scholarly journals Numerosity representations in crows obey the Weber–Fechner law

2016 ◽  
Vol 283 (1827) ◽  
pp. 20160083 ◽  
Author(s):  
Helen M. Ditz ◽  
Andreas Nieder
Keyword(s):  
Weber Fraction ◽  
Discrimination Performance ◽  
Brain Structures ◽  
Number System ◽  
Number Line ◽  
Just Noticeable Difference ◽  
Match To Sample ◽  
Close Phylogenetic Relationship ◽  
The One ◽  
Numerical Representations

The ability to estimate number is widespread throughout the animal kingdom. Based on the relative close phylogenetic relationship (and thus equivalent brain structures), non-verbal numerical representations in human and non-human primates show almost identical behavioural signatures that obey the Weber–Fechner law. However, whether numerosity discriminations of vertebrates with a very different endbrain organization show the same behavioural signatures remains unknown. Therefore, we tested the numerical discrimination performance of two carrion crows ( Corvus corone ) to a broad range of numerosities from 1 to 30 in a delayed match-to-sample task similar to the one used previously with primates. The crows' discrimination was based on an analogue number system and showed the Weber-fraction signature (i.e. the ‘just noticeable difference’ between numerosity pairs increased in proportion to the numerical magnitudes). The detailed analysis of the performance indicates that numerosity representations in crows are scaled on a logarithmically compressed ‘number line’. Because the same psychophysical characteristics are found in primates, these findings suggest fundamentally similar number representations between primates and birds. This study helps to resolve a classical debate in psychophysics: the mental number line seems to be logarithmic rather than linear, and not just in primates, but across vertebrates.

Analog Numerical Representations in Rhesus Monkeys: Evidence for Parallel Processing

2004 ◽  
Vol 16 (5) ◽  
pp. 889-901 ◽  
Author(s):  
Andreas Nieder ◽  
Earl K. Miller
Keyword(s):  
Rhesus Monkeys ◽  
Neuronal Response ◽  
Weber Fraction ◽  
Discrimination Performance ◽  
Model Organisms ◽  
Response Latencies ◽  
Numerosity Discrimination ◽  
Numerical Information ◽  
Match To Sample ◽  
Numerical Representations

Monkeys have been introduced as model organisms to study neural correlates of numerical competence, but many of the behavioral characteristics of numerical judgments remain speculative. Thus, we analyzed the behavioral performance of two rhesus monkeys judging the numerosities 1 to 7 during a delayed match-to-sample task. The monkeys showed similar discrimination performance irrespective of the exact physical appearance of the stimuli, confirming that performance was based on numerical information. Performance declined smoothly with larger numerosities, and reached discrimination threshold at numerosity “4.” The nonverbal numerical representations in monkeys were based on analog magnitudes, object tracking process (“subitizing”) could not account for the findings because the continuum of small and large numbers shows a clear Weber fraction signature. The lack of additional scanning eye movements with increasing set sizes, together with indistinguishable neuronal response latencies for neurons with different preferred numerosities, argues for parallel encoding of numerical information. The slight but significant increase in reaction time with increasing numerosities can be explained by task difficulty and consequently time-consuming decision processes. The behavioral results are compared to single-cell recordings from the prefrontal cortex in the same subjects. Models for numerosity discrimination that may account for these results are discussed.

scholarly journals Testing Economic Theory

2018 ◽  
pp. 303-313
Author(s):  
Christopher P. Guzelian
Keyword(s):  
A Priori ◽  
Number System ◽  
Physical Reality ◽  
Economic Research ◽  
Economic Knowledge ◽  
Fractional Reserve Banking ◽  
Physical Universe ◽  
And Mathematics ◽  
Reserve Banking ◽  
The One

Two years ago, Bob Mulligan and I empirically tested whether the Bank of Amsterdam, a prototypical central bank, had caused a boom-bust cycle in the Amsterdam commodities markets in the 1780s owing to the bank’s sudden initiation of low-fractional-re-serve banking (Guzelian & Mulligan 2015).1 Widespread criticism came quickly after we presented our data findings at that year’s Austrian Economic Research Conference. Walter Block representa-tively responded: «as an Austrian, I maintain you cannot «test» apodictic theories, you can only illustrate them».2 Non-Austrian, so-called «empirical» economists typically have no problem with data-driven, inductive research. But Austrians have always objected strenuously on ontological and epistemolog-ical grounds that such studies do not produce real knowledge (Mises 1998, 113-115; Mises 2007). Camps of economists are talking past each other in respective uses of the words «testing» and «eco-nomic theory». There is a vital distinction between «testing» (1) an economic proposition, praxeologically derived, and (2) the rele-vance of an economic proposition, praxeologically derived. The former is nonsensical; the latter may be necessary to acquire eco-nomic theory and knowledge. Clearing up this confusion is this note’s goal. Rothbard (1951) represents praxeology as the indispensible method for gaining economic knowledge. Starting with a Aristote-lian/Misesian axiom «humans act» or a Hayekian axiom of «humans think», a voluminous collection of logico-deductive eco-nomic propositions («theorems») follows, including theorems as sophisticated and perhaps unintuitive as the one Mulligan and I examined: low-fractional-reserve banking causes economic cycles. There is an ontological and epistemological analog between Austrian praxeology and mathematics. Much like praxeology, we «know» mathematics to be «true» because it is axiomatic and deductive. By starting with Peano Axioms, mathematicians are able by a long process of creative deduction, to establish the real number system, or that for the equation an + bn = cn, there are no integers a, b, c that satisfy the equation for any integer value of n greater than 2 (Fermat’s Last Theorem). But what do mathematicians mean when they then say they have mathematical knowledge, or that they have proven some-thing «true»? Is there an infinite set of rational numbers floating somewhere in the physical universe? Naturally no. Mathemati-cians mean that they have discovered an apodictic truth — some-thing unchangeably true without reference to physical reality because that truth is a priori.

Response Differences in Monkey TE and Perirhinal Cortex: Stimulus Association Related to Reward Schedules

2000 ◽  
Vol 83 (3) ◽  
pp. 1677-1692 ◽  
Author(s):  
Zheng Liu ◽  
Barry J. Richmond
Keyword(s):  
Visual Cues ◽  
Perirhinal Cortex ◽  
Reward Schedule ◽  
Behavioral Context ◽  
Related Activity ◽  
Match To Sample ◽  
Behavioral Evidence ◽  
The One ◽  
Different Response ◽  
Visual Cortical

Anatomic and behavioral evidence shows that TE and perirhinal cortices are two directly connected but distinct inferior temporal areas. Despite this distinctness, physiological properties of neurons in these two areas generally have been similar with neurons in both areas showing selectivity for complex visual patterns and showing response modulations related to behavioral context in the sequential delayed match-to-sample (DMS) trials, attention, and stimulus familiarity. Here we identify physiological differences in the neuronal activity of these two areas. We recorded single neurons from area TE and perirhinal cortex while the monkeys performed a simple behavioral task using randomly interleaved visually cued reward schedules of one, two, or three DMS trials. The monkeys used the cue's relation to the reward schedule (indicated by the brightness) to adjust their behavioral performance. They performed most quickly and most accurately in trials in which reward was immediately forthcoming and progressively less well as more intermediate trials remained. Thus the monkeys appeared more motivated as they progressed through the trial schedule. Neurons in both TE and perirhinal cortex responded to both the visual cues related to the reward schedules and the stimulus patterns used in the DMS trials. As expected, neurons in both areas showed response selectivity to the DMS patterns, and significant, but small, modulations related to the behavioral context in the DMS trial. However, TE and perirhinal neurons showed strikingly different response properties. The latency distribution of perirhinal responses was centered 66 ms later than the distribution of TE responses, a larger difference than the 10–15 ms usually found in sequentially connected visual cortical areas. In TE, cue-related responses were related to the cue's brightness. In perirhinal cortex, cue-related responses were related to the trial schedules independently of the cue's brightness. For example, some perirhinal neurons responded in the first trial of any reward schedule including the one trial schedule, whereas other neurons failed to respond in the first trial but respond in the last trial of any schedule. The majority of perirhinal neurons had more complicated relations to the schedule. The cue-related activity of TE neurons is interpreted most parsimoniously as a response to the stimulus brightness, whereas the cue-related activity of perirhinal neurons is interpreted most parsimoniously as carrying associative information about the animal's progress through the reward schedule. Perirhinal cortex may be part of a system gauging the relation between work schedules and rewards.

The number line in the primary grades

1961 ◽  
Vol 8 (2) ◽  
pp. 75-76
Author(s):  
Robert B. Ashlock
Keyword(s):  
Primary Grades ◽  
Number System ◽  
Number Line

If new arithmetic processes are to be introduced successfully in the primary grades, pupils must understand the number system thoroughly. The number line can often be used to deepen understandings, through experiences which follow initial concrete manipulation but precede the more abst ract computational problems.

A better understanding of our number system

1962 ◽  
Vol 9 (2) ◽  
pp. 71-73
Author(s):  
Eileen K. Claspill
Keyword(s):  
Number System ◽  
Place Value ◽  
The Third ◽  
The One

Most students and many adults have difficulty understanding our number system. The Hindu-Arabic system which we use is a tens system. The place value, beyond the units place, that each number holds is based on the power of the number ten. For example, the nine in 1906 is in the hundreds place which means that its value is nine times ten squared. The one is in the thousands place which means that its value is one times ten cubed or one times ten to the third power (10×l0×l0).

MULTIFUNCTION RNS MODULO 2n ± 1 MULTIPLIERS

2012 ◽  
Vol 21 (04) ◽  
pp. 1250027 ◽  
Author(s):  
TSO-BING JUANG ◽  
CHAO-TSUNG KUO ◽  
GO-LONG WU ◽  
JIAN-HAO HUANG
Keyword(s):  
Real Time ◽  
Number System ◽  
Residue Number System ◽  
Real Time Processing ◽  
Time Processing ◽  
The Real ◽  
Delay Constraints ◽  
Residue Number ◽  
The One

In this paper, multifunction residue number system (RNS) modulo (2n ± 1) multipliers are proposed. By adopting common circuits for summing up the partial products with extra controls, our proposed multipliers could perform both modulo (2n + 1) and (2n - 1) multiplications. The levels for summation of partial products are n + 1, which are same as the conventional modulo multipliers which with only one kind of modulo multiplications. The proposed multifunction modulo (2n ± 1) multipliers can save at least about 42.5% area under the same delay constraints and above 65.8% Area × Delay Product (ADP) compared with the one composed of modulo (2n + 1) and modulo (2n - 1) multiplication operations. Our proposed multipliers could be applied to ease the tremendous computation overload in the real-time processing applications.

scholarly journals Developmental Continuity in the Link Between Sensitivity to Numerosity and Physical Size

2015 ◽  
Vol 1 (1) ◽  
pp. 7-20 ◽  
Author(s):  
Ariel Starr ◽  
Elizabeth M. Brannon
Keyword(s):  
Individual Differences ◽  
Weber Fraction ◽  
Number Line ◽  
Line Length ◽  
Stimulus Dimension ◽  
Significant Relation ◽  
General Representation ◽  
Physical Size ◽  
Magnitude Comparison ◽  
Developmental Continuity

Converging evidence suggests that representations of number, space, and other dimensions depend on a general representation of magnitude. However, it is unclear whether there exists a privileged relation between certain magnitude dimensions or if all continuous magnitudes are equivalently related. Four-year-old children and adults were tested with three magnitude comparison tasks – nonsymbolic number, line length, and luminance – to determine whether individual differences in sensitivity are stable across dimensions. A Weber fraction (w) was calculated for each participant in each stimulus dimension. For both children and adults, accuracy and w values for number and line length comparison were significantly correlated, whereas neither accuracy nor w was correlated for number and luminance comparison. However, although line length and luminance comparison performance were not correlated in children, there was a significant relation in adults. These results suggest that there is a privileged relation between number and line length that emerges early in development and that relations between other magnitude dimensions may be later constructed over the course of development.

scholarly journals ‘Difference Is the One Thing That We Have in Common’

Tripodos ◽  
2021 ◽  
pp. 37-56
Author(s):  
Chiara Modugno ◽  
Tonny Krijnen
Keyword(s):  
Focus Groups ◽  
Photo Elicitation ◽  
Audience Research ◽  
Television Production ◽  
Television Shows ◽  
Fair Representation ◽  
Intersectional Representation ◽  
Research Interviews ◽  
The One ◽  
Numerical Representations

Television production is championing diversity in representation with record numbers compared to previous years. Netflix’s Sense8 is definitely amongst the highest scoring shows as concerns intersectional representation. Such remarkable representation was worth the 2016 GLAAD Outstanding Drama Series award, a prize granted to the most diverse television shows. However, this applause is geared solely to numerical representations while current academic discussion focuses more on the concept of fair representations. Not only is being represented of importance, but how one is represented. The present paper employs photovoice and photo elicitation to investigate how Sense8 fans articulate what constitutes a fair representation of queer gender identities within the show. The present research addresses two gaps in the literature. First, a methodological one: the employment of creative visual methodologies to transcend the limitations of the most common methods used for audience research —interviews and focus groups. Secondly, this study follows the contemporary conversation around fair representation by addressing what is now a gap in the existing literature on queer television: what is fair representation from an audience perspective? The results of this study show how audiences’ perspectives on fair representation differ from those formulated in public and academic debates.

Spatial–numerical associations: Shared symbolic and non-symbolic numerical representations

2019 ◽  
Vol 72 (10) ◽  
pp. 2423-2436 ◽  
Author(s):  
Stefan Buijsman ◽  
Carlos Tirado
Keyword(s):  
Theoretical Perspective ◽  
Number System ◽  
Snarc Effect ◽  
Approximate Number System ◽  
Symbolic Processing ◽  
Shared Representations ◽  
Theoretical Perspectives ◽  
Number Representations ◽  
Neuroimaging Data ◽  
Numerical Representations

During the last decades, there have been a large number of studies into the number-related abilities of humans. As a result, we know that humans and non-human animals have a system known as the approximate number system that allows them to distinguish between collections based on their number of items, separately from any counting procedures. Dehaene and others have argued for a model on which this system uses representations for numbers that are spatial in nature and are shared by our symbolic and non-symbolic processing of numbers. However, there is a conflicting theoretical perspective in which there are no representations of numbers underlying the approximate number system, but only quantity-related representations. This perspective would then suggest that there are no shared representations between symbolic and non-symbolic processing. We review the evidence on spatial biases resulting from the activation of numerical representations, for both non-symbolic and symbolic tests. These biases may help decide between the theoretical differences; shared representations are expected to lead to similar biases regardless of the format, whereas different representations more naturally explain differences in biases, and thus behaviour. The evidence is not yet decisive, as the behavioural evidence is split: we expect bisection tasks to eventually favour shared representations, whereas studies on the spatial–numerical association of response codes (SNARC) effect currently favour different representations. We discuss how this impasse may be resolved, in particular, by combining these behavioural studies with relevant neuroimaging data. If this approach is carried forward, then it may help decide which of these two theoretical perspectives on number representations is correct.

The Relationship Between Approximate Number System and Mathematics Competency in Indonesian Primary School Children

2018 ◽  
Vol 24 (8) ◽  
pp. 6259-6264
Author(s):  
Kevin Wijaya ◽  
Fransiskus X Ivan ◽  
Adre Mayza
Keyword(s):  
Response Time ◽  
Primary School ◽  
School Children ◽  
Weber Fraction ◽  
Number System ◽  
Primary School Children ◽  
Methodological Aspect ◽  
Approximate Number System ◽  
And Mathematics ◽  
The Relationship

The purpose of this study is to investigate the relationship between Approximate Number System (ANS), a cognitive system which represents and estimates the cardinality of a set, and mathematics competency of primary school children. Many findings on ANS and its relations with mathematics competency showed inconsistency. This research is the first of its kind in Indonesia. 318 fourth and fifth-grade primary school students were instructed to perform non-symbolic (dots) comparison task to measure their Weber fraction (w), accuracy (percentage correct), and response time (ms) which are the measurement for ANS acuity. Mathematics competencies of the students were taken from school’s report card and the data were standardized for each school separately. Correlation and regression linear analysis were conducted to find the relationship between ANS acuity and mathematics’ competency. Analysis showed there was a weak but significant (p < 0.05) correlation between two measurements of ANS acuity, namely the Weber fraction and accuracy, with mathematics competency, but not response time (p > 0.05). Further analysis with linear regression showed there was no relationship between the two variables and mathematics score, which disproves this correlation. This study shows that there is no relationship between children’s ANS acuity and mathematics competency. Intrinsic factors such as children’s attention, engagement, and motivation, also methodological aspect needed further consideration. Future studies are needed to investigate the methodological aspect related to the measurement of ANS and mathematics’ competency as there is no ‘gold standard’ yet to measure ANS.

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