  We present a novel and global three-dimensional reduction of a
non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations
[J. Rubin and M. Wechselberger, Giant squid--hidden canard: the 3D geometry of
the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), pp. 5--32] that is
based on geometric singular perturbation theory (GSPT). We investigate the
dynamics of the resulting three-dimensional system in two parameter regimes in
which the flow evolves on three distinct timescales. Specifically, we
demonstrate that the system exhibits bifurcations of oscillatory dynamics and
complex mixed-mode oscillations (MMOs), in accordance with the geometric
mechanisms introduced in [P. Kaklamanos, N. Popovi\'c, and K. U. Kristiansen,
Bifurcations of mixed--mode oscillations in three--timescale systems: An
extended prototypical example, Chaos: An Interdisciplinary Journal of Nonlinear
Science, 32 (2022), p. 013108], and we classify the various firing patterns in
terms of the external applied current. While such patterns have been documented
in [S. Doi, S. Nabetani, and S. Kumagai, Complex nonlinear dynamics of the
Hodgkin-Huxley equations induced by time scale changes, Biological Cybernetics,
85 (2001), pp. 51--64] for the multiple-timescale Hodgkin-Huxley equations, we
elucidate the geometry that underlies the transitions between them, which had
not been previously emphasised.
