  The study of synchronization in populations of coupled biological oscillators
is fundamental to many areas of biology to include neuroscience, cardiac
dynamics and circadian rhythms. Studying these systems may involve tracking the
concentration of hundreds of variables in thousands of individual cells
resulting in an extremely high-dimensional description of the system. However,
for many of these systems the behaviors of interest occur on a collective or
macroscopic scale. We define a new macroscopic reduction for networks of
coupled oscillators motivated by an elegant structure we find in experimental
measurements of circadian gene expression and several mathematical models for
coupled biological oscillators. We characterize the emergence of this structure
through a simple argument and demonstrate its applicability to stochastic and
heterogeneous systems of coupled oscillators. Finally, we perform the
macroscopic reduction for the heterogeneous stochastic Kuramoto equation and
compare the low-dimensional macroscopic model with numerical results from the
high-dimensional microscopic model.
