  In this work we consider a general class of $2$-dimensional hybrid systems.
Assuming that the system possesses an attracting equilibrium point, we show
that, when periodically driven with a square-wave pulse, the system possesses a
periodic orbit which may undergo smooth and nonsmooth grazing bifurcations. We
perform a semi-rigorous study of the existence of periodic orbits for a
particular model consisting of a leaky integrate-and-fire model with a dynamic
threshold. We use the stroboscopic map, which in this context is a
$2$-dimensional piecewise-smooth discontinuous map. For some parameter values
we are able to show that the map is a quasi-contraction possessing a (locally)
unique maximin periodic orbit. We complement our analysis using advanced
numerical techniques to provide a complete portrait of the dynamics as
parameters are varied. We find that for some regions of the parameter space the
model undergoes a cascade of gluing bifurcations, while for others the model
shows multistability between orbits of different periods.
