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1.  Sensitivity to numerosity is not a unique visuospatial psychophysical predictor of mathematical ability☆ 
Vision Research  2013;89(100):1-9.
•Individual differences in numerosity acuity predict mathematical ability.•We tested 300+ participants to see if this relationship is unique to numerosity.•Visual numerosity and orientation task performance predicted mathematics scores.•Performance improved with age, and males significantly outperformed females.•This highlights links between mathematics and multiple visuospatial abilities.
Sensitivity to visual numerosity has previously been shown to predict human mathematical performance. However, it is not clear whether it is discrimination of numerosity per se that is predictive of mathematical ability, or whether the association is driven by more general task demands. To test this notion we had over 300 participants (ranging in age from 6 to 73 years) perform a symbolic mathematics test and 4 different visuospatial matching tasks. The visual tasks involved matching 2 clusters of Gabor elements for their numerosity, density, size or orientation by a method of adjustment. Partial correlation and regression analyses showed that sensitivity to visual numerosity, sensitivity to visual orientation and mathematical education level predict a significant proportion of shared as well as unique variance in mathematics scores. These findings suggest that sensitivity to visual numerosity is not a unique visual psychophysical predictor of mathematical ability. Instead, the data are consistent with mathematics representing a multi-factorial process that shares resources with a number of visuospatial tasks.
PMCID: PMC3748346  PMID: 23820087
Number; Density; Size; Orientation; Mathematics; Spatial vision; IPS, intraparietal sulcus
2.  The Brightness of Colour 
PLoS ONE  2009;4(3):e5091.
The perception of brightness depends on spatial context: the same stimulus can appear light or dark depending on what surrounds it. A less well-known but equally important contextual phenomenon is that the colour of a stimulus can also alter its brightness. Specifically, stimuli that are more saturated (i.e. purer in colour) appear brighter than stimuli that are less saturated at the same luminance. Similarly, stimuli that are red or blue appear brighter than equiluminant yellow and green stimuli. This non-linear relationship between stimulus intensity and brightness, called the Helmholtz-Kohlrausch (HK) effect, was first described in the nineteenth century but has never been explained. Here, we take advantage of the relative simplicity of this ‘illusion’ to explain it and contextual effects more generally, by using a simple Bayesian ideal observer model of the human visual ecology. We also use fMRI brain scans to identify the neural correlates of brightness without changing the spatial context of the stimulus, which has complicated the interpretation of related fMRI studies.
Rather than modelling human vision directly, we use a Bayesian ideal observer to model human visual ecology. We show that the HK effect is a result of encoding the non-linear statistical relationship between retinal images and natural scenes that would have been experienced by the human visual system in the past. We further show that the complexity of this relationship is due to the response functions of the cone photoreceptors, which themselves are thought to represent an efficient solution to encoding the statistics of images. Finally, we show that the locus of the response to the relationship between images and scenes lies in the primary visual cortex (V1), if not earlier in the visual system, since the brightness of colours (as opposed to their luminance) accords with activity in V1 as measured with fMRI.
The data suggest that perceptions of brightness represent a robust visual response to the likely sources of stimuli, as determined, in this instance, by the known statistical relationship between scenes and their retinal responses. While the responses of the early visual system (receptors in this case) may represent specifically the statistics of images, post receptor responses are more likely represent the statistical relationship between images and scenes. A corollary of this suggestion is that the visual cortex is adapted to relate the retinal image to behaviour given the statistics of its past interactions with the sources of retinal images: the visual cortex is adapted to the signals it receives from the eyes, and not directly to the world beyond.
PMCID: PMC2659800  PMID: 19333398
3.  Floral Morphogenesis: Stochastic Explorations of a Gene Network Epigenetic Landscape 
PLoS ONE  2008;3(11):e3626.
In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5–10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.
PMCID: PMC2572848  PMID: 18978941
4.  What Are Lightness Illusions and Why Do We See Them? 
PLoS Computational Biology  2007;3(9):e180.
Lightness illusions are fundamental to human perception, and yet why we see them is still the focus of much research. Here we address the question by modelling not human physiology or perception directly as is typically the case but our natural visual world and the need for robust behaviour. Artificial neural networks were trained to predict the reflectance of surfaces in a synthetic ecology consisting of 3-D “dead-leaves” scenes under non-uniform illumination. The networks learned to solve this task accurately and robustly given only ambiguous sense data. In addition—and as a direct consequence of their experience—the networks also made systematic “errors” in their behaviour commensurate with human illusions, which includes brightness contrast and assimilation—although assimilation (specifically White's illusion) only emerged when the virtual ecology included 3-D, as opposed to 2-D scenes. Subtle variations in these illusions, also found in human perception, were observed, such as the asymmetry of brightness contrast. These data suggest that “illusions” arise in humans because (i) natural stimuli are ambiguous, and (ii) this ambiguity is resolved empirically by encoding the statistical relationship between images and scenes in past visual experience. Since resolving stimulus ambiguity is a challenge faced by all visual systems, a corollary of these findings is that human illusions must be experienced by all visual animals regardless of their particular neural machinery. The data also provide a more formal definition of illusion: the condition in which the true source of a stimulus differs from what is its most likely (and thus perceived) source. As such, illusions are not fundamentally different from non-illusory percepts, all being direct manifestations of the statistical relationship between images and scenes.
Author Summary
Sometimes the best way to understand how the visual brain works is to understand why it sometimes does not. Thus, visual illusions have been central to the science and philosophy of human consciousness for decades. Here we explain the root cause of brightness illusions, not by modelling human perception or its assumed physiological substrate (as is more typically done), but by modelling the basic challenge that all visual animals must resolve if they are to survive: the inherent ambiguity of sensory data. We do this by training synthetic neural networks to recognise surfaces under different lights in scenes with naturalistic structure. The result is that the networks not only solve this task robustly (i.e., they exhibit “lightness constancy”), they also—as a consequence—exhibit the same illusions of lightness that humans also see. In short, these synthetic systems not only get it right like we do, but also get it wrong like we do, too. This emergent coincidence strongly provides causal evidence that illusions (and by extension all percepts) represent the probable source of images in past visual experience, which has fundamental consequences for explaining how and why we see what we do. The study also suggests the first formal definition of what an illusion is: The condition in which the actual source of a stimulus differs from its most likely source.
PMCID: PMC1994982  PMID: 17907795

Results 1-4 (4)