Models of neural computation: Difference between revisions

Revision as of 01:39, 8 July 2016

Revision as of 18:35, 31 July 2016

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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. 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>

==Examples==

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===Models of information transfer in neurons===

{{main article|biological neuron model}}

The most widely used models of information transfer in biological neurons are based on analogies with electrical circuits. The equations to be solved are time-dependent differential equations with electro-dynamical variables such as current, conductance or resistance, capacitance and voltage.

====Hodgkin–Huxley model and its derivatives====

{{main article|Hodgkin–Huxley model}}

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.

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====Cross-correlation in sound localization: Jeffress model====

According to [[Lloyd A. Jeffress|Jeffress]],<ref>{{cite journal | last1 = Jeffress, | first1 = L.A., | year = 1948. | title = A place theory of sound localization. ''| url = | journal = Journal of Comparative and Physiological Psychology | volume = 41'', | issue = | pages = 35–39. }}</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>Brian J. Fischer and Charles H. Anderson, 2004. A computational model of sound localization in the barn owl ''Neurocomputing" 58–60 (2004) 1007–1012</ref> the response of the postsynaptic neuron to the signals from the left and right ears is given by

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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 Neuroscience" 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 = 1 | last2 = et al. (| year = 2008) | title = Response properties of motion-sensitive visual interneurons in the lobula plate of Drosophila melanogaster. | url = | journal = Curr. Biol. | volume = 18, | issue = | pages = 368–374 }}</ref>

The master equation for response is

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===Anti-Hebbian adaptation: spike-timing dependent plasticity===

<ref>{{cite journal | last1 = Tzounopoulos | first1 = T, | last2 = Kim | first2 = Y, | last3 = Oertel | first3 = D, | last4 = Trussell | first4 = LO (| year = 2004) | title = Cell-specific, spike timing-dependent plasticities in the dorsal cochlear nucleus. | url = | journal = Nat Neurosci | volume = 7: | issue = | pages = 719–725 }}</ref>

<ref>{{cite journal | last1 = Roberts | first1 = Patrick D. Roberts,| last2 = Portfors | first2 = Christine V. Portfors| year = (2008) | title = Cell-specific, spike timing-dependent plasticities in the dorsal cochlear nucleus. | url = | journal = Biological Cybernetics | volume = 98: | issue = | pages = 491–507 }}</ref>

===Feedback and control: models of flight control in the fly===

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===Neural networks===

{{main article|neural network}}

In this approach the strength and type, excitatory or inhibitory, of synaptic connections are represented by the magnitude and sign of weights, that is, numerical [[coefficients]] <math>w'</math> in front of the inputs ''x'' to a particular neuron. The response of the ''j''-th neuron is given by a sum of nonlinear, usually "[[sigmoid function|sigmoidal]]" functions <math>g</math> of the inputs as: