Astero Provata

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators.

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Oscillatory dynamics in low-dimensional supports: A lattice Lotka–Volterra model

The effects of low-dimensional supports (one and two dimensions) on the steady state and the dynamics of open reactive systems capable of giving rise to oscillatory behavior are studied. A lattice Lotka–Volterra model involving reaction, adsorption, and desorption mechanisms is developed for which mean-field behavior predicts a continuum of closed trajectories around a center. It is shown that the spatial constraints of the support radically change this behavior. Specifically, while in one dimension, oscillations are suppressed, in two dimensions, the system selects a preferred oscillation frequency depending on the intrinsic parameters and the lattice geometry.

Chimera states in population dynamics: networks with fragmented and hierarchical connectivities

We study numerically the development of chimera states in networks of nonlocally coupled oscillators whose limit cycles emerge from a Hopf bifurcation. This dynamical system is inspired from population dynamics and consists of three interacting species in cyclic reactions. The complexity of the dynamics arises from the presence of a limit cycle and four fixed points. When the bifurcation parameter increases away from the Hopf bifurcation the trajectory approaches the heteroclinic invariant manifolds of the fixed points producing spikes, followed by long resting periods. We observe chimera states in this spiking regime as a coexistence of coherence (synchronization) and incoherence (desynchronization) in a one-dimensional ring with nonlocal coupling and demonstrate that their multiplicity depends on both the system and the coupling parameters. We also show that hierarchical (fractal) coupling topologies induce traveling multichimera states. The speed of motion of the coherent and incoherent parts along the ring is computed through the Fourier spectra of the corresponding dynamics.

Nonlinear chemical dynamics in low dimensions: An exactly soluble model

Restricting space to low dimensions can cause deviations from the mean-field behavior in certain statistical systems. We investigate, both numerically and analytically, the behavior of the chemical reaction A+2X⇌3X in one and two dimensions. In one dimension, we produce exact results showing that the trimolecular reaction system stabilizes in a nonequilibrium, locally frozen, asymptotic state in which the ratior of A to X particles is a constant number,r=0.38, quite different from the mean-field ratio,rMF=1. The same trimolecular model, however, reaches the mean-field limit in two dimensions. In contrast, the bimolecular chemical reaction A+X⇌2X is shown to agree with the mean-field predictions in all dimensions. For both models, we show that the adoption of certain types of transition rules in the laws of evolution can lead to oscillatory steady states.

Controlling chimera states: The influence of excitable units.

We explore the influence of a block of excitable units on the existence and behavior of chimera states in a nonlocally coupled ring-network of FitzHugh-Nagumo elements. The FitzHugh-Nagumo system, a paradigmatic model in many fields from neuroscience to chemical pattern formation and nonlinear electronics, exhibits oscillatory or excitable behavior depending on the values of its parameters. Until now, chimera states have been studied in networks of coupled oscillatory FitzHugh-Nagumo elements. In the present work, we find that introducing a block of excitable units into the network may lead to several interesting effects. It allows for controlling the position of a chimera state as well as for generating a chimera state directly from the synchronous state.

Modeling chemical reactions by forced limit-cycle oscillator: Synchronization phenomena and transition to chaos

Abstract The lattice limit-cycle (LLC) model is introduced as a minimal mean-field scheme which can model reactive dynamics on lattices (low dimensional supports) producing non-linear limit cycle oscillations. Under the influence of an external periodic force the dynamics of the LLC may be drastically modified. Synchronization phenomena, bifurcations and transitions to chaos are observed as a function of the strength of the force. Taking advantage of the drastic change on the dynamics due to the periodic forcing, it is possible to modify the output/product or the production rate of a chemical reaction at will, simply by applying a periodic force to it, without the need to change the support properties or the experimental conditions.

Chimera patterns in two-dimensional networks of coupled neurons

We discuss synchronization patterns in networks of FitzHugh-Nagumo and leaky integrate-and-fire oscillators coupled in a two-dimensional toroidal geometry. A common feature between the two models is the presence of fast and slow dynamics, a typical characteristic of neurons. Earlier studies have demonstrated that both models when coupled nonlocally in one-dimensional ring networks produce chimera states for a large range of parameter values. In this study, we give evidence of a plethora of two-dimensional chimera patterns of various shapes, including spots, rings, stripes, and grids, observed in both models, as well as additional patterns found mainly in the FitzHugh-Nagumo system. Both systems exhibit multistability: For the same parameter values, different initial conditions give rise to different dynamical states. Transitions occur between various patterns when the parameters (coupling range, coupling strength, refractory period, and coupling phase) are varied. Many patterns observed in the two models follow similar rules. For example, the diameter of the rings grows linearly with the coupling radius.

Evolutionary Constraints Favor a Biophysical Model Explaining Hox Gene Collinearity

The Hox gene collinearity enigma has often been approached using models based on biomolecular mechanisms. The biophysical model is an alternative approach based on the hypothesis that collinearity is caused by physical forces pulling the Hox genes from a territory where they are inactive to a distinct spatial domain where they are activated in a step by step manner. Such Hox gene translocations have recently been observed in support of the biophysical model. Genetic engineering experiments, performed on embryonic mice, gave rise to several unexpected mutant expressions that the biomolecular models cannot predict. On the contrary, the biophysical model offers convincing explanation. Evolutionary constraints consolidate the Hox clusters and as a result, denser and well organized clusters may create more efficient physical forces and a more emphatic manifestation of gene collinearity. This is demonstrated by stochastic modeling with white noise perturbing the expression of Hox genes. As study cases the genomes of mouse and amphioxus are used. The results support the working hypothesis that vertebrates have adopted their comparably more compact Hox clustering as a tool needed to develop more complex body structures. Several experiments are proposed in order to test further the physical forces hypothesis.

The “clustered structure” of the purines/pyrimidines distribution in DNA distinguishes systematically between coding and non-coding sequences

A method allowing to measure the inhomogeneous distribution of purines/pyrimidines in nucleotide sequences is developed. We show that this measure relates to the coding or non-coding character of the considered sequence. Coding sequences present a near to the random Pu or Py distribution. This property is shared by both protein-coding DNA and functional RNA-coding DNA. Non-coding sequences present a highly clustered inhomogeneity. We propose the hypothesis, corroborated with appropriate computer simulations, that this is due to the action of various transposition events accumulated for long time periods.

DNA viewed as an out-of-equilibrium structure

The complexity of the primary structure of human DNA is explored using methods from nonequilibrium statistical mechanics, dynamical systems theory, and information theory. A collection of statistical analyses is performed on the DNA data and the results are compared with sequences derived from different stochastic processes. The use of χ^{2} tests shows that DNA can not be described as a low order Markov chain of order up to r=6. Although detailed balance seems to hold at the level of a binary alphabet, it fails when all four base pairs are considered, suggesting spatial asymmetry and irreversibility. Furthermore, the block entropy does not increase linearly with the block size, reflecting the long-range nature of the correlations in the human genomic sequences. To probe locally the spatial structure of the chain, we study the exit distances from a specific symbol, the distribution of recurrence distances, and the Hurst exponent, all of which show power law tails and long-range characteristics. These results suggest that human DNA can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment. Based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the Monte Carlo rejection sampling method, we construct a model DNA sequence. This method allows us to keep both long- and short-range statistical characteristics of the native DNA data. The model sequence presents the same characteristic exponents as the natural DNA but fails to capture spatial correlations and point-to-point details.