Sensorimotor mechanisms selective to numerosity: evidence from individual differences

Reviewer #1 Public Review

Anobile and colleagues present a manuscript detailing an account of numerosity processing with an appeal to a two-channel model. Specifically, the authors propose that the perception of numerosity relies on (at least) two distinct channels for small and large numerosities, which should be evident in subject reports of perceived numerosity. To do this, the authors had subjects reproduce visual dot arrays of numerosities ranging from 8 to 32 dots, by having subjects repetitively press a response key at a pre-instructed rate (fast or slow) until the number of presses equaled the number of perceived dots. The subjects performed the task remarkably well, yet with a general bias to overestimate the number of presented dots. Further, no difference was observed in the precision of responses across numerosities, providing evidence for a scalar system. No differences between fast and slow tapping were observed. For behavioral analysis, the authors examined correlations between the Weber fractions for all presented numerosities. Here, it was found that the precision at each numerosity was similar to that at neighboring numerosities, but less similar to more distant ones. The authors then went on to conduct PCA and clustering analyses on the weber fractions, finding that the first two components exhibited an interaction with the presented numerosity, such that each was dominant at distinct lower and upper ranges and further well-fit by a log-Gaussian model consistent with the channel explanation proposed at the beginning.

Overall, the authors provide compelling evidence for a two-channel system supporting numerosity processing that is instantiated in sensorimotor processes. A strength of the presented work is the principled approach the authors took to identify mechanisms, as well as the controls put in place to ensure adequate data for analysis. Some questions do remain in the data, and there are aspects of the presentation that could be adjusted.

-The use of a binary colormap for the correlation matrix seems unnecessary. Binary colormaps between two opposing colors (with white in the middle) are best for results spanning positive and negative values (say, correlation values between -1 and +1), but the correlations here are all positive, so a uniform colormap should be applied. I can appreciate that the authors were trying to emphasize that a 2+ channel system would lead to lower correlations at larger ratios, but that's emphasized better in the numerical ratio line plots.

-In Figure 1, the correlation matrices in Figure 1 appear blurred out. I am not sure if this was intentional but suspect it was not, and so they should appear like those presented in Figure 3.

-It's notable that the authors also collected data on a timing task to rule out a duration-based strategy in the numerosity task. If possible, it would be great to have the author also conduct the rest of the analyses on the duration task as well; that is, to look at WF correlation matrices/ratios as well as PCA. There is evidence that duration processing is also distinctly sensorimotor, and may also rely on similar channels. Evidence either for or against this would likely be of great interest.

-For the duration task, there was no fast tapping condition. Why not? Was this to keep the overall task length short?

-The number of subjects/trials seems a bit odd. Why did some subjects perform both and not others? The targets say they were presented "between 25 and 30 times", but why was this variable at all?

-For the PCA analysis, my read of the methods and results is that this was done on all the data, across subjects. If the data were run on individual subjects and the resulting PCA components averaged, would the same results be found?

-For the data presented in Figure 2, it would be helpful to also see individual subject data underlaid on the plots to get a sense of individual differences. For the reproduced number, these will likely be clustered together given how small the error bars are, but for the WF data it may show how consistently "flat" the data are. Indeed, in other magnitude reproduction tasks, it is not uncommon to see the WF decrease as a function of target magnitude (or even increase). It may be possible that the reason for the observed findings is that some subjects get more variable (higher WFs) with larger target numbers and others get less variable (lower WFs).

-Regarding the two-channel model, I wonder how much the results would translate to different ranges of numerosities? For example, are the two channels supported here specific to these ranges of low and high numbers, or would there be a re-mapping to a higher range (say, 32 to 64 dots) or to a narrower range (say 16 to 32 dots). It would be helpful to know if there is any evidence for this kind of remapping.

Reviewer #2 Public Review

The authors wish to apply established psychophysical methods to the study of number. Specifically, they wish to test the hypothesis - supported by their previous work - that human sensorimotor processes are tuned to specific number ranges. In a novel set of tasks, they ask participants to tap a button N times (either fast or slow), where N varies between 8 and 32 across trials. As I understood it, they then computed the Weber fraction (WF) for each participant for each number and correlated those values across participants and numbers. They find stronger correlations for nearby numbers than for distant numbers and interpret this as evidence of sensorimotor tuning functions. Two other analyses - cluster analyses and principal component analyses (PCA) - suggest that participants' performance relied on at least 2 mechanisms, one for encoding low numbers of taps (around 10) and another for encoding larger numbers (around 27).

Strengths

Individual differences can be a rich source of scientific insight and I applaud the authors for taking them seriously, and for exploring new avenues in the study of numerical cognition.

Weaknesses

Inter-subject-correlation
The experiment "is based on the idea that interindividual variability conveys information that can reveal common sensory processes (Peterzell & Kennedy, 2016)" but I struggled to understand the logic of this technique. The authors explain it most clearly when they write "Regions of high intercorrelation between neighbouring stimuli intensity can be interpreted to imply that sets of stimuli are processed by the same (shared) underlying channel. This channel, while responding relatively more to its preferred stimulus, will also be activated by neighbouring stimuli that although slightly different from the preferred intensity, are nevertheless included in the same response distribution." As I understood it, the correlations are performed "between participants, for all targets values" - meaning that they are measuring the extent to which different participants' WFs vary together. But why is this a good measure of channels? This analysis seems to assume that if people have channels for numerical estimation, they will have the same channels, tuned to the same numerical ranges. But this is an empirical question - individual participants could have wildly different channels, and perhaps different numbers of channels (even in the tested range). If they do, then this between-subject analysis would mask these individual differences (despite the subtitle).

Different channels
I had trouble understanding much of the analyses, and this may account for at least some of my confusion. That said, as I understand it, the results are meant to provide "evidence that tuned mechanisms exist in the human brain, with at least two different tunings" because of the results of the clustering analysis and PCA. However, as the authors acknowledge, "PCA aims to summarize the dataset with the minimal number of components (channels). We can therefore not exclude the possible existence of more than two (perhaps not fully independent) channels." So I believe this technique does not provide more evidence for the existence of 2 channels as for the existence of 4 or 8 or 11 channels, the upper bound for a task testing 11 different numbers. If we can conclude that people may have one channel per number, what does "channel" mean?

Several other questions arose for me when thinking through this technique. If people did have two channels (at least in this range), why would they be so broad? Why would they be centered so near the ends of the tested range? Can such effects be explained by binning on the part of the participants, who might have categorized each number (knowingly or not) as either "small" or "large"? Whereas the experiment tested numbers 8-32, numbers are infinite - How could a small number of channels cover an infinite set? Or even the set 8-10,000? More broadly, I was unsure what advantages channels would have - that is - how in principle would having distinct channels for processing similar stimuli improve (rather than impede) discrimination abilities?

No number perception
I was uncertain about the analogy to studies of other continuous dimensions like spatial frequency, motion, and color. In those studies, participants view images with different spatial frequency, motion, or color - the analogy would be to see dot arrays containing different numbers of dots. Instead, here participants read written numerals (like "19"), symbols which themselves do not have any numerical properties to perceive. How does that difference change the interpretation of the effects? One disadvantage of using numerals is that they introduce a clear discontinuity: Our base-10 numerical system artificially chunks integers into decades, potentially causing category-boundary effects in people's reproductions.

Sensorimotor
The authors wished to test for "sensorimotor mechanisms selective to numerosity" but it's not clear what makes their effects sensorimotor (or selective to numerosity, see below). It's true they found effects using a tapping task (which like all behavior is sensorimotor), but it's not clear that this effect is specific to sensorimotor number reproduction. They might find similar effects for numerical comparison or estimation tasks. Such findings would suggest the effect may be a general feature of numerical cognition across modalities.

Specific to numbers
The authors argue that their effects are "number selective" but they do not provide compelling evidence for this selectivity. In principle, their main findings could be explained by the duration of tapping rather than the number of taps. They argue this is unlikely for two reasons. The first reason is that the overall pattern of results was unchanged across the fast and slow tapping conditions, but differences in duration were confounded with numerosity in both conditions, so the comparison is uninformative. (Given this, I am not sure what we stand to learn by comparing the two tapping speeds.) The second reason is that temporal reproduction was less precise in their control condition than numerical reproduction, but this logic is unclear: Participants could still use duration (or some combination of speed and duration) as a helpful cue to numerosity, even if their duration reproductions were imperfect.

If the authors wish to test the role of duration, they might consider applying the same analytical techniques they use for numbers to their duration data. Perhaps participants show similar evidence for duration-selective channels, in the absence of number, as they do for other non-numerical domains (like spatial frequency).

Theories of numerical cognition. An expansive literature on numerical cognition suggests that many animals, human children, and adults across cultures have two systems for representing numerosity without counting - one that can represent the exact cardinality of sets smaller than about 4 and another that represents the approximate number of larger sets (but see Cheyette & Piantadosi, 2020). The current paper would benefit from better relating its findings to this long lineage of theories and findings in numerical approximation across cultures, ages, and species.