Chapter 3:
Computing in Reaction-Diffusion and Excitable Media—Case Studies of Unconventional Processors
Andrew Adamatzky
Overview
Recently, we have seen an explosive growth of interest in nonstandard computing architectures, materials, and algorithms: molecular electronics, quantum computation, genetic algorithms,
membrane
computing, DNA computing, and many others. Computation based on wave dynamics and reaction-diffusion processes in physical, chemical, and biological systems (
generally
classified
as nonlinear media) is one of the new approaches being followed (Adamatzky 2001). In this chapter, we will provide a brief account of the subject.
Target waves, spiral waves, and self-localized mobile excitations, to mention a few, are typical space-time patterns in active nonlinear media. Is it possible to employ these phenomena to carry out something useful, to process images, to compute logical functions, to control robots? The answer is yes; we shall
prove
it in this chapter. Consider, for example, an active chemical medium and a change in the concentration of reagents at a few sites: Diffusive or phase waves are generated and spread, they interact with each other and form dynamic or stationary patterns as a result of their interactions. The medium's microvolumes update their states
simultaneously
. Aside from a small degree of asynchronous acting that can be neglected, molecules also diffuse and
react
in parallel. Thus the medium can be thought of as a massive parallel processor. In this wet processor, data and the results of a computation are encoded as concentration profiles of reagents, while the computation is achieved by the spreading and interaction of the waves. In this chapter, we show how these wet processors work and how they
employ
space-time dynamics in the form of activity patterns to perform useful
computations
. We
demonstrate
how various problems are
solved
in active nonlinear media, where data and results are given by spatial defects, and information processing is implemented by the spreading and interaction of phase or diffusive waves.
The field of reaction-diffusion and excitable computing is
rapidly
expanding. It has already affected domains as diverse as smart materials, computational complexity, theory of computation,
robotics
, logic, and mathematical physics. Lack of space
prevents
us from exposing the full spectrum of results obtained in the field; we would rather refer the reader to present
textbooks
(Adamatzky 2001, 2002), where every element of this unconventional computing has been scrutinized. This chapter is restricted to discussing classical examples of wet processors, exemplifying critical issues of the research, and outlining our perspective for further studies.
3.1
Reaction-Diffusion and
Excitation
A great variety of natural processes can be described in terms of propagating fronts. Well-known phenomena include: the dynamics of excitation in heart and neural
tissue
, calcium waves in
cell
cytoplasm, the spreading of genes in population dynamics, and forest fires. All these systems, and many more, are capable of implementing some basic computational operations. In this chapter, we consider mostly those based on wave dynamics in nonlinear chemical systems.
A nonlinear chemical medium is bistable: Each microvolume of the medium has at least two steady stable states, and the microvolume switches between these states. In the chemical medium, fronts of
diffusing
reactants propagate with constant velocity and wave form; the reagents of the wave front convert reagents ahead of the front into products left behind (Epstein and Showalter 1996). In an excitable chemical medium, the wave propagation occurs because of coupling between diffusion and autocatalytic
reactions
. When autocatalytic species are produced in one microvolume of the medium, they diffuse to the neighboring microvolumes and thus trigger an autocatalytic reaction there. That is why an excitable medium responds to perturbations that exceed a excitation threshold by generating excitation waves (Epstein and Showalter 1996; Adamatzky 2001).
Why are excitation waves so good for computing? Unlike mechanical waves, excitation waves do not conserve energy but
conserve
waveform and amplitude; they do not interfere, and
generally
do not reflect (Krinsky 1984). Because of these properties, excitation waves can play an essential role of information transmission in active nonlinear media processors.