The cerebral cortex is the biological substrate for a great deal of information processing in the brain. It is characterized by a high degree of structural and dynamical complexity, as reflected in the large number of parameters in cortical models. A basic task in computational neuroscience is to constrain model parameters based on available data, and to rationalize the impact of parameters on network dynamics. In this talk, I will report on a computational study of these problems in the context of the primate primary visual cortex. Borrowing ideas from multiscale modeling and statistical mechanics, we construct simple, interpretable coarse-grained models that nevertheless reproduce the results of realistic network models. Using this, we are able to map cortical parameter space and resolve a conundrum: while neuronal networks can be highly sensitive to small changes in parameters, they are often quite robust despite natural biological variability.