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The well-studied Hindmarsh-Rose model of neural action potential is revisitedfrom the point of view of global bifurcation analysis. This slow-fast system ofthree paremeterised differential equations is arguably the simplest reductionof Hodgkin-Huxley models capable of exhibiting all qualitatively importantdistinct kinds of spiking and bursting behaviour. First, keeping the singularperturbation parameter fixed, a comprehensive two-parameter bifurcation diagramis computed by brute force. Of particular concern is the parameter regime wherelobe-shaped regions of irregular bursting undergo a transition to stripe-shapedregions of periodic bursting. The boundary of each stripe represents a foldbifurcation that causes a smooth spike-adding transition where the number ofspikes in each burst is increased by one. Next, numerical continuation studiesreveal that the global structure is organised by various curves of homoclinicbifurcations.  In particular the lobe to stripe transition is organised by a sequence ofcodimension-two orbit- and inclination-flip points that occur along {\em each}homoclinic branch. Each branch undergoes a sharp turning point and henceapproximately has a double-cover of the same curve in parameter space. Thesharp turn is explained in terms of the interaction between a two-dimensionalunstable manifold and a one-dimensional slow manifold in the singular limit.Finally, a new local analysis is undertaken using approximate Poincar\'{e} mapsto show that the turning point on each homoclinic branch in turn induces aninclination flip that gives birth to the fold curve that organises thespike-adding transition. Implications of this mechanism for explainingspike-adding behaviour in other excitable systems are discussed.

Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh-Rose burster