  Torus canards are solutions of slow/fast systems that alternate between
attracting and repelling manifolds of limit cycles of the fast subsystem. A
relatively new dynamic phenomenon, torus canards have been found in neural
applications to mediate the transition from spiking to bursting via
amplitude-modulated spiking. In $\mathbb{R}^3$, torus canards are degenerate:
they require one-parameter families of 2-fast/1-slow systems in order to be
observed and even then, they only occur on exponentially thin parameter
intervals. The addition of a second slow variable unfolds the torus canard
phenomenon, making them generic and robust. That is, torus canards in slow/fast
systems with (at least) two slow variables occur on open parameter sets. So
far, generic torus canards have only been studied numerically, and their
behaviour has been inferred based on averaging and canard theory. This
approach, however, has not been rigorously justified since the averaging method
breaks down near a fold of periodics, which is exactly where torus canards
originate. In this work, we combine techniques from Floquet theory, averaging
theory, and geometric singular perturbation theory to show that the average of
a torus canard is a folded singularity canard. In so doing, we devise an
analytic scheme for the identification and topological classification of torus
canards in $\mathbb{R}^4$. We demonstrate the predictive power of our results
in a model for intracellular calcium dynamics, where we explain the mechanisms
underlying a novel class of elliptic bursting rhythms, called
amplitude-modulated bursting, by constructing the torus canard analogues of
mixed-mode oscillations. We also make explicit the connection between our
results here with prior studies of torus canards and torus canard explosion in
$\mathbb{R}^3$, and discuss how our methods can be extended to slow/fast
systems of arbitrary (finite) dimension.
