Critical phenomena in cellular automata: perturbing the update, the transitions, the topology

International audienceWe survey the effect of perturbing the regular structure of a cellular automaton. We are interested in critical phenomena, i.e., when a continuous variation in the local rules of a cellular automaton triggers a qualitative change of its global behaviour. We focus on three types of perturbations: (a) when the updating is made asynchronous, (b) when the transition rule is made stochastic, (c) when the topological defects are introduced. It is shown that although these perturbations have various effects on CA models, they generally produce the same effects, which are identified as first-order or second-order phase transitions. We present open questions related to this topic and discuss implications on the activity of modelling

Does Life resist asynchrony ?

We study Conway's Game of Life with an asynchronous updating

Continuous cellular automata on irregular tessellations : mimicking steady-state heat flow

Leaving a few exceptions aside, cellular automata (CA) and the intimately related coupled-map lattices (CML), commonly known as continuous cellular automata (CCA), as well as models that are based upon one of these paradigms, employ a regular tessellation of an Euclidean space in spite of the various drawbacks this kind of tessellation entails such as its inability to cover surfaces with an intricate geometry, or the anisotropy it causes in the simulation results. Recently, a CCA-based model describing steady-state heat flow has been proposed as an alternative to Laplace's equation that is, among other things, commonly used to describe this process, yet, also this model suffers from the aforementioned drawbacks since it is based on the classical CCA paradigm. To overcome these problems, we first conceive CCA on irregular tessellations of an Euclidean space after which we show how the presented approach allows a straightforward simulation of steady-state heat flow on surfaces with an intricate geometry, and, as such, constitutes an full-fledged alternative for the commonly used and easy-to-implement finite difference method, and the more intricate finite element method

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897–923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters

Modélisation des systèmes complexes

Michel Morvan et Henri Berestycki, directeurs d’études Émergence dans les systèmes complexes : des cas réels aux modèles formels L’émergence dans les systèmes complexes a fait l’objet d’une double interrogation dans ce séminaire mettant en jeu deux points de vue complémentaires. D’un côté, nous avons étudié les phénomènes d’émergence tels qu’ils peuvent être vus par un philosophe. Nous avons tenté de dégager les invariants caractéristiques des situations d’émergence. D’un autre côté, nous nou..

Stochastic Minority on Graphs

Cellular automata have been mainly studied on very regular graphs carrying the cells (like lines or grids) and under synchronous dynamics (all cells update simultaneously). In this paper we study how the asynchronism and the topology of cells act upon the dynamics of the classical Minority rule. Beyond its apparent simplicity, this rule yields complex behaviors which are clearly linked to the structure of the graph carrying the cells

A study of stochastic 2D Minority CA : would wearing stripes be a fatality for snob people ?

Cellular automata are usually associated with synchronous deterministic dynamics, and their asynchronous or stochastic versions have been far less studied although relevant for modeling purposes. The study of their asynchronous dynamics is all the more needed that their asynchronous behaviors are drastically different from their synchronous ones. This paper analyzes the dynamics of a two-dimensional cellular automaton, 2D Minority, under fully asynchronous dynamics, where only one random cell updates at each time step. This cellular automaton is of particular interest in computer science, biology or social science for instance, and already presents a rich variety of behaviors although the apparent simplicity of its transition rule. In particular, it captures some important features, like the emergence of striped patterns, which are common, according to experiments, to other important automata, such as Game of Life. In this paper, we present a mathematical analysis of the first steps and the last steps of the asynchronous dynamics of 2D Minority. Our results are based on the definition of an interaction energy and rely on the analysis of the dynamics of the borders between competing regions. Our results are a first step towards a complete analysis of this stochastic cellular automaton. Many questions remain open: in particular describing mathematically the middle part of the evolution of 2D Minority where many regions compete with each other, or defining similar parameters (energy, borders,...) for other automata (such as Game of Life) that present similarities with 2D Minority in their asynchronous behaviors

Attraction Basins as Gauges of Robustness against Boundary Conditions in Biological Complex Systems

One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally

Patterning by cell-to-cell communication

This thesis addresses the question of how patterning may arise through cell-to-cell communication. It combines quantitative data analysis with computational techniques to understand biological patterning processes. The fi�rst section describes an investigation into the robustness of an evolved arti�ficial patterning system. Cellular automata rules were implemented sequentially according to the instructions in a simple `genome'. In this way, a set of target patterns could be evolved using a genetic algorithm. The patterning systems were tested for robustness by perturbing cell states during their development. This exposed how certain types of patterning rule had very di�fferent levels of robustness to perturbations. Rules that generated patterns with complex divergent patterns were more likely to amplify the e�ffect of a perturbation. When smaller genomes, comprising less individual rules, were evolved to match certain target patterns, these were shown to be more likely to select complex patterning rules. As a result, the developmental systems based on smaller genomes were less robust than those with larger genome sizes. Section two provides an analysis of the patterning of microchaetes in the epithelial layer of the notum of Drosophila flies. It is shown that the pattern spacing is not sufficiently described by a model of lateral inhibition through Delta-Notch signalling between adjacent cells. A computational model is used to demonstrate the viability of long range signalling through a dynamic network of �filopodia, observed in the basal layer of the epithelium. In-vivo experiments con�rm that when fi�lopodia lengths are effected by mutations the pattern spacing reduces in accordance with the model. In the fi�nal section the behaviour of simple asynchronous cellular automata are analysed. It is shown how these diff�er to the synchronous cellular automata used in the fi�rst section. A set of rules are identifi�ed whose emergent behaviour is similar to the lateral inhibition patterning process established by the Delta-Notch signalling system. Among these rules a particular subset are found to produce patterns that adjust their spacing, over the course of their development, towards a more ordered and densely packed state. A re-examination of the Delta-Notch signalling model reveals that this type of packing optimisation could take place with either dynamic �filopodial signalling, or as an alternative, transient Delta signalling at each cell. Under certain parameter regimes the patterns become more densely packed over time, whilst maintaining a minimum zone of inhibition around each Delta expressing cell. The asynchronous CA are also used to demonstrate how stripes can be formed by cell-to-cell signalling and optimised, under certain conditions, so that they align in a single direction. This is presented as a possible novel alternative to the reaction-di�ffusion mechanism that is commonly used to model the patterning of spots and stripes