Introduction: Spontaneous activity in visual cortex is high-dimensional and state-dependent, yet its organizing principles are unclear (Stringer et al. Science 2019; Berkes et al. Science 2011). We combine large-scale imaging, modelling, and theory to ask: 1. Does spontaneous activity share the structure of evoked activity? 2. Is spontaneous activity organized spatially? Methods: We constructed a modified Light Beads Microscope (Demas et al. Nature Methods 2021) to image Ca2+ transients in excitatory neurons from large volumes. We developed an open-source, GPU-accelerated volumetric cell extraction pipeline. These methods enabled simultaneous recording of 30,000+ cells from awake mice during spontaneous activity and visual stimulation. Results: Cells with similar visual preferences did not have higher spontaneous correlations. At the population level, the covariance structure of spontaneous activity was not predicted by that of evoked activity, showing that evoked and spontaneous activity do not share the same structure. Spontaneous activity exhibited a weakly spatial structure, with pairwise correlations decaying over ~1mm. We developed Spatial Shared Variance Component Analysis (extending Stringer et al. Science 2019) to quantify shared activity between spatially separate populations of cells. The magnitude and dimensionality of shared activity decayed with distance between populations. Cortical correlations could thus be characterized by a few, strong, global dimensions and many weaker local dimensions. We modelled spontaneous activity using a linear RNN with distance-dependent connectivity driven by low-dimensional inputs, which was sufficient to produce high-dimensional activity with spatiotemporal structure similar to that of neural data. We showed analytically that this model can produce high dimensional activity from low-dimensional inputs. Conclusions: Spontaneous activity does not share its structure with evoked activity. It follows a spatial organization with a few global and many spatially localized modes. The global modes may correspond to a low-dimensional arousal state, while remaining dimensions may be “reverberations” predicted by our model.