In
automata theory (a subfield of
computer science), 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 numbers. 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 equations proposed by
Alan Turing 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 equations. 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.
See also
*
Analog computer
*
Coupled map lattice
A coupled map (mathematics), map lattice (group), lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the Chaos theory, c ...
References
{{Reflist
Cellular automata