factors for the ANOVAs with the BayesFactor package for R (

The reach trajectories revealed clear ratio-based gradients during facial expression and numerosity judgments in the functional ANOVAs testing the effect of Ratio across the reach trajectories for each task (

Reaching trajectories in the number and face task. Negative values are for the right side. The heat vectors above and below represent point wise significance levels for left and right ratio-dependent effects in reaching, respectively. Error shading is 2 s.e.m. As indicated by the effect of ratio on trajectory, both the number and face judgments elicited similarity-dependent spatial gradients in motor responses.

To characterize the pattern of spatial positioning we tested for compressive scaling in the modulation of reach trajectories by the stimuli. Compressive effects are known to emerge during numerical comparison (

In Equation 1, slope is the rate of growth of logarithmic compression and baseline is an offset that determines how close horizontal positions are to the midline (e.g. if ratio = 0, then baseline is the maximal distance to the midline in trials with perfect information about numerical difference). An example application of this function to a reach trajectory is shown in ^{2} number task left: ^{2} face task left:

Spatial pattern of movement as a function of ratio along the average trajectory. A: Example of the fitting procedure for the left target in the number task. Right plot is a zoom of positions at the dashed rectangle (time slice 50). Circles are data and the solid line is the best-fit function. B: Main summary measures of the fitting procedure at each point of time for the left and right target. The heat bar at the right of each plot is for the R^{2} measure of fit. Significance is colored with orange, non-significant with white.

Even though both tasks were well characterized by the same ratio-modulated reach pattern, the average slope of logarithmic positioning over time was larger in the number task than the face task. The baseline, which determines the maximal distance from midline at each point of time, did not differ between tasks. This was confirmed with unpaired t-tests comparing the average parameters from Equation 1 across time for each of the tasks (

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Slight differences in logarithmic slope between tasks can be explained by the fact that the number task had, on average, greater spread in the reach trajectories across ratios. A repeated measures ANOVA of Task x Ratio x Side over subjects’ mean horizontal reach positions revealed an interaction between Task and Ratio (^{2}_{g} = .006;) as well as a main effect of Ratio (^{2}_{g} = .124). The main effects of Task (^{2}_{g} = .008), Side (^{2}_{g} = .036), and the remaining interactions were not significant (all p values for the interactions > .07). This indicates that trajectories were more extended in space across ratios during numerical judgments than facial judgments and this affected the slope. Because mean horizontal positions measure how attractive the other option was, the greater spread in the number task suggests that participants were overall more confident at lift-off during the face judgments compared to the number judgments. This interpretation also is consistent with the slightly higher accuracy reported for the face task. However, the important point is that the relative positions of the reach trajectories are both explained by a common compressive model in the face and number tasks (

One final observation about the logarithmic model is that it gave on average slightly better logarithmic fits to the left than to the right in the number task (R^{2}: L: ^{2}: L: ^{2} measure. The cause of the greater spread on the left side during the number task is unclear but it is unlikely to be related to number line representation (

We found that motor output was similarly modulated by stimulus similarity during judgments in two distinct domains: numerical and facial processing. The speed, timing, and spatial pattern of subjects' motor responses were similar in almost every way during facial and numerical judgments. A Bayes factor analysis showed equivalent velocity and movement time patterns for the face and number tasks, reach trajectories in both tasks were significantly modulated by stimulus difficulty ratios, and a model comparison analysis over reach patterns showed that a compressive, logarithmic model best explains behavior for both tasks. Exceptions to the pattern of similarity included differences between the number and face tasks in the relative degree of compression between response sides and differences in the overall speed and accuracy of responding. However, we showed that the similarities between conditions far outweighed the differences and that the observed differences between conditions do not distinguish the direct mapping and response competition hypotheses. Overall, the data support the conclusion that a domain-general decision-making mechanism modulates communication between cognitive computations, such as numerical and emotional intensity, and motor planning processes. We now discuss the results in light of prior studies claiming domain-specific representations directly mapping to reach.

In the number domain, the view that domain-specific representations can be captured with reach dynamics has been considered in at least three studies (

The second study arguing for a direct mapping of number to reach showed that reaching gestures reflect logarithmically compressed representations of numbers (

A third study reported that reach was a window to symbolic and non-symbolic number formats (^{ii}. In one condition these numbers were Arabic numerals, and in another condition they were array of dots. The main result was that Arabic numerals always produced a decisive reach towards the side with larger target probability while the reach elicited by the array of dots scaled with numerical distance. Because of this difference it was claimed that number formats project to the motor system and influence reaching trajectories.

This interpretation hinged on the notion that reaching trajectories are modulated by number formats. However, the effect of format was weak because reaching trajectories were similar in symbolic and non-symbolic formats for low numbers and only differed at larger numbers (further details in

The results of the present study are most consistent with the response competition hypothesis (

Recent research on the mental number line reported that logarithmic compression can be explained by adaptive decision-making (

An open question is whether reach modulation by degree of cognitive difficulty reflects pre-decision (encoding + pre-threshold accumulation of evidence) or post-decision processing (motor parameter settings + post-threshold accumulation of evidence) (see similar debate in

To answer the question of what is in a reach, we argue that motor plans inherit normalized information from domain-general mental comparison processes. This can explain the similarity of reach behaviors in the number and face tasks, and explain why trajectories are modulated by comparison difficulty across different cognitive domains. Reach patterns from mental comparison do not uniquely reveal a direct mapping of number to space but instead reveal how mental comparisons evolve in the mind.

A critical test for strong claims of domain-specific number effects in reach is to contrast results between numerical tasks and a comparable tasks in a non-numeric domain. This type of approach has been implemented to study the SNARC effect: the finding that in the Western educated mind low numbers are placed to the left and large numbers to the right (

In both tasks, ratio effects were mostly noticeable in the horizontal coordinate (first row, color conventions for ratios as in main text). Also, reach was roughly optimal in the horizontal and depth axis i.e. the orange trace shows a trajectory that minimizes jerk (

Funding was provided by the James S. McDonnell Foundation (220020300) and the National Science Foundation (Education Core Research Grant DRL-1459625) to JC.

η^{2}_{g} is the generalized eta squared, a measure of effect size. Criteria: .02 < η^{2}_{g} < .13 small effect; .13 < η^{2}_{g} < 0.26 medium effect; η^{2}_{g} > .26 large effect (

The probability was uniformly determined based on the numbers. In the example, the left side had 3/8 probability and the right 5/8 of probability. More details in Chapman et al., 2014.

The authors have declared that no competing interests exist.

The authors have no support to report.