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.

Continuation-Based Numerical Detection of After-Depolarization and Spike-Adding Thresholds

“…In contrast to periodic bursting, transients bursts are challenging to address with standard bifurcation methods, which are designed for longterm behavior. Periodic bursters, also called bursting oscillators, have been studied extensively (see Rinzel, 1987;Terman, 1991;Guckenheimer, Gueron, & Harris-Warrick, 1993;Izhikevich, 2000;Govaerts & Dhooge, 2002;Guckenheimer, Tien, & Willms, 2005; Tsaneva-Atanasova, Osinga, Riess, & Sherman, 2010;Ermentrout & Terman, 2010). Here, a parameter-driven modification of firing patterns, such as spike adding, is associated with so-called fold bifurcations of these periodic orbits (Terman, 1991;Tsaneva-Atanasova et al, 2010).…”

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

“…Note that there are other Hopf bifurcations on Z, but these have been found (numerically) to occur on the repelling branch Z r of Z, and so we will concentrate only on those Hopfs that are O(ε) close to the fold surface L. The criticality of the fast subsystem Hopf typically differentiates between plateau and pseudoplateau bursting [47,39,56,54]. In our model system, Z H has always been found (numerically) to be subcritical so that the associated bursts are of pseudo-plateau type.…”

“…Historically, the analysis of bursting in slow/fast systems was pioneered by [40], and several treatments of pseudoplateau bursting followed suit [38,39,47,56]. In this traditional 3-fast/1-slow approach, the small oscillations are born from a slow passage through a dynamic Hopf bifurcation [36,37,3] and the MMOs are hysteresis loops that alternately jump at a fold and a subcritical Hopf (fold/sub-Hopf bursts) [41,24].…”

“…Characteristics of the burst pattern such as frequency and duration determine how much calcium enters the cell, which in turn determines the level of hormone secretion [59]. One particular type of bursting that has been the focus of recent modeling efforts is pseudo-plateau bursting, which features small amplitude oscillations or spikes in the active phase superimposed on relaxation type oscillations [51,47,39,56]. Pseudo-plateau bursts are distinguished from plateau bursts, which feature large amplitude fast spiking in the active phase [40,4,32,57].…”

Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

“…The coupling effect of these different fastslow time scales generally results in a system that exhibits periodic motion characterized by a combination of relatively large amplitude and nearly harmonic small amplitude oscillations [9,10], conventionally denoted by with and corresponding to the numbers of the large and small amplitude oscillations, respectively [11]. Variables that exhibit large amplitude oscillations generate a spiking state (SP) [12], whereas the systems may be in a quiescent state (QS) when all the variables exhibit small amplitude oscillations [13].…”