High Dimensional Geometry of Brain Representations

[madness]

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?

 

[jvicens]

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.