Dynamical Phase Transitions in Graph Cellular Automata

Discrete dynamical systems can exhibit complex behaviour from the iterative application of straightforward local rules. A famous example are cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method (DCM) and its backtracking version (BDCM), we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with ``conforming non-conformist'' update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighbourhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit non-conformist behaviour. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.Comment: 15 page

Boolean Delay Equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time. They represent therewith metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil's staircases and ``fractal sunbursts``. All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid earth problems. The former have used small systems of BDEs, while the latter have used large networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular the discussion on partial BDEs is updated and enlarge

Modeling Fluid Flow In Heterogeneous And Anisotropic Porous Media

Permeability distribution in reservoirs is very important for the flow of water or oil and gas. In this study, the effects of various heterogeneous permeability distributions on the flow field are simulated using the finite difference technique. We have simulated the flow for two types of heterogeneous distributions, one is Gaussian and the other is self-similar or fractal, the latter being much rougher than the former. The results show that the flow is not sensitive to the roughness of the distribution. In the case of lineated heterogeneities, anisotropy in the flow properties occurs. The anisotropy is not very significant if the lineated highly permeable regions are surrounded by less permeable regions. However, in the case of lineated fractures, where the background permeability is small, the flow is very sensitive to the direction of the lineation, such anisotropy can produce orders of magnitude difference in permeability. Furthermore, it is shown that the degree of anisotropy depends on the connectivity of the fractures. The anisotropy decreases with decreasing connectivity.Massachusetts Institute of Technology. Full Waveform Acoustic Logging ConsortiumUnited States. Dept. of Energy (Grant DE-FG02-86ER13636

Predictability: a way to characterize Complexity

Different aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing different kind of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to finite-time and finite-resolution effects on predictability, which can be accounted with suitable generalization of the standard indicators. The problems involved in systems with intrinsic randomness is discussed, with emphasis on the important problems of distinguishing chaos from noise and of modeling the system. The characterization of irregular behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports. Related information at this http://axtnt2.phys.uniroma1.i

Aerospace medicine and biology: A continuing bibliography with indexes (supplement 331)

This bibliography lists 129 reports, articles and other documents introduced into the NASA Scientific and Technical Information System during December, 1989. Subject coverage includes: aerospace medicine and psychology, life support systems and controlled environments, safety equipment, exobiology and extraterrestrial life, and flight crew behavior and performance

Incorporating calcium signalling in Vertex models of neural tube closure

Recent experiments have shown that apical constriction (AC) during neural tube closure (NTC) is driven by cell contractions preceded by asynchronous and cell-autonomous Ca2+ flashes. Disruption of these Ca2+ signals and contractions leads to neural tube defects, such as anencephaly. However, the inherent two-way mechanochemical coupling of Ca2+ signaling and mechanics is poorly understood, and live-cell imaging is difficult. Thus, models can help greatly but the few available partially reproduce experimental findings. We first study a modified implementation of the mechanochemical vertex model of Suzuki et al [196]; the modified Suzuki model. We numerically implement it by developing CelluLink, a new opensource (Python), user-friendly software package for vertex modelling. CelluLink’s parallel processing enables fast yet thorough parameter sweeps, guided by an analytically derived bifurcation diagram. CelluLink can be adapted to study other multicellular challenges. Subsequently, in close collaboration with experimentalists, we develop a one-way mechanochemical model to study the effect of Ca2+ on mechanics. This model significantly improves upon the Suzuki model, reproducing several experimental observations. We incorporate, for the first time, the surface ectoderm and the experimental Ca2+ flash amplitude and frequency profiles. Furthermore, guided by experiments, we model the damping coefficient of the vertices and cell-cell adhesion as functions of actomyosin concentration and cell size. The one-way model successfully reproduces the significant reduction in neural plate size during AC, within 2%-8% of the initial area. We then develop a two-way mechanochemical model which captures the two-way coupling between Ca2+ signals and mechanics. We incorporate stretch-sensitive Ca2+ channels, enabling the cell to respond to mechanical stimuli. The model reproduces the results of the one-way model, but more accurately, the Ca2+ frequency and amplitude arise from the interaction between the cells and are not imposed. We leverage our models to propose a series of hypotheses for future experiments

On Self-Organized Criticality and Synchronization in Lattice Models of Coupled Dynamical Systems

Lattice models of coupled dynamical systems lead to a variety of complex behaviors. Between the individual motion of independent units and the collective behavior of members of a population evolving synchronously, there exist more complicated attractors. In some cases, these states are identified with self-organized critical phenomena. In other situations, with clusterization or phase-locking. The conditions leading to such different behaviors in models of integrate-and-fire oscillators and stick-slip processes are reviewed.Comment: 41 pages. Plain LaTeX. Style included in main file. To appear as an invited review in Int. J. Modern Physics B. Needs eps