Synchronization of Goodwin's oscillators under boundedness and nonnegativeness constraints for solutions
This work addresses a theoretical gap in synchronizing biologically relevant Goodwin's oscillators, which are used to model circadian rhythms and metabolic pathways, by providing a rigorous synchronization method that respects physical constraints.
The paper identifies that existing synchronization criteria for cyclic feedback systems cannot be directly applied to Goodwin's oscillators due to boundedness and nonnegativity constraints, and proposes a nonlinear synchronization protocol with saturated control inputs that guarantees these constraints and achieves synchronization under sufficiently strong coupling.
In the recent paper by Hamadeh et al. (2012) an elegant analytic criterion for incremental output feedback passivity (iOFP) of cyclic feedback systems (CFS) has been reported, assuming that the constituent subsystems are incrementally output strictly passive (iOSP). This criterion was used to prove that a network of identical CFS can be synchronized under sufficiently strong linear diffusive coupling. A very important class of CFS consists of biological oscillators, named after Brian Goodwin and describing self-regulated chains of enzymatic reactions, where the product of each reaction catalyzes the next reaction, while the last product inhibits the first reaction in the chain. Goodwin's oscillators are used, in particular, to model the dynamics of genetic circadian pacemakers, hormonal cycles and some metabolic pathways. In this paper we point out that for Goodwin's oscillators, where the individual reactions have nonlinear (e.g. Mikhaelis-Menten) kinetics, the synchronization criterion, obtained by Hamadeh et al., cannot be directly applied. This criterion relies on the implicit assumption of the solution boundedness, dictated also by the chemical feasibility (the state variables stand for the concentrations of chemicals). Furthermore, to test the synchronization condition one needs to know an explicit bound for a solution, which generally cannot be guaranteed under linear coupling. At the same time, we show that these restrictions can be avoided for a nonlinear synchronization protocol, where the control inputs are "saturated" by a special nonlinear function (belonging to a wide class), which guarantees nonnegativity of the solutions and allows to get explicit ultimate bounds for them. We prove that oscillators synchronize under such a protocol, provided that the couplings are sufficiently strong.