Continuing advances in neural recording technologies and ambitions for their use necessitate improvements in our ability to meaningfully model and characterize activity at the scale of neural populations. Here I contribute two theoretical advances in this field. First, I theoretically characterize critical properties of certain focal seizures, prove that the existing standard Wilson-Cowan model (WCM) cannot generate activity with these properties, and then extend the WCM to have those properties. This enables modeling average population activity in focal seizures with the WCM. Second, I introduce a novel approach to numerically characterize any population activity in both time and space. This is simply third-order correlation (triple correlation) which I prove uniquely characterizes any finite dataset. Moreover I show triple correlation can be summarized by partitioning its underlying motifs into fourteen equivalence classes that embody well-known computational properties (e.g. synchrony, feedback). These motif classes reflect underlying changes in the structure of spatiotemporal neural activity. Finally, I apply this latter theoretical advance, using these motif classes to automate detection of seizures in newborn infants.