How complex, natural signals are represented in the activity (spiking) patterns of neural populations is not well understood. For this talk, I will describe data from a series of experiments that examine the spatiotemporal pattern of song-evoked spiking in populations of simultaneously recorded neurons in the secondary auditory cortices of European starlings (Sturnus vulgaris). Single neurons in these regions display composite receptive fields that incorporate large numbers (a dozen or more) orthogonal features matched to the acoustics of species typical song. Considered independently, the spiking response of a given neuron at a given point in time is therefore ambiguous with respect to the stimulus. Applied topology provides a promising tool to resolve this ambiguity and capture invariant structure in the spiking coactivity in arbitrarily large neural populations. I will show that the topology of the population spike train carries stimulus-specific structure that is not reducible to that of individual neurons. I then introduce a topology-based similarity measure for population coactivity that is sensitive to invariant stimulus structure and show that this measure captures invariant neural representations tied to the learned relationships between natural vocalizations.