Continuous spatial automaton: Difference between revisions

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{{short description|Abstract machines which have a continuum of locations with finite states}}

'''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.

A '''continuous spatial automaton''' is a type of computer model studied in [[automata theory]], a subfield of [[computer science]]. It is similar to a [[cellular automaton]], in that it models the evolution of a set of many states over time. Unlike a cellular automaton, which has a discrete grid of states, a continuous spatial automaton has a continuum of locations in one or more dimensions. The state at each ___location may be chosen from a discrete set of numbers, or from a continuous interval of [[real number]]s. The states may also vary continuously in time, and in this case the state evolves according to a [[differential equation]].

One important example is [[reaction-diffusion]] textures, differential equations 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, continuum limit [[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 formation]], localized states and travelling waves.<ref>S Amari. Dynamics of pattern formation in lateral inhibition type neural fields. Biological Cybernetics, 27:77–87, 1977.</ref><ref>http://www.scholarpedia.org/article/Neural_fields</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>

One important example is [[reaction-diffusionreaction–diffusion]] textures, differential equations proposed by [[Alan Turing]] to explain how chemical reactions could create the stripes on [[zebra]]s and spots on leopards[[leopard]]s. When these are approximated by CAcellular automata, such CAscellular automata often yield similar patterns. Another important example is neural fields, which are the [[continuum limit]] of [[Neural network (biology)|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 waveswave]]s.<ref>S. Amari. "Dynamics of pattern formation in lateral inhibition type neural fields." ''Biological Cybernetics'', 27:77–87, 1977.</ref><ref>http://www{{Cite journal|doi = 10.4249/scholarpedia.org/article/Neural_fields1373|title = Neural fields|year = 2006|last1 = Coombes|first1 = Stephen|journal = Scholarpedia|volume = 1|issue = 6|page = 1373|bibcode = 2006SchpJ...1.1373C| doi-access=free| s2cid=33785752 }}</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, 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. http://web.eecs.utk.edu/~mclennan/papers/ut-cs-93-206.pdf</ref>

Bruce 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. http://web.eecs.utk.edu/~mclennan/papers/ut-cs-93-206.pdf</ref>

==References==

{{Reflist}}

==See also==

*[[Analog computer]]

*[[Coupled map lattice]]

==References==

{{Reflist}}

[[Category:Cellular automata]]