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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|last=Singh|first=Chandan|last2=Levy|first2=William B.|date=2017-07-13|title=A consensus layer V pyramidal neuron can sustain interpulse-interval coding
==Examples==
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| pmid = 11341579
| jstor = 1543311
| citeseerx = 10.1.1.116.5190
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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 = 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 | url = | journal = Neurocomputing | volume = 58–60 | issue = | pages = 1007–1012 | doi=10.1016/j.neucom.2004.01.159}}</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 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 = 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 = 5| pages = 368–374 | doi=10.1016/j.cub.2008.02.022| pmid = 18328703 }}</ref>
The master equation for response is
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===Anti-Hebbian adaptation: spike-timing dependent plasticity===
* {{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 = 7| pages = 719–725 | doi=10.1038/nn1272| pmid = 15208632 }}
* {{cite journal | last1 = Roberts | first1 = Patrick D. | last2 = Portfors | first2 = Christine V. | year = 2008 | title = Design principles of sensory processing in cerebellum-like structures| doi = 10.1007/s00422-008-0217-1 | pmid = 18491162 | journal = Biological Cybernetics | volume = 98 | issue = 6| pages = 491–507 }}
===Feedback and control: models of flight control in the fly===
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===Genetic algorithms===
[[Genetic algorithms]] are used to evolve neural (and sometimes body) properties in a model brain-body-environment system so as to exhibit some desired behavioral performance. The evolved agents can then be subjected to a detailed analysis to uncover their principles of operation. Evolutionary approaches are particularly useful for exploring spaces of possible solutions to a given behavioral task because these approaches minimize a priori assumptions about how a given behavior ought to be instantiated. They can also be useful for exploring different ways to complete a computational neuroethology model when only partial neural circuitry is available for a biological system of interest.<ref>{{cite journal
===NEURON===
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Nervous systems differ from the majority of silicon-based computing devices in that they resemble [[analog computer]]s (not [[digital data]] processors) and massively [[parallel computing|parallel]] processors, not [[von Neumann architecture|sequential]] processors. To model nervous systems accurately, in real-time, alternative hardware is required.
The most realistic circuits to date make use of [[analogue electronics|analog]] properties of existing [[digital electronics]] (operated under non-standard conditions) to realize Hodgkin–Huxley-type models ''in silico''.<ref>L. Alvadoa, J. Tomasa, S. Saghia, S. Renauda, T. Balb, A. Destexheb, G. Le Masson, 2004. Hardware computation of conductance-based neuron models. Neurocomputing 58–60 (2004) 109–115</ref><ref>{{cite journal
===Retinomorphic chips===
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