Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.


Introduction
Numerical abilities are crucial for several species.In modern human societies, they are necessary for many basic and advanced everyday activities, such as managing money, telling the time, making a phone call, using passwords, performing measurements, etc.However, human adults share with young infants (e.g., Benavides-Varela and Reoyo-Serrano, 2021;de Hevia, 2016;Feigenson et al., 2002;Hyde and Spelke, 2011) and non-human animals (e.g., Agrillo et al., 2006;Agrillo and Bisazza, 2018;Rugani, Cavazzana et al., 2013;Rugani et al., 2008;Rugani, Vallortigara et al., 2013) a subset of basic numerical skills that are considered the evolutionary foundation of more complex numerical reasoning (Dehaene, 2011).For example, young infants discriminate sequences with small numbers of items (e.g. 1 vs 2, 2 vs 3 crackers) and fail with larger ones (3 vs. 4) in both the visual (Feigenson et al., 2002) and auditory domain, and with both linguistic or non-linguistic stimuli (Benavides-Varela and Reoyo-Serrano, 2021).Behavioral research is supported by neural data revealing dissociable neural signatures responding to small (1− 3) and large (8− 32) numerical representations in 6-7.5-month-old children (deHevia, 2016;Hyde and Spelke, 2011).Similarly, comparative studies with non-human animals have evidenced, for example, the mosquitofish capacity to discriminate shoals with a 1:2 ratio in the large number range (2 vs. 4, 4 vs. 8 and 8 vs.16) but not at a 2:3 ratio (8 vs. 12).Instead, in the small number range, when the sets differed by one element, fish were able to discriminate 2 vs. 3 or 3 vs. 4 fish.Similar results were reported by Rugani et al. (2008) in chicks, even after controlling for other quantitative cues such as area and perimeter (Rugani, Vallortigara et al., 2013).
These sub-set of numerical abilities inform individuals and favor rapid and adaptive behaviors associated with the number of conspecifics in the environment, food, predators, or preys, thus providing survival and reproductive advantages (Butterworth, 2022;Butterworth et al., 2018;Nieder, 2020).From a human lifespan perspective, when these abilities are not adequately developed, they can give rise to a neurodevelopmental disorder called dyscalculia (Ardila and Rosselli, 2002;Butterworth, 2019).
Basic number processing abilities have been often attested in human adults, infants, and non-human species by assessing their abilities to represent and discriminate non-symbolical quantities (e.g., sets of dots, objects, sounds, etc.) and compute simple operations on these representations.Such number processing abilities have been reported in species of mammals (Brannon and Terrace, 1998;Cantlon and Brannon, 2006;Merten and Nieder, 2009) birds (Kirschhock et al., 2021;Pepperberg, 2006;Rugani et al., 2009Rugani et al., , 2015) ) fish (Agrillo et al., 2017;Potrich et al., 2015aPotrich et al., , 2019Potrich et al., , 2022) ) reptiles (Gazzola et al., 2018;Miletto Petrazzini et al., 2018) amphibians (Stancher et al., 2015) and invertebrates (Bortot et al., 2019(Bortot et al., , 2020(Bortot et al., , 2021;;Dacke and Srinivasan, 2008;Giurfa, 2019).These basic numerical processing abilities are underpinned by two systems (Benavides-Varela et al., 2018;Hyde, 2011).First, a non-symbolic and language-independent mechanism, the Approximate Number System (ANS), which supports analogical and rough representations of the number of objects in a set and allows individuals to discriminate between sets of different numerosity (Feigenson et al., 2004).The sense of approximate number is not especially precise, but rather relatively noisy.Participants' accuracy on ANS tasks is ratio-dependent and follows the Weber-Fechner's Law, which means that it becomes increasingly difficult for an individual to discriminate between two numbers as the difference between these numbers decreases (Dehaene, 2011;Ditz and Nieder, 2016;Mechner, 1958;Moyer and Landauer, 1967;Piazza et al., 2010;Potrich et al., 2015a;Skorupski et al., 2018).A second system supporting numerical abilities is the object tracking system, which is devoted to precise and accurate perception, but it is limited to small numbers (i.e., 1 to 4).This system supports the capacity to rapidly grasp the number of objects in a set (also known as subitizing) without going through any process of counting or estimating (Atkinson et al., 1976;Cantrell and Smith, 2013;Feigenson et al., 2004).
In the last decades, studies have assessed and studied these basic numerical processing abilities based on both behavioral and brain activity measures.However, it is only in the last few years that studies have begun to employ techniques based on molecular and genetic mechanisms to study numerical cognition.Similarly, research has just recently embarked on exploring how changes in the concentration of neurotransmitters in specific areas of the brain can modulate numerical learning capacities.Even if just starting to develop, this innovative research has expanded the current understanding of how numerical abilities developed and function throughout both evolution and development.In addition, they have opened new perspectives to understand individual differences and mathematical learning disabilities, as well as potential applications to improve mathematical competencies in clinical groups and educational settings.
This review aims to offer an overview of these promising lines of ongoing research.The final goal is to advance tentative links across studies that identified functional brain regions, variations in the concentrations of signaling molecules (i.e., neurotransmitters), and candidate genes that contribute to the risk of developing numerical deficits, in order to contribute a comprehensive neuroscience view of the neural basis of mathematical learning.

Non-symbolic numerical processing and brain activation in human and non-human animals
Numerical cognition involves a large variety of processes ranging from basic mechanisms of mental representation of quantities to complex arithmetic procedures.This section provides a brief overview of the main areas involved in humans' basic non-symbolic number processing (e.g., representing a given number of dots and comparing this to another set of dots).It is not a comprehensive review of the neuroimaging of numerical cognition; for more background information the reader is advised to consult other recent articles (e.g., Ferrigno and Cantlon, 2017;Hyde, 2021;Moeller et al., 2015).
Functional brain systems related to basic non-symbolic number processing have been well described over the past 30 years (Dehaene, 1997;Dehaene et al., 1999Dehaene et al., , 2003)).Pallial regions in the posterior parietal and prefrontal cortices have been identified as two of the main areas subserving quantity processing (e.g., Dehaene, 1996;Piazza et al., 2004Piazza et al., , 2007;;Pinel et al., 2001).In particular, neuroimaging studies have repeatedly reported bilateral intraparietal sulcus (IPS) activation in human adults in a variety of numerical tasks, for instance when participants are exposed to a stream of sets of dots which occasionally change in numerosity (Piazza et al., 2004) or when they are asked to estimate the numerosity of sets (e.g., Castelli et al., 2006;Piazza et al., 2006).The IPS is involved when the stimuli are presented in an auditory (series of tones) and a visual modality (series of dots) (Piazza et al., 2006), and also when individuals perform numerical tasks that require the use of symbols, for instance when reading Arabic numbers (Dehaene et al., 2008) or performing calculations (Dehaene et al., 2003).This suggests that the IPS is dedicated to process numerical information, independently of stimuli presentation modality and task demands.
Interestingly, the IPS -particularly in the right hemisphereappears to be devoted to process numerical information from the first stages of development.This area is preferentially activated when children are exposed to a stream of dots changing in numerosity, but not when the number stays constant in the stream and the identity of the objects change (Cantlon et al., 2006;Kersey and Cantlon, 2017).Studies on 4 years old children, using functional magnetic resonance imaging (fMRI), have rather identified a double dissociation with dorsal intraparietal regions sensitive to numerical variations, and a ventral occipitotemporal system responsive to object identity.This dissociation has been observed even earlier in development, at 3 and 6 months of age (Hyde et al., 2010;Izard et al., 2008), when infants have limited experience with the world.These findings highlight that the activation of right parietal areas in response to numerical transformations is biologically rooted, preserved in ontogeny, and already functional prior to symbolic education in humans.
Besides IPS involvement, some studies have reported prefrontal cortex (PFC) activation related to basic number processing (e.g., Cantlon et al., 2006;Izard et al., 2008;Jacob and Nieder, 2009;Piazza et al., 2007).This area has been identified as involved in numerical estimation both in human and non-human primates (Nieder et al., 2002;Piazza et al., 2007).A fMRI study by Piazza et al. (2007) using an adaptation paradigm demonstrated that the PFC and inferior frontal cortices were activated when human adults were exposed to numerically deviant stimuli.These results were successively replicated in another fMRI adaptation study (Jacob and Nieder, 2009).The latter study not only confirmed the PFC response to numerical changes but also provided evidence that it was specific and not a generic response to deviancy, since the area was not responsive to changes in non-numerical dimensions such as color.Similar findings have been reported in fMRI studies with preschool (Cantlon et al., 2006) and school children (Ansari and Dhital, 2006).Another study by Ansari and Dhital (2006) also showed that the activation of in PFC would be subject to a distance effect, revealing higher activation for more difficult comparisons separated by small as opposed to large numerical distances.
Interestingly, various factors such as age, task demands, and practice may interact and influence the involvement of the PFC for processing numerical information in humans.Age-related changes in neural activation were evident from a stronger recruitment of frontal regions in school-age children compared to adults performing the same nonsymbolic magnitude comparisons (Ansari and Dhital, 2006), but also in studies testing symbolic number processing and arithmetic.Moreover, in both normally developing children and adults, the gain of arithmetic competence determines a shift of activation from frontal brain areas to more specific functional response in the IPS (e.g., Ansari et al., 2005;Arsalidou and Taylor, 2011;Kaufmann et al., 2006;Kucian et al., 2008;Zamarian et al., 2009).It thus seems that the automatization of arithmetic procedures leads to a reduced involvement of the PFC.
Besides parietal and frontal areas, recent studies have identified brain activity -in particular early event related potentials (ERPs) derived from primary sensory systems-associated with basic visual numerical processing (e.g., Arsalidou and Taylor, 2011;DeWind et al., 2019;Fornaciai et al., 2017;Park et al., 2016).However, compared to other brain areas, the study of primary sensory areas activation in response to numerical information is still at its dawn.Park et al. (2016) compared ERPs responses to numerical variability in occipito-parietal areas in both E. Visibelli et al. hemispheres and in the occipital cortex.The authors found an ERP response modulated by the numerical quantity that peaked at around 180 ms in the occipito-parietal areas.Moreover, in the medial occipital cortex, they observed an even earlier response at around 75 ms, suggesting that the numerical dimension is perceived early on at a visual level, in the occipital cortex.Subsequently, Fornaciai et al. (2017) replicated Park et al. (2016) results showing an effect of numerosity on the ERPs that peaked at 175-275 ms post stimulus onset in the occipito-parietal regions and earlier at 75-125 ms in the medial occipital areas.Additionally, they found polarity inversion of ERPs associated with upper and lower presentation of stimuli in the medial occipital cortex early at 55 ms and 90 ms.The polarity inversion reflects the involvement of early visual cortices: V1, V2 and V3 in numerical processing.Thus, primary visual areas might be the first cortical stage that extracts visual numerical information because the related ERPs require shorter latencies than do ERPs from parietal regions.More recently, DeWind et al. ( 2019) extended the electroencephalography (EEG) findings using fMRI with a whole-brain approach, therefore adding to the previous research that was limited to the study of only occipito-parietal and occipital electrical responses.Results replicated the study by Fornaciai et al., showing an effect of numerosity changes on signal intensity.In fact, blood-oxygen-level dependent (BOLD) signal intensity in early visual cortices increased with the array's numerosity independently from other non-numerical visual features.It is important to note that this path presumably supports only visual representations of numerosity.If so, future studies have yet to address how the flow of numerical information might be conveyed in the brain when other sensory modalities are involved.In addition, these studies have focused on adult representations of numerosity.An intriguing question regards to what extent primary sensory processes are involved early in development.
Another area of the primary sensory path i.e., the thalamus, has also been implicated in the processing of numerical information in humans.An adaptation study in adults found decreased activation in thalamic regions in response to the repeated presentation of sets with the same numerosity (Piazza et al., 2007), suggesting that this area might be implicated in encoding numerical information.Left thalamic activation has been also identified when adults perform approximate judgment tasks, independently of stimulus modality (Stanescu-Cosson et al., 2000).Likewise, a study in children found significant involvement of the thalamic areas in a numerosity estimation task.Moreover, the study also showed that right thalamic responses were significantly stronger in low mathematical ability children compared to high ability peers (Kovas et al., 2009).This suggests that a lateralized activation of the thalamus in childhood might be considered a potential signature of individual differences in basic numerical abilities.
It is also worth mentioning that some studies have not found numerosity-related activation in core numerosity areas, and that other areas of the brain, like the cerebellum and the insula, have also been found to be involved in basic numerical tasks (Kovas et al., 2009;Shuman and Kanwisher, 2004).Kovas et al. (2009) and Shuman and Kanwisher (2004) for example, challenged the hypothesis that the IPS specifically represents abstract numerical magnitude.More specifically, Kovas et al. (2009) divided 10-year-old participants into two groups according to whether they had high or low mathematical abilities.These two groups showed similar and different brain activation patterns.Similar responses in the groups in specific brain areas (e.g., the insula) were interpreted as this area being functional for numerical abilities in all children.Response differences between high ability children and low ability children in areas such as the cerebellum, left claustrum, right calcarine sulcus, left lingual gyrus, and right thalamic area were instead suggestive of an involvement of a widely distributed network outside the core numerosity areas.According to the authors, these networks might determine different mathematical abilities.Shuman and Kanwisher (2004) investigated whether non-symbolic number processing elicited hIPS activation by conducting three experiments in adult participants: a number comparison task, an adaptation paradigm, and a numerical task with manipulated difficulty ("distance effect").The findings from these three experiments went in the opposite direction to previous literature and predictions.Firstly, results showed no adaptation effect for numerosity repetition in any of the ROIs considered, including the hIPS region.Secondly, the IPS did not respond more to number than to color (control condition) and there was no variation in activation with task difficulty.The authors hypothesized that the failure to demonstrate domain-specific activation for non-symbolic number in the IPS was due to this region being involved in both numerical processing and other processes that do not involve number (i.e., color).
To summarize, these studies have greatly contributed to identifying the brain areas supporting the human ability to represent and manipulate numerical quantities.Several studies using different neuroimaging techniques and paradigms in adults report that posterior parietal and prefrontal cortices support basic numerical skills in humans.In addition to the IPS and prefrontal cortices, researchers have recently highlighted the involvement of primary visual cortices and thalamic regions when people process numerical information, but additional studies are yet to confirm these recent discoveries.From a developmental perspective, studies indicate that similar areas support numerical processing abilities early in life.However, compared to adults, children show a more rightlateralized activation of the posterior parietal areas and an increased activation in prefrontal regions (See also Salillas et al., 2023 for a recent review).
What could have caused numerical cognition and the specialization of the subserving brain systems to emerge?What problems was it designed to solve in the first place?Numerical representation capacities are not limited to the human species.As it will be outlined in the next section, research on the neural basis of numerical representations abilities in non-human animals shed light on the evolutionary origins of these abilities and contribute to understanding the basic computations behind numerical reasoning.

Are the neural mechanisms supporting basic numerical processing shared with other species?
Human and non-human animals share a sense of number as they demonstrate comparable responses when processing non-symbolic quantities, for example they display approximate numerical capacities, distance effects, ratio-dependent performances, etc. (e.g., Beran and Beran, 2004;Brannon and Terrace, 1998;Cantlon andBrannon, 2005, 2006;Miletto Petrazzini and Wynne, 2016;Nieder, 2005).These relevant behavioral findings evidence that numerical competence is a phylogenetically early faculty.However, they leave open the question of whether similar behavioral patterns across species have their roots in brain homologies or whether the observed capacities arrived via independent brain functions and structures throughout evolution.In response to this key challenge, comparative studies in neuroscience have used different techniques to directly explore how the activation of specific brain areas and the firing of individual neurons in those areas underlie and link to the animals' numerical skills (see Nieder, 2021 for a review).
Early neurophysiological studies in monkeys used a delayed matchto-sample paradigm in which animals were trained to judge, after a short delay, whether two consecutive visual displays contained the same number of items.Findings showed that the monkeys were able to make these judgments in spite that the elements to be judged varied considerably in their physical appearance.More importantly, the studies also revealed the presence of neurons that encoded for specific quantities.These neurons were tuned to the number of items in a visual display: they reached their firing peak when the preferred quantity was presented and their response would decrease as the distance from the target number increased (Nieder et al., 2002;Nieder and Merten, 2007;Viswanathan and Nieder, 2015).
Where were these neurons situated?Neurons tuned to numerosity E. Visibelli et al. were identified in two subregions of the posterior parietal cortex, namely the lateral intraparietal (LIP; e.g., Nieder and Miller, 2004;Roitman et al., 2007) and the ventral intraparietal areas (VIP; e.g., Nieder and Miller, 2004;Tudusciuc and Nieder, 2007).Thus, the localization of these neurons revealed a plausible homology with the intraparietal activation commonly reported in human neuroimaging studies.Additional evidence of the posterior parietal areas being involved in numerical processing comes from electrophysiological and fMRI studies in different mammal species, such as cats and dogs (e.g., Thompson et al., 1970).For instance, Aulet et al. (2019) used fMRI in awake dogs passively attending to visual arrays of dots varying in numerosity (2, 4, 6, 8 and 10 dots).Results showed that the spontaneous ability to discriminate numerical information in this species is also supported by parieto-temporal lobes activation, which increased as a function of numerical ratio.Thus, the similarity between human and nonhuman brain responses points to an evolutionary precursor in the parietal association cortex region which could potentially help to decipher codes that give rise to a sense of number across phylogeny (Nieder, 2021) and to elucidate the roots of more complex numerical abilities in humans (Nieder and Dehaene, 2009).Like in humans, numerosity-related information in monkeys is not only processed in the parietal cortices, but also represented in the frontal lobe, particularly in the lateral prefrontal cortex (Nieder, 2012;Nieder et al., 2002;Nieder and Merten, 2007;Nieder andMiller, 2003, 2004).PFC number-selective neuron activations seem to be independent from continuous variables such as physical changes of the arrays (Nieder et al., 2002;Nieder and Miller, 2004).In addition, PFC activation in the primate is also observed when numerosities are presented in auditory format or as motor actions (Sawamura et al., 2002).Thus, it has been suggested that abstract numerical information, first encoded in the parietal cortex, might be transmitted to the PFC where it is maintained to obtain control over behavior (Nieder and Dehaene, 2009).Therefore, this parietal-to-frontal flow of information supporting basic numerical representations in the brain appears to be common to humans and other primates.
Recent research has aimed to investigate whether animals phylogenetically distant from humans show comparable neural correlates and present similar brain activation in response to numerical information.Interesting studies on birds, specifically corvids, have identified the avian caudolateral nidopallium (NCL) as a possible homologous structure to mammals' PFC.In particular, single cell recording studies observed number-selective neurons activation in the NCL of crows trained to discriminate quantities in a delayed match to sample task similar to the ones observed in the intraparietal cortex of non-human primates (Ditz and Nieder, 2015;Kirschhock et al., 2021) and young chicks (Kobylkov et al., 2022).Moreover, activation of the same number neurons was found in the NCL of crows that did not receive training, confirming a spontaneous predisposition of this pallial area in number processing (Wagener et al., 2018).However, as noted by Stacho et al. (2020), this similarity might be considered as a homoplasy rather than a homology since the NCL does not show the characteristic layers of mammals PFC.
To summarize, studies on human and non-human primates have demonstrated that common neural underpinnings of numerical processing are present among these species.In fact, different species show neural activation of parieto-prefrontal cortices when they are exposed to numbers or involved in discrimination and estimation tasks.Moreover, parietal activation was observed even in mammals phylogenetically more distant to humans than primates, such as dogs.Finally, comparable brain activation related to numerical processing has been observed in birds: the avian NCL appears to be a promising area of homoplasy with mammals' PFC.
It is important to note that most of the evidence of brain activation and neural correlates so far comes from vertebrates, especially nonhuman primates, fish, and birds.Comparative analyses with different vertebrates and invertebrates are needed to better understand the neural origins of number sense.For example, promising studies on reptiles (Miletto Petrazzini et al., 2017), honeybees and spiders have highlighted a sense of number and sophisticated numerical skills in these species (e. g., Bortot et al., 2019;Cross and Jackson, 2017;Giurfa, 2019;Giurfa et al., 2022).Bees, for example, would be able to enumerate the number of landmarks necessary to land on a feeding place basing their judgements on visual patterns information (Chittka and Geiger, 1995).Specific brain regions of these animals, namely the bilateral mushroom bodies and the central complex, have been hypothesized to be the possible centers of numerical processing and to be homologous to the mammalian pallial areas activated in numerical tasks (Tomer et al., 2010).This hypothesis has been recently assessed by Vasas and Chittka (2019) using an insect-inspired neural network model, however no data has been gathered so far regarding brain activation.Studying the brain activity of these species phylogenetically distant from vertebrates might open new insights on the evolutionary basis of numerical processing.It might also contribute to the open debate on whether similar behavioral evidence of numerical abilities comes from an ancestor common to all vertebrates (e.g., Cantlon, 2012) or that numerical competence rises from selection pressures animals had to face to survive during their evolutionary path.The first hypothesis would imply that all species are endowed with the same brain structures from which numerical abilities developed, while the second suggests that competencies would rise from similar challenges and would be passed down genetically to future generations for specie's survival (e.g., Nieder, 2021).

Contribution of molecular techniques to understanding neural correlates of numerical processing
Implementing more refined molecular techniques has the potential to open new directions for studying the neural correlates underpinning numerical processes.This has been the focus of innovative ongoing research in animal models, particularly domestic chicks and zebrafish.Numerical abilities in fish had been extensively documented (e.g.Agrillo et al., 2012;Miletto Petrazzini et al., 2012;Petrazzini et al., 2015reviewed in Agrillo et al., 2017;Agrillo and Bisazza, 2018).Behavioral studies conducted by Hager and Helfman (1991), and Mehlis et al. (2015) had shown, for example, that, when placed in an unfamiliar environment, individual fish tend to join the larger of two shoals.Additionally, fish exhibit -like other species-ratio-dependent discrimination which implies that their performance aligns with Weber's law (Agrillo et al., 2008;Potrich et al., 2015b).
Recently, technical advances have been explored by Messina et al. (Messina et al., 2020(Messina et al., , 2022b) ) to investigate the brain regions involved in discrete quantity estimation in zebrafish.A habituation/dishabituation behavioral paradigm was implemented, in combination with molecular biology analyses.Fish were first habituated to a set of visual stimuli (either 3 or 9 small red dots).Five hours later, during the dishabituation phase, fish were exposed to a novel stimulus that changed in numerosity (from 3 to 9 or vice versa), shape, surface area, or nothing (control groups).This paradigm involves an elementary form of learning, namely decreasing a behavioral response when habituated to a repeatedly presented stimulus and increasing it in response to a novel stimulus (Gerlai, 2016).Zebrafish showed a behavioral response to numerical variations by increasing the time close to the novel stimulus.Thirty minutes after the dishabituation phase, Messina et al. (2020) dissected zebrafish's brains to investigate the brain regions where changes in neuronal activation occurred in response to numerical novelty, namely neural correlates associated with quantity discrimination processes.Neuronal activity was traced by looking at the expression of the immediate early genes (IEGs; c-fos and egr-1).C-fos is an indicator of neuronal activation associated with numerical performance and egr-1 has been associated with more general learning and memory processes (Gallo et al., 2018).To quantify IEG expression in response to numerical novelty and evaluate its changes within the major brain regions in the fish (telencephalon, thalamus, optic tectum, retina, cerebellum, and medulla oblongata), Messina and colleagues used reverse transcription-quantitative polymerase chain reaction (RT-qPCR).
The authors found the most consistent changes in IEG expression concerning the elaboration of numerical dishabituation in the telencephalon and the thalamus, and to a limited extent in the medulla oblongata.However, the level of IEG expression in these areas and in other areas -including the retina, optic tectum, and cerebellum-was also modulated by surface changes.The latter finding suggests that exploring specific sections within the telencephalon and thalamus would be necessary to provide a more precise localization of the brain areas coding for number representations in zebrafish.Thus, a subsequent study by the same group attempted such refined exploration by performing both qPCR and in-situ hybridization (Messina et al., 2022b).In that study, after the dishabituation phase, fish were sacrificed and their brains were dissected into pallial (dorsal) sub-regions of the telencephalon, including the central, medial, and lateral sections (Ganz et al., 2014;Nieuwenhuys, 2009) and subpallial or ventral telencephalic nuclei (V; Ganz et al., 2012).Once again, qPCR analyzed the variation in the expression of egr-1 after the dishabituation phase in these brain regions.The signal for c-fos positive cell detection was also measured but not analyzed because it was too weak (Messina et al., 2022b).In this case, the dorso-central was the only brain region within the pallium that showed a modulation in IEG expression in relation to numerical discrimination processes.However, once again, IEG expression in this area was not only modulated by numerical changes but also by shape changes.Therefore, in situ hybridization assays were instrumental to investigate which specific area differentially responded to each of those variables.The findings revealed that the most caudal parts of the dorso-central pallium responded to numerical changes whereas the rostral parts were sensitive to shape changes.This study points to a pallial mechanism involved in numerical discrimination in zebrafish, which is in line with the location of number-sensitive neurons found with in-vivo whole-brain functional imaging (Messina et al., 2022a).At the same time, the study leaves open the question of whether specific areas in the thalamus (structure identified in the research conducted by Messina et al., 2020in fish, and by Piazza et al., 2007and Stanescu--Cosson et al., 2000 in humans, but not assessed in this in-situ hybridization analysis) might also support these abilities.Finding a recruitment of specific areas both in telencephalic and in thalamic areas for numerical representations would be in line with the sensory input path for discrete magnitude representation outlined by Lorenzi et al. (2021) and Vallortigara et al. (2022), where numerosity is first extracted in primary sensory systems and then processed in associative telencephalic regions.Also in humans, numerical information seems to be encoded in early visual cortices first and then in associative regions as mentioned in the sections above (Fornaciai et al., 2017;Park et al., 2016;Zamarian et al., 2009).Further confirmation in other species would evidence a brain mechanism for extracting visual approximate numerical quantities that shares other similarities across species.
Other parallelisms with the brain mechanisms observed in humans have also been reported in studies using molecular techniques.In the study by Boschetti et al. (2015), for example, the authors trained domestic chicks to pay attention to a specific number of objects on a screen and habituated them to this task for several days.During the test phase, the chicks were divided into two groups: a Number group, which saw a new and variable quantity of objects, and a Control group, which saw new objects in the same quantity as during familiarization.Neuronal activation was traced by analyzing the expression of c-fos protein in different brain regions, including the hippocampus, septum, visual wulst, and the caudolateral nidopallium.Results showed that, compared to the control group, the Number group had a significantly higher number of immunoreactive cells in the dorsolateral hippocampus in the right hemisphere and the dorsal septum in the left hemisphere.It's worth noting that in birds, the dorsolateral hippocampus is considered homologous to the entorhinal cortex in mammals, which is part of the parahippocampus.This finding, the authors claimed, is interesting insofar as the parahippocampal areas in humans are connected to the intraparietal cortex, which plays a crucial role in numerical processing (Boschetti et al., 2015).Notably, however, the localization of these areas conflicts with those reported in previous studies in corvids, in which number-related neurons were rather found in the caudolateral nidopallium, a high-level association area of the avian telencephalon.Further studies are thus needed to extend and confirm these results.
In conclusion, this section reviewed studies unveiling areas associated with numerical processing and using advanced and appropriate techniques in different species.Evidence was gathered with different methods including fMRI, EEG and functional near infrared spectroscopy (fNIRS) in humans, single cell recording in non-human vertebrates, and molecular biology techniques in zebrafish and chicks.The studies consistently show the recruitment of telencephalic areas for processing numerical information (humans: Piazza et al., 2002Piazza et al., , 2004Piazza et al., , 2007;;non-human primates: Nieder, 2012;Nieder et al., 2002;Nieder and Merten, 2007;Nieder and Miller, 2004;Roitman et al., 2007;Viswanathan and Nieder, 2013;corvids: Ditz et al., 2018;Ditz andNieder, 2015, 2016;zebrafish: Messina et al., 2020zebrafish: Messina et al., , 2022b;;chicks: Kobylkov et al., 2022).Specifically, frontal and parietal regions are recruited in numerical processing in humans and non-human primates while homoplasies have been found in the neural activation of other species such as birds and fishes.
We expect that forthcoming research in these species will yield a better comprehension of numerical cognition at different levels of complexity, from gene to behavior.This might open new directions for understanding the functional role of genes associated with mathematical abilities, which have been so far studied almost exclusively in humans and in association with more complex mathematical skills.The studies of numerical learning addressing genetic and molecular levels of analysis will be reviewed in the following sections.

Genetic contributions to numerical cognition and developmental dyscalculia (DD)
Individual differences in mathematical performance have been known for a long time (Dowker, 2005;Dowker et al., 2019).While some individuals excel in math achievement, have an extreme interest for numbers, and are exceptionally rapid and efficient calculators (Lubinski and Benbow, 2006), others struggle with persistent and severe difficulties in learning even the most basic aspects of arithmetic.Low mathematical achievement negatively impacts children's school attainment, mental health, and even self-esteem (Fritz et al., 2019).In adulthood, it reduces the range of working opportunities and it compromises an individual's independence in activities of the everyday life (e.g., Arcara et al., 2017;Benavides-Varela et al., 2015, 2017, 2020;Semenza et al., 2014;Vigna et al., 2022).
These inter-individual differences in numerical capacities have been increasingly studied from the point of view of their relation to domainspecific (i.e., basic numerical capacities; e.g., Butterworth et al., 2017;Passolunghi and Lanfranchi, 2012), domain-general cognitive abilities (e.g., working memory and other executive functions; Alloway and Passolunghi, 2011;Macchitella et al., 2023;Passolunghi et al., 2014), and contextual factors, such as social, economic, and parental influences (e.g., Benavides-Varela et al., 2016;Stevenson et al., 1993).However, experimental approaches have also consistently demonstrated the importance of considering genetic factors in explaining individual variations, both in normal and in pathological groups.
Familial aggregation studies, for example, inform about the degree to which an individual exhibits the same trait or disorder as their parents, siblings, etc. and are a common first step in the identification of genetic determinants of a given phenotype or disease.Early familial aggregation studies of developmental dyscalculia (a specific learning disability that results in impaired arithmetic skills (American Psychiatric Association, 2013; Shalev, 2004), indicated that it tends to occur in members of the same family (Shalev et al., 2001).Shalev et al.'s study reported, in a sample composed of 39 children with dyscalculia, that 66% of the mothers, 40% of the fathers, 53% of the siblings, and 44% of second-degree relatives also had developmental dyscalculia.However, the clustering of such a difficulty in close family members may be explained by sharing of either environmental or genetic factors or both.Indeed, numerical home activities (Benavides-Varela et al., 2016), cultural and parental attitudes towards mathematics (Stevenson et al., 1993), etc. can also explain familial segregation effects (Szűcs and Goswami, 2013).Relatively more solid genetic evidence about individual differences in human behavior comes from twin and adoption studies which will be discussed next.

Twin studies
Twin studies provide a quantitative measure of heritability by comparing effects between monozygotic (MZ) and dizygotic (DZ) twins.Because MZ twins share 100% of their genetic polymorphism while DZ twins share only 50%, any greater similarity in MZ compared to DZ should reflect a genetic contribution, given that both MZ and DZ pairs are supposed to share equally similar environments (Pinel and Dehaene, 2013).Early twin and adoption studies of mathematics performance reported a wide range of heritabilities from 0.20 to 0.90 (e.g., Alarcón et al., 1997;Kovas et al., 2007;Light et al., 1998;Loehlin and Nichols, 1976;Oliver et al., 2004;Thompson et al., 1991;Tosto et al., 2014).The high variability across studies is possibly due to differences in sample sizes, ages, and performance measures employed (Oliver et al., 2004).Twin studies particularly assessing basic number processing and discrimination capacities have reported a modest heritability of 0.32 in 16-year-old twin pairs (Tosto et al., 2014).
Large twin studies have also made important efforts to quantify shared (e.g., 0.16 to 0.20 in Kovas et al., 2007) and non-shared environmental influences over individual phenotypes (e.g., 0.68 in Tosto et al., 2014) informing the relationship between nature and nurture over mathematical abilities in children.To our knowledge, however, no study has provided genetic models to estimate the relative contribution of genetic and environmental factors longitudinally.Such studies would be informative to understand how the separate factors (i.e.phenotype-environment) as well as the interaction between them, can have different weights across development.
Another important area of investigation concerns the extent to which these genetic influences pertain to domain-specific abilities in mathematics or to domain-general cognitive capacities.Twin studies have been also informative in this regard.Studies have reported genetic correlations (ranging from 0.40 to 0.98) between reading and mathematical abilities in unselected samples (Knopik and DeFries, 1999;Kovas et al., 2005;Light et al., 1998;Markowitz et al., 2005;Thompson et al., 1991).A partial genetic overlap between reading and mathematics has been also reported in selected groups with learning disabilities (e.g., Knopik et al., 1997;Kovas et al., 2007;Light and DeFries, 1995).As a result, the 'Generalist Genes' Hypothesis, has been put forward suggesting that genes that affect one area of learning, such as mathematics performance, are largely the same genes that affect other abilities (Plomin and Kovas, 2005), although some genetic effects that are specific to each ability have been also acknowledged (Plomin and Kovas, 2005).
An fMRI study by Pinel and Dehaene (Pinel and Dehaene, 2013) in 19 pairs of monozygotic (MZ) and 13 pairs of dizygotic (DZ) adult twins provided a relevant piece of evidence for linking genetic correlations (higher similarity between MZ than between DZ twins) and functional activation in areas of the brain that support numerical processing.In the study, participants performed a subtraction task, a numerical control task, and a motor control task.The results evidenced that the bilateral posterior superior parietal lobes and the right intraparietal sulcus -which, as explained in the first section, support core number processing-showed a significant genetic contribution.This points to a plausible physiological basis for the reported heritability.However, this study only recruited adults, thus leaving open the possibility that genetic links and the contribution to brain responses may be modulated differently during early phases of arithmetical learning.
Altogether twin studies suggest a pervasive influence of genetic variation in numerical capacities.In the age of molecular genetics, however, the classical twin study design is only one aspect of genetics research.As illustrated above, twin studies estimate the heritability of a trait (in this case mathematical abilities/disabilities), while molecular genetics attempts to locate and identify the effects of candidate genes associated with performance tests.The studies reported below used genetic-molecular approaches to identify specific genes in Developmental Dyscalculia (DD) and in other genetic syndromes showing mathematical difficulties.

Molecular-genetic approaches
Genetic variants can take multiple forms, including variation at the level of a single nucleotide (i.e., single-nucleotide polymorphisms or SNPs), variation in the number of nucleotide repeats (i.e., variable number of tandem repeats or VNTRs), or chromosomal reorganization.Each form of variation can potentially alter genomic function and thus phenotype (Zhang and Meaney, 2010).One of the best-known applications of the information obtained by genome sequences are the genome-wide association studies (abbreviated GWAS).This research approach is used to identify genomic variants that are statistically associated with a risk for a disease or a particular trait.The most common approach of GWAS is the case-control design, which entails the evaluation of two large groups of individuals, looking for genomic variants that occur more frequently in a case group affected with a specific disease or trait (like dyscalculia) compared to a healthy or normally developing control group.
The studies by Docherty et al. (2010), Baron-Cohen et al. (2014) and Chen et al. (2017) are GWAS that used SNPs to identify possible genes affecting mathematical ability and disability.Docherty et al. (2010) first collected 5019 10-year-old twins' mathematical scores.Then, they pooled the DNA from individuals with the highest and lowest mathematical performance (300 individuals each).Docherty et al. (2010) isolated 10 SNPs (rs11225308, rs363449, rs17278234, rs11154532, rs12199332, rs12613365, rs6588923, rs2300052, rs6947045 and rs1215603) that could explain individual differences in mathematical abilities.The nearest genes to these genetic variants were the MMP7, GRIK1, DNAH5, SMAD3, ARID1B, FLJ20160, GUCY1A2, NRCAM, DLD, and NUAK1, making them potential markers for mathematical ability and disability.These 10 SNPs accounted for 2.9% of phenotypic variance in mathematics ability.It is important to note that such a modest fraction of the phenotypic variation explained is common for most complex cognitive traits with polygenic heritability and varies depending on the statistical analyses implemented (Kutalik et al., 2011).In Docherty's study, significant (Bonferroni corrected) associations were found only for three genes (MMP7, GRIK1, and DNAH5).These genes encode for proteins with various functions.MMP7 encodes a protein that degrades components of the extracellular matrix.GRIK1 encodes the neuronal glutamate receptor subunit GluR-5, and DNAH5 encodes the dynein axonemal heavy chain 5 protein and its mutations cause primary ciliary dyskinesia (Carvalho and Haase, 2019).Notably, studies have also related them with embryonic growth and tissue repair (Chakraborti et al., 2003;Hornef et al., 2006), neurotransmission and synaptic plasticity (Bortolotto et al., 1999;Huettner, 2003), and normal brain development (Ibañez-Tallon et al., 2004).Baron-Cohen et al. (2014) GWAS also looked for candidate genes associated with mathematical ability in 16 to 18-year-old participants who differed in having either high (n = 419) or low (n = 183) mathematical ability.Differently from Docherty et al. (2010), they controlled for verbal ability and the number of SNPs assessed (N = 906,600) was nearly double of that in the previous study.Although no SNP reached genome-wide significance, individual genotyping -in a confirmatory sample-indicated that only one SNPs was associated with mathematical abilities (rs789859).This SNP is located on the long arm of chromosome 3 -3q29 within the promoter of FAM43A (family with sequence similarity 43 member A).This gene codes for a protein whose functions are not fully elucidated.However, genetic variants in this region had also been associated with autism, schizophrenia and learning difficulties (Mulle et al., 2010;Nava et al., 2014;Sagar et al., 2013;Willatt et al., 2005).
Another GWAS study was carried out in 1622 7-13 year-old Chinese children (Chen et al., 2017), using the grades on midterm and final exams at the school as the quantitative measures.Genotyping was carried out using a 1.2 million SNP platform which identified four SNPs (rs1012694, rs11743006, rs17778739 and rs17777541) on the SPOCK1 gene.This gene encodes a highly conserved glycoprotein the Testican-1, whose function is under research.It appears to be involved in regulating proliferation, cell-cycle progression, apoptosis, adhesion, and cell-matrix interaction, and it is most prominently expressed in the thalamus of the brain (Edgell et al., 2004).Chen and collaborators suggest that the SPOCK1 gene may be a potential candidate underlying mathematical development.
More recently, Skeide et al. ( 2020) explored the associations between brain structure (grey matter volume) and 18 SNPs located on 10 candidate genes identified in previous studies (RP11-815M8.1,FLJ20160, ROBO1, FAM43A/LSG1, SFT2D1, DLD, NRCAM, NUAK1, C14orf64, and GRIK1; Docherty et al., 2010;Chen et al., 2017;Baron--Cohen et al., 2014;Mascheretti et al., 2014).The authors investigated 3-6-year-old unschooled children (exploration sample: n = 101; replication sample: n = 77) and longitudinally assessed whether the brain-gene associations predicted mathematical performance in school.A significant association between math candidate genes and grey matter volume was detected only for ROBO1.The effect of ROBO1 on grey matter volume was localized in the right parietal cortex (right intraparietal sulcus) and extended to the ventral superior right parietal gyrus.Additionally, grey matter volume associated with ROBO1 within these regions at 3-6 years of age was significantly associated with math scores at 7-9 years of age.This result is compatible with previously mentioned studies, showing that the parietal cortex contributed to mathematical cognition from infancy (e.g., Cantlon et al., 2006), and suggests that individual differences in right parietal cortex volume related to ROBO1 can predict mathematical variability in children.ROBO1 is a member of the immunoglobulin gene superfamily and encodes a protein important for axon guidance and cell migration (National Library of Medicine, n. d.).ROBO1 has been also linked to processes of neurogenesis and proliferation of neurons in the developing neocortex (Yeh et al., 2014) and may be involved in the regulation of the structure and connectivity of the corpus callosum (Darki et al., 2017).
To recap, a few GWAS have specifically focused on identifying genetic variants associated with mathematics ability (see Table 1).Some of these variants are located on genes that are expressed in nerve cell tissue and that regulate neurogenesis, cell functioning, migration and more generally neurodevelopment and function.Moreover, a few of them seem to be associated with cerebral cortical volume and activation in brain areas deputed to quantitative processing.However, there is little overlap across studies.This might be partly explained by the quantitative trait locus hypothesis, indicating that the phenotype of a complex trait -such as the mathematical ability-might be produced by a large number of genes, each one with a small effect (Plomin et al., 2009).Individual differences in mathematical performance might also imply different trait architectures, namely that dyscalculia in one individual might be caused by a microduplication, while in another individual by a deletion of a given sequence on a different gene, and yet in another by no detectable gene, etc. (Carvalho and Haase, 2019).Other factors that might explain the heterogeneity of the results obtained concern methodological variations such as differences in the threshold used across studies to define the low and high mathematical achievement, variations in the performance measures, as well as in the extension/presence of Given the heterogeneous results obtained across studies, and that many of these genetic variants appear associated with neurodevelopment, one intriguing question concerns whether they would also influence individual differences in other cognitive functions.Studies addressing this question and using molecular-genetic approaches will be briefly reviewed in the next section.

Shared genetic components between mathematics and other cognitive abilities
Another strategy that has been often used in the literature to address individual differences consists in estimating the amount of genetic variation associated with one trait and that correlates with variations in other traits.This approach has been used to search for candidate genes associated with phenotypes that tend to co-occur with developmental dyscalculia and dyslexia (Knopik and DeFries, 1999;Light and DeFries, 1995;Markowitz et al., 2005).For example, Ludwig et al.'s (2013) GWA study assessed participants with dyslexia and other groups of unaffected samples.They found that the genetic variant rs133885 within the MYO18B gene was correlated with mathematics ability across samples, suggesting that this SNP contributes to mathematical abilities in children independently of their reading capacities.In addition, using structural MRI in healthy adults, this study showed that this genetic variant contributes to the IPS morphology, namely that carriers displayed a decreased depth of the right intra-parietal sulcus (Ludwig et al., 2013 see also Molko et al., 2003).However, the contribution of this SNP to mathematical abilities remains inconclusive as a subsequent replication study with a larger number of participants failed to find the association to this genetic variant in both dyslexic and general populations (Pettigrew et al., 2015).
In a subsequent study, Mascheretti et al. (2014) investigated the role of eight SNPs within KIAA0319 and ROBO1 genes, in a cohort of 493 Italian nuclear families of probands affected by developmental dyslexia.The aim was to explore the contribution of these genes to mathematical difficulties observed in some dyslexic phenotypes.The findings showed a significant association between ROBO1 and mathematical skills (i.e., mental calculation), supporting the role of this gene in mathematical skills of dyslexic individuals.A related study of the same group (Marino et al., 2011) explored the association between mathematical traits and two developmental dyslexia susceptibility genes (DCDC2 and DYX1C1).The study revealed an association between DCDC2 and "Numerical Facts" and between DYX1C1 and "mental calculation".Together, these studies suggest that some common genetic basis might modulate language and mathematical traits in dyslexic individuals, particularly on those mathematical tasks that entail verbal processes, such as multiplication tables.Stefansson et al. (2014) scanned for Copy Number Variants (CNVs) associated with schizophrenia and/or autism.CNVs are duplications and deletions of genomic regions (Niemi et al., 2018) which turn a specific segment of DNA different among individuals' genomes.Such regions may or may not contain a gene(s) and they are common causes of intellectual disability, autism, and schizophrenia (Carvalho and Haase, 2019).In their study, Stefansson et al. observed that forty-seven controls who did not suffer from either disease or intellectual disability, had a history of reading and mathematical difficulties, and also showed a chromosome deletion of 15q11.2 between breakpoints 1 and 2 (15q11.2(BP1-BP2)).This deletion is known to affect four genes (NIPA1, NIPA2, CYFIP1, and TUBGCP5) and increases the risk for different disorders (Farrell et al., 2020), including a four times higher risk for the dyslexia and dyscalculia phenotype.In addition, structural MRI showed that 15q11.2(BP1-BP2)deletion carriers have reduced gray and white matter in brain areas linked to reading and writing.This finding was supported by Ulfarsson et al. (2017) who confirmed that 15q11.2(BP1-BP2)deletion increased the risk of combined dyslexia and dyscalculia.In addition, by using MRI they showed volume changes in gray matter structures in the left fusiform gyrus, an area that has a major role in both reading and mathematical processing (Butterworth et al., 2011).With fMRI they found decreased activation of the left fusiform and left angular gyri during a linguistic (phonological lexical decision) and mathematical (multiplication verification) task, suggesting the role of this CNV on both structural and functional brain features underlying dyslexia and dyscalculia.Together, the results of the studies addressing mathematical difficulties in children with dyslexia suggest that a shared set of genetic factors may influence both reading and mathematical functions (Kovas and Plomin, 2006).However, in many of these studies the sample constitutes a quite heterogeneous group.It is thus open to discussion whether these results apply equally to unselected samples of the general population, to individuals with mathematical and other cognitive disabilities, and to selected or clinical groups diagnosed with dyscalculia only (Benavides-Varela, 2020;Butterworth, 2019).

Mathematical learning disabilities in genetic syndromes
Individuals with other genetic syndromes that typically affect mathematical functions (e.g., Fragile X Syndrome, Prader-Willi Syndrome, William's syndrome, 22q11.2deletion syndrome or Turner syndrome) have been considered when investigating genetic components underlying mathematical ability (Messina et al., 2022a;Molko et al., 2003;Reiss et al., 2000;Rivera et al., 2002).Some of these studies will be illustrated in this section to gain a better understanding of the heterogeneity observed in mathematical abilities (Brankaer et al., 2017).A summary of their findings is presented in Table 2.
One case of particular interest is Turner syndrome (Baker and Reiss, 2016).This genetic condition affects approximately 1 in 2000 females.It results from a sporadic partial or complete absence of one of the two X chromosomes in females (Ranke and Saenger, 2001).Females with Turner Syndrome present a 50% prevalence of dyscalculia (Murphy et al., 2006).The syndrome typically produces processing deficits (particularly slower reaction times) in the domain of numbers.Arithmetic difficulties in subjects with Turner syndrome are particularly evident on subtractions and operations with large numbers, subitizing and cognitive estimation (Bruandet et al., 2004;Simon et al., 2008), as well as symbolic number processing (Brankaer et al., 2017;Simon et al., 2008).Unlike other genetic conditions, these deficits are often manifested in the absence of intellectual disability and verbal disability (Reiss et al., 2000).
An fMRI study by Molko and colleagues showed an abnormal modulation of intraparietal activations in Turner syndrome as a function of number size during exact and approximate calculation.Morphological analysis also revealed an abnormal length, depth and geometry of the right IPS, suggesting an important disorganization of this brain region in this genetic condition.This pioneering study confirmed a crucial role of the right IPS in the development of arithmetic abilities and pointed to a genetic form of numerical difficulties, which can be related to both functional and structural anomalies in the brain areas processing numerical features (Molko et al., 2003).
Other syndromes caused by different genetic mechanisms including chromosomal abnormalities, microdeletions, microduplications, etc. display more complex profiles characterized by numerical difficulties, together with other cognitive deficits.This is the case of Klinefelter syndrome, a genetic condition distinguished by the presence of an additional X chromosome and which affects 0.1 to 0.2% of males.It causes impairments in arithmetic tasks (Ross et al., 2009;Rovet et al., 1996) but also prominent verbal and reading deficits (Bender et al., 1986 see Karipidis and Hong, 2020 for a review).Mathematical impairments have been also reported in individuals with velocardiofacial syndrome who showed difficulties in symbolic (De Smedt et al., 2008) and basic number processing skills (Attout et al., 2017;De Smedt et al., 2009;Oliveira et al., 2014;Simon et al., 2005), as well as mathematical reasoning tasks.This condition is caused by deletions on the 22 chromosome (22q11.2) which also result in congenital heart malformations, psychiatric disorders, intellectual disability and language delay (for a review see Carvalho et al., 2014;Carvalho and Haase, 2019).A functional brain imaging study linked the numerical deficits observed in individuals with velocardiofacial syndrome with reduced temporal lobe gray matter volume (Eliez et al., 2001).A subsequent study also indicated that visuospatial and numerical information processing difficulties in affected individuals are associated with abnormalities in posterior parietal brain areas, the corpus callosum, and closely related areas in the thalamic and cingulate regions (Simon et al., 2005).
Mathematical learning impairments are an important characteristic also of Williams syndrome (O'Hearn and Landau, 2007), which is caused by deletions in 7q11.23.Individuals with Williams syndrome tend to fail in numerosity as well as number line estimations (Ansari et al., 2007;Opfer and Martens, 2012) and enumeration in the subitizing range (O'Hearn et al., 2011 but see also Paterson et al., 1999Paterson et al., , 2006)).Studies have separately reported deficits in symbolic and non-symbolic modalities (Krajcsi et al., 2009;Rousselle et al., 2013), as well as dissociations between both number processing systems (Libertus et al., 2014).People with Williams syndrome also showed a complex condition from a neuropsychological and medical viewpoint, with common impairments in motor, executive, and visuospatial abilities (Vandeweyer et al., 2012).
Some forms of dyscalculia have also been described in monogenic conditions like neurofibromatosis type 1 (Burgio et al., 2017;Mazzocco, 2001), Prader-Willi syndrome (Bertella et al., 2005;Semenza et al., 2008), and fragile X syndrome.The latter genetic condition is caused by the expansion of a cytosine-guanine-guanine repeat on the X mental retardation gene (FMR1) and currently provides the best evidence of a monogenic component linked to dyscalculic traits (Carvalho and Haase, 2019).Eighty-seven percent of girls with fragile X syndrome show significant deficits in mathematical abilities (Murphy et al., 2006) with reported difficulties in basic numerical comprehension, magnitude estimation and mental number line judgements (e.g., Murphy and Mazzocco, 2008).Full mutations of this gene may also cause other cognitive deficits such as intellectual disabilities, autism and ADHD, while premutations may sometimes affect working memory, executive functions and visuospatial perception.Interestingly, Semenza and colleagues (2012) found that FMR1 premutation carriers, while cognitively spared and capable of carrying out calculation tasks, show subtle mathematical weaknesses in basic number understanding like dealing with analogue scales and certain aspects of number transcoding.The contribution of domain specific (i.e.mathematical) and domain general cognitive factors on the emergence of these deficits constitutes a fertile area of current and future research.

Neurotransmitters' concentrations and math
As illustrated in the previous sections, the main brain areas and networks that underpin numerical cognition in human and non-human species have been extensively studied.Many studies have also addressed the genetic components associated with numerical abilities, particularly in humans.In contrast, to date, only a few studies have explored the contribution of molecular mechanisms to numerical learning.We are not aware of any research that investigated the relationship between basic numerical representations (see Section 1) and neurotransmitter concentrations in different areas of the brain.However, four studies recently focused on the role of neurochemicals in higher human mathematical abilities, such as (symbolic) arithmetic and numerical reasoning (Krause et al., 2018;Zacharopoulos et al., 2022Zacharopoulos et al., , 2021aZacharopoulos et al., , 2021b)).All these studies used proton magnetic resonance spectroscopy (H-MRS), sometimes in combination with other neuroscientific methods.H-MRS is an MRI-based neuroimaging approach that measures in vivo metabolites in a predefined brain region of interest (Egerton, 2021).Specifically, it uses resonance signals from protons of hydrogen to produce spectral data representing the chemical composition of molecules in the brain (Martín Noguerol et al., 2016;Tognarelli et al., 2015).MRS is the first noninvasive technique that assays the products of gene expression or metabolism in the human brain (Roy et al., 2022), even if positron emission tomography (PET) is another possibility to trace neurotransmitters (Ceccarini et al., 2020;Sander and Hesse, 2017).Notably, in contrast to PET and single-photon emission computed tomography (SPECT), MRS does not require the injection of radioactive tracers and, for this reason, might be considered as less invasive.
MRS has been widely used over the past 10-15 years to measure creatine, myoinositol, n-acetyl aspartate, choline, and lactate, which are important for the diagnosis of different neurological disorders (Tran et al., 2009).With the advent of more recent signal acquisition sequences, scientists have started to use MRS to investigate also glutamate and gamma-aminobutyric acid (GABA), which are, respectively, the brain's major excitatory and inhibitory neurotransmitters (Krause, 2015).These neurotransmitters are particularly of interest because they

Table 2
Summary of the genetic syndromes and mechanisms and the associated numerical Impairments.

Fragile X Syndrome
Expansion of a cytosineguanine-guanine repeat on the X mental retardation gene (FMR1) Basic numerical comprehension, magnitude estimation and mental number line judgements and spare calculation abilities ( Murphy and Mazzocco, 2008;Semenza et al., 2012) E. Visibelli et al. are abundant in the brain and affect its activity and connectivity.For example, they have been shown to play a role in cortical excitation and inhibition, perceptual training, visual perception, attention, and cognitive skill acquisition (Cohen Kadosh et al., 2015;Frangou et al., 2018;Kihara et al., 2016;Lunghi et al., 2015;Terhune et al., 2015).Furthermore, these neurotransmitters seem also to be largely involved in neuroplasticity (Hassan, 2021).As far as numerical cognition is concerned, researchers have recently analyzed how resting concentrations of glutamate and GABA, as measured with MRS, possibly relate to different mathematical skills.In detail, the concentrations of these neurotransmitters were measured in brain regions that previous research indicated as relevant for both basic and more advanced numerical tasks, namely the intraparietal sulcus (IPS) and the middle frontal gyrus (MFG).Despite similarities in methods and theoretical frameworks, the MRS studies that relate neurotransmitters' concentrations to mathematical skills can be hardly summarized together, because each of them focused on different research questions.Still, some general considerations can be drawn.

Concentrations of GABA and Glutamate in parietal and frontal brain regions and its association with math abilities
A first study that explored how neurotransmitters subserve numerical skills was conducted by Zacharopoulos et al. (2021b), who analyzed the relation between different numerical skills and resting concentrations of GABA and glutamate in the left IPS and left MFG.Fifty-four university first-year undergraduates completed the following tests: the numerical operations and the mathematical reasoning subtests of the Wechsler Individual Achievement Test (WIAT-II UK), the Tempo-Test-Rekenen (de Vos, 1992), the computational estimation, the numerical agility tests, and the number sequences subtest of the numerical aptitude test (which requires to identify, choosing from a pool of five numbers, the number that correctly completes a sequence; e.g., 1 3 6 10 15 21 _; Choices: 26 27 28 29 30).Additionally, structural MRI and MRS data were collected while participants were watching a movie, and baseline-level neurochemical concentrations were measured.Results showed a negative association between the performance in the number sequences task and the resting concentration of GABA within the left IPS.The relation between GABA in the IPS and the number sequences task was specific to parietal but not frontal regions, and to GABA but not glutamate.Moreover, no significant relationship was found between baseline-level of neurotransmitter concentrations and the performance in the other numerical tasks (Zacharopoulos et al., 2021b).Interestingly, the number sequences task requires the identification of a logical rule that regulates the relations between numbers in a sequence.Some of the abilities this task required -such as numerical procedures, numerical flexibility, or the seeking of a general logic rulewere shared with other tested tasks.As a consequence, the authors concluded that GABA concentration in the left IPS is associated with a unique and specific set of skills related to the identification of a logical rule in the numerical domain that involves numerical flexibility.Notably, in this study neurotransmitter levels were measured at baseline, so their contribution might vary when the person is actively involved in solving a numerical task.

Age-modulated GABA and Glutamate concentrations in the IPS and changes of math abilities
In another study, the same research group examined whether the relationship between resting GABA and glutamate concentrations in the left IPS and the left MFG and math abilities change across development.Previous research had shown that these neurotransmitters play a role in plasticity and learning, but the implications of this fact on a complex ability that emerges slowly (such as arithmetic, mathematical reasoning, etc.) were still unknown.Therefore, Zacharopoulus et al. (2021c) analyzed how the baseline concentration of these neurotransmitters is linked to current and future mathematical abilities.A cross-sectional longitudinal design was adopted, and 255 students from primary school to university were examined twice, with approximately 1.5 years of distance.The mathematical abilities were assessed by z-scoring and averaging the results obtained in three tests, namely the numerical operations and the mathematical reasoning subtests of the Wechsler Individual Achievement Test (WIAT), and the tempo test Rekenen.Moreover, the general cognitive abilities were assessed with the matrix reasoning subtest of the Wechsler Abbreviated Scale of Intelligence (WASI II).Structural MRI and MRS data were obtained, assessing resting GABA and glutamate concentration in the left IPS and left MFG.Findings revealed that resting glutamate and GABA in the left IPS explained unique variance both in current and future mathematical achievement.However, the relation between IPS neurotransmitters concentration and mathematical achievement was moderated by age: adults showed a positive association between glutamate concentration and mathematical abilities and a negative association between GABA concentration and mathematical abilities, while the opposite was found in younger participants.Of note, these associations were not confounded by general cognitive ability.Concerning the left MFG, glutamate concentration was found to be negatively associated with mathematical abilities in younger participants but positively associated with mathematical abilities in mature participants.Contrary to the IPS, however, age did not moderate the relationship between GABA concentration and mathematical abilities in the left MFG.These results were interpreted considering the concept of plasticity: GABA and glutamate concentrations enhance or constrain the plasticity of a given cognitive function depending on the sensitive period for that cognitive function.Specifically, it is welldocumented that, as the person becomes more proficient in solving arithmetic problems, the reliance on frontopariental network is reduced, while the reliance on other brain regions that support fact retrieval and semantic memory (such as the angular gyrus and the hippocampus) increases (Bailey et al., 2012;Geary et al., 2007;Menon, 2016).Therefore, it might be that the developmental increment in glutamate and decrement in GABA from childhood to adulthood, particularly in the IPS, represents one of the mechanisms that underpin this switch.

GABA and Glutamate concentrations in case of math education withdrawal
The same research group additionally investigated whether the relationship between neurotransmitter concentrations in left MFG and left IPS and math abilities is affected by a lack of math education (Zacharopoulos et al., 2021a).This question is appropriate because in some countries, such as the UK, 16-y-old adolescents can decide to stop studying math.Therefore, English adolescents from similar educational systems and levels might differ specifically in their math education.Individuals with such different mathematical backgrounds were recruited for this research.Like in previous studies, structural MRI and MRS were used to assess GABA and glutamate concentrations in left IPS and left MFG, while resting fMRI was adopted to examine the frontoparietal connectivity.Mathematical abilities (namely the performance in numerical operation and the mathematical reasoning subtests of the Wechsler Individual Achievement Test), math anxiety, and general cognitive abilities were assessed.This study consisted of two experiments.In the first one, 87 A-level students were tested to see whether the neurotransmitter concentrations in their brains could classify whether they were lacking mathematical education.Results showed that decreased GABA concentration within the left MFG successfully classified whether an adolescent was studying math and was negatively associated with frontoparietal connectivity.The second experiment, instead, was aimed at clarifying whether differences in neurotransmitter concentrations exist before the continuation of math education, thus impacting the decision or likelihood of continuing math education (biomarker account), or whether the differences are due to the lack of math education itself (plasticity account).In this case, 42 pre-A-level E. Visibelli et al. students were tested.Results did not support the biomarker account, indicating that previous findings were not due to preexisting differences before mathematical education ceased.In addition, it was observed that left MFG GABA also predicted the changes in mathematical reasoning ~19 months later.

GABA and Glutamate concentrations associated with exceptional math skills
Another study measuring neurotransmitter levels and that is worth mentioning compared the neurobiochemical characteristics of a math genius to those of four gender-, age-and education-matched highfunctioning expert controls without prodigious abilities (Krause et al., 2018).Indeed, understanding the neurobiological factors that underlie exceling skills can be relevant for the description of the individual variability observed in the numerical domain.Recently, Krause and colleagues (2018) studied the neurobiological characteristics of a math genius with no known history of neurological or psychiatric conditions.Whenever presented with complex arithmetic problems (often multiplication, root calculation, or prime factorization), the tested genius provided the answer within only a few seconds and his standardized mathematical abilities ranked above the 99.8th percentile.During the study, the performance in numerical operations and mathematical reasoning was assessed for both the genius and the controls by using the standardized Wechsler Individual Achievement Test.In a separate session, MRS was used to measure glutamate and GABA levels in the right MFG; the primary visual cortex was additionally assessed as a control region.The choice of focusing on the right MFG was motivated because previous studies had found that this area is active during mental calculation in expert calculators (Fehr et al., 2010;Pesenti et al., 2001).When compared to controls, the genius showed a lower ratio of glutamate and GABA concentration in the right middle frontal gyrus (MFG), but not glutamate or GABA individually.Importantly, caution is needed before coming to any conclusion about excitation/inhibition balance in a given area of the brain.The MRS measures of glutamate and GABA concentrations are likely to consist of a combination of neurotransmitters as well as precursors that could be converted into other metabolic entities (Krause et al., 2018;Tognarelli et al., 2015;Vandeweyer et al., 2012).Notably, Krause and colleagues (Krause et al., 2018) did not investigate the concentrations of glutamate and GABA in the IPS, nor in the left MFG.Future research could additionally investigate the ratio of glutamate and GABA in left and right IPS, as well as left MFG, to better understanding the balance of the concentration of these neurotransmitters in different areas of the brain relevant for numerical cognition.The concentrations of glutamate and GABA were measured also in the primary visual cortex; in this case, no significant differences were detected.

The role of dopaminergic modulation in mathematical abilities
Besides GABA and glutamate, dopamine is another neurotransmitter that might regulate numerical cognition.To date, some studies have analyzed the role of dopamine in executive functions and goal-directed behavior in primates (Ott et al., 2014(Ott et al., , 2018;;Ott and Nieder, 2019;Vijayraghavan et al., 2017).However, studies about the relationship between dopamine and numerical cognition are scarce.A study that is worth mentioning was conducted by Juĺio-Costa et al. (2013), who identified a possible relation between dopamine metabolism and numerical cognition in 155 8-12-year-old children.One of the enzymes that regulate dopamine's metabolism is Catechol-O-methyltransferase (COMT; Karoum et al., 2002).Valine158methionine (val158met) is a polymorphism of the COMT enzyme, generated by nucleotide substitution of the COMT gene (Weinshilboum et al., 1999).Tetra-primer amplification refractory mutation system-polymerase chain reaction (ARMS-PCR), a genotyping method, was used to detect SNPs (Ruiz-Sanz et al., 2007), in the COMT val158met polymorphism.Numerical cognition was assessed through symbolic numerical tasks (i.e., simple operations, transcoding tasks from verbal to Arabic digits and vice versa) and non-symbolic numerical tasks (i.e., magnitude estimation and comparison tasks).Juĺio-Costa et al. (2013) found an association between Val158met and number processing tasks.Moreover, results showed a difference between children homozygous for valine allele (COMT val/val) and children heterozygous plus methionine homozygous (met+).Specifically, the groups differed regarding numerical non-symbolic measures, namely non-symbolic magnitude comparison task and non-symbolic magnitude estimation task.This study adds on and links to the previous section on SNPs associated with mathematical abilities in humans and encourages future studies to analyze the role of dopamine in numerical cognition more in-depth.

Summary
Table 3 summarizes the overall findings of the studies presented in this section.Results showed that concentrations of GABA in the left IPS are negatively associated with overall math abilities (Zacharopoulos et al., 2021c) and specially with number sequence tasks (Zacharopoulos et al., 2021b).Notably, when participants lack math education, no significant associations with GABA concentrations in IPS are found.Conversely, reduction of GABA levels in another brain area, namely the left MFG, predicted withdrawal from math education (Zacharopoulos et al., 2021a).On the other hand, concentrations of glutamate in the left IPS were associated with overall mathematical skills.This association appeared to be modulated by age (Zacharopoulos et al., 2021c).Similarly, glutamate concentration in the left MFG was found to be negatively associated with mathematical abilities in younger participants but positively associated with mathematical abilities in mature participants (Zacharopoulos et al., 2021c).Additional research would be necessary to confirm the latter findings, which were only verified in one the studies carried out so far in this emerging field.

Modulation of mathematical brain functions using transcranial stimulation
New techniques based on transcranial electrical stimulation (tES) are promising for the enhancement of mathematical cognition and learning (Lazzaro et al., 2022;Schroeder et al., 2017).tES is an umbrella term that includes different non-invasive techniques, such as the transcranial direct current stimulation (tDCS), the high-definition tDCS (HD-tDCS), the transcranial random noise stimulation (tRNS), and the transcranial alternating current stimulation (tACS) (Lazzaro et al., 2022).All these techniques involve a weak electrical current applied through electrodes placed on the scalp.
Although the neurobiological bases of these techniques are not fully understood, different molecular and cellular mechanisms, including changes in membrane potential and neural synchronization, protein synthesis, and glial cells are likely to be involved (Korai et al., 2021).
Notably, some studies have shown that anodal and cathodal tDCS affect neurotransmitters concentrations and activity in the stimulated cerebral cortex.Specifically, a MRS study showed that anodal stimulation reduces local GABA concentration, while cathodal tDCS reduces local glutamatergic neuronal activity with a highly correlated reduction in GABA (Stagg et al., 2009).These results were confirmed by a study conducted by Antonenko et al. (2017), who observed a significant reduction of GABA levels after anodal tDCS compared with sham tDCS.
Another research found that, upon administration of GABA antagonists, anodal tDCS produces delayed but enhanced excitability increase in cortical or subcortical areas (Nitsche et al., 2004).More recently, Alvarez-Alvarado et al. ( 2021) measured Glutamine/glutamate (Glx) and GABA concentrations via proton magnetic resonance spectroscopy at baseline and at the end of a 2-week intervention that consisted in tDCS stimulation over the prefrontal cortex combined with a cognitive training.Preliminary evidence suggested that this intervention E. Visibelli et al. increased excitatory neurotransmitter concentrations in frontal cortices.All in all, research indicates that tDCS affects different molecular and cellular mechanisms in the brain, including local neurotransmitters concentrations.
Similarly, in a recent study it was observed that repeated tRNS stimulation on prefrontal cortex of 62 mice influenced the excitatoryinhibitory ratio in this brain region by reducing GABAergic activity (Sánchez-León et al., 2021).Therefore, it is possible that also tRNS's efficacy is at least partly related to the modulation of neurotransmitter concentrations.
Possible changes in GABA and glutamate by means of tES directly link to the roles of these neurotransmitters in mathematical abilities as illustrated above, and to the reported effectiveness of this technique for improving some aspects of numerical cognition (Lazzaro et al., 2022;Schroeder et al., 2017).For example, Lazzaro et al. (2022) reviewed 26 studies, the majority of which had used tDCS and tRNS to enhance non-symbolic and symbolic number processes (N = 9), arithmetic processes (N = 16), or both number and arithmetic processes (N = 1).The review concluded that most studies (20 out of 26) found that tES is effective in improving some aspects of numerical cognition.Specifically, anodal tDCS over parietal regions and bilateral tDCS over frontal regions strongly enhance number processing; bilateral tDCS over parietal and frontal regions and left anodal tDCS over frontal regions improve arithmetic skills; bilateral tRNS regardless of brain regions (parietal or frontal) is effective at improving number and arithmetic processes (Lazzaro et al., 2022).
Similarly, other review articles (Schroeder et al., 2017;Simonsmeier et al., 2018) evidenced the potential of using tES in educational settings and highlighted the conditions under which these techniques might be more effective.The study of Simonsmeier et al. (2018), for instance, suggests that tES mainly affect neurotransmitters and synaptic mechanisms during the learning process, while their role on recalling information is limited.Indeed, their findings revealed that participants perform better in a test when the stimulation is administered during the learning intervention, than during the test itself.
Although these techniques represent promising tools in clinical and school settings, all the reviews noted a high methodological heterogeneity across studies, as well as a lack of research in clinical populations.In the future, a deeper knowledge of these biological mechanisms, together with a better understanding of the impact of the levels of these neurotransmitters on mathematical skills in typical and atypical populations, will possibly result in a widespread use of tES techniques.

Conclusions
The studies reviewed here provide a comprehensive systems neuroscience view of the neural basis of mathematical learning (see Fig. 1).In the first section, we summarize the main areas involved in humans' basic non-symbolic number processing.Studies converge in identifying numerosity-related activation in two brain areas, namely associative posterior parietal regions and the prefrontal cortices.Parietal and frontal areas have been also identified in young children.However, the studies indicate that, compared to adults, developmental populations show a more right-lateralized activation of the posterior parietal areas and an increased activation in prefrontal regions.This suggests that maturational, as well as cultural or educational factors influence the organization of the brain systems that subserve numerical capacities.
From an evolutionary perspective, studies have verified common neural underpinnings in human and non-human primates.For example monkeys, like humans, show neural activation of parieto-prefrontal cortices when they represent numerical information in different sensory modalities (e.g., visual, auditory, etc.).Parietal, but not frontal, activation has also been attested in other mammals such as dogs.In addition, researchers have identified brain regions that are sensitive to numerical changes in species phylogenetically more distant to humans, such as corvids, domestic chicks, and fish.Overall, the study of these species has contributed to the open debate on whether numerical abilities emerged from an ancestor common to all vertebrates.In birds, the avian caudolateral nidopallium has been identified as a possible homologous structure to mammals' PFC.In fish, like in primates,

Table 3
Summary of the main findings concerning the relationship between resting neurotransmitters concentrations and mathematical skills.Significant associations are in bold.The symbol "⊕" indicates positive associations, while the symbol "⊖" indicates negative associations.Intraparietal sulcus (IPS), middle frontal gyrus (MFG), gamma-aminobutyric acid (GABA).associative pallial regions (i.e., caudal parts of the dorsocentral pallium) and primary sensory systems (i.e., the thalamus) have been associated with numerical discrimination capacities.Crucially, parallelisms with the brain mechanisms observed in humans have been possible by the recent inclusion of more refined molecular techniques to explore these animal models.On the other hand, studies exploring inter-individual differences in numerical capacities in humans converge on the importance of considering the genetic contributions to numerical cognition.Different approaches have been implemented.For example, measures of concordance between monozygotic and dizygotic twins suggest a pervasive influence of genetic variation in numerical capacities.Moreover, higher similarity between monozygotic than dizygotic twin pairs in the functional activation of core number processing areas i.e., parietal brain regions, points to a plausible physiological basis for the reported heritability.However, there is large variability in the genetic correlations reported across heritability and twin studies.
A few GWAS have specifically focused on identifying genetic variants associated with mathematical difficulties.Although several candidate genes have been identified, there is little overlap across studies.This, on the one hand, reflects the nature of the mathematical abilities tested which involved a wide variety of numerical skills across studies (not E. Visibelli et al.only basic numerical processing).On the other hand, it indicates that a given complex phenotype -such as mathematical learning disabilities-is likely to result from a large number of genes and their interactions.The differential roles of all these genes probably explain the heterogeneity of the profiles of mathematical learning difficulties.Interestingly, only one candidate gene, namely ROBO1, has been so far linked to numerical disabilities in more than one study.This gene has been recently associated with individual differences in the structure (volume) of the right parietal cortex in young children.In addition, these individual differences predicted mathematical scores of the children, years later, when they were in primary school.Skeide et al. (2020) is the first study to provide such straight brain-gene associations.This opened new avenues for fundamental research.A recent study also examined the distribution of ROBO1 -among other genes previously associated with developmental dyscalculia-in the left and right pallium of the zebrafish (Messina et al., 2021).The study found that ROBO1 (along with NIPA1, NIPA2) was predominantly expressed in the right hemisphere of the animals' brain.This suggests a basic homology in the expression of this gene in human a non-human animals and highlights the relevance of assessing cerebral lateralization to further understanding the neurobiological mechanisms involved in the insurgence of developmental dyscalculia (Miletto Petrazzini et al., 2020;Shalev et al., 1995).Remarkably, the expression of ROBO1 is also altered in dyslexic individuals.This aligns with the view that genes that affect one area of learning, such as mathematics performance, might also affect other learning abilities, such as reading (Kovas et al., 2005) and, therefore cause the co-morbidity between dyscalculia and dyslexia observed in some individuals.
Studies in persons with a known genetic syndrome further illustrate the heterogeneity of mathematics learning disabilities and the wide range of genetic and chromosomic variations that could be involved in determining such difficulties.Distinct cognitive and developmental profiles in these genetic syndromes and their association with poor mathematics achievement have relevant implications for diagnosis and intervention.In other words, because there are many different factors that cause mathematics difficulties, the approaches to effectively supporting these children have been possibly be tailored to their specific profiles (Mazzocco, 2015).The independent contribution of mathematical and other cognitive factors is difficult to tell apart in these genetic syndromes and represents a central issue of current and future research.
An innovative set of studies explored mathematical learning from yet a different level of analysis.They focused on understanding how specific neurotransmitters might be engaged in mathematical functioning.A pioneer study showed that COMT-related dopaminergic modulation may be related to the development of magnitude processing and magnitude representations (Júlio-Costa et al., 2013).More recent studies investigated the roles of GABA and glutamate concentrations in brain regions relevant for numerical cognition, namely the intra-parietal sulcus and medial frontal gyrus, and its relation to math abilities.They showed how neurotransmitters are regulated and balanced and how this balance changes as a function of age, mathematical learning exposure, or proficiency.However, some limitations prevent final conclusions at the current state of the research.First, significant correlations were found with mathematical tasks that differed across studies.Second, neurotransmitter concentrations were acquired in baseline conditions rather than when participants were carrying out the arithmetic tasks.Third, the studies investigated neurotransmitter concentrations in a variety of brain regions (e.g., only the left or the right IPS but not both), which prevents an integrative framework of the neurochemical regulation in the brain and its associations with math abilities.Still, these studies constitute a breakthrough and represent an opportunity for better understanding the regulation of specific neurotransmitters as possible markers of developmental disorders, individual performance, and brain plasticity associated with math learning capacities across the lifespan.This would, in turn, open new perspectives in what interventions to improve numerical abilities may concern.Transcranial stimulation over parietal and frontal regions, for example, have been successful for improving some aspects of numerical cognition (Lazzaro et al., 2022;Schroeder et al., 2017).Although the neurobiological bases of these techniques are not fully understood, their effects over GABA and glutamate regulation processes might be possibly involved in their effectiveness.
In short, research exploring the neurobiology of mathematical learning at different levels of analysis translates into an opportunity to elucidate a wide range of issues from system brain mechanisms, their evolutionary roots, and the implications for educational and clinical practices.This has also the potential to shed light on the possible causes of mathematical learning difficulties, on early diagnostic markers, and on intervention programs for people suffering from them (for examples of intervention programs see Benavides-Varela et al., 2020, 2023;Re et al., 2020).We anticipate that forthcoming research within only a few years will yield a clearer characterization of the downstream steps of these complex abilities and will build up clearer connections between cellular, molecular, genetic, brain functions and the related behavioral outcomes.This requires major, collaborative, cross-level research, and urges investigators not only to continue but to extend the scope of this integrative approach.

Table 1
Summary of the studies exploring candidate genes and genetic variants associated with individual differences in mathematical abilities.