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I have a friend who wants to study something applied to neurosciences. He is going to begin his grad studies in mathematics. He asked me which areas of mathematics could be applied to neurosciences. Since I don't know the answer, I thought mathoverflow would be the right place to ask. I mean, there are many areas of mathematics that could be applied to neurosciences. But the question is the following: which are the fields that have already been applied to neurosciences? Are there areas related to dynamical systems, stochastic process, probability, topology, analysis, PDE or algebra applied to neurosciences? Articles are welcome.

Thank you in advance

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My wife is a neuroscientist. I can tell you what she uses:

$\bullet$ a LOT of statistics.
$\bullet$ signal processing (such as wavelet transform).
$\bullet$ some baysian probability theory.

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(just my two cents, but too long for a comment)

Does your friend care more about the tools he uses in his research (e.g. theorem-proving vs. developing models vs. using known mathematics to analyze data and/or models) or the community in which his work has impact (e.g. mathematicians vs. applied scientists).

If the former I would advise following the branch of mathematics he most enjoys. If he is hoping to maintain a connection to neuroscience he might sample from statistics and applied analysis (by which I mean probability, stochastic processes, information theory, PDE and dynamical systems) while noting that other possibilities for application exist (e.g. computational homology). That said, he should be aware that direct and immediate impact in the neuroscience community will likely require skills other than theorem-proving.

If the latter I would advise devoting considerable energy to learning about the applied science which he hopes to advance in parallel with a broad mathematical education and to be open to learning whatever tools necessary (likely not just theorem-proving) to advance that science.

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The site scholarpedia.org is a good start. http://www.scholarpedia.org/article/Encyclopedia:Computational_neuroscience

It was started by a computational/mathematical neuroscientist, and has good articles on mathemtics relevant to neuroscience.

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Non-negative Matrix Factorization (cf e.g. http://en.wikipedia.org/wiki/Non-negative_matrix_factorization) plays a role in knowledge representation and extraction.

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