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The Neurogeometry of Perception: A Journey into Geometric Cognition

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.

Geometric cognition is a fundamental aspect of human intelligence. From the earliest civilizations crafting tools, building structures, and charting stars, to modern-day architects, engineers, and designers, our ability to understand and manipulate geometric shapes has been pivotal. Neurogeometry posits that this isn’t just a learned skill but rather an innate aspect of our cognitive makeup. The brain’s neurogeometrical structures enable us to intuitively grasp concepts like distance, proximity, similarity, and symmetry. This inherent geometric intuition suggests that our brains are, in essence, hard-wired to perceive and understand the world through geometric lenses.

What makes neurogeometry particularly captivating is its interdisciplinary nature. It stands at the crossroads of neuroscience, mathematics, psychology, and even philosophy. By exploring how the brain encodes geometric information, we gain insights not just into neural processes but also into the very nature of cognition and consciousness. Our perceptions, memories, and even dreams are replete with geometric motifs. As we delve deeper into the neurogeometrical framework, we begin to appreciate the profound symbiosis between the brain’s neural networks and the geometric patterns that pervade our experiences.

Both Sarti and Citti have made significant contributions to the field, often collaborating to explore the vast landscape of neurogeometry. Their research delves deep into the origins and nature of neurogeometrical patterns, seeking to understand how these patterns model neural connectivity and association fields. Their inquisitiveness has led them to investigate the statistics of edges in natural images, shedding light on how our brain processes and interprets visual stimuli.

A Lie Group is a set that combines the properties of both a differentiable manifold and a group in a compatible manner. In simpler terms, it’s a continuous group of symmetries. For instance, the set of all rotation operations in three-dimensional space forms a Lie Group. These groups are essential in understanding the symmetries underlying various physical systems, from the motion of planets to the behavior of subatomic particles.

Central to neurogeometry is the model based on the Lie group, which offers a mathematical framework for understanding the relationship between neural connections and geometric perception. “The neurogeometrical model based on Lie group,” as described by Sarti and Citti, provides a brief yet comprehensive overview of this profound relationship.

Lie groups are used in various areas of science, especially in physics, to describe symmetries. For instance, the symmetries in the laws of physics, which remain invariant under certain transformations (like rotations), can be described using Lie groups. This concept of symmetry and transformation is crucial when you think about perception, especially visual perception.

Neurogeometry, in its essence, tries to model the architecture and functional patterns of the visual cortex. One of the primary goals is to understand how the brain can recognize patterns, shapes, and structures, even when they are transformed (e.g., rotated, scaled, or translated). In other words, our brain has an in-built symmetry in recognizing patterns, much like the symmetries described by Lie groups.

When Sarti and Citti refer to the “neurogeometrical model based on Lie group,” they are hinting at a mathematical model where the symmetries in our visual processing, the way our brain handles various visual transformations, can be described using the principles of Lie groups. This offers a consistent and elegant framework to understand the complex neural processes involved in visual perception.

In practical terms, imagine seeing a cat. Whether the cat is sitting, lying down, far, near, or even partially obscured, our brain can still recognize it as a cat. This ability to recognize patterns amidst transformations (like changes in position, scale, or orientation) hints at an underlying symmetry in our visual processing. And this symmetry, when described mathematically, can be captured using the principles of Lie groups.

The origins of geometric perception can be traced back to the pioneering work of Gestalt psychologists such as Wertheimer, Kohler, and Köhler. They were among the first to formalize rules of perception, grounding them in geometric laws. These laws not only took into account position but also factors like brightness, orientation, and scale. Drawing from this foundation, Sarti and Citti emphasize the importance of normalized differences in these features, which play a crucial role in our perception.

One intriguing aspect of neurogeometry is the phenomenon of simultaneous contrast. Here, two circles with the same luminance, reflecting the same amount of light, can be perceived differently based on their backgrounds. The circle on a darker background appears brighter than the one on a lighter background. This phenomenon, deeply rooted in our visual processing, offers a glimpse into the intricate neurogeometrical patterns at play.

Figure 1: Left: Two circles with the same intensity of grey. Middle: the same circles are represented with a nonconstant background. As a result, the circle on the right appears to be brighter than the other one which has a darker background. Right: The Kanizsa square.

At first glance, simultaneous contrast might be dismissed as a mere optical illusion. However, its significance extends far beyond simple visual trickery. This phenomenon underscores the adaptability and sensitivity of our visual system. The contrast isn’t just a passive observation of luminance; it’s a dynamic interpretation based on context. By adjusting perceptions based on surrounding stimuli, our brains ensure that we discern details even in varying lighting conditions or amidst visual noise. This adaptability, facilitated by neurogeometrical structures, is essential for tasks ranging from discerning patterns in the dark to recognizing faces in a crowd.

Simultaneous contrast is more than just an interesting visual quirk; it’s a direct reflection of the underlying neural dynamics at work. The phenomenon provides a tangible demonstration of how neurons in the visual cortex interact, emphasizing the importance of inhibitory and excitatory neural networks. As the brain processes visual information, it doesn’t just register raw data. Instead, it contrasts and compares, enhancing certain signals while dampening others based on their relative importance and context. This intricate dance of neural activity, governed by neurogeometrical principles, ensures that our perception is both accurate and adaptive.

Figure 3: Kanizsa squares and diamonds with progressively more misaligned edges. From left to right the misalignment is 0, 6 and 12 degrees.
Figure 17 —The reconstruction made by authors’ model. In good agreement with the experiments, with edge misalignment of 0, degrees both the square and the diamond are reconstructed (left images), for misalignment of 6 degree the square is not reconstructed, while the diamond is reconstructed (middle images). Finally for 12 degrees neither the square not the diamond are reconstructed (right

In their figure 17, the authors showcase the capabilities of their model through a compelling demonstration. When presented with images of a square and a diamond, the model’s response varies based on the edge misalignment. With an edge misalignment of 0 degrees, both geometric shapes are effectively reconstructed by the model, mirroring our brain’s ability to recognize patterns in perfect alignment. However, as the misalignment increases to 6 degrees, the model struggles to reconstruct the square, though it still successfully identifies the diamond. At a more pronounced misalignment of 12 degrees, the model fails to reconstruct both shapes.

This experiment underscores the model’s sensitivity to edge alignments, a feature reminiscent of the human visual system. It’s a testament to the model’s robustness and its ability to mimic the intricate processes of visual perception.

The experiment with the square and diamond offers profound insights into the phenomenon of simultaneous contrast. The fact that the model can reconstruct certain shapes based on alignment hints at the brain’s innate preference or sensitivity to specific geometric configurations. Just as the human visual system might perceive two identically luminous circles differently based on their backgrounds, it might also be more attuned to recognizing certain shapes over others based on their alignment.

The Sarti-Citti model, through its elegant mathematical framework, provides a tangible means to study such perceptual biases and sensitivities. By understanding how the model reacts to various stimuli, researchers can glean insights into the neurogeometrical structures governing our visual perception.

At its core, the model constructed by Sarti and Citti epitomizes the marriage of geometry and neurology. It doesn’t just provide a theoretical foundation; it offers a practical tool to simulate and study the brain’s visual processing mechanisms. By simulating scenarios like the one described in figure 17, the model becomes a sandbox, allowing researchers to tweak variables, introduce new stimuli, and observe the resulting perceptual outcomes. Such experiments pave the way for a deeper understanding of simultaneous contrast and the holistic nature of visual perception.

The insights gleaned from simultaneous contrast extend beyond the realm of vision. They hint at a broader principle of perceptual relativity, where our perceptions are continually shaped and reshaped by context. Whether it’s the perceived loudness of a sound based on background noise or the taste of food influenced by prior meals, our sensory experiences are relative, not absolute. The neurogeometrical patterns that govern visual contrast may well find parallels in other sensory modalities, suggesting a universal principle of contextual perception deeply embedded in our neural architecture.

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Diving deeper into the nature of visual processing, Sarti and Citti’s work underscores its holistic nature. They argue that the

“nature of visual processing is intrinsically holistic and that the globality of the process emerges when the neurogeometric structure assumes an operational role”.

This perspective sheds light on the interconnectedness of visual stimuli and our brain’s intricate processing mechanisms. It’s tempting to think of visual perception as a mere aggregation of individual stimuli, where each visual cue is processed in isolation. However, the holistic approach suggests otherwise. Our brains do not operate as a collection of disjointed modules, each processing a specific visual element. Instead, there’s a symphony of interconnected processes that collectively interpret the visual scene. Elements like color, shape, movement, and depth don’t exist in isolation; they inform and influence each other, resulting in a cohesive perceptual experience. This holistic processing ensures that we don’t just see fragments of the world but comprehend it in its entirety.

Imagine a tapestry, woven with threads of various colors and textures. Each thread contributes to the overall image, but it’s the interplay of these threads that brings the picture to life. Similarly, our visual experiences are a tapestry of sensory inputs, intricately interwoven. The brain’s neurogeometrical structures facilitate this weaving, integrating diverse visual cues into a unified perception. Shadows give depth to objects, colors set the mood of a scene, and movement provides context. Separately, these elements tell partial stories, but together, they narrate a complete and immersive tale of our visual world.

Recognizing the holistic nature of visual processing has profound implications for cognitive science and neuroscience. It challenges traditional models that compartmentalize sensory processing and emphasizes the need for a more integrative approach. By understanding how different visual elements interplay within the neurogeometrical framework, researchers can gain deeper insights into cognitive disorders, develop more effective neural prosthetics, and even enhance visual experiences in virtual realities. In essence, the holistic perspective not only enriches our understanding of visual perception but also paves the way for advancements that bridge the gap between the brain and the external world.

As we journey through the landscape of neurogeometry, the insights and research of Alessandro Sarti and Giovanna Citti lay the groundwork for future endeavors in this domain. With each revelation and discovery, we inch closer to understanding the enigmatic relationship between our brain and the geometric world it perceives.


We remind some basic principles of the neurogeometrical approach as it has been proposed by D. Hoffmann, J. Petitot, and G. Citti-A. Sarti to model the functional architecture of the primary visual cortex. The neurogeometrical model based on Lie group is briefly sketched. Then we investigate the origin of the neurogeometrical pattern modeling neural connectivity and association fields by comparing it with the statistics of edges in natural images. Finally, we remark that the nature of visual processing is intrinsically holistic and that the globality of the process emerges when the neurogeometric structure assumes an operatorial role. Global emergence of perceptual units is then observed as spectral decomposition of the neurogeometrical operator.

On the origin and nature of neurogeometry, Alessandro Sarti and Giovanna Citti
Published: 2011

In this paper we first recall the definition of the geometrical model of the visual cortex, focusing in particular on the geometrical properties of horizontal cortical connectivity. Then we recognize that histograms of edges – co-occurrences are not isotropic distributed, and are strongly biased in horizontal and vertical directions of the stimulus. Finally, we introduce a new model of nonisotropic cortical connectivity modeled on the histogram of edges – co-occurrences. Using this kernel we are able to justify oblique phenomena comparable with experimental findings.

Neurogeometry of perception: isotropic and anisotropic aspects, Giovanna Citti, Alessandro Sarti
Published: 8 June 2019

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