  The well-studied Hindmarsh-Rose model of neural action potential is revisited
from the point of view of global bifurcation analysis. This slow-fast system of
three paremeterised differential equations is arguably the simplest reduction
of Hodgkin-Huxley models capable of exhibiting all qualitatively important
distinct kinds of spiking and bursting behaviour. First, keeping the singular
perturbation parameter fixed, a comprehensive two-parameter bifurcation diagram
is computed by brute force. Of particular concern is the parameter regime where
lobe-shaped regions of irregular bursting undergo a transition to stripe-shaped
regions of periodic bursting. The boundary of each stripe represents a fold
bifurcation that causes a smooth spike-adding transition where the number of
spikes in each burst is increased by one. Next, numerical continuation studies
reveal that the global structure is organised by various curves of homoclinic
bifurcations.
  In particular the lobe to stripe transition is organised by a sequence of
codimension-two orbit- and inclination-flip points that occur along {\em each}
homoclinic branch. Each branch undergoes a sharp turning point and hence
approximately has a double-cover of the same curve in parameter space. The
sharp turn is explained in terms of the interaction between a two-dimensional
unstable manifold and a one-dimensional slow manifold in the singular limit.
Finally, a new local analysis is undertaken using approximate Poincar\'{e} maps
to show that the turning point on each homoclinic branch in turn induces an
inclination flip that gives birth to the fold curve that organises the
spike-adding transition. Implications of this mechanism for explaining
spike-adding behaviour in other excitable systems are discussed.
