A cellular automaton model for collective neural dynamics

A stochastic epidemic model for the collective behaviour of a large set of Boolean automata placed upon the sites of a complete graph is revisited. In this paper we study the generalisation of the model to take into account inhibitory neurons. The resulting stochastic cellular automata are completely defined by five parameters: the number of excitatory neurons, N, the number of inhibitory neurons, M, the probabilities of excitation, α, and inhibition, γ, among neurons and the spontaneous transition rate from the firing to the quiescent state, β.We propose that the background of the electroencephalographic signals could be mimicked by the fluctuations in the total number of firing neurons in the excitatory subnetwork. These fluctuations are Gaussian and the mean-square displacement from an initial state displays a strongly subdiffusive behaviour approximately given by , where NA=β/(β+Mγ), τ=2(Nα−β). Comparison with real EEG records exhibits good agreement with these predictions.     

A new cellular automaton model for traffic flow

Establishment of effective traffic models to reveal fundamental traffic characteristics is an essential requirement in the design, planning and operation of transportation systems. In 1992 Nagel and Schreckenberg presented a cellular automaton model describing traffic flow of N cars on a single lane and applied it in the famous project TRANSIMS on transportation simulation. In this paper, the author proposes a new model for the same problem and gives a comparison of simulation results with the former ones. The comparison shows that the new model works better under the condition of high traffic density.     

A novel fast overrelaxation updating method for continuous-discontinuous cellular automaton

Because of its local property, cellular automaton method has been widely applied in different subjects, but the main problem is that the cellular updating is time-consuming. In order to improve its calculation efficiency, a fast successive relaxation updating method is proposed in this paper. Firstly, an accelerating factor ω is defined, and a fast successive relaxation updating theory and its corresponding convergence conditions are developed. In each updating step, the displacement increment is enlarged ω times as a new increment to replace the old one, similarly, the nodal forces for its neighbors caused by this displacement increment are also enlarged by the same accelerating factor, and do those updating operations until the convergence is achieved. By this method, the convergence rate is greatly improved, by a suitable accelerating factor, 90 to 98% of iteration steps are decreased compared to that of the traditional method. Besides, the influence factors for the accelerating factor are studied, and numerical studies show that the suitable accelerating factor is 1.85 < ω < 1.99, which is greatly influenced by cell stiffness, and the optimal accelerating factor is 1.96 < ω < 1.99. Finally, numerical examples are given to illustrate that the present method is effective and high convergence rate compared to the traditional method.     

The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics

Consider a cellular automaton defined on where each lattice site may take on one ofN values, referred to as colors. TheN colors are arranged in a cyclic hierarchy, meaning that colork follows colork–1 modN (k=0,...,N–1). Any two colors that are not adjacent in this hierarchy form an inert pair. In this scheme, there is symmetry in theN colors. Initialized the cellular automaton with product measure, and let time pass in discrete units. To get the configuration at timet+1 from the one at timet, each lattice site looks at the colors of its two nearest neighbors, and if it sees the color that follows its own color, then that site changes color to the color that follows; otherwise, that site does not change color. All such updates occur synchronously at timet+1. For each value ofN2, the fundamental question is whether each site in the cellular automaton changes color infinitely often (fluctuation) or only finitely often (fixation). We prove here that ifN4, then fluctuation occurs, and ifN5, then fixation occurs.     

A cellular automaton model for boundary friction of PEEK and PEEK6

Nowadays a common material for journal bearings is PEEK (Polyetheretherketon) which is increasingly used under boundary friction. A Cellular Automaton model is developed to describe the boundary layer dynamics of PEEK and its composite PEEK6 sliding against a steel surface. Whereas PEEK is a homogeneous synthetic material, PEEK6 contains also carbon fibres for stability and fillers working as solid lubricants. The automaton shows a representative area of the real surface. The processes in the interface are described by a set of rules based on physical assumptions and measurements. Based on these rules the simulations show the dependency of friction and wear on the load spectrum. The discrete simulation allows to observe the topographic development of the PEEK and PEEK6 surface as well as the build-up of a transfer film on the steel surface. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast method for computing time-dependent normal pressure distributions with cellular automaton

When two bodies slide against each other, one part of the dissipated energy causes a topography change. Tribological research on brake bads shows a rich dynamic of the boundary layer: plateau-like structures of a typical length scale grow with time due to agglomerating wear particles and collapse spontaneously at a stability limit [4], [1]. This time-dependent behaviour can be modeled with cellular automata, which consider local resolution of temperature, wear particle density and frictional power [4]. Beside this the instationary normal pressure distribution and the distinction between areas with and without contact is expected to have a significant influence [3]. This paper derives a fast scheme to estimate the time-variant pressure distribution of a deterministic and dynamic topography by a cellular automaton. The approach is discussed in the light of computational performance and the solution's characteristics. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational methods for Conway's Game of Life cellular automaton

The cellular automaton model of computation has drawn the interest of researchers from different disciplines including computer science, biology, mathematics, economy, biochemistry and philosophy. Although a cellular automaton is based on a set of simple rules, over time complex patterns may evolve. We present computational methods for implementing and optimizing a well known two-state cellular automaton, Conway's Game of Life, on a 16-core Intel Xeon. The evaluation is based on three multicore algorithms. The first algorithm is coherent and utilizes shared memory and barrier synchronization. The remaining two algorithms are distributed and utilize private memories and explicit core-to-core message passing. We provide a link to our open source simulation software.     

A non-ergodic probabilistic cellular automaton with a unique invariant measure

We exhibit a Probabilistic Cellular Automaton (PCA) on {0,1}^Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Discontinuous cellular automaton method for crack growth analysis without remeshing

A numerical technical of discontinuous cellular automaton method for crack growth analysis without remeshing is developed. In this method, the level set method is employed to track the crack location and its growth path, where the level set functions and calculation grids are independent, so no explicit meshing for crack surface and no remeshing for crack growth are needed. Then, the discontinuous enrichment shape functions which are enriched by the Heaviside function and the exact near-tip asymptotic field functions are constructed to model the discontinuity of cracks. Finally, a discontinuous cellular automaton theory is proposed, which are composed of cell, neighborhood and updating rules for discontinuous case. There is an advantage that the calculation is only applied on local cell, so no assembled stiffness matrix but only cell stiffness is needed, which can overcome the stiffness matrix assembling difficulty caused by unequal degrees of nodal freedom for different cells, and much easier to consider the local properties of cells. Besides, the present method requires much less computer memory than that of XFEM because of it local property.     

A new mechanism of aggregation in a lattice-gas cellular automaton model

A cellular automaton with moving and resting “particles” is introduced as a model of swarming and aggregation. No external (indirect) sources of information (e.g., chemoattractants) are needed. Instead, “particles” are capable of (directly) evaluating orientations as well as the rest particle density in their local vicinity. Simulations exhibit transitions from random movement to collective motion and, furthermore, to aggregation. Mean-field analysis of the “aggregation instability” provides an explanation of spatial pattern formation.     

The mathematical solution of a cellular automaton model which simulates traffic flow with a slow-to-start effect

In this paper we investigate a cellular automaton model associated with traffic flow and of which the mathematical solution is unknown before. We classify all kinds of stationary states and show that every state finally evolves to a stationary state. The obtained flow-density relation shows multiple branches corresponding to the stationary states in congested phases, which are essentially due to the slow-to-start effect introduced into this model. The stability of these states is formulated by a series of lemmas, and an algorithm is given to calculate the stationary state that the current state finally evolves to. This algorithm has a computational requirement in proportion to the number of cars.     

The flow complex: A data structure for geometric modeling

We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in . The fixed points of this flow are exactly the critical points of the distance function. Our main result is a mathematical characterization and algorithms to compute the stable manifolds, i.e., the inflow regions, of the fixed points. It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set. We call the latter complex the flow complex of the point set. The flow complex is suited for geometric modeling tasks like surface reconstruction.     

Lk (2)-sufficient subspaces for families with shift and scale parameters

In the paper we describe L_k ⁽²⁾-sufficient subspaces of dimension k + 1 for the families mentioned in the title. We give necessary conditions for the presence of these subspaces, expressed in terms of the moments of a distribution function.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 279–291, August, 1976.     

Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media

In this paper, we prove the uniqueness of weak solutions for a pseudo-parabolic equation modeling two-phase flow in a porous medium, where dynamic effects are included in the capillary pressure. We transform the equation into an equivalent system, and then prove the uniqueness of weak solutions to the system which leads to the uniqueness of weak solutions for the original model.     

A Model of the joint motion of agents with a three-level hierarchy based on a cellular automaton

The collective interaction of agents for jointly overcoming (negotiating) obstacles is simulated. The simulation uses a cellular automaton. The automaton’s cells are filled with agents and obstacles of various complexity. The agents' task is to negotiate the obstacles while moving to a prescribed target point. Each agent is assigned to one of three levels, which specifies a hierarchy of subordination between the agents. The complexity of an obstacle is determined by the amount of time needed to overcome it. The proposed model is based on the probabilities of going from one cell to another.     

Interval modeling of dynamics for multibody systems

Modeling of multibody systems is an important though demanding field of application for interval arithmetic. Interval modeling of dynamics is particularly challenging, not least because of the differential equations which have to be solved in the process. Most modeling tools transform these equations into a (non-autonomous) initial value problem, interval algorithms for solving of which are known. The challenge then consists in finding interfaces between these algorithms and the modeling tools. This includes choosing between “symbolic” and “numerical” modeling environments, transforming the usually non-autonomous resulting system into an autonomous one, ensuring conformity of the new interval version to the old numerical, etc. In this paper, we focus on modeling multibody systems’ dynamics with the interval extension of the “numerical” environment MOBILE, discuss the techniques which make the uniform treatment of interval and non-interval modeling easier, comment on the wrapping effect, and give reasons for our choice of MOBILE by comparing the results achieved with its help with those obtained by analogous symbolic tools.     

Skyrmion dynamics in (2 + 1) dimensions

We review our construction, reported in previous papers, of some models in (2 + 1) dimensions that have topologically stable solutions. We reproduce our arguments which have allowed us to construct a Skyrme-like S² model and we show that its classical solutions are stable. We also review the scattering properties of these solutions.     

Coupled problems in transient fluid and structural dynamics in nuclear engineering: Part 2: A discrete model for squeeze flow between two adjacent LMFBR subassemlies

The paper deals with a detailed investigation of the interaction between sub-assemblies and coolant dynamics within an LMFBR core under transient pressure loading. The one-dimensional discrete model approximates the axial squeeze flow by a combined Lagrangian particle Euler cell approach. The coupling between the fluid and the structure is achieved by a step by step scheme. An energy balance criterion is used to improve its computational efficiency. An extension for two-dimensional squeeze flow, where axial and transverse flow around the subassembly may occur, completes the mathematical model.     

A two step algorithm for solving a large scale semi-definite logit model

This paper proposes a two step algorithm for solving a large scale semi-definite logit model, which is appreciated as a powerful model in failure discriminant analysis. This problem has been successfully solved by a cutting plane (outer approximation) algorithm. However, it requires much more computation time than the corresponding linear logit model. A two step algorithm to be proposed in this paper is intended to reduce the amount of computation time by eliminating a certain portion of the data based on the information obtained by solving an associated linear logit model. It will be shown that this algorithm can generate a solution with almost the same quality as the solution obtained by solving the original large scale semi-definite model within a fraction of computation time.