Content deleted Content added
m →Geometrization of the brain and tensor network theory: minor wording edit |
Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5 |
||
| (40 intermediate revisions by 24 users not shown) | |||
Line 1:
{{short description|Theory of brain function}}
'''Tensor network theory''' is a theory of [[brain]] function (particularly that of the [[cerebellum]]) that provides a mathematical model of the [[transformation geometry|transformation]] of sensory [[space-time]] coordinates into motor coordinates and vice versa by cerebellar [[neuronal networks]]. The theory was developed by Andras Pellionisz and [[Rodolfo Llinas]] in the 1980s as a [[geometrization]] of brain function (especially of the [[central nervous system]]).▼
{{For|the tensor network theory used in quantum physics|Tensor network}}
▲'''Tensor network theory''' is a theory of [[brain]] function (particularly that of the [[cerebellum]]) that provides a mathematical model of the [[transformation geometry|transformation]] of sensory [[space-time]] coordinates into motor coordinates and vice versa by cerebellar [[neuronal networks]]. The theory was developed by Andras Pellionisz and [[Rodolfo Llinas]] in the 1980s as a [[geometrization]] of brain function (especially of the [[central nervous system]]) using [[tensor]]s.<!--
--><ref name="
--><ref name="Neuroscience1985-Pellionisz">{{Cite journal| author = Pellionisz, A., Llinás, R. | year =1985
[[File:Metrictensor.svg|thumb|Metric tensor that
==History==
[[File:Neuronal Network scheme.JPG|thumb|300px|right|Neuronal network schematic. The sensory inputs get transformed by the hidden layer representing the central nervous system which in turn outputs a motor response.]]
===Geometrization movement of the mid-20th century===
The mid-20th century saw a concerted movement to quantify and provide geometric models for various fields of science, including biology and physics.<ref name=GeoBio>{{cite journal|last=Rashevsky|first=N|title=The Geometrization of Biology|journal=Bulletin of Mathematical Biophysics|date=1956|volume=18|
The geometrization of biology in parallel with the geometrization of physics covered a multitude of fields, including populations, disease outbreaks, and evolution, and continues to be an active field of research even today.<ref name=epidemicmodels>{{cite journal|last=Kahil|first=M|title=Geometrization of Some Epidemic Models|journal=Wseas Transactions on Mathematics|date=2011|volume=10|issue=12|
▲The geometrization of biology in parallel with the geometrization of physics covered a multitude of fields, including populations, disease outbreaks, and evolution, and continues to be an active field of research even today.<ref name=epidemicmodels>{{cite journal|last=Kahil|first=M|title=Geometrization of Some Epidemic Models|journal=Wseas Transactions on Mathematics|date=2011|volume=10|issue=12|page=454-462|accessdate=November 18, 2013}}</ref> <ref name=evolutionmodels>{{cite journal|last=Nalimov|first=W|title=Geometrization of biological ideas: probablistic model of evolution|journal=Zhurnal Obshchei Biologii|date=2011|volume=62|issue=5|page=437-448|accessdate=November 16, 2013}}</ref> By developing geometric models of populations and disease outbreaks, it is possible to predict the extent of the epidemic and allow public health officials and medical professionals to control disease outbreaks and better prepare for future epidemics. <ref name=epidemicmodels>{{cite journal|last=Kahil|first=M|title=Geometrization of Some Epidemic Models|journal=Wseas Transactions on Mathematics|date=2011|volume=10|issue=12|page=454-462|accessdate=November 18, 2013}}</ref> Likewise, there is work being done to develop geometric models for the evolutionary process of species in order to study the process of evolution, the space of morphological properties, the diversity of forms and spontaneous changes and mutations. <ref name=evolutionmodels>{{cite journal|last=Nalimov|first=W|title=Geometrization of biological ideas: probablistic model of evolution|journal=Zhurnal Obshchei Biologii|date=2011|volume=62|issue=5|page=437-448|accessdate=November 16, 2013}}</ref>
===Geometrization of the brain and tensor network theory===
Around the same time as all of the developments in the geometrization of biology and physics, some headway was made in the geometrization of neuroscience. At the time, it became more and more necessary for brain functions to be quantified in order to study them more rigorously. Much of the progress can be attributed to the work of Pellionisz and Llinas and their associates who developed the tensor network theory in order to give researchers a means to quantify and model central nervous system activities.
--><ref name="Neuroscience1985-Pellionisz"/>
[[File:VOR coordinates.PNG|thumb|300px|Six rotational axes about which the extraocular muscles turn the eye and the three rotational axes about which the vestibular semicircular canals measure head-movement. According to tensor network theory, a metric tensor can be determined to connect the two coordinate systems.]]▼
▲Around the same time as all of the developments in the geometrization of biology and physics, some headway was made in the geometrization of neuroscience. At the time, it became more and more necessary for brain functions to be quantified in order to study them more rigorously. Much of the progress can be attributed to the work of Pellionisz and Llinas and their associates who developed the tensor network theory in order to give researchers a means to quantify and model central nervous system activities. <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><!--
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"/> 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"/> By treating the brain as a "geometrical object" and assuming that (1) neuronal network activity is [[Vector (mathematics and physics)|vectorial]] and (2) that the networks themselves are organized [[tensor]]ially, brain function could be quantified and described simply as a network of tensors.<ref name="Neuroscience1980-Pellionisz"/><ref name="Neuroscience1985-Pellionisz"/>
▲ --><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
Line 26 ⟶ 24:
==Example==
▲[[File:VOR coordinates.PNG|thumb|300px|Six rotational axes about which the extraocular muscles turn the eye and the three rotational axes about which the vestibular semicircular canals measure head-movement. According to tensor network theory, a metric tensor can be determined to connect the two coordinate systems.]]
===Vestibulo-ocular reflex===
In 1986, Pellionisz described the [[geometrization]] of the "three-neuron [[vestibulo-ocular reflex]] arc" in a cat using tensor network theory.
<br>
<br>
Here, Pellionisz described the analysis of the sensory input into the [[vestibular system|vestibular canals]] as the [[covariance and contravariance of vectors|covariant]] vector component of tensor network theory. Likewise, the synthesized motor response ([[reflex
The resulting metric tensor allowed for accurate predictions of the neuronal connections between the three intrinsically orthogonal [[vestibular system|vestibular canals]] and the six [[extraocular muscles]] that control the [[eye movement (sensory)|movement of the eye]].<ref name="VOR arc"
▲The resulting metric tensor allowed for accurate predictions of the neuronal connections between the three intrinsically orthogonal [[vestibular system|vestibular canals]] and the six [[extraocular muscles]] that control the [[eye movement (sensory)|movement of the eye]].<ref name="VOR arc">{{cite journal|last=Pellionisz|first=Andras|coauthors=Werner Graf|title=Tensor Network Model of the "Three-Neuron Vestibulo-Ocular Reflex-Arc" in Cat|journal=Journal of Theoretical Neurobiology|date=1986|year=1986|month=October|volume=5|page=127-151|accessdate=November 17, 2013}}</ref>
==Applications==
===Neural Networks and Artificial Intelligence===
Neural networks modeled after the activities of the central nervous system have allowed researchers to solve problems impossible to solve by other means. [[Artificial neural networks]] are now being applied in various applications to further research in other fields.
One notable non-biological application of the tensor network theory was the simulated automated landing of a damaged F-15 fighter jet on one wing using a "Transputer parallel computer neural network".<ref name=flightcontrol>{{cite journal|last=Pellionisz|first=Andras|title=Flight Control by Neural Nets: A Challenge to Government/Industry/Academia|journal=International Conference on Artificial Neural Networks|date=1995
==References==
{{Reflist}}
==External Links==▼
* [https://scholar.google.com/citations?user=oZioQ_MAAAAJ&hl=en Andras Pellionisz Google Scholar page Page]
[[Category:
| |||