Continuous spatial automaton: Difference between revisions

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'''Continuous spatial automata''', unlike [[cellular automata]], have a continuum of locations., Thewhile statestates of alocations ___locationstill isare a finite[[discrete number of real numbers|discrete]]. Time can also be continuous, and in this case the state evolves according to differential equations.

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|integro-differential equations]]<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>