From plateau to pseudo-plateau bursting: making the transition

Abstract

Bursting electrical activity is ubiquitous in excitable cells such as neurons and many endocrine cells. The technique of fast/slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the design principles for bursting in endocrine cells are just quantitative variations of those for bursting in neurons.

Figure 1

Bursting oscillations and bifurcation structures for plateau bursting (left) simulated using the Chay-Keizer model, and pseudo-plateau bursting (right) simulated using a model for pituitary lactotrophs (Tabak et al., 2007) with υ_n = −9.5 mV. (A) plateau bursting oscillations. (B) Slow dynamics of Ca²⁺ concentration. (C) The fast/slow analysis. (D) Pseudo-plateau bursting oscillations. (E) Ca²⁺ dynamics are faster than in plateau bursting, producing a much shorter burst period. (F) The fast/slow analysis. In panels C and F, LSN, USN, and HM are lower saddle-node, upper saddle-node and homoclinic bifurcations, respectively. The Hopf bifurcations are supHB (supercritical) in panel C and subHB (subcritical) in panel F. V_max and V_min correspond to the maximum and minimum voltage of the fast subsystem periodic solution for a range of c values. Default parameter values are used for both models, with f = 0.00025 for plateau and f = 0.01 for pseudo-plateau bursting.

Figure 2

Pseudo-plateau bursting generated with the Chay-Keizer model by increasing the speed of K⁺ current activation. (A) When τ_n is decreased to 17.1 ms and f is increased to 0.01, pseudo-plateau bursting is produced. (B) The fast-subsystem bifurcation structure, in which the upper branch is stabilized. There are two supercritical Hopf bifurcations connected by a branch of stable periodic solutions, which differs from Fig. 1F.

Figure 3

Steady state functions for the K⁺ current activation variable n (the n_∞ (V) curve, blue) and the Ca²⁺ current activation variable m (the m_∞ curve, red). When υ_n is increased from −16 mV (point a) to −14.5 mV (point b) or −12 mV (point c), the n_∞ curve shifts to the right. This makes the delayed rectifier channels activate at higher voltages. When υ_m is decreased from −20 mV (point d) to −22.5 mV (point e) or −26 mV (point f), the m_∞ curve shifts to the left. This makes the Ca²⁺ channels activate at lower voltages. When the m_∞ and n_∞ curves are sufficiently far apart, pseudo-plateau bursting is produced.

Figure 4

Effects of shifting the n_∞(V) curve rightward. (A) The bifurcation structure for plateau bursting using default values (Table 1). (B) A transitional bifurcation structure obtained by increasing υ_n to −14 mV. (C) The bifurcation structure for pseudo-plateau bursting obtained with υ_n = −12 mV.

Figure 5

Bursting oscillations generated with the Chay-Keizer model and different values of f and υ_n. For the left panels υ_n= −16 mV, and for the right υ_n= −12 mV. (A), (B) f = 0.00025. (C), (D) f = 0.005. (E), (F) f = 0.01.

Figure 6

Two-parameter bifurcation diagrams illustrating regions of plateau and pseudo-plateau bursting. The green (LSN), blue (HB), red (HM), and black (USN) curves represent the lower saddle node, Hopf, homoclinic, and upper saddle node bifurcation points, respectively. (A), (C) The midpoints of activation curves are varied. (B), (D) Current conductances are varied. In panel D, the region below the bottom dotted curve is neither plateau nor pseudo-plateau bursting since the order is c_LSN < c_HB < c_HM < c_USN.

Figure 7

Cumulative properties of parameter variations. (A) Changes in υ_n and υ_m that convert plateau (top left square, (a, d)) to pseudo-plateau bursting (points on or below the curve). a, b and c are midpoints of the n_∞(V) curves, and d, e and f are midpoints of the m_∞(V) curves (see Fig. 3). (B) Changes in υ_n and g_K that convert plateau to pseudo-plateau bursting (on or below the curve).

Figure 8

Bursting oscillations using the lactotroph model with parameter values in Table 2. (A) Pseudo-plateau bursting with υ_n = −9.5 mV. (B) Plateau bursting with υ_n = −15 mV and f = 0.00025.

Figure 9

Two-parameter bifurcation diagrams for a model of the pituitary lactotroph illustrating regions of plateau and pseudo-plateau bursting. (A), (B) Using the parameter values given in Table 2. (C) Using g_A = 4 nS, τ_h = 20 ms and λ = 1.2. (D) Using υ_m = −27 mV, g_K = 7.2 nS, g_A = 2.5 nS, τ_h = 20 ms and λ = 1.2.