The interface between the mathematical sciences and the biosciences is two-way and may be summarized as "math -> bio" and "bio -> math." This talk will have two parts. First, I will describe previous work on gaits of four-legged animals (based on distinguishing gaits, such as walk, trot, and pace, by their spatio-temporal symmetries). Second, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations.