  Excitatory and inhibitory nonlinear noisy leaky integrate and fire models are
often used to describe neural networks. Recently, new mathematical results have
provided a better understanding of them. It has been proved that a fully
excitatory network can blow-up in finite time, while a fully inhibitory network
has a global in time solution for any initial data. A general description of
the steady states of a purely excitatory or inhibitory network has been also
given. We extend this study to the system composed of an excitatory population
and an inhibitory one. We prove that this system can also blow-up in finite
time and analyse its steady states and long time behaviour. Besides, we
illustrate our analytical description with some numerical results. The main
tools used to reach our aims are: the control of an exponential moment for the
blow-up results, a more complicate strategy than that considered in C\'aceres,
Carrillo and Perthame (2011), for studying the number of steady states, entropy
methods combined with Poincar\'e inequalities for the long time behaviour and,
finally, high order numerical schemes together with parallel computation
techniques in order to obtain our numerical results.
