The phase diagram of random threshold networks

Threshold networks are used as models for neural or gene regulatory networks. They show rich dynamical behaviour with a transition between a frozen phase and a chaotic phase. We investigate the phase diagram of randomly connected threshold networks with real-valued thresholds h and a fixed number of inputs per node. The nodes are updated according to the same rules as in a model of the cell-cycle network of Saccharomyces cereviseae (Li et al 2004 Proc. Natl Acad. Sci. USA 101 4781–6). Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the ‘on’ and ‘off’ states, and for the sensitivity λ. The results are compared with simulations of quenched networks. We find that for integer values of h the simulations show marked deviations from the annealed approximation even for large networks. This can be attributed to the particular choice of the updating rule.