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I need some advice about my career. Currently, I'm an undergraduate student of math. Since I can remember, I wanted to be a scientist, so I decided to go for applied math. The field that I'm interested in is Mathematical Neuroscience.
I want to know more precisely what mathematicians can do in this field and what mathematicals tools do they need.
Finally, if you could recommend me some readings, pages and maybe reserch groups that you know about, I will be very thankful.
That's it. Thanks.
There are many different models for neural dynamics. It is common to see a lot of stressing of neural networks, and there are certainly a lot of interesting questions you can ask about these mathematical structures (recurrent networks can be Turing complete, nonrecurrent networks can approximate a function to any given delta in a given range, given sufficient nodes, etc.).
But the brain operates on many different levels. At the cellular level, much of the operation is chemical reaction networks (metabolism), which are differential equations with very simple interpretations in terms of reactants and products, but when combined in large metabolic graphs can demonstrate many interesting phenomena. Neurons group into larger modular architectonic structures that fulfill larger functional computation roles.
At a higher level, dynamic epistemic logics and other temporal logics can be used to describe beliefs and belief revision in the face of sensory input. Rewriting logics have been used to great extent here. Building effective state machines and the automata of thought is still in it's infancy but shows a lot of promise in bringing the semantic layer into machine learning. These kinds of approaches also do not show as much reliance on the abstract statistical partitioning one sees in a lot of the pattern recognition literature, if that is anathema.
I'd recommend taking a look at Arbib, Erdi, and Szentagothai's seminal "Neural Organization: Structure, Function, and Dynamics" if you are interested in these approach to mathematical modelling of neural ontology.
There is a relatively new field called Applied Topology, which has been developing in recent years. It mostly involves applications of Homology theory to various areas in engineering and science. In particular, it has recently been used to study connections in brain networks.
Here is an article on the topic (They study the functional patterns of a human brain under the influence of psilocybin ("Magic Mushrooms"), compared to the patterns without drug influence): http://rsif.royalsocietypublishing.org/content/11/101/20140873.full.pdf+html
And here's a poster summarizing their research: http://www.math.ku.dk/english/research/conferences/2014/dcat2014/petri.pdf
A lot of articles in this field are a result of collaboration between researchers from different areas of science, and even different areas within mathematics (for example, pure topologists together with probabilists), who usually don't have a vast background in the fields of their collaborators. That relates to the concern you're expressing in your second paragraph.
EDIT: Here are some introductory papers which you might want to look at to get an idea of what tools are used in this field (although those papers are more general and don't deal specifically with neuroscience):
http://arxiv.org/pdf/1003.5175.pdf http://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf (in general, Ghrist has several good introductory papers on applied topology.)
And here are some papers that I haven't read, so I don't know whether they are understandable or contain any basic introductory chapters, but they might give you an idea of additional applications to neuroimaging:
http://www.fil.ion.ucl.ac.uk/spm/doc/biblio/Keyword/RFT.html http://projecteuclid.org/euclid.aoas/1287409373 https://maia-2.biostat.wisc.edu/sites/default/files/tr_228.pdf
neural networks are in the intersection of those two fields(maths and neuroscience). It's a lot of info about it but it involves statistics. A lot of math background is needed to understand neural networks like: Computational theory, graph theory, coding theory, matching and flow theory, information theory, symbolic dynamics..etc.
This seems to be a pretty cool introduction. http://page.mi.fu-berlin.de/rojas/neural/neuron.pdf
