Weaker number sense accounts for impaired numerosity perception in dyscalculia: Behavioral and computational evidence.

Impaired numerosity perception in developmental dyscalculia (low "number acuity") has been interpreted as evidence of reduced representational precision in the neurocognitive system supporting non-symbolic number sense. However, recent studies suggest that poor numerosity judgments might stem from stronger interference from non-numerical visual information, in line with alternative accounts that highlight impairments in executive functions and visuospatial abilities in the etiology of dyscalculia. To resolve this debate, we used a psychophysical method designed to disentangle the contribution of numerical and non-numerical features to explicit numerosity judgments in a dot comparison task and we assessed the relative saliency of numerosity in a spontaneous categorization task. Children with dyscalculia were compared to control children with average mathematical skills matched for age, IQ, and visuospatial memory. In the comparison task, the lower accuracy of dyscalculics compared to controls was linked to weaker encoding of numerosity, but not to the strength of non-numerical biases. Similarly, in the spontaneous categorization task, children with dyscalculia showed a weaker number-based categorization compared to the control group, with no evidence of a stronger influence of non-numerical information on category choice. Simulations with a neurocomputational model of numerosity perception showed that the reduction of representational resources affected the progressive refinement of number acuity, with little effect on non-numerical bias in numerosity judgments. Together, these results suggest that impaired numerosity perception in dyscalculia cannot be explained by increased interference from non-numerical visual cues, thereby supporting the hypothesis of a core number sense deficit. RESEARCH HIGHLIGHTS: A strongly debated issue is whether impaired numerosity perception in dyscalculia stems from a deficit in number sense or from poor executive and visuospatial functions. Dyscalculic children show reduced precision in visual numerosity judgments and weaker number-based spontaneous categorization, but no increasing reliance on continuous visual properties. Simulations with deep neural networks demonstrate that reduced neural/computational resources affect the developmental trajectory of number acuity and account for impaired numerosity judgments. Our findings show that weaker number acuity in developmental dyscalculia is not necessarily related to increased interference from non-numerical visual cues.

Measuring temporal bias in sequential numerosity comparison.

While several methods have been proposed to assess the influence of continuous visual cues in parallel numerosity estimation, the impact of temporal magnitudes on sequential numerosity judgments has been largely ignored. To overcome this issue, we extend a recently proposed framework that makes it possible to separate the contribution of numerical and non-numerical information in numerosity comparison by introducing a novel stimulus space designed for sequential tasks. Our method systematically varies the temporal magnitudes embedded into event sequences through the orthogonal manipulation of numerosity and two latent factors, which we designate as "duration" and "temporal spacing". This allows us to measure the contribution of finer-grained temporal features on numerosity judgments in several sensory modalities. We validate the proposed method on two different experiments in both visual and auditory modalities: results show that adult participants discriminated sequences primarily by relying on numerosity, with similar acuity in the visual and auditory modality. However, participants were similarly influenced by non-numerical cues, such as the total duration of the stimuli, suggesting that temporal cues can significantly bias numerical processing. Our findings highlight the need to carefully consider the continuous properties of numerical stimuli in a sequential mode of presentation as well, with particular relevance in multimodal and cross-modal investigations. We provide the complete code for creating sequential stimuli and analyzing participants' responses.

Severe Developmental Dyscalculia Is Characterized by Core Deficits in Both Symbolic and Nonsymbolic Number Sense.

A long-standing debate concerns whether developmental dyscalculia is characterized by core deficits in processing nonsymbolic or symbolic numerical information as well as the role of domain-general difficulties. Heterogeneity in recruitment and diagnostic criteria make it difficult to disentangle this issue. Here, we selected children (n = 58) with severely compromised mathematical skills (2 SD below average) but average domain-general skills from a large sample referred for clinical assessment of learning disabilities. From the same sample, we selected a control group of children (n = 42) matched for IQ, age, and visuospatial memory but with average mathematical skills. Children with dyscalculia showed deficits in both symbolic and nonsymbolic number sense assessed with simple computerized tasks. Performance in the digit-comparison task and the numerosity match-to-sample task reliably separated children with developmental dyscalculia from controls in cross-validated logistic regression (area under the curve = .84). These results support a number-sense-deficit theory and highlight basic numerical abilities that could be targeted for early identification of at-risk children as well as for intervention.

Post-stroke deficit prediction from lesion and indirect structural and functional disconnection.

Behavioural deficits in stroke reflect both structural damage at the site of injury, and widespread network dysfunction caused by structural, functional, and metabolic disconnection. Two recent methods allow for the estimation of structural and functional disconnection from clinical structural imaging. This is achieved by embedding a patient's lesion into an atlas of functional and structural connections in healthy subjects, and deriving the ensemble of structural and functional connections that pass through the lesion, thus indirectly estimating its impact on the whole brain connectome. This indirect assessment of network dysfunction is more readily available than direct measures of functional and structural connectivity obtained with functional and diffusion MRI, respectively, and it is in theory applicable to a wide variety of disorders. To validate the clinical relevance of these methods, we quantified the prediction of behavioural deficits in a prospective cohort of 132 first-time stroke patients studied at 2 weeks post-injury (mean age 52.8 years, range 22-77; 63 females; 64 right hemispheres). Specifically, we used multivariate ridge regression to relate deficits in multiple functional domains (left and right visual, left and right motor, language, spatial attention, spatial and verbal memory) with the pattern of lesion and indirect structural or functional disconnection. In a subgroup of patients, we also measured direct alterations of functional connectivity with resting-state functional MRI. Both lesion and indirect structural disconnection maps were predictive of behavioural impairment in all domains (0.16 < R2 < 0.58) except for verbal memory (0.05 < R2 < 0.06). Prediction from indirect functional disconnection was scarce or negligible (0.01 < R2 < 0.18) except for the right visual field deficits (R2 = 0.38), even though multivariate maps were anatomically plausible in all domains. Prediction from direct measures of functional MRI functional connectivity in a subset of patients was clearly superior to indirect functional disconnection. In conclusion, the indirect estimation of structural connectivity damage successfully predicted behavioural deficits post-stroke to a level comparable to lesion information. However, indirect estimation of functional disconnection did not predict behavioural deficits, nor was a substitute for direct functional connectivity measurements, especially for cognitive disorders.

Extraordinary claims, extraordinary evidence? A discussion.

Roberts (2020, Learning & Behavior, 48[2], 191-192) discussed research claiming honeybees can do arithmetic. Some readers of this research might regard such claims as unlikely. The present authors used this example as a basis for a debate on the criterion that ought to be used for publication of results or conclusions that could be viewed as unlikely by a significant number of readers, editors, or reviewers.

Sensorimotor, Attentional, and Neuroanatomical Predictors of Upper Limb Motor Deficits and Rehabilitation Outcome after Stroke.

The rehabilitation of motor deficits following stroke relies on both sensorimotor and cognitive abilities, thereby involving large-scale brain networks. However, few studies have investigated the integration between motor and cognitive domains, as well as its neuroanatomical basis. In this retrospective study, upper limb motor responsiveness to technology-based rehabilitation was examined in a sample of 29 stroke patients (18 with right and 11 with left brain damage). Pretreatment sensorimotor and attentional abilities were found to influence motor recovery. Training responsiveness increased as a function of the severity of motor deficits, whereas spared attentional abilities, especially visuospatial attention, supported motor improvements. Neuroanatomical analysis of structural lesions and white matter disconnections showed that the poststroke motor performance was associated with putamen, insula, corticospinal tract, and frontoparietal connectivity. Motor rehabilitation outcome was mainly associated with the superior longitudinal fasciculus and partial involvement of the corpus callosum. The latter findings support the hypothesis that motor recovery engages large-scale brain networks that involve cognitive abilities and provides insight into stroke rehabilitation strategies.

Learning Numerosity Representations with Transformers: Number Generation Tasks and Out-of-Distribution Generalization.

One of the most rapidly advancing areas of deep learning research aims at creating models that learn to disentangle the latent factors of variation from a data distribution. However, modeling joint probability mass functions is usually prohibitive, which motivates the use of conditional models assuming that some information is given as input. In the domain of numerical cognition, deep learning architectures have successfully demonstrated that approximate numerosity representations can emerge in multi-layer networks that build latent representations of a set of images with a varying number of items. However, existing models have focused on tasks requiring to conditionally estimate numerosity information from a given image. Here, we focus on a set of much more challenging tasks, which require to conditionally generate synthetic images containing a given number of items. We show that attention-based architectures operating at the pixel level can learn to produce well-formed images approximately containing a specific number of items, even when the target numerosity was not present in the training distribution.

Sparse DCM for whole-brain effective connectivity from resting-state fMRI data.

Contemporary neuroscience has embraced network science and dynamical systems to study the complex and self-organized structure of the human brain. Despite the developments in non-invasive neuroimaging techniques, a full understanding of the directed interactions in whole brain networks, referred to as effective connectivity, as well as their role in the emergent brain dynamics is still lacking. The main reason is that estimating brain connectivity requires solving a formidable large-scale inverse problem from indirect and noisy measurements. Building on the dynamic causal modelling framework, the present study offers a novel method for estimating whole-brain effective connectivity from resting-state functional magnetic resonance data. To this purpose sparse estimation methods are adapted to infer the parameters of our novel model, which is based on a linearized, region-specific haemodynamic response function. The resulting algorithm, referred to as sparse DCM, is shown to compare favorably with state-of-the art methods when tested on both synthetic and real data. We also provide a graph-theoretical analysis on the whole-brain effective connectivity estimated using data from a cohort of healthy individuals, which reveals properties such as asymmetry in the connectivity structure as well as the different roles of brain areas in favoring segregation or integration.

Making Sense of Number Words and Arabic Digits: Does Order Count More?

The ability to choose the larger between two numbers reflects a mature understanding of the magnitude associated with numerical symbols. The present study explores how the knowledge of the number sequence and memory capacity (verbal and visuospatial) relate to number comparison skills while controlling for cardinal knowledge. Preschool children's (N = 140, Mage-in-months  = 58.9, range = 41-75) knowledge of the directional property of the counting list as well as the spatial mapping of digits on the visual line were assessed. The ability to order digits on the visual line mediated the relation between memory capacity and number comparison skills while controlling for cardinal knowledge. Beyond cardinality, the knowledge of the (spatial) order of numbers marks the understanding of the magnitude associated with numbers.

Understanding Dyslexia Through Personalized Large-Scale Computational Models.

Learning to read is foundational for literacy development, yet many children in primary school fail to become efficient readers despite normal intelligence and schooling. This condition, referred to as developmental dyslexia, has been hypothesized to occur because of deficits in vision, attention, auditory and temporal processes, and phonology and language. Here, we used a developmentally plausible computational model of reading acquisition to investigate how the core deficits of dyslexia determined individual learning outcomes for 622 children (388 with dyslexia). We found that individual learning trajectories could be simulated on the basis of three component skills related to orthography, phonology, and vocabulary. In contrast, single-deficit models captured the means but not the distribution of reading scores, and a model with noise added to all representations could not even capture the means. These results show that heterogeneity and individual differences in dyslexia profiles can be simulated only with a personalized computational model that allows for multiple deficits.

Spatial order relates to the exact numerical magnitude of digits in young children.

Spatial representation of numbers has been repeatedly associated with the development of numerical and mathematical skills. However, few studies have explored the contribution of spatial mapping to exact number representation in young children. Here we designed a novel task that allows a detailed analysis of direction, ordinality, and accuracy of spatial mapping. Preschool children, who were classified as competent counters (cardinal principle knowers), placed triplets of sequentially presented digits on the visual line. The ability to correctly order triplets tended to decrease with the larger digits. When triplets were correctly ordered, the direction of spatial mapping was predominantly oriented from left to right and the positioning of the target digits was characterized by a pattern of underestimation with no evidence of logarithmic compression. Crucially, only ordinality was associated with performance in a digit comparison task. Our results suggest that the spatial (ordinal) arrangement of digits is a powerful source of information that young children can use to construct the representation of exact numbers. Therefore, digits may acquire numerical meaning based on their spatial order on the number line.

Spatial grounding of symbolic arithmetic: an investigation with optokinetic stimulation.

Growing evidence suggests that mental calculation might involve movements of attention along a spatial representation of numerical magnitude. Addition and subtraction on nonsymbolic numbers (numerosities) seem to induce a "momentum" effect, and have been linked to distinct patterns of neural activity in cortical regions subserving attention and eye movements. We investigated whether mental arithmetic on symbolic numbers, a cornerstone of abstract mathematical reasoning, can be affected by the manipulation of overt spatial attention induced by optokinetic stimulation (OKS). Participants performed additions or subtractions of auditory two-digit numbers during horizontal (experiment 1) or vertical OKS (experiment 2), and eye movements were concurrently recorded. In both experiments, the results of addition problems were underestimated, whereas results of subtractions were overestimated (a pattern that is opposite to the classic Operational Momentum effect). While this tendency was unaffected by OKS, vertical OKS modulated the occurrence of decade errors during subtractions (i.e., fewer during downward OKS and more frequent during upward OKS). Eye movements, on top of the classic effect induced by OKS, were affected by the type of operation during the calculation phase, with subtraction consistently leading to a downward shift of gaze position and addition leading to an upward shift. These results highlight the pervasive nature of spatial processing in mental arithmetic. Furthermore, the preeminent effect of vertical OKS is in line with the hypothesis that the vertical dimension of space-number associations is grounded in universal (physical) constraints and, thereby, more robust than situated and culture-dependent associations with the horizontal dimension.

Warnings and caveats in brain controllability.

A recent article by Gu et al. (Nat. Commun. 6, 2015) proposed to characterize brain networks, quantified using anatomical diffusion imaging, in terms of their "controllability", drawing on concepts and methods of control theory. They reported that brain activity is controllable from a single node, and that the topology of brain networks provides an explanation for the types of control roles that different regions play in the brain. In this work, we first briefly review the framework of control theory applied to complex networks. We then show contrasting results on brain controllability through the analysis of five different datasets and numerical simulations. We find that brain networks are not controllable (in a statistical significant way) by one single region. Additionally, we show that random null models, with no biological resemblance to brain network architecture, produce the same type of relationship observed by Gu et al. between the average/modal controllability and weighted degree. Finally, we find that resting state networks defined with fMRI cannot be attributed specific control roles. In summary, our study highlights some warning and caveats in the brain controllability framework.

Computational foundations of the visual number sense.

We provide an emergentist perspective on the computational mechanism underlying numerosity perception, its development, and the role of inhibition, based on our deep neural network model. We argue that the influence of continuous visual properties does not challenge the notion of number sense, but reveals limit conditions for the computation that yields invariance in numerosity perception. Alternative accounts should be formalized in a computational model.

Spontaneous non-verbal counting in toddlers.

A wealth of studies have investigated numerical abilities in infants and in children aged 3 or above, but research on pre-counting toddlers is sparse. Here we devised a novel version of an imitation task that was previously used to assess spontaneous focusing on numerosity (i.e. the predisposition to grasp numerical properties of the environment) to assess whether pre-counters would spontaneously deploy sequential (item-by-item) enumeration and whether this ability would rely on the object tracking system (OTS) or on the approximate number system (ANS). Two-and-a-half-year-olds watched the experimenter performing one-by-one insertion of 'food tokens' into an opaque animal puppet and then were asked to imitate the puppet-feeding behavior. The number of tokens varied between 1 and 6 and each numerosity was presented many times to obtain a distribution of responses during imitation. Many children demonstrated attention to the numerosity of the food tokens despite the lack of any explicit cueing to the number dimension. Most notably, the response distributions centered on the target numerosities and showed the classic variability signature that is attributed to the ANS. These results are consistent with previous studies on sequential enumeration in non-human primates and suggest that pre-counting children are capable of sequentially updating the numerosity of non-visible sets through additive operations and hold it in memory for reproducing the observed behavior.

Voluntary eye movements direct attention on the mental number space.

Growing evidence suggests that orienting visual attention in space can influence the processing of numerical magnitude, with leftward orienting speeding up the processing of small numbers relative to larger ones and the converse for rightward orienting. The manipulation of eye movements is a convenient way to direct visuospatial attention, but several aspects of the complex relationship between eye movements, attention orienting and number processing remain unexplored. In a previous study, we observed that inducing involuntary, reflexive eye movements by means of optokinetic stimulation affected number processing only when numerical magnitude was task relevant (i.e., during magnitude comparison, but not during parity judgment; Ranzini et al., in J Cogn Psychol 27, 459-470, (2015). Here, we investigated whether processing of task-irrelevant numerical magnitude can be modulated by voluntary eye movements, and whether the type of eye movements (smooth pursuit vs. saccades) would influence this interaction. Participants tracked with their gaze a dot while listening to a digit. The numerical task was to indicate whether the digit was odd or even through non-spatial, verbal responses. The dot could move leftward or rightward either continuously, allowing tracking by smooth pursuit eye movements, or in discrete steps across a series of adjacent locations, triggering a sequence of saccades. Both smooth pursuit and saccadic eye movements similarly affected number processing and modulated response times for large numbers as a function of direction of motion. These findings suggest that voluntary eye movements redirect attention in mental number space and highlight that eye movements should play a key factor in the investigation of number-space interactions.

Number-space interactions in the human parietal cortex: Enlightening the SNARC effect with functional near-infrared spectroscopy.

Interactions between numbers and space have become a major issue in cognitive neuroscience, because they suggest that numerical representations might be deeply rooted in cortical networks that also subserve spatial cognition. The spatial-numerical association of response codes (SNARC) is the most robust and widely replicated demonstration of the link between numbers and space: in magnitude comparison or parity judgments, participants' reaction times to small numbers are faster with left than right effectors, whereas the converse is found for large numbers. However, despite the massive body of research on number-space interactions, the nature of the SNARC effect remains controversial and no study to date has identified its hemodynamic correlates. Using functional near-infrared spectroscopy, we found a hemodynamic signature of the SNARC effect in the bilateral intraparietal sulcus, a core region for numerical magnitude representation, and left angular gyrus (ANG), a region implicated in verbal number processing. Activation of intraparietal sulcus was also modulated by numerical distance. Our findings point to number semantics as cognitive locus of number-space interactions, thereby revealing the intrinsic spatial nature of numerical magnitude representation. Moreover, the involvement of left ANG is consistent with the mediating role of verbal/cultural factors in shaping interactions between numbers and space.

Extra-large letter spacing improves reading in dyslexia.

Although the causes of dyslexia are still debated, all researchers agree that the main challenge is to find ways that allow a child with dyslexia to read more words in less time, because reading more is undisputedly the most efficient intervention for dyslexia. Sophisticated training programs exist, but they typically target the component skills of reading, such as phonological awareness. After the component skills have improved, the main challenge remains (that is, reading deficits must be treated by reading more--a vicious circle for a dyslexic child). Here, we show that a simple manipulation of letter spacing substantially improved text reading performance on the fly (without any training) in a large, unselected sample of Italian and French dyslexic children. Extra-large letter spacing helps reading, because dyslexics are abnormally affected by crowding, a perceptual phenomenon with detrimental effects on letter recognition that is modulated by the spacing between letters. Extra-large letter spacing may help to break the vicious circle by rendering the reading material more easily accessible.