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What sorts of mathematical tools, models and methods and theoretical frameworks do people use to simulate the function of the brain's neural networks? What mathematical properties do different brains have?

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One of my professors from undergrad was using partial differential equations to study the firing of neurons. –  Joe Johnson 126 Jan 13 '11 at 15:49
    
ooh that's awesome! –  Snipergirl Jan 13 '11 at 16:01
    
Kloeden and Platen's book on numerical solution of stochastic differential equations describes a model of neuron firing based on hitting times of diffusions. –  Simon Jan 13 '11 at 16:33
    
You may be able to find more resources on the UK Mathematical Neuroscience Network, but just for starters, I've recently read a favourable review of the new textbook Mathematical Foundations of Neuroscience by Ermentrout and Terman, so you may consider looking there. –  Willie Wong Jan 13 '11 at 16:49
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I don't either. But it's awesome. –  TonyK Jan 13 '11 at 19:12

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From what I understand there are broadly two distinct applications of mathematics to neuroscience. One uses mathematics to study the biological/chemical/physical aspects of the mechanisms in the brain, such as action potentials and the interactions between neurons. The type of math used here is differential equations/dynamic systems. Relevant wikipedia articles are http://en.wikipedia.org/wiki/Biological_neuron_models and http://en.wikipedia.org/wiki/Biological_neural_network

The other application is more abstract and concerns itself more with the computational aspects of neurons instead of the biophysical mechanisms by which they operate. This area is more discrete and more aligned with subjects like computer science, artificial intelligence, and statistical learning. Relevant wikipedia articles are http://en.wikipedia.org/wiki/Artificial_neural_network and http://en.wikipedia.org/wiki/Machine_learning

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Commonly used models in mathematical physics for collections of neurons are so-called neural networks.

Here is some general explanation of the models.

Some popular models are the Hopfield model, Ashkin-Teller neural networks, Blume-Emery-Griffiths neural networks,...

Techniques used to solve them involve statistical mechanics which result in partition sums (often handled with path integrals), differential equations to model dynamics, stochastic processes, renormalization theory, etc...

Just realize that these models are usually rudementary in the sense that they can't model a fully functioning brain but just a "small" collection of interconnected neurons.

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See for example this. A lot of information theory is used.

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