Oscillatory dynamics of the mammalian circadian clock induced by the core delayed negative feedback loop

The mammalian circadian clock plays a crucial role in regulating the 24-h physiological and behavioral processes. This work presents an improved model for the mammalian circadian clock by integrating physiologically plausible modifications into the Kim–Forger model, and investigates the oscillatory dynamics of the mammalian circadian clock induced by the core delayed negative feedback loop using bifurcation theory. The modifications involve introducing time delays in PER protein transcription, translation, and translocation, replacing the first-order law for PER degradation rate with the Michaelis-Menten law, and revising the transcription rate of Per genes. The modified model accurately captures circadian oscillations and overcomes the unrealistic constraints of the Kim–Forger model. In addition, by utilizing the time delay as a control parameter, the theoretical results reveal that the delay induces a supercritical Hopf bifurcation, leading to the emergence of circadian oscillations with enhanced robustness, longer periods, increased amplitudes, and phase delays. Finally, numerical simulations are presented to assess the efficacy of the modified model and to validate the theoretical analysis.