Experimental neuroscience is undergoing a period of rapid progress in the collection of neural activity and connectivity data. This promises to allow more direct testing of a variety of theoretical ideas, and thus advance our understanding of "how the brain works." Detecting meaningful structure in neural data, however, remains a significant challenge. A major obstacle is that these data often measure quantities that are related to more "fundamental" variables by an unknown nonlinear transformation. This transformation obscures the underlying structure, diminishing the power of traditional linear algebra-flavored tools. Methods from computational topology, however, are often capable of detecting the hidden structure. We adapt these methods for the analysis of correlation matrices, and illustrate their use for testing the "coding space" hypothesis on neural data.