Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK.
J Theor Biol. 2010 Jun 21;264(4):1133-46. doi: 10.1016/j.jtbi.2010.03.030. Epub 2010 Mar 20.
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.
高原爆裂是许多可兴奋细胞的典型特征，例如分泌激素的内分泌细胞和一些分泌神经递质的神经元。尽管在许多这些细胞类型中，爆发模式是由电压门控钙通道和钙敏钾通道的相互作用调节的，但它们可能非常不同。我们研究了在快速子系统中分别具有超临界或亚临界 Hopf 分岔的内分泌细胞模型中发现的所谓方波和拟高原爆发模式。通过使用 Hindmarsh 和 Rose（英国皇家学会会刊 B 221（1222）87-102）的多项式模型，该模型保留了我们考虑的生物物理模型类的主要特性，我们对两种爆发模式的完整快速-缓慢系统进行了详细的分岔分析。我们发现这两种情况都导致了爆发的两种可能途径，即 Hopf 分岔的临界性对于表征爆发途径并不重要。实际的途径取决于整个系统的固定点相对于快速子系统的同宿分岔的相对位置。我们的全系统分岔分析揭示了内分泌爆发的一些特性，这些特性无法通过标准的快速-缓慢分析来捕捉。
