Striving to make contact with mathematics and to be consistent with neuroanatomy at the same time, I propose an idealized picture of the cerebral cortex consisting of a hierarchical network of brain regions each further subdivided into interconnecting layers not unlike those in artificial neural networks. Each layer is idealized as a 2D sheet of neurons, spatially homogeneous with primarily local interactions, a setup reminiscent of that in statistical mechanics. Zooming into local circuits, one gets into the domain of dynamical systems. Here the dynamics are characterized by competition and balance between two "opposing" groups of agents (Excitatory and Inhibitory neurons), trying to negotiate local dynamic equilibria in response to spatially inhomogeneous external stimuli. I will illustrate some of these ideas using a biologically realistic model of the monkey visual cortex, built and benchmarked to reproduce visual phenomena with the ultimate aim of explaining cortical mechanisms. The modeling work discussed is in collaboration with Bob Shapley and Logan Chariker.