Dissociations and interactions between time, numerosity and space processing

Impact of macular scotoma and tubular vision on oculomotor behavior and performance in visuospatial comparison tasks

Our aim in this study was to understand how we perform visuospatial comparison tasks by analyzing ocular behavior and to examine how restrictions in macular or peripheral vision disturb ocular behavior and task performance. Two groups of 18 healthy participants with normal or corrected visual acuity performed visuospatial comparison tasks (computerized version of the elementary visuospatial perception [EVSP] test) (Pisella et al., 2013) with a gaze-contingent mask simulating either tubular vision (first group) or macular scotoma (second group). After these simulations of pathological conditions, all participants also performed the EVSP test in full view, enabling direct comparison of their oculomotor behavior and performance. In terms of oculomotor behavior, compared with the full view condition, alternation saccades between the two objects to compare were less numerous in the absence of peripheral vision, whereas the number of within-object exploration saccades decreased in the absence of macular vision. The absence of peripheral vision did not affect accuracy except for midline judgments, but the absence of central vision impaired accuracy across all visuospatial subtests. Besides confirming the crucial role of the macula for visuospatial comparison tasks, these experiments provided important insights into how sensory disorder modifies oculomotor behavior with or without consequences on performance accuracy.

Auditory time perception impairment in children with developmental dyscalculia

Developmental dyscalculia (DD) is a specific learning disability which prevents children from acquiring adequate numerical and arithmetical competences. We investigated whether difficulties in children with DD spread beyond the numerical domain and impact also their ability to perceive time. A group of 37 children/adolescent with and without DD were tested with an auditory categorization task measuring time perception thresholds in the sub-second (0.25-1 s) and supra-second (0.75-3 s) ranges. Results showed that auditory time perception was strongly impaired in children with DD at both time scales. The impairment remained even when age, non-verbal reasoning, and gender were regressed out. Overall, our results show that the difficulties of DD can affect magnitudes other than numerical and contribute to the increasing evidence that frames dyscalculia as a disorder affecting multiple neurocognitive and perceptual systems.

Memory capacity as the core mechanism of the development of space-time interferences in children

This study investigated the development of spatiotemporal perceptual interactions in 5-to-7 years old children. Participants reproduced the temporal and spatial interval between sequentially presented visual stimuli. The time and spacing between stimuli were experimentally manipulated. In addition, cognitive capacities were assessed using neuropsychological tests. Results revealed that starting at 5 years old, children exhibited spatial biases in their time estimations and temporal biases in their spatial estimations, pointing at space-time interference. In line with developmental improvement of temporal and spatial abilities, these spatiotemporal biases decreased with age. Importantly, short-term memory capacity was a predictor of space-time interference pointing to shared cognitive mechanisms between time and space processing. Our results support the symmetrical hypothesis that proposes a common neurocognitive mechanism for processing time and space.

Duration, numerosity and length processing in healthy ageing and Parkinson's disease

People constantly process temporal, numerical, and length information in everyday activities and interactions with the environment. However, it is unclear whether quantity perception changes during ageing. Previous studies have provided heterogeneous results, sometimes showing an age-related effect on a particular quantity, and other times reporting no differences between young and elderly samples. However, three dimensions were never compared within the same study. Here, we conducted two experiments with the aim of investigating the processing of duration, numerosity and length in both healthy and pathological ageing. The experimental paradigm consisted of three bisection tasks in which participants were asked to judge whether the presented stimulus (i.e. a time interval, a group of dots, or a line) was more similar to the short/few or long/many standards. The first study recruited healthy young and elderly participants, while the second recruited healthy elderly participants and patients with Parkinson's disease, a clinical condition commonly associated with temporal impairments. The results of both experiments showed that discrimination precision differed between domains in all groups, with higher precision in the numerosity task and lower sensitivity in judging duration. Furthermore, while discrimination abilities were affected in healthy elderly and, even more so, in Parkinson's disease group, no domain-specific impairments emerged. According to our research, reduced discrimination precision might be explained by an alteration of a single system for all quantities or by an age-related general cognitive decline.

Cortical quantity representations of visual numerosity and timing overlap increasingly into superior cortices but remain distinct

Many sensory brain areas are organized as topographic maps where neural response preferences change gradually across the cortical surface. Within association cortices, 7-Tesla fMRI and neural model-based analyses have also revealed many topographic maps for quantities like numerosity and event timing, often in similar locations. Numerical and temporal quantity estimations also show behavioral similarities and even interactions. For example, the duration of high-numerosity displays is perceived as longer than that of low-numerosity displays. Such interactions are often ascribed to a generalized magnitude system with shared neural responses across quantities. Anterior quantity responses are more closely linked to behavior. Here, we investigate whether common quantity representations hierarchically emerge by asking whether numerosity and timing maps become increasingly closely related in their overlap, response preferences, and topography. While the earliest quantity maps do not overlap, more superior maps overlap increasingly. In these overlapping areas, some intraparietal maps have consistently correlated numerosity and timing preferences, and some maps have consistent angles between the topographic progressions of numerosity and timing preferences. However, neither of these relationships increases hierarchically like the amount of overlap does. Therefore, responses to different quantities are initially derived separately, then progressively brought together, without generally becoming a common representation. Bringing together distinct responses to different quantities may underlie behavioral interactions and allow shared access to comparison and action planning systems.

Top-down determinants of the numerosity-time interaction

Previous studies have reported that larger visual stimuli are perceived as lasting longer than smaller ones. However, this effect disappears when participants provide a qualitative judgment, by stating whether two stimuli have the "same or different" duration, instead of providing an explicit quantitative judgment (which stimulus lasts longer). Here, we extended these observations to the interaction between the numerosity of visual stimuli, i.e. clouds of dots, and their duration. With "longer vs shorter" responses, participants judged larger numerosities as lasting longer than smaller ones, both when the responses were related to the order (Experiment 1) or color (Experiment 4) of stimuli. In contrast, no similar effect was found with "same vs different" responses (Experiment 2) and in a time motor reproduction task (Experiment 3). The numerosity-time interference in Experiment 1 and Experiment 4 was not due to task difficulty, as sensory precision was equivalent to that of Experiment 2. We conclude that in humans the functional interaction between numerosity and time is not guided, in the main, by a shared bottom-up mechanism of magnitude coding. Rather, high-level and top-down processes involved in decision-making and guided by the use of "magnitude-related" response codes play a crucial role in triggering interference among different magnitude domains.

The more numerous the longer: how the integration between numerosity and time leads to a common neural response

If you are stuck in a traffic jam, the more numerous the queuing cars are, the longer you expect to wait. Time and numerosity are stimulus dimensions often associated in the same percept and whose interaction can lead to misjudgements. At brain level it is unclear to which extent time and numerosity recruit same/different neural populations and how their perceptual integration leads to changes in these populations' responses. Here we used high-spatial-resolution functional magnetic resonance imaging with neural model-based analyses to investigate how the topographic representations of numerosity and time change when these dimensions are varied together on the same visual stimulus in a congruent (the more numerous the items, the longer the display time) or incongruent manner. Compared to baseline conditions, where only one dimension was changed at a time, the variation of both stimulus dimensions led to changes in neural population responses that became more sensitive either to the two features or to one of them. Magnitude integration led also to degradation of topographies and shifts in response preferences. These changes were more pronounced in the comparison between parietal and frontal maps. Our results while pointing to partially distinct representations of time and numerosity show a common neural response to magnitude integration.

The nature of magnitude integration: Contextual interference versus active magnitude binding

Magnitude dimensions such as duration and numerosity have been shown to systematically interact, biasing each other in a congruent fashion: the more numerous a set of items is, the longer it is perceived to last in time. This integration between dimensions plays an important role in defining how we perceive magnitude. So far, however, the nature of magnitude integration remains unclear. Is magnitude integration a contextual interference, occurring whenever different types of information are concurrently available in the visual field, or does it involve an active “binding” of the different dimensions of the same object? To address these possibilities, we measured the integration bias induced by numerosity on perceived duration, in two cases: with duration and numerosity conveyed by distinct stimuli, or by the same stimulus. We show that a congruent integration effect can be observed only when the two magnitudes belong to the same stimulus. Instead, when the two magnitudes are conveyed by distinct stimuli, we observed an opposite effect. These findings demonstrate for the first time that a congruent integration occurs only between the dimensions of the same stimulus, suggesting the involvement of an active mechanism integrating the different dimensions of the same object in a unified percept.

The specious interaction of time and numerosity perception

Magnitude information is essential to create a representation of the external environment and successfully interact with it. Duration and numerosity, for example, can shape our predictions and bias each other (i.e. the greater the number of people queuing, the longer we expect to wait). While these biases suggest the existence of a generalized magnitude system, asymmetric effects (i.e. numerosity affecting duration but not vice versa) challenged this idea. Here, we propose that such asymmetric integration depends on the stimuli used and the neural processing dynamics they entail. Across multiple behavioural experiments employing different stimulus presentation displays (static versus dynamic) and experimental manipulations known to bias numerosity and duration perceptions (i.e. connectedness and multisensory integration), we show that the integration between numerosity and time can be symmetrical if the stimuli entail a similar neural time-course and numerosity unfolds over time. Overall, these findings support the idea of a generalized magnitude system, but also highlight the role of early sensory processing in magnitude representation and integration.

Spatial attention shifts contribute to the size congruity effect

The size congruity effect in a numerical Stroop task shows that magnitude judgments of two numbers are faster and more accurate when the numerically larger number also appears in a physically larger size, indicating the interaction between numerical and physical magnitudes. It has recently been suggested that spatial shifts of attention between the two numbers may contribute to the size congruity effect. However, a complete line of evidence for the attentional attribution to the size congruity effect remains to be established. Therefore, the present study aimed to provide further demonstrations for the idea that spatial shifts of attention contribute to the size congruity effect during magnitude judgments regarding either the numerical or physical dimension of two numbers. Participants were sequentially or simultaneously presented with a pair of single-digit Arabic numbers whose numerical and physical magnitudes varied independently. They were instructed to perform a magnitude judgment regarding the numerical value or physical size of the paired numbers. Across three experiments, we consistently found that the size congruity effect was reduced or eliminated when number pairs were presented sequentially compared to when they were presented simultaneously. Because in the sequential presentation mode the paired numbers were successively presented at central fixation and therefore spatial attention shifts should be completely precluded by the central presentation of number stimuli, the present findings support the notion that spatial shifts of attention between numbers in the simultaneous presentation mode play an important role in generating the size congruity effect for both numerical and physical tasks.

Cross-dimensional magnitude interaction is modulated by representational noise: evidence from space-time interaction

Magnitudes along different dimensions (e.g., space and time) tend to interact with each other in perception, with some magnitude dimensions more susceptible to cross-dimensional interference than others. What causes such asymmetries in cross-dimensional magnitude interaction is being debated. The current study investigated whether the representational noise of magnitudes modulates the (a)symmetry in space-time interaction. In three experiments using different formats of length, we showed that dynamic unfilled lengths resulted in a higher representational noise than either static unfilled length or static filled length. Correspondingly, we observed that the time-on-space effect was larger for dynamic unfilled lengths than for static unfilled length or static filled length (and it did not differ between the latter two). Further correlational analyses showed that the susceptibility of a target dimension to the influence of a concurrent dimension increased as a function of participants' representational noise in the target dimension (e.g., the noisier length representations, the larger the time-on-space effect). In all, our study showed that the representational noise of space and time modulates the way the two dimensions interact. These findings suggest that cross-dimensional magnitude interactions arise as a result of memory interference, with noisier magnitudes being more prone to being nudged by concurrent magnitudes in other dimensions. Such memory interference can be seen as a result of Bayesian inference with correlated priors between magnitude dimensions.

The specific role of the striatum in interval timing: The Huntington’s disease model

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Patients with Huntington’s Disease (HD) report a temporal deficit in daily life.

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We tested HD gene carriers and controls in spatial (cm) and temporal (s) tasks.

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Early stage HD patients, but not presymptomatic carriers, were more impaired in time.

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Striatal volume was associated with the temporal deficit in gene carriers.

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Evaluation of interval timing processing should be used as a clinical tool.

Time processing over intervals of hundreds of milliseconds to minutes, also known as interval timing, is associated with the striatum. Huntington’s disease patients (HD) with striatal degeneration have impaired interval timing, but the extent and specificity of these deficits remain unclear. Are they specific to the temporal domain, or do they extend to the spatial domain too? Do they extend to both the perception and production of interval timing? Do they appear before motor symptoms in Huntington’s disease (Pre-HD)?

We addressed these issues by assessing both temporal abilities (in the seconds range) and spatial abilities (in the cm range) in 20 Pre-HD, 25 HD patients, and 25 healthy Controls, in discrimination, bisection and production paradigms. In addition, all participants completed a questionnaire assessing temporal and spatial disorientation in daily life, and the gene carriers (i.e., HD and Pre-HD participants) underwent structural brain MRI.

Overall, HD patients were more impaired in the temporal than in the spatial domain in the behavioral tasks, and expressed a greater disorientation in the temporal domain in the daily life questionnaire. In contrast, Pre-HD participants showed no sign of a specific temporal deficit. Furthermore, MRI analyses indicated that performances in the temporal discrimination task were associated with a larger striatal grey matter volume in the striatum in gene carriers.

Altogether, behavioral, brain imaging and questionnaire data support the hypothesis that the striatum is a specific component of interval timing processes. Evaluations of temporal disorientation and interval timing processing could be used as clinical tools for HD patients.

The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception

A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.

Number is special: time, space, and number interact in a temporal reproduction task

Space, time, and number are among fundamental aspects of behavior and reasoning about the environment. Recent studies have shown that these dimensions highly interact with each other. To explain such interaction, two theories have been proposed: A Theory of Magnitude (ATOM), which posits the existence of a common magnitude system, and Conceptual Metaphor Theory (CMT), which proposes abstract domains such as time and number are mapped through more concrete domains such as space. The present study investigates the interaction of number, time and space in a single experimental paradigm using a temporal reproduction task with a visuospatial component. We also investigated whether mathematical education and continuous involvement with calculations and numbers change the processing precision related to number, time, and space. Two groups of students in mathematics (n = 28) and Persian literature (n = 28) participated in a time reproduction task. The stimuli included Arabic numbers 1, 2, 8, and 9, which were presented to the participants over short (300, 400, 500 ms) and long durations (1000, 1100, 1200 ms) on both sides (left and right) of the monitor. The interaction effect of spatialـnumerical and temporal-numerical was found to be significant. There was no overall time-space interaction, but the triple interaction effect between number, time, and space was significant suggesting the existence of a common representational system. This main result was slightly in line with recent proposed theories. Furthermore, the results showed that the main effect of group was not significant. In addition, we found that among the three factors (number, time, and space) the effect of number is more prominent, i.e., when number disappeared the interaction effect was not observed. The results also suggest that the nature of interactions between these factors is not influenced by cognitive and educational factors. The findings of the study are finally discussed in terms of symmetrical or asymmetrical cross-dimensional influences within the frameworks of ATOM and CMT theories.

Can Implicit or Explicit Time Processing Impact Numerical Representation? Evidence From a Dual Task Paradigm

Whether the human brain processes various types of magnitude, such as numbers and time, through a shared representation or whether there are different representations for each type of magnitude is still debated. Here, we investigated two aspects of number-time interaction: the effects of implicit and explicit processing of time on numbers and the bi-directional interaction between time and number processing. Thirty-two participants were randomly assigned into two experimental groups that performed, respectively, a Single task (number comparison, with implicit time processing) and a Dual task (number comparison as a primary task, with explicit time processing as a secondary task). Results showed that participants, only in the Dual task, were faster and more accurate when processing large numbers paired with long rather than short durations, whereas the opposite pattern was not evident for small numbers. Moreover, participants were more accurate when judging long durations after having processed large rather than small numbers, whereas the opposite pattern emerged for short durations. We propose that number processing influences time processing more than vice versa, suggesting that numbers and time might be at least partially independently represented. This finding can pave the way for investigating the hierarchical representation of space, numbers, and time.

Number, time, and space are not singularly represented: Evidence against a common magnitude system beyond early childhood

Our ability to represent temporal, spatial, and numerical information is critical for understanding the world around us. Given the prominence of quantitative representations in the natural world, numerous cognitive, neurobiological, and developmental models have been proposed as a means of describing how we track quantity. One prominent theory posits that time, space, and number are represented by a common magnitude system, or a common neural locus (i.e., Bonn & Cantlon in Cognitive Neuropsychology, 29(1/2), 149-173, 2012; Cantlon, Platt, & Brannon in Trends in Cognitive Sciences, 13(2), 83-91, 2009; Meck & Church in Animal Behavior Processes, 9(3), 320, 1983; Walsh in Trends in Cognitive Sciences, 7(11), 483-488, 2003). Despite numerous similarities in representations of time, space, and number, an increasing body of literature reveals striking dissociations in how each quantity is processed, particularly later in development. These findings have led many researchers to consider the possibility that separate systems may be responsible for processing each quantity. This review will analyze evidence in favor of a common magnitude system, particularly in infancy, which will be tempered by counter evidence, the majority of which comes from experiments with children and adult participants. After reviewing the current data, we argue that although the common magnitude system may account for quantity representations in infancy, the data do not provide support for this system throughout the life span. We also identify future directions for the field and discuss the likelihood of the developmental divergence model of quantity representation, like that of Newcombe (Ecological Psychology, 2, 147-157, 2014), as a more plausible account of quantity development.

Bidirectional examination of the interaction between time and numerosity corroborates that numerosity influences time estimation but not vice versa

There has been great interest in the idea that time, number, and space share a common magnitude system. However, only a handful of studies examined bidirectional interaction between time and number and the results varied depending on the specifics of the methods and stimulus properties of each study. The present study investigated bidirectional interaction between time and number using estimation tasks. We used duration (Experiment 1) and numerosity (Experiment 2) estimation tasks to investigate the effect of numerosity-on-duration and duration-on-numerosity estimation. The results from the two experiments demonstrated that numerosity influences duration processing but not vice versa; that is, there was unidirectional interaction between numerosity and time. The duration of stimulus presentation was overestimated for stimuli larger in (task-irrelevant) numerosity. Possible mechanisms underlying the unidirectional interaction between time and number are discussed.

Evaluation of the Triple Code Model of numerical processing-Reviewing past neuroimaging and clinical findings

This review reconciles past findings on numerical processing with key assumptions of the most predominant model of arithmetic in the literature, the Triple Code Model (TCM). This is implemented by reporting diverse findings in the literature ranging from behavioral studies on basic arithmetic operations over neuroimaging studies on numerical processing to developmental studies concerned with arithmetic acquisition, with a special focus on developmental dyscalculia (DD). We evaluate whether these studies corroborate the model and discuss possible reasons for contradictory findings. A separate section is dedicated to the transfer of TCM to arithmetic development and to alternative accounts focusing on developmental questions of numerical processing. We conclude with recommendations for future directions of arithmetic research, raising questions that require answers in models of healthy as well as abnormal mathematical development.

WHAT THIS PAPER ADDS

This review assesses the leading model in the field of arithmetic processing (Triple Code Model) by presenting knowledge from interdisciplinary research. It assesses the observed contradictory findings and integrates the resulting opposing viewpoints. The focus is on the development of arithmetic expertise as well as abnormal mathematical development. The original aspect of this article is that it points to a gap in research on these topics and provides possible solutions for future models.

Developmental Dyscalculia in Adults: Beyond Numerical Magnitude Impairment

Numerous studies have tried to identify the core deficit of developmental dyscalculia (DD), mainly by assessing a possible deficit of the mental representation of numerical magnitude. Research in healthy adults has shown that numerosity, duration, and space share a partly common system of magnitude processing and representation. However, in DD, numerosity processing has until now received much more attention than the processing of other non-numerical magnitudes. To assess whether or not the processing of non-numerical magnitudes is impaired in DD, the performance of 15 adults with DD and 15 control participants was compared in four categorization tasks using numerosities, lengths, durations, and faces (as non-magnitude-based control stimuli). Results showed that adults with DD were impaired in processing numerosity and duration, while their performance in length and face categorization did not differ from controls' performance. Our findings support the idea of a nonsymbolic magnitude deficit in DD, affecting numerosity and duration processing but not length processing.

Evidence for different processes involved in the effects of nontemporal stimulus size and numerical digit value on duration judgments

Perceived duration has been shown to be positively related to task-irrelevant, nontemporal stimulus magnitude. To account for this finding, Walsh's (2003) A Theory of Magnitude (ATOM) model suggests that magnitude of time is not differentiated from magnitude of other nontemporal stimulus characteristics and collectively processed by a generalized magnitude system. In Experiment 1, we investigated the combined effects of stimulus size and numerical quantity, as two nontemporal stimulus dimensions covered by the ATOM model, on duration judgments. Participants were required to reproduce the duration of target intervals marked by Arabic digits varying in physical size and numerical value. While the effect of stimulus size was effectively moderated by target duration, the effect of numerical value appeared to require attentional resources directed to the numerical value in order to become effective. Experiment 2 was designed to further elucidate the mediating influence of attention on the effect of numerical value on duration judgments. An effect of numerical value was only observed when participants' attention was directed to digit value, but not when participants were required to pay special attention to digit parity. While the ATOM model implies a common metrics and generalized magnitude processing for time, size, and quantity, the present findings provided converging evidence for the notion of two qualitatively different mechanisms underlying the effects of nontemporal stimulus size and numerical value on duration judgments. Furthermore, our data challenge the implicit common assumption that the effect of numerical value on duration judgments represents a continuously increasing function of digit magnitude.

Quantifying a threat: Evidence of a numeric processing bias

Humans prioritize the processing of threats over neutral stimuli; thus, not surprisingly, the presence of threats has been shown to alter performance on both perceptual and cognitive tasks. Yet whether the quantification process is disrupted in the presence of threat is unknown. In three experiments, we examined numerical estimation and discrimination abilities in adults in the context of threatening (spiders) and non-threatening (e.g., flowers) stimuli. Results of the numerical estimation task (Experiment 1) showed that participants underestimated the number of threatening relative to neutral stimuli. Additionally, numerical discrimination data reveal that participants' abilities to discriminate between the number of entities in two arrays were worsened when the arrays consisted of threatening entities versus neutral entities (Experiment 2). However, discrimination abilities were enhanced when threatening content was presented immediately before neutral dot arrays (Experiment 3). Together, these studies suggest that threats impact our processing of visual numerosity via changes in attention to numerical stimuli, and that the nature of the threat (intrinsic or extrinsic to the stimulus) is vital in determining the direction of this impact. Intrinsic threat content in stimuli impedes its own quantification; yet threat that is extrinsic to the sets to be enumerated enhances numerical processing for subsequently presented neutral stimuli.

Innate and Cultural Spatial Time: A Developmental Perspective

We reviewed literature to understand when a spatial map for time is available in the brain. We carefully defined the concepts of metrical map of time and of conceptual representation of time as the mental time line (MTL) in order to formulate our position. It is that both metrical map and conceptual representation of time are spatial in nature. The former should be innate, related to motor/implicit timing, it should represent all magnitudes with an analogic and bi-dimensional structure. The latter MTL should be learned, available at about 8-10 years-old and related to cognitive/explicit time. It should have uni-dimensional, linear and directional structure (left-to-right in Western culture). We bear the centrality of the development of number cognition, of time semantic concepts and of reading/writing habits for the development of ordinality and linearity of the MTL.

Cross-Representational Interactions: Interface and Overlap Mechanisms

A crucial question facing cognitive science concerns the nature of conceptual representations as well as the constraints on the interactions between them. One specific question we address in this paper is what makes cross-representational interplay possible? We offer two distinct theoretical scenarios: according to the first scenario, co-activated knowledge representations interact with the help of an interface established between them via congruent activation in a mediating third-party general cognitive mechanism, e.g., attention. According to the second scenario, co-activated knowledge representations interact due to an overlap between their features, for example when they share a magnitude component. First, we make a case for cross-representational interplay based on grounded and situated theories of cognition. Second, we discuss interface-based interactions between distinct (i.e., non-overlapping) knowledge representations. Third, we discuss how co-activated representations may share their architecture via partial overlap. Finally, we outline constraints regarding the flexibility of these proposed mechanisms.

Heterogeneity of Developmental Dyscalculia: Cases with Different Deficit Profiles

Developmental Dyscalculia (DD) has long been thought to be a monolithic learning disorder that can be attributed to a specific neurocognitive dysfunction. However, recent research has increasingly recognized the heterogeneity of DD, where DD can be differentiated into subtypes in which the underlying cognitive deficits and neural dysfunctions may differ. The aim was to further understand the heterogeneity of developmental dyscalculia (DD) from a cognitive psychological perspective. Utilizing four children (8–9 year-old) we administered a comprehensive cognitive test battery that shed light on the cognitive-behavioral profile of each child. The children were compared against norm groups of aged-matched peers. Performance was then contrasted against predominant hypotheses of DD, which would also give insight into candidate neurocognitive correlates. Despite showing similar mathematical deficits, these children showed remarkable interindividual variability regarding cognitive profile and deficits. Two cases were consistent with the approximate number system deficit account and also the general magnitude-processing deficit account. These cases showed indications of having domain-general deficits as well. One case had an access deficit in combination with a general cognitive deficit. One case suffered from general cognitive deficits only. The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits.

Binding space and time through action

Space and time are intimately coupled dimensions in the human brain. Several lines of evidence suggest that space and time are processed by a shared analogue magnitude system. It has been proposed that actions are instrumental in establishing this shared magnitude system. Here we provide evidence in support of this hypothesis, by showing that the interaction between space and time is enhanced when magnitude information is acquired through action. Participants observed increases or decreases in the height of a visual bar (spatial magnitude) while judging whether a simultaneously presented sequence of acoustic tones had accelerated or decelerated (temporal magnitude). In one condition (Action), participants directly controlled the changes in bar height with a hand grip device, whereas in the other (No Action), changes in bar height were externally controlled but matched the spatial/temporal profile of the Action condition. The sign of changes in bar height biased the perceived rate of the tone sequences, where increases in bar height produced apparent increases in tone rate. This effect was amplified when the visual bar was actively controlled in the Action condition, and the strength of the interaction was scaled by the magnitude of the action. Subsequent experiments ruled out that this was simply explained by attentional factors, and additionally showed that a monotonic mapping is also required between grip force and bar height in order to bias the perception of the tones. These data provide support for an instrumental role of action in interfacing spatial and temporal quantities in the brain.

Numerical Context and Time Perception: Contrast Effects and the Perceived Duration of Numbers

In the current study, we examined how the contextual repetition of magnitude information presented in either symbolic (Arabic digits) or nonsymbolic (numerosities) formats impacted on the perceived duration of a later occurring target number. The results of the current study demonstrated a time-magnitude bias in which, on average, large magnitude target numbers were judged to last for longer durations relative to small magnitude target numbers, regardless of notation (symbolic number and numerosity). Furthermore, context effects were found, in which a greater discrepancy in the target's magnitude from the initial context led to longer perceived duration ratings. However, this was found to be asymmetrical, occurring only for large magnitude targets. Additionally, the type of context effect was shown to be determined by whether the context was presented in the same notation as the target or a different notation.

Exploring the Boundaries of the Number–Temporal Order Association

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

Combined effects of motor response, sensory modality, and stimulus intensity on temporal reproduction

The ability to estimate a filled interval of time is affected by numerous non-temporal factors, such as the sensory modality, duration, and the intensity of the stimulus. Here we explore the role of modality (auditory or visual), stimulus intensity (low vs. high), and motor response speed on the ability to reproduce the duration of short (<1 s) filled intervals. In accordance with the literature, the reproduced duration was affected by both the modality and the intensity of the stimulus; longer reproduction times were generally observed for visual as compared to auditory stimuli, and for low as compared to high-intensity stimuli. We used general estimating equations in order to determine whether these factors independently affected participants' ability to reproduce a given duration, after eliminating the variability associated with reaction time, since it covaried with the reproduced durations. This analysis revealed that stimulus duration, modality, and intensity were all significant independent predictors of the reproduced durations. Additionally, duration interacted with intensity when reproducing auditory intervals. That is, after taking into account the general speeding-up effect that high-intensity stimuli have on responses, they seem to have an additional effect on the rate of the internal clock. These results support previous evidence suggesting that auditory and visual clocks run at different speeds.

Learning to integrate versus inhibiting information is modulated by age

Cognitive training aiming at improving learning is often successful, but what exactly underlies the observed improvements and how these differ across the age spectrum are currently unknown. Here we asked whether learning in young and older people may reflect enhanced ability to integrate information required to perform a cognitive task or whether it may instead reflect the ability to inhibit task-irrelevant information for successful task performance. We trained 30 young and 30 aging human participants on a numerosity discrimination task known to engage the parietal cortex and in which cue-integration and inhibitory abilities can be distinguished. We coupled training with parietal, motor, or sham transcranial random noise stimulation, known for modulating neural activity. Numerosity discrimination improved after training and was maintained long term, especially in the training + parietal stimulation group, regardless of age. Despite the quantitatively similar improvement in the two age groups, the content of learning differed remarkably: aging participants improved more in inhibitory abilities, whereas younger subjects improved in cue-integration abilities. Moreover, differences in the content of learning were reflected in different transfer effects to untrained but related abilities: in the younger group, improvements in cue integration paralleled improvements in continuous quantity (time and space), whereas in the elderly group, improvements in numerosity-based inhibitory abilities generalized to other measures of inhibition and corresponded to a decline in space discrimination, possibly because conflicting learning resources are used in numerosity and continuous quantity processing. These results indicate that training can enhance different, age-dependent cognitive processes and highlight the importance of identifying the exact processes underlying learning for effective training programs.

Of magnitudes and metaphors: explaining cognitive interactions between space, time, and number

Space, time, and number are fundamental to how we act within and reason about the world. These three experiential domains are systematically intertwined in behavior, language, and the brain. Two main theories have attempted to account for cross-domain interactions. A Theory of Magnitude (ATOM) posits a domain-general magnitude system. Conceptual Metaphor Theory (CMT) maintains that cross-domain interactions are manifestations of asymmetric mappings that use representations of space to structure the domains of number and time. These theories are often viewed as competing accounts. We propose instead that ATOM and CMT are complementary, each illuminating different aspects of cross-domain interactions. We argue that simple representations of magnitude cannot, on their own, account for the rich, complex interactions between space, time and number described by CMT. On the other hand, ATOM is better at accounting for low-level and language-independent associations that arise early in ontogeny. We conclude by discussing how magnitudes and metaphors are both needed to understand our neural and cognitive web of space, time and number.

Time estimation deficits in childhood mathematics difficulties

Time perception has not been comprehensively examined in mathematics difficulties (MD). Therefore, verbal time estimation, production, and reproduction were tested in 13 individuals with MD and 16 healthy controls, matched for age, sex, and intellectual skills. Individuals with MD performed comparably to controls in time reproduction, but showed a tendency to be less accurate on tasks of verbal time estimation and time production. More specifically, these individuals overestimated the duration of a time interval in the verbal time estimation task and showed underproduction when required to produce a time sample. All previous significant comparisons remained significant after controlling for the effects of interval duration, working memory, attention allocation, and quantity estimation. These findings lead us to suggest that time estimation, and more specifically the "internal clock," is abnormally fast in individuals with MD. Results are discussed in terms of Meck and Church's model of temporal processing and Dehaene's triple code model for number processing.

Commonalities for Numerical and Continuous Quantity Skills at Temporo-parietal Junction

How do our abilities to process number and other continuous quantities such as time and space relate to each other? Recent evidence suggests that these abilities share common magnitude processing and neural resources, although other findings also highlight the role of dimension-specific processes. To further characterize the relation between number, time, and space, we first examined them in a population with a developmental numerical dysfunction (developmental dyscalculia) and then assessed the extent to which these abilities correlated both behaviorally and anatomically in numerically normal participants. We found that (1) participants with dyscalculia showed preserved continuous quantity processing and (2) in numerically normal adults, numerical and continuous quantity abilities were at least partially dissociated both behaviorally and anatomically. Specifically, gray matter volume correlated with both measures of numerical and continuous quantity processing in the right TPJ; in contrast, individual differences in number proficiency were associated with gray matter volume in number-specific cortical regions in the right parietal lobe. Together, our new converging evidence of selective numerical impairment and of number-specific brain areas at least partially distinct from common magnitude areas suggests that the human brain is equipped with different ways of quantifying the outside world.

Time counts: bidirectional interaction between time and numbers in human adults

Number is known for influencing time processing, but to what extent time influences number in human adults is unclear. We investigated possible bidirectional interactions (number on time and time on number) using a novel Stroop-like task; participants compared numbers or temporal durations in congruent (larger number presented for longer duration) or incongruent conditions (smaller number presented for longer duration). Time and number tasks were presented in different blocks (Experiment 1) or within the same block of trials with task instructions provided at the offset of the stimuli (Experiment 2). Analyses of response times (RTs) and their distribution revealed that number affected time from early RTs, and time affected number at late RTs - an asymmetry observed only when time and number tasks were presented in separate blocks. Thus, carefully chosen tasks and appropriate data analysis can reveal bidirectionality between time and number, consistent with shared magnitude or decision mechanisms.

Age does not count: resilience of quantity processing in healthy ageing

Quantity skills have been extensively studied in terms of their development and pathological decline. Recently, numerosity discrimination (i.e., how many items are in a set) has been shown to be resilient to healthy ageing despite relying on inhibitory skills, but whether processing continuous quantities such as time and space is equally well-maintained in ageing participants is not known. Life-long exposure to quantity-related problems may progressively refine proficiency in quantity tasks, or alternatively quantity skills may decline with age. In addition, is not known whether the tight relationship between quantity dimensions typically shown in their interactions is preserved in ageing. To address these questions, two experimental paradigms were used in 38 younger and 32 older healthy adults who showed typical age-related decline in attention, executive function and memory tasks. In both groups we first assessed time and space discrimination independently using a two-choice task (i.e., "Which of two horizontal lines is longer in duration or extension?"), and found that time and space processing were equally accurate in younger and older participants. In a second paradigm, we assessed the relation between different quantity dimensions which were presented as a dynamic pattern of dots independently changing in duration, spatial extension and numerosity. Younger and older participants again showed a similar profile of interaction between number, cumulative area and duration, although older adults showed a greater sensitivity to task-irrelevant information than younger adults in the cumulative area task but lower sensitivity in the duration task. Continuous quantity processing seems therefore resilient to ageing similar to numerosity and to other non-quantity skills like vocabulary or implicit memory; however, ageing might differentially affect different quantity dimensions.

Differences in the acuity of the Approximate Number System in adults: the effect of mathematical ability

It is largely admitted that processing numerosity relies on an innate Approximate Number System (ANS), and recent research consistently observed a relationship between ANS acuity and mathematical ability in childhood. However, studies assessing this relationship in adults led to contradictory results. In this study, adults with different levels of mathematical expertise performed two tasks on the same pairs of dot collections, based either on numerosity comparison or on cumulative area comparison. Number of dots and cumulative area were congruent in half of the stimuli, and incongruent in the other half. The results showed that adults with higher mathematical ability obtained lower Weber fractions in the numerical condition than participants with lower mathematical ability. Further, adults with lower mathematical ability were more affected by the interference of the continuous dimension in the numerical comparison task, whereas conversely higher-expertise adults showed stronger interference of the numerical dimension in the continuous comparison task. Finally, ANS acuity correlated with arithmetic performance. Taken together, the data suggest that individual differences in ANS acuity subsist in adulthood, and that they are related to mathematical ability.

Transfer of cognitive training across magnitude dimensions achieved with concurrent brain stimulation of the parietal lobe

Improvement in performance following cognitive training is known to be further enhanced when coupled with brain stimulation. Here we ask whether training-induced changes can be maintained long term and, crucially, whether they can extend to other related but untrained skills. We trained overall 40 human participants on a simple and well established paradigm assessing the ability to discriminate numerosity--or the number of items in a set--which is thought to rely on an "approximate number sense" (ANS) associated with parietal lobes. We coupled training with parietal stimulation in the form of transcranial random noise stimulation (tRNS), a noninvasive technique that modulates neural activity. This yielded significantly better and longer lasting improvement (up to 16 weeks post-training) of the precision of the ANS compared with cognitive training in absence of stimulation, stimulation in absence of cognitive training, and cognitive training coupled to stimulation to a control site (motor areas). Critically, only ANS improvement induced by parietal tRNS + Training transferred to proficiency in other parietal lobe-based quantity judgment, i.e., time and space discrimination, but not to quantity-unrelated tasks measuring attention, executive functions, and visual pattern recognition. These results indicate that coupling intensive cognitive training with tRNS to critical brain regions resulted not only in the greatest and longer lasting improvement of numerosity discrimination, but importantly in this enhancement being transferable when trained and untrained abilities are carefully chosen to share common cognitive and neuronal components.

Perceptual and motor congruency effects in time-space association

It is well established that temporal events are represented on a spatially oriented mental time line from left to right. Depending on the task characteristics, the spatial representation of time may be linked to different types of dimensions, including manual response codes and physical space codes. The aim of the present study was to analyze whether manual response and physical space codes are independent of each other or whether they interact when both types of information are involved in the task. The participants performed a temporal estimation task with two lateralized response buttons in four experiments. In the first experiment, the target stimuli were presented on the left side, at the center, or on the right side of the space, whereas the reference stimuli were always presented centrally. The reverse situation was presented in the second experiment. In the third experiment, both stimuli were presented in opposite spatial positions (e.g., left-right), whereas in the last experiment, both stimuli were presented in the same spatial position (e.g., left-left). In all experiments, perceptual and motor congruency effects were found, but no modulation of the congruency effects was found when both the perceptual and motor components were congruent. The results indicated that physical, spatial, and manual response codes are independent from each other for time-space associations, even when both codes are involved in the task. These results are discussed in terms of the "intermediate-coding" account.

A common metric magnitude system for the perception and production of numerosity, length, and duration

Numerosity, length, and duration processing may share a common functional mechanism situated within the parietal cortex. A strong parallelism between the processing of these three magnitudes has been revealed by similar behavioral signatures (e.g., Weber–Fechner's law, the distance effect) and reciprocal interference effects. Here, we extend the behavioral evidence for a common magnitude processing mechanism by exploring whether the under- and overestimation patterns observed during numerical perception and production tasks are also present in length and duration perception and production. In a first experiment, participants had to perform two estimation tasks (i.e., perception and production) on three magnitudes (i.e., numerosities, lengths, and durations). The results demonstrate similar patterns for the three magnitudes: underestimation was observed in all perception tasks, whereas overestimation was found in all production tasks. A second experiment ensured that this pattern of under- and over-estimation was not solely generated by the mere process of perceiving or producing something. Participants were required to estimate the alphabetical position of a letter (i.e., perception task) or to produce the letter corresponding to a given position (i.e., production task). No under- or overestimation were observed in this experiment, which suggests that the process of perceiving or producing something alone cannot explain the systematic pattern of estimation observed on magnitudes. Together, these findings strengthen the idea that magnitude estimations share a common metric system, requiring similar mechanisms and/or representations.

Training enhances the interference of numerosity on duration judgement

The interference of magnitudes in different dimensions has been demonstrated previously, but the effect of training in one dimension on judgment of another has yet to be examined. The present study aimed to investigate the effect of training in numerosity judgment on judgment of duration. 32 participants took part in two sessions, 12 days apart, and had to judge which of two successive sets of items was presented longer. Half of the participants (training group) were additionally trained in 11 sessions to judge which one of the two successive sets of items was more numerous. It was found that the participants in the training group became more prone to the interference of numerosity on judging duration after training, when compared to the control group. Thus, being trained to more easily perceive the difference in number of items in the two sets affected the perception of duration. On the 3-month follow up session, no effect was found with 20 participants (n = 10 for each group). These findings indicate that the interference of magnitudes in different dimensions can be modulated by training. We discuss that this modulatory effect might be due to neural changes in shared brain regions between interfering magnitudes and/or is mediated by higher levels of perception.

Neuro-anatomical correlates of a number bisection bias: A neuropsychological voxel-based morphometry study

The number bisection tasks, whereby participants estimate the midpoint of a given number interval, is frequently used to explore the idea that numbers are spatially represented within the brain across a 'mental number line'. Some neuropsychological research supports the argument that number bisection is a spatial task, recruiting parietal brain regions, whereas other data suggest that number bisection is dissociable from spatial processing and is instead dependent on working memory in the prefrontal cortices. This study explored the anatomical correlates of deficits in the number bisection task, using voxel-based morphometry in a sample of 25 neuropsychological patients with both left and right hemisphere damage. Interestingly, impairments in number bisection were strongly associated with grey matter lesions in the left hemisphere including both frontal and prefrontal cortices, extending to inferior parietal cortex. Similar prefrontal and frontal grey matter areas were found to be associated with increased leftward deviations (underestimations of the midpoint), whereas no suprathreshold clusters were observed for rightward deviations from the midpoint. Analysis of white matter integrity revealed that lesions in the tracts connecting the parietal and frontal cortices (i.e. the superior longitudinal fasciculus) were highly associated with leftward deviation impairments in number bisection. The data suggest that there is a common parieto-frontal number processing network underlying performances on number bisection, with larger numbers represented on the left side.

A test revealing the slow acquisition and the dorsal stream substrate of visuo-spatial perception

We propose a battery of simple clinical tests to assess the development of elementary visuo-spatial perception. We postulate that most of the tasks we selected rely on the visual dorsal stream, although the dual-stream theory (Milner & Goodale, 1995) discards the role of the dorsal stream for visual perception. In order to test the contribution of this anatomical substrate in visuo-spatial perception, we evaluated the performance of two adult patients with acquired bilateral occipito-parietal (dorsal stream) damage. Additionally, the developmental evolution was assessed by testing 96 children from 4 to 12 years old (4 two-year age groups of 24 children). In order to determine the point at which children achieved adult performance, and to provide a control group for the two patients, we also tested a group of 14 healthy adults. The results highlighted the necessity for age-dependent normative values: adult performance was achieved only at the age of 8 for length and size comparisons and at 12 for dot localisation. In contrast, the ability to judge angles and midlines did not reach adult performance even in the oldest group of children, suggesting further acquisition through adolescence. Occipito-parietal lesions strongly and differentially affected elementary visuo-spatial tasks. In overall scores, the two adult patients were approximately at the level of 6-year olds, below the outlier limit of the adult group. They were on average within the adult interquartile range for processing length and size but clearly outside for the 4 other subtests (Angle, Midline, Position perception and Position selection). As a whole, these data both shed light on the neuroanatomical bases of visuo-spatial perception and allow for age-specific comparisons in children with developmental disorders potentially linked to visuo-spatial and/or attentional defects.

The modulation of implicit magnitude on time estimates

Studies in time and quantity have shown that explicit magnitude (e.g. Arabic numerals, luminance, or size) modulates time estimates with smaller magnitude biasing the judgment of time towards underestimation and larger magnitude towards overestimation. However, few studies have examined the effect of implicit magnitude on time estimates. The current study used a duration estimation task to investigate the effects of implicit magnitude on time estimation in three experiments. During the duration estimation task, the target words named objects of various lengths (Experiment 1), weights (Experiment 2) and volumes (Experiment 3) were presented on the screen and participants were asked to reproduce the amount of time the words remained on the screen via button presses. Results indicated that the time estimates were modulated by the implicit magnitude of the word's referent with words named objects of smaller magnitude (shorter, lighter, or smaller) being judged to last a shorter time, and words named objects of greater magnitude (longer, heavier, or bigger) being judged to last a longer time. These findings were consistent with previous studies examining the effect of implicit spatial length on time estimates. More importantly, current results extended the implicit magnitude of length to the implicit magnitude of weight and volume and demonstrated a functional interaction between time and implicit magnitude in all three aspects of quantity, suggesting a common generalized magnitude system. These results provided new evidence to support a theory of magnitude (ATOM).

When time is space: evidence for a mental time line

Time and space are tightly linked in the physical word. Recently, several lines of evidence have suggested that the mental representation of time might be spatial in nature. For instance, time-space interactions have been described as a strong preference to associate the past with the left space and the future with the right space. Here we review the growing evidence of interactions between time and space processing, systematized according to the type of interaction being investigated. We present the empirical findings supporting the possibility that humans represent the subjective time flow on a spatially oriented "mental time line" that is accessed through spatial attention mechanisms. The heterogeneous time-space interactions are then compared with the number-space interactions described in the numerical cognition literature. An alternative hypothesis, which maintains a common system for magnitude processing, including time, space, and number, is also discussed. Finally, we extend the discussion to the more general issue of how the representation of these concepts might be grounded into the cortical circuits that support spatial attention and sensorimotor transformations.