We can analyze the role of Rev-Erb-α on the stability of circadian oscillations using a model consisting of 19 ODEs. The parameter v_sB describes the coupling of the protein REV-ERB-α on the other cellular circadian clock components. Using the same values of the parameters as in Figure 8 of the same paper, we can analyze the bifurcation due to the coupling parameter v_sB. A bifurcation point close to v_sB^*=1.61 nM/h exists at which oscillations change from damped to a limit cycle. Figure 1 shows the amplitude of oscillations of Bmal1 mRNA at various values around the bifurcation point. This result implies that the type of bifurcation is that of supercritical Hopf bifurcation, which also predicts that the period of oscillations is unaffected by the bifurcation [result not shown]. This result may be experimentally confirmed by manipulating in a continuous fashion the coupling strength of REV-ERBα on the transcription of Bmal1. If the coupling strength is not strong enough—i.e. below the bifurcation point—then the period of the circadian rhythm will not change but the rhythmicity of Bmal1 mRNA will weaken and eventually be undetectable.

ARTICLE REVIEW
Reference
A. Granada, R.M. Hennig, B. Ronacher, A. Kramer, and H. Herzel, Phase response curves: elucidating the dynamics of coupled oscillators, Methods in Enzymology 454, Chapter 1 (2009)

Abstract
Phase response curves (PRCs) are widely used in circadian clocks, neuroscience, and heart physiology. They quantify the response of an oscillator to pulse-like perturbations. Phase response curves provide valuable information on the properties of oscillators and their synchronization. This chapter discusses biological self-sustained oscillators (circadian clock, physiological rhythms, etc.) in the context of nonlinear dynamics theory. Coupled oscillators can synchronize with different frequency ratios, can generate toroidal dynamics (superposition of independent frequencies), and may lead to deterministic chaos. These nonlinear phenomena can be analyzed with the aid of a phase transition curve, which is intimately related to the phase response curve. For illustration purposes, this chapter discusses a model of circadian oscillations based on a delayed negative feedback. In a second part, the chapter provides a step-by-step recipe to measure phase response curves. It discusses specifications of this recipe for circadian rhythms, heart rhythms, neuronal spikes, central pattern generators, and insect communication. Finally, it stresses the predictive power of measured phase response curves. PRCs can be used to quantify the coupling strength of oscillations, to classify oscillator types, and to predict the complex dynamics of periodically driven oscillations.

Error/Inaccuracy

Remarks
I haven't read the entire chapter yet, only the Introduction part where the basic concepts and definition were discussed.

The mammalian circadian clock is believed to be performed by a hypothalamic tissue called suprachiasmatic nucleus (SCN) which is called as such because it resides right next to the optic chiasm. It is composed of tens of thousands of interconnected neurons, a fraction of which each exhibit circadian oscillation in electrical activity and expression of some genes. Even though these neurons do not have similar period, their interaction causes them to synchronize with each other thereby producing a stable circadian period of the whole SCN. The mechanism behind the individual oscillations and how they interact with each other is still not completely known. Watch the following 2-min lecture on the SCN from the Howard Hughes Medical Institute.

Recent article (PDF) by Alexander Skupin and Michael Falcke showed that calcium ions are not cellular oscillators. Their result also proved that limit cycle oscillations are not enough to describe experimental data, suggesting that spatial components must be accounted for in the model. Click the picture below to read the caption.