Abstract
Amplitude changes caused by environmental cues are universal in the circadian clock and associated with various diseases. Singularity behavior, characterized by the disruption of circadian rhythms due to critical stimuli, has been observed across various species. Several mathematical models of the circadian clock have replicated this phenomenon. A comprehensive understanding of the amplitude response remains elusive due to experimental limitations. In this study, we address this question by utilizing a simple normal form model that accurately fits previous experimental data, thereby presenting a general mechanism. We employ a geometric framework to illustrate the dynamics in different stimuli of light-induced transcription (LIT) and light-induced degradation (LID), highlighting the core role of invisible instability in amplitude response. Our model systematically elucidates how stimulus mode, phase, and strength determine amplitude responses. The results show that external stimuli induce alterations in both the amplitudes of individual oscillators and the synchronization among oscillators, collectively influencing the overall amplitude response. While experimental methods impose constraints resulting in limited outcomes under specific conditions, our model provides a comprehensive and three-dimensional mechanistic explanation. A comparison with existing experimental findings demonstrates the consistency of our proposed mechanism. Considering the response direction, the framework enables the identification of phases that lead to increased circadian amplitude. Based on this mechanism derived from the framework, stimulus strategies for resetting circadian rhythms with reduced side effects could be designed. Our results demonstrate that the framework has great potential for understanding and applying stimulus responses in the circadian clock and other limit cycle oscillations.