Time-dependent changes in membrane excitability during glucose-induced bursting activity in pancreatic β cells

Abstract

In our companion paper, the physiological functions of pancreatic β cells were analyzed with a new β-cell model by time-based integration of a set of differential equations that describe individual reaction steps or functional components based on experimental studies. In this study, we calculate steady-state solutions of these differential equations to obtain the limit cycles (LCs) as well as the equilibrium points (EPs) to make all of the time derivatives equal to zero. The sequential transitions from quiescence to burst-interburst oscillations and then to continuous firing with an increasing glucose concentration were defined objectively by the EPs or LCs for the whole set of equations. We also demonstrated that membrane excitability changed between the extremes of a single action potential mode and a stable firing mode during one cycle of bursting rhythm. Membrane excitability was determined by the EPs or LCs of the membrane subsystem, with the slow variables fixed at each time point. Details of the mode changes were expressed as functions of slowly changing variables, such as intracellular [ATP], [Ca(2+)], and [Na(+)]. In conclusion, using our model, we could suggest quantitatively the mutual interactions among multiple membrane and cytosolic factors occurring in pancreatic β cells.

Figure 1.

Changes in EPs and LCs in the whole β-cell model by varying [G]. Four bifurcation diagrams showing continuous changes in EPs or LCs for V_m (A), [ATP] (B), [Ca²⁺]_i (C), and [Na⁺]_i (D) with respect to [G]. Stable EPs, unstable EPs, stable LCs, and unstable LCs are indicated by black, red, blue, and yellow lines, respectively. For LCs, the maximum and minimum values in oscillations were plotted. The amplitude of oscillation in [ATP] and [Na⁺]_i was small, and the maximum and minimum curves of LCs fused with each other. AUTO failed to find unstable LCs at [G] < 9 mM. Open circles indicate bifurcation points; HB, Hopf bifurcation from a stable EP to an unstable EP (at 6.9 mM [G]); TR, Torus bifurcation from a stable LC to an unstable LC (at 18.84 mM [G]).

Figure 2.

Mode changes of membrane excitability by varying P_CaV. (A) Bifurcation diagrams showing continuous changes in V_m of EPs or LCs as a function of P_CaV. Stable EPs, unstable EPs, stable LCs, and unstable LCs are indicated by black, red, blue, and yellow lines, respectively. The black line for a stable EP corresponds to the resting membrane potential, and the two blue lines for a stable LC correspond to the amplitude of the action potentials. The slow variables were fixed to the following values: [ATP] = 2.64 mM, [MgADP] = 0.0591 mM, [Re] = 0.61 mM, [Na⁺]_i = 5.87 mM, [K⁺]_i = 127 mM, [Ca²⁺]_i = 0.124 µM, [Ca²⁺]_ER = 0.0247 mM, f_us = 0.843, and I₁ = 0.152. Open circles indicate bifurcation points; LP, LP bifurcation; HB, Hopf bifurcation; PD, period doubling bifurcation. Black vertical lines with the blue italic numeral indicate control values of P_CaV (48.9 pA mM⁻¹) in the β-cell model. EP₁, EP₂, and EP₃ (gray dots) are the intersections of the EP curve with the black vertical line. Gray vertical lines passing through the corresponding bifurcation points separate individual modes, as indicated at the top. (B) Steady-state I-V relationship. Zero current potentials correspond to EP₁, EP₂, and EP₃ in A.

Figure 3.

Representative membrane responses to current injections in different modes. The amplitudes of injected current pulses are indicated inside panels. Duration of the pulse was 5 ms, except 15 ms in Mode E. In each panel, a different value of P_CaV was used for simulation (units in pA mM⁻¹): Mode A, P_CaV was 30; Mode B, 45; Mode C, 48.9; Mode D, 60; Mode E, 65; Mode F, 69. The same values were used for the slow variables as in Fig. 2. The time axes were identical for all panels.

Figure 4.

Time-dependent mode changes in membrane excitability during one burst–interburst cycle. (A) Time-based simulation of V_m at 8 mM [G] (gray continuous line). Colored dots are EP₁, EP₂, EP₃, and LC (min and max) in the membrane system, which were measured from the bifurcation diagrams calculated with fixed values of [S]_is, [Ca²⁺]_ER, f_us, and I₁ at corresponding time points. Stable EPs, unstable EPs, stable LCs, and unstable LCs are indicated by black, red, blue, and yellow dots, respectively. The unstable LC (yellow) was only observed at the moment of mode change from Mode B to C (19.5 s) during <0.1 s, whereas it was not observed at the switch from Mode C to B during the burst (15.5 s). During the burst period, two sets of EPs and LCs were demonstrated at the sequential maximum or minimum of [Ca²⁺]_i. These two EP₃s are almost superimposed in the figure. The mode of membrane excitability at each moment is indicated at the top. (Inset) Trace of [Ca²⁺]_i during the same burst–interburst cycle. (B) Bifurcation diagrams at six representative time points in A, as indicated in the top left part of the figure. During the burst, three time points were selected from those of sequential minimums of [Ca²⁺]_i (11, 14.6, and 15.5 s). The same color codes were used for the dots as those in A. Black vertical lines were drawn at P_CaV = 48.9 pA mM⁻¹, the standard value in the β-cell model.

Figure 5.

Effects of individual slow variables on the mode change toward termination of the burst. Each bifurcation diagram was obtained by varying single slow variable, [ATP] (A), f_us (B), [Ca²⁺]_i (C), [Na⁺]_i (D), [K⁺]_i (E), and I₁ (F) as a bifurcation parameter, with the remaining slow variables fixed at their values at 0.9 s before the LP bifurcation (open circles). Stable EPs, unstable EPs, stable LCs, and unstable LCs are indicated by black, red, blue, and yellow lines, respectively. Six gray vertical lines in A–D indicate the values of the corresponding slow variables sampled at an interval of 1 s (from 11 to 16 s in Fig. 4 A). In C, the sequential values of the minimum [Ca²⁺]_i were sampled. In E and F, two gray lines indicate the corresponding values at 11 and 16 s. Arrows represent the sampling sequence. We confirmed that the bifurcation diagrams were qualitatively the same when slow variables were fixed at different time points during the burst.