Abstract: We present a new method for comparing topological features using dissimilarity matrices obtained from observing activity in distinct complex systems. Our method uses the Dowker complex of a cross-dissimilarity matrix to identify all possible ways a common feature could be represented by the barcodes of activity within the individual systems. This method can be used to study both how distinct systems respond to the same stimuli and how behavior in one system drives behavior in another. Motivated by questions in neuroscience, our framework will allow researchers to investigate open problems such as how neural systems code for complex stimuli and how such coding structures propagate and evolve through different neural systems without direct reference to external correlates. The same tools can also be applied more generally to explore two-dimensional persistence and to identify which topological features are preserved after dimensionality reduction. This is joint work with Chad Giusti (University of Delaware) and Robert Ghrist (University of Pennsylvania)