Autocatalator as the source of instability in the complex non-linear neuroendocrine model

Abstract

The mathematical analogy between properties of simple and complex models of non-linear reactions was used to determine reaction steps in the complex model, necessary to generate instability region and appropriate type of bifurcations on the border between stable and unstable non-equilibrium stationary state. The autocatalator was recognized as the simple prototype two-variable non-linear model practical for examination of the complex four-variable non-linear neuroendocrine system known as the Hypothalamic-Pituitary-Adrenal (HPA) axis. In both cases, we derived the instability criteria by stoichiometric network analysis (SNA), determine conditions under which dynamic transitions, i.e. bifurcations occur, and identify the type of bifurcation. The supercritical Andronov-Hopf bifurcation was found in both cases whereas saddle-node bifurcation was detected only in the model for HPA axis. Thus, by stoichiometric network analysis, the mathematical analogy is found between two different models with same autocatalytic steps, that is, between two models with easily recognizable qualitative analogy.