I will report on recent work which proposes that the network dynamics of the mammalian visual cortex are neither homogeneous nor synchronous but highly structured and strongly shaped by temporally localized barrages of excitatory and inhibitory firing we call `multiple-firing events' (MFEs). Our proposal is based on careful study of a network of spiking neurons built to reflect the coarse physiology of a small patch of layer 2/3 of V1. When appropriately benchmarked this network is capable of reproducing the qualitative features of a range of phenomena observed in the real visual cortex, including orientation tuning, spontaneous background patterns, surround suppression and gamma-band oscillations. Detailed investigation into the relevant regimes reveals causal relationships among dynamical events driven by a strong competition between the excitatory and inhibitory populations. Testable predictions are proposed; challenges for mathematical neuroscience will also be discussed. This is joint work with Aaditya Rangan.