  Kinetically Constrained Models (KCMs) have been widely studied in the context
of glassy dynamics, focusing on the influence of dynamical constraints on the
slowing down of the dynamics of a macroscopic system. In these models, it has
been shown using the thermodynamic formalism for histories, that there is a
coexistence between an active and an inactive phase. This coexistence can be
described by a first-order transition, and a related discontinuity in the
derivative of the large deviation function for the activity. We show that
adding a driving field to a KCM model does not destroy this first-order
transition for the activity. Moreover, a singularity is also found in the large
deviation function of the current at large fields. We relate for the first time
this property to microscopic structures, in particular the heterogeneous,
intermittent dynamics of the particles, transient shear-banding and blocking
walls. We describe both the shear-thinning and the shear-thickening regimes,
and find that the behaviour of the current is well reproduced by a simple
model.
