Leonardo Gregory Brunnet

Long-range order with local chaos in lattices of diffusively coupled ODEs

Abstract The existence of non-trivial collective behavior in lattices of diffusively coupled differential equations is investigated. For a two-dimensional square lattice of Rossler systems, a rotating long-range order is observed. This case is best described in terms of a complex Ginzburg-Landau (CGL) equation submitted to the local noise produced by the chaotic Rossler units. The parameters of this CGL equation are estimated to be in the so-called “Benjamin-Feir stable” region. The collective oscillation regime thus corresponds to the linearly-stable, spatially-homogeneous solution of the equivalent CGL equation. The possibility of more complex collective behavior in similar systems is discussed.

Cellular automaton block model of traffic in a city

We present a cellular automaton model of traffic in a city where cars sit between crossings so that they never block the transversal movements. They turn with probability γ, 0⩽γ⩽1. The model is presented in two variants depending on the direction of the flow on the different streets. We numerically find that the mean velocity of traffic continuously decreases with increasing concentration of cars. For a given concentration the mean velocity is minimum for γ=0.5 in both variants of the model. Exact expressions for γ=0,0.5,1 are found for an infinite city and a global picture emerges in terms of asymptotic order, local jam and fluctuations.

Differential adhesion between moving particles as a mechanism for the evolution of social groups.

The evolutionary stability of cooperative traits, that are beneficial to other individuals but costly to their carrier, is considered possible only through the establishment of a sufficient degree of assortment between cooperators. Chimeric microbial populations, characterized by simple interactions between unrelated individuals, restrain the applicability of standard mechanisms generating such assortment, in particular when cells disperse between successive reproductive events such as happens in Dicyostelids and Myxobacteria. In this paper, we address the evolutionary dynamics of a costly trait that enhances attachment to others as well as group cohesion. By modeling cells as self-propelled particles moving on a plane according to local interaction forces and undergoing cycles of aggregation, reproduction and dispersal, we show that blind differential adhesion provides a basis for assortment in the process of group formation. When reproductive performance depends on the social context of players, evolution by natural selection can lead to the success of the social trait, and to the concomitant emergence of sizeable groups. We point out the conditions on the microscopic properties of motion and interaction that make such evolutionary outcome possible, stressing that the advent of sociality by differential adhesion is restricted to specific ecological contexts. Moreover, we show that the aggregation process naturally implies the existence of non-aggregated particles, and highlight their crucial evolutionary role despite being largely neglected in theoretical models for the evolution of sociality.

Preferential Duplication of Intermodular Hub Genes: An Evolutionary Signature in Eukaryotes Genome Networks

Whole genome protein-protein association networks are not random and their topological properties stem from genome evolution mechanisms. In fact, more connected, but less clustered proteins are related to genes that, in general, present more paralogs as compared to other genes, indicating frequent previous gene duplication episodes. On the other hand, genes related to conserved biological functions present few or no paralogs and yield proteins that are highly connected and clustered. These general network characteristics must have an evolutionary explanation. Considering data from STRING database, we present here experimental evidence that, more than not being scale free, protein degree distributions of organisms present an increased probability for high degree nodes. Furthermore, based on this experimental evidence, we propose a simulation model for genome evolution, where genes in a network are either acquired de novo using a preferential attachment rule, or duplicated with a probability that linearly grows with gene degree and decreases with its clustering coefficient. For the first time a model yields results that simultaneously describe different topological distributions. Also, this model correctly predicts that, to produce protein-protein association networks with number of links and number of nodes in the observed range for Eukaryotes, it is necessary 90% of gene duplication and 10% of de novo gene acquisition. This scenario implies a universal mechanism for genome evolution.

EXPLORING COLLECTIVE BEHAVIORS WITH A MULTI-ATTRACTOR QUARTIC MAP

We simulate a 2D coupled map lattice formed by individual units consisting of a multi-attractor quartic map. We show that the interesting recently discovered non-trivial collective behaviors (where macroscopic quantities show well-defined, usually regular, temporal evolution in spite of the presence of local disorder in space and time) also exist, over wide parameter domains, in the presence of local periodic order, in systems having more realistic units allowing coexistence of more than one attractor.

Phase coherence in chaotic oscillatory media

Collective oscillations of lattices of locally coupled chaotic Rossler oscillators are studied with regard to the dynamical scaling of their phase interfaces. Using analogies with the complex Ginzburg–Landau and the Kardar–Parisi–Zhang equations, we argue that phase coherence should be lost in the infinite-size limit. Our numerical results, however, indicate possible discrepancies with a Langevin-like description using an effective white-noise term.

Multi-state coupled map lattices

We investigate a two-dimensional locally coupled map lattice (CML) with the local dynamics driven by the multi-attractor quartic map. In particular, we explore a region where two local fixed points exist, one being periodic and the other chaotic. Different sets of initial conditions such as random initial values for each site or arrangements favoring equal weights to the different local attractors were used. The system reaches different asymptotic states as the intensity or the topology of the local coupling is varied. Among the asymptotic states, we find either homogeneous collective behavior or mixtures of these with synchronized states. These states are characterized and interpreted throughout this work by the distributions of the values of the maps and by the average roughness over the lattice.

Analytic solutions for links and triangles distributions in finite Barabási–Albert networks

Barabasi–Albert model describes many different natural networks, often yielding sensible explanations to the subjacent dynamics. However, finite size effects may prevent from discerning among different underlying physical mechanisms and from determining whether a particular finite system is driven by Barabasi–Albert dynamics. Here we propose master equations for the evolution of the degrees, links and triangles distributions, solve them both analytically and by numerical iteration, and compare with numerical simulations. The analytic solutions for all these distributions predict the network evolution for systems as small as 100 nodes. The analytic method we developed is applicable for other classes of networks, representing a powerful tool to investigate the evolution of natural networks.

Dynamic spectral analysis of phase turbulence

We analyze the time fluctuations associated to the power spectrum of a finite system governed by the complex Ginzburg–Landau equation (CGLE) in the phase turbulence region. It is shown that, for any given value of the parameters of the CGLE, these fluctuations follow an exponential law with the wavenumber. The exponent, α, is such that α→0 indicating a critical behavior when the system is approaching the defect turbulence region. On the contrary α→∞ near the Benjamin–Feir line.

Mean-cluster approach indicates cell sorting time scales are determined by collective dynamics

Cell migration is essential to cell segregation, playing a central role in tissue formation, wound healing, and tumor evolution. Considering random mixtures of two cell types, it is still not clear which cell characteristics define clustering time scales. The mass of diffusing clusters merging with one another is expected to grow as t^{d/d+2} when the diffusion constant scales with the inverse of the cluster mass. Cell segregation experiments deviate from that behavior. Explanations for that could arise from specific microscopic mechanisms or from collective effects, typical of active matter. Here we consider a power law connecting diffusion constant and cluster mass to propose an analytic approach to model cell segregation where we explicitly take into account finite-size corrections. The results are compared with active matter model simulations and experiments available in the literature. To investigate the role played by different mechanisms we considered different hypotheses describing cell-cell interaction: differential adhesion hypothesis and different velocities hypothesis. We find that the simulations yield normal diffusion for long time intervals. Analytic and simulation results show that (i) cluster evolution clearly tends to a scaling regime, disrupted only at finite-size limits; (ii) cluster diffusion is greatly enhanced by cell collective behavior, such that for high enough tendency to follow the neighbors, cluster diffusion may become independent of cluster size; (iii) the scaling exponent for cluster growth depends only on the mass-diffusion relation, not on the detailed local segregation mechanism. These results apply for active matter systems in general and, in particular, the mechanisms found underlying the increase in cell sorting speed certainly have deep implications in biological evolution as a selection mechanism.