We introduce a random network model in which one can prescribe the frequency of second order edge motifs. We derive effective equations for the activity of spiking neuron models coupled via such networks. A key consequence of the motif-induced edge correlations is that one cannot derive closed equations for average activity of the nodes (the average firing rate neurons) but instead must develop the equations in terms of the average activity of the edges (the synaptic drives). As a result, the network topology increases the dimension of the effective dynamics and allows for a larger repertoire of behavior. We demonstrate this behavior through simulations of spiking neuronal networks.