Models of neural computation: Difference between revisions

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A '''linear''' system is one whose response in a specified unit of measure, to a set of inputs considered at once, is the sum of its responses due to the inputs considered individually.

[[Linear algebra|Linear]] systems are easier to analyze mathematically and are a persuasive assumption in many models including the McCulloch and Pitts neuron, population coding models, and the simple neurons often used in [[Artificial neural network]]s. Linearity may occur in the basic elements of a neural circuit such as the response of a postsynaptic neuron, or as an emergent property of a combination of nonlinear subcircuits.<ref name="MolnarHsueh2009">{{cite journal|last1=Molnar|first1=Alyosha|last2=Hsueh|first2=Hain-Ann|last3=Roska|first3=Botond|last4=Werblin|first4=Frank S.|title=Crossover inhibition in the retina: circuitry that compensates for nonlinear rectifying synaptic transmission|journal=Journal of Computational Neuroscience|volume=27|issue=3|year=2009|pages=569–590|issn=0929-5313|doi=10.1007/s10827-009-0170-6 | pmid = 19636690|pmc=2766457}}</ref> Though linearity is often seen as incorrect, there has been recent work suggesting it may, in fact, be biophysically plausible in some cases.<ref>{{Cite journal|last1=Singh|first1=Chandan|last2=Levy|first2=William B.|date=2017-07-13|title=A consensus layer V pyramidal neuron can sustain interpulse-interval coding|journal=PLOS OneONE|volume=12|issue=7|pages=e0180839|doi=10.1371/journal.pone.0180839|pmid=28704450|pmc=5509228|arxiv=1609.08213|bibcode=2017PLoSO..1280839S|issn=1932-6203|doi-access=free}}</ref><ref>{{Cite journal|last1=Cash|first1=Sydney|last2=Yuste|first2=Rafael|date=1998-01-01|title=Input Summation by Cultured Pyramidal Neurons Is Linear and Position-Independent|journal=Journal of Neuroscience|language=en|volume=18|issue=1|pages=10–15|issn=0270-6474|pmid=9412481|doi=10.1523/JNEUROSCI.18-01-00010.1998|pmc=6793421|doi-access=free}}</ref>

==Examples==

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The Hodgkin–Huxley model, widely regarded as one of the great achievements of 20th-century biophysics, describes how [[action potential]]s in neurons are initiated and propagated in axons via [[voltage-gated ion channel]]s. It is a set of [[nonlinearity|nonlinear]] [[ordinary differential equation]]s that were introduced by [[Alan Lloyd Hodgkin]] and [[Andrew Huxley]] in 1952 to explain the results of [[voltage clamp]] experiments on the [[squid giant axon]]. Analytic solutions do not exist, but the [[Levenberg–Marquardt algorithm]], a modified [[Gauss–Newton algorithm]], is often used to [[curve fitting|fit]] these equations to voltage-clamp data.

The [[FitzHugh–Nagumo model]] is a simplicationsimplification of the Hodgkin–Huxley model. The [[Hindmarsh–Rose model]] is an extension which describes neuronal spike bursts. The Morris–Lecar model is a modification which does not generate spikes, but describes slow-wave propagation, which is implicated in the inhibitory synaptic mechanisms of [[central pattern generator]]s.

====Solitons====

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According to [[Lloyd A. Jeffress|Jeffress]],<ref>{{cite journal | last1 = Jeffress | first1 = L.A. | year = 1948 | title = A place theory of sound localization | journal = Journal of Comparative and Physiological Psychology | volume = 41 | issue = 1| pages = 35–39 | doi=10.1037/h0061495 | pmid=18904764}}</ref> in order to compute the ___location of a sound source in space from [[interaural time difference]]s, an auditory system relies on [[Analog delay line|delay lines]]: the induced signal from an [[ipsilateral]] auditory receptor to a particular neuron is delayed for the same time as it takes for the original sound to go in space from that ear to the other. Each postsynaptic cell is differently delayed and thus specific for a particular inter-aural time difference. This theory is equivalent to the mathematical procedure of [[cross-correlation]].

Following Fischer and Anderson,<ref>{{cite journal | last1 = Fischer | first1 = Brian J. | last2 = Anderson | first2 = Charles H. | year = 2004 | title = A computational model of sound localization in the barn owl | journal = Neurocomputing | volume = 58–60 | pages = 1007–1012 | doi=10.1016/j.neucom.2004.01.159| s2cid = 31927198 }}</ref> the response of the postsynaptic neuron to the signals from the left and right ears is given by

<math>y_{R}\left(t\right) - y_{L}\left(t\right)</math>

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====Cross-correlation for motion detection: Hassenstein–Reichardt model====

A motion detector needs to satisfy three general requirements: pair-inputs, asymmetry and nonlinearity.<ref>Borst A, Egelhaaf M., 1989. Principles of visual motion detection. "Trends in Neurosciences" 12(8):297–306</ref> The cross-correlation operation implemented asymmetrically on the responses from a pair of photoreceptors satisfies these minimal criteria, and furthermore, predicts features which have been observed in the response of neurons of the lobula plate in bi-wing insects.<ref>{{cite journal | last1 = Joesch | first1 = M. |display-authors=etal | year = 2008 | title = Response properties of motion-sensitive visual interneurons in the lobula plate of Drosophila melanogaster | journal = Curr. Biol. | volume = 18 | issue = 5| pages = 368–374 | doi=10.1016/j.cub.2008.02.022| pmid = 18328703 | s2cid = 18873331 | doi-access = free | bibcode = 2008CBio...18..368J }}</ref>

The master equation for response is

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====Cerebellum sensory motor control====

[[Tensor network theory]] is a theory of [[cerebellum|cerebellar]] function 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="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 | url= https://www.academia.edu/download/31409354/pellionisz_1980_cerebellar_coordination_via_a_metric_tensor_fullpaper.pdf | 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| doi = 10.1016/0306-4522(85)90001-6 | pmid = 4080158| s2cid =10747593 }}</ref>

==Software modelling approaches and tools==