Gibo Shingo, Ito Hiroshi
Graduate School of Design, Kyushu University, 4-9-1, Shiobaru Minami-ku, Fukuoka 815-8540, Japan.
Faculty of Design, Kyushu University, 4-9-1, Shiobaru Minami-ku, Fukuoka 815-8540, Japan.
J Theor Biol. 2015 Aug 7;378:89-95. doi: 10.1016/j.jtbi.2015.04.024. Epub 2015 Apr 30.
Many biological rhythms are generated by negative feedback regulation. Griffith (1968) proved that a negative feedback model with two variables expressed by ordinary differential equations do not generate self-sustained oscillations. Kurosawa et al. (2002) expanded Griffith׳s result to the general type of negative feedback model with two variables. In this paper, we propose discrete and ultradiscrete feedback models with two variables that exhibit self-sustained oscillations. To obtain the model, we applied tropical discretization and ultradiscretization to a continuous model with two variables and then investigated its bifurcation structures and the conditions of parameters for oscillations. We found that when the degradation rate of the variables is lower than their synthesis rate, the proposed models generate oscillations by Neimark-Sacker bifurcation. We further demonstrate that the ultradiscrete model can be reduced to a Boolean system under some conditions.
许多生物节律是由负反馈调节产生的。格里菲思（1968年）证明，一个由常微分方程表示的具有两个变量的负反馈模型不会产生自持振荡。黑泽等人（2002年）将格里菲思的结果扩展到具有两个变量的一般类型的负反馈模型。在本文中，我们提出了具有两个变量的离散和超离散反馈模型，它们表现出自持振荡。为了得到该模型，我们对一个具有两个变量的连续模型应用了热带离散化和超离散化，然后研究了其分岔结构和振荡的参数条件。我们发现，当变量的降解速率低于其合成速率时，所提出的模型通过奈马克 - 萨克分岔产生振荡。我们进一步证明，在某些条件下，超离散模型可以简化为一个布尔系统。
