Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems

Most current methods for identifying coherent structures in spatially-extended systems rely on prior information about the form which those structures take. Here we present two new approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages, and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively), and provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv requirements; write first author for higher-resolution version

Modeling tumor cell migration: from microscopic to macroscopic

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several millions of cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in details the conditions of validity of the approximations used in the derivation and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids and jamming systems.Comment: Final published version; 14 pages, 7 figure

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling $p$) and deterministic critical slope processes with internal correlation time $t_c$ equal to the avalanche lifetime, in Model A, and $t_c\equiv 1$, in Model B. In both cases nonuniversal scaling properties of avalanche distributions are found for $p\ge p^\star$, where $p^\star$ is related to directed percolation threshold in $d=3$. Distributions of avalanche durations for $p\ge p^\star$ are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of $p$. At $p=p^\star$ a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at $p^\star$ approaches the parity conserving universality class in Model A, and the mean-field universality class in Model B. We also estimate roughness exponent at the transition