On the decomposition of stochastic cellular automata

In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular automaton can be decomposed into a set of deterministic cellular automata, each of which contributes to the behavior of the stochastic cellular automaton.Comment: Submitted to Journal of Computation Science, Special Issue on Cellular Automata Application

Applying a Dynamical Systems Model and Network Theory to Major Depressive Disorder

Mental disorders like major depressive disorder can be seen as complex dynamical systems. In this study we investigate the dynamic behaviour of individuals to see whether or not we can expect a transition to another mood state. We introduce a mean field model to a binomial process, where we reduce a dynamic multidimensional system (stochastic cellular automaton) to a one-dimensional system to analyse the dynamics. Using maximum likelihood estimation, we can estimate the parameter of interest which, in combination with a bifurcation diagram, reflects the expectancy that someone has to transition to another mood state. After validating the proposed method with simulated data, we apply this method to two empirical examples, where we show its use in a clinical sample consisting of patients diagnosed with major depressive disorder, and a general population sample. Results showed that the majority of the clinical sample was categorized as having an expectancy for a transition, while the majority of the general population sample did not have this expectancy. We conclude that the mean field model has great potential in assessing the expectancy for a transition between mood states. With some extensions it could, in the future, aid clinical therapists in the treatment of depressed patients.Comment: arXiv admin note: text overlap with arXiv:1610.0504

Coarse-Grained Analysis of Microscopic Neuronal Simulators on Networks: Bifurcation and Rare-events computations

We show how the Equation-Free approach for mutliscale computations can be exploited to extract, in a computational strict and systematic way the emergent dynamical attributes, from detailed large-scale microscopic stochastic models, of neurons that interact on complex networks. In particular we show how the Equation-Free approach can be exploited to perform system-level tasks such as bifurcation, stability analysis and estimation of mean appearance times of rare events, bypassing the need for obtaining analytical approximations, providing an "on-demand" model reduction. Using the detailed simulator as a black-box timestepper, we compute the coarse-grained equilibrium bifurcation diagrams, examine the stability of the solution branches and perform a rare-events analysis with respect to certain characteristics of the underlying network topology such as the connectivity degre

Large deviations for mean field models of probabilistic cellular automata

Probabilistic cellular automata form a very large and general class of stochastic processes. These automata exhibit a wide range of complex behavior and are of interest in a number of fields of study, including mathematical physics, percolation theory, computer science, and neurobiology. Very little has been proved about these models, even in simple cases, so it is common to compare the models to mean field models. It is normally assumed that mean field models are essentially trivial. However, we show here that even the mean field models can exhibit surprising behavior. We prove some rigorous results on mean field models, including the existence of a surrogate for the energy in certain non-reversible models. We also briefly discuss some differences that occur between the mean field and lattice models. © 2006 Wiley Periodicals, Inc

Phase transitions in probabilistic cellular automata

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of A. Toom. They are defined as stochastic perturbations of cellular automata with a binary state space and a monotonic transition function and possessing a property of erosion. These models were studied by A. Toom, who gave both a criterion for erosion and a proof of the stability of homogeneous space-time configurations. Basing ourselves on these major findings, we prove, for a set of initial conditions, exponential convergence of the induced processes toward the extremal invariant measure with a highly predominant state. We also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing. This result is due to joint work with A. de Maere. For the two-dimensional probabilistic cellular automata in the same class and for the same extremal invariant measure, we give an upper bound to the probability of a block of cells with the opposite state. The upper bound decreases exponentially fast as the diameter of the block increases. This upper bound complements, for dimension 2, a lower bound of the same form obtained for any dimension greater than 1 by R. Fern\'andez and A. Toom. In order to prove these results, we use graphical objects that were introduced by A. Toom and we give a review of their construction.Comment: PhD thesis, 229 pages. The author was supported by a grant from the Belgian F.R.S.-FNRS (Fonds de la Recherche Scientifique) as Aspirant FNR