Andrew Wuensche

Classifying cellular automata automatically: finding gliders, filtering, and relating space-time patterns, attractor basins, and the Z parameter

Cellular automata (CA) rules can be classified automatically for a spectrum of ordered, complex, and chaotic dynamics by a measure of the variance of input-entropy over time. Rules that support interacting gliders and related complex dynamics can be identified, giving an unlimited source for further study. The distribution of rule classes in rule-space can be shown. A byproduct of the method allows the automatic “filtering” of CA space-time patterns to show up gliders and related emergent configurations more clearly. The classification seems to correspond to our subjective judgment of space-time dynamics. There are also approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of in-degree sizes in their branching structure, and to the rule parameter, Z. Based on computer experiments using the software Discrete Dynamics Lab (DDLab), this article explains the methods and presents results for 1D CA.

The gene expression matrix: towards the extraction of genetic network architectures

Understanding of complex biological processes requires knowledge of the component molecularelements, as well as the principles that govern the interactions between them in forming higherordered structures. We are founding our laboratory studies of CNS development and celldifferentiation on the integrative concept of a genetic network, based on the tenets of geneticinformation flow. But first it is important to establish the intellectually challenging principles bywhich complex networks of functionally cross-linked elements lead to predictable, higher-orderedbehaviors. To this end we are studying Boolean network models, which exhibit dynamicproperties similar to those of living systems, such as self-organization and cycling. In this model,genes are conceptualized as binary (on/off) elements interacting within a freely cross-wirednetwork. The on/off pattern, or state, of the entire network of genes updates itself as the genesinteract, until the system reaches a final state, the attractor. This process of updating representsthe pattern, or trajectory, of gene expression which results in the mature organism ordifferentiated cell type, representing analogies of the attractor.Since trajectories and attractors are specific expressions of the architecture of a particularsystem, any experimental strategy must gain access to the states of the biological network. In thatcontext, PCR (polymerase chain reaction) is being used to measure the expression of a largevariety genes at different time points in a tissue or experimental cell system in order to gainaccess to data on trajectories. While many alternative trajectories may be obtainedexperimentally during cell and tissue differentiation or responses to perturbation, it is equallyimportant to development the computational tools to infer genetic network architectures fromsuch data sets. Here we discuss a heuristic approach to this problem using examples fromBoolean networks as illustrations. Finally, analysis of experimental data is expected to providetestable hypotheses concerning further interconnections, some of which might not otherwise bepredicted by strict molecular/mechanistic approaches. Especially within light of the massivegenetic tool set generated by the genome projects, one may anticipate that a strategy of large scalegene expression mapping and genetic signaling network inference may become essential to thestudy of complex medical problems such as cancer or tissue regeneration.

On Spiral Glider-Guns In Hexagonal Cellular Automata: Activator-Inhibitor Paradigm

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

Discrete Dynamics Lab: Tools for Investigating Cellular Automata and Discrete Dynamical Networks

DDLab is interactive graphics software for creating and visualizing discrete dynamical networks, and studying their behavior in terms of both space‐time patterns and basins of attraction. The networks can range from cellular automata to random Boolean networks. This article provides some general background, and gives the flavor of DDLab with a range of examples. Further details can be found at www.ddlab.comwww.ddlab.com

Finding gliders in cellular automata

Gliders, or localized propagating structures, play a role of autonomous signals in cellular-automata models of collision-based computing devices. A method is described for automatically classifying cellular automata rules for a spectrum of ordered, complex and chaotic dynamics, and thus identify rules that support interacting gliders. This is achieved by a measure of the variance of input-entropy over time. The distribution of rule classes in rule-space is discovered. The method also allows automatic “filtering” of cellular automata space-time patterns to show up gliders and related emergent configurations more clearly. Cellular automata dynamics is shown to exhibit some approximate correlations with global measures on convergence in attractor basins, characterized by the distribution of in-degree sizes in their branching structure, and to the rule parameter Z. The research is based on computer experiments using the software Discrete Dynamics Lab (DDLab) [26]

Operating binary strings using gliders and eaters in reaction-diffusion cellular automaton

We study transformations of 2-, 4- and 6-bit numbers in interactions between traveling and stationary localizations in the Spiral Rule reaction-diffusion cellular automaton. The Spiral Rule automaton is a hexagonal ternary-state two-dimensional cellular automaton - a finite-state machine imitation of an activator-inhibitor reaction-diffusion system. The activator is self-inhibited in certain concentrations. The inhibitor dissociates in the absence of the activator. The Spiral Rule cellular automaton has rich spatio-temporal dynamics of traveling (glider) and stationary (eater) patterns. When a glider brushes an eater the eater may slightly change its configuration, which is updated once more every next hit. We encode binary strings in the states of eaters and sequences of gliders. We study what types of binary compositions of binary strings are implementable by sequences of gliders brushing an eater. The models developed will be used in future laboratory designs of reaction-diffusion chemical computers.

Is it Art or Science

Is the art of cellular automata (CA) a legitimate subject of preoccupation? — yes, of course! Although the study of CA is an exercise in experimental dynamics by computer algorithms on top of which mathematical theories and conjectures are superimposed, the graphic representations themselves confer intuitive subjective impressions that are inescapably art. To the simple art lover these are intriguing immediate images which imagination may strive to interpret or merely accept. To the CA theorist and practitioner the “art” is imbued with layers of deeper meaning, just as Zen art can be experienced either on the surface or by the Zen master.

On spiral glider–guns in hexagonal cellular automata: activator–inhibitor paradigm

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

Basins of Attraction in Disordered Networks

Abstract An explicit portrait of basins of attraction in disordered cellular automata networks are accessible for the first time. Such networks are discrete generalisations of connectionist models. Basin of attraction (fields) may serve as the basis of a mind model, and sculpting basin fields may offer a new approach for brain-like computation.