Simulations of large neural networks have the potential to contribute uniquely to the study of epilepsy, from the effects of extremely local changes in neuron environment and behavior, to the effects of large scale wiring anomalies. Currently, simulations with sufficient detail in the neuron model, however, are limited to cell counts that are far smaller than scales measured by typical probes. Furthermore, it is likely that future simulations will follow the path that large-scale simulations in other fields have and include hierarchically interacting components covering different scales and different biophysics. The resources needed for problem solving in this domain call for petascale computing--computing with supercomputers capable of 10(15) operations a second and holding datasets of 10(15) bytes in memory. We will lay out the structure of our simulation of epileptiform electrical activity in the neocortex, describe experiments and models of its scaling behavior in large cluster supercomputers, identify tight spots in this behavior, and project the performance onto a candidate next generation computing platform.