  This contribution reviews the parallel dynamics of Q-Ising neural networks
for various architectures: extremely diluted asymmetric, layered feedforward,
extremely diluted symmetric, and fully connected. Using a probabilistic
signal-to-noise ratio analysis, taking into account all feedback correlations,
which are strongly dependent upon these architectures the evolution of the
distribution of the local field is found. This leads to a recursive scheme
determining the complete time evolution of the order parameters of the network.
Arbitrary Q and mainly zero temperature are considered. For the asymmetrically
diluted and the layered feedforward network a closed-form solution is obtained
while for the symmetrically diluted and fully connected architecture the
feedback correlations prevent such a closed-form solution. For these symmetric
networks equilibrium fixed-point equations can be derived under certain
conditions on the noise in the system. They are the same as those obtained in a
thermodynamic replica-symmetric mean-field theory approach.
