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Following an electrical stimulus, the transmembrane voltage of cardiac tissuerises rapidly and remains at a constant value before returning to the restingvalue, a phenomenon known as an action potential. When the pacing rate of aperiodic train of stimuli is increased above a critical value, the actionpotential undergoes a period-doubling bifurcation, where the resultingalternation of the action potential duration is known as alternans in themedical literature. Existing cardiac models treat alternans either as a smoothor as a border-collision bifurcation. However, recent experiments in pacedcardiac tissue reveal that the bifurcation to alternans exhibits hybridsmooth/nonsmooth behaviors, which can be qualitatively described by a model ofso-called unfolded border-collision bifurcation. In this paper, we obtainanalytical solutions of the unfolded border-collision model and use it toexplore the crossover between smooth and nonsmooth behaviors. Our analysisshows that the hybrid smooth/nonsmooth behavior is due to large variations inthe system's properties over a small interval of the bifurcation parameter,providing guidance for the development of future models.

Cardiac Alternans Arising From an Unfolded Border-Collision Bifurcation