Perceptual axioms are irreconcilable with Euclidean geometry.

There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so-called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other.

A clinico-anatomical dissection of the magnocellular and parvocellular pathways in a patient with the Riddoch syndrome.

KEY MESSAGE: The Riddoch syndrome is thought to be caused by damage to the primary visual cortex (V1), usually following a vascular event. This study shows that damage to the anatomical input to V1, i.e., the optic radiations, can result in selective visual deficits that mimic the Riddoch syndrome. The results also highlight the differential susceptibility of the magnocellular and parvocellular visual systems to injury. Overall, this study offers new insights that will improve our understanding of the impact of brain injury and neurosurgery on the visual pathways. The Riddoch syndrome, characterised by the ability to perceive, consciously, moving visual stimuli but not static ones, has been associated with lesions of primary visual cortex (V1). We present here the case of patient YL who, after a tumour resection surgery that spared his V1, nevertheless showed symptoms of the Riddoch syndrome. Based on our testing, we postulated that the magnocellular (M) and parvocellular (P) inputs to his V1 may be differentially affected. In a first experiment, YL was presented with static and moving checkerboards in his blind field while undergoing multimodal magnetic resonance imaging (MRI), including structural, functional, and diffusion, acquired at 3 T. In a second experiment, we assessed YL's neural responses to M and P visual stimuli using psychophysics and high-resolution fMRI acquired at 7 T. YL's optic radiations were partially damaged but not severed. We found extensive activity in his visual cortex for moving, but not static, visual stimuli, while our psychophysical tests revealed that only low-spatial frequency moving checkerboards were perceived. High-resolution fMRI revealed strong responses in YL's V1 to M stimuli and very weak ones to P stimuli, indicating a functional P lesion affecting V1. In addition, YL frequently reported seeing moving stimuli and discriminating their direction of motion in the absence of visual stimulation, suggesting that he was experiencing visual hallucinations. Overall, this study highlights the possibility of a selective loss of P inputs to V1 resulting in the Riddoch syndrome and in hallucinations of visual motion.

Neural patterns of conscious visual awareness in the Riddoch syndrome.

The Riddoch syndrome is one in which patients blinded by lesions to their primary visual cortex can consciously perceive visual motion in their blind field, an ability that correlates with activity in motion area V5. Our assessment of the characteristics of this syndrome in patient ST, using multimodal MRI, showed that: 1. ST's V5 is intact, receives direct subcortical input, and decodable neural patterns emerge in it only during the conscious perception of visual motion; 2. moving stimuli activate medial visual areas but, unless associated with decodable V5 activity, they remain unperceived; 3. ST's high confidence ratings when discriminating motion at chance levels, is associated with inferior frontal gyrus activity. Finally, we report that ST's Riddoch Syndrome results in hallucinatory motion with hippocampal activity as a correlate. Our results shed new light on perceptual experiences associated with this syndrome and on the neural determinants of conscious visual experience.

The neural determinants of abstract beauty.

We have enquired into the neural activity which correlates with the experience of beauty aroused by abstract paintings consisting of arbitrary assemblies of lines and colours. During the brain imaging experiments, subjects rated abstract paintings according to aesthetic appeal. There was low agreement on the aesthetic classification of these paintings among participants. Univariate analyses revealed higher activity with higher declared aesthetic appeal in both the visual areas and the medial frontal cortex. Additionally, representational similarity analysis (RSA) revealed that the experience of beauty correlated with decodable patterns of activity in visual areas. These results are broadly similar to those obtained in previous studies on facial beauty. With abstract art, it was the involvement of visual areas implicated in the processing of lines and colours while with faces it was of visual areas implicated in the processing of faces. Both categories of aesthetic experience correlated with increased activity in medial frontal cortex. We conclude that the sensory areas participate in the selection of stimuli according to aesthetic appeal and that it is the co-operative activity between the sensory areas and the medial frontal cortex that is the basis for the experience of abstract visual beauty. Further, this co-operation is enabled by "experience dependent" functional connections, in the sense that currently the existence and high specificity of these connections can only be demonstrated during certain experiences.

The Paton prize lecture 2021: A colourful experience leading to a reassessment of colour vision and its theories.

In this lecture, given in honour of Sir William Paton, a brilliant scientist and one of Britain's great patrons of biology, I give a personal account of the fundamental issues in colour vision that I have tackled since 1973, when I discovered a cortical zone lying outside the primary visual cortex that is rich in cells with chromatic properties. I do not provide an exhaustive review of colour vision but summarise how my views on colour vision and theories surrounding it have changed in light of that discovery.

Human faces and face-like stimuli are more memorable.

We have previously suggested a distinction in the brain processes governing biological and artifactual stimuli. One of the best examples of the biological category consists of human faces, the perception of which appears to be determined by inherited mechanisms or ones rapidly acquired after birth. In extending this work, we inquire here whether there is a higher memorability for images of human faces and whether memorability declines with increasing departure from human faces; if so, the implication would add to the growing evidence of differences in the processing of biological versus artifactual stimuli. To do so, we used images and memorability scores from a large data set of 58,741 images to compare the relative memorability of the following image categories: real human faces versus buildings, and extending this to a comparison of real human faces with five image categories that differ in their grade of resemblance to a real human face. Our findings show that, in general, when we compare the biological category of faces to the artifactual category of buildings, the former is more memorable. Furthermore, there is a gradient in which the more an image resembles a real human face the more memorable it is. Thus, the previously identified differences in biological and artifactual images extend to the field of memory.

Judgments of mathematical beauty are resistant to revision through external opinion.

We here address the question of the extent to which judgments of mathematical beauty (which we categorize as biological beauty) are resistant to revision through external opinion. A total of 100 mathematicians of different national and ethnic origins were asked to rate 60 mathematical equations for their beauty; after being presented a fictitious "expert rating," they were asked to re-rate the same equations. Results showed that the judgments of mathematical beauty had a high level of resistance to external opinion. This is in line with the resistance to revision of a judgments for other categories of biological beauty.

On the necessity of importing neurobiology into mathematics.

In Gödel's Incompleteness Theorems, for every mathematical system there are correct statements that cannot be proven to be correct within that system. We here extend this to address the question of axiomatic statements that are perceived (or known) to be correct but which mathematics, as presently constituted, cannot prove. We refer to these as perceptual axioms.

The power of external influences to modify judgments of facial and moral beauty.

Empirical evidence shows that the often-made positive correlation between human physical and moral beauty is tenuous. In this study, we aimed to learn whether facial and moral beauty can be psychophysically separated. Participants (n = 95) provided beauty and goodness (i.e., trustworthiness) ratings for pictures of faces, after which they were presented with a fictitious peer rating for the same face and asked to re-rate the face. We used the difference between the initial and final ratings to quantify the degree of resistance to external influence. We found that judgments of facial beauty were more resistant to external influence than judgments of facial "goodness"; in addition, there was significantly higher agreement within beauty ratings than within goodness ratings. These findings are discussed in light of our Bayesian-Laplacian classification of priors, from which we conclude that moral beauty relies more upon acquired "artifactual" priors and facial beauty more on inherited biological priors.

The neural determinants of beauty.

The perception of faces correlates with activity in a number of brain areas, but only when a face is perceived as beautiful is the medial orbitofrontal cortex (mOFC) also engaged. Here, we enquire whether it is the emergence of a particular pattern of neural activity in face perceptive areas during the experience of a face as beautiful that determines whether there is, as a correlate, activity in mOFC. Seventeen subjects of both genders viewed and rated facial stimuli according to how beautiful they perceived them to be while the activity in their brains was imaged with functional magnetic resonance imaging. A univariate analysis revealed parametrically scaled activity within several areas, including the occipital face area (OFA), fusiform face area (FFA) and the cuneus; the strength of activity in these areas correlated with the declared intensity of the aesthetic experience of faces; multivariate analyses showed strong patterns of activation in the FFA and the cuneus and weaker patterns in the OFA and the posterior superior temporal sulcus (pSTS). The mOFC was only engaged when specific patterns of activity emerged in these areas. A psychophysiological interaction analysis with mOFC as the seed area revealed the involvement of the right FFA and the right OFA. We conjecture that it is the collective specific pattern-based activity in these face perceptive areas, with activity in the mOFC as a correlate, that constitutes the neural basis for the experience of facial beauty, bringing us a step closer to understanding the neural determinants of aesthetic experience.

The striate cortex and hemianopia.

We begin with the functions of the striate cortex (area V1 of the visual cortex) and end with a review of the effects of damage to striate cortex or its inputs; namely, homonymous hemifield defects. Clinical and anatomical studies accrued over the past 25 years have modified our understanding of the role of V1 in vision. We discuss the evidence that V1 is not the sole recipient of visual signals; is not the earliest recipient of visual signals; and is not essential for conscious vision. In the second section, we give a brief history of how the visual field was found to be represented in striate cortex, then cover the work that has demonstrated the overrepresentation of the central region of vision in humans. The common patterns of visual field disturbance caused by damage to the retrochiasmal visual system are discussed, with some less common examples shown as brief case studies.

The differential power of extraneous influences to modify aesthetic judgments of biological and artifactual stimuli.

We addressed the question of the extent to which external information is capable of modifying aesthetic ratings given to two different categories of stimuli-images of faces (which belong to the biological category) and those of abstract paintings with no recognizable objects (which sit in the artifactual category). A total of 51 participants of different national origins rated the beauty of both sets of stimuli, indicating the certainty of their rating; they then re-rated them after being exposed to the opinions of others on their aesthetic status. Of these 51 participants, 42 who met our criteria were selected to complete the experiment. The results showed that individuals were less prone to modifying their ratings of stimuli belonging to the biological category compared to those falling into the artifactual category. We discuss this finding in light of our theoretical Bayesian-Laplacian model and on the evidence given by previous empirical research.

"Multiplexing" cells of the visual cortex and the timing enigma of the binding problem.

In this opinion essay, I address the perennial binding problem, that is to say of how independently processed visual attributes such as form, colour and motion are brought together to give us a unified and holistic picture of the visual world. A solution to this central issue in neurobiology remains as elusive as ever. No one knows today how it is implemented. The issue is not a new one and, though discussed most commonly in the context of the visual brain, it is not unique to it either. Karl Lashley summarized it well years ago when he wrote that a critical problem for brain studies is to understand how "the specialized areas of the cerebral cortex interact to provide the integration evident in thought and behaviour" (Lashley, 1931).

The Bayesian-Laplacian brain.

We outline what we believe could be an improvement in future discussions of the brain acting as a Bayesian-Laplacian system. We do so by distinguishing between two broad classes of priors on which the brain's inferential systems operate: in one category are biological priors (ß priors) and in the other artefactual ones (α priors). We argue that ß priors, of which colour categories and faces are good examples, are inherited or acquired very rapidly after birth, are highly or relatively resistant to change through experience, and are common to all humans. The consequence is that the probability of posteriors generated from ß priors having universal assent and agreement is high. By contrast, α priors, of which man-made objects are examples, are acquired post-natally and modified at various stages throughout post-natal life; they are much more accommodating of, and hospitable to, new experiences. Consequently, posteriors generated from them are less likely to find universal assent. Taken together, in addition to the more limited capacity of experiment and experience to alter the ß priors compared with α priors, another cardinal distinction between the two is that the probability of posteriors generated from ß priors having universal agreement is greater than that for α priors. The two categories are distinct at the extremes; there is, however, a middle range where they merge into one another to varying extents, resulting in posteriors that draw upon both categories.

The biological basis of the experience and categorization of colour.

We used the Land Colour Mondrian experiments in a Bayesian context to test the degree to which subjects vary in categorizing the colour of different patches, when each patch is made to reflect light of the identical wavelength-energy composition. The brain uses a ratio-taking mechanism to determine the ratio of light of every waveband reflected from a surface and from its surrounds. Our (Bayesian) hypothesis was that this ratio-taking mechanism is similar in all humans and therefore leads to a constant categorization of colours that differs little between them. The similarly categorized colours are the initial priors, with initial hues attached to them. Twenty subjects of different ethnic and cultural backgrounds, for all but one of whom English was not the primary language, viewed eight patches of different colour in two Mondrian displays; each patch, when viewed, was made to reflect identical ratios of long-, middle- and short-wave light. Subjects were asked to match the colour of the viewed patch with that of the Munsell chip coming closest in colour to that of the viewed patch, without using language. In terms of hue, there was less variability in matching warm hues than cool ones. In terms of colour categorization, there was little variability overall. We take the lack of significant variability between subjects in the matches made as a pointer to similar computational mechanisms being employed in different subjects to perceive colours, thus permitting them to assume that their categorization of colours has universal agreement and assent.

Corrigendum: The Biological Basis of Mathematical Beauty.

[This corrects the article DOI: 10.3389/fnhum.2018.00467.].

The Biological Basis of Mathematical Beauty.

Our past studies have led us to divide sensory experiences, including aesthetic ones derived from sensory sources, into two broad categories: biological and artifactual. The aesthetic experience of biological beauty is dictated by inherited brain concepts, which are resistant to change even in spite of extensive experience. The experience of artifactual beauty on the other hand is determined by post-natally acquired concepts, which are modifiable throughout life by exposure to different experiences (Zeki, 2009; Zeki and Chén, 2016). Hence, in terms of aesthetic rating, biological beauty (in which we include the experience of beautiful faces or human bodies) is characterized by less variability between individuals belonging to different ethnic origins and cultural backgrounds or the same individual at different times. Artifactual beauty (in which we include the aesthetic experience of human artifacts, such as buildings and cars) is characterized by greater variability between individuals belonging to different ethnic and cultural groupings and by the same individual at different times. In this paper, we present results to show that the experience of mathematical beauty (Zeki et al., 2014), even though it constitutes an extreme example of beauty that is dependent upon (mathematical) culture and learning, is consistent with one of the characteristics of the biological categories, namely a lesser variability in terms of the aesthetic ratings given to mathematical formulae experienced as beautiful.

The Rough Seas of Cortical Cartography.

Efforts to parcellate the cortex into areas based on fine-scale anatomical signatures (e.g., cytoarchitectonics) date back to the early years of the last century. In the ensuing decades, rapidly growing knowledge of cortical connections encouraged neurobiologists to search for connectivity-based principles underlying the organization of the cerebral cortex. Using such an approach, Felleman and Van Essen presented, in their 1991 paper, a connectivity-based hierarchical principle dictating the relationship between cortical areas. This helped invigorate debates on the principles underlying cortical organization, including searches for alternative, or complementary, principles.