The phase diagram of random threshold networks

Threshold networks are used as models for neural or gene regulatory networks. They show a rich dynamical behaviour with a transition between a frozen 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 [PNAS 101, 4781 (2004)]. Using the annealed approximation, we derive expressions for the time evolution of the proportion of nodes in the "on" and "off" state, and for the sensitivity $\lambda$. 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.Comment: 8 pages, 6 figure

Response of Boolean networks to perturbations

We evaluate the probability that a Boolean network returns to an attractor after perturbing h nodes. We find that the return probability as function of h can display a variety of different behaviours, which yields insights into the state-space structure. In addition to performing computer simulations, we derive analytical results for several types of Boolean networks, in particular for Random Boolean Networks. We also apply our method to networks that have been evolved for robustness to small perturbations, and to a biological example