  Circle maps frequently arise in mathematical models of physical or biological
systems. Motivated by Cherry flows and `threshold' systems such as integrate
and fire neuronal models, models of cardiac arrhythmias, and models of
sleep/wake regulation, we consider how structural transitions in circle maps
occur. In particular we describe how maps evolve near the creation of a
discontinuity.
  We show that the natural way to create discontinuities in the maps associated
with both threshold systems and Cherry flows results in a square root
singularity in the map as the discontinuity is approached from either one or
both sides. We analyse the generic properties of maps with gaps displaying this
local behaviour, showing how border collisions and saddle-node bifurcations are
interspersed. This highlights how the Arnold tongue picture for tongues
bordered by saddle-node bifurcations is amended once gaps are present.
  For the threshold models we also show that a loss of injectivity naturally
results in the creation of multiple gaps giving rise to a novel codimension two
bifurcation.
