# High Dimensional Geometry of Brain Representations

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In summary, the paper discusses the relationship between fractal geometry and the representation of natural images in the brain's visual cortex. The supplementary material presents interesting mathematical results, particularly Theorems 3-5, which establish a relationship between the upper Minkowski dimension of a manifold and the decay of its eigenspectrum. This is a significant development in mathematics and has potential implications for understanding the brain's complexity and information processing abilities.
Very cool paper just out relating fractal geometry to the representation of natural images in visual cortex. The main argument is that brain's representations are as high dimensional as they could possibly be without becoming fractal.

https://www.nature.com/articles/s41586-019-1346-5
I'm curious what the geometers here make of the mathematical results in the supplementary (https://static-content.springer.com/esm/art:10.1038/s41586-019-1346-5/MediaObjects/41586_2019_1346_MOESM1_ESM.pdf). There's a set of theorems relating upper Minkowski dimension to eigenspectrum decay of a manifold (perhaps most interestingly Theorems 3-5). Does this constitute an interesting mathematical development in its own right?

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berkeman

As a scientist with a background in geometry, I find this paper and its supplementary material to be extremely interesting and thought-provoking. The idea that the brain's representations are as high dimensional as possible without becoming fractal is a fascinating concept that could have significant implications for our understanding of how the brain processes and represents information.

The mathematical results presented in the supplementary material, particularly Theorems 3-5, are indeed very intriguing. These theorems establish a relationship between the upper Minkowski dimension of a manifold and the decay of its eigenspectrum, which is a measure of the distribution of eigenvalues. This provides a mathematical framework for understanding the relationship between the complexity of a manifold (such as the brain) and the decay of its eigenspectrum.

In my opinion, this is a significant mathematical development in its own right. It not only sheds light on the structure of the brain and its representations, but also has potential applications in other fields such as image processing and signal analysis. Furthermore, these results could potentially pave the way for future research on the fractal nature of the brain's representations and how it relates to its information processing capabilities.

Overall, I believe that this paper and its supplementary material are a valuable contribution to the field of neuroscience and mathematics. It opens up new avenues for exploration and provides a deeper understanding of the brain's intricate functioning. I look forward to seeing further developments and applications of these mathematical results in the future.

## 1. What is high dimensional geometry of brain representations?

The high dimensional geometry of brain representations is a field of study that focuses on understanding the structure and organization of neural activity in the brain. It involves analyzing the patterns of activity in large sets of neurons and how they relate to different cognitive processes and behaviors.

## 2. Why is high dimensional geometry of brain representations important?

Studying the high dimensional geometry of brain representations can provide insights into how the brain processes information and how different brain regions communicate with each other. This can help us understand the neural basis of various cognitive functions and potentially lead to new treatments for neurological disorders.

## 3. How is high dimensional geometry of brain representations studied?

This field of study uses a combination of techniques, including neuroimaging, machine learning, and mathematical modeling, to analyze and interpret patterns of neural activity. Researchers may also use advanced statistical methods to identify specific features or dimensions of brain representations.

## 4. What are some applications of high dimensional geometry of brain representations?

Understanding the high dimensional geometry of brain representations has numerous potential applications, including improving brain-computer interfaces, developing more precise diagnostic tools for neurological disorders, and identifying biomarkers for predicting and monitoring cognitive decline.

## 5. What are some current challenges in studying high dimensional geometry of brain representations?

One of the main challenges in this field is the complexity of the brain and the vast amount of data that must be analyzed. Additionally, there is still much to be learned about how different brain regions interact and how to accurately interpret patterns of neural activity. Further advancements in technology and methods will be crucial for making progress in this area of research.

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