How auditory neurons count temporal intervals and decode information.

The numerical sense of animals includes identifying the numerosity of a sequence of events that occur with specific intervals, e.g., notes in a call or bar of music. Across nervous systems, the temporal patterning of spikes can code these events, but how this information is decoded (counted) remains elusive. In the anuran auditory system, temporal information of this type is decoded in the midbrain, where "interval-counting" neurons spike only after at least a threshold number of sound pulses have occurred with specific timing. We show that this decoding process, i.e., interval counting, arises from integrating phasic, onset-type and offset inhibition with excitation that augments across successive intervals, possibly due to a progressive decrease in "shunting" effects of inhibition. Because these physiological properties are ubiquitous within and across central nervous systems, interval counting may be a general mechanism for decoding diverse information coded/encoded in temporal patterns of spikes, including "bursts," and estimating elapsed time.

Financial and numerical abilities: patterns of dissociation in neurological and psychiatric diseases.

The present work investigates whether financial abilities can be associated with numerical abilities and with general cognitive abilities. We compared performance on numerical and financial tests, and on tests routinely used to measure general cognitive performance, in healthy controls and in a group of people with heterogeneous pathological conditions including mild cognitive impairment, amyotrophic lateral sclerosis, traumatic brain injury, and schizophrenia. Patients showed lower performances in both numerical and financial abilities compared to controls. Numerical and financial skills were positively correlated in both groups, but they correlated poorly with measures of general cognitive functioning. Crucially, only basic financial tasks -such as counting currencies- but not advanced ones -like financial judgments- were associated with numerical or general cognitive functioning in logistic regression analyses. Conversely, advanced financial abilities, but not basic ones, were associated with abstract reasoning. At a qualitative analysis, we found that deficits in numerical and financial abilities might double dissociate. Similarly, we observed double dissociations between difficulties in financial abilities and cognitive deficits. In conclusion, financial abilities may be independent of numerical skills, and financial deficits are not always related to the presence of cognitive difficulties. These findings are important for both clinical and legal practice.

The influence of visual illusion perception on numerosity estimation could be evolutionarily conserved: exploring the numerical Delboeuf illusion in humans (Homo sapiens) and fish (Poecilia reticulata).

Discriminating between different quantities is an essential ability in daily life that has been demonstrated in a variety of non-human vertebrates. Nonetheless, what drives the estimation of numerosity is not fully understood, as numerosity intrinsically covaries with several other physical characteristics. There is wide debate as to whether the numerical and spatial abilities of vertebrates are processed by a single magnitude system or two different cognitive systems. Adopting a novel approach, we aimed to investigate this issue by assessing the interaction between area size and numerosity, which has never been conceptualized with consideration for subjective experience in non-human animals. We examined whether the same perceptual biases underlying one of the best-known size illusions, the Delboeuf illusion, can be also identified in numerical estimation tasks. We instructed or trained human participants and guppies, small teleost fish, to select a target numerosity (larger or smaller) of squares between two sets that actually differed in their numerosity. Subjects were also presented with illusory trials in which the same numerosity was presented in two different contexts, against a large and a small background, resembling the Delboeuf illusion. In these trials, both humans and fish demonstrated numerical biases in agreement with the perception of the classical version of the Delboeuf illusion, with the array perceived as larger appearing more numerous. Thus, our results support the hypothesis of a single magnitude system, as perceptual biases that influence spatial decisions seem to affect numerosity judgements in the same way.

Quantity discrimination by kittens of the domestic cat (Felis silvestris catus).

Quantitative abilities are well described in many species and in diverse life situations, including in the adult domestic cat. However, such abilities have been much less studied during ontogeny. In the present study we examined spontaneous quantity discrimination by pre-weaning age kittens in two-way food choice experiments. In Experiment 1, 26 kittens performed 12 trials with different ratios between the number of same-size food items. In Experiment 2, 24 other kittens performed eight trials with different ratios between the size of two food items. We found, in general, that the kittens discriminated between the different amounts of food and spontaneously chose the larger one, but that their choice was influenced by the ratio of difference. The kittens in Experiment 1 chose the larger number of same-size food items if the ratio was smaller than 0.4 and in Experiment 2 they chose the larger pieces of food if the ratio between the items was smaller than 0.5. Because the kittens' choice was not influenced by the absolute number of food items or the numerical difference between them in Experiment 1, it suggests that their cognitive performance relied on an analog magnitude system rather than on an object file system during the quantity discrimination tasks. We discuss our results considering the ecological and social background of cats and compare it with the performance of previously studied species.

Neurocognitive correlates of numerical abilities in Parkinson's disease.

People with Parkinson's disease (PD) experience functional limitations early in the progression of the disease, showing, among other cognitive deficits, difficulties in mathematical abilities. The neural correlates of such abilities have been scarcely investigated in PD, and it is not known whether patients may exhibit difficulties only in formal numerical tasks (e.g., mental multiplications), or also in everyday activities involving numbers (i.e., informal numerical abilities such as time estimates). The present study investigated formal and informal numerical abilities in PD patients and explored their relationship with cortical and subcortical brain volume. We examined patients with PD and mild cognitive impairment (PD-MCI) and age-matched healthy controls (HCs) using the numerical activities of daily living (NADL) battery, assessing both scholastic numerical abilities (formal test), and the ability to use numbers in everyday life (informal test). We compared NADL performances in both groups. Within the PD group, we investigated the association between NADL and cortical and subcortical volumes using multiple linear regressions. The correlation with other cognitive tests was also explored. PD-MCI performed worse than HC in the formal NADL test. In PD-MCI patients, brain-behavior correlations showed two distinct patterns: cortical volumes correlated positively, while striatal volumes correlated negatively with NADL formal tasks. Both formal and informal tests correlated with measures of cognitive functioning. Our results suggest specific impairments in formal numerical abilities in PD-MCI, but not in everyday activities. While cortical atrophy is associated with worse performance, the negative correlations with subcortical regions suggest that their activation may reflect potential compensatory mechanisms.

Towards a standardization of non-symbolic numerical experiments: GeNEsIS, a flexible and user-friendly tool to generate controlled stimuli.

Several studies have suggested that vertebrate and invertebrate species may possess a number sense, i.e. an ability to process in a non-symbolic and non-verbal way the numerousness of a set of items. However, this hypothesis has been challenged by the presence of other non-numerical continuous physical variables, which vary along with numerosity (i.e., any change in the number of visual physical elements in a set naturally involves a related change in visual features such as area, density, contour length and convex hull of the stimulus). It is therefore necessary to control and manipulate the continuous physical information when investigating the ability of humans and other animals to perceive numerousness. During decades of research, different methods have been implemented in order to address this issue, which has implications for experiment replicability and inter-species comparisons, since no general standardized procedure is currently being used. Here we present the 'Generation of Numerical Elements Images Software' (GeNEsIS) for the creation of non-symbolic numerical arrays in a standardized and user-friendly environment. The main aim of this tool is to provide researchers in the field of numerical cognition a manageable and precise instrument to produce visual numerical arrays controlled for all the continuous variables. Additionally, we implemented the ability to actively guide stimuli presentation during habituation/dishabituation and dual-choice comparison tasks used in human and comparative research.

A sense of number in invertebrates.

Non-symbolic numerical abilities are widespread among vertebrates due to their important adaptive value. Moreover, these abilities were considered peculiar of vertebrate species as numerical competence is regarded as cognitively sophisticated. However, recent evidence convincingly showed that this is not the case: invertebrates, with their limited number of neurons, proved able to successfully discriminate different quantities (e.g., of prey), to use the ordinal property of numbers, to solve arithmetic operations as addition and subtraction and even to master the concept of zero numerosity. To date, though, the debate is still open on the presence and the nature of a «sense of number¼ in invertebrates. Whether this is peculiar for discrete countable quantities (numerosities) or whether this is part of a more general magnitude system dealing with both discrete and continuous quantities, as hypothesized for humans and other vertebrates. Here we reviewed the main studies on numerical abilities of invertebrates, discussing in particular the recent findings supporting the hypothesis of a general mechanism that allows for processing of both discrete (i.e., number) and continuous dimensions (e.g., space).

Giraffes go for more: a quantity discrimination study in giraffes (Giraffa camelopardalis).

Many species, including humans, rely on an ability to differentiate between quantities to make decisions about social relationships, territories, and food. This study is the first to investigate whether giraffes (Giraffa camelopardalis) are able to select the larger of two sets of quantities in different conditions, and how size and density affect these decisions. In Task 1, we presented five captive giraffes with two sets containing a different quantity of identical foods items. In Tasks 2 and 3, we also modified the size and density of the food reward distribution. The results showed that giraffes (i) can successfully make quantity judgments following Weber's law, (ii) can reliably rely on size to maximize their food income, and (iii) are more successful when comparing sparser than denser distributions. More studies on different taxa are needed to understand whether specific selective pressures have favored the evolution of these skills in certain taxa.

Aphasia and Math: Deficits with Basic Number Comprehension and in Numerical Activities of Daily Living.

OBJECTIVE: In the present study, we explored numerical problems in individuals with aphasia. We investigate whether numerical deficits, usually accompanying aphasia, can be observed on number comprehension tasks that do not necessarily require an oral response. METHOD: Individuals with aphasia were classified into anterior, posterior, and global subgroups according to the lesion type. To investigate numerical cognition, we used a relatively recent tool, the Numerical Activities of Daily Living (NADL). RESULTS: The results showed that individuals with aphasia have problems with tasks of basic number comprehension as well as in most NADL. In the formal part of the NADL, anterior aphasic patients made comparatively more errors than the posterior aphasic patients. Global aphasic patients presented an invariably poor performance on almost all tasks. CONCLUSION: The results provide insight into how numerical deficits may impair an individual with aphasia in activities of daily living. This study is a preliminary attempt to start the validation process of the NADL for the Greek population.

Mastery of structured quantities like finger or dice patterns predict arithmetic performance.

In the present study, we investigated whether structured quantities like finger or dice patterns are enumerated better than unstructured quantities because they may not require counting. Moreover, we hypothesized children's mastery of structured quantities to predict their later arithmetic performance longitudinally. In particular, we expected that children more proficient in enumerating structured quantities early in their numerical development, should develop more effective calculation strategies later because they may rely on counting less. Therefore, we conducted a longitudinal study (including 116 children, 58 girls) over the course of about 7 months from preschool (at about 6 years of age) to the middle of first grade. Results showed that structured quantities were indeed enumerated more accurately and faster than unstructured quantities in preschool. Additionally, we observed significant associations of enumeration of structured and unstructured with children's addition performance in first grade. However, regression analysis indicated only enumeration of structured but not unstructured quantities to significantly predict later addition performance. In sum, this longitudinal study clearly indicates that mastery of structured quantities seems to be beneficial for children's development of basic arithmetic abilities.

The experience-based format of probability improves probability estimates: The moderating role of individual differences in numeracy.

People with low statistical numeracy have difficulties understanding numerical information. For instance, they often misunderstand the probability of experiencing side effects, which could reduce adherence to medical treatments. We investigated whether presenting information about probability using a method based on the direct experience of events influences the accuracy of probability estimates compared to viewing a static numerical description of the same information. Participants completed a numeracy test and were randomly assigned to one of two conditions. In the description-based probability condition, participants were presented with 24 binomial distributions consisting of a target stimulus "X" and a distractor stimulus "·" in the form of odds (the distribution "7 × 13 ·" is an example of a 35% probability: here the target [distractor] stimulus was present 7[13] times in a 20-stimulus distribution). In the experience-based probability condition, participants observed the same information but the stimuli were randomly arranged and displayed sequentially. Participants in both conditions estimated the probability of the target stimulus in each trial. In the experience-based format participants with low numeracy made more accurate probability estimates in comparison to the description-based format. In contrast, accuracy in participants with high numeracy was similar in the two formats.

Discrimination of group numerousness under predation risk in anuran tadpoles.

For social animals, group size discrimination may play a major role in setting the trade-off between the costs and benefits of membership. Several anuran tadpoles show different degrees of social aggregation when exposed to the risk of predation. Despite the importance of aggregative behaviour as an anti-predatory response, the mechanism underlying tadpole choice of the group to join to has not been sufficiently investigated. To establish whether visual cues provide sufficient information to enable tadpoles to choose between aggregations differing in size, we explored the abilities of the larvae of two anuran species (green toad Bufotes balearicus and edible frog Pelophylax esculentus) to discriminate among four numerical combinations of conspecific tadpoles (1 vs. 4, 3 vs. 4, 4 vs. 6 and 4 vs. 8), either in the presence or absence of predatory cues. Our results suggest that in anuran larvae the capacity to discriminate between quantities is limited to small numbers (1 vs. 4 for B. balearicus and both 1 vs. 4 and 3 vs. 4 for P. esculentus). Predator-exposed toad tadpoles stayed longer close to the larger group, supporting aggregation as a major anti-predator behaviour in bufonids, while frog tadpoles showed a preference for the smaller groups, though in predator-free trials only, probably associated with lower intra-specific competition.

Continuous and discrete quantity discrimination in tortoises.

The ability to estimate quantity, which is crucially important in several aspects of animal behaviour (e.g. foraging), has been extensively investigated in most taxa, with the exception of reptiles. The few studies available, in lizards, report lack of spontaneous discrimination of quantity, which may suggest that reptiles could represent an exception in numerical abilities among vertebrates. We investigated the spontaneous ability of Hermann's tortoises ( Testudo hermanni) to select the larger quantity of food items. Tortoises were able to choose the larger food item when exposed to two options differing in size, but equal in numerousness (0.25, 0.50, 0.67 and 0.75 ratio) and when presented with two groups differing in numerousness, but equal in size (1 versus 4, 2 versus 4, 2 versus 3 and 3 versus 4 items). The tortoises succeeded in both size and numerousness discrimination, and their performance appeared to depend on the ratio of items to be discriminated (thus following Weber's Law). These findings in chelonians provide evidence of an ancient system for the extrapolation of numerical magnitudes from given sets of elements, shared among vertebrates.

Experimental setting affects the performance of guppies in a numerical discrimination task.

A recent study found that guppies (Poecilia reticulata) can be trained to discriminate 4 versus 5 objects, a numerical discrimination typically achieved only by some mammals and birds. In that study, guppies were required to discriminate between two patches of small objects on the bottom of the tank that they could remove to find a food reward. It is not clear whether this species possesses exceptional numerical accuracy compared with the other ectothermic vertebrates or whether its remarkable performance was due to a specific predisposition to discriminate between differences in the quality of patches while foraging. To disentangle these possibilities, we trained guppies to the same numerical discriminations with a more conventional two-choice discrimination task. Stimuli were sets of dots presented on a computer screen, and the subjects received a food reward upon approaching the set with the larger numerosity. Though the cognitive problem was identical in the two experiments, the change in the experimental setting led to a much poorer performance as most fish failed even the 2 versus 3 discrimination. In four additional experiments, we varied the duration of the decision time, the type of stimuli, the length of training, and whether correction was allowed in order to identify the factors responsible for the difference. None of these parameters succeeded in increasing the performance to the level of the previous study, although the group trained with three-dimensional stimuli learned the easiest numerical task. We suggest that the different results with the two experimental settings might be due to constraints on learning and that guppies might be prepared to accurately estimate patch quality during foraging but not to learn an abstract stimulus-reward association.

Development and testing of a rapid method for measuring shoal size discrimination.

The shoal-choice test is a popular method to investigate quantity discrimination in social fish based on their spontaneous preference for the larger of two shoals. The shoal-choice test usually requires a long observation time (20-30 min), mainly because fish switch between the two shoals with low frequency, thus reducing the possibilities of comparison. This duration limits the use of the shoal-choice test for large-scale screenings. Here, we developed a new version of the shoal-choice test in which the subject was confined in a large transparent cylinder in the middle of the tank throughout the experiment to bound the minimum distance from the stimulus shoals and favour switching. We tested the new method by observing guppies (Poecilia reticulata) in a 4 versus 6 fish discrimination (experiment 1). The new method allowed for a faster assessment of the preference for the larger shoal (<5 min), resulting in potential application for large population screenings. Guppies switched five times more frequently between the two shoals and remained close to the first chosen shoal ten times less compared to experiments with the old method. In experiment 2, we found that with the new method guppies were able to discriminate up to 5 versus 6 fish, a discrimination that was not achieved with the classical method. This last result indicates that minor methodological modifications can lead to very different findings in the same species and suggests that caution should be exercised when interpreting inter-specific differences in quantitative abilities.

Sex differences in discrimination reversal learning in the guppy.

In several mammalian and avian species, females show a higher performance than males in tasks requiring cognitive flexibility such as the discrimination reversal learning. A recent study showed that female guppies are twice as efficient as males in a reversal learning task involving yellow-red discrimination, suggesting a higher cognitive flexibility in female guppies. However, the possibility exists that the superior performance exhibited by females does not reflect a general sex difference in cognitive abilities, but instead, is confined to colour discrimination tasks. To address this issue, we compared male and female guppies in two different discrimination reversal learning tasks and we performed a meta-analysis of these experiments and the previous one involving colour discrimination. In the first experiment of this study, guppies were tested in a task requiring them to learn to select the correct arm of a T-maze in order to rejoin a group of conspecifics. In experiment 2, guppies were observed in a numerical task requiring them to discriminate between 5 and 10 dots in order to obtain a food reward. Although females outperformed males in one condition of the T-maze, we did not find any clear evidence of females' greater reversal learning performance in either experiment. However, the meta-analysis of the three experiments supported the hypothesis of females' greater reversal learning ability. Our data do not completely exclude the idea that female guppies have a generally higher cognitive flexibility than males; however, they suggest that the size of this sex difference might depend on the task.

Quantitative abilities in a reptile (Podarcis sicula).

The ability to identify the largest amount of prey available is fundamental for optimizing foraging behaviour in several species. To date, this cognitive skill has been observed in all vertebrate groups except reptiles. In this study we investigated the spontaneous ability of ruin lizards to select the larger amount of food items. In Experiment 1, lizards proved able to select the larger food item when presented with two alternatives differing in size (0.25, 0.50, 0.67 and 0.75 ratio). In Experiment 2 lizards presented with two groups of food items (1 versus 4, 2 versus 4, 2 versus 3 and 3 versus 4 items) were unable to select the larger group in any contrast. The lack of discrimination in the presence of multiple items represents an exception in numerical cognition studies, raising the question as to whether reptiles' quantitative abilities are different from those of other vertebrate groups.

Early numerical abilities and cognitive skills in kindergarten children.

In this study, a unitary path analysis model was developed to investigate the relationship between cognitive variables (derived from published studies) and early numerical abilities in children attending the last year of kindergarten. We tested 100 children starting their last year of kindergarten on the following cognitive abilities: intelligence, phonological abilities, counting, verbal and visuospatial short-term memory and working memory, processing speed, and early numerical abilities. The same children were tested again on early numerical abilities at the end of the same year. The children's early numerical abilities at the beginning of the final year of kindergarten were found to be directly related to their verbal intelligence, phonological abilities, processing speed, and working memory and to be indirectly related to their nonverbal intelligence. Early numerical abilities at the end of the same year are directly related not only to early numerical abilities assessed at the beginning of the year but also to working memory and phonological abilities as well as have an indirect relationship with verbal and nonverbal intelligence. Overall, our results showed that both general and specific abilities are related to early mathematic learning in kindergarten-age children.

Longitudinal development of number line estimation and mathematics performance in primary school children.

Children's ability to relate number to a continuous quantity abstraction visualized as a number line is widely accepted to be predictive of mathematics achievement. However, a debate has emerged with respect to how children's placements are distributed on this number line across development. In the current study, different models were applied to children's longitudinal number placement data to get more insight into the development of number line representations in kindergarten and early primary school years. In addition, longitudinal developmental relations between number line placements and mathematical achievement, measured with a national test of mathematics, were investigated using cross-lagged panel modeling. A group of 442 children participated in a 3-year longitudinal study (ages 5-8 years) in which they completed a number-to-position task every 6 months. Individual number line placements were fitted to various models, of which a one-anchor power model provided the best fit for many of the placements at a younger age (5 or 6 years) and a two-anchor power model provided better fit for many of the children at an older age (7 or 8 years). The number of children who made linear placements also grew with age. Cross-lagged panel analyses indicated that the best fit was provided with a model in which number line acuity and mathematics performance were mutually predictive of each other rather than models in which one ability predicted the other in a non-reciprocal way. This indicates that number line acuity should not be seen as a predictor of math but that both skills influence each other during the developmental process.

Changes in the superior longitudinal fasciculus and anterior thalamic radiation in the left brain are associated with developmental dyscalculia.

Developmental dyscalculia is a neurodevelopmental disorder specific to arithmetic learning even with normal intelligence and age-appropriate education. Difficulties often persist from childhood through adulthood lowering the individual's quality of life. However, the neural correlates of developmental dyscalculia are poorly understood. This study aimed to identify brain structural connectivity alterations in developmental dyscalculia. All participants were recruited from a large scale, non-referred population sample in a longitudinal design. We studied 10 children with developmental dyscalculia (11.3 ± 0.7 years) and 16 typically developing peers (11.2 ± 0.6 years) using diffusion-weighted magnetic resonance imaging. We assessed white matter microstructure with tract-based spatial statistics in regions-of-interest tracts that had previously been related to math ability in children. Then we used global probabilistic tractography for the first time to measure and compare tract length between developmental dyscalculia and typically developing groups. The high angular resolution diffusion-weighted magnetic resonance imaging and crossing-fiber probabilistic tractography allowed us to evaluate the length of the pathways compared to previous studies. The major findings of our study were reduced white matter coherence and shorter tract length of the left superior longitudinal/arcuate fasciculus and left anterior thalamic radiation in the developmental dyscalculia group. Furthermore, the lower white matter coherence and shorter pathways tended to be associated with the lower math performance. These results from the regional analyses indicate that learning, memory and language-related pathways in the left hemisphere might be related to developmental dyscalculia in children.