What the Solitaire illusion tells us about perception of numerosity.

In four experiments we investigated the Solitaire illusion. In this illusion, most observers see as more numerous a set of dots that forms a single central group, compared to dots on the outside of that group. We confirmed and extended the effect to configurations with much higher numerosity than the original and of various colours. Contrary to prediction, separating the two groups, so that they are presented side by side, reduced but did not abolish or reverse the illusion. In this illusion, therefore, neither total size of the region (area), not average distance of the elements has the expected effect. In Experiments 3 and 4 we eliminated the regularity of the pattern, by sampling 50% (Exp 3) or only a 10% (Exp 4) of the elements. These produces quasi-random configurations. For these configurations the bias for the inner groups was still present, and it was only eliminated when the groups were shown as separate. However, the effect never reversed (no bias for the outer group, despite its larger area). We conclude that the Solitaire illusion is evidence of a strong bias in favour of centrally located elements, a bias that can overcome other factors.

On the usefulness of graph-theoretic properties in the study of perceived numerosity.

Observers can quickly estimate the quantity of sets of visual elements. Many aspects of this ability have been studied and the underlying system has been called the Approximate Number Sense (Dehaene, 2011). Specific visual properties, such as size and clustering of the elements, can bias an estimate. For intermediate numerical quantities at low density (above five, but before texturization), human performance is predicted by a model based on the region of influence of elements (occupancy model: Allïk & Tuulmets, 1991). For random 2D configurations we computed ten indices based on graph theory, and we compared them with the occupancy model: independence number, domination, connected components, local clustering coefficient, global clustering coefficient, random walk, eigenvector centrality, maximum clique, total degree of connectivity, and total edge length. We made comparisons across a range of parameters, and we varied the size of the region of influence around each element. The analysis of the pattern of correlations suggests two main groups of graph-based measures. The first group is sensitive to the presence of local clustering of elements, the second seems more sensitive to density and the way information spreads in graphs. Empirical work on perception of numerosity may benefit from comparing, or controlling for, these properties.

The effect of clustering on perceived quantity in humans (Homo sapiens) and in chicks (Gallus gallus).

Animals can perceive the numerosity of sets of visual elements. Qualitative and quantitative similarities in different species suggest the existence of a shared system (approximate number system). Biases associated with sensory properties are informative about the underlying mechanisms. In humans, regular spacing increases perceived numerosity (regular-random numerosity illusion). This has led to a model that predicts numerosity based on occupancy (a measure that decreases when elements are close together). We used a procedure in which observers selected one of two stimuli and were given feedback with respect to whether the choice was correct. One configuration had 20 elements and the other 40, randomly placed inside a circular region. Participants had to discover the rule based on feedback. Because density and clustering covaried with numerosity, different dimensions could be used. After reaching a criterion, test trials presented two types of configurations with 30 elements. One type had a larger interelement distance than the other (high or low clustering). If observers had adopted a numerosity strategy, they would choose low clustering (if reinforced with 40) and high clustering (if reinforced with 20). A clustering or density strategy predicts the opposite. Human adults used a numerosity strategy. Chicks were tested using a similar procedure. There were two behavioral measures: first approach response and final circumnavigation (walking behind the screen). The prediction based on numerosity was confirmed by the first approach data. For chicks, one clear pattern from both responses was a preference for the configurations with higher clustering. (PsycINFO Database Record

An Electrophysiological Index of Perceptual Goodness.

A traditional line of work starting with the Gestalt school has shown that patterns vary in strength and salience; a difference in "Perceptual goodness." The Holographic weight of evidence model quantifies goodness of visual regularities. The key formula states that W = E/N, where E is number of holographic identities in a pattern and N is number of elements. We tested whether W predicts the amplitude of the neural response to regularity in an extrastriate symmetry-sensitive network. We recorded an Event Related Potential (ERP) generated by symmetry called the Sustained Posterior Negativity (SPN). First, we reanalyzed the published work and found that W explained most variance in SPN amplitude. Then in four new studies, we confirmed specific predictions of the holographic model regarding 1) the differential effects of numerosity on reflection and repetition, 2) the similarity between reflection and Glass patterns, 3) multiple symmetries, and 4) symmetry and anti-symmetry. In all cases, the holographic approach predicted SPN amplitude remarkably well; particularly in an early window around 300-400 ms post stimulus onset. Although the holographic model was not conceived as a model of neural processing, it captures many details of the brain response to symmetry.

A quantum theory for the irreplaceable role of docosahexaenoic acid in neural cell signalling throughout evolution.

Six hundred million years ago, the fossil record displays the sudden appearance of intracellular detail and the 32 phyla. The "Cambrian Explosion" marks the onset of dominant aerobic life. Fossil intracellular structures are so similar to extant organisms that they were likely made with similar membrane lipids and proteins, which together provided for organisation and specialisation. While amino acids could be synthesised over 4 billion years ago, only oxidative metabolism allows for the synthesis of highly unsaturated fatty acids, thus producing novel lipid molecular species for specialised cell membranes. Docosahexaenoic acid (DHA) provided the core for the development of the photoreceptor, and conversion of photons into electricity stimulated the evolution of the nervous system and brain. Since then, DHA has been conserved as the principle acyl component of photoreceptor synaptic and neuronal signalling membranes in the cephalopods, fish, amphibian, reptiles, birds, mammals and humans. This extreme conservation in electrical signalling membranes despite great genomic change suggests it was DHA dictating to DNA rather than the generally accepted other way around. We offer a theoretical explanation based on the quantum mechanical properties of DHA for such extreme conservation. The unique molecular structure of DHA allows for quantum transfer and communication of π-electrons, which explains the precise depolarisation of retinal membranes and the cohesive, organised neural signalling which characterises higher intelligence.