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Oscillatory behavior has been found in different specialized genetic networks. Previous work has demonstrated nonsynchronous, erratic single-cell oscillations in a genetic network composed of nonspecialized regulatory components and based entirely on negative feedback. Here, we present the construction of a more robust, hysteresis-based genetic relaxation oscillator and provide a theoretical analysis of the conditions necessary for single-cell and population synchronized oscillations. The oscillator is constructed by coupling two subsystems that have previously been implemented experimentally. The first subsystem is the toggle switch, which consists of two mutually repressive genes and can display robust switching between bistable expression states and hysteresis. The second subsystem is an intercell communication system involved in quorum-sensing. This sub-system drives the toggle switch through a hysteresis loop in single cells and acts as a coupling between individual cellular oscillators in a cell population. We demonstrate the possibility of both population synchronization and suppression of oscillations (cluster formation), depending on diffusion strength and other parameters of the system. We also propose the optimal choice of the parameters and small variations in the architecture of the gene regulatory network that substantially expand the oscillatory region and improve the likelihood of observing oscillations experimentally. [ABSTRACT FROM AUTHOR]

In this paper we extend the results of Frankel and Kiemel [SIAM J. Appl. Math, 53 (1993), pp. 1436-1446] to a network of slowly coupled oscillators. First, we use Malkin's theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupled oscillators (1) near Andronov-Hopf bifurcations, (2) near saddle-node on invariant circle bifurcations, and (3) near relaxation oscillations. We compare and contrast synchronization properties of slowly and weakly coupled oscillators. [ABSTRACT FROM AUTHOR]

We use the Malkin theorem to derive phase equations for networks of weakly connected relaxation oscillators. We find an explicit formula for the connection functions when the oscillators have one-dimensional slow variables. The functions are discontinuous in the relaxation limit μ → 0, which provides a simple alternative illustration to the major conclusion of the fast threshold modulation (FTM) theory by Somers and Kopell [Biological Cybernetics, 68 (1993), pp. 393–407] that synchronization of relaxation oscillators has properties that are quite different from those of smooth (nonrelaxation) oscillators. We use Bonhoeffer­Van Der Pol relaxation oscillators to illustrate the theory numerically. [ABSTRACT FROM AUTHOR]

We give the periodic solution of the van der Poi equation &xuml; + x = ε(1 - x²)x in the form of a series converging for all values of the damping parameter ε. For ε small the series solution reduces to a perturbation series in powers of ε and is obtained from this series, essentially by the analytical continuation method, using a suitable transformation of the parameter. The coefficients of the perturbation series and, by virtue of the simplicity of the transformation proposed, those of the transformed series as well are given in explicit form by recurrent analytical formulae, so that both series can be developed in principle up to any order in ε. Actually, the coefficients in the expansions can be obtained by symbolic computation in exact rational number form up to a very high order, the only limitation being the available computing resources. [ABSTRACT FROM AUTHOR]