Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior

Networks of excitable nodes have recently attracted much attention particularly in regards to neuronal dynamics, where criticality has been argued to be a fundamental property. Refractory behavior, which limits the excitability of neurons is thought to be an important dynamical property. We therefore consider a simple model of excitable nodes which is known to exhibit a transition to instability at a critical point ($\lambda=1$), and introduce refractory period into its dynamics. We use mean-field analytical calculations as well as numerical simulations to calculate the activity dependent branching ratio that is useful to characterize the behavior of critical systems. We also define avalanches and calculate probability distribution of their size and duration. We find that in the presence of refractory period the dynamics stabilizes while various parameter regimes become accessible. A sub-critical regime with $\lambda<1.0$, a standard critical behavior with exponents close to critical branching process for $\lambda=1$, a regime with $1<\lambda<2$ that exhibits an interesting scaling behavior, and an oscillating regime with $\lambda>2.0$. We have therefore shown that refractory behavior leads to a wide range of scaling as well as periodic behavior which are relevant to real neuronal dynamics.