Numbers and the ability to count and calculate are an essential part of human culture. They are part of everyday life, featuring in calendars, computers or the weekly shop, but also in some of humanity’s biggest achievements: without them the pyramids or space travel would not exist. A precursor of sophisticated mathematical skill could reside in a simpler mental ability: the capacity to assess numerical quantities at a glance. This ‘number sense’ appears in humans in early childhood and it is also present in other animals, but it is still poorly understood.

Brain imaging techniques have identified the parts of the brain that are active when perceiving numbers or making calculations. As techniques have advanced, it has become possible to resolve fine differences in brain activity that occur when people switch their attention between different visual tasks. But how exactly does the human brain process visual information to make sense of numbers? One theory suggests that humans use visual cues, such as the size of a group of objects or how densely packed objects are, to estimate numbers. On the other hand, it is also possible that humans can sense number directly, without reference to other properties of the group being observed.

Castaldi et al. presented twenty adult volunteers with groups of dots and asked them to focus either on the number of dots or on the size of the dots during a brain scan. This approach allowed the separation of brain signals specific to number from signals corresponding to other visual cues, such as size or density of the group. The experiment revealed that brain activity changed depending on the number of dots displayed. The signal related to number became stronger when people focused on the number of dots, while signals related to other properties of the group remained unchanged. Moreover, brain signals for number were observed at the very early stages of visual processing, in the parts of the brain that receive input from the eyes first.

These results suggest that the human visual system perceives number directly, and not by processing information about the size or density of a group of objects. This finding provides insights into how human brains encode numbers, which could be important to understand disorders where number sense can be impaired leading to difficulties learning math and operating with numbers.

One largely debated theme in cognitive neuroscience is how the human brain developed the ability to perform mathematics. While mathematical skills certainly rely on the interplay of a wide range of cognitive functions (De Smedt et al., 2013; Fias, 2016; Iuculano and Menon, 2018), an influential theory in the field proposes that a necessary prerequisite to develop such a sophisticated uniquely human ability resides in the ‘number sense’ (Dehaene, 1997). This is a phylogenetically ancient competence that enables humans and other animals to assess and mentally manipulate the approximate number of objects in sets. In humans the precision of the number sense (or ‘numerical acuity’, typically measured by visual number discrimination) sharpens with age and with the acquisition of formal mathematical education (Piazza et al., 2013), and correlates with arithmetical skills throughout the life-span (Halberda et al., 2008; Libertus et al., 2011; Libertus et al., 2013; Chen and Li, 2014; Anobile et al., 2016a; Anobile et al., 2018). Deviations from the typical developmental trend of numerical acuity can be a symptom of developmental dyscalculia (Piazza et al., 2010), a neurodevelopmental disorder that causes specific mathematical learning difficulties.

The neural substrate subtending this sense of numerical quantity is thought to be shared across species and has been linked to a network of areas in the frontal and parietal cortices sensitive to changes in numerosity since very early in life (Izard et al., 2008; Hyde and Spelke, 2011; see for reviews: Cantlon, 2012; de Hevia et al., 2017). In these areas electrophysiological recordings in monkeys identified single neurons tuned to specific numerosities of visual arrays (Nieder et al., 2002; Nieder and Miller, 2004; Roitman et al., 2007; Nieder, 2016) and fMRI studies in humans found activation in these areas to be modulated during quantity perception as well as during calculation (for reviews see: Arsalidou and Taylor, 2011; Eger, 2016; Piazza and Eger, 2016). While the first imaging studies in humans were limited by the low spatial resolution and univariate subtraction-based analyses, fMRI adaptation and multivariate pattern analysis methods provide higher sensitivity to finer-scale activity differences (Kourtzi and Grill-Spector, 2005; Norman et al., 2006; Tong and Pratte, 2012). These methods allowed researchers to study the representation of individual numbers by recording the distance-dependent signal release from adaptation (Piazza et al., 2004), or reading out patterns of number-related activity across multiple voxels of the frontal and parietal cortex (Eger et al., 2009). Moreover, population-receptive field mapping (pRF) methods identified individual locations tuned to specific numerosities arranged in spatially organized maps in the parietal cortex (Harvey et al., 2013).

While these earlier findings mostly pointed at the key role of parietal and frontal areas in numerical representation, some recent studies found that it is possible to decode the number of items seen by the subjects from the fMRI activity patterns in early visual areas (Bulthé et al., 2014; Eger et al., 2015; Bulthé et al., 2015; DeWind et al., 2019, but see Castaldi et al., 2016). Moreover, spatially organized numerosity maps were recently claimed to extend to the occipital cortex (Harvey and Dumoulin, 2017a) and early ERP components compatible with generators in early visual areas responded to variations in the numerosity of visual arrays (Park et al., 2015; Fornaciai et al., 2017; Fornaciai and Park, 2017).

Several properties characterizing numerosity perception, such as being ratio-dependent (Weber’s law) and being susceptible to adaptation, led some authors to suggest that number is a ‘primary’ visual property of the image that is directly perceived through specialized and dedicated mechanisms (Burr and Ross, 2008; Ross, 2010; Anobile et al., 2016b). However, in spite of dedicated efforts on modeling the extraction of numerosity from the visual image (Dehaene and Changeux, 1993; Verguts and Fias, 2004; Dakin et al., 2011; Stoianov and Zorzi, 2012; Morgan et al., 2014), the detailed neural processing mechanisms used by the brain to arrive at a representation of numerosity from the visual input remain little understood, and much less understood than the ones for other basic visual features such as orientation, colour, motion, etc. Numerosity is a notoriously difficult feature to study since changes in numerosity tend to be associated with changes in other quantitative features of the sets during natural viewing conditions (e.g., more items tend to occupy a larger area, or be spaced more densely), and it appears impossible to control for all of these associated quantities at the same time. For this reason, in spite of a large body of behavioral and neuroscientific work on this topic, it still remains debated whether the available evidence supports a sensory extraction mechanism directly sensitive to numerosity. Some have argued instead that numerosity might be judged indirectly by weighing a combination of other, non-numerical, quantitative features of the stimuli (Gebuis and Reynvoet, 2012; Gebuis et al., 2014; Leibovich et al., 2016a). For example, numerosity can be mathematically defined as the product of density (number of items per unit of area) by field area; or by the total surface area divided by mean item size. Thus, decisions on numerical quantity could be taken merely indirectly, on the basis of representations of these non-numerical properties, without numerosity being encoded directly by perceptual systems.

While this possibility is interesting, several behavioral findings argue against it: (1) the discrimination of numerosity and of one often correlated non-numerical feature (item density) follow different psychophysical laws (Anobile et al., 2016b), and (2) at least for relatively small numbers of not too densely spaced items, perceptual thresholds for numerosity discrimination are typically much smaller than the ones predicted from the thresholds for density and field area together (Cicchini et al., 2016), making it unlikely that estimates of numerosity are based on the latter. For what concerns the neuronal level, a few recent studies have started to directly quantify the effects of non-numerical dimensions of non-symbolic numerical stimuli (e.g. Park et al., 2015; Fornaciai et al., 2017; Harvey and Dumoulin, 2017b; Fornaciai and Park, 2018; DeWind et al., 2019). Those studies found that activity in earlier (occipital) or later (parietal) brain regions appeared to be linked to the numerical content of sets after taking into account effects of certain non-numerical dimensions. However, they mostly only considered the effect of one non-numerical variable at the time and compare it to that of number, without taking into account effects explained by all relevant non-numerical dimensions together. Thus, it still remains unclear to what extent activity evoked by non-symbolic numerical stimuli within early and later regions can be explained by a mechanism that encodes numerosity in itself, or by the ensemble of responses to the different non-numerical dimensions of the stimuli.

Here, we implement a new approach to separate brain signals related to numerical and non-numerical quantities and test for a neuronal mechanism directly sensitive to the numerosity of visual sets along the dorsal visual stream hierarchy. We propose that the following signatures would advocate for the existence of such a mechanism:

First, information on numerosity should be detectable in regional activity patterns after multiple important non-numerical quantities are simultaneously (and not only individually) taken into account. Second, and importantly, it should be possible to selectively amplify this numerical information depending on whether the numerical dimension of the stimuli is task relevant, similar to the attentional amplification that has been previously shown for other task-relevant primary features, such as orientation, contrast, color, direction etc. (Jehee et al., 2011; Ester et al., 2016). If a brain area encodes numerical information in a way that is separable from associated non-numerical dimensions, tasks involving selective attention to number should enhance the information about numerosity, without affecting the level of information on associated non-numerical dimensions. Thus, we propose that the presence of such independent attentional amplification is a key criterion in order to identify which brain areas explicitly encode information on numerosity.

On the contrary, if activity patterns could be entirely accounted for by the combination of responses to multiple non-numerical dimensions of the stimuli, no information specifically related to number should be found in the patterns of activity once accounting for the other (non-numerical) dimensions simultaneously. Furthermore, if numerosity was not directly encoded but only indirectly inferred from percepts of non-numerical properties, attentional enhancement should not occur for signals related to numerosity, but if anything, only for other properties (e.g., density and field area) that can jointly define it.

To test these predictions, we created a novel stimulus space to disentangle the contribution of numerical and non-numerical dimensions to brain activity patterns, and designed a task where attention is selectively directed towards either of two orthogonal quantitative dimensions of the visual array (number or item size). We exploited the enhanced sensitivity achieved by fMRI at ultra-high field (7 Tesla) and specific multivariate pattern analyses to simultaneously model and separate the contributions of numerical and different non-numerical quantities to fine-scale activity patterns within multiple regions defined by a probabilistic atlas based on visual topography.

We scanned twenty healthy adult volunteers while they performed two tasks on arrays of dots varying orthogonally in numerosity (6, 10, or 17 items), average item size (0.04, 0.07, or 0.12 visual square degrees - vd²) and total field area (44 or 20 vd²) (Figure 1A). Participants alternated between a ‘number’ and a ‘size’ task in different blocks: during the ‘number’ blocks they had to direct attention to the numerosity of each sample stimulus and keep it in memory for comparison with an occasionally following match stimulus, while during ‘size’ blocks they performed the equivalent task on the average item size of the arrays (Figure 1B). When a match stimulus appeared (indicated by a change in color of the fixation point), participants had to decide whether the match stimulus was larger or smaller on the attended dimension than the previous sample held in memory and to respond by button press.

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Behavioral performance and univariate fMRI activation effects

Response accuracies for comparison of match stimuli were overall high and not significantly different across tasks (86% for the number task and 85% for the average size task, t(19) = 0.46, p = 0.65), suggesting that subjects attended to the correct stimulus dimension and the difficulty was on average successfully matched across tasks (Figure 2A).

Figure 2

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We started the analysis of the functional imaging data by evaluating overall regional activation effects during both tasks. Surface-based random-effects group analysis identified similar bilateral activations in the occipito-parietal and frontal cortex during both tasks for sample stimuli against the implicit baseline where participants were just looking at the fixation point with blank screen and without performing any task (Figure 2B and C, thresholded at p < 0.001 uncorrected). In both tasks the activity covered a wide occipito-parietal area starting from the superior occipital and transverse occipital sulci and extending throughout the intraparietal sulcus (IPS) up to the post-central sulcus.

The frontal activity mainly covered the superior frontal gyrus. The direct contrast of sample stimulus-related activity during the number versus the size task revealed no area with significantly stronger activation for either of the two, despite the uncorrected significance threshold (Figure 2D). Altogether, these results suggest that task difficulty was successfully matched and that under these conditions attending to different quantitative dimensions leads to equivalent overall activation of the brain regions involved in the task. Differences in overall activation level can therefore not confound the following more specific results on the within-dimension discriminability of quantitative features.

Multivariate fMRI Pattern Analyses

Read-out of sample numerosity is modulated by task

Given that the whole brain univariate contrasts had confirmed equivalent activations across the two tasks, we further investigated, using multivariate classification, what was the degree of discriminability of activity patterns evoked by different sample numerosities across different regions of the dorsal visual stream and during the number and size task. In each subject we identified several regions of interest (ROIs) derived from a surface-based probabilistic atlas based on visual topography (Wang et al., 2015), (Figure 3A). Within each region, we used an equivalent number of most activated voxels (in the orthogonal contrast ‘all sample stimuli > baseline’) to train and test multivariate classifiers to discriminate between numerosities for each task (for the ability of the classifier to discriminate task see Figure 3—figure supplement 1 and Supplementary file 3 of the Supplementary Material). Figure 3B shows the across-subject overlap map for the included voxels which mainly highlight the foveal portion of the different ROIs, in line with the central presentation of the dot arrays. We first compared decoding accuracies in three large regions corresponding to early, intermediate and higher-level areas (including areas from V1 to V3, from V3AB to V7 and from IPS1 to IPS5, respectively). Then, to track the presence of information discriminative of numerosity across the dorsal visual stream more in detail, we further compared the classification accuracies across seven contiguous ROIs from V1 up to IPS345 (for results concerning the intraparietal sulcus excluding those regions defined by visual topography see Figure 3—figure supplement 2A and Supplementary file 4a and 4b in Supplementary Material). Figure 3C shows the performance of the classifiers trained to discriminate between different numerosities as a function of task. Overall, the presented sample numerosity could be decoded significantly above chance in all the ROIs and during both the number and size task, however with important differences. When explicitly attending to numerosity, the classification accuracy gradually increased across the dorsal stream (starting to be enhanced from intermediate areas, specifically from V3AB on), and was highest in parietal areas. During the size task, when attention was not explicitly directed towards the numerical aspect of the stimuli, the different numerosities were still decodable, however the classification accuracies were reduced in intermediate and higher regions, while they remained almost unchanged in early visual areas (specifically in V1, V2 and V3).

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The task-driven modulation of decoding accuracies across the three major ROIs is confirmed by a significant interaction between ROI and task (F(1.69,32.11) = 9.81, p = 0.0008). For the number task, the classification accuracy progressively increased from the early visual areas (slightly above 60%) to intermediate and higher level regions where it reached almost 70% correct. Post-hoc tests showed that the classification accuracy increase in intermediate and higher areas with respect to early areas was very close to or clearly significant (p = 0.075, Cohen’s d = 0.52, and p = 0.028, Cohen’s d = 0.57 respectively). During the size task, the classification accuracy in the intermediate and higher regions dropped down to 61% and 60% respectively (yet remaining highly significantly above chance in both cases, see p-values in Supplementary file 1). The change in classification accuracy across tasks was highly significant both for the intermediate and higher areas, (p = 0.001, Cohen’s d = 0.88 and p = 0.00001, Cohen’s d = 0.98). On the other hand, the classification accuracy in the early visual areas remained nearly constant (62%) and was not significantly modulated by task (p = 0.5).

The significant interaction between ROI and task was confirmed when testing the seven individual regions (F(4.10,77.97) = 7.17, p = 0.00005). Although the post-hoc tests did not show significant differences in classification accuracies across individual ROIs, for the number task the decoding accuracies progressively increased across the visual hierarchy and varied from slightly above 60% in the primary visual areas (V1 = 62%, V2 = 62%, V3 = 61%) up to almost 70% in the intermediate and higher ROIs (V3AB = 64%, V7 = 65%; IPS12 = 67%, IPS345 = 67%). During the size task, decoding accuracies were much reduced in intermediate and higher regions (V3AB = 58%, V7 = 58%; IPS12 = 59%, IPS345 = 58%, yet still significantly above chance in all ROIs, see p-values in Supplementary file 1), while they remained almost unchanged in the primary visual areas (V1 = 61%, V2 = 62%, V3 = 60%). Accordingly, the post-hoc tests indicated that the classification accuracy in individual regions significantly changed across tasks only from V3AB on (V1: p = 0.28, Cohen’s d = 0.24; V2: p = 0.83, Cohen’s d = 0.05; V3: p = 0.55, Cohen’s d = 0.14; V3AB p = 0.0002, Cohen’s d = 0.98; V7: p = 0.00003, Cohen’s d = 1.20; IPS12: p = 0.00001, Cohen’s d = 1.03; IPS345: p = 0.00005, Cohen’s d = 1.26).

In sum, multivariate classification analyses revealed that the sample numerosity presented could be read out from brain activity patterns in all ROIs tested during both tasks, although accuracy was enhanced in mid-to-higher level but not in earlier regions when number was the attended feature. However, since in this analysis activations for all sample stimuli for a given numerosity were pooled together, the decoding performance obtained could still be partly driven by features other than numerosity per se.

Multiple regression RSA to disentangle the contributions of quantitative dimensions

As a critical test of whether the representations of numerical and non-numerical features of the stimuli could be dissociated across the dorsal visual stream, we performed Representational Similarity Analysis (Kriegeskorte, 2008; Kriegeskorte and Kievit, 2013) which, unlike classification-based decoding, allows to assess the effect of multiple quantitative dimensions on activity patterns simultaneously. For each ROI and task, we obtained a neural representational dissimilarity matrix (neural RDM, Figure 4A) by computing the correlation distance between activation patterns for each possible pair of conditions. We then applied multiple regression analysis to test in how far the fMRI pattern dissimilarity structure could be explained by multiple predictor matrices reflecting the stimuli’s dissimilarity along several important quantitative dimensions: numerosity, average item size, total field area, total surface area and density (Figure 4B). Of note, our design orthogonally manipulating numerosity, average item size and total field area ensured that numerosity was also partly decorrelated from density and total surface area (as shown by the correlation values in the Predictor Correlation matrix, Figure 4B), yet not completely (correlation between number and density predictors = 0.43; between number and total surface area predictors = 0.33). Correlations between predictors in a multiple regression lead to a reduction of the unique variance attributable to each one of them, and to a greater variability of the estimated betas. An estimation of variance inflation factors (VIF) for each predictor in our case revealed that these remained reasonably low (corresponding to 1.4874, 1.1957, 1.2048, 1.3238 and 1.4591 for number, average item size, total field area, total surface area and density, respectively). By using a multiple regression approach we capitalize on the fact that the resulting beta weights reflect only the part of the variance that each one of these stimulus descriptors uniquely explained in the pattern of activity of a given ROI on top of all the others. Thus, by entering numerical and non-numerical dimensions together into a multiple regression, a significantly above zero beta for number would imply that the numerical information is contributing to the pattern of activity within a given ROI, over and above the contribution of the other non-numerical quantitative dimensions.

Figure 4

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Figure 5 displays the results of the estimated beta weights for various ROIs separately for the number (Figure 5A) and size tasks (Figure 5B). Beta weights for the effect of number independent of the other dimensions (black triangles) were generally positive and progressively explained the activity patterns better when proceeding from lower to higher-level regions when task relevant. The evolution of the numerical information across the visual stream was attenuated during the size task, yet betas remained significantly above zero in all regions (see p-values in Supplementary file 2). Beta weights for the non-numerical dimensions (other shapes in Figure 5) were pronounced predominantly in the earlier visual areas and, importantly, they appeared to be not clearly affected by task.

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Our work exploited the enhanced spatial resolution provided by ultra-high field fMRI to reveal how the human brain represents multiple quantitative dimensions of non-symbolic numerical stimuli. Furthermore, we tested whether and at what cortical level the numerical information can be represented and specifically modulated by attention independently of non-numerical visual properties of the image.

At the level of overall regional activity, attending to the numerosity or to the average size of the dots in the array recruited largely overlapping occipital and parietal areas, as also previously observed for perception and comparison of different types of quantities (Pinel et al., 2004; Dormal and Pesenti, 2009; Borghesani et al., 2019); for a meta-analysis on other non-numerical representations see: Sokolowski et al., 2017). Two previous fMRI studies have investigated task-related differences in univariate regional activity elicited by numbers of items either in the subitizing (Leibovich et al., 2015) or in the estimation (Leibovich et al., 2016b) range when participants attended to either the numerosity or to the total surface area of dot arrays while these dimensions varied congruently or incongruently with each other. During comparison of dot arrays within the subitizing range, the overall regional brain activity in the cingulate and right superior frontal gyrus was shown to be modulated by task order (Leibovich et al., 2015). When testing larger numerosities, the activation in the right temporal parietal junction was modulated by the congruency between dimensions only when number, but not when total surface area, was the relevant dimension in a comparison task and several regions, including cingulate gyrus, anterior insula, superior frontal gyrus, orbitofrontal and inferior parietal cortex did either respond preferentially during the numerical or the non-numerical task (Leibovich et al., 2016b).

While the fact that in the current experiment we do not observe significant differences between tasks at the univariate level might seem to contrast with the results by Leibovich et al. (2015) and Leibovich et al. (2016b), several methodological differences are worth noting: first, the non-numerical dimensions investigated are different, being average item size in our case and total surface area in theirs. Second, in the current experiment we made the changes in number and in average item size equally discriminable and we successfully matched task difficulty as demonstrated by the equal percentage of correct responses across tasks. The numerical and non-numerical dimensions were not matched for perceptual discriminability in the studies by Leibovich et al. (2015) and Leibovich et al. (2016b), leaving open the possibility that some of the differences in cortical activation may reflect the higher difficulty associated with the numerical as opposed to the non-numerical task. This possibility seems likely also because the experiments by Leibovich et al. (2015) and Leibovich et al. (2016b) required participants to perform a comparison on every trial, thus the measured response not only reflects the perceptual process, but also the decision/comparison component. In the current experiment instead, our contrast is related to the sample trials, where participants were only required to perceive/memorize the value on the relevant stimulus dimension, without performing a comparison (which was required only in the occasional match trials). Overall, having balanced task difficulty and excluded the contribution of the comparison process from the response, the current study found no differences between tasks at the univariate level. Only multivariate pattern analysis could detect differences in the way information along the different dimensions was encoded as a function of task in our study.

Multivariate decoding analyses showed that the sample numerosity presented could be read out from brain activity significantly above chance all along the visual stream, however with important differences across regions. When explicitly attended, the numerical information could be read out with gradually higher accuracy following an occipital-parietal gradient, up to a maximum level in the parietal cortices. The effect of attention strongly affected the accuracy of the numerical discrimination in intermediate and higher regions while leaving the accuracy in the early visual areas unaffected. The successful read-out of information related to numerosity from parietal cortices in the current experiment contrasts with some previous studies where fMRI signals discriminative of numerosity could not be detected in the parietal regions (DeWind et al., 2019; Fornaciai and Park, 2018). Differences in paradigms and sensitivity of the scanners used may account for this discrepancy. Most crucially, in those studies, participants were shown different numerosities and the task required detecting changes in the colour of the dots. Thus, participants’ attention may have not been directed to the numerosity of the visual arrays in that case, and the numerical information may have been reduced when focussing on the dots’ colour, similarly to what was observed for the size task in the current experiment. Although in the present study we could still read out numerical information even when it was irrelevant for the task, this signal may have remained undetected by less sensitive MRI scanners.

Above chance decoding of numerosity during both tasks was observed here within both the medial IPS parts organized according to visual topography, as well as more lateral/inferior IPS parts outside the visual field maps, to a similar extent. Both regions also allowed to successfully decode the task carried out by the subject, with slightly higher accuracy in the most lateral/inferior IPS part with respect to the most medial IPS part. It has been recently proposed that there might be a sub-regional specialization along the intra-parietal sulcus, with the most medial/superior regions supporting processing of physical quantities and the lateral/inferior part supporting numerical operations, such as numerical comparison and calculation (Harvey et al., 2017; see also Castaldi et al., 2019). Our multivariate decoding results only marginally allow us to support distinct roles for these regions, however, addressing this precise question was not an explicit aim of the present study. It remains possible that the representation of numerical information in these different subparts supports different cognitive processes which cannot be differentiated here. For example, while the pattern of activity read out from the IPS field maps might underly the perception of numerosity, the numerical information reflected by the pattern of activity of the more lateral/task-responsive regions of IPS might provide input to internal manipulations of quantity during numerical operations, but further studies are necessary to explore this possibility.

In our study the number presented could be decoded not only in parietal cortex, but already from the earliest stages of visual processing. However, since the multivariate classification analysis collapsed across the non-numerical dimensions of our stimulus set it is unclear whether the information underlying successful decoding was strictly numerical, especially in earlier regions. Some previous studies have dealt with the problem of correlations between numerical and non-numerical stimulus dimensions by controlling for non-numerical features one at the time and testing for fMRI adaptation effects, or replicability of decoding performance or layouts across conditions where individual non-numerical features where controlled for (Piazza et al., 2004; Eger et al., 2009; Harvey et al., 2013; Harvey and Dumoulin, 2017a; DeWind et al., 2019). When the effects of non-numerical dimensions were measured directly, this was done in some studies by computing the explained variance or classification performance for each feature in isolation and comparing it to the one for number (Harvey and Dumoulin, 2017b; Cavdaroglu and Knops, 2019), leaving unspecified the degree to which the simultaneous contribution of several non-numerical dimensions could account for the findings (Gebuis et al., 2014). Some other previous studies have taken a different approach, by modeling jointly the effects of numerosity and two non-numerical dimensions (termed ‘size in area’ and ‘spacing’) which were designed to be orthogonal to numerosity but do not necessarily constitute natural, perceptually relevant feature dimensions, but rather mathematically defined constructs (DeWind et al., 2015; Park et al., 2015; DeWind et al., 2019; Fornaciai and Park, 2018). This design also allowed the authors to estimate, from the combined beta weights of numerosity and the mentioned two orthogonal dimensions, which feature represented by different directions in their stimulus space most accounted for the effects in a given ERP component or brain area. However, brain signals can reflect a combination of responses to multiple quantitative dimensions, and this approach does not permit to distinguish, for example, a modulation by numerosity from two independent modulations by field area and density. In our study, on the contrary, we separated the contributions of numerical and non-numerical stimulus dimensions by applying multiple regression to representational distance matrices which allowed us to test for the extent to which numerosity could explain the pattern of activity while taking into account simultaneously the variability explained by several important natural non-numerical features. Indeed, estimating significantly above zero beta values for number implies that information about numerosity is present in the pattern of activity over and above the contributions of all the other included non-numerical features. We found that information specific to number was detectable beyond the information related to the other dimensions, and that the numerical information was gradually enhanced when progressing along the visual stream when explicitly task relevant, and less strongly represented, although still detectable, when not task-relevant. Importantly, the level of information on other quantitative but non-numerical properties of the image, such as total field area, total surface area and density, although reliably detected especially in earlier brain regions, was not altered when explicitly attending to the numerical quantity. Given that numerical information becomes available in parallel with the other features and independently selectable by attention from early processing stages on, it appears that the human visual system has the capacity to detect signals related to numerosity and separate these from other quantitative dimensions, starting from very basic visual primitives. This supports the existence of a sensory processing mechanisms from which numerosity could be derived directly, rather than making numerical judgements merely indirectly on the basis of percepts of associated non-numerical quantities. Of course, the question of why numerical judgements are nevertheless often influenced by non-numerical quantities, and how these interactions might arise from the involved neuronal populations, remains an important one that deserves further study. Here we modeled the influence of several important natural non-numerical quantitative dimensions on brain activity, though our selection is necessarily non-exhaustive. We acknowledge the fact that we cannot formally rule out that a feature of a different type than those considered by us may have contributed to the effects observed here, and our conclusions hold to the extent of the non-numerical features tested.

The enhancement of numerical information in activation patterns found here when number was the relevant stimulus dimension is extending a growing body of work on the neuronal correlates of feature-based attention. Neurophysiological studies have shown that attention to basic visual features either increases the gain or sharpens responses of neuronal populations preferentially responsive to these features in different visual areas (e.g. Treue and Martínez Trujillo, 1999; McAdams and Maunsell, 2000; Reynolds et al., 2000; Martinez-Trujillo and Treue, 2004; David et al., 2008), see also: Carrasco (2011) for a review). Correspondingly, fMRI decoding studies have found that directing attention to one feature dimension such as orientation, motion direction or color or to particular values within one given dimension improves the read-out of these features from brain activity in early sensory regions (Kamitani and Tong, 2005; Kamitani and Tong, 2006; Serences and Boynton, 2007; Jehee et al., 2011) but in some cases also in higher-level areas (Liu et al., 2011; Ester et al., 2016). According to one influential account, higher-level fronto-parietal areas such as the lateral intraparietal area (LIP) implement spatial ‘priority maps’ in which the level of activity at individual locations depends jointly on the different features of objects at these locations as well as on top-down factors such as their task relevance, associated reward, etc (Itti and Koch, 2001; Thompson and Bichot, 2005; Gottlieb, 2007; Sapountzis et al., 2018). Independent of spatial priority, LIP neurons have also been found to represent higher-level factors such as learned category membership and other non-spatial information (Freedman and Assad, 2009) and to flexibly switch between encoding of different visual features, such as color or motion, depending on the task (Toth and Assad, 2002; Ibos and Freedman, 2014). The idea of a role for intraparietal areas as mere ‘priority maps’ or reflecting entirely flexible encoding of information on task-relevant features (without intrinsic selectivity) can insufficiently account for our results, since it would predict an equivalent amplification of the representation of average size when this is the attended feature instead of number. This is not what we observed. Our results are thus more compatible with an enhancement of the responses of neuronal populations with intrinsic selectivity to the feature numerosity in these areas (comparable to the one observed for other features in lower-level visual regions).

While the existence of individual neurons tuned to different numbers of items in intraparietal cortex is well established (Nieder and Miller, 2004; Roitman et al., 2007), the only electrophysiological study that recorded from neurons in the ventral intraparietal (VIP) cortex in macaque monkeys under changing task conditions (Viswanathan and Nieder, 2015) found that neurons encoded numerosity to the same extent, regardless of whether the task required to attend to the number or the color of the items. This differs from our results which show a clear attentional amplification of numerosity information. Given that the human IPS 1–5 investigated in the current work is usually considered to be the equivalent of the macaque LIP/VIP complex (Kastner et al., 2017), the difference between results may be due to a difference across species, but differences in paradigms and in the nature of the signal recorded in the two studies make it difficult to directly relate the two findings. For example, monkeys were trained initially with the color match to sample task, then re-trained to respond to number, thus implying comparisons across an extended time period and different context, whereas our participants switched between the two tasks within the same scanning session. In addition, it is possible that the color task with a single color per stimulus and a small number of highly distinguishable alternatives placed lower demands on attentional load compared to our average size task, therefore leaving number processing unaltered. Nevertheless, as a common denominator both studies agree on pointing to some degree of spontaneous encoding of numerosity in intraparietal areas under conditions of attention to an orthogonal stimulus dimension.

The gradual enhancement of numerosity information observed by us in the number task when progressing along the dorsal visual stream is compatible with a multi-stage process of the extraction of numerosity where attention may operate at multiple levels over which attentional enhancements accumulate. If numerosity information can be retrieved from multiple levels of the cortical hierarchy, this does not need to imply that this feature is encoded by individual neurons at all these levels, but it may be detectable by multivariate methods even if it existed only in distributed form across the population of neurons. As one speculative interpretation, the numerical information read out from early visual areas could reflect a location map (Dehaene and Changeux, 1993), or the process of object segmentation where different individual items start to be separately represented, but this representation may not yet be in a form that is most easily read out for numerical discrimination. Higher areas may progressively transform and concentrate the initially distributed information onto individual neurons, which most likely constitute the base on which we operate when comparing numbers. This interpretation is in line with a recent study showing that although different numerosities could be discriminated based on the pattern of activity in early visual areas and parietal cortex, the behavioral precision of numerical discrimination was correlated with the decoding accuracy only in the latter region (Lasne et al., 2019).

While earlier behavioral research suggested that the precision of numerical representation is predictive of formal arithmetic and gets refined with development and mathematical learning (Halberda and Feigenson, 2008; Nys et al., 2013; Piazza, 2010; Piazza et al., 2010; Piazza et al., 2013), other recent evidence has led to a slightly different view: what might be changing with development and mathematical competence could be the ability to focus on numerical information while filtering out non-numerical dimensions during a numerical comparison task (Castaldi et al., 2018; Piazza et al., 2018; Starr et al., 2017; Wilkey et al., 2018). The influence of non-numerical dimensions on numerical judgments decreases over normotypical development and both dyscalculic children (Bugden and Ansari, 2016; Piazza et al., 2018; Szucs et al., 2013; Wilkey et al., 2018) and adults (Castaldi et al., 2018) seem to be disproportionately affected by non-numerical dimensions during numerical comparison tasks. In light of these behavioral findings, it would be interesting to see whether in dyscalculic subjects, numerical information at the neuronal level is less precisely encoded overall or merely less accessible to attentional selection. It is possible that the capacity to selectively focus on the numerical information and to enhance it already from early levels of visual analysis on, as shown in the current study, is learned or emerging over development, and future studies should directly test this hypothesis.

A surprising result of the current experiment is that we could not find information about average item size in the pattern of activity in any of the regions examined, even though this feature’s perceptual discriminability was equated with the one of numerosity. This suggests that the neural mechanisms supporting average size representation may differ from those engaged during single object size analysis which has been shown to overlap partly with numerosity maps in parietal regions (Harvey et al., 2015). Mechanisms for average size perception, and in general for ensemble statistics are still unclear. It has been previously suggested that average item size perception, like density perception, may rely on texture processing mechanisms rather than individual item identification (Im and Halberda, 2013). Various regions along the ventral visual stream have been implicated in texture perception. In particular, adaptation studies have identified recovery of fMRI signal in the medial part of the posterior collateral sulcus that was selective for texture as opposed to color or shape of 3D irregular objects (Cavina-Pratesi et al., 2010) and the parahippocampal place area (PPA) showed equal release from adaptation for object ensemble and surface textures, suggesting that ensembles and textures are processed similarly (Cant and Xu, 2012). It is possible that average size is also represented in the ventral stream which was not covered here, and future studies should focus on these regions to try to detect a representation of average size. What we observed, however, was that beta weights for density obtained from RSA regression became significant in the parietal regions during the size task, suggesting that texture processing mechanisms may be automatically activated during the average size task. This interpretation, however, has to remain speculative and future studies should investigate neural mechanisms relating texture, density and average size processing.

In conclusion, with this study using high-resolution, high-field fMRI we provide direct neuroscientific evidence for a sensory processing mechanism capable of disentangling signals related to visual numerosity from the ones related to associated non-numerical quantities from early stages of cortical processing on, which can then be independently and progressively amplified across the dorsal visual stream when numerical information is explicitly task-relevant. An important goal for the future will be to better understand what are the processing steps and transformations occurring at the different levels of the cortical hierarchy that allow the human brain to isolate numerical information, for example by comparing fMRI data against computational models simulating the visual extraction of numerosity. In addition, it will be important to understand how neuronal representations of numerosity are shaped developmentally and at which cortical levels they can be perturbed to given rise to impaired behavior.

Individual subjects' data points for behavioural and fMRI results for all regions of interest, corresponding to Figures 2A, 3C, 5, Figure 3—figure supplements 1 and 2, and Figure 5—figure supplements 1 and 2 as. cvs files. The maps displayed in figure 2B-D and 3B are provided in a format readable with Freesurfer/Freeview, one of the most widely used free neuroimaging softwares. The functional imaging dataset is available via the Open Science Framework (osf.io/6zch2).

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Author details

1. Elisa Castaldi

   Cognitive Neuroimaging Unit, CEA DRF/JOLIOT, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin Center, Gif-sur-Yvette, France

   Contribution

   Conceptualization, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   elisa.castaldi@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-0327-6697

2. Manuela Piazza

   Center for Mind/Brain Sciences, University of Trento, Trento, Italy

   Contribution

   Conceptualization, Supervision, Validation, Visualization, Methodology, Writing—review and editing

   Competing interests

   No competing interests declared

3. Stanislas Dehaene

   Cognitive Neuroimaging Unit, CEA DRF/JOLIOT, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin Center, Gif-sur-Yvette, France

   Contribution

   Conceptualization, Supervision, Validation, Methodology, Writing—review and editing

   Competing interests

   No competing interests declared

4. Alexandre Vignaud

   UNIRS, CEA DRF/JOLIOT, Université Paris-Saclay, NeuroSpin Center, Gif-sur-Yvette, France

   Contribution

   Validation, Investigation, Methodology, Writing—review and editing

   Competing interests

   No competing interests declared

5. Evelyn Eger

   Cognitive Neuroimaging Unit, CEA DRF/JOLIOT, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin Center, Gif-sur-Yvette, France

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing—original draft, Project administration, Writing—review and editing

   Competing interests

   No competing interests declared

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