It is well-known that a network of excitatory and inhibitory neurons with recurrent excitation and lateral inhibition can generate a stationary localized structure. In this talk, we explore the effects of: first, weak noisy inputs where we are able to compute the diffusive drift of these structures, something that is important for the maintenance of working memory; and second, we look at what the effects of slow weak negative feedback are on these systems. In the latter case, we study phenomena in one- and two- spatial dimensions. We are able to reduce the dynamics to one or two Volterra integral equations, from which we can analyze instabilities and bifurcations to various types of complex dynamics. This work is done in collaboration with Zack Kilpatrick, Brent Doiron, and Youngmin Park.