Critical branching dynamics in excitable network without refractory periods

Critical state plays an important role in emerged behavior of neural networks. Excitable networks are widely used in simulation studies. However, the simulation of critical state is sophisticated because the state sensitively depends on dynamical and topological features. We study the effect of refractory period of dynamical units and connection symmetry on the excitable network with critical branching ratio. In bidirectional random networks with refractory period, the critical branching ratio is well known. Here, we show that the collective behavior changes from critical state into ceaseless activity, when refractory period vanishes. Instead of mean degree of all nodes, we show that the control parameter is determined by the mean degree of active nodes which is larger than the average degree and induces the transition. When refractory period is present, the number of effective links connected to resting nodes is equal to average degree. The mechanism ensures the critical branching ratio is unity. In directed networks, collective behavior keeps the critical state when refractory period vanished. However, a few reciprocal links make the network without refractory period becoming ceaseless.