We investigate the influence of the network topology on the asymptotic dynamical patterns, attractors, in a general model of excitable dynamics on signed directed graphs. In this framework, network topology manifests itself as an interplay of positive and negative feedback loops. A small change in a feedback loop, by addition or removal of edges in the graph, can drastically change the dynamical patterns in the network, characterized by the appearance and disappearance of attractors from the attractor space of the network. We identify the determinants of such events via a systematic set of numerical experiments. As application examples, we discuss the basal ganglia network that is relevant in the context of Parkinson's disease and the two-compartment cortico-thalamic network thought to be related to generating epileptic seizures, showing that a given attractor in the attractor space of a network can be induced or destroyed via a specific set of topological manipulations. Thus, we propose a systematic way to alter the dynamical landscape of the system via changes in its topology and hence for perturbation therapies like deep brain stimulation.