Continuous spatial automaton: Difference between revisions

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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 number]]s. Time can also be continuous, and in this case the state evolves according to [[differential equations]].

One important example is [[reaction–diffusion]] textures, [[differential equation]]s proposed by [[Alan Turing]] to explain how chemical reactions could create the stripes on [[zebra]]s 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]] where average firing rates evolve based on [[integro-differential equation]]s.<ref>H. R. Wilson and J. D. Cowan. "Excitatory and inhibitory interactions in localized populations of model neurons" ''Biophysical Journal'', 12:1–24, 1972.</ref><ref>H. R. Wilson and J. D. Cowan. "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue" ''Kybernetik'', 13:55–80, 1973.</ref> Such models demonstrate [[spatiotemporal pattern]] [[pattern formation|formation]], localized states and [[travelling wave]]s.<ref>S. Amari. "Dynamics of pattern formation in lateral inhibition type neural fields" ''Biological Cybernetics'', 27:77–87, 1977.</ref><ref>{{Cite journal|url=http://www.scholarpedia.org/article/Neural_fields|doi = 10.4249/scholarpedia.1373|title = Neural fields|year = 2006|last1 = Coombes|first1 = Stephen|journal = Scholarpedia|volume = 1|issue = 6|page = 1373|bibcode = 2006SchpJ...1.1373C| doi-access=free }}</ref> They have been used as models for cortical memory states and visual hallucinations.<ref>G. B. Ermentrout and J. D. Cowan. "A mathematical theory of visual hallucination patterns" ''Biological Cybernetics'', 34:137–150, 1979.</ref>

MacLennan [http://www.cs.utk.edu/~mclennan/contin-comp.html] considers continuous spatial automata as a model of computation, and demonstrated that they can implement Turing-universality.<ref>David H. Wolpert and Bruce J. MacLennan, [http://web.eecs.utk.edu/~mclennan/papers/ut-cs-93-206.pdf "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. </ref>