Tensor network theory: Difference between revisions

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Revision as of 17:22, 19 November 2013

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In 1980, Pellionisz and Llinas introduced their tensor network theory to describe the behavior of the cerebellum in transforming afferent sensory inputs into efferent motor outputs.<ref name="Neuroscience1980-Pellionisz">{{Cite journal| author =Pellionisz, A., Llinás, R. | year =1980 | month = | title =Tensorial Approach To The Geometry Of Brain Function: Cerebellar Coordination Via A Metric Tensor | journal = Neuroscience | volume =5 | issue = 7| pages = 1125––1136 | id = | url= http://usa-siliconvalley.com/inst/pellionisz/80_metric/80_metric.html | doi = 10.1016/0306-4522(80)90191-8 | pmid=6967569}}</ref> They proposed that intrinsic multidimensional central nervous system space could be described and modeled by an extrinsic network of tensors that together describe the behavior of the central nervous system. <ref name="Neuroscience1980-Pellionisz">{{Cite journal| author =Pellionisz, A., Llinás, R. | year =1980 | month = | title =Tensorial Approach To The Geometry Of Brain Function: Cerebellar Coordination Via A Metric Tensor | journal = Neuroscience | volume =5 | issue = 7| pages = 1125––1136 | id = | url= http://usa-siliconvalley.com/inst/pellionisz/80_metric/80_metric.html | doi = 10.1016/0306-4522(80)90191-8 | pmid=6967569}}</ref> By treating the brain as a "geometrical object" and assuming that (1) neuronal network activity is [[vectorVector (mathematics and physics)|vectorial]] and (2) that the networks themselves are organized [[tensor|tensorially]], brain function could be quantified and described simply as a network of tensors. <ref name="Neuroscience1980-Pellionisz">{{Cite journal| author =Pellionisz, A., Llinás, R. | year =1980 | month = | title =Tensorial Approach To The Geometry Of Brain Function: Cerebellar Coordination Via A Metric Tensor | journal = Neuroscience | volume =5 | issue = 7| pages = 1125––1136 | id = | url= http://usa-siliconvalley.com/inst/pellionisz/80_metric/80_metric.html | doi = 10.1016/0306-4522(80)90191-8 | pmid=6967569}}</ref><ref name="Neuroscience1985-Pellionisz">{{Cite journal| author = Pellionisz, A., Llinás, R. | year =1985 | month = | title= Tensor Network Theory Of The Metaorganization Of Functional Geometries In The Central Nervous System | journal = Neuroscience | volume =16 | issue =2 | pages = 245–273| url = http://usa-siliconvalley.com/inst/pellionisz/85_metaorganization/85_metaorganization.html | doi = 10.1016/0306-4522(85)90001-6 | pmid = 4080158}}</ref>

*Sensory input = [[covariance and contravariance of vectors|covariant]] tensor

*Motor output = [[covariance and contravariance of vectors|contravariant]] tensor