Tensor network theory: Difference between revisions

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{{short description|Theory of brain function}}

:''{{For |the tensor network theory used in quantum physics,|Tensor see [[matrix product state]]''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.<!--

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--><ref name="Neuroscience1980-Pellionisz">{{Cite journal| author =Pellionisz, A., Llinás, R. | year =1980 | 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= httphttps://usa-siliconvalleywww.com/instacademia.edu/pellioniszdownload/80_metric31409354/80_metricpellionisz_1980_cerebellar_coordination_via_a_metric_tensor_fullpaper.htmlpdf | doi = 10.1016/0306-4522(80)90191-8 | pmid=6967569| s2cid =17303132 }}{{dead link|date=July 2022|bot=medic}}{{cbignore|bot=medic}}</ref><!--

--><ref name="Neuroscience1985-Pellionisz">{{Cite journal| author = Pellionisz, A., Llinás, R. | year =1985 | 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| s2cid =10747593 }}{{dead link|date=May 2021|bot=medic}}{{cbignore|bot=medic}}</ref>

[[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.]]

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===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|journal=Lecture Notes in Mathematics|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|accessdate=November 18, 2013}}</ref><ref name=evolutionmodels>{{cite journal|last=Nalimov|first=W|title=Geometrization of biological ideas: probablisticprobabilistic model of evolution|journal=Zhurnal Obshchei Biologii|date=2011|volume=62|issue=5|pages=437–448|accessdatepmid=November 16, 201311605554}}</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===

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==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|accessdate=November 17, 2013}}</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"/>

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===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|accessdate=September 9, 2013}}</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==

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==External links==

* [httphttps://wwwscholar.usa-siliconvalleygoogle.com/citations?user=oZioQ_MAAAAJ&hl=en Andras Pellionisz ProfessionalGoogle Scholar page Page]

[[Category:NeuroscienceComputational neuroscience]]