Neural mean-field models attempt to describe the electrical signals of networks of neurons at length scales beyond the extent of a given neuron. We investigate periodic solutions to a particular mean-field model of the cortex (Liley et. al, Network-Comp Neural (2002) 13, 67-113) with certain spatial symmetries. The solutions we study originate from Turing-Hopf bifurcations, where spatially homogeneous equilibria destabilize into periodic solutions with some spatial dependence. The detection of these bifurcations can be done simply from analysis of the (low dimensional) spatially homogeneous reduction of the equations, but studying the spatiotemporal solutions that develop must be done with the full PDE model. We will present our method, and its implementation using PETSc, for finding and continuing these solutions using distributed finite difference discretization of the model. With submillimetre discretization, the centimetre length scales we are interested in result in millions of unknowns to be solved.