Neuronal variability is a reflection of recurrent circuitry and cellular physiology. The modulation of neuronal variability is a reliable signature of cognitive and processing state. A pervasive yet puzzling feature of cortical circuits is that despite their complex wiring, population-wide shared spiking variability is low dimensional with all neurons fluctuating en masse. We show that the spatiotemporal dynamics in a spatially structured network produce large population-wide shared variability. When the spatial and temporal scales of inhibitory coupling match known physiology, model spiking neurons naturally generate low dimensional shared variability that captures in vivo population recordings along the visual pathway. Further, we show that firing rate models with spatial coupling can also generate chaotic and low-dimensional rate dynamics. The chaotic parameter region expands when the network is driven by correlated noisy inputs, while being insensitive to the intensity of independent noise.