Developmental change in numerical estimation

Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a “representational shift.” This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as number-line estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude. The present study assessed these 2 theories\u27 abilities to account for the development of numerical estimation in 5- through 10-year-olds. The proportional account explained estimation patterns better than the logarithmic-to-linear-shift account for all age groups, at both group and individual levels. These findings contribute to our understanding of the nature and development of the mental representation of number and have more general implications for theories of cognitive developmental change

How numbers mean : Comparing random walk models of numerical cognition varying both encoding processes and underlying quantity representations

How do people derive meaning from numbers? Here, we instantiate the primary theories of numerical representation in computational models and compare simulated performance to human data. Specifically, we fit simulated data to the distributions for correct and incorrect responses, as well as the pattern of errors made, in a traditional “relative quantity” task. The results reveal that no current theory of numerical representation can adequately account for the data without additional assumptions. However, when we introduce repeated, error-prone sampling of the stimulus (e.g., Cohen, 2009) superior fits are achieved when the underlying representation of integers reflects linear spacing with constant variance. These results provide new insights into (i) the detailed nature of mental numerical representation, and, (ii) general perceptual processes implemented by the human visual system

A Physiologically-Inspired Model of Numerical Classification Based on Graded Stimulus Coding

In most natural decision contexts, the process of selecting among competing actions takes place in the presence of informative, but potentially ambiguous, stimuli. Decisions about magnitudes – quantities like time, length, and brightness that are linearly ordered – constitute an important subclass of such decisions. It has long been known that perceptual judgments about such quantities obey Weber's Law, wherein the just-noticeable difference in a magnitude is proportional to the magnitude itself. Current physiologically inspired models of numerical classification assume discriminations are made via a labeled line code of neurons selectively tuned for numerosity, a pattern observed in the firing rates of neurons in the ventral intraparietal area (VIP) of the macaque. By contrast, neurons in the contiguous lateral intraparietal area (LIP) signal numerosity in a graded fashion, suggesting the possibility that numerical classification could be achieved in the absence of neurons tuned for number. Here, we consider the performance of a decision model based on this analog coding scheme in a paradigmatic discrimination task – numerosity bisection. We demonstrate that a basic two-neuron classifier model, derived from experimentally measured monotonic responses of LIP neurons, is sufficient to reproduce the numerosity bisection behavior of monkeys, and that the threshold of the classifier can be set by reward maximization via a simple learning rule. In addition, our model predicts deviations from Weber Law scaling of choice behavior at high numerosity. Together, these results suggest both a generic neuronal framework for magnitude-based decisions and a role for reward contingency in the classification of such stimuli

A Neural Model of How the Brain Represents and Compares Multi-Digit Numbers: Spatial and Categorical Processes

Both animals and humans are capable of representing and comparing numerical quantities, but only humans seem to have evolved multi-digit place-value number systems. This article develops a neural model, called the Spatial Number Network, or SpaN model, which predicts how these shared numerical capabilities are computed using a spatial representation of number quantities in the Where cortical processing stream, notably the Inferior Parietal Cortex. Multi-digit numerical representations that obey a place-value principle are proposed to arise through learned interactions between categorical language representations in the What cortical processing stream and the Where spatial representation. It is proposed that learned semantic categories that symbolize separate digits, as well as place markers like "tens," "hundreds," "thousands," etc., are associated through learning with the corresponding spatial locations of the Where representation, leading to a place-value number system as an emergent property of What-Where information fusion. The model quantitatively simulates error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. In the Where cortical process, it is proposed that transient responses to inputs are integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence. Neural mechanisms are defined which give rise to an ordered spatial numerical map ordering and Weber law characteristics as emergent properties. The dynamics of numerical comparison are encoded in activity pattern changes within this spatial map. Such changes cause a "directional comparison wave" whose properties mimic data about numerical comparison. These model mechanisms are variants of neural mechanisms that have elsewhere been used to explain data about motion perception, attention shifts, and target tracking. Thus, the present model suggests how numerical representations may have emerged as specializations of more primitive mechanisms in the cortical Where processing stream. The model's What-Where interactions can explain human psychophysical data, such as error rates and reaction times, about multi-digit (base 10) numerical stimuli, and describe how such a competence can develop through learning. The SpaN model and its explanatory range arc compared with other models of numerical representation.Defense Advanced Research Projects Agency and the Office of Naval Research (N00014-95-1-0409); National Science Foundation (IRI-97-20333

A Neural Model of How The Brain Represents and Compares Numbers

Many psychophysical experiments have shown that the representation of numbers and numerical quantities in humans and animals is related to number magnitude. A neural network model is proposed to quantitatively simulate error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. Transient responses to inputs arc integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence. The dynamics of numerical comparison are encoded in activity pattern changes within this spatial map. Such changes cause a "directional comparison wave" whose properties mimic data about numerical comparison. These model mechanisms are variants of neural mechanisms that have elsewhere been used to explain data about motion perception, attention shifts, and target tracking. Thus, the present model suggests how numerical representations may have emerged as specializations of more primitive mechanisms in the cortical Where processing stream.National Science Foundation (IRI-97-20333); Defense Advanced research Projects Agency and the Office of Naval Research (N00014-95-1-0409); National Institute of Health (1-R29-DC02952-01

A simple derivation and classification of common probability distributions based on information symmetry and measurement scale

Commonly observed patterns typically follow a few distinct families of probability distributions. Over one hundred years ago, Karl Pearson provided a systematic derivation and classification of the common continuous distributions. His approach was phenomenological: a differential equation that generated common distributions without any underlying conceptual basis for why common distributions have particular forms and what explains the familial relations. Pearson's system and its descendants remain the most popular systematic classification of probability distributions. Here, we unify the disparate forms of common distributions into a single system based on two meaningful and justifiable propositions. First, distributions follow maximum entropy subject to constraints, where maximum entropy is equivalent to minimum information. Second, different problems associate magnitude to information in different ways, an association we describe in terms of the relation between information invariance and measurement scale. Our framework relates the different continuous probability distributions through the variations in measurement scale that change each family of maximum entropy distributions into a distinct family.Comment: 17 pages, 0 figure

Dwarf Galaxy Dark Matter Density Profiles Inferred from Stellar and Gas Kinematics

We present new constraints on the density profiles of dark matter (DM) halos in seven nearby dwarf galaxies from measurements of their integrated stellar light and gas kinematics. The gas kinematics of low mass galaxies frequently suggest that they contain constant density DM cores, while N-body simulations instead predict a cuspy profile. We present a data set of high resolution integral field spectroscopy on seven galaxies and measure the stellar and gas kinematics simultaneously. Using Jeans modeling on our full sample, we examine whether gas kinematics in general produce shallower density profiles than are derived from the stars. Although 2/7 galaxies show some localized differences in their rotation curves between the two tracers, estimates of the central logarithmic slope of the DM density profile, gamma, are generally robust. The mean and standard deviation of the logarithmic slope for the population are gamma=0.67+/-0.10 when measured in the stars and gamma=0.58+/-0.24 when measured in the gas. We also find that the halos are not under concentrated at the radii of half their maximum velocities. Finally, we search for correlations of the DM density profile with stellar velocity anisotropy and other baryonic properties. Two popular mechanisms to explain cored DM halos are an exotic DM component or feedback models that strongly couple the energy of supernovae into repeatedly driving out gas and dynamically heating the DM halos. We investigate correlations that may eventually be used to test models. We do not find a secondary parameter that strongly correlates with the central DM density slope, but we do find some weak correlations. Determining the importance of these correlations will require further model developments and larger observational samples. (Abridged)Comment: 29 pages, 18 figures, 10 tables, accepted for publication in Ap