The capacity for sensory systems to encode relevant information that is invariant to many stimulus changes is central to normal, real-world, cognitive function. This invariance is thought to be reflected in the complex spatiotemporal activity patterns of neural populations, but our understanding of population-level representational invariance remains coarse. Applied topology is a promising tool to discover invariant structure in large datasets. Here, we use topological techniques to characterize and compare the spatiotemporal pattern of coactive spiking within populations of simultaneously recorded neurons in the secondary auditory region NCM of European starlings ( Sturnus vulgaris ). We show that the pattern of population spike train coactivity carries stimulus specific structure that is not reducible to that of individual neurons. We then introduce a topology-based similarity measure for population coactivity that is sensitive to invariant stimulus structure and show that this measure captures invariant...