## Discrete Dynamics Lab (DDLab)

** Category** Cross-Omics>Pathway Analysis/Gene Regulatory
Networks/Tools and Intelligent Software>Neural Network Systems/Tools

** Abstract** DDLab is interactive graphics software for creating, visualizing,
and analyzing many aspects of Cellular Automata (CA), Random Boolean
Networks (RBNs), and Discrete Dynamical Networks (DDNs) in general,
and studying their behavior, both from the time-series perspective - space-
time patterns, and from the state-space perspective - ‘Attractor Basins’.

DDLab is relevant to research, applications and education in the fields of complexity, self-organization, emergent phenomena, chaos, collision based computing, neural networks (NNs), content addressable memory, genetic regulatory networks (GRNs), dynamical encryption, generative art and music, and the study of the abstract mathematical/physical/dynamical phenomena in their own right.

Networks of sparsely inter-connected elements with discrete values and ‘updating in parallel’ are central to a wide range of natural and ‘artificial phenomena’ drawn from many areas of science; from physics to biology to cognition; to social and economic organization; to parallel computation and artificial life; to complex systems in general.

“Decision making” networks like this are applied as idealized models in the study of complexity and emergence, and in the behavior of networks in general, including ‘biomolecular networks’ such as Neural and Genetic networks. The networks themselves have intrinsic interest as mathematical, physical, dynamical and computational systems with a large body of literature devoted to their study.

Because the dynamics is difficult to describe by classical mathematics, ‘computer simulation’ is required, and there is a need for ‘simulation software’ for non-experts in programming to ‘model networks’ in their particular fields.

DDLab is able to construct these networks and investigate many aspects of their dynamical behavior.

As well as generating space-time patterns in one, two or three dimensions, DDLab is able to construct ‘Attractor Basins’, graphs that link network states according to their transitions, analogous to Poincare's “phase portrait” which provided advanced insights into ‘continuous dynamics’.

A key insight is that the dynamics on the networks converge, thus they fall into a number of ‘basins of attraction’. This is the “network's memory”, its ability to ‘hierarchically categorize’ its patterns of activation (state-space), as a function of the ‘precise network architecture’.

Relating this to space-time patterns in CA, high convergence implies order; low convergence implies disorder or chaos. The most interesting emergent structures occur at the transition, sometimes called the “edge of chaos”.

DDLab has recently been generalized for multi-value logic. Up to 8 values (or colors) are now possible, instead of just Boolean logic (two values - 0, 1). Of course, with just 2 values selected, DDLab can behave as before. Multi-values open up new possibilities for ‘dynamical behavior’ and modeling.

Another major update is an option to constrain DDLab to run forward only, to generate ‘space-time patterns’ for various types of totalistic rules, reducing memory load by cutting out all basins of attraction functions.

This allows larger neighborhoods. In 2d the neighborhoods are predefined to make hexagonal as well square lattices. Many interesting cellular automaton rules with “life”-like and other complex dynamics can be found in totalistic multi-value rule-space, in 3d as well as 2d.

DDLab is an applications program; it does Not require writing code. Network parameters and the graphics presentation can be flexibly set, reviewed and altered interactively, including changes on-the-fly. There are built in tools for constructing and manipulating networks. A wide variety of measures, data, analysis and statistics are available.

For small networks, it’s possible to compute and draw ‘basins of attraction’, and measure their convergence and stability to perturbation. For larger networks, basins of attraction can be investigated statistically.

Discrete dynamical networks (DDNs) --

A discrete dynamical network in DDLab can be imagined as a “software simulation” of a collection light bulbs which transmit information to each other about their color state (on/off for binary), and change color according to the arriving signals.

More abstractly, ‘the network’ is made up of elements or “cells”, connected to each other by directed links or “wires”, where a wire has an input and output terminal. A cell takes on a value (or color), and transmits this value down its output wires.

Its value is updated as a function of the values on its input wires. Updating is usually done in parallel, in discrete “time-steps”, but may also be sequential in a predetermined order.

This is the system in a nutshell. What remains is to set up the network according to its various parameters:

1) The value-range, v. The range of values that is available to a cell. The number of possible internal states of the cell, or colors, or letters, in its “alphabet”.

2) The number of network elements, the system size, n.

3) How the elements are arranged in space: in a 1d, 2d or 3d lattice with axial dimensions i; j; h, or some other arrangement.

4) The number of input wires, k, to each cell, or the “k-mix” if k is Not homogeneous. k, may vary from 0 to 25. Maximum k is reduced for greater value-range v.

5) The “wiring scheme”: defining the location of the output terminals of each cell's input wires, the element's “neighborhood”.

6) The “rule scheme”: the rules or ‘logical functions’ in the network. Each element applies a rule to its inputs to compute its output. Usually this is made into a look-up table, the “rule-table”, listing the outputs of all possible input patterns.

CA have a homogeneous ‘rule scheme’, the same rule throughout the network. RBN and DDN may have a completely arbitrary, heterogeneous, rule scheme, or again, it may be biased in some way.

DDlab is able to ‘create networks’ with any combination of these parameters, and graphically represent and analyze both the networks themselves and the dynamics resulting from the ‘changing patterns’ as the complex feedback web unfolds.

Network updating may be sequential as well as parallel, noisy as well as deterministic.

Space-time patterns and Basins of attraction --

DDLab has two (2) alternative ways of looking at ‘network dynamics’. Local dynamics - running the ‘network forwards’; and Global dynamics - which entails running the ‘network backwards’.

Running forwards - generates the network's ‘space-time patterns’ from a given initial state. Many alternative graphical representations of ‘space- time patterns’, and methods for gathering and analyzing data, are available to illustrate different aspects of local network dynamics, including “filtering” to show up ‘emergent structures’ more clearly.

Running “backwards” - generates multiple predecessors rather than a trajectory of unique successors. This procedure reconstructs the branching sub-tree of ancestor patterns rooted on a particular state. States without predecessors are disclosed, the so called “garden-of-Eden” states, the leaves of the sub-trees.

Sub-trees, basins of attraction (with a topology of trees rooted on attractor cycles), or the entire basin of attraction field can be displayed as ‘directed graphs’ in real time, with many presentation options, and methods for gathering/analyzing data. The Attractor Basins of ‘random maps’ may be generated with or without some bias in the mapping.

‘Attractor Basins’ represent the network's “memory” by their hierarchical categorization of state-space; each basin is categorized by its attractor and each sub-tree by its root. ‘Learning/forgetting’ algorithms allow attaching/detaching sets of states as predecessors of a given state by automatically mutating rules or changing connections.

DDLab user interface --

DDLab is an interactive applications program that does Not require writing code (as stated above...). The graphical user interface (GUI) allows setting, viewing and amending ‘network parameters’, and the various presentation and analysis functions, by responding to prompts or accepting defaults.

*System Requirements*

Web-based.

*Manufacturer*

- Andy Wuensche
- Flat 6, 3 Grape Street
- London WC2 H8DX, UK
- E-mail: andy@ddlab.org

** Manufacturer Web Site** Andy Wuensche DDLab

** Price** Contact manufacturer.

** G6G Abstract Number** 20575

** G6G Manufacturer Number** 104179