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Concept: Weber–Fechner law


In the past years, a few methods have been developed to translate human EEG to music. In 2009, PloS One 4 e5915, we developed a method to generate scale-free brainwave music where the amplitude of EEG was translated to music pitch according to the power law followed by both of them, the period of an EEG waveform is translated directly to the duration of a note, and the logarithm of the average power change of EEG is translated to music intensity according to the Fechner’s law. In this work, we proposed to adopt simultaneously-recorded fMRI signal to control the intensity of the EEG music, thus an EEG-fMRI music is generated by combining two different and simultaneous brain signals. And most importantly, this approach further realized power law for music intensity as fMRI signal follows it. Thus the EEG-fMRI music makes a step ahead in reflecting the physiological process of the scale-free brain.

Concepts: Cognitive science, Electroencephalography, Medical tests, Sound, Functional magnetic resonance imaging, Pitch, Weber–Fechner law, Scientific pitch notation


  The effect of enzymatic deamidation by protein-glutaminase (PG) on protein solubility and flavor binding potential of soymilk was studied. Treatment of soymilk with PG for 2 h (temperature of 44 °C and enzyme:substrate ratio (E/S) of 40 U/g protein) resulted in high degree of protein deamidation (66.4% DD) and relatively low degree of protein hydrolysis (4.25% DH). Deamidated (DSM) and control soymilks (CSM) did not differ with respect to aroma, but differed in taste characteristics by sensory evaluation. Protein solubility in DSM was enhanced at weakly acidic conditions (pH 5.0), but did not differ from non-deamidated soymilk at pH values of 3.0 and 7.0. Odor detection thresholds for the flavor compounds vanillin and maltol were approximately 5 and 3 fold lower, respectively, in DSM than in CSM. Dose-response curves (Fechner’s law plots and n exponents from Stevens’s power law) further demonstrated that DSM had a lower flavor binding potential than CSM. PG deamidation has the potential to reduce flavor binding problems encountered in high protein-containing foods and beverages. Practical Application:  The findings of this study can help lead to the development of technology to produce protein-containing foods with improved functional properties, especially protein solubility, and potentially decreased flavor fade problems associated with flavor-protein interactions, especially with carbonyl containing flavor compounds.

Concepts: Acid, Enzyme, PH, Olfaction, Flavor, Weber–Fechner law, Psychophysics, Stevens' power law


One foundation of numerical cognition is that discrimination accuracy depends on the proportional difference between compared values, closely following the Weber-Fechner discrimination law. Performance in non-symbolic numerical discrimination is used to calculate individual Weber fraction, a measure of relative acuity of the approximate number system (ANS). Individual Weber fraction is linked to symbolic arithmetic skills and long-term educational and economic outcomes. The present findings suggest that numerical discrimination performance depends on both the proportional difference and absolute value, deviating from the Weber-Fechner law. The effect of absolute value is predicted via computational model based on the neural correlates of numerical perception. Specifically, that the neural coding “noise” varies across corresponding numerosities. A computational model using firing rate variation based on neural data demonstrates a significant interaction between ratio difference and absolute value in predicting numerical discriminability. We find that both behavioral and computational data show an interaction between ratio difference and absolute value on numerical discrimination accuracy. These results further suggest a reexamination of the mechanisms involved in non-symbolic numerical discrimination, how researchers may measure individual performance, and what outcomes performance may predict.

Concepts: Psychology, Mathematics, Sociology, Perception, Number, Weber–Fechner law


It has been proposed that the ability of humans to quickly perceive numerosity involves a visual sense of number. Different paradigms of enumeration and numerosity comparison have produced a gamut of behavioral and neuroimaging data, but there has been no unified conceptual framework that can explain results across the entire range of numerosity. The current work tries to address the ongoing debate concerning whether the same mechanism operates for enumeration of small and large numbers, through a computational approach. We describe the workings of a single-layered, fully connected network characterized by self-excitation and recurrent inhibition that operates at both subitizing and estimation ranges. We show that such a network can account for classic numerical cognition effects (the distance effect, Fechner’s law, Weber fraction for numerosity comparison) through the network steady state activation response across different recurrent inhibition values. The model also accounts for fMRI data previously reported for different enumeration related tasks. The model also allows us to generate an estimate of the pattern of reaction times in enumeration tasks. Overall, these findings suggest that a single network architecture can account for both small and large number processing.

Concepts: Mathematics, Neuroscience, Cognitive science, Perception, Number, Number names, Weber–Fechner law, Gustav Fechner


Models of human perception - including perceptual “laws” - can be valuable tools for deriving visualization design recommendations. However, it is important to assess the explanatory power of such models when using them to inform design. We present a secondary analysis of data previously used to rank the effectiveness of bivariate visualizations for assessing correlation (measured with Pearson’s r) according to the well-known Weber-Fechner Law. Beginning with the model of Harrison et al. [1], we present a sequence of refinements including incorporation of individual differences, log transformation, censored regression, and adoption of Bayesian statistics. Our model incorporates all observations dropped from the original analysis, including data near ceilings caused by the data collection process and entire visualizations dropped due to large numbers of observations worse than chance. This model deviates from Weber’s Law, but provides improved predictive accuracy and generalization. Using Bayesian credibility intervals, we derive a partial ranking that groups visualizations with similar performance, and we give precise estimates of the difference in performance between these groups. We find that compared to other visualizations, scatterplots are unique in combining low variance between individuals and high precision on both positively- and negatively-correlated data. We conclude with a discussion of the value of data sharing and replication, and share implications for modeling similar experimental data.

Concepts: Scientific method, Perception, Difference, Model, Pitch, Philosophy of perception, Weber–Fechner law


In recent research, a systematic association of musical pitch with space has been described in the so-called Spatial-Pitch-Association-of-Response Codes-effect (SPARC). Typically, high pitch is associated with upper/right and low pitch with lower/left space. However, a theoretical classification of these associations regarding their experiential sources is difficult. Therefore, we applied a theoretical framework of numerical cognition classifying similar Space-Associated Response Codes (SARC) effects according to their groundedness, embodiedness and situatedness. We tested these attributes with a group of non-musicians and with a group of highly skilled cello players playing high tones with lower hand positions (i.e., reverse SPARC alignment) in a standard SPARC context of a piano and a reversed SPARC context of a cello. The results showed that SPARC is grounded, in general. However, for cello player SPARC is also situated and embodied. We conclude that groundedness, embodiedness and situatedness provide general characteristics of mapping cognitive representations to space.

Concepts: Cognitive psychology, Pitch, Weber–Fechner law, Musical tuning, Cello, Pitch accent, A440, Fingerboard


Although visual working memory (VWM) has been studied extensively, it is unknown how people form confidence judgments about their memories. Peirce (1878) speculated that Fechner’s law-which states that sensation is proportional to the logarithm of stimulus intensity-might apply to confidence reports. Based on this idea, we hypothesize that humans map the precision of their VWM contents to a confidence rating through Fechner’s law. We incorporate this hypothesis into the best available model of VWM encoding and fit it to data from a delayed-estimation experiment. The model provides an excellent account of human confidence rating distributions as well as the relation between performance and confidence. Moreover, the best-fitting mapping in a model with a highly flexible mapping closely resembles the logarithmic mapping, suggesting that no alternative mapping exists that accounts better for the data than Fechner’s law. We propose a neural implementation of the model and find that this model also fits the behavioral data well. Furthermore, we find that jointly fitting memory errors and confidence ratings boosts the power to distinguish previously proposed VWM encoding models by a factor of 5.99 compared to fitting only memory errors. Finally, we show that Fechner’s law also accounts for metacognitive judgments in a word recognition memory task, which is a first indication that it may be a general law in metacognition. Our work presents the first model to jointly account for errors and confidence ratings in VWM and could lay the groundwork for understanding the computational mechanisms of metacognition. (PsycINFO Database Record

Concepts: Scientific method, Psychology, Memory, Law, Theory, Logarithmic scale, Weber–Fechner law, Accounts receivable


Neurons in the avian nidopallium caudolaterale (NCL), an endbrain structure that originated independently from the mammalian neocortex, process visual numerosities. To clarify the code for number in this anatomically distinct endbrain area in birds, neuronal responses to a broad range of numerosities were analyzed. We recorded single-neuron activity from the NCL of crows performing a delayed match-to-sample task with visual numerosities as discriminanda. The responses of >20% of randomly selected neurons were modulated significantly by numerosities ranging from one to 30 items. Numerosity-selective neurons showed bell-shaped tuning curves with one of the presented numerosities as preferred numerosity regardless of the physical appearance of the items. The resulting labeled-line code exhibited logarithmic compression obeying the Weber-Fechner law for magnitudes. Comparable proportions of selective neurons were found, not only during stimulus presentation, but also in the delay phase, indicating a dominant role of the NCL in numerical working memory. Both during sensory encoding and memorization of numerosities in working memory, NCL activity predicted the crows' number discrimination performance. These neuronal data reveal striking similarities across vertebrate taxa in their code for number despite convergently evolved and anatomically distinct endbrain structures.

Concepts: Nervous system, Brain, Evolution, Action potential, Cerebral cortex, Memory, Dopamine, Weber–Fechner law


We investigated, in a university student population, spontaneous (non-speeded) fast and slow number-to-line mapping responses using non-symbolic (dots) and symbolic (words) stimuli. Seeking for less conventionalized responses, we used anchors 0-130, rather than the standard 0-100. Slow responses to both types of stimuli only produced linear mappings with no evidence of non-linear compression. In contrast, fast responses revealed distinct patterns of non-linear compression for dots and words. A predicted logarithmic compression was observed in fast responses to dots in the 0-130 range, but not in the reduced 0-100 range, indicating compression in proximity of the upper anchor 130, not the standard 100. Moreover, fast responses to words revealed an unexpected significant negative compression in the reduced 0-100 range, but not in the 0-130 range, indicating compression in proximity to the lower anchor 0. Results show that fast responses help revealing the fundamentally distinct nature of symbolic and non-symbolic quantity representation. Whole number words, being intrinsically mediated by cultural phenomena such as language and education, emphasize the invariance of magnitude between them-essential for linear mappings, and therefore, unlike non-symbolic (psychophysical) stimuli, yield spatial mappings that don’t seem to be influenced by the Weber-Fechner law of psychophysics. However, high levels of education (when combined with an absence of standard upper anchors) may lead fast responses to overestimate magnitude invariance on the lower end of word numerals.

Concepts: Mathematics, Education, Student, Logarithmic scale, Anchor, Elementary mathematics, Weber–Fechner law, Compression


It is well-known in numerical cognition that higher numbers are represented with less absolute fidelity than lower numbers, often formalized as a logarithmic mapping. Previous derivations of this psychological law have worked by assuming that relative change in physical magnitude is the key psychologically-relevant measure (Fechner, 1860; Sun et al., 2012; Portugal & Svaiter, Minds and Machines, 21(1), 73-81, 2011). Ideally, however, this property of psychological scales would be derived from more general, independent principles. This paper shows that a logarithmic number line is the one which minimizes the error between input and representation relative to the probability that subjects would need to represent each number. This need probability is measured here through natural language and matches the form of need probabilities found in other literatures. The derivation does not presuppose anything like Weber’s law and makes minimal assumptions about both the nature of internal representations and the form of the mapping. More generally, the results prove in a general setting that the optimal psychological scale will change with the square root of the probability of each input. For stimuli that follow a power-law need distribution this approach recovers either a logarithmic or power-law psychophysical mapping (Stevens, 1957, 1961, 1975).

Concepts: Psychology, Mind, Philosophy, Number, Real number, Elementary mathematics, Weber–Fechner law, Scales