Networks of interconnected neurons display diverse patterns of collective spiking activity. Relating this collective activity to the network's connectivity structure is a key goal of computational neuroscience. We approach this question for clustered networks, which can form via biologically realistic learning rules and allow for the re-activation of previously evoked patterns. Previous studies of clustered networks have focused on metastabilty between fixed points corresponding to individual clusters, leaving open the question of whether clustered spiking networks can display more rich dynamics---and if so, how these are related to clustered connectivity. Here, we show that the combinatorial threshold linear network (CTLN) model is a mean field theory for our clustered spiking networks in the limit of large population size and fast inhibition. By applying the large body of ``graph rules" for CTLNs, we can predict the dynamic attractors of our clustered spiking networks from the structure of between-cluster connectivity. This allows us to construct networks displaying a diverse array of nonlinear cluster dynamics, including metastable periodic orbits and chaotic attractors.