If you want to do quickly learn about applications of some neural network concepts on a real simulator, there is a great online book (now wiki) called 'Computational Cognitive Neuroscience' at http://grey.colorado.edu/CompCogNeuro/index.php/CCNBook/Main
The book is used at schools as a textbook, and takes you through lots of different brain areas, from individual neurons all the way to higher-order executive functioning.
In addition, each section is augmented with homework 'projects' that are already down for you. Just download, follow the steps, and simulate everything that the chapter talked about. The software they use, Emergent, is a little finnicky but incredibly robust: its the product of more than 10 years of work I believe.
I went through it in an undergrad class this past semester, and it was great. Walks you through everything step by step