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{{short description|Theory of brain function}}
'''Tensor network theory''' is a theory of brain function (specifically in the [[cerebellum]]) by Llinas and Pellionisz which provides a mathematical model of transformation of sensory (covariant) space-time coordinates into motor (contravariant) coordinates by cerebellar neuronal networks.<!--▼
{{For|the tensor network theory used in quantum physics|Tensor network}}
--><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 = | pages = 1125—-1136 | id = | url= http://usa-siliconvalley.com/inst/pellionisz/80_metric/80_metric.html | doi = 10.1016/0306-4522(80)90191-8}}</ref><!--▼
▲'''Tensor network theory''' is a theory of [[brain]] function (
--><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>▼
▲ --><ref name="Neuroscience1980-Pellionisz">{{
▲ --><ref name="Neuroscience1985-Pellionisz">{{
[[File:Metrictensor.svg|thumb|Metric tensor that transforms input covariant tensors into output contravariant tensors. These tensors can be used to mathematically describe cerebellar neuronal network activities in the central nervous system.]]
==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|pages=31–54|doi=10.1007/bf02477842}}</ref><ref name=GeoPhysics>{{cite journal|last=Palais|first=Richard|title=The Geometrization of Physics|date=1981|pages=1–107|url=http://vmm.math.uci.edu/GeometrizationOfPhysics.pdf|archive-date=2020-02-29|access-date=2020-02-29|archive-url=https://web.archive.org/web/20200229175011/http://vmm.math.uci.edu/GeometrizationOfPhysics.pdf|url-status=dead}}</ref><ref name=physicstoday>{{cite journal|last=Mallios|first=Anastasios|title=Geometry and physics of today|journal=International Journal of Theoretical Physics|date=August 2006|volume=45|issue=8|doi=10.1007/s10773-006-9130-3|pages=1552–1588|arxiv=physics/0405112|bibcode=2006IJTP...45.1552M |s2cid=17514844 }}</ref> The [[geometrization]] of biology began in the 1950s in an effort to reduce concepts and principles of biology down into concepts of geometry similar to what was done in physics in the decades before.<ref name="GeoBio"/> In fact, much of the geometrization that took place in the field of biology took its cues from the geometrization of contemporary physics.<ref name=BioPhysics>{{cite book|last=Bailly|first=Francis|title=Mathematics and the Natural Sciences: The Physical Singularity of Life|year=2011|publisher=Imperial College Press|isbn=978-1848166936}}</ref> One major achievement in [[general relativity]] was the geometrization of [[gravity|gravitation]].<ref name="BioPhysics"/> This allowed the trajectories of objects to be modeled as [[geodesic curvature|geodesic curves]] (or optimal paths) in a [[Riemannian manifold|Riemannian space manifold]].<ref name="BioPhysics"/> During the 1980s, the field of [[theoretical physics]] also witnessed an outburst of geometrization activity in parallel with the development of the [[Unified Field Theory]], the [[Theory of Everything]], and the similar [[Grand Unified Theory]], all of which attempted to explain connections between known physical phenomena.<ref name=GeoUnity>{{cite journal|last=KALINOWSKI|first=M|title=The Program of Geometrization of Physics: Some Philosophical Remarks|journal=Synthese|date=1988|pages=129–138| doi = 10.1007/bf00869432|volume=77|issue=2|s2cid=46977351}}</ref>
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|pages=454–462}}</ref><ref name=evolutionmodels>{{cite journal|last=Nalimov|first=W|title=Geometrization of biological ideas: probabilistic model of evolution|journal=Zhurnal Obshchei Biologii|date=2011|volume=62|issue=5|pages=437–448|pmid=11605554}}</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"/> 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"/>
===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="Neuroscience1980-Pellionisz"/><!--
--><ref name="Neuroscience1985-Pellionisz"/>
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"/>
*Sensory input = [[covariance and contravariance of vectors|covariant]] tensor
*Motor output = [[covariance and contravariance of vectors|contravariant]] tensor
*Cerebellar neuronal network = [[metric tensor]] that transforms the sensory input into the motor output
==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.<ref name="VOR arc">{{cite journal|last=Pellionisz|first=Andras|author2=Werner Graf |title=Tensor Network Model of the "Three-Neuron Vestibulo-Ocular Reflex-Arc" in Cat|journal=Journal of Theoretical Neurobiology|date=October 1986|volume=5|pages=127–151}}</ref> The "three-neuron [[vestibulo-ocular reflex]] arc" is named for the three neuron circuit the arc comprises. Sensory input into the [[vestibular system]] ([[angular acceleration]] of the head) is first received by the primary vestibular neurons which subsequently [[synapse]] onto secondary vestibular neurons.<ref name="VOR arc"/> These secondary neurons carry out much of the signal processing and produce the efferent signal heading for the [[oculomotor nerve|oculomotor neurons]].<ref name="VOR arc"/> Prior to the publishing of this paper, there had been no quantitative model to describe this "classic example of a basic [[sensory-motor coupling|sensorimotor]] transformation in the [[central nervous system]]" which is precisely what tensor network theory had been developed to model.<ref name="VOR arc"/>
<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]]ive [[eye movement (sensory)|eye movement]]) is described as the [[covariance and contravariance of vectors|contravariant]] vector component of the theory. By calculating the [[neuronal network]] transformations between the sensory input into the [[vestibular system]] and the subsequent motor response, a [[metric tensor]] representing the [[neuronal network]] was calculated.<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"/>
==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}}</ref> The fighter jet's sensors fed information into the flight computer which in turn transformed that information into commands to control the plane's wing-flaps and ailerons to achieve a stable touchdown. This was synonymous to sensory inputs from the body being transformed into motor outputs by the cerebellum. The flight computer's calculations and behavior was modeled as a metric tensor taking the covariant sensor readings and transforming it into contravariant commands to control aircraft hardware.<ref name="flightcontrol"/>
==References==
{{
==External links==
* [https://scholar.google.com/citations?user=oZioQ_MAAAAJ&hl=en Andras Pellionisz Google Scholar page Page]
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