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22
Asynchronism induces second order phase transitions in elementary cellular automata
 Journal of Cellular Automata
"... Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the ..."
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Cited by 22 (8 self)
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Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using MonteCarlo simulations. We show that this phenomenon is a secondorder phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics.
Asynchronous Behavior of Doublequiescent Elementary Cellular Automata
"... Abstract. In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) ↦ → 0 and (1, 1, 1) ↦ → 1), under αasynchronous dynamics (i.e., each cell ..."
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Cited by 17 (1 self)
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Abstract. In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) ↦ → 0 and (1, 1, 1) ↦ → 1), under αasynchronous dynamics (i.e., each cell is updated at each time step independently with probability 0 < α � 1). This work generalizes previous work in [1], in the sense that we study here a continuous range of asynchronism that goes from full asynchronism to full synchronism. We characterize formally the sensitivity to asynchronism of the relaxation times for 52 of the 64 considered automata. Our work relies on the design of probabilistic tools that enable to predict the global behaviour by counting local configuration patterns. These tools may be of independent interest since they provide a convenient framework to deal exhaustively with the tedious case analysis inherent to this kind of study. The remaining 12 automata (only 5 after symmetries) appear to exhibit interesting complex phenomena (such as polynomial/exponential/infinite phase transitions). 1
Robustness in regulatory networks: a multidisciplinary approach
 Acta Biotheoretica
, 2008
"... Abstract. We give in this paper indications about the dynamical impact (as phenotypic changes) coming from the main sources of perturbation in biological regulatory networks. First, we define the boundary of the interaction graph expressing the regulations between the main elements of the network ..."
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Cited by 15 (5 self)
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Abstract. We give in this paper indications about the dynamical impact (as phenotypic changes) coming from the main sources of perturbation in biological regulatory networks. First, we define the boundary of the interaction graph expressing the regulations between the main elements of the network (genes, proteins, metabolites,...). Then, we search what changes in the state values on the boundary could cause some changes of states in the core of the system (robustness to boundary conditions). After, we analyse the role of the mode of updating (sequential, block sequential or parallel) on the asymptotics of the network, essentially on the occurrence of limit cycles (robustness to updating methods). Finally, we show the influence of some topological changes (e.g. suppression or addition of interactions) on the dynamical behaviour of the system (robustness to topology perturbations).
Progresses in the Analysis of Stochastic 2D Cellular Automata: a Study of Asynchronous 2D Minority
, 706
"... Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of as ..."
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Cited by 14 (4 self)
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Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on onedimensional probabilistic cellular automata, either on single examples or on speci c classes. As for other classic dynamical systems in physics, extending known methods from one to twodimensional systems is a long lasting challenging problem. In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when red, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial con guration satis es some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on de ning an energy function from the transition table of the automaton may be extended to the wider class of threshold automata. 1
S.: Boundary Conditions and Phase Transitions in Neural Networks. Theoretical Results
"... Abstract. This paper gives new simulation results on the asymptotic behaviour of theoretical neural networks on Z and Z2 following an extended Hopfield law. It specifically focuses on the influence of fixed boundary conditions on such networks. First, we will generalise the theoretical results alr ..."
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Cited by 10 (8 self)
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Abstract. This paper gives new simulation results on the asymptotic behaviour of theoretical neural networks on Z and Z2 following an extended Hopfield law. It specifically focuses on the influence of fixed boundary conditions on such networks. First, we will generalise the theoretical results already obtained for attractive networks in one dimension to more complicated neural networks. Then, we will focus on twodimensional neural networks. Theoretical results have already been found for the nearest neighbours Ising model in 2D with translationinvariant local isotropic interactions. We will detail what happens for this kind of interaction in neural networks and we will also focus on more complicated interactions, i.e., interactions that are not local, neither isotropic, nor translationinvariant. For all these kinds of interactions, we will show that fixed boundary conditions have significant impacts on the asymptotic behaviour of such networks. These impacts result in the emergence of phase transitions whose geometric shape will be numerically characterised.
Directed Percolation arising in Stochastic Cellular Automata Analysis
, 2008
"... Cellular automata are both seen as a model of computation and as tools to model real life systems. Historically they were studied under synchronous dynamics where all the cells of the system are updated at each time step. Meanwhile the question of probabilistic dynamics emerges: on the one hand, to ..."
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Cited by 8 (2 self)
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Cellular automata are both seen as a model of computation and as tools to model real life systems. Historically they were studied under synchronous dynamics where all the cells of the system are updated at each time step. Meanwhile the question of probabilistic dynamics emerges: on the one hand, to develop cellular automata which are capable of reliable computation even when some random errors occur [24,14,13]; on the other hand, because synchronous dynamics is not a reasonable assumption to simulate real life systems. Among cellular automata a specific class was largely studied in synchronous dynamics: the elementary cellular automata (ECA). These are the "simplest" cellular automata. Nevertheless they exhibit complex behaviors and even Turing universality. Several studies [20,7,8,5] have focused on this class under αasynchronous dynamics where each cell has a probability α to be updated independently. It has been shown that some of these cellular automata exhibit interesting behavior such as phase transition when the asynchronicity rate α varies. Due to their richness of behavior, probabilistic cellular automata are also very hard to study. Almost nothing is known of their behavior [20]. Understanding these "simple " rules is a key step to analyze more complex systems. We present here a coupling between oriented percolation and ECA 178 and confirms observations made in [5] that percolation may arise in cellular automata. As a consequence this coupling shows that there is a positive probability that the ECA 178 does not reach a stable configuration with positive probability as soon as the initial configuration is not a stable configuration and α> 0.996. Experimentally, this result seems to stay true as soon as α> αc ≈ 0.5.
S.: Robustness of Dynamical Systems Attraction Basins Against State Perturbations: Theoretical Protocol and Application in Systems Biology
 In: Proceedings of the 2nd International Conference on Complex Intelligent and Software Intensive Systems, IEEE Computer
, 2008
"... Abstract. This paper aims at giving a general and precise method to achieve a good understanding of discrete dynamical systems by focusing on their attraction basins. This work is the result of a previous one which has permitted to show that the structural changes introduced by fixed boundary condit ..."
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Cited by 7 (7 self)
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Abstract. This paper aims at giving a general and precise method to achieve a good understanding of discrete dynamical systems by focusing on their attraction basins. This work is the result of a previous one which has permitted to show that the structural changes introduced by fixed boundary conditions on regulatory networks could directly and strongly influence the properties of their attraction basins. In this paper, we give an exhaustive stochastic study protocol to understand what happens on attraction basins of dynamical systems when the latter are subjected to state perturbations. Then, we give an application of this protocol by giving the results obtained on a specific system which is a model of the Arabidopsis thaliana flower’s morphogenesis, depending on a specific boundary condition.
Probing robustness of cellular automata through variations of asynchronous updating
 Natural Computing (Online First
, 2012
"... Typically viewed as a deterministic model of spatial computing, cellular automata are here considered as a collective system subject to the noise inherent to natural computing. The classical updating scheme is replaced by stochastic versions which either randomly update cells or disrupt the cellto ..."
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Cited by 3 (2 self)
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Typically viewed as a deterministic model of spatial computing, cellular automata are here considered as a collective system subject to the noise inherent to natural computing. The classical updating scheme is replaced by stochastic versions which either randomly update cells or disrupt the celltocell transmission of information. We then use the novel updating schemes to probe the behaviour of Elementary Cellular Automata, and observe a wide variety of results. We study these behaviours in the scope of macroscopic statistical phenomena and microscopic analysis. Finally, we discuss the possibility to use updating schemes to probe the robustness of complex systems.
Coalescing Cellular Automata
, 2006
"... Abstract. We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We pr ..."
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Abstract. We say that a Cellular Automata (CA) is coalescing when its execution on two distinct (random) initial configurations in the same asynchronous mode (the same cells are updated in each configuration at each time step) makes both configurations become identical after a reasonable time. We prove coalescence for two elementary rules and show that there exists infinitely many coalescing CA. We then conduct an experimental study on all elementary CA and show that some rules exhibit a phase transition, which belongs to the universality class of directed percolation. 1
Directed percolation in asynchronous elementary cellular automata: a detailed study
, 2007
"... Cellular automata are widely used to model natural systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rat ..."
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Cited by 2 (0 self)
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Cellular automata are widely used to model natural systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a discontinuity in the behaviour of the cellular automaton. This works aims at investigating the nature of this change of behaviour using intensive numerical simulations. We apply a twostep protocol to show that the phenomenon is a phase transition whose critical exponents are in good agreement with the predicted values of directed percolation.