In automata theory (a subfield of

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

), continuous spatial automata, unlike cellular automata, have a continuum of locations, while the state of a location still is any of a finite number of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s. Time can also be continuous, and in this case the state evolves according to

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

. One important example is reaction–diffusion textures,

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s proposed by

Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical com ...

to explain how chemical reactions could create the stripes on zebras and spots on leopards. When these are approximated by CA, such CAs often yield similar patterns. Another important example is neural fields, which are the continuum limit of

neural networks A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

where average firing rates evolve based on

integro-differential equation In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. General first order linear equations The general first-order, linear (only with respect to the term involving derivative ...

s. Such models demonstrate

spatiotemporal pattern Spatiotemporal patterns are patterns that occur in a wide range of natural phenoma and are characterized by a spatial and a temporal patterning. The general rules of pattern formation hold. In contrast to "static", pure spatial patterns, the ...

formation Formation may refer to: Linguistics * Back-formation, the process of creating a new lexeme by removing or affixes * Word formation, the creation of a new word by adding affixes Mathematics and science * Cave formation or speleothem, a secondar ...

, localized states and travelling waves. They have been used as models for cortical memory states and visual hallucinations. MacLenna

considers continuous spatial automata as a model of computation, and demonstrated that they can implement Turing-universality.David H. Wolpert and Bruce J. MacLennan
"A Universal Field Computer That is Purely Linear"
University of Tennessee, Knoxville, Department of Computer Science Technical Report CS-93-206, September 14, 1993, 28 pp.

References

{{Reflist Cellular automata

See also

References