Weak Chimeras Research Articles

Synchronization, clustering, and weak chimeras in a densely coupled transcription-based oscillator model for split circadian rhythms.

The synchronization dynamics for the circadian gene expression in the suprachiasmatic nucleus is investigated using a transcriptional circadian clock gene oscillator model. With global coupling in constant dark (DD) conditions, the model exhibits a one-cluster phase synchronized state, in dim light (dim LL), bistability between one- and two-cluster states and in bright LL, a two-cluster state. The two-cluster phase synchronized state, where some oscillator pairs synchronize in-phase, and some anti-phase, can explain the splitting of the circadian clock, i.e., generation of two bouts of daily activities with certain species, e.g., with hamsters. The one- and two-cluster states can be reached by transferring the animal from DD or bright LL to dim LL, i.e., the circadian synchrony has a memory effect. The stability of the one- and two-cluster states was interpreted analytically by extracting phase models from the ordinary differential equation models. In a modular network with two strongly coupled oscillator populations with weak intragroup coupling, with appropriate initial conditions, one group is synchronized to the one-cluster state and the other group to the two-cluster state, resulting in a weak-chimera state. Computational modeling suggests that the daily rhythms in sleep-wake depend on light intensity acting on bilateral networks of suprachiasmatic nucleus (SCN) oscillators. Addition of a network heterogeneity (coupling between the left and right SCN) allowed the system to exhibit chimera states. The simulations can guide experiments in the circadian rhythm research to explore the effect of light intensity on the complexities of circadian desynchronization.

Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators.

Despite their simplicity, networks of coupled phase oscillators can give rise to intriguing collective dynamical phenomena. However, the symmetries of globally and identically coupled identical units do not allow solutions where distinct oscillators are frequency-unlocked-a necessary condition for the emergence of chimeras. Thus, forced symmetry breaking is necessary to observe chimera-type solutions. Here, we consider the bifurcations that arise when full permutational symmetry is broken for the network to consist of coupled populations. We consider the smallest possible network composed of four phase oscillators and elucidate the phase space structure, (partial) integrability for some parameter values, and how the bifurcations away from full symmetry lead to frequency-unlocked weak chimera solutions. Since such solutions wind around a torus they must arise in a global bifurcation scenario. Moreover, periodic weak chimeras undergo a period-doubling cascade leading to chaos. The resulting chaotic dynamics with distinct frequencies do not rely on amplitude variation and arise in the smallest networks that support chaos.

Existence and stability of chimera states in a minimal system of phase oscillators.

We examine partial frequency locked weak chimera states in a network of six identical and indistinguishable phase oscillators with neighbour and next-neighbour coupling and two harmonic coupling of the form . We limit to a specific partial cluster subspace, reduce to a two dimensional system in terms of phase differences, and show that this has an integral of motion for and . By careful analysis of the phase space, we show that there is a continuum of neutrally stable weak chimera states in this case. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about the stability and bifurcation of weak chimeras for small and that agree with numerical path-following of the solutions.

Robust Weak Chimeras in Oscillator Networks with Delayed Linear and Quadratic Interactions.

We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two populations of (at least) two elements using a general method based on a delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model-based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge chimera dynamics in phase models and real-world oscillator networks.

Isotropy of Angular Frequencies and Weak Chimeras with Broken Symmetry

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here, we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector—for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling.

Chaotic weak chimeras and their persistence in coupled populations of phase oscillators

Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov exponents constructed so far, we show that weak chimeras that are chaotic can exist in the limit of vanishing coupling between coupled populations of phase oscillators. We present numerical evidence that positive Lyapunov exponents can persist for a positive measure set of this inter-population coupling strength.

Weak chimeras in minimal networks of coupled phase oscillators.

We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six, and ten indistinguishable oscillators, where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks.

Microinjection of Cre recombinase protein into zygotes enables specific deletion of two eukaryotic selection cassettes and enhances the expression of a DsRed2 reporter gene in Ccr2/Ccr5 double‐deficient mice

The chemokine receptors CCR2 and CCR5 represent potential novel therapeutic targets to treat important inflammatory and infectious diseases, including atherosclerosis and HIV infection. To study the functions of both receptors in vivo, we aimed to generate Ccr2/Ccr5 double-deficient mice. As these genes are separated by <20 kb, they were inactivated consecutively by two rounds of gene targeting in embryonic stem (ES) cells. Thereby neomycin and hygromycin selection cassettes flanked by four identical loxP recognition sequences for Cre recombinase were integrated into the ES cell genome together with EGFP and DsRed2 reporter genes. Both selection cassettes could be deleted in vitro by transiently transfecting ES cells with Cre expression vectors. However, after blastocyst microinjection these cells yielded only weak chimeras, and germline transmission was not achieved. Therefore, Ccr2/Ccr5 double-deficient mice were generated from ES cells still carrying both selection cassettes. Microinjection of zygotes with a recombinant fusion protein consisting of maltose-binding protein and Cre (MBP-Cre) allowed the selective deletion of both cassettes. All sequences in between and both reporter genes were left intact. Deletion of both selection cassettes resulted in enhanced DsRed2 reporter gene expression. Cre protein microinjection of zygotes represents a novel approach to perform complex recombination tasks.