  Neural networks of the brain form one of the most complex systems we know.
Many qualitative features of the emerging collective phenomena, such as
correlated activity, stability, response to inputs, chaotic and regular
behavior, can, however, be understood in simple models that are accessible to a
treatment in statistical mechanics, or, more precisely, classical statistical
field theory.
  This tutorial presents the fundamentals behind contemporary developments in
the theory of neural networks of rate units that are based on methods from
statistical mechanics of classical systems with a large number of interacting
degrees of freedom. In particular we will focus on a relevant class of systems
that have quenched (time independent) disorder. In neural networks, the main
source of disorder arises from random synaptic couplings between neurons. These
systems are in many respects similar to spin glasses. The tutorial therefore
also explains the methods for these disordered systems as far as they are
applied in neuroscience.
  The presentation consists of two parts. In the first part we introduce
stochastic differential equations in the Martin - Siggia - Rose - De Dominicis
- Janssen path integral formalism. In the second part we employ this language
to derive the dynamic mean-field theory for deterministic random networks, the
basis of the seminal work by Sompolinsky, Crisanti, Sommers 1988, as well as a
recent extension to stochastic dynamics.
