Stability and 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. The associated stationary problem was reduced to a nonlinear system in only two unknowns, the transmembrane potential V and the intracellular calcium concentration [Ca]( i ). By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For a variable steady depolarizing or hyperpolarizing current (I (st)), the corresponding projection of the static bifurcation diagram in the (I (st), V) plane is complex, featuring three branches of stationary solutions joined by two limit points. On the upper branch oscillations can occur, being either damped at a stable focus or diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle, thus resulting in early after-depolarizations (EADs). 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 (g (si) and g (K)), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances sustained oscillations tending to a limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods, leading to identification of Hopf bifurcation points, limit points of cycles and period doubling bifurcations. In particular settings, e.g. one-fifth-of-normal g (si), EADs and sustained high amplitude oscillations due to an unstable resting state may occur simultaneously.

Relationships between kinetic constants and the amino acid composition of enzymes from the yeast Saccharomyces cerevisiae glycolysis pathway

“…It should be noted that this study well corresponds to a certain line of research in recent years where the set of primary structure-derived features [31,32] or integral physico-chemical indices of proteins [33] have been used to predict the values of kinetic constants for particular enzymes.…”

Action Potential Characterization of Human Induced Pluripotent Stem Cell–Derived Cardiomyocytes Using Automated Patch-Clamp Technology

“…There is considerable experimental support and a solid theoretical background justifying this method of external current injection to suppress spontaneous rhythmic firing. Thus, previous studies based on dynamic system analysis and bifurcation theory methods applied to various models of sinoatrial or ventricular CMs, [42][43][44][45][46][47] taking a constant externally applied current as parameter and assessing its influence over a wide range of values, have shown distinct intervals of stable dynamic equilibrium at a steady V m or stable limit cycles representing sustained oscillations, that is, spontaneous pace-making. The transitions between these regimes occur at Hopf bifurcation points in the parameter space of the model.…”

Dynamical mechanisms of phase-2 early afterdepolarizations in human ventricular myocytes: insights from bifurcation analyses of two mathematical models

“…The membrane current system of the K05 model includes I Ca,L, IKr, IKs, 4-aminopyridinesensitive transient outward current (Ito), Na ϩ channel current (INa), inward-rectifier K ϩ channel current (IK1), background Na ϩ (INab) and Ca 2ϩ (ICab) currents, INaK, INCX, and Ca 2ϩ pump current (IpCa). Time-dependent changes in the membrane potential (Vm) are described by the equation, dV m ⁄ dt ϭ I stim Ϫ ͑ICaL ϩ I Kr ϩ I Ks ϩ I to ϩ I Na ϩ I K1 ϩ I Nab ϩ I Cab ϩ I NaK ϩ I NCX ϩ I pCa͒ (1) where Istim represents the stimulus current (in pA/pF). The O11 model further incorporates the Ca 2ϩ activated and background Cl Ϫ currents.…”

“…Tran et al (60) also explored the dynamical mechanisms of EAD formation with particular focus on the roles of I CaL and I Ks by utilizing the "slow-fast decomposition analysis" (14) in which a bifurcation diagram for the subsystem consisting of fast variables was constructed as a function of the slow variable (activation gating variable for I Ks ) regarded as a parameter for the fast subsystem. Furthermore, bifurcation analyses have been used for elucidating how individual components contribute to normal pacemaking in SAN cells (18, 27, 30 -33, 36), generation of abnormal oscillations or biological pacemaker activity in ventricular myocytes (1,5,8,28,29,62), and EADs in ventricular myocytes (1,5,15,53,54). These theoretical studies have clearly demonstrated the significance of bifurcation analyses for general understanding and systematic description of the dynamical mechanisms of normal and abnormal oscillatory behaviors.…”