Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh--Rose Burster

The Hindmarsh--Rose model of neural action potential is revisited from the point of view of global bifurcation analysis, with the singular perturbation parameter held fixed. Of particular concern is a 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. It is shown via numerical path-following that the lobe-to-stripe transition is organized by a sequence of codimension-one and -two homoclinic bifurcations. Specifically, each of a sequence of homoclinic bifurcation curves in the parameter plane is found to undergo a sharp turn, due to interaction between a two-dimensional unstable manifold and the one-dimensional slow manifold that persists from the singular limit. Local analysis using approximate Poincaré maps shows that each turning point induces an inclination-flip bifurcation that gives birth to the fold curve that organizes the spike adding transition. Implications of this mechanism for other excitable systems are discussed.