We investigate epileptiform activity of excitatory-inhibitory networks that is initiated when inhibitory neurons cease firing due to depolarization block. We identify different transitions to seizure-like activity in a simplified rate model and demonstrate that these transitions do occur in a network of spiking neurons. By incorporating the effects of complex network structure into the rate model, we analyze the effects of the overrepresentation of different network motifs (subgraphs of the larger network) on the bifurcations leading to seizure-like activity. We find that, in the presence of depolarization block, the frequency of particular convergent motifs (subgraphs with edges converging onto a single node) can strongly influence the transition to synchronous activity.