Humans and many other animals have the ability to make spontaneous and rapid estimates of the approximate number of items that they can see. This sense of number, or “numbersense”, is particularly important in humans, as evidence suggests that it lays the groundwork for acquiring mathematical skills.

Researchers have many questions about numbersense. Is it a kind of perception? Or does it require more active thought, like counting? Do people have the same sense of number when they view, hear or touch items that depict the same number? Having a sense of number is essential for carrying out certain actions, like the following the steps in a dance, but the connection between action and numbersense is not entirely clear.

A process called adaptation means that viewing specific stimuli for a period of time can affect what people think they see subsequently. For example, viewing large numbers of dots makes subsequent smaller groups of dots seem like they contain fewer dots than they actually do. Anobile, Arrighi et al. have now investigated the link between action and numbersense by asking volunteers to tap one hand either rapidly or slowly in one spot for a short time. The volunteers were then shown a series flashes or a cloud of dots in the region where they had been tapping and asked to estimate the number of flashes or dots.

After fast tapping, the volunteers greatly underestimated the numbers of flashes or dots that they saw; after slow tapping, they overestimated the numbers. However, if the images were shown far away from where the volunteers had been tapping, their estimates were more accurate.

Overall, the results suggest that adaptation is controlled by space-specific sensory mechanisms rather than some kind of active counting. Furthermore, numbersense appears to have a generalized form that is shared by the brain regions responsible for perception and action. Because numbersense and mathematical ability are linked, this strong connection between action and number perception may have important implications for understanding and treating math-related learning disabilities. Anobile, Arrighi et al. next plan to study how movement-driven adaptation affects numbersense in children and adults with these conditions.

Animals, including humans, estimate spontaneously and reasonably accurately the approximate quantity of arrays of objects, without recourse to other forms of representation, such as density (Cicchini et al., 2016). Even newborn infants of less than 3 days show selective habituation to number (Izard et al., 2009). There is now very good evidence in both human and non-human primates that number is encoded by intraparietal and prefrontal cortex (Castelli et al., 2006; Dehaene et al., 2003; Harvey et al., 2013; Nieder, 2005, 2012, 2016; Nieder et al., 2006; Nieder and Miller, 2004; Piazza and Eger, 2016; Piazza et al., 2004, 2007), even in numerically naive monkeys (Viswanathan and Nieder, 2013). All these studies point to the existence of a visual sense of number within a parietal–frontal network (Dehaene, 2011).

A truly abstract sense of number should be capable of encoding the numerosity of any set of discrete elements, displayed simultaneously or sequentially, in whatever sensory modality. Some evidence exists for such a generalized number sense. Neurons in the lateral prefrontal cortex (lPFC) of behaving monkeys encode numerosity for both auditory and visual sensory modalities, suggesting supra-modal numerosity processing (Nieder, 2012). Another study reported separate populations of neurons in the intraparietal sulcus (IPS) responding selectively to sequential or simultaneous numerical displays, while a third set of neurons showed numerosity selectivity for both simultaneous and sequential presentations, suggesting that the information about spatial and temporal numerosity converges to a more abstract representation (Nieder et al., 2006). There is also evidence from functional imaging in humans for a right lateralized fronto-parietal circuit activated by both auditory and visual number sequences, and that right IPS is involved in processing both sequential and simultaneous numerosity formats (Castelli et al., 2006; Piazza et al., 2006).

Psychophysical evidence showing little cost in cross-modal or cross-format matching also points to a common number sense spanning sensory modalities and formats. For example, human adults are very efficient in making cross-modal and cross-format judgments, with very little cost in either accuracy or reaction times when comparing auditory with visual temporal sequences or dot arrays (Barth et al., 2003; Brannon, 2003). Developmental work also show similar accuracy in pre-schoolers for comparing spatial array of dots either with other spatial arrays, or with sequences of sounds (Barth et al., 2005). Preferential-looking studies show infants prefer to look at screens displaying adults faces numerically matched with the soundtrack of adult voices (Jordan and Brannon, 2006), and abstract visual ensembles (shapes) numerically matched with on-going sequence of sounds (Izard et al., 2009). However, not all agree: Tokita & Ishiguchi (Tokita and Ishiguchi, 2012) reported significantly lower precision for cross-format number comparisons in adults, than for within format comparisons.

That there is little or no cost in these matches is certainly indicative of efficient transfer of information between senses, but says little about the mechanisms involved. The match is made at the decision level, so the interaction could be at any stage up to and including decision mechanisms. One of the more powerful psychophysical techniques to probe mechanisms is adaptation (Mollon, 1974; Thompson and Burr, 2009). Number, like most other primary visual attributes, is also highly susceptible to adaptation (Schwiedrzik et al., 2016; Burr and Ross, 2008): visually inspecting for a few seconds a large number of items results in the perceived numerosity of a subsequent ensemble to be strongly underestimated, and vice-versa after adaptation to low numbers (Burr and Ross, 2008). More recently we have shown that adaptation to numerosity also occurs with sequentially presented stimuli, and that the adaptation effects are both cross-modal and cross-format (Arrighi et al., 2014): adapting to sequences of tones affects the perceived numerosity of a subsequently presented series of flashes (and vice versa), and adapting to sequences of flashes affects the perceived numerosity of spatial arrays of items. Importantly, the adaptation was spatially selective, in external rather than eye-centered coordinates, suggesting it has a perceptual rather than cognitive basis. fMRI studies have demonstrated BOLD activity selective for adaptation to numerosity in the human parietal sulcus (Piazza et al., 2004, 2007; Castaldi et al., 2014; He et al., 2015).

All these studies strongly suggest that the number sense is a high-generalized system, capable of combining numerical information from different senses, and across different formats. Numerical information is also relevant for the production of specific action sequences, from dance routines to more simple repetitive behavioral tasks. A few studies point to an interconnection between numerosity and motor control. For example, neurons in area 5 of the superior parietal lobule of monkey show a clear selectivity for the number of self-produced actions, and inactivation of the area impedes number-based tasks (Sawamura et al., 2002, 2010). Other work has shown that the left ventral premotor cortex is activated by counting successive sensory stimuli (Kansaku et al., 2007), and that the human cerebellum shows strong activation for simple numerical calculations (Arsalidou and Taylor, 2011).

The existence of anatomical and functional connections between number and action-generation systems raise the possibility that number-for-action could be encoded within a truly abstract numerosity mechanism. To test this idea, we measured cross-adaptation between motor repetitions and perception of numerosity. The results show that adapting to self-generated action does affect the representations of numerosity of external events, both sequential (series of flashes) and simultaneous (dots ensembles), and that the adaptation is spatially selective in external, not hand-centered coordinates.

Each trial began with a motor adaptation phase in which participants performed tapping movements for six seconds, under two different conditions (tested on separate sessions): 'high adaptation', where participants were asked to tap as quickly as possible (average 5–6 taps/s); and 'low adaptation' where they tapped more slowly (average 1.12 taps/s: see Materials and methods and Figure 5). After the adaptation phase, the test stimulus – either a sequence of flashes or a cloud of dots (tested on separate sessions) – was randomly displayed either to the same side of the screen where the hand had been tapping, or to the symmetrically opposite side. Participants estimated the numerosity of the test stimulus, which varied randomly from trial to trial within the range 6–14. To minimize sensory feedback, participants were placed in a dark room and wore soundproof headphones, and tapped in mid-air behind the computer screen without touching any surface.

The results are shown in Figure 1. Panels A & B show numerosity estimates averaged over all subjects as a function of the physical numerosities displayed. When the test stimulus was displayed on the right side of the screen (where the adaptation had occurred), rapid tapping caused a consistent underestimation of the numerosity of the test, while slow tapping caused an overestimation. The adaptation effects were similarly strong for when the test was a sequence of flashes (Figure 1A) as when it was an array of dots presented simultaneously (Figure 1B). Interestingly, the effect occurred only when the stimuli were presented on the same side as the tapping hand (the right side): when presented on the other (left) side, adaptation produced no consistent effect (Figure 1A and B open symbols).

Figure 1

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We defined adaptation magnitude as the percentage difference in perceived numerosity after adaptation to fast or slow tapping, averaged over all numerosities. For sequential and simultaneous presentations, the adaptation magnitude averaged across subjects (filled symbols in Figure 1C and D) was around 20 and 25% respectively for stimuli presented to the adapted location, a very strong effect. For stimuli presented to the unadapted location, the average effect was only 4 & 2%.

We also calculated adaptation magnitude for individual subjects. Figure 1C and D plot adaptation magnitudes for the congruent condition (where the visual stimuli were presented to the right side), against the incongruent condition (stimuli to the left side). All subjects showed a significant effect in the congruent condition (error bars 1 sem), but very little effect in the incongruent condition. ANOVA showed that the congruent conditions were highly significant (F_(1,32) = 70.219, p = 0.001, η² = 0.29, Cohen’s d = 1.278 and F_(1,48) = 47.176, p = 0.0004, η² = 0.217, Cohen’s d = 1.062 for sequential and simultaneous condition respectively), while the non-congruent conditions were weak and insignificant (≈ 4% effect, F_(1,32) = 1.403, p = 0.302, η² = 0.007, Cohen’s d = 0.167 and ≈ 2% effect, F_(1,48) = 0.919, p = 0.375, η² = 0.008, Cohen’s d = 0.179). That the adaptation is spatially specific suggests it is of a perceptual rather than cognitive nature, and unlikely to result from a response bias or any other generalized artifact.

This first experiment revealed two clear results: that motor adaptation affects visual estimates of numerosity, for both sequential and simultaneous displays; and that the adaptation is spatially specific. The spatial specificity suggests that the effect is not a high-level, cognitive phenomenon (such as 'internal counting'), but perceptual in nature, mediated by neural mechanisms with circumscribed receptive fields. To verify the robustness of the spatial selectivity, and to understand it better, we repeated the experiment with a new subject pool, changing the tapping hand and location. In this experiment we tested only the simultaneous presentation, as this is the most revealing (and surprising) result.

The violet symbols of Figure 2A replicate the results of the previous experiment, tapping with the right (dominant) hand and testing on both right and left sides (randomly interleaved): the adaptation effect was again strong for stimuli presented on the same side (filled symbols), and non-existent for stimuli on the other side (open symbols) (F_(1,40) = 70.207, p = 0.000397, η² = 0.116, Cohen’s d = 0.724; F_(1,40) = 2.036, p=0.213, η² = 0.0019, Cohen’s d = 0.0873; for adapted and unadapted location respectively). The red symbols of Figure 2B show the results for tapping on the left with the left (non-dominant) hand: again the effects occurred only for visual stimuli presented on the congruent side (left), although they were somewhat weaker (F_(1,40) = 9.305, p = 0.028, η² = 0.05, Cohen’s d = 0.4588; F_(1,40) = 0.265, p = 0.629, η² = 0.001, Cohen’s d = 0.0633; for adapted and unadapted location respectively). Figure 2C shows results for tapping with the dominant (right) hand on the left side of the screen. Here, adaptation was found only for stimuli presented to the left side of the screen, suggesting that it is spatially selective in external rather than hand-centered coordinates (F_(1,40) = 36.840, p = 0.002, η² = 0.104, Cohen’s d = 0.6814; F_(1,40) = 1.380, p = 0.293, η² = 0.0023, Cohen’s d = 0.096; for adapted and unadapted location respectively). Figure 2D shows the results for all six subjects. There is some variability between subjects, particular in the crossed condition, where one subject showed adaptation to stimuli on the right after tapping on the left with the right hand, but by and large the individual data reinforce the group data.

Figure 2

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In the previous experiment, subjects tapped in mid air to minimize sensory feedback. In the next series of experiments we manipulated the amount of sensory feedback in the adaptation phase to examine interactions between sensory and motor signals. In the first condition (tactile only), subjects tapped a mouse behind the monitor, allowing for tactile feedback (Figure 3A). The adaptation effect in this condition was strong, around 20% (F_(1,40) = 743.738, p = 0.0001, η² = 0.203, Cohen’s d = 1.009). In the next condition (visual and tactile), the monitor accompanied each mouse-tap with a flash, to give visual as well as tactile feedback. Despite the extra feedback, adaptation remained around 20%, (F_(1,40) = 36.746, p = 0.002, η² = 0.184, Cohen’s d = 0.949) as shown in panel B of Figure 3. The last adaptation condition (visual only) comprised a sequence of visual flashes whose rates were determined by the adapting motor routine of the previous conditions (visual and tactile). Again, the adaptation effect was found to be strong (F_(1,40) = 61.740, p = 0.001, η² = 0.230, Cohen’s d = 1.093), and similar to the other conditions, around 20% (Figure 3C), making these three adaptation conditions equally effective as tapping in mid-air (F_(4,29) = 0.475, p = 0.754, η² = 0.07, Cohen’s d = 0.548: see Figure 3D).

Figure 3

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We also verified the results with a two-alternative forced-choice technique. Subjects adapted to high or low tapping rates, as in the first experiment (no tactile or visual feedback), then two clouds of dots were simultaneously presented to the right (adapted) and left (unadapted) positions. The numerosity of each stimulus varied from trial to trial over the range 5–20, and subjects indicated which stimulus appeared more numerous. Figure 4A plots average responses as a function of the difference between the right and the left stimulus (normalized to the average of the two numerosities), to yield psychometric functions. The effect of adaptation was again clear: adapting to low tapping rates shifts the curve to the left (compared with baseline), consistent with an overestimation of the perceived numerosity (t₍₅₎ = 3.285, p = 0.021, Cohen’s d = 1.101) and high tapping rates caused the opposite effect, even if weaker (t₍₅₎ = 1.237, p = 0.27, Cohen’s d = 0.558). The differences in the points of subjective equality (PSEs, given by the 50% point of the curves) of the two adapting conditions again gives an index of magnitude of adaptation, around 15%. Figure 4B shows the PSEs for adaptation to the two conditions. Despite some variability amongst subjects the effects are quite robust and statistical significant as shown by a two-tailed paired t-test: t₍₅₎ = 3.56, p = 0.029, Cohen’s d = 1.612. This experiment confirms the main results with a different technique, and also confirms the spatial selectivity of the adaptation: if adaptation was not spatially selective, it would work equally on the presentations to the left and right sides, annulling the effect.

Figure 4

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This study shows that estimates of numerosity, both sequential and simultaneous, are strongly biased after adapting to repetitive finger tapping: rapid tapping decreases apparent numerosity, slow tapping increases it. The effect is spatially selective, primarily in external rather than hand-centered coordinates.

There has been a long-standing debate as to whether adaptation effects operate on numerosity per se, or via texture-density mechanisms (Burr and Ross, 2008; Anobile et al., 2014, 2015, 2016; Bell et al., 2015; Dakin et al., 2011; Durgin, 1995, 2008; Morgan et al., 2014; Ross and Burr, 2012; Ross, 2010; Tibber et al., 2012, 2013). A similar argument could be made here: that the adaptation was to temporal frequency, rather than to numerosity. As with spatial adaptation, there are many reasons to suggest that this is unlikely. However, the cross-format adaptation (adapt to tapping sequence and test on dot array) clearly rules out this possibility: the spatial arrays are not temporally modulated. It is numerosity that is being adapted, not temporal frequency.

The current results reinforce the many previous studies (Izard et al., 2009; Nieder, 2012; Nieder et al., 2006; Barth et al., 2003; Brannon, 2003; Barth et al., 2005; Jordan and Brannon, 2006; Arrighi et al., 2014; Jordan et al., 2005) discussed in the introduction that point to the existence of a generalized sense of number. Most of these studies relied principally on cross-modal comparisons of number, which could occur at any processing stage, up to and including decision mechanisms. The spatial selectivity shown in our study suggests that the interaction is perceptual rather than cognitive: adapting on the left side did not affect stimuli on the right, and vice versa. Importantly, the specificity was in external coordinates, as adapting the left field with the right hand caused adaptation for visual stimuli presented to the left, not the right visual field. This complements nicely the result of our previous study (Arrighi et al., 2014), where we showed that adaptation to visual sequences affects number perception of both sequential and simultaneous presentations, in a spatially selective manner. Interspersing an eye-movement between adaptation and test showed that the adaptation was spatially specific in external rather than eye-centered coordinates: as the current study shows the selectivity is external, not hand-centered. It would be interesting to look at the spatial tuning of the adaptation on a finer grain, to define the size of the adaptation field. The present study shows that the adaptation is at least broadly tuned, confined to a particular hemifield. It would be very informative to determine whether there was also selectivity within each hemifield, and on how fine a grain.

Some may find the spatial selectivity of the adaptation difficult to reconcile with the concept of a generalized, abstract sense of number. However, cross-modal effects can also show spatial selectivity. For example, cross-modal integration of visual and auditory (or tactile) information occurs only if the stimuli are spatially coincident (within certain bounds) (Slutsky and Recanzone, 2001). Similarly event time, which certainly transcends modalities, and also seems to be coded in parietal cortex (Leon and Shadlen, 2003), is affected by motion adaptation, in a spatially selective manner (Burr et al., 2007; Fornaciai et al., 2016). Interestingly, the spatial selectivity of the adaptation is in external – not eye-based – coordinates as we observed for number, here and in the previous study (Arrighi et al., 2014).

We tested adaptation to action under various feedback conditions: visual and tactile, only visual, only tactile, and minimal feedback. All conditions produced similar amounts of adaptation. In the 'minimal feedback' conditions, where subjects tapped in mid-air, there was no tactile feedback from hitting a surface. We could not, however, remove all forms of kinaesthetic feedback, and therefore cannot be certain whether the adaptation signal was the intension to move, or the sensory proprioceptive feedback from the finger. But both are signals about action, whether they are 'inflow' or 'outflow'. It is interesting that this condition with reduced perceptual feedback produced the same amount of adaptation, as did the conditions with visual and/or tactile feedback. It is also interesting that the vision-only condition produced similar adaptation.

Many studies have suggested that vision and action are linked (Arrighi et al., 2011; Goodale and Milner, 1992). This study is a further clear example of their interconnection, in the encoding the numerosity of internally generated actions and externally generated events.

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Author details

1. Giovanni Anobile

   1. Department of Neuroscience, Psychology, Pharmacology and Child Health, University of Florence, Florence, Italy
   2. Department of Developmental Neuroscience, Stella Maris Scientific Institute, Pisa, Italy

   Contribution

   GA, Wrote the paper, Analysis and interpretation of data, Performed research, Designed the research, Conception and design, Acquisition of data

   Contributed equally with

   Roberto Arrighi

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0003-2796-0661

2. Roberto Arrighi

   Department of Neuroscience, Psychology, Pharmacology and Child Health, University of Florence, Florence, Italy

   Contribution

   RA, Wrote the paper, Performed research, Designed the research, Conception and design, Acquisition of data, Analysis and interpretation of data

   Contributed equally with

   Giovanni Anobile

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0002-5435-6729

3. Irene Togoli

   Department of Neuroscience, Psychology, Pharmacology and Child Health, University of Florence, Florence, Italy

   Contribution

   IT, Performed research, Designed the research, Acquisition of data, Analysis and interpretation of data

   Competing interests

   The authors declare that no competing interests exist.

4. David Charles Burr

   1. Department of Neuroscience, Psychology, Pharmacology and Child Health, University of Florence, Florence, Italy
   2. Institute of Neuroscience, National Research Council, Pisa, Italy
   3. School of Psychology, University of Western Australia, Perth, Australia

   Contribution

   DCB, Wrote the paper, Designed the research, Conception and design, Analysis and interpretation of data

   For correspondence

   dave@in.cnr.it

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0003-1541-8832

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