  Following an electrical stimulus, the transmembrane voltage of cardiac tissue
rises rapidly and remains at a constant value before returning to the resting
value, a phenomenon known as an action potential. When the pacing rate of a
periodic train of stimuli is increased above a critical value, the action
potential undergoes a period-doubling bifurcation, where the resulting
alternation of the action potential duration is known as alternans in the
medical literature. Existing cardiac models treat alternans either as a smooth
or as a border-collision bifurcation. However, recent experiments in paced
cardiac tissue reveal that the bifurcation to alternans exhibits hybrid
smooth/nonsmooth behaviors, which can be qualitatively described by a model of
so-called unfolded border-collision bifurcation. In this paper, we obtain
analytical solutions of the unfolded border-collision model and use it to
explore the crossover between smooth and nonsmooth behaviors. Our analysis
shows that the hybrid smooth/nonsmooth behavior is due to large variations in
the system's properties over a small interval of the bifurcation parameter,
providing guidance for the development of future models.
