  We introduce simple models of genetic regulatory networks and we proceed to
the mathematical analysis of their dynamics. The models are discrete time
dynamical systems generated by piecewise affine contracting mappings whose
variables represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.
