State space of a Boolean Network with N=4nodes and K=1links per node. Nodes can be either switched on (red) or off (blue). Thin (black) arrows symbolise the inputs of the Boolean function which is a simple "copy"-function for each node. The thick (grey) arrows show what a synchronous update does. Altogether there are 6 (orange) attractors, 4 of them are fixed points.
A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.[1]
Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly convey the correct pattern of expressed and suppressed genes.[2][3]
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.[4]
