  Boolean Networks and their dynamics are of great interest as abstract
modeling schemes in various disciplines, ranging from biology to computer
science. Whereas parallel update schemes have been studied extensively in past
years, the level of understanding of asynchronous updates schemes is still very
poor. In this paper we study the propagation of external information given by
regulatory input variables into a random Boolean network. We compute both
analytically and numerically the time evolution and the asymptotic behavior of
this propagation of external regulation (PER). In particular, this allows us to
identify variables which are completely determined by this external
information. All those variables in the network which are not directly fixed by
PER form a core which contains in particular all non-trivial feedback loops. We
design a message-passing approach allowing to characterize the statistical
properties of these cores in dependence of the Boolean network and the external
condition. At the end we establish a link between PER dynamics and the full
random asynchronous dynamics of a Boolean network.
