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Roles of L-type Ca²⁺ and delayed-rectifier K⁺ currents in sinoatrial node pacemaking: insights from stability and bifurcation analyses of a mathematical model

¹Department of Physiology, Kanazawa Medical University, Ishikawa 920-0293; and ²First Department of Internal Medicine, Tottori University School of Medicine, Yonago 683-0826, Japan

Submitted 4 December 2002 ; accepted in final form 7 August 2003

To elucidate the dynamical mechanisms of the sinoatrial (SA) node pacemaker activity, we investigated the roles of L-type Ca²⁺ (I_Ca,L) and delayed-rectifier K⁺ (I_Kr) currents in pacemaking by stability and bifurcation analyses of our rabbit SA node model (Kurata Y, Hisatome I, Imanishi S, and Shibamoto T. Am J Physiol Heart Circ Physiol 283: H2074–H2101, 2002). Equilibrium points (EPs), periodic orbits, stability of EPs, and Hopf bifurcation points were calculated as functions of conductance or gating time constants of the currents for constructing bifurcation diagrams. Structural stability (robustness) of the system was also evaluated by computing stability and dynamics during applications of constant bias currents (I_bias). Blocking I_Ca,L or I_Kr caused stabilization of an EP and cessation of pacemaking via a Hopf bifurcation. The unstable zero-current potential region determined with I_bias applications, where spontaneous oscillations appear, shrunk and finally disappeared as I_Ca,L diminished, but shrunk little when I_Kr was eliminated. The reduced system, including no time-dependent current except I_Ca,L, exhibited pacemaker activity. These results suggest that I_Ca,L is responsible for EP instability and pacemaker generation, whereas I_Kr is not necessarily required for constructing a pacemaker cell system. We further explored the effects of various K⁺ currents with different kinetics on stability and dynamics of the model cell. The original I_Kr of delayed activation and inward rectification appeared to be most favorable for generating large-amplitude oscillations with stable frequency, suggesting that I_Kr acts as an oscillation amplifier and frequency stabilizer. I_Kr may also play an important role in preventing bifurcation to quiescence of the system.

rabbit; primary pacemaker cell; nonlinear dynamics; bifurcation diagram; computer simulation

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