Stability and Self-Sustained Oscillations in a Ventricular Cardiomyocyte Model

Abstract

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For different values of a steady depolarizing/hyperpolarizing current (Ist), the corresponding projection of the static bifurcation diagram in the (Ist, V) plane is complex, featuring three branches of stationary solutions delimited by two saddle-node bifurcation points. In addition, on the upper branch oscillations can occur within an Ist range [−4.45, −0.51 μA/cm2] where the Jacobian of the linearized system features two complex conjugated eigenvalues. Oscillations are either damped at a stable focus or amplified until the system trajectory is diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle. The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (gsi and gK), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances self-sustained oscillations tending to a stable limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods based on numerical continuation algorithms. Both unsustained oscillations, resulting in early after-depolarizations, and sustained oscillations may trigger dangerous ventricular arrhythmias by multiple mechanisms. In particular settings, e.g. for a normal gK and one-fifth-of-normal gsi, these two arrhythmia-threatening conditions may occur simultaneously.