#geometriccognition

[geometrymatters]

The Neurogeometry of Perception: A Journey into Geometric Cognition

In the realm of cognitive science and neurology, there exists a fascinating intersection where geometry meets perception, aptly termed “neurogeometry”. This interdisciplinary field seeks to understand how our brains process and interpret the visual world through geometric structures and patterns. Alessandro Sarti and Giovanna Citti, prominent figures in this domain, have extensively explored the fundamental principles of neurogeometry, uncovering the intricate relationship between the architecture of our brains and the geometric forms we perceive.

“Neurogeometry” is not merely a fusion of “neuroscience” and “geometry”. It’s an ambitious endeavor to model the functional architecture of the primary visual cortex and understand how geometric patterns underpin our visual processing. As described by Sarti and Citti,

“We remind some basic principles of the neurogeometrical approach as it has been proposed by various researchers to model the functional architecture of the primary visual cortex.”

This statement underscores the comprehensive nature of the approach and its foundational importance in cognitive science. The very essence of neurogeometry lies in its quest to unravel the architectural blueprint of our perceptual processes. Our brains, complex and intricate, are not just passive receivers of visual stimuli. Instead, they actively construct a coherent understanding of the world through geometric frameworks. Every curve we perceive, every angle we discern, and every spatial relationship we recognize is a testament to the brain’s inherent ability to process the world geometrically. Neurogeometry, therefore, serves as a bridge, connecting the abstract realm of geometric shapes and patterns to the tangible reality of neural processes.

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[geometrymatters]

Tatyana O. Sharpee’s work in the realm of geometric cognition—a field dedicated to understanding how the brain interprets and represents spatial information—has yielded groundbreaking insights into how our minds map the world around us. Continuing her previous argument for hyperbolic geometry in neural circuits, the recent paper, co-authored with Huanqiu Zhang, P. Dylan Rich, and Albert K. Lee, sheds light on the hyperbolic geometry of hippocampal spatial representations.

The hippocampus, a crucial part of the brain involved in memory and spatial navigation, houses ‘place cells’—neurons that fire when an animal is in a particular location. The geometry of these spatial representations, however, has remained largely unknown. Breaking new ground, Sharpee and her team reveal that the hippocampus does not represent space according to a linear geometry, as might be expected. Instead, they discovered a hyperbolic representation.

We investigated whether hyperbolic geometry underlies neural networks by analyzing the responses of sets of neurons from the dorsal CA1 region of the hippocampus. This region is considered essential for spatial representation and trajectory planning.

Imagine holding a map of your city. A linear representation would be akin to the map’s scale, where an inch on paper corresponds to a fixed number of miles in reality. A hyperbolic representation, in contrast, changes the scale depending on where you are on the map. The implications of this discovery are profound. A hyperbolic representation provides more positional information than a linear one, potentially aiding in complex navigation tasks.

This hyperbolic representation isn’t static—it dynamically expands with experience. As an animal spends more time exploring its environment, the spatial representations in its brain expand. The expansion is proportional to the logarithm of time spent exploring, suggesting our brains continually refine our spatial maps based on our experiences. As we spend more time in an area, our mental map of that area becomes more detailed and expansive. This dynamic updating could be crucial for efficiently navigating familiar environments.

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[geometrymatters]

The human brain is a complex organ responsible for various cognitive functions. Scientists from the University of Sydney and Fudan University have made a significant discovery regarding brain signals that traverse the outer layer of neural tissue and form spiral patterns. These spirals, observed during both resting and cognitive states, have been found to play a crucial role in organizing brain activity and cognitive processing.

The research study, published in Nature Human Behaviour, focuses on the identification and analysis of spiral-shaped brain signals and their implications for understanding brain dynamics and functions. The study utilized functional magnetic resonance imaging (fMRI) brain scans of 100 young adults to observe and analyze these brain signals. By adapting methods used in understanding complex wave patterns in turbulence, the researchers successfully identified and characterized the spiral patterns observed on the cortex.

Our study suggests that gaining insights into how the spirals are related to cognitive processing could significantly enhance our understanding of the dynamics and functions of the brain. —Associate Professor Pulin Gong

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[geometrymatters]

In the growing field of geometric cognition, one name stands out due to her significant contributions: Elizabeth Spelke. A Professor of Psychology at Harvard University, Spelke is known for her pioneering work on the cognitive development of infants and children. Her research has provided fundamental insights into how young minds perceive and understand the world around them.

One of Spelke’s most influential research areas involves understanding how human beings, from infancy, comprehend the geometric properties of the world. This work has been instrumental in shaping the field of geometric cognition, a branch of cognitive science that studies how people understand, perceive, and reason about geometry and space.

Spelke and her collaborators have proposed a fascinating theory: that humans possess innate geometric abilities that emerge early in infancy and form the foundation of our understanding of the physical world. This suggests that geometry is not merely a mathematical concept taught in school but a universal mental construction inherent to the human brain.

In the “Geometry as a Universal Mental Construction,” chapter (from “Stanislas Dehaene & Elizabeth Brannon, Space, Time and Number in the Brain ) co-authored with Véronique Izard, Pierre Pica, Stanislas Dehaene, and Danielle Hinchey, Spelke delves deeper into this theory. They present compelling evidence supporting their idea, drawn from studies conducted across different cultures and age groups.

The team begins their exploration by discussing the broad concept of geometry and its universal presence. They state, “The study of geometry concerns the properties of space, as revealed through the shapes and relative positions of objects”. This statement indicates that geometry is not limited to academic contexts but is deeply interwoven with our day-to-day perception of the world.

The researchers argue that humans and many animal species share an intuitive understanding of basic geometric principles. This understanding guides their interactions with the environment and aids in navigation. Furthermore, this geometric intuition seems to be innate, surfacing in infancy long before formal education begins.

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[geometrymatters]

For over a century, wave equations of physical systems have played a pivotal role in understanding diverse phenomena, ranging from Schrödinger’s model of the atom to the neural field theory of the brain. These mathematical representations predict that the expressed energy of a system is confined into natural modes or ‘eigenstates’, largely determined by the system’s geometry. The implication of these eigenstates transcends the realm of atomic structures, providing a theoretical framework that finds resonance in the dynamics of the human brain.

In a groundbreaking study by Pang et al. (2023), solutions to the Helmholtz equation applied on the cortical geometry have been shown to outperform competing approaches across a wide array of tasks and resting state fMRI data. This comes as a significant development, as the Helmholtz equation, originally formulated in the context of wave propagation and harmonics, finds a novel application in the realm of neuroscience. The promise it holds is to illuminate the complex structure and function of the brain through a fresh lens, one that emphasizes geometric constraints.

This paper shows that solutions of the Helmholtz equation on the cortical geometry outperform competing approaches across a diverse range of task & resting state fMRI data. — Michael Breakspear

Remarkably, the application of geometric modes is not confined to the cortical surface alone. They extend into subcortical structures, presenting a striking correspondence with patterns of subcortical functional connectivity. This uncovers a new facet of the brain’s functional network, suggesting that the geometric principles that govern cortical activity may also apply to deeper, subcortical regions. This is a potential game-changer for understanding the intricate dynamics of the human brain.

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[geometrymatters]

Some community parks are square, reflecting the city block where they are located — but irregularly shaped parks reduce the mortality risk of residents who live nearby, according to a study conducted by Huaquing Wang, a Ph.D. student in Urban and Regional Sciences, and Lou Tassinary, professor of visualization.

“Nearly all studies investigating the effects of natural environments on human health are focused on the amount of a community’s green space,” said the scholars in a paper describing their project. “We found that the shape or form of green space has an important role in this association.”

Wang and Tassinary conducted statistical studies on Philadelphia land cover data in order to examine the relationships between landscape spatial indicators and health outcomes. Residents in census tracts with more linked, aggregated, and complex-shaped greenspaces had a reduced mortality risk, according to the researchers.

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[geometrymatters]

Scientists Sue Yeon Chung and L.F. Abbott from Columbia University developed an approach for understanding neural networks, by analyzing the geometric properties of neural populations and understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. When a group of neurons demonstrates variability in response to stimuli or through internal recurrent dynamics, manifold-like representations emerge.

In their work, the team highlights the important examples of how geometrical techniques and the insights they provide have aided the understanding of biological and artificial neural networks.

They investigate the geometry of these high-dimensional representations, i.e., neuronal population geometry, using mathematical and computational tools. Their review includes a variety of geometrical approaches that provide insight into the function of the networks.

An important aspect of their research is the insight that dividing sets of neural population activities is more difficult when their representation patterns are on a curved surface, instead of a linear one – a concept that plays an important role in the analysis of neural population geometry and goes back to the beginnings of artificial neural networks.

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[geometrymatters]

After noticing that weather data from roughly circular cities like Dallas and London often show more rain than triangular cities like Chicago and Los Angeles, Dev Niyogi and his colleagues at the University of Texas at Austin decided to investigate the link between the shape of an urban area and its rainfall. The authors argue that how wind and weather interact with shape in urban environments should be taken into account when building future urban spaces that must be more resilient to the effects of climate change.

Circular cities, square cities, and triangular cities are rated in order of rainfall amount and intensity from largest to smallest. The findings are important for city development that is both sustainable and resilient, especially for those that are expanding.

This study provides the first investigation of the impact of city shape on urban rainfall in inland and coastal environments. Under calm synoptic conditions, the city shape impact is much more evident in coastal environments.

The team idealized large eddy simulations coupled with the Weather Research and Forecasting model. In the inland vs coastal environment, there are changes in the timing of urban-induced rainfall. This is linked to the land-sea breeze’s differing diurnal cycles of vertical velocity and cloud water mixing ratio. The effect of city shape on rainfall is especially visible at the coast, where buoyant flows from cities modify the interactions between urban-rural circulation and sea breeze.

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[geometrymatters]

Infants may recognize regular sound sequences during their first year of life. As we grow older, we gain the capacity to recognize increasingly complicated patterns in streams of words and musical sounds. Traditionally, cognitive scientists thought that the brain used a complex algorithm to discover connections between dissimilar concepts, resulting in a higher-level comprehension.

Christopher Lynn, Ari Kahn, and Danielle Bassett of the University of Pennsylvania are developing an altogether new model, showing that our capacity to identify patterns may be influenced in part by the brain’s drive to encode things in the simplest way possible.

The brain does more than just process incoming information, said Lynn, a physics graduate student. “It constantly tries to predict what’s coming next. If, for instance, you’re attending a lecture on a subject you know something about, you already have some grasp of the higher-order structure. That helps you connect ideas together and anticipate what you’ll hear next.”

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[geometrymatters]

For neuroscientists studying complex systems, patterns exhibit valuable data that may or may not correspond to higher levels of cognitive processes. Tyler Millhouse proposes a criterion evaluating just how real a pattern is likely to be, improving a SFI External Professor Daniel Dennett’s 1991 explanation, which utilized ‘compressibility’ to determine how genuine a pattern is likely to be. Dennett characterized genuine patterns by whether complicated scientific data can be properly represented by smaller scientific models, and just how extremely detailed pictures may be compressed into JPEG files that capture the important elements of the original image. Millhouse further argues that the more complex the interpretation required, the less real the pattern is likely to be.

My aim is to argue for a new and more demanding criterion for the reality of patterns. This criterion is inspired by real patterns, both in its original form and in later interpretations, but it also builds on algorithmic information theory and on similarity criteria of model fidelity.

To understand how Dennett makes philosophical use of the connection between regularity and compression, it is worth revisiting one of his central examples—image compression. The presence of a pattern in data is a matter of degree, a philosophical insight that is exemplified by the checkerboard compression test.

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[geometrymatters]

Is it possible for a concept to be beautiful? Mathematicians frequently describe arguments as “beautiful” or “dull,” and prominent scientists have argued that mathematical beauty serves as a pointer to the truth. Do laypeople, like mathematicians and scientists, have an aesthetic experience with mathematics? According to three studies, they do. When participants assessed the resemblance of basic mathematical arguments to landscape paintings or pieces of classical piano music, their rankings were internally consistent.

The study, co-authored by a Yale mathematician and a psychologist from the University of Bath, demonstrates that typical Americans can evaluate mathematical arguments for beauty just as they do pieces of art or music. The math’s beauty, they discovered, was not one-dimensional either: using nine criteria for beauty — such as grace, complexity, universality, and so on — 300 people agreed on the particular ways that four distinct proofs were beautiful.

Johnson split the research into three sections. The first task required a sample of people to match the four math proofs to the four landscape paintings based on how aesthetically similar they found them; the second task required a different sample to do the same but compare the proofs to sonatas, and the third task required another unique sample of people to independently rate each of the four artworks and math proofs on a scale of zero to ten.

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[geometrymatters]

In her 2019 review, Tatyana O Sharpee connects several lines of research to argue that hyperbolic geometry should be broadly applicable to neural circuits as well as other biological circuits. Networks with hyperbolic geometry are most sensitive to both internal and external disturbances, which explains why. Moreover, these networks enable effective communication whether nodes are added or withdrawn.

The well-known Zipf’s rule, which is also commonly referred to as the Pareto distribution and asserts that the likelihood of observing a certain pattern is inversely linked to its rank, is another characteristic of hyperbolic geometry, according to the author. Several biological systems, including protein sequences, brain networks, and economics, exhibit Zipf’s law. These discoveries give more evidence for the universality of networks having an underlying hyperbolic metric structure.

A three-dimensional hyperbolic space is relevant for neuronal signaling, according to recent discoveries in neuroscience. Compared to other dimensions, the three-dimensional hyperbolic space could offer more resilience. The paper provides an example of how the olfactory system’s new topographic arrangement was discovered using hyperbolic coordinates. The adoption of such coordinates could make it easier for pertinent signals to be represented elsewhere in the brain.

The link between hyperbolic geometry, Zipf’ law and maximally informative representations clarifies why hyperbolic geometry is simultaneously relevant for both the olfactory system and word distribution. In the case of olfaction, plants and animals have to produce chemical signals that will be discernible by other animals. — Tatyana O. Sharpee

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[geometrymatters]

Although consciousness is a multifaceted phenomena, important aspects like awareness and alertness have only been theoretically rather than neurobiologically defined. According to a novel theory, different neurofunctional dimensions of the brain contain aspects of consciousness that may be detected and measured by variations in blood flow across time.

“Consciousness is complex and studying it is like solving a scrambled Rubik’s cube,” [..] “If you look at just a single surface, you may be confused by the way it is organized. You need to work on the puzzle looking at all dimensions.” —  Zirui Huang, Ph.D., research assistant professor in the University of Michigan Medical School Department of Anesthesiology.

Some examples of these dimensions when it comes to consciousness include arousability, or the brain’s capacity to be awake, awareness, or what we actually experience—such as the color of a rose—and sensory organization, or how sights, sounds, and feelings come to be woven together to create our seamless conscious experience.

The researchers looked for such dimensions of the mind in the brain’s geometry. They looked at the topology or gradients across brain areas rather than at discrete brain regions. The scientists used fMRI data from study participants who were awake, anesthetized, in a sort of coma, or who had mental illnesses like schizophrenia to construct a map of these so-called cortical gradients of awareness.

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[geometrymatters]

“The human function is to ‘discover or observe’ mathematics,” said twentieth-century British mathematician G. H. Hardy. Humanity has been searching for beauty and order in the arts and in nature for generations, dating back to the ancient Greeks. This search for mathematical beauty has led to the discovery of recurring mathematical structures such as the golden ratio, Fibonacci numbers, and Lucas numbers, which have captivated the interest of artists and scientists alike.

This quest’s enchantment comes with significant stakes. In truth, art is the ultimate expression of human ingenuity, and comprehending it mathematically would provide us with the keys to decoding human civilization and evolution. However, it wasn’t until later that the scope and size of humanity’s pursuit of mathematical beauty was dramatically broadened by the convergence of three distinct inventions.

The development of robust statistical approaches to capture hidden patterns in massive amounts of data, as well as the mass digitization of large art archives, have made it feasible to disclose the—otherwise imperceptible to the human eye—mathematics hidden in large artistic corpora.

Music, storytelling, language phonology, comedy in jokes, and even equations have all been included in the current growth. Lee et al. extend this quest by looking for statistical signatures of compositional proportions in a quasi-canonical dataset of 14,912 landscape paintings spanning the period from Western renaissance to contemporary art (from 1500 CE to 2000 CE).

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[geometrymatters]

Nikos Salingaros is a mathematician and architectural theorist who has developed a theory of architecture that emphasizes the importance of socially-organized housing and the geometry of control. Salingaros argues that traditional architecture, which is based on symmetrical and hierarchical designs, is inherently oppressive and creates a sense of alienation among residents. In contrast, he promotes the use of complex and adaptive designs that are based on the principles of self-organization and emergent behavior found in natural systems.

One of the key concepts in Salingaros’ theory of architecture is the idea of socially-organized housing, which refers to the use of architecture to create a sense of community and social cohesion among residents. This is achieved by designing housing units that are connected to one another through a network of shared spaces, such as courtyards, gardens, and walkways. These shared spaces serve as the “social glue” that binds residents together and creates a sense of belonging.

These shared spaces would also counter-act the psychological process of control that, according to the author, the geometry of buildings and the design of cities can exert on the human mind. Individual buildings and urban areas are shaped by a hard, mechanical geometry, but the interaction between individual structures and the layout’s geometry influence the shape of the street network. In urban and architectural terms, there are various ways to demonstrate authority, and we find them all in government-built social housing.

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[geometrymatters]

Physicists at Washington University in St. Louis researching the brain have demonstrated that monitoring signals from a single neuron may be as effective as gathering information from numerous neurons at once using large, expensive arrays of electrodes.A key topic in neuroscience is what information single neurons get about overall brain network activity. For years, researchers in the laboratory of Ralf Wessel, professor of physics at Arts & Sciences, have used modern neurotechnology and physics-inspired data analysis to investigate sensory information processing in the brain.

“We know that in critical systems you can zoom in or out really far, and get the same statistical patterns. This property is called scale-freeness — or fractalness — and criticality may explain the origins of widely observed fractal activity in the brain,” said James K. Johnson, first author of the paper and a graduate student in the Wessel laboratory.

The researchers sought to zoom all the way down for their new experiment. They explained that evidence for criticality has been found at all greater scales. “The scale of the single cell was the last frontier,” Johnson said. “We cheated a bit, though. The statistical patterns used to evince criticality in the brain are called neuronal avalanches. Essentially, it’s just a spurt of ‘spiking,’ or messaging between neurons.”

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