Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop.
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.
Graduate School of Design, Kyushu University, 4-9-1, Shiobaru Minami-ku, Fukuoka 815-8540, Japan. Electronic address: shingo.gibo@riken.jp.
Shingo
S
Gibo
Faculty of Design, Kyushu University, 4-9-1, Shiobaru Minami-ku, Fukuoka 815-8540, Japan. Electronic address: hito@design.kyushu-u.ac.jp.
Hiroshi
H
Ito
D000465:Algorithms; D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D025461:Feedback, Physiological / Q000502:physiology*; D008954:Models, Biological
Biological rhythms; Boolean system; Neimark–Sacker bifurcation; Self-sustained oscillations; Tropical discretization