Arithmetic Memory Is Modality Specific

In regards to numerical cognition and working memory, it is an open question as to whether numbers are stored into and retrieved from a central abstract representation or from separate notation-specific representations. This study seeks to help answer this by utilizing the numeral modality effect (NME) in three experiments to explore how numbers are processed by the human brain. The participants were presented with numbers (1–9) as either Arabic digits or written number words (Arabic digits and dot matrices in Experiment 2) at the first (S1) and second (S2) stimuli. The participant’s task was to add the first two stimuli together and verify whether the answer (S3), presented simultaneously with S2, was correct. We hypothesized that if reaction time (RT) at S2/S3 depends on the modality of S1 then numbers are retrieved from modality specific memory stores. Indeed, RT depended on the modality of S1 whenever S2 was an Arabic digit which argues against the concept of numbers being stored and retrieved from a central, abstract representation.     

Numerical magnitude in the human parietal lobe; tests of representational generality and domain specificity

Behavioral evidence suggests that human adults have a single system for representing the numerical magnitude of both symbolic numbers (e.g., Arabic digits) and nonsymbolic number stimuli (e.g., dot arrays). Brain imaging studies have implicated a specific parietal region in symbolic number processing, leading to the influential hypothesis that this region is the locus of a dedicated, domain-specific number system. Here we evaluated a prediction of this hypothesis, that this region should be activated not only by symbolic but also nonsymbolic number processing. Using nonsymbolic stimuli, we tested for higher parietal activations for number than for nonnumber comparison tasks (experiment 1), fMRI adaptation for numerosity repetition (experiment 2), and greater fMRI increases with increasing task difficulty for number than nonnumber tasks (experiment 3). None of these predictions were supported by the data, posing a serious challenge to the hypothesis that a single, domain-specific parietal region underlies both symbolic and nonsymbolic number representation.     

Free energy gap and statistical thermodynamic fidelity of DNA codes.

DNA nanotechnology often requires collections of oligonucleotides called "DNA free energy gap codes" that do not produce erroneous crosshybridizations in a competitive muliplexing environment. This paper addresses the question of how to design these codes to accomplish a desired amount of work within an acceptable error rate. Using a statistical thermodynamic and probabilistic model of DNA code fidelity and mathematical random coding theory methods, theoretical lower bounds on the size of DNA codes are given. More importantly, DNA code design parameters (e.g., strand number, strand length and sequence composition) needed to achieve experimental goals are identified.     

Visual evoked potential in guinea pigs

The present study was designed to establish visual evoked potential (VEP) as one of clinical tests for veterinary medicine. Experiments were carried out on eight adult male guinea pigs weighed 350 to 750 g. We investigated influences of click sound, luminous intensity and habituation on VEP patterns. The VEP of the guinea pig was composed of primary (P 10, N 20, P 30, N 40) and secondary (P 55, N 75, P 100, N 140) components, followed by a rhythmic after-discharge. Click sound with flash produced some unclear peaks in VEP, while click sound without flash elicited clear six peaks. These different components of the response to stimulation suggested that the acoustically evoked potential induced some peaks in VEP. With the intensity used in the present study, changes in luminous intensity resulted in unrecognizable difference among the VEPs. Early components of VEP were not clearly influenced by the habituation to stimulation. As the stimulation was repeated, rhythmic after-discharge seemed to be suppressed in the half of experiments.     

A magnitude code common to numerosities and number symbols in human intraparietal cortex

Activation of the horizontal segment of the intraparietal sulcus (hIPS) has been observed in various number-processing tasks, whether numbers were conveyed by symbolic numerals (digits, number words) or by nonsymbolic displays (dot patterns). This suggests an abstract coding of numerical magnitude. Here, we critically tested this hypothesis using fMRI adaptation to demonstrate notation-independent coding of numerical quantity in the hIPS. Once subjects were adapted either to dot patterns or to Arabic digits, activation in the hIPS and in frontal regions recovered in a distance-dependent fashion whenever a new number was presented, irrespective of notation changes. This remained unchanged when analyzing the hIPS peaks from an independent localizer scan of mental calculation. These results suggest an abstract coding of approximate number common to dots, digits, and number words. They support the idea that symbols acquire meaning by linking neural populations coding symbol shapes to those holding nonsymbolic representations of quantities.     

Operational Momentum in Multiplication and Division?

Biases are commonly seen in numerical cognition. The operational momentum (OM) effect shows that responses to addition and subtraction problems are biased in the whole-number direction of the operation. It is not known if this bias exists for other arithmetic operations. To determine whether OM exists in scalar operations, we measured response bias in adults performing symbolic (Arabic digits) and non-symbolic (dots) multiplication and division problems. After seeing two operands, with either a multiplication (×) or division (÷) sign, participants chose among five response choices. Both non-random performance profiles and the significant contribution of both operands in a multiple regression analysis predicting the chosen values, suggest that adults were able to use numerical information to approximate the outcomes in both notations, though they were more accurate on symbolic problems. Performance on non-symbolic problems was influenced by the size of the correct choice relative to alternatives. Reminiscent of the bias in addition and subtraction, we found a significant response bias for non-symbolic problems. Non-symbolic multiplication problems were overestimated and division problems were underestimated. These results indicate that operational momentum is present in non-symbolic multiplication and division. Given the influence of the size of the correct choice relative to alternatives, an interaction between heuristic bias and approximate calculation is possible.     

The conduction of optic nerve fibers using different visual patterns

The Authors have studied the behaviour of checkerboard pattern visual evoked potential (VEP) latencies by using different spatial frequency stimuli and different stimulating visual fields in order to demonstrate whether spatial frequency might constitute a parameter capable of exciting different retinal regions like different stimulus fields. According to the recent literature low spatial frequency stimuli generate VEP with latencies which are significantly shorter than high spatial frequency stimuli, making this method more reliable for the differentiation of macular and peripheral retinal fields.     

Multi-representation of symbolic and nonsymbolic numerical magnitude in Chinese number processing

Numerical information can be conveyed by either symbolic or nonsymbolic representation. Some symbolic numerals can also be identified as nonsymbolic quantities defined by the number of lines (e.g., I, II, III in Roman and -, =, ≡ in Japanese Kanji and Chinese). Here we report that such multi-representation of magnitude can facilitate the processing of these numerals under certain circumstances. In a magnitude comparison task judging 1 to 9 (except 5) Chinese and Arabic numerals presented at the foveal (at the center) or parafoveal (3° left or right of the center) location, multi-representational small-value Chinese numerals showed a processing advantage over single-representational Arabic numerals and large-value Chinese numerals only in the parafoveal condition, demonstrated by lower error rates and faster reaction times. Further event-related potential (ERP) analysis showed that such a processing advantage was not reflected by traditional ERP components identified in previous studies of number processing, such as N1 or P2p. Instead, the difference was found much later in a N400 component between 300-550 msec over parietal regions, suggesting that those behavioral differences may not be due to early processing of visual identification, but later processing of subitizing or accessing mental number line when lacking attentional resources. These results suggest that there could be three stages of number processing represented separately by the N1, P2p and N400 ERP components. In addition, numerical information can be represented simultaneously by both symbolic and nonsymbolic systems, which will facilitate number processing in certain situations.     

The role of visual information in numerosity estimation

Mainstream theory suggests that the approximate number system supports our non-symbolic number abilities (e.g. estimating or comparing different sets of items). It is argued that this system can extract number independently of the visual cues present in the stimulus (diameter, aggregate surface, etc.). However, in a recent report we argue that this might not be the case. We showed that participants combined information from different visual cues to derive their answers. While numerosity comparison requires a rough comparison of two sets of items (smaller versus larger), numerosity estimation requires a more precise mechanism. It could therefore be that numerosity estimation, in contrast to numerosity comparison, might rely on the approximate number system. To test this hypothesis, we conducted a numerosity estimation experiment. We controlled for the visual cues according to current standards: each single visual property was not informative about numerosity. Nevertheless, the results reveal that participants were influenced by the visual properties of the dot arrays. They gave a larger estimate when the dot arrays consisted of dots with, on average, a smaller diameter, aggregate surface or density but a larger convex hull. The reliance on visual cues to estimate numerosity suggests that the existence of an approximate number system that can extract numerosity independently of the visual cues is unlikely. Instead, we propose that humans estimate numerosity by weighing the different visual cues present in the stimuli.     

Revisiting the operational RNA code for amino acids: Ensemble attributes and their implications

It has been suggested that tRNA acceptor stems specify an operational RNA code for amino acids. In the last 20 years several attributes of the putative code have been elucidated for a small number of model organisms. To gain insight about the ensemble attributes of the code, we analyzed 4925 tRNA sequences from 102 bacterial and 21 archaeal species. Here, we used a classification and regression tree (CART) methodology, and we found that the degrees of degeneracy or specificity of the RNA codes in both Archaea and Bacteria differ from those of the genetic code. We found instances of taxon-specific alternative codes, i.e., identical acceptor stem determinants encrypting different amino acids in different species, as well as instances of ambiguity, i.e., identical acceptor stem determinants encrypting two or more amino acids in the same species. When partitioning the data by class of synthetase, the degree of code ambiguity was significantly reduced. In cryptographic terms, a plausible interpretation of this result is that the class distinction in synthetases is an essential part of the decryption rules for resolving the subset of RNA code ambiguities enciphered by identical acceptor stem determinants of tRNAs acylated by enzymes belonging to the two classes. In evolutionary terms, our findings lend support to the notion that in the pre-DNA world, interactions between tRNA acceptor stems and synthetases formed the basis for the distinction between the two classes; hence, ambiguities in the ancient RNA code were pivotal for the fixation of these enzymes in the genomes of ancestral prokaryotes.     

Codes, operations, measurements and neural networks

Numerous neural codes and primary neural operations (logical and arithmetical ones, mappings, transformations) were listed [e.g. Perkel, D., Bullock, T.H., 1968. Neurosci. Res. Program Bull 6, 221-348] during the past decades. None of them is ubiquitous or universal. In reality neural operations take place in continuous time and working with unreliable elements, but they still can be simulated with synchronized discrete time scales and chaotic models. Here, a possible neural mechanism, called 'measure like' code is introduced and examined. The neurons are regarded as measuring devices, dealing with 'measures', more or less in mathematical sense. The subadditivity--eminent property of measures--may be implemented with neuronal refractoriness and such synapses operate like particle counters with dead time. This hypothetical code is neither ubiquitous, nor universal, e.g. temporal summation (multiplication) causes just the opposite phenomenon, the supra-additivity also with respect to the number of spikes (anti-measures). This is a cause of more difficult neural implementation of OR gate, than that of the AND. Possibilities for transitional mechanisms (e.g. between traditional logical gates, etc.) are stressed here. Parameter tuning might change either code or operation.     

Graphical Aids to the Estimation and Discrimination of Uncertain Numerical Data

This research investigates the performance of graphical dot arrays designed to make discrimination of relative numerosity as effortless as possible at the same time as making absolute (quantitative) numerosity estimation as effortful as possible. Comparing regular, random, and hybrid (randomized regular) configurations of dots, the results indicate that both random and hybrid configurations reduce absolute numerosity estimation precision, when compared with regular dots arrays. However, discrimination of relative numerosity is significantly more accurate for hybrid dot arrays than for random dot arrays. Similarly, human subjects report significantly lower levels of subjective confidence in judgments when using hybrid dot configurations as compared with regular configurations; and significantly higher levels of subjective confidence as compared with random configurations. These results indicate that data graphics based on the hybrid, randomized-regular configurations of dots are well-suited to applications that require decisions to be based on numerical data in which the absolute quantities are less certain than the relative values. Examples of such applications include decision-making based on the outputs of empirically-based mathematical models, such as health-related policy decisions using data from predictive epidemiological models.     

Early fixation of an optimal genetic code

The evolutionary forces that produced the canonical genetic code before the last universal ancestor remain obscure. One hypothesis is that the arrangement of amino acid/codon assignments results from selection to minimize the effects of errors (e.g., mistranslation and mutation) on resulting proteins. If amino acid similarity is measured as polarity, the canonical code does indeed outperform most theoretical alternatives. However, this finding does not hold for other amino acid properties, ignores plausible restrictions on possible code structure, and does not address the naturally occurring nonstandard genetic codes. Finally, other analyses have shown that significantly better code structures are possible. Here, we show that if theoretically possible code structures are limited to reflect plausible biological constraints, and amino acid similarity is quantified using empirical data of substitution frequencies, the canonical code is at or very close to a global optimum for error minimization across plausible parameter space. This result is robust to variation in the methods and assumptions of the analysis. Although significantly better codes do exist under some assumptions, they are extremely rare and thus consistent with reports of an adaptive code: previous analyses which suggest otherwise derive from a misleading metric. However, all extant, naturally occurring, secondarily derived, nonstandard genetic codes do appear less adaptive. The arrangement of amino acid assignments to the codons of the standard genetic code appears to be a direct product of natural selection for a system that minimizes the phenotypic impact of genetic error. Potential criticisms of previous analyses appear to be without substance. That known variants of the standard genetic code appear less adaptive suggests that different evolutionary factors predominated before and after fixation of the canonical code. While the evidence for an adaptive code is clear, the process by which the code achieved this optimization requires further attention.     

Functional imaging of numerical processing in adults and 4-y-old children

Adult humans, infants, pre-school children, and non-human animals appear to share a system of approximate numerical processing for non-symbolic stimuli such as arrays of dots or sequences of tones. Behavioral studies of adult humans implicate a link between these non-symbolic numerical abilities and symbolic numerical processing (e.g., similar distance effects in accuracy and reaction-time for arrays of dots and Arabic numerals). However, neuroimaging studies have remained inconclusive on the neural basis of this link. The intraparietal sulcus (IPS) is known to respond selectively to symbolic numerical stimuli such as Arabic numerals. Recent studies, however, have arrived at conflicting conclusions regarding the role of the IPS in processing non-symbolic, numerosity arrays in adulthood, and very little is known about the brain basis of numerical processing early in development. Addressing the question of whether there is an early-developing neural basis for abstract numerical processing is essential for understanding the cognitive origins of our uniquely human capacity for math and science. Using functional magnetic resonance imaging (fMRI) at 4-Tesla and an event-related fMRI adaptation paradigm, we found that adults showed a greater IPS response to visual arrays that deviated from standard stimuli in their number of elements, than to stimuli that deviated in local element shape. These results support previous claims that there is a neurophysiological link between non-symbolic and symbolic numerical processing in adulthood. In parallel, we tested 4-y-old children with the same fMRI adaptation paradigm as adults to determine whether the neural locus of non-symbolic numerical activity in adults shows continuity in function over development. We found that the IPS responded to numerical deviants similarly in 4-y-old children and adults. To our knowledge, this is the first evidence that the neural locus of adult numerical cognition takes form early in development, prior to sophisticated symbolic numerical experience. More broadly, this is also, to our knowledge, the first cognitive fMRI study to test healthy children as young as 4 y, providing new insights into the neurophysiology of human cognitive development.     

Visual Evoked Potentials and Disorders of Visual Function in Children Suffering from Cerebral Palsy

Using recording of visual evoked potentials (VEP), we examined 126 children (from 1 to 14 years old) suffering from spastic forms of cerebral palsy. In the overwhelming majority of cases (84.1%), we found in these patients significant modifications of the VEP related to dysfunctions of the visual analyzer system at different levels of the latter. We analyzed correlation of clinical manifestations of the disease with deviations of the VEP characteristics from the age norm. We conclude that an EEG examination, including VEP recording, allows one to obtain objective estimates of disorders of the visual function in the studied population of patients.     

A Two-Minute Paper-and-Pencil Test of Symbolic and Nonsymbolic Numerical Magnitude Processing Explains Variability in Primary School Children's Arithmetic Competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.     

Categorical Biases in Perceiving Spatial Relations

We investigate the effect of spatial categories on visual perception. In three experiments, participants made same/different judgments on pairs of simultaneously presented dot-cross configurations. For different trials, the position of the dot within each cross could differ with respect to either categorical spatial relations (the dots occupied different quadrants) or coordinate spatial relations (the dots occupied different positions within the same quadrant). The dot-cross configurations also varied in how readily the dot position could be lexicalized. In harder-to-name trials, crosses formed a “+” shape such that each quadrant was associated with two discrete lexicalized spatial categories (e.g., “above” and “left”). In easier-to-name trials, both crosses were rotated 45° to form an “×” shape such that quadrants were unambiguously associated with a single lexicalized spatial category (e.g., “above” or “left”). In Experiment 1, participants were more accurate when discriminating categorical information between easier-to-name categories and more accurate at discriminating coordinate spatial information within harder-to-name categories. Subsequent experiments attempted to down-regulate or up-regulate the involvement of language in task performance. Results from Experiment 2 (verbal interference) and Experiment 3 (verbal training) suggest that the observed spatial relation type-by-nameability interaction is resistant to online language manipulations previously shown to affect color and object-based perceptual processing. The results across all three experiments suggest that robust biases in the visual perception of spatial relations correlate with patterns of lexicalization, but do not appear to be modulated by language online.     

Children’s Mapping between Non-Symbolic and Symbolic Numerical Magnitudes and Its Association with Timed and Untimed Tests of Mathematics Achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.     

Different Stimuli,Different Spatial Codes: A Visual Map and an Auditory Rate Code for Oculomotor Space in the Primate Superior Colliculus

Maps are a mainstay of visual, somatosensory, and motor coding in many species. However, auditory maps of space have not been reported in the primate brain. Instead, recent studies have suggested that sound location may be encoded via broadly responsive neurons whose firing rates vary roughly proportionately with sound azimuth. Within frontal space, maps and such rate codes involve different response patterns at the level of individual neurons. Maps consist of neurons exhibiting circumscribed receptive fields, whereas rate codes involve open-ended response patterns that peak in the periphery. This coding format discrepancy therefore poses a potential problem for brain regions responsible for representing both visual and auditory information. Here, we investigated the coding of auditory space in the primate superior colliculus(SC), a structure known to contain visual and oculomotor maps for guiding saccades. We report that, for visual stimuli, neurons showed circumscribed receptive fields consistent with a map, but for auditory stimuli, they had open-ended response patterns consistent with a rate or level-of-activity code for location. The discrepant response patterns were not segregated into different neural populations but occurred in the same neurons. We show that a read-out algorithm in which the site and level of SC activity both contribute to the computation of stimulus location is successful at evaluating the discrepant visual and auditory codes, and can account for subtle but systematic differences in the accuracy of auditory compared to visual saccades. This suggests that a given population of neurons can use different codes to support appropriate multimodal behavior.