Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems

Abstract

Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase-response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical analysis tools are illustrated on several system-theoretic questions and models arising in the biology of cellular rhythms.

Figures (13)

• Figure 1: Schematic representation of a phase-resetting experiment. (a) In isolated conditions (closed system), the observable event (vertical arrow) occurs every $T$ units of time. (b) Following a phase-resetting stimulus at time $(t_s - \hat{t}_0)$ after an event (open system), the successive observable event times $\hat{t}_i$, for $i\in\mathbb{N}_{>0}$, are altered. $\hat{T} \mathrel{:=} \hat{t}_1 - \hat{t}_0$ denotes the time interval between the pre- and post-stimulus events.
• Figure 2: Graphical representation of the wrap-to-$[-\pi,\pi)$ operation. Given a real number $x$ in radians, $x \if@display\mkern18mu\mkern8mu({\operator@font wrap\;to}\mkern6mu[-\pi,\pi)) \equiv \left[x + \pi \pmod{2\pi}\right] - \pi$ wraps $x$ to the interval $[-\pi,\pi)$. It adds or subtracts an integer multiple of $2\pi$ such that the result belongs to $[-\pi,\pi)$. (A solid dot indicates that the endpoint is included in the set; whereas an open dots indicates that the endpoint is excluded from the set.)
• Figure 3: Diagram of the quantitative model for circadian oscillations in mammals involving interlocked negative and positive regulations of Per, Cry, and Bmal1 genes by their protein products. (Figure is modified, with permission, from Leloup:2003cp. © (2003) National Academy of Sciences, U.S.A.)
• Figure 4: Local robustness analysis to parameter variations in the space of infinitesimal phase response curves. Normalized robustness measures $\rho^\omega$ (angular frequency) and $\rho^{q}$ (infinitesimal phase response curve) reveal the distinct sensitivity of three distinct genetic circuits (Cry, Per, and Bmal1). Each point is associated with a particular parameter. The three lines are regressions over the parameters of the three gene loops. The dashed bisector indicates the positions at which the two measures of robustness are identical. Only parameters associated with the Cry-loop exhibit a low influence on the period and a high influence on the infinitesimal phase response curve. The color code corresponds to different subsets of parameters associated with different loops: Per-loop in blue, Cry-loop in red, and Bmal1-loop in green. Parameters associated with interlocked loops are represented in black.
• Figure 5: (Caption on next page.)

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Remark
• Remark
• Remark
• Remark
• Remark
• Remark