Regional difference in dynamical property of sinoatrial node pacemaking: role of na+ channel current

Abstract

To elucidate the regional differences in sinoatrial node pacemaking mechanisms, we investigated 1), bifurcation structures during current blocks or hyperpolarization of the central and peripheral cells, 2), ionic bases of regional differences in bifurcation structures, and 3), the role of Na(+) channel current (I(Na)) in peripheral cell pacemaking. Bifurcation analyses were performed for mathematical models of the rabbit sinoatrial node central and peripheral cells; equilibrium points, periodic orbits, and their stability were determined as functions of parameters. Structural stability against applications of acetylcholine or electrotonic modulations of the atrium was also evaluated. Blocking L-type Ca(2+) channel current (I(Ca,L)) stabilized equilibrium points and abolished pacemaking in both the center and periphery. Critical acetylcholine concentration and gap junction conductance for pacemaker cessation were higher in the periphery than in the center, being dramatically reduced by blocking I(Na). Under hyperpolarized conditions, blocking I(Na), but not eliminating I(Ca,L), abolished peripheral cell pacemaking. These results suggest that 1), I(Ca,L) is responsible for basal pacemaking in both the central and peripheral cells, 2), the peripheral cell is more robust in withstanding hyperpolarizing loads than the central cell, 3), I(Na) improves the structural stability to hyperpolarizing loads, and 4), I(Na)-dependent pacemaking is possible in hyperpolarized peripheral cells.

FIGURE 1

Simulated effects of blocking individual sarcolemmal currents or SR Ca²⁺ release on pacemaker activity of the Zhang et al. (A) and Kurata et al. (B) models for the central and peripheral cells. Membrane potential behaviors on decreasing one of I_Ca,L, I_Na, I_Kr, I_h, I_Ca,T, and the SR Ca²⁺ release current I_rel (from top to bottom) are shown. Effects of the complete block of each current (solid lines) and 50% block of I_Ca,L, I_Na, or I_Kr (dashed lines) were tested. The SR Ca²⁺ release rate (P_rel) was reduced to 1% of the control value. Control APs are superimposed in each panel (dotted lines). Differential equations of the model systems were numerically solved for 11 s (A) or 301 s (B), with an initial condition appropriate to an EP and a 1-ms stimulus of 1 pA/pF; the membrane potential behaviors during the last 1 s (starting from a maximum diastolic potential) were depicted.

FIGURE 2

Effects of reducing g_Ca,L on the EP stability and oscillation dynamics of the Zhang et al. (A) and Kurata et al. (B) models for central and peripheral cells. One-parameter bifurcation diagrams for g_Ca,L with the steady-state branch as a locus of V_E and the periodic branches as the potential extrema of LCs (V_min, V_max) are shown (top); the CL of LCs is also plotted as a function of g_Ca,L (bottom). The steady-state and periodic branches consist of the stable (thick lines) and unstable (thin lines) segments. The normalized g_Ca,L value was reduced at an interval of 0.001 for calculations of EPs and stable LCs. The points of Hopf bifurcation of EPs, saddle-node bifurcation of LCs, period-doubling bifurcation of LCs, and Neimark-Sacker bifurcation of LCs are located, as labeled H_S, SN_P, PD, and NS, respectively: (A) H_S at g_Ca,L = 0.6178 (central) and 0.4912 (peripheral); SN_P at g_Ca,L = 0.5890 (central) and 0.2337 (peripheral). (B) H_S at g_Ca,L = 0.2457 (central) and 0.1891 (peripheral); SN_P at g_Ca,L = 0.085 (peripheral); PD at g_Ca,L = 0.257 (central); and NS at g_Ca,L = 0.312 (central).

FIGURE 3

Effects of reducing g_Kr on the EP stability and oscillation dynamics of the Zhang et al. (A) and Kurata et al. (B) models. One-parameter bifurcation diagrams for g_Kr with the steady-state branch (V_E) and the periodic branches (V_min, V_max) are shown (top); the CL of LCs is also plotted as a function of g_Kr (bottom). The normalized g_Kr value was reduced at an interval of 0.001 for calculations of EPs and stable LCs. The points of Hopf bifurcation of EPs (H_S) and saddle-node bifurcation of LCs (SN_P) are located: (A) H_S at g_Kr = 0.3384 (central) and 0.0228 (peripheral); SN_P at g_Kr = 0.3298 (central). (B) H_S at g_Kr = 0.3440 (central) and 0.1867 (peripheral).

FIGURE 4

Changes in EPs and their stability during I_bias applications of the normal (control), I_Ca,L-removed (g_Ca,L = 0), and I_Kr-removed (g_Kr = 0) versions of the Kurata et al. central (A) and peripheral (B) models. Shown are the I_bias-V_E curves illustrating how V_E and its stability changed when I_bias was applied (top). V_E was systematically changed by applying hyperpolarizing (negative) or depolarizing (positive) I_bias of various amplitudes; the resulting relationship between V_E (abscissa) and I_bias amplitude (ordinate) is depicted. Stability of EPs was evaluated by the stability analysis based on the linear stability theory (see Appendix B); real parts of eigenvalues (Re[λ]) of a Jacobian matrix for each EP (V_E) representing the rate constants (ms⁻¹) of a system to be attracted to or go away from an EP are shown (bottom). In the I_bias-V_E curve, the segments of thin lines designate the loci of unstable EPs with two positive Re[λ] values; those of thick lines are the stable EPs for which Re[λ] values are all negative. There are 2–4 Hopf bifurcation points (H1–H4), corresponding to the zero crossings of Re[λ]. Saddle-node bifurcation points are also located (labeled SN). Individual time-dependent current components in steady state are superimposed to show the contribution of each current to the stability of EPs: a, I_Ca,L; b, I_Ca,T; c, I_h; d, I_st; e, I_Na; f, I_Kr; g, I_to + I_sus; and h, I_Ks.

FIGURE 5

Oscillation dynamics during I_bias applications of the normal (control), I_Ca,L-reduced (g_Ca,L = 0.2 or 0), and I_Kr-removed (g_Kr = 0) versions of the Kurata et al. central (A) and peripheral (B) models. A bifurcation diagram for I_bias with the steady-state (V_E) and stable periodic (V_min, V_max) branches is shown; CL is also plotted against I_bias. Various amplitudes of hyperpolarizing (negative) or depolarizing (positive) I_bias were applied at an interval of 0.001 pA/pF. Hopf (H1–H3) or saddle-node (SN1, SN2) bifurcation of EPs, and saddle-node (SN_P), period-doubling (PD), or Neimark-Sacker (NS) bifurcation of LCs are located.

FIGURE 6

Effects of reducing g_Ca,L on bifurcation structures (stability and bifurcation of EPs) during I_bias applications of the Zhang et al. (A) and Kurata et al. (B) models for central and peripheral cells. Displacements of Hopf (H1–H4) and saddle-node (SN1, SN2) bifurcation points in the I_bias-V_E curve with decreasing g_Ca,L at an interval of 0.001 are shown on the potential (V_E) coordinates (top) and current (I_bias) coordinates (bottom). The path of the control V_E at I_bias = 0 is also shown on the potential coordinates (V₀), intersecting the locus of H1 or H2 at the Hopf bifurcation points (H_S) as shown in Fig. 2. In the insets, V_E and its stability, as well as bifurcation points, are shown as functions of I_bias for g_Ca,L = 1, 0.75, and 0.5 (A, central); g_Ca,L = 1, 0.55, and 0 (A, peripheral); or g_Ca,L = 1, 0.2, and 0 (B), as indicated by the dashed lines. The unstable regions where spontaneous oscillations occur are as follows: A, between H1 and H2 (or H3 and H4 in the periphery); B (central), between H1 and H2 (or H3 and H4); B (peripheral), between H1 and H2, or above H3 (or above H1 at g_Ca,L = 0.221–0.605). A codimension-two saddle-node bifurcation point at which the loci of H1 and H2 merge together, i.e., the unstable region disappears, is labeled SN_S2 (codimension-two saddle-node bifurcation of EPs) at g_Ca,L = 0.6131 (A, central) and 0.1382 (B, central).

FIGURE 7

Effects of reducing g_Kr on bifurcation structures during I_bias applications of the Zhang et al. (A) and Kurata et al. (B) models. Displacements of Hopf (H1–H4) and saddle-node (SN1, SN2) bifurcation points in the I_bias-V_E curve with decreasing g_Kr at an interval of 0.001 are shown on the potential (top) and current (bottom) coordinates. The path of V₀ is also shown on the potential coordinates, intersecting the locus of H1 or H2 at the Hopf bifurcation points (H_S) as shown in Fig. 3. In the insets, V_E and its stability, as well as bifurcation points, are shown as functions of I_bias for g_Kr = 1, 0.5, and 0 (A) or g_Kr = 1, 0.2, and 0 (B), as indicated by the dashed lines. The unstable regions are as follows: A, between H1 and H2; B (central), between H1 and H2 (or SN1 and H2 at g_Kr < 0.318), and H3 and H4 (or H3 and SN1 at g_Kr < 0.090); B (peripheral), between H1 and H2, and above H3.

FIGURE 8

Effects of ACh applications (increases in [ACh]) on the stability and dynamics of the Zhang et al. (A) and Kurata et al. (B) models. One-parameter bifurcation diagrams for [ACh] (log[M]) depicting a locus of V_E and the potential extrema of LCs (V_min, V_max) are shown (top); the CL of LCs is also shown as functions of log[ACh] (bottom). For calculations of EPs and stable LCs, the log[ACh] values were increased at the interval of 0.001. Hopf bifurcation points on the steady-state branches are labeled H_S: A, at log[ACh] = −7.3332 (central) and −6.6059 (peripheral); B, at log[ACh] = −7.2218 (central) and −6.7545 (peripheral). The points at which a saddle-node (SN_P) or period-doubling (PD) bifurcation of LCs occurs with disappearance of stable LCs are located on the periodic branches: A, SN_P at log[ACh] = −7.3239 (central); PD at log[ACh] = −6.7148 (peripheral); B, PD at log[ACh] = −7.329 (central) and −6.874 (peripheral). Potential extrema and CLs of the irregular dynamics emerging between PD and H_S are also superimposed.

FIGURE 9

Effects of electrotonic loads of the atrium (increases in G_C) on the stability and dynamics of the Zhang et al. (A) and Kurata et al. (B) models. One-parameter bifurcation diagrams for G_C (nS) depicting a locus of V_E and the potential extrema of LCs (V_min, V_max) are shown (top); the CL of LCs is also shown as functions of G_C (bottom). G_C values for the central and peripheral cells were increased at the interval of 0.001 and 0.01 (nS), respectively. Hopf bifurcation points are labeled H_S: A, at G_C = 0.2067 (central) and 11.953 (peripheral); B, at G_C = 0.2925 (central) and 8.0688 (peripheral). The points at which a saddle-node (SN_P), period-doubling (PD), or Neimark-Sacker (NS) bifurcation of LCs occurs are located: A, SN_P at G_C = 0.2152 (central) and 10.484 (peripheral); B, PD at G_C = 0.230 (central); NS at G_C = 5.51 (peripheral). Potential extrema and CLs of the irregular dynamics emerging between the bifurcation of LCs and H_S are also superimposed.

FIGURE 10

Effects of blocking I_Na or I_h on bifurcation structures during I_bias applications of the I_Ca,L-removed Zhang et al. (A) and Kurata et al. (B) peripheral models. Displacements of Hopf (H1–H4) and saddle-node (SN1–SN4) bifurcation points in the I_bias-V_E curve with decreasing g_Na or g_h at an interval of 0.001 are shown on the potential (top) and current (bottom) coordinates. The path of V₀ is also shown on the potential coordinates. In the insets, V_E and its stability, as well as bifurcation points, are shown as functions of I_bias for g_Ca,L = 1, 0.5, and 0, as indicated by the dashed lines. The unstable regions where spontaneous oscillations occur are as follows: A, between H1 and H2 (or H3 and H4); B, between H1 (or SN1) and H2, or above H3. A codimension-two saddle-node bifurcation point at which the loci of H1 and H2 merge together is labeled SN_S2 at g_Na = 0.4470 (A) and 0.388 (B).

FIGURE 11

Effects of blocking individual currents on bifurcations during ACh applications of the Zhang et al. (A) and Kurata et al. (B) peripheral models. Two-parameter bifurcation diagrams for the normalized maximum current conductance (abscissa) and log[ACh] (ordinate) are shown; the critical [ACh] values at Hopf bifurcation points were plotted as functions of the conductance, which was decreased at an interval of 0.001. The values of [ACh] (log[M]) were increased at an interval of 0.001 for searching for bifurcation points. In the insets, one-parameter bifurcation diagrams for log[ACh] used for construction of the two-parameter diagrams are shown for g_Na, g_h, and g_Ca,L at 1, 0.5, and 0, as indicated by the dashed lines in the two-parameter diagrams.

FIGURE 12

Effects of blocking individual currents on bifurcations during electrotonic modulations of the Zhang et al. (A) and Kurata et al. (B) peripheral models. Two-parameter bifurcation diagrams for the normalized maximum current conductance (abscissa) and G_C (ordinate) are shown; the critical G_C values at Hopf bifurcation points were plotted as functions of the conductance, which was decreased at an interval of 0.001. The values of G_C (nS) were increased at an interval of 0.01 for searching for bifurcation points. In the insets, one-parameter bifurcation diagrams for G_C used for construction of the two-parameter diagrams are shown for g_Na, g_h, and g_Ca,L at 1, 0.5, and 0, as indicated by the dashed lines in the two-parameter diagrams.

FIGURE 13

Bifurcation structures during ACh applications of the normal (control) and I_Na-removed (0 I_Na) versions of the Zhang et al. (A) and Kurata et al. (B) peripheral models. One-parameter bifurcation diagrams for log[ACh] depicting the steady-state (V_E) and periodic (V_min, V_max) branches are shown (top); the CL of LCs is also shown as functions of log[ACh] (bottom). Procedures of the calculations are the same as for Fig. 8. Hopf (H_S) and period-doubling (PD) bifurcation points are located: A, H_S at log[ACh] = −6.6059 (control) and −7.3586 (0 I_Na); PD at log[ACh] = −6.7148 (control) and −7.4250 (0 I_Na); B, H_S at log[ACh] = −6.7545 (control) and −7.2241 (0 I_Na); PD at log[ACh] = −6.874 (control) and −7.930 (0 I_Na).

FIGURE 14

Bifurcation structures during electrotonic modulations of the normal (control) and I_Na-removed (0 I_Na) versions of the Zhang et al. (A) and Kurata et al. (B) peripheral models. One-parameter bifurcation diagrams for G_C depicting the steady-state (V_E) and periodic (V_min, V_max) branches are shown (top); the CL of LCs is also shown as functions of G_C (bottom). Procedures of the calculations are the same as for Fig. 9. Hopf (H_S), saddle-node (SN_P), period-doubling (PD), and Neimark-Sacker (NS) bifurcation points are located: A, H_S at G_C = 11.953 (control) and 1.3152 (0 I_Na); SN_P at G_C = 10.484 (control); PD at G_C = 1.1380 (0 I_Na); B, H_S at G_C = 8.0688 (control) and 1.3170 (0 I_Na); PD at G_C = 0.15 (0 I_Na); NS at G_C = 5.51 (control).

FIGURE 15

EP stability and bifurcation during decreases in g_Ca,L and/or g_Na of the Zhang et al. (A) and Kurata et al. (B) peripheral models in the presence of ACh at various concentrations. Each panel shows a two-parameter bifurcation diagram for g_Ca,L and g_Na, depicting the loci of Hopf bifurcation points. Values of [ACh] are shown at the top of each panel in log[M]. The normalized g_Ca,L value was decreased from 1 to 0 at an interval of 0.001 for searching for bifurcations, with g_Na fixed at various values; the g_Na value was decreased from 1 to 0 at an interval of 0.001 (see Appendix C). Critical g_Ca,L values at which Hopf bifurcations occurred were determined and plotted as functions of g_Na. The symbols S and US represent the stable and unstable EP regions, respectively.

FIGURE 16

Bifurcation structures of the normal and hyperpolarized peripheral cells during the block of I_Ca,L or I_Na. Both the Zhang et al. (A) and Kurata et al. (B) models were tested. One-parameter bifurcation diagrams for g_Ca,L or g_Na depicting steady-state (V_E) and periodic (V_min, V_max) branches were constructed for the peripheral cells in the absence (left) and presence (right) of ACh (top); the CL of LCs is also shown as functions of g_Ca,L or g_Na (bottom). The [ACh] values for the Zhang et al. and Kurata et al. models are 1 × 10^−6.8 and 1 × 10^−7.0 M (158.5 and 100 nM), respectively. Procedures of the calculations are the same as for Fig. 2. Hopf (H_S), saddle-node (SN_P), period-doubling (PD), and Neimark-Sacker (NS) bifurcations are located: A, H_S at g_Ca,L = 0.4912 (normal) and g_Na = 0.5917 (hyperpolarized); SN_P at g_Ca,L = 0.2337 (normal); PD at g_Na = 0.7073 (hyperpolarized); B, H_S at g_Ca,L = 0.1891 (normal) and g_Na = 0.3178 (hyperpolarized); SN_P at g_Ca,L = 0.085 (normal); PD at g_Na = 0.074 (normal); NS at g_Na = 0.084 (normal) and 0.595 (hyperpolarized).

FIGURE 17

Steady-state behaviors of spontaneous APs, underlying transmembrane ionic currents, and intracellular Ca²⁺ dynamics simulated by the central and peripheral versions of the Zhang et al. (A) and Kurata et al. (B) models with the standard parameter values. Differential equations were numerically solved for 11 s (A) or 181 s (B) with initial conditions appropriate to an EP and a 1-ms stimulus of 1 pA/pF. Model cell behaviors during the last 1 s starting from maximum diastolic potential are depicted. Intracellular Ca²⁺ dynamics include the changes of Ca²⁺ concentrations in subspace ([Ca²⁺]_S) and myoplasm ([Ca²⁺]_i).

FIGURE 18

How to construct the two-parameter bifurcation diagram for g_Ca,L and g_Na (Fig. 15) from one-parameter bifurcation diagrams for g_Ca,L or g_Na. (Left) The two-parameter bifurcation diagram for the Zhang et al. peripheral model with the [ACh] of 1 × 10^−7.32 M (47.863 nM). The point labeled C corresponds to the control condition. The leftward and downward shifts from C indicate the block of I_Na and I_Ca,L, respectively (shown by the arrows). Dashed lines designate the points of g_Na (top) or g_Ca,L (bottom) for the construction of the one-parameter bifurcation diagrams shown on the right. (Right) One-parameter bifurcation diagrams depicting V_E and its stability as functions of g_Ca,L (top) or g_Na (bottom). The normalized g_Ca,L (top) or g_Na (bottom) value as the first parameter was decreased from 1 to 0 at an interval of 0.001 for searching for Hopf bifurcations, with the second parameter g_Na (top) or g_Ca,L (bottom) fixed at various values as shown on the right of each panel. Critical values of the first parameter g_Ca,L (top) or g_Na (bottom) at which Hopf bifurcations occurred were determined by the stability analysis. The thick and thin segments are the stable and unstable V_E regions, respectively; the boundaries between the two segments correspond to the bifurcation points. The two-parameter bifurcation diagram can be constructed by plotting these bifurcation values of the first parameter g_Ca,L (or g_Na) as functions of the second parameter g_Na (or g_Ca,L).