A.S. Kuznetsov

SYNCHRONY IN A POPULATION OF HYSTERESIS-BASED GENETIC OSCILLATORS ∗

Oscillatory behavior has been found in different specialized genetic networks. Pre- vious 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 pro- vide 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 im- plemented 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.

EXISTENCE OF LIMIT CYCLES IN THE REPRESSILATOR EQUATIONS

The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.

Bistable phase synchronization and chaos in a system of coupled van der Pol-Duffing oscillators

Analysis of numerical solutions for a system of two van der Pol-Duffing oscillators with nonlinear coupling showed that there exist chaotic switchings (occurring at irregular time intervals) between two oscillatory regimes differing by nearly time-constant phase shifts between the coupled subsystems. The analysis includes the investigation of bifurcations of the periodic motions corresponding to synchronization of two subsystems, finding stability regions of synchronization regimes and scenarios of the transitions to chaos.

Stationary spatial patterns in CNN of bistable cells with nonlinear couplings

We consider pattern formation in a chain of nonlinearly coupled bistable cells in cellular neural nets. It is shown that the spatial distribution in a wide region of coupling parameter is contrasted initial distribution. It is revealed that description of stationary spatial distributions in a chain with unidirectional couplings reduces to construction of the corresponding mapping trajectory.

CONTROLLING PATTERN FORMATION IN A CNN OF CHUA’S CIRCUITS

In this letter we consider the nonlinear dynamics of a 2-dimensional CNN (cellular neural networks) made of a two-dimensional array of Chua’s oscillators, interconnected via nonlinear coupling. We focus our attention on the possibility of intelligently controlling the pattern formation process by applying an external signal from independent current sources to the cells of the CNN, or by an intelligent choice of initial conditions.

Analysis of regularization of dynamics in a circular chain of bistable chaotic elements with variable number of couplings

Dynamics of a circular chain of coupled bistable active elements is investigated for different number of couplings in an ensemble. It is found that dependences of the boundaries of existence domains for all the considered modes are characterized by a sharp change in the region with the smallest number of couplings.

Chaotic synchronization of long chains by a small number of couplings

Analysis of numerical solutions for systems consisting of coupled van der Pol-Duffing oscillators with nonlinear coupling is presented. We study how drive and response systems in the form of long oscillator chains can be synchronized by a small number of interconnections between chains. We show that oscillations of the drive system oscillators and response system oscillators are synchronized in pairs, whereas the oscillators inside each chain remain unsynchronized.