  Rapid action potential generation --- spiking --- and alternating intervals
of spiking and quiescence --- bursting --- are two dynamic patterns observed in
neuronal activity. In computational models of neuronal systems, the transition
from spiking to bursting often exhibits complex bifurcation structure. One type
of transition involves the torus canard, which was originally observed in a
simple biophysical model of a Purkinje cell. In this article, we expand on that
original result by showing that torus canards arise in a broad array of
well-known computational neuronal models with three different classes of
bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting,
and fold/fold cycle bursting. The essential features that these models share
are multiple time scales leading naturally to decomposition into slow and fast
systems, a saddle-node of periodic orbits in the fast system, and a torus
bifurcation in the full system. We show that the transition from spiking to
bursting in each model system is given by an explosion of torus canards. Based
on these examples, as well as on emerging theory, we propose that torus canards
are a common dynamic phenomenon separating the regimes of spiking and bursting
activity.
