  Morphogenesis depends crucially on the complex rheological properties of cell
tissues and on their ability to maintain mechanical integrity while rearranging
at long times. In this paper, we study the rheology of polygonal cellular
networks described by a vertex model in the presence of fluctuations. We use a
triangulation method to decompose shear into cell shape changes and cell
rearrangements. Considering the steady-state stress under constant shear, we
observe nonlinear shear-thinning behavior at all magnitudes of the
fluctuations, and an even stronger nonlinear regime at lower values of the
fluctuations. We successfully capture this nonlinear rheology by a mean-field
model that describes the tissue in terms of cell elongation and cell
rearrangements. We furthermore introduce anisotropic active stresses in the
vertex model and analyze their effect on rheology. We include this anisotropy
in the mean-field model and show that it recapitulates the behavior observed in
the simulations. Our work clarifies how tissue rheology is related to
stochastic cell rearrangements and provides a simple biophysical model to
describe biological tissues. Further, it highlights the importance of
nonlinearities when discussing tissue mechanics.
