Roles of hyperpolarization-activated current I_f in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

Abstract

(27,35,41,55), yielding four principal hypotheses that attribute pacemaker depolarization primarily to 1) activation of the L-type Ca 2ϩ channel current (I Ca,L ) (58); 2) deactivation of the delayedrectifier K ϩ current (I K ) (39); 3) development of the hyperpolarization-activated cation current (I f ) (9); and 4) spontaneous Ca 2ϩ release from sarcoplasmic reticulum (SR) and subsequent activation of the Na ϩ /Ca 2ϩ exchanger current (I NaCa ) (28,30,33,34,45). Nevertheless, there is no dominant pacemaker mechanism, and the roles of individual time-dependent, voltage-gated channel currents in pacemaker generation remained to be clarified. From the theoretical studies using mathematical models for rabbit SAN cells, we have provided significant insights into the dynamic mechanisms of the natural SAN pacemaking and the roles of I Ca,L , I K , and Na ϩ channel current (I Na ) in basal pacemaking (22,25). We have suggested that 1) the instability of an equilibrium point (EP) is essentially important for the generation of robust pacemaking without annihilation; 2) I Ca,L is responsible for EP instability and generation of pacemaker activity; 3) I K makes the action potential amplitude larger, eases changes in oscillation frequency during hyperpolarization or depolarization, and plays a pivotal role in preventing bifurcation to quiescence; and 4) I Na contributes to EP instability and robust pacemaking against hyperpolarizing loads in the peripheral SAN. However, the roles of I f in SAN pacemaking, remaining the subject of controversy (27), have not yet been determined by the theoretical approach.I f , encoded by the hyperpolarization-activated cyclic nucleotide-gated (HCN) channel gene, is known to contribute to pacemaker depolarization as the pacemaker current (9, 27) and is a key current for engineering of biological pacemakers (43,46,48,53,56). I f contributes to prevention of excess hyperpolarization and excessively slow pacemaking against hyperpolarizing loads (16,27,35), autonomic regulations of spontaneous activity (7), stabilization of pacemaker frequency (16,33,40), and generation of phase 4 depolarization in the periphery of intact SAN, suffering electrotonic influences of the atrium (38). In isolated SAN cells or the center of intact SAN, however, I f does not appear to be indispensable for generation of spontaneous oscillations under normal conditions because 1) I f blockers did not abolish spontaneous activity of real SAN cells or the intact SAN, although the block of I f might be incomplete at pacemaker potentials (16,27,38,41); 2) any model SAN cells did not undergo a bifurcation to quiescence during inhibition of I f , exhibiting automaticity, even in the absence of I f (21, 55); and 3) in our preliminary study, the effects of reducing I f on stability and oscillation dynamics of model cells were much smaller than those of reducing I Ca,L or I K . Nevertheless, the most prominent electrophysiological property of SAN cells is the possession of relatively large I f , which is very small or negligible ...

Roles of sarcoplasmic reticulum Ca2+ cycling and Na+/Ca2+ exchanger in sinoatrial node pacemaking: Insights from bifurcation analysis of mathematical models

“…The theory and methods for bifurcation analyses are essentially the same as described previously (20,21). Bifurcation parameters chosen in this study included the following: 1) P up; 2) kNCX, indicating NCX density (given as percentages of the standard value of 187.5 pA/pF); 3) maximum conductance of I CaL (gCaL), ICaT (gCaT), and IbCa (gbCa); and 4) the amplitude of the constant bias current (I bias).…”

“…As a role of SR Ca 2ϩ cycling and NCX in SAN pacemaking, they may enhance the robustness of pacemaker activity against hyperpolarizing or depolarizing loads, like sarcolemmal ionic channels (17,20,21). As mentioned in METHODS, a convenient way of evaluating robustness of pacemaker systems is to determine the critical I bias value for bifurcations to quiescence or arrhythmic firing.…”

“…Bifurcation diagrams can be constructed by the ordinary numerical integration of differential equations if the concepts of bifurcation theory are applied, which is the brute-force method (7,39). However, strict bifurcation (stability) analysis, called the "continuation" method (5,39,47), can determine the stability and bifurcation of EPs (and LCs) more rapidly and accurately even when a system exhibits multistability (coexistence of multiple EPs and LCs) with hysteresis, as suggested for nonlinear systems, including cardiac myocytes (10,21,22). It is difficult to detect such multistability by numerical simulation.…”

“…In fact, our previous studies (17,20,21) have indicated that significant insights into the roles of individual components in SAN cell pacemaking can be obtained from bifurcation analyses. This study further provides novel insights into the roles of SR Ca 2ϩ cycling and NCX in SAN cell pacemaking from the aspect of EP (in)stability.…”

“…The term "bifurcation" is defined as a qualitative change in solutions of a nonlinear dynamical system, e.g., in physiological terms, the initiation or cessation of pacemaker activity and emergence of arrhythmic firings. Bifurcation analysis using mathematical models of rabbit SAN cells has enabled us to gain significant insights into the roles of membrane clock components, i.e., sarcolemmal I CaL , I K , Na ϩ channel current (I Na ), and I f , in basal pacemaking (17,20,21). We have suggested that 1) the instability of an equilibrium point (EP) as a stationary state is of fundamental importance for the generation of robust pacemaking without annihilation (17); 2) I CaL is responsible for EP instability (17); 3) I K plays a pivotal role in preventing bifurcation to quiescence (17); and 4) I Na and I f contribute to EP instability and robust pacemaking against hyperpolarizing loads in peripheral SAN (20,21).…”