Advances in experimental techniques, including fMRI, optical imaging, multi-electrode recordings, and optogenetics, combined with sophisticated data analytic tools, are beginning to shed light on the intricate functional architecture of specific brain regions, and how this provides a substrate for both spontaneous and stimulus-evoked neural activity patterns. In this talk, I use the mathematical theory of neural fields (spatially-structured neural networks) to explore the relationship between structure and dynamics in various models of visual cortex. I begin by showing how the common types of geometric hallucinations arise dynamically through the spontaneous formation of neural activity patterns within the cortical network. This suggests that the cortical mechanisms that generate geometric visual hallucinations are closely related to those used to process edges and contours in normal vision, which is consistent with recent findings by experimental collaborators. I then show how stochastic neural fields can be used to investigate the stimulus-dependent suppression of neural variability in multiple-attractor networks. The underlying topology of the network is taken to be the circle S1 or the sphere S2 and the corresponding neural field equations are taken to have rotation symmetry. Both the underlying symmetry group and the curvature of the attractor manifold (in the case of S2) play a crucial role in understanding the effects of noise.