Slow variable dominance and phase resetting in phantom bursting

Abstract

Bursting oscillations are common in neurons and endocrine cells. One type of bursting model with two slow variables has been called 'phantom bursting' since the burst period is a blend of the time constants of the slow variables. A phantom bursting model can produce bursting with a wide range of periods: fast (short period), medium, and slow (long period). We describe a measure, which we call the 'dominance factor', of the relative contributions of the two slow variables to the bursting produced by a simple phantom bursting model. Using this tool, we demonstrate how the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. We then show that the dominance curves obtained as a parameter is varied can be useful in making predictions about the resetting properties of the model cells. Finally, we demonstrate two mechanisms by which phase-independent resetting of a burst can be achieved, as has been shown to occur in the electrical activity of pancreatic islets.

Figure 1

(A) Fast bursting, g_s1 = 20 pS. (B) Bursting is driven by s₁ (dashed), while s₂ is nearly constant (solid). (C,D) Medium bursting driven by both s₁ and s₂, g_s1 = 7 pS. (E, F) Slow bursting, is driven by s₂, g_s1 = 2 pS.

Figure 2

(A) Fast subsystem bifurcation diagram of fast bursting (g_s1 = 20 pS) with s₁ as the bifurcation parameter and s₂ = 0.436. The s₁-nullcline and burst trajectory are superimposed on the bifurcation diagram. The circle represents a Hopf bifurcation, the square represents a homoclinic bifurcation, and the triangles represent saddle node bifurcations. (B) Fast/slow analysis of medium bursting (g_s1 = 7 pS). There are two bifurcation diagrams with s₁ as the bifurcation parameter. The curve on the left has s₂ fixed at its maximum value (0.633) achieved during the bursting, while the curve on the right has s₂ fixed at is minimum value (0.600). The burst trajectory is superimposed on the diagram. Arrows indicate direction of movement of the z-curve driven by variations in s₂.

Figure 3

Phase plane analysis of fast and medium relaxation oscillations. (A) The V-nullcline (z-shaped curve) and s₁-nullcline (dotted curve) for fast oscillations with g_s1 = 40 pS and s₂ fixed at 0.436. The trajectory (heavy solid curve) follows the upper and lower branches of the V-nullcline. (B) The V-nullcline and s₁-nullcline for medium oscillations with g_s1 = 20 pS. The V-nullcline on the left has s₂ at it maximum value (0.619), while the V-nullcline on the right has s₂ at its minimum value (0.591). (C) Fast relaxation oscillations driven by s₁. (D) Medium relaxation oscillations driven by s₁ and s₂.

Figure 4

Measuring the effect of a slow variable on the duration of the active phase. The time constant of the slow variable, τ, is increased by δτ at the beginning of the active phase (arrow). This causes the slow variable to slow down and the active phase duration to increase by δAP (bold curve).

Figure 5

Interpretation of the dominance factor, DF = cosθ−sinθ. When θ = 0, DF = 1 and the oscillation is fast. When $\mathrm{\theta }=\frac{\mathrm{\pi }}{2}$, DF = −1 and the oscillation is slow. Medium frequency oscillations occur when $\mathrm{\theta }\in (0,\frac{\mathrm{\pi }}{2})$ and DF ∈ (−1, 1).

Figure 6

Results of the quantification method on the phantom relaxation oscillator. δτ=τ here and in other figures that follow. The results obtained using δτ = 0.05τ are similar for the relaxation case. (A) Oscillation period decreases with g_s1. (B) C values for active and silent phases and for s₁ and s₂. (C) For low values of g_s1 the DF is close to −1 indicating that s₂ is the variable driving slow oscillations, while for high values of g_s1 DF is close to 1 indicating that s₁ is the variable driving slow oscillations.

Figure 7

Results for phantom bursting. (A) Burst period decreases with g_s1. (B) C values for active and silent phases and for s₁ and s₂. (C) For low values of g_s1, DF is close to −1 indicating that s₂ is the variable driving slow bursting, while for high values of g_s1, DF is close to 1 indicating that s₁ is the variable driving fast bursting. The type of bursting can be defined in terms of the dominance factors.

Figure 8

As s₁ rises during an active phase, I_s1 increases, which promotes the termination of the AP. However, I_s1 starts to decline toward the end of the burst, leading to burst prolongation. Therefore, an increase in the time constant for s₂ (τ_s2), leads to a longer decline in s₁ (bold part of curve), which acts to increase AP duration.

Figure 9

Results for phantom bursting with g_s2 as the varying parameter and g_s1 = 8.5 pS. (A) Burst period decreases with g_s2. For g_s2 < 19 pS, the system spikes continuously. (B) C values for active and silent phases and for s₁ and s₂. (C) For low values of g_s2, DF_AP is close to −1 and DF_SP is close to 1 indicating that s₂ drives the active phase, while s₁ drives the silent phase. However, for high values of g_s2 DF_AP is close to 1 and DF_SP is close to −1 indicating that s₁ drives the AP, while s₂ drives the SP.

Figure 10

Bifurcation diagrams with g_s1 = 8.5 pS. The two dashed curves are the bifurcation diagrams for the extreme values of s₂. (A)For g_s2 = 100 pS, the phase point gets stuck in the SP. (B)For g_s2 = 40 pS, the phase point does not get stuck. (C)For g_s2 = 20 pS, the phase point gets stuck in the AP.

Figure 11

Resetting with g_s2 = 27 pS and g_s1 = 8.5 pS. In this case, s₁ is in control of the SP (DF = 0.85), while s₂ is in control of the AP (DF = −0.99). (A) Half-way through the AP the system was reset to the SP (arrow), which has full length. s₁ has reached its maximum at the time of resetting (bottom curve). The V and s₁ time courses have been scaled to facilitate comparison. (B) Half-way through the SP the system was reset to the AP (arrow), which is reduced. s₂ is in the middle of decreasing to its minimum value at the time of resetting (bottom curve). (C) The duration of the induced AP is phase dependent. (D) The duration of the induced SP is close to the duration of the unperturbed SP if the resetting occurs after s₁ reaches its maximum value.

Figure 12

Resetting with g_s2 = 97 pS and g_s1 = 8.5 pS. In this case, s₁ is in control of the AP (DF = 0.79), while s₂ is in control of the SP (DF = −0.68). (A) Half-way through the AP the system was reset to the SP (arrow), which is reduced. s₂ is midway to its maximum value at the time of resetting (bottom curve). (B) Half-way through the SP the system was reset to the AP (arrow), which has full length. s₁ has reached its minimum value at the time of reseting (bottom curve). (C) The duration of the induced AP is close to the width of the unperturbed AP if the resetting occurs after s₁ reaches its minimum value. (D) The width of the induced SP is phase dependent.

Figure 13

Bursting produced by the model with V-dependent s₁ and s₂ time scales (Eqs. 18, 19), g_s1 = 16 pS and g_s2 = 30 pS. (A) Voltage time course. (B) τ_s1 ≈ 10 sec during the active phase and ≈ 100 msec during the silent phase. s₁ is in control of the active phase. (C) τ_s2 ≈ 100 msec during the active phase and ≈ 10 sec during the silent phase. s₂ is in control of the silent phase.

Figure 14

Bidirectional resetting produced by the bursting model with V-dependent s₁ and s₂ time scales, g_s1 = 16 pS, and g_s2 = 30 pS. (A) Silent-active phase-independent resetting. (B) Active-silent phase-independent resetting.