Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)