  We study a class of planar integrate and fire (IF) models called adaptive
integrate and fire (AIF) models, which possesses an adaptation variable on top
of membrane potential, and whose subthreshold dynamics is piecewise linear
(PWL). These AIF models therefore have two reset conditions, which enable
bursting dynamics to emerge for suitable parameter values. Such models can be
thought of as hybrid dynamical systems. We consider a particular slow dynamics
within AIF models and prove the existence of bursting cycles with $N$ resets,
for any integer $N$. Furthermore, we study the transition between $N$- and
$(N+1)$-reset cycles upon vanishingly small parameter variations and prove (for
$N=2$) that such transitions are organised by canard cycles. Finally, using
numerical continuation we compute branches of bursting cycles, including
canard-explosive branches, in these AIF models, by suitably recasting the
periodic problem as a two-point boundary-value problem.
