Full system bifurcation analysis of endocrine bursting models

Abstract

Plateau bursting is typical of many electrically excitable cells, such as endocrine cells that secrete hormones and some types of neurons that secrete neurotransmitters. Although in many of these cell types the bursting patterns are regulated by the interplay between voltage-gated calcium channels and calcium-sensitive potassium channels, they can be very different. We investigate so-called square-wave and pseudo-plateau bursting patterns found in endocrine cell models that are characterized by a super- or subcritical Hopf bifurcation in the fast subsystem, respectively. By using the polynomial model of Hindmarsh and Rose (Proceedings of the Royal Society of London B 221 (1222) 87-102), which preserves the main properties of the biophysical class of models that we consider, we perform a detailed bifurcation analysis of the full fast-slow system for both bursting patterns. We find that both cases lead to the same possibility of two routes to bursting, that is, the criticality of the Hopf bifurcation is not relevant for characterizing the route to bursting. The actual route depends on the relative location of the full-system's fixed point with respect to a homoclinic bifurcation of the fast subsystem. Our full-system bifurcation analysis reveals properties of endocrine bursting that are not captured by the standard fast-slow analysis.

Figure 1

Bifurcation diagrams of the fast subsystem in the generic endocrine model (Appendix A.1) showing the bifurcations associated with square-wave (fold-homoclinic) bursting (panel a) and pseudo-plateau (fold-subHopf) bursting (panel b). (a) V_mL = −22.5 mV; (b) V_mL = −27.5 mV; HB — Hopf bifurcation; SN — saddle-node bifurcation; SNP — saddle-node of periodics; HC — homoclinic bifurcation point. Dashed lines denote instability. Sample bursting trajectories are superimposed on the bifurcation diagrams for each of the models.

Figure 2

Bifurcation diagrams of the fast subsystem in the polynomial model Eqns. (1)–(2) showing the bifurcations associated with the transition from square-wave (fold-homoclinic) to pseudo-plateau (fold-subHopf) bursting as well as the z-nullclines (diagonal purple, blue and green lines) for three different values of the parameter b₁ and with (a) s = −1.61, or (b) s = −2.6; HB — Hopf bifurcation; SN — saddle-node bifurcation; HC — homoclinic bifurcation point. Dashed lines denote instability.

Figure 3

(a) Three-dimensional view (ε, z, x) of the one-parameter bifurcation diagram with respect to ε of the full polynomial model Eqns. (1)–(3) in the case of square-wave bursting (s = −1.61, b₁ = −0.01); HB — Hopf bifurcation; SNP — saddle-node of periodics; FP — fixed point; HC — homo-clinic bifurcation point. Dashed lines denote instability; (b) Sample bursting trajectories with increasing number of spikes, i.e., decreasing values of ε = 0.009; 0.005; 0.004; 0.0035; 0.0027; 0.0024; 0.002 are superimposed on the bifurcation diagram.

Figure 4

(a) Bifurcation diagram of the full polynomial model Eqns. (1)–(3) with respect to ε in the case of pseudo-plateau bursting (s = −2.6, b₁ = −0.01); HB — Hopf bifurcation; SNP — saddle-node of periodics; FP — fixed point; PD — period-doubling bifurcation; HC — homoclinic bifurcation point. Dashed lines denote instability; (b) Sample bursting trajectories with increasing number of spikes, i.e., decreasing values of ε = 0.08; 0.06; 0.035; 0.023 are superimposed on the bifurcation diagram.

Figure 5

(a) Bifurcation of the full polynomial model Eqns. (1)–(3) in the case of square-wave bursting (s = −1.61, b₁ = −0.045); HB — Hopf bifurcation; PD — period-doubling bifurcation; FP — fixed point; HC — homoclinic bifurcation point. Dashed lines denote instability; (b) Sample bursting trajectories with increasing number of spikes, i.e., decreasing values of ε = 0.07; 0.04; 0.03; 0.024; 0.018; 0.015; 0.014; 0.012 are superimposed on the bifurcation diagram.

Figure 6

(a) Bifurcation diagram of the full polynomial model Eqns. (1)–(3) in the case of pseudo-plateau bursting (s = −2.6, b₁ = −0.21); HB — Hopf bifurcation; PD — period-doubling bifurcation; FP — fixed point; SNP —saddle-node of periodics; HC — homoclinic bifurcation point. Dashed lines denote instability; (b) Sample bursting trajectories with increasing number of spikes, i.e., decreasing values of ε = 1.105; 0.9; 0.6; 0.4; 0.25; 0.15 are superimposed on the bifurcation diagram.

Figure 7

Simulations of the polynomial model showing the apparently chaotic spike-adding transition in the pseudo-plateau bursting regime in the case when (a) FP is well below the HC (s = −2.6, b₁ = −0.01); and (b) when FP is well above the HC (s = −2.6, b₁ = −0.21).

Figure 8

Bifurcation diagrams of the full polynomial model Eqns. (1)–(3) in the cases of (a) square-wave bursting (s = −1.61, b₁ = −0.024), where FP lies just below the HC; and (b) pseudo-plateau bursting (s = −2.6, b₁ =−0.066), where FP lies slightly above the HC; HB — Hopf bifurcation; PD — period-doubling bifurcation; FP — fixed point; SNP — saddle-node of periodics; HC — homoclinic bifurcation point. Dashed lines denote instability.

Figure 9

Simulations showing the behavior in the pseudo-plateau bursting regime for small ε in the cases (a) when FP is well below the HC (s = −2.6, b₁ =−0.01, ε = 0.0001); and (b) when FP is above the HC but below SN₁ (s =−2.6, b₁ = −0.12, ε = 0.001).

Figure 10

Three-dimensional view (ε, z, −b₁) of the two-parameter bifurcation diagram of the full polynomial model with respect to ε and b₁ showing the loci of HB₂ for s = −1.61 (blue) and s = −2.6 (red); HB — Hopf bifurcation; SN — saddle-node bifurcation. The vertical, dashed lines show the z-values of the hopf bifurcation HB₁ of the fast subsystem (ε = 0).