No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Symbolic and non-symbolic representations of numerical zero in the human brain
... Recently, neural correlates of zero in the human brain have been measured using magnetoencephalography (MEG), 47 which detects the collective magnetic fields generated by the synchronized electrical activity of large groups of neurons. Similar to our findings from single-neuron recordings, the MEG representations of zero were positioned along a graded neural number line shared with other countable numbers. ...
The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills necessitate a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of non-symbolic numerical quantity, then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. While it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.
Some conscious experiences are more vivid than others. Although perceptual vividness is a key component of human consciousness, how variation in this magnitude property is registered by the human brain is unknown. A striking feature of neural codes for magnitude in other psychological domains such as number or reward is that the magnitude property is represented independently of its sensory features. To test whether perceptual vividness also covaries with neural codes that are invariant to sensory content, we reanalysed existing MEG and fMRI data from two distinct studies which quantified perceptual vividness via subjective ratings of awareness and visibility. Using representational similarity and decoding analyses, we find evidence for content-invariant neural signatures of perceptual vividness distributed across visual, parietal, and frontal cortices. Our findings indicate that the neural correlates of subjective vividness may share similar properties to magnitude codes in other cognitive domains.
Whether small numerical quantities are represented by a special subitizing system that is distinct from a large-number estimation system has been debated for over a century. Here we show that two separate neural mechanisms underlie the representation of small and large numbers. We performed single neuron recordings in the medial temporal lobe of neurosurgical patients judging numbers. We found a boundary in neuronal coding around number 4 that correlates with the behavioural transition from subitizing to estimation. In the subitizing range, neurons showed superior tuning selectivity accompanied by suppression effects suggestive of surround inhibition as a selectivity-increasing mechanism. In contrast, tuning selectivity decreased with increasing numbers beyond 4, characterizing a ratio-dependent number estimation system. The two systems with the coding boundary separating them were also indicated using decoding and clustering analyses. The identified small-number subitizing system could be linked to attention and working memory that show comparable capacity limitations.
Auditory perception is traditionally conceived as the perception of sounds-a friend's voice, a clap of thunder, a minor chord. However, daily life also seems to present us with experiences characterized by the absence of sound-a moment of silence, a gap between thunderclaps, the hush after a musical performance. In these cases, do we positively hear silence? Or do we just fail to hear, and merely judge or infer that it is silent? This longstanding question remains controversial in both the philosophy and science of perception, with prominent theories holding that sounds are the only objects of auditory experience and thus that our encounter with silence is cognitive, not perceptual. However, this debate has largely remained theoretical, without a key empirical test. Here, we introduce an empirical approach to this theoretical dispute, presenting experimental evidence that silence can be genuinely perceived (not just cognitively inferred). We ask whether silences can "substitute" for sounds in event-based auditory illusions-empirical signatures of auditory event representation in which auditory events distort perceived duration. Seven experiments introduce three "silence illusions"-the one-silence-is-more illusion, silence-based warping, and the oddball-silence illusion-each adapted from a prominent perceptual illusion previously thought to arise only from sounds. Subjects were immersed in ambient noise interrupted by silences structurally identical to the sounds in the original illusions. In all cases, silences elicited temporal distortions perfectly analogous to the illusions produced by sounds. Our results suggest that silence is truly heard, not merely inferred, introducing a general approach for studying the perception of absence.
Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus’s retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity’s straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing. The authors show that spatial frequency domain Fourier power closely but nonlinearly follows numerosity, simplifying computing numerosity from early visual responses. Monotonic early visual cortex and neural network responses follow Fourier power, while later tuned responses follow numerosity.
Representing the absence of objects is psychologically demanding. People are slower, less confident and show lower metacognitive sensitivity (the alignment between subjective confidence and objective accuracy) when reporting the absence compared with presence of visual stimuli. However, what counts as a stimulus absence remains only loosely defined. In this Registered Report, we ask whether such processing asymmetries extend beyond the absence of whole objects to absences defined by stimulus features or expectation violations. Our pre-registered prediction was that differences in the processing of presence and absence reflect a default mode of reasoning: we assume an absence unless evidence is available to the contrary. We predicted asymmetries in response time, confidence, and metacognitive sensitivity in discriminating between stimulus categories that vary in the presence or absence of a distinguishing feature, or in their compliance with an expected default state. Using six pairs of stimuli in six experiments, we find evidence that the absence of local and global stimulus features gives rise to slower, less confident responses, similar to absences of entire stimuli. Contrary to our hypothesis, however, the presence or absence of a local feature has no effect on metacognitive sensitivity. Our results weigh against a proposal of a link between the detection metacognitive asymmetry and default reasoning, and are instead consistent with a low-level visual origin of metacognitive asymmetries for presence and absence.
While knowledge on the development of understanding positive integers is rapidly growing, the development of understanding zero remains not well-understood. Here, we test several components of preschoolers’ understanding of zero: Whether they can use empty sets in numerical tasks (as measured with comparison, addition, and subtraction tasks); whether they can use empty sets soon after they understand the cardinality principle (cardinality-principle knowledge is measured with the give-N task); whether they know what the word “zero” refers to (tested in all tasks in this study); and whether they categorize zero as a number (as measured with the smallest-number and is-it-a-number tasks). The results show that preschoolers can handle empty sets in numerical tasks as soon as they can handle positive numbers and as soon as, or even earlier than, they understand the cardinality principle. Some also know that these sets are labeled as “zero.” However, preschoolers are unsure whether zero is a number. These results identify three components of knowledge about zero: operational knowledge, linguistic knowledge, and meta-knowledge. To account for these results, we propose that preschoolers may understand numbers as the properties of items or objects in a set. In this view, zero is not regarded as a number because an empty set does not include any items, and missing items cannot have any properties, therefore, they cannot have the number property either. This model can explain why zero is handled correctly in numerical tasks even though it is not regarded as a number.
People have better metacognitive sensitivity for decisions about the presence compared to the absence of objects. However, it is not only objects themselves that can be present or absent, but also parts of objects and other visual features. Asymmetries in visual search indicate that a disadvantage for representing absence may operate at these levels as well. Furthermore, a processing advantage for surprising signals suggests that a presence/absence asymmetry may be explained by absence being passively represented as a default state, and presence as a default-violating surprise. It is unknown whether the metacognitive asymmetry for judgments about presence and absence extends to these different levels of representation (object, feature, and default violation). To address this question and test for a link between the representation of absence and default reasoning more generally, here we measure metacognitive sensitivity for discrimination judgments between stimuli that are identical except for the presence or absence of a distinguishing feature, and for stimuli that differ in their compliance with an expected default state.
[This corrects the article DOI: 10.1371/journal.pone.0232551.].
Different species of animals can discriminate numerosity, the countable number of objects in a set. The representations of countable numerosities have been deciphered down to the level of single neurons. However, despite its importance for human number theory, a special numerical quantity, the empty set (numerosity zero), has remained largely unexplored. We explored the behavioral and neuronal representation of the empty set in carrion crows. Crows were trained to discriminate small numerosities including the empty set. Performance data showed a numerical distance effect for the empty set in one crow, suggesting that the empty set and countable numerosities are represented along the crows' "mental number line." Single-cell recordings in the endbrain region nidopallium caudolaterale (NCL) showed a considerable proportion of NCL neurons tuned to the preferred numerosity zero. As evidenced by neuronal distance and size effects, NCL neurons integrated the empty set in the neural number line. A subsequent neuronal population analysis using a statistical classifier approach showed that the neuronal numerical representations were predictive of the crows' success in the task. These behavioral and neuronal data suggests that the conception of the empty set as a cognitive precursor of a zero-like number concept is not an exclusive property of the cerebral cortex of primates. Zero as a quantitative category cannot only be implemented in the layered neocortex of primates, but also in the anatomically distinct endbrain circuitries of birds that evolved based on convergent evolution.SIGNIFICANCE STATEMENT The conception of "nothing" as number "zero" is celebrated as one of the greatest achievements in mathematics. To explore whether precursors of zero-like concepts can be found in vertebrates with a cerebrum that anatomically differs starkly from our primate brain, we investigated this in carrion crows. We show that crows can grasp the empty set as a null numerical quantity that is mentally represented next to number one. Moreover, we show that single neurons in an associative avian cerebral region specifically respond to the empty set and show the same physiological characteristics as for countable quantities. This suggests that zero as a quantitative category can also be implemented in the anatomically distinct endbrain circuitries of birds that evolved based on convergent evolution.
Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1–9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.
The question whether human beings process empty sets as zero has received little research attention. In this study, we used the distance and end effects as indicators for treating empty sets as a numerical entity that represents an absence of quantity. In a series of experiments, participants performed a magnitude comparison task. They were presented with empty sets and other numerosities from 1 to 9, presented as dot arrays. We manipulated task instructions relevant to the target (i.e., “choose the target that contains more/less dots” in Experiment 1) or the given numerical range mentioned in the instructions (i.e., 0–9 or 1–9 in Experiment 2) to create conditions in which an empty set would be perceived as the smallest value of the experimental numerical range. The results revealed distance effects for comparisons to empty sets, irrespective of task instructions. In Experiment 3, we manipulated the response mode. Two groups of participants responded to target location, one group with a key-press and the other vocally, while the third group responded vocally to target color. The results revealed distance effects for comparisons to empty sets only when responding to target location, regardless of the response mode, indicating that spatial features should be primed in order to perceive an empty set as a numerical entity. These findings show that perceiving an empty set as nothing or as zero depends on the context in which it is presented.
A core feature of human cognition is an ability to separate private states of mind – what we think or believe – from public actions – what we say or do. This ability is central to successful social interaction – with different social contexts often requiring different mappings between private states and public actions in order to minimise conflict and facilitate communication. Here we investigated how the human brain supports private-public mappings, using an interactive task which required subjects to adapt how they communicated their confidence about a perceptual decision to the social context. Univariate and multivariate analysis of fMRI data revealed that a private-public distinction is reflected in a medial-lateral division of prefrontal cortex – with lateral frontal pole (FPl) supporting the context-dependent mapping from a private sense of confidence to a public report. The concept of private-public mappings provides a promising framework for understanding flexible social behaviour.
MVPA-Light is a MATLAB toolbox for multivariate pattern analysis (MVPA). It provides native implementations of a range of classifiers and regression models, using modern optimization algorithms. High-level functions allow for the multivariate analysis of multi-dimensional data, including generalization (e.g., time x time) and searchlight analysis. The toolbox performs cross-validation, hyperparameter tuning, and nested preprocessing. It computes various classification and regression metrics and establishes their statistical significance, is modular and easily extendable. Furthermore, it offers interfaces for LIBSVM and LIBLINEAR as well as an integration into the FieldTrip neuroimaging toolbox. After introducing MVPA-Light, example analyses of MEG and fMRI datasets, and benchmarking results on the classifiers and regression models are presented.
Being confident in whether a stimulus is present or absent (a detection judgment) is qualitatively distinct from being confident in the identity of that stimulus (a discrimination judgment). In particular, in detection, evidence can only be available for the presence, not the absence, of a target object. This asymmetry suggests that higher-order cognitive and neural processes may be required for confidence in detection, and more specifically, in judgments about absence. In a within-subject, pre-registered and performance-matched fMRI design, we observed quadratic confidence effects in frontopolar cortex for detection but not discrimination. Furthermore, in the right temporoparietal junction, confidence effects were enhanced for judgments of target absence compared to judgments of target presence. We interpret these findings as reflecting qualitative differences between a neural basis for metacognitive evaluation of detection and discrimination, potentially in line with counterfactual or higher-order models of confidence formation in detection.
Humans can learn abstract concepts that describe invariances over relational patterns in data. One such concept, known as magnitude, allows stimuli to be compactly represented on a single dimension (i.e. on a mental line). Here, we measured representations of magnitude in humans by recording neural signals whilst they viewed symbolic numbers. During a subsequent reward-guided learning task, the neural patterns elicited by novel complex visual images reflected their payout probability in a way that suggested they were encoded onto the same mental number line, with 'bad' bandits sharing neural representation with 'small' numbers and 'good' bandits with 'large' numbers. Using neural network simulations, we provide a mechanistic model that explains our findings and shows how structural alignment can promote transfer learning. Our findings suggest that in humans, learning about reward probability is accompanied by structural alignment of value representations with neural codes for the abstract concept of magnitude.
Understanding zero
It has been said that the development of an understanding of zero by society initiated a major intellectual advance in humans, and we have been thought to be unique in this understanding. Although recent research has shown that some other vertebrates understand the concept of the “empty set,” Howard et al. now show that an understanding of this concept is present in untrained honey bees (see the Perspective by Nieder). This finding suggests that such an understanding has evolved independently in distantly related species that deal with complexity in their environments, and that it may be more widespread than previously appreciated.
Science , this issue p. 1124 ; see also p. 1069
Numerical format describes the way magnitude is conveyed, for example, as a digit (“3”) or Roman numeral (“III”). In the field of numerical cognition, there is an ongoing debate of whether magnitude representation is independent of numerical format. Here, we examine the time course of magnitude processing when using different symbolic formats. We presented participants with a series of digits and dice patterns corresponding to the magnitudes of 1 to 6 while performing a 1-back task on magnitude. Magnetoencephalography offers an opportunity to record brain activity with high temporal resolution. Multivariate pattern analysis applied to magnetoencephalographic data allows us to draw conclusions about brain activation patterns underlying information processing over time. The results show that we can cross-decode magnitude when training the classifier on magnitude presented in one symbolic format and testing the classifier on the other symbolic format. This suggests a similar representation of these numerical symbols. In addition, results from a time generalization analysis show that digits were accessed slightly earlier than dice, demonstrating temporal asynchronies in their shared representation of magnitude. Together, our methods allow a distinction between format-specific signals and format-independent representations of magnitude showing evidence that there is a shared representation of magnitude accessed via different symbols.
Neurons in the primate parieto-frontal network represent the number of visual items in a collection, but it is unknown whether this system encodes empty sets as conveying null quantity. We recorded from the ventral intraparietal area (VIP) and the prefrontal cortex (PFC) of monkeys performing a matching task including empty sets and countable numerosities as stimuli. VIP neurons encoded empty sets predominantly as a distinct category from numerosities. In contrast, PFC neurons represented empty sets more similarly to numerosity one than to larger numerosities, exhibiting numerical distance and size effects. Moreover, prefrontal neurons represented empty sets abstractly and irrespective of stimulus variations. Compared to VIP, the activity of numerosity neurons in PFC correlated better with the behavioral outcome of empty-set trials. Our results suggest a hierarchy in the processing from VIP to PFC, along which empty sets are steadily detached from visual properties and gradually positioned in a numerical continuum. These findings elucidate how the brain transforms the absence of countable items, nothing, into an abstract quantitative category, zero.
Zero is a fundamental concept in mathematics and modern science. Empty sets are considered a precursor of the concept of numerosity zero and a part of numerical continuum. How is numerosity zero (the absence of visual items) represented in the primate cortex? To address this question, we trained monkeys to perform numerical operations including numerosity zero. Here we show a group of neurons in the posterior parietal cortex of the monkey activated in response to numerosity 'zero'. 'Zero' neurons are classified into exclusive and continuous types; the exclusive type discretely encodes numerical absence and the continuous type encodes numerical absence as a part of a numerical continuum. "Numerosity-zero" neurons enhance behavioral discrimination of not only zero numerosity but also non-zero numerosities. Representation of numerosity zero in the parietal cortex may be a precursor of non-verbal concept of zero in primates.
Neuronal population codes are increasingly being investigated with multivariate pattern-information analyses. A key challenge is to use measured brain-activity patterns to test computational models of brain information processing. One approach to this problem is representational similarity analysis (RSA), which characterizes a representation in a brain or computational model by the distance matrix of the response patterns elicited by a set of stimuli. The representational distance matrix encapsulates what distinctions between stimuli are emphasized and what distinctions are de-emphasized in the representation. A model is tested by comparing the representational distance matrix it predicts to that of a measured brain region. RSA also enables us to compare representations between stages of processing within a given brain or model, between brain and behavioral data, and between individuals and species. Here, we introduce a Matlab toolbox for RSA. The toolbox supports an analysis approach that is simultaneously data- and hypothesis-driven. It is designed to help integrate a wide range of computational models into the analysis of multichannel brain-activity measurements as provided by modern functional imaging and neuronal recording techniques. Tools for visualization and inference enable the user to relate sets of models to sets of brain regions and to statistically test and compare the models using nonparametric inference methods. The toolbox supports searchlight-based RSA, to continuously map a measured brain volume in search of a neuronal population code with a specific geometry. Finally, we introduce the linear-discriminant t value as a measure of representational discriminability that bridges the gap between linear decoding analyses and RSA. In order to demonstrate the capabilities of the toolbox, we apply it to both simulated and real fMRI data. The key functions are equally applicable to other modalities of brain-activity measurement. The toolbox is freely available to the community under an open-source license agreement (http://www.mrc-cbu.cam.ac.uk/methods-and-resources/toolboxes/license/).
Parsing a cognitive task into a sequence of operations is a central problem in cognitive neuroscience. We argue that a major advance is now possible owing to the application of pattern classifiers to time-resolved recordings of brain activity [electroencephalography (EEG), magnetoencephalography (MEG), or intracranial recordings]. By testing at which moment a specific mental content becomes decodable in brain activity, we can characterize the time course of cognitive codes. Most importantly, the manner in which the trained classifiers generalize across time, and from one experimental condition to another, sheds light on the temporal organization of information-processing stages. A repertoire of canonical dynamical patterns is observed across various experiments and brain regions. This method thus provides a novel way to understand how mental representations are manipulated and transformed.
The increase in spatiotemporal resolution of neuroimaging devices is accompanied by a trend towards more powerful multivariate analysis methods. Often it is desired to interpret the outcome of these methods with respect to the cognitive processes under study. Here we discuss which methods allow for such interpretations, and provide guidelines for choosing an appropriate analysis for a given experimental goal: For a surgeon who needs to decide where to remove brain tissue it is most important to determine the origin of cognitive functions and associated neural processes. In contrast, when communicating with paralyzed or comatose patients via brain-computer interfaces, it is most important to accurately extract the neural processes specific to a certain mental state. These equally important but complementary objectives require different analysis methods. Determining the origin of neural processes in time or space from the parameters of a data-driven model requires what we call a forward model of the data; such a model explains how the measured data was generated from the neural sources. Examples are general linear models (GLMs). Methods for the extraction of neural information from data can be considered as backward models, as they attempt to reverse the data generating process. Examples are multivariate classifiers. Here we demonstrate that the parameters of forward models are neurophysiologically interpretable in the sense that significant nonzero weights are only observed at channels the activity of which is related to the brain process under study. In contrast, the interpretation of backward model parameters can lead to wrong conclusions regarding the spatial or temporal origin of the neural signals of interest, since significant nonzero weights may also be observed at channels the activity of which is statistically independent of the brain process under study. As a remedy for the linear case, we propose a procedure for transforming backward models into forward models. This procedure enables the neurophysiological interpretation of the parameters of linear backward models. We hope that this work raises awareness for an often encountered problem and provides a theoretical basis for conducting better interpretable multivariate neuroimaging analyses.
The cognitive concept of representation plays a key role in theories of brain information processing. However, linking neuronal activity to representational content and cognitive theory remains challenging. Recent studies have characterized the representational geometry of neural population codes by means of representational distance matrices, enabling researchers to compare representations across stages of processing and to test cognitive and computational theories. Representational geometry provides a useful intermediate level of description, capturing both the information represented in a neuronal population code and the format in which it is represented. We review recent insights gained with this approach in perception, memory, cognition, and action. Analyses of representational geometry can compare representations between models and the brain, and promise to explain brain computation as transformation of representational similarity structure.
Nine experiments of timed odd–even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial–Numerical Association of Response Codes effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss. The results supported a modular architecture for number processing, with distinct but interconnected Arabic, verbal, and magnitude representations. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Models of human number representation are based mainly on evidence from indirect sources such as number comparison tasks and findings on acquired dyscalculia. Researchers have rarely looked at the processing times of individual numbers. The experiments described in this article indicate that this neglect may have been unwarranted because number reading times considerably constrain the range of acceptable theoretical models. In particular, it is found that the time to process an Arabic integer from 1 to 99 is a function of the logarithm of the number magnitude, the frequency of the number, and sometimes the syllable length of the number name. In addition, processing a number facilitates the processing of a subsequent number with a close value. The effects of number magnitude and number priming are found for number naming as well, indicating that phonological recoding in silent reading (as evidenced by the syllable-length effect) happens after the internal semantic numerical representation has been accessed. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The cognitive representation of negative numbers has recently been studied with magni-tude classification and variable standard (Fischer, 2003b). Results suggested that negative numbers are cognitively represented on a "mental number line". Conflicting with this obser-vation we report here that response biases for parity classification of negative numbers are related to absolute digit magnitude only. A control experiment with magnitude classification relative to a fixed standard shows that our result does not reflect mere inattention to the minus sign. Together, the available evidence suggests that we process negative numbers less automatically compared to positive numbers.
Judging the presence or absence of a stimulus is likely the most basic perceptual decision. A fundamental difference of detection tasks in contrast to discrimination tasks is that only the stimulus presence decision can be inferred from sensory evidence, whereas the alternative decision about stimulus absence lacks sensory evidence by definition. Detection decisions have been studied in an intentional, action-based framework, in which decisions were regarded as intentions to pursue particular actions. These studies have found that only stimulus-present decisions are actively encoded by neurons, whereas the decision about the absence of a stimulus does not affect default neuronal responses. We tested whether this processing mechanism also holds for abstract detection decisions that are dissociated from motor preparation. We recorded single-neuron activity from the prefrontal cortex (PFC) of monkeys performing a visual detection task that forced a report-independent decision. We not only found neurons that actively encoded the subjective decision of monkeys about the presence of a stimulus, but also cells responding actively for the decision about the absence of stimuli. These results suggest that abstract detection decisions are processed in a different way compared with the previously reported action-based decisions. In a report-independent framework, neuronal networks seem to generate a second set of neurons actively encoding the absence of sensory stimulation, thus translating decisions into abstract categories. This mechanism may allow the brain to "buffer" a decision in a nonmovement-related framework.
The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation.
This paper describes FieldTrip, an open source software package that we developed for the analysis of MEG, EEG, and other electrophysiological data. The software is implemented as a MATLAB toolbox and includes a complete set of consistent and user-friendly high-level functions that allow experimental neuroscientists to analyze experimental data. It includes algorithms for simple and advanced analysis, such as time-frequency analysis using multitapers, source reconstruction using dipoles, distributed sources and beamformers, connectivity analysis, and nonparametric statistical permutation tests at the channel and source level. The implementation as toolbox allows the user to perform elaborate and structured analyses of large data sets using the MATLAB command line and batch scripting. Furthermore, users and developers can easily extend the functionality and implement new algorithms. The modular design facilitates the reuse in other software packages.
Neuropsychology and human functional neuroimaging have implicated human parietal cortex in numerical processing, and macaque electrophysiology has shown that intraparietal areas house neurons tuned to numerosity. Yet although the areas responding overall during numerical tasks have been well defined by neuroimaging, a direct demonstration of individual number coding by spatial patterns has thus far been elusive.
We used multivariate pattern recognition on high-resolution functional imaging data to decode the information content of fine-scale signals evoked by different individual numbers. Parietal activation patterns for individual numerosities could be accurately discriminated and generalized across changes in low-level stimulus parameters. Distinct patterns were evoked by symbolic and nonsymbolic number formats, and individual digits were less accurately decoded (albeit still with significant accuracy) than numbers of dots. Interestingly, the numerosity of dot sets could be predicted above chance from the brain activation patterns evoked by digits, but not vice versa. Finally, number-evoked patterns changed in a gradual fashion as a function of numerical distance for the nonsymbolic notation, compatible with some degree of orderly layout of individual number representations.
Our findings demonstrate partial format invariance of individual number codes that is compatible with more numerous but more broadly tuned populations for nonsymbolic than for symbolic numbers, as postulated by recent computational models. In more general terms, our results illustrate the potential of functional magnetic resonance imaging pattern recognition to understand the detailed format of representations within a single semantic category, and beyond sensory cortical areas for which columnar architectures are well established.
Number symbols have allowed humans to develop superior mathematical skills that are a hallmark of technologically advanced cultures. Findings in animal cognition, developmental psychology, and anthropology indicate that these numerical skills are rooted in nonlinguistic biological primitives. Recent studies in human and nonhuman primates using a broad range of methodologies provide evidence that numerical information is represented and processed by regions of the prefrontal and posterior parietal lobes, with the intraparietal sulcus as a key node for the representation of the semantic aspect of numerical quantity.
In this study, subjects were asked to judge which of two digits (e.g., 3 5) was larger either in physical or in numerical
size. Reaction times were facilitated when the irrelevant dimension was congruent with the relevant dimension and were inhibited
when the two were incongruent (size congruity effect). Although judgments based on physical size were faster, their speed
was affected by the numerical distance between the members of the digit pair, indicating that numerical distance is automatically
computed even when it is irrelevant to the comparative judgment being required by the task. This finding argues for parallel
processing of physical and semantic information in this task.
A spatial filtering method for localizing sources of brain electrical activity from surface recordings is described and analyzed. The spatial filters are implemented as a weighted sum of the data recorded at different sites. The weights are chosen to minimize the filter output power subject to a linear constraint. The linear constraint forces the filter to pass brain electrical activity from a specified location, while the power minimization attenuates activity originating at other locations. The estimated output power as a function of location is normalized by the estimated noise power as a function of location to obtain a neural activity index map. Locations of source activity correspond to maxima in the neural activity index map. The method does not require any prior assumptions about the number of active sources of their geometry because it exploits the spatial covariance of the source electrical activity. This paper presents a development and analysis of the method and explores its sensitivity to deviations between actual and assumed data models. The effect on the algorithm of covariance matrix estimation, correlation between sources, and choice of reference is discussed. Simulated and measured data is used to illustrate the efficacy of the approach.
Previous research has demonstrated how children develop the ability to use notational representations to indicate simple quantities. These studies have shown a developmental shift from the use of idiosyncratic, to analogical, to conventional, numerical notations. The present paper extends these findings by reporting the results from a study in which children from 3 to 7 years old were asked to write a representation to indicate a quantity presented in a game-like scenario. Three kinds of quantities were included: whole numbers, zeros, and fractions. The children's notations were shown to them shortly after they were produced and then again two weeks later to see if children could interpret them. The results showed the familiar developmental pattern towards increased use of conventional notations for all quantities. The ability to read the notations was greatest for conventional numbers where performance was at ceiling, lower for analogue representations, and very poor for idiosyncratic global recordings. Children's choice of a notational format was influenced almost entirely by their age and not by the quantity being represented. Children were able to solve the zero problems almost as well as they could the whole numbers, but their understanding and use of representations for fractions was very limited.
Three experiments were performed to test whether infants show a bias for detecting the presence of a feature in a stimulus rather than its absence. In the 1st experiment, 24 16-week-old infants were given 3 paired-comparison problems, each of which included a 25-s familiarization phase followed by 2 test trials. Infants were familiarized to 1 member of a set of capital alphabetical letters (E-F; Q-O; B-R). Then they were given a paired-comparison recognition test under 1 of 2 conditions. In the feature-present condition, the familiar letter (e.g., F) was paired with a novel letter containing the addition of a distinguishing element (e.g., E). In the feature-absent condition, infants were presented with a familiar letter (e.g., E) paired with a novel letter in which 1 element was removed (e.g., F). Infants showed a novelty preference to the letter in which the distinguishing feature was present, but there was no preference for novelty in the feature-absent condition. The 2nd experiment showed that infants' fixation to the letter containing the presence of the feature was not due to a simple preference for the letter with the greater number of elements. Finally, to test whether infants' failure to discriminate the absence was due to insufficient encoding time, 36 infants were tested in a 3rd experiment in which familiarization time was varied. After 20 s of familiarization, no evidence of discrimination was observed in either the feature-present or feature-absent condition. After 30 s, however, infants could discriminate the novel letter in the feature-present condition but not in the feature-absent condition. The significance of these results is discussed in terms of theoretical explanations for the development of the feature-presence bias.
Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In the present study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, while no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as “zero”, but rather as “nothing”, when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.
The brain's function is to enable adaptive behavior in the world. To this end, the brain processes information about the world. The concept of representation links the information processed by the brain back to the world and enables us to understand what the brain does at a functional level. The appeal of making the connection between brain activity and what it represents has been irresistible to neuroscience, despite the fact that representational interpretations pose several challenges: We must define which aspects of brain activity matter, how the code works, and how it supports computations that contribute to adaptive behavior. It has been suggested that we might drop representational language altogether and seek to understand the brain, more simply, as a dynamical system. In this review, we argue that the concept of representation provides a useful link between dynamics and computational function and ask which aspects of brain activity should be analyzed to achieve a representational understanding. We peel the onion of brain representations in search of the layers (the aspects of brain activity) that matter to computation. The article provides an introduction to the motivation and mathematics of representational models, a critical discussion of their assumptions and limitations, and a preview of future directions in this area.
Cluster‐based permutation tests are gaining an almost universal acceptance as inferential procedures in cognitive neuroscience. They elegantly handle the multiple comparisons problem in high‐dimensional magnetoencephalographic and EEG data. Unfortunately, the power of this procedure comes hand in hand with the allure for unwarranted interpretations of the inferential output, the most prominent of which is the overestimation of the temporal, spatial, and frequency precision of statistical claims. This leads researchers to statements about the onset or offset of a certain effect that is not supported by the permutation test. In this article, we outline problems and common pitfalls of using and interpreting cluster‐based permutation tests. We illustrate these with simulated data in order to promote a more intuitive understanding of the method. We hope that raising awareness about these issues will be beneficial to common scientific practices, while at the same time increasing the popularity of cluster‐based permutation procedures. Cluster‐based permutation tests are a powerful solution to the multiple comparisons problem in EEG and MEG data. We report on extremely common, yet inapplicable interpretations of this procedure, suggesting unwarranted precision of the actual underlying test statistic and leading to strong, but unsubstantiated claims. In this article, we outline problems and common pitfalls of using and interpreting cluster‐based permutation tests. Accurate interpretations of cluster‐based permutation tests will contribute to the adequate utilization, as well as the popularity, of this powerful method.
Our human-specific symbolic number skills that underpin science and technology spring from nonsymbolic set size representations. Despite the significance of numerical competence, its single-neuron mechanisms in the human brain are unknown. We therefore recorded from single neurons in the medial temporal lobe of neurosurgical patients that performed a calculation task. We found that distinct groups of neurons represented either nonsymbolic or symbolic number, but not both number formats simultaneously. Numerical information could be decoded robustly from the population of neurons tuned to nonsymbolic number and with lower accuracy also from the population of neurons selective to number symbols. The tuning characteristics of selective neurons may explain why set size is represented only approximately in behavior, whereas number symbols allow exact assessments of numerical values. Our results suggest number neurons as neuronal basis of human number representations that ultimately give rise to number theory and mathematics.
Sensory and motor cortices each contain multiple topographic maps with the structure of sensory organs (such as the retina or cochlea) mapped onto the cortical surface. These sensory maps are hierarchically organized. For example, visual field maps contain neurons that represent increasingly large parts of visual space with increasingly complex responses 1 . Some visual neurons respond to stimuli with a particular numerosity — the number of objects in a set. We recently discovered a parietal topographic numerosity map in which neural numerosity preferences progress gradually across the cortical surface 2 , analogous to sensory maps. Following this analogy, we hypothesized that there may be multiple numerosity maps. Numerosity perception is implicated in many cognitive functions, including foraging 3 , multiple object tracking 4 , dividing attention 5 , decision-making 6 and mathematics 7, 8, 9 . Here we use ultra-high-field (7 Tesla, 7T) functional magnetic resonance imaging (fMRI) and neural-model-based analyses to reveal numerosity-selective neural populations organized into six widely separated topographic maps in each hemisphere. Although we describe subtle differences between these maps, their properties are very similar, unlike in sensory map hierarchies. These maps are found in areas implicated in object recognition, motion perception, attention control, decision-making and mathematics. Multiple numerosity maps may allow interactions with these cognitive systems, suggesting a broad role for quantity processing in supporting many perceptual and cognitive functions.
Zero stands for emptiness, for nothing, and yet it is considered to be one of the greatest achievements of humankind. This review first recapitulates the discovery of the number zero in human history, then follows its progression in human development, traces its evolution in the animal kingdom, and finally elucidates how the brain transforms 'nothing' into an abstract zero category. It is argued that the emergence of zero passes through four corresponding representations in all of these interrelated realms: first, sensory 'nothing'; then categorical 'something'; then quantitative empty sets; and finally the number zero. The concept of zero shows how the brain, originally evolved to represent stimuli ('something'), detaches from empirical properties to achieve ultimate abstract thinking.
Machine learning or MVPA (Multi Voxel Pattern Analysis) studies have shown that the neural representation of quantities of objects can be decoded from fMRI patterns, in cases where the quantities were visually displayed. Here we apply these techniques to investigate whether neural representations of quantities depicted in one modality (say, visual) can be decoded from brain activation patterns evoked by quantities depicted in the other modality (say, auditory). The main finding demonstrated, for the first time, that quantities of dots were decodable by a classifier that was trained on the neural patterns evoked by quantities of auditory tones, and vice-versa. The representations that were common across modalities were mainly right-lateralized in frontal and parietal regions. A second finding was that the neural patterns in parietal cortex that represent quantities were common across participants. These findings demonstrate a common neuronal foundation for the representation of quantities across sensory modalities and participants and provide insight into the role of parietal cortex in the representation of quantity information. Hum Brain Mapp, 2016. © 2016 Wiley Periodicals, Inc.
Two groups of children were trained to discriminate between two displays which could only be differentiated by a single distinctive feature located on one of the displays. Subjects trained with the distinctive feature located on the positive display learned the simultaneous discrimination while feature negative subjects did not. Recording of finger location of the subjects indicated a localization on the distinctive feature by feature positive subjects while feature negative subjects did not localize on the distinctive feature.
Do young children understand the numerical value of empty sets prior to developing a concept of symbolic zero? Are empty sets represented as mental magnitudes? In order to investigate these questions, we tested 4-year old children and adults with a numerical ordering task in which the goal was to select two stimuli in ascending numerical order with occasional empty set stimuli. Both children and adults showed distance effects for empty sets. Children who were unable to order the symbol zero (e.g., 0<1), but who successfully ordered countable integers (e.g., 2<4) nevertheless showed distance effects with empty sets. These results suggest that empty sets are represented on the same numerical continuum as non-empty sets and that children represent empty sets numerically prior to understanding symbolic zero.
Conducted 2 studies to investigate children's conceptions of the number zero. In Exp I, 57 3–7 yr olds were tested. Findings show that Ss understood that zero is a number among other numbers with its own unique value, namely nothing. Ss' achievement of this understanding occurred in 3 phases. At each phase, understanding of zero lagged behind comparable understanding of other small numbers. Exp II investigated the developing conception of simple algebraic rules, such as
a + 0 =
a in 48 children between the ages of 5½ and 10 yrs. Results show that even the younger Ss had some understanding of several algebraic rules. The older Ss had acquired more such knowledge, but at all ages algebraic understanding was advanced for rules pertaining to zero, in comparison to those pertaining to other small numbers. Results suggest that zero plays a special role in children's increasingly algebraic knowledge of number. It is concluded that since zero is difficult to conceive of and use originally (Exp I), children develop special rules for its use; this provides a 1st step toward their formulation of more general algebraic rules (Exp II) and toward an expanded conception of number and mathematics. (14 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Many applied problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator—the sample covariance matrix—is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte Carlo confirm that the asymptotic results tend to hold well in finite sample.
Most of us use numbers daily for counting, estimating quantities or formal mathematics, yet despite their importance our understanding of the brain correlates of these processes is still evolving. A neurofunctional model of mental arithmetic, proposed more than a decade ago, stimulated a substantial body of research in this area. Using quantitative meta-analyses of fMRI studies we identified brain regions concordant among studies that used number and calculation tasks. These tasks elicited activity in a set of common regions such as the inferior parietal lobule; however, the regions in which they differed were most notable, such as distinct areas of prefrontal cortices for specific arithmetic operations. Given the current knowledge, we propose an updated topographical brain atlas of mental arithmetic with improved interpretative power.
The Psychophysics Toolbox is a software package that supports visual psychophysics. Its routines provide an interface between a high-level interpreted language (MATLAB on the Macintosh) and the video display hardware. A set of example programs is included with the Toolbox distribution.
There is evidence to suggest that animals, young infants and adult humans possess a biologically determined, domain-specific representation of number and of elementary arithmetic operations. Behavioral studies in infants and animals reveal number perception, discrimination and elementary calculation abilities in non-verbal organisms. Lesion and brain-imaging studies in humans indicate that a specific neural substrate, located in the left and right intraparietal area, is associated with knowledge of numbers and their relations ('number sense'). The number domain is a prime example where strong evidence points to an evolutionary endowment of abstract domain-specific knowledge in the brain because there are parallels between number processing in animals and humans.The numerical distance effect, which refers to the finding that the ability to discriminate between two numbers improves as the numerical distance between them increases, has been demonstrated in humans and animals, as has the number size effect,which refers to the finding that for equal numerical distance,discrimination of two numbers worsens as their numerical size increases.