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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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
Commonly used models in mathematical physics for collections of neurons are so-called neural networks.
Here is some general explanation of the models.
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.