Open AccessMultistability, oscillations and bifurcations in feedback loops
Author(s) -
Maria C. A. Leite,
Yunjiao Wang
Publication year - 2010
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2010.7.83
Subject(s) - multistability , feedback loop , negative feedback , nonlinear system , control theory (sociology) , positive feedback , loop (graph theory) , computer science , dynamics (music) , topology (electrical circuits) , physics , mathematics , artificial intelligence , control (management) , computer security , quantum mechanics , voltage , combinatorics , acoustics , electrical engineering , engineering
Feedback loops are found to be important network structures in regulatory networks of biological signaling systems because they are responsible for maintaining normal cellular activity. Recently, the functions of feedback loops have received extensive attention. The existing results in the literature mainly focus on verifying that negative feedback loops are responsible for oscillations, positive feedback loops for multistability, and coupled feedback loops for the combined dynamics observed in their individual loops. In this work, we develop a general framework for studying systematically functions of feedback loops networks. We investigate the general dynamics of all networks with one to three nodes and one to two feedback loops. Interestingly, our results are consistent with Thomas' conjectures although we assume each node in the network undergoes a decay, which corresponds to a negative loop in Thomas' setting. Besides studying how network structures influence dynamics at the linear level, we explore the possibility of network structures having impact on the nonlinear dynamical behavior by using Lyapunov-Schmidt reduction and singularity theory.