Dynamics and formation of oscillatory patterns in complex directed networks consisting of excitable nodes

Self-sustained oscillations of excitable networks have been extensively studied. However, many properties of directed networks have still not been understood clearly. In this paper we explore the dynamics and pattern formation of strongly connected complex directed Boolean map (BM) networks with excitatory interactions by using synchronous updating policy. Several theorems on the oscillatory dynamics are proposed and rigorously proven by logical analysis. Based on these theorems, we can successfully predict the global asymptotic behaviors of networks from local initial conditions of few nodes in a single loop, and effectively control and modulate the oscillations of networks by modifying few links.