Editing Neuromorphic computing (section)

==Examples==
{{Very long section|date=September 2022}}
As early as 2006, researchers at [[Georgia Tech]] published a field programmable neural array.<ref>{{Cite book|last1 = Farquhar|first1 = Ethan|date = May 2006|pages = 4114–4117|last2 = Hasler|first2 = Paul.| title=2006 IEEE International Symposium on Circuits and Systems | chapter=A Field Programmable Neural Array |doi = 10.1109/ISCAS.2006.1693534|isbn = 978-0-7803-9389-9|s2cid = 206966013}}</ref> This chip was the first in a line of increasingly complex arrays of floating gate transistors that allowed programmability of charge on the gates of [[MOSFET]]s to model the channel-ion characteristics of neurons in the brain and was one of the first cases of a silicon programmable array of neurons.

In November 2011, a group of [[MIT]] researchers created a computer chip that mimics the analog, ion-based communication in a synapse between two neurons using 400 transistors and standard [[CMOS]] manufacturing techniques.<ref>{{cite web|title=MIT creates "brain chip"|date=November 15, 2011 |url=http://www.extremetech.com/extreme/105067-mit-creates-brain-chip|access-date=4 December 2012}}</ref><ref name="Neuromorphic silicon paper">{{cite journal|title=Neuromorphic silicon neurons and large-scale neural networks: challenges and opportunities|doi=10.3389/fnins.2011.00108|pmid=21991244|pmc=3181466|volume=5|page=108|journal=Frontiers in Neuroscience|year=2011|last1=Poon|first1=Chi-Sang|last2=Zhou|first2=Kuan|doi-access=free}}</ref>

In June 2012, [[spintronic]] researchers at [[Purdue University]] presented a paper on the design of a neuromorphic chip using [[Spin valve|lateral spin valve]]s and [[memristor]]s. They argue that the architecture works similarly to neurons and can therefore be used to test methods of reproducing the brain's processing. In addition, these chips are significantly more energy-efficient than conventional ones.<ref name="Spin Devices Prop">{{Cite arXiv|title=Proposal For Neuromorphic Hardware Using Spin Devices|eprint=1206.3227|last1=Sharad|first1=Mrigank|last2=Augustine|first2=Charles|last3=Panagopoulos|first3=Georgios|last4=Roy|first4=Kaushik|class=cond-mat.dis-nn|year=2012}}</ref>

Research at [[HP Labs]] on Mott memristors has shown that while they can be non-[[Volatile memory|volatile]], the volatile behavior exhibited at temperatures significantly below the [[phase transition]] temperature can be exploited to fabricate a [[neuristor]],<ref name=":0" /> a biologically-inspired device that mimics behavior found in neurons.<ref name=":0">{{Cite journal | doi = 10.1038/nmat3510| pmid = 23241533| title = A scalable neuristor built with Mott memristors| journal = Nature Materials| volume = 12| issue = 2| pages = 114–7| year = 2012| last1 = Pickett | first1 = M. D. | last2 = Medeiros-Ribeiro | first2 = G. | last3 = Williams | first3 = R. S. | bibcode = 2013NatMa..12..114P| s2cid = 16271627}}</ref> In September 2013, they presented models and simulations that show how the spiking behavior of these neuristors can be used to form the components required for a [[Turing machine]].<ref>{{cite journal|doi=10.1088/0957-4484/24/38/384002|title=Phase transitions enable computational universality in neuristor-based cellular automata|author1=Matthew D Pickett|author2=R Stanley Williams|name-list-style=amp|date=September 2013|publisher=IOP Publishing Ltd|journal=Nanotechnology|volume=24|issue=38|pmid=23999059|bibcode=2013Nanot..24L4002P|s2cid=9910142 |at=384002}}</ref>

[[Neurogrid]], built by ''Brains in Silicon'' at [[Stanford University]],<ref>{{cite journal|last1=Boahen|first1=Kwabena|title=Neurogrid: A Mixed-Analog-Digital Multichip System for Large-Scale Neural Simulations|journal=Proceedings of the IEEE|date=24 April 2014|volume=102|issue=5|pages=699–716|doi=10.1109/JPROC.2014.2313565|s2cid=17176371}}</ref> is an example of hardware designed using neuromorphic engineering principles. The circuit board is composed of 16 custom-designed chips, referred to as NeuroCores. Each NeuroCore's analog circuitry is designed to emulate neural elements for 65536 neurons, maximizing energy efficiency. The emulated neurons are connected using digital circuitry designed to maximize spiking throughput.<ref>{{cite journal|doi=10.1038/503022a|pmid = 24201264|title = Neuroelectronics: Smart connections|journal = Nature|volume = 503|issue = 7474|pages = 22–4|year = 2013|last1 = Waldrop|first1 = M. Mitchell|bibcode = 2013Natur.503...22W|doi-access = free}}</ref><ref>{{cite journal|doi=10.1109/JPROC.2014.2313565|title = Neurogrid: A Mixed-Analog-Digital Multichip System for Large-Scale Neural Simulations|journal = Proceedings of the IEEE|volume = 102|issue = 5|pages = 699–716|year = 2014|last1 = Benjamin|first1 = Ben Varkey|last2 = Peiran Gao|last3 = McQuinn|first3 = Emmett|last4 = Choudhary|first4 = Swadesh|last5 = Chandrasekaran|first5 = Anand R.|last6 = Bussat|first6 = Jean-Marie|last7 = Alvarez-Icaza|first7 = Rodrigo|last8 = Arthur|first8 = John V.|last9 = Merolla|first9 = Paul A.|last10 = Boahen|first10 = Kwabena|s2cid = 17176371}}</ref>

A research project with implications for neuromorphic engineering is the Human Brain Project that is attempting to simulate a complete human brain in a supercomputer using biological data. It is made up of a group of researchers in neuroscience, medicine, and computing.<ref>{{cite web|title=Involved Organizations|url=http://www.humanbrainproject.eu/partners.html|access-date=22 February 2013|archive-url=https://web.archive.org/web/20130302142627/http://www.humanbrainproject.eu/partners.html|archive-date=2 March 2013}}</ref> [[Henry Markram]], the project's co-director, has stated that the project proposes to establish a foundation to explore and understand the brain and its diseases, and to use that knowledge to build new computing technologies. The three primary goals of the project are to better understand how the pieces of the brain fit and work together, to understand how to objectively diagnose and treat brain diseases and to use the understanding of the human brain to develop neuromorphic computers. Since the simulation of a complete human brain will require a powerful supercomputer,  the current focus on neuromorphic computers is being encouraged.<ref>{{cite web|title=Human Brain Project|url=http://www.humanbrainproject.eu|access-date=22 February 2013}}</ref> $1.3{{Nbsp}}billion has been allocated to the project by The [[European Commission]].<ref>{{cite web|title=The Human Brain Project and Recruiting More Cyberwarriors|url=http://www.marketplace.org/topics/tech/human-brain-project-and-recruiting-more-cyberwarriors|access-date=22 February 2013|date=January 29, 2013}}</ref>

Other research with implications for neuromorphic engineering involve the [[BRAIN Initiative]]<ref name="economist">[https://www.economist.com/news/science-and-technology/21582495-computers-will-help-people-understand-brains-better-and-understanding-brains Neuromorphic computing: The machine of a new soul], The Economist, 2013-08-03</ref> and the [[TrueNorth]] chip from [[IBM]].<ref>{{cite journal|last1=Modha|first1=Dharmendra|title=A million spiking-neuron integrated circuit with a scalable communication network and interface|journal=Science|date=Aug 2014|volume=345|issue=6197|pages=668–673|doi=10.1126/science.1254642|pmid=25104385|bibcode=2014Sci...345..668M|s2cid=12706847}}</ref> Neuromorphic devices have also been demonstrated using nanocrystals, nanowires, and conducting polymers.<ref>{{Cite web|url=http://jessamynfairfield.com/wp-content/uploads/2017/03/PWMar17Fairfield.pdf|title=Smarter Machines|last=Fairfield|first=Jessamyn|date=March 1, 2017}}</ref> There also is development of a memristive device for quantum neuromorphic architectures.<ref>{{cite journal |last1=Spagnolo |first1=Michele |last2=Morris |first2=Joshua |last3=Piacentini |first3=Simone |last4=Antesberger |first4=Michael |last5=Massa |first5=Francesco |last6=Crespi |first6=Andrea |last7=Ceccarelli |first7=Francesco |last8=Osellame |first8=Roberto |last9=Walther |first9=Philip |title=Experimental photonic quantum memristor |journal=Nature Photonics |date=April 2022 |volume=16 |issue=4 |pages=318–323 |doi=10.1038/s41566-022-00973-5 |arxiv=2105.04867 |bibcode=2022NaPho..16..318S |s2cid=234358015 |language=en |issn=1749-4893}}<br/>News article: {{cite news |title=Erster "Quanten-Memristor" soll KI und Quantencomputer verbinden |url=https://www.derstandard.de/consent/tcf/story/2000134458057/erster-quanten-memristor-sollki-und-quantencomputer-verbinden |access-date=28 April 2022 |work=DER STANDARD |language=de-AT}}<br/>Lay summary report: {{cite news |title=Artificial neurons go quantum with photonic circuits |url=https://phys.org/news/2022-03-artificial-neurons-quantum-photonic-circuits.html |access-date=19 April 2022 |work=[[University of Vienna]] |language=en}}</ref> In 2022, researchers at MIT have reported the development of brain-inspired [[Physical neural network|artificial synapses]], using [[Proton#Hydrogen ion|the ion proton]] ({{chem|H|+}}), for 'analog [[deep learning]]'.<ref>{{cite news |title='Artificial synapse' could make neural networks work more like brains |url=https://www.newscientist.com/article/2331368-artificial-synapse-could-make-neural-networks-work-more-like-brains/ |access-date=21 August 2022 |work=New Scientist}}</ref><ref>{{cite journal |last1=Onen |first1=Murat |last2=Emond |first2=Nicolas |last3=Wang |first3=Baoming |last4=Zhang |first4=Difei |last5=Ross |first5=Frances M. |last6=Li |first6=Ju |last7=Yildiz |first7=Bilge |last8=del Alamo |first8=Jesús A. |title=Nanosecond protonic programmable resistors for analog deep learning |journal=Science |date=29 July 2022 |volume=377 |issue=6605 |pages=539–543 |doi=10.1126/science.abp8064 |pmid=35901152 |bibcode=2022Sci...377..539O |s2cid=251159631 |url=http://li.mit.edu/Archive/Papers/22/Onen22EmondScience.pdf |language=en |issn=0036-8075}}</ref>

[[Intel]] unveiled its neuromorphic research chip, called "[[Intel Loihi|Loihi]]", in October 2017. The chip uses an asynchronous [[spiking neural network]] (SNN) to implement adaptive self-modifying event-driven fine-grained parallel computations used to implement learning and inference with high efficiency.<ref>{{cite journal |last1=Davies |first1=Mike |title=Loihi: A Neuromorphic Manycore Processor with On-Chip Learning |journal=IEEE Micro |date=January 16, 2018 |volume=38 |issue=1 |pages=82–99 |display-authors=etal|doi=10.1109/MM.2018.112130359 |s2cid=3608458 }}</ref><ref name="Morris2017">{{cite web |last1=Morris |first1=John |title=Why Intel built a neuromorphic chip |url=https://www.zdnet.com/article/why-intel-built-a-neuromorphic-chip/ |website=ZDNet |access-date=17 August 2018 |language=en}}</ref>

[[IMEC]], a Belgium-based nanoelectronics research center, demonstrated the world's first self-learning neuromorphic chip. The brain-inspired chip, based on OxRAM technology, has the capability of self-learning and has been demonstrated to have the ability to compose music.<ref>{{cite web |title=Imec demonstrates self-learning neuromorphic chip that composes music |url=https://www.imec-int.com/en/articles/imec-demonstrates-self-learning-neuromorphic-chip-that-composes-music |website=IMEC International |access-date=1 October 2019}}</ref> IMEC released the 30-second tune composed by the prototype. The chip was sequentially loaded with songs in the same time signature and style. The songs were old Belgian and French flute minuets, from which the chip learned the rules at play and then applied them.<ref>{{cite web|last1=Bourzac|first1=Katherine|title=A Neuromorphic Chip That Makes Music|url=https://spectrum.ieee.org/a-neuromorphic-chip-that-makes-music|access-date=1 October 2019|website=IEEE Spectrum|date=May 23, 2017}}</ref>

The [[Blue Brain Project]], led by Henry Markram, aims to build biologically detailed digital reconstructions and simulations of the mouse brain. The Blue Brain Project has created in silico models of rodent brains, while attempting to replicate as many details about its biology as possible. The supercomputer-based simulations offer new perspectives on understanding the structure and functions of the brain.

The European Union funded a series of projects at the University of Heidelberg, which led to the development of [[BrainScaleS]] (brain-inspired multiscale computation in neuromorphic hybrid systems), a hybrid analog [[neuromorphic]] supercomputer located at Heidelberg University, Germany. It was developed as part of the Human Brain Project neuromorphic computing platform and is the complement to the [[SpiNNaker]] supercomputer (which is based on digital technology). The architecture used in BrainScaleS mimics biological neurons and their connections on a physical level; additionally, since the components are made of silicon, these model neurons operate on average 864 times (24 hours of real time is 100 seconds in the machine simulation) faster than that of their biological counterparts.<ref>{{Cite web|date=2016-03-21|title=Beyond von Neumann, Neuromorphic Computing Steadily Advances|url=https://www.hpcwire.com/2016/03/21/lacking-breakthrough-neuromorphic-computing-steadily-advance/|access-date=2021-10-08|website=HPCwire|language=en-US}}</ref>

In 2019, the European Union funded the project "Neuromorphic quantum computing"<ref>{{Cite journal |title=Neuromrophic Quantum Computing {{!}} Quromorphic Project {{!}} Fact Sheet {{!}} H2020 |url=https://cordis.europa.eu/project/id/828826 |access-date=2024-03-18 |website=CORDIS {{!}} European Commission |language=en |doi=10.3030/828826|url-access=subscription }}</ref> exploring the use of neuromorphic computing to perform quantum operations. Neuromorphic quantum computing<ref>{{Citation |last1=Pehle |first1=Christian |title=Neuromorphic quantum computing |date=2021-03-30 |arxiv=2005.01533 |last2=Wetterich |first2=Christof|journal=Physical Review E |volume=106 |issue=4 |page=045311 |doi=10.1103/PhysRevE.106.045311 |pmid=36397478 |bibcode=2022PhRvE.106d5311P }}</ref> (abbreviated as 'n.quantum computing') is an [[unconventional computing]] type of computing that uses neuromorphic computing to perform quantum operations.<ref>{{Cite journal |last=Wetterich |first=C. |date=2019-11-01 |title=Quantum computing with classical bits |url=https://www.sciencedirect.com/science/article/pii/S0550321319302627 |journal=Nuclear Physics B |volume=948 |page=114776 |doi=10.1016/j.nuclphysb.2019.114776 |issn=0550-3213|arxiv=1806.05960 |bibcode=2019NuPhB.94814776W }}</ref><ref>{{Citation |last1=Pehle |first1=Christian |title=Emulating quantum computation with artificial neural networks |date=2018-10-24 |arxiv=1810.10335 |last2=Meier |first2=Karlheinz |last3=Oberthaler |first3=Markus |last4=Wetterich |first4=Christof}}</ref> It was suggested that [[quantum algorithm]]s, which are algorithms that run on a realistic model of [[Quantum computing|quantum computation]], can be computed equally efficiently with neuromorphic quantum computing.<ref>{{Cite journal |last1=Carleo |first1=Giuseppe |last2=Troyer |first2=Matthias |date=2017-02-10 |title=Solving the quantum many-body problem with artificial neural networks |url=https://www.science.org/doi/10.1126/science.aag2302 |journal=Science |language=en |volume=355 |issue=6325 |pages=602–606 |doi=10.1126/science.aag2302 |pmid=28183973 |issn=0036-8075|arxiv=1606.02318 |bibcode=2017Sci...355..602C }}</ref><ref>{{Cite journal |last1=Torlai |first1=Giacomo |last2=Mazzola |first2=Guglielmo |last3=Carrasquilla |first3=Juan |last4=Troyer |first4=Matthias |last5=Melko |first5=Roger |last6=Carleo |first6=Giuseppe |date=May 2018 |title=Neural-network quantum state tomography |url=https://www.nature.com/articles/s41567-018-0048-5 |journal=Nature Physics |language=en |volume=14 |issue=5 |pages=447–450 |doi=10.1038/s41567-018-0048-5 |issn=1745-2481|arxiv=1703.05334 |bibcode=2018NatPh..14..447T }}</ref><ref>{{Cite journal |last1=Sharir |first1=Or |last2=Levine |first2=Yoav |last3=Wies |first3=Noam |last4=Carleo |first4=Giuseppe |last5=Shashua |first5=Amnon |date=2020-01-16 |title=Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems |url=https://link.aps.org/doi/10.1103/PhysRevLett.124.020503 |journal=Physical Review Letters |volume=124 |issue=2 |page=020503 |doi=10.1103/PhysRevLett.124.020503|pmid=32004039 |arxiv=1902.04057 |bibcode=2020PhRvL.124b0503S }}</ref><ref>{{Citation |last1=Broughton |first1=Michael |title=TensorFlow Quantum: A Software Framework for Quantum Machine Learning |date=2021-08-26 |arxiv=2003.02989 |last2=Verdon |first2=Guillaume |last3=McCourt |first3=Trevor |last4=Martinez |first4=Antonio J. |last5=Yoo |first5=Jae Hyeon |last6=Isakov |first6=Sergei V. |last7=Massey |first7=Philip |last8=Halavati |first8=Ramin |last9=Niu |first9=Murphy Yuezhen}}</ref><ref name="Di Ventra">{{Citation |last=Di Ventra |first=Massimiliano |title=MemComputing vs. Quantum Computing: some analogies and major differences |date=2022-03-23 |arxiv=2203.12031}}</ref> Both, traditional quantum computing and neuromorphic quantum computing are physics-based unconventional computing approaches to computations and do not follow the [[von Neumann architecture]]. They both construct a system (a circuit) that represents the physical problem at hand, and then leverage their respective physics properties of the system to seek the "minimum". Neuromorphic quantum computing and quantum computing share similar physical properties during computation.<ref name="Di Ventra"/><ref>{{Cite journal |last1=Wilkinson |first1=Samuel A. |last2=Hartmann |first2=Michael J. |date=2020-06-08 |title=Superconducting quantum many-body circuits for quantum simulation and computing |journal=Applied Physics Letters |volume=116 |issue=23 |doi=10.1063/5.0008202 |issn=0003-6951|arxiv=2003.08838 |bibcode=2020ApPhL.116w0501W }}</ref>

[[Brainchip]] announced in October 2021 that it was taking orders for its Akida AI Processor Development Kits<ref>{{Cite web|url=https://cdn-api.markitdigital.com/apiman-gateway/ASX/asx-research/1.0/file/2924-02438858-2A1332482?access_token=83ff96335c2d45a094df02a206a39ff4|title=Taking Orders of Akida AI Processor Development Kits|date=21 October 2021}}</ref> and in January 2022 that it was taking orders for its Akida AI Processor PCIe boards,<ref>{{cite web | url=https://www.electronics-lab.com/first-mini-pciexpress-board-with-spiking-neural-network-chip/ | title=First mini PCIexpress board with spiking neural network chip | date=January 19, 2022 }}</ref> making it the world's first commercially available neuromorphic processor.

===Neuromemristive systems===
Neuromemristive systems are a subclass of neuromorphic computing systems that focuses on the use of memristors to implement [[neuroplasticity]]. While neuromorphic engineering focuses on mimicking biological behavior, neuromemristive systems focus on abstraction.<ref>{{Cite web|url=https://digitalops.sandia.gov/Mediasite/Play/a10cf6ceb55d47608bb8326dd00e46611d|title=002.08 N.I.C.E. Workshop 2014: Towards Intelligent Computing with Neuromemristive Circuits and Systems – Feb. 2014|website=digitalops.sandia.gov|access-date=2019-08-26}}</ref> For example, a neuromemristive system may replace the details of a [[Cerebral cortex|cortical]] microcircuit's behavior with an abstract neural network model.<ref>C. Merkel and D. Kudithipudi, "Neuromemristive extreme learning machines for pattern classification," ISVLSI, 2014.</ref>

There exist several neuron inspired threshold logic functions<ref name="Maan 1–13"/> implemented with memristors that have applications in high level [[pattern recognition]] applications. Some of the applications reported recently include [[speech recognition]],<ref>{{Cite journal|title = Memristor pattern recogniser: isolated speech word recognition|journal = Electronics Letters|pages = 1370–1372|volume = 51|issue = 17|doi = 10.1049/el.2015.1428|first1 = A.K.|last1 = Maan|first2 = A.P.|last2 = James|first3 = S.|last3 = Dimitrijev|year = 2015|bibcode = 2015ElL....51.1370M|hdl = 10072/140989|s2cid = 61454815|hdl-access = free}}</ref> [[face recognition]]<ref>{{Cite journal|title = Memristive Threshold Logic Face Recognition|journal = Procedia Computer Science|date = 2014-01-01|pages = 98–103|volume = 41|series = 5th Annual International Conference on Biologically Inspired Cognitive Architectures, 2014 BICA|doi = 10.1016/j.procs.2014.11.090|first1 = Akshay Kumar|last1 = Maan|first2 = Dinesh S.|last2 = Kumar|first3 = Alex Pappachen|last3 = James|doi-access = free|hdl = 10072/68372|hdl-access = free}}</ref> and [[object recognition]].<ref>{{Cite journal|title = Memristive Threshold Logic Circuit Design of Fast Moving Object Detection|journal = IEEE Transactions on Very Large Scale Integration (VLSI) Systems|date = 2015-10-01|issn = 1063-8210|pages = 2337–2341|volume = 23|issue = 10|doi = 10.1109/TVLSI.2014.2359801|first1 = A.K.|last1 = Maan|first2 = D.S.|last2 = Kumar|first3 = S.|last3 = Sugathan|first4 = A.P.|last4 = James|arxiv = 1410.1267|s2cid = 9647290}}</ref> They also find applications in replacing conventional digital logic gates.<ref>{{Cite journal|title = Resistive Threshold Logic|journal = IEEE Transactions on Very Large Scale Integration (VLSI) Systems|date = 2014-01-01|issn = 1063-8210|pages = 190–195|volume = 22|issue = 1|doi = 10.1109/TVLSI.2012.2232946|first1 = A.P.|last1 = James|first2 = L.R.V.J.|last2 = Francis|first3 = D.S.|last3 = Kumar|arxiv = 1308.0090|s2cid = 7357110}}</ref><ref>{{Cite journal|title = Threshold Logic Computing: Memristive-CMOS Circuits for Fast Fourier Transform and Vedic Multiplication|journal = IEEE Transactions on Very Large Scale Integration (VLSI) Systems|date = 2015-11-01|issn = 1063-8210|pages = 2690–2694|volume = 23|issue = 11|doi = 10.1109/TVLSI.2014.2371857|first1 = A.P.|last1 = James|first2 = D.S.|last2 = Kumar|first3 = A.|last3 = Ajayan|arxiv = 1411.5255|s2cid = 6076956}}</ref>

For (quasi)ideal passive memristive circuits, the evolution of the memristive memories can be written in a closed form ([[Caravelli-Traversa-Di Ventra equation|Caravelli–Traversa–Di Ventra equation]]):<ref>{{cite journal |last=Caravelli  |display-authors=etal|arxiv=1608.08651 |title=The complex dynamics of memristive circuits: analytical results and universal slow relaxation |year=2017 |doi=10.1103/PhysRevE.95.022140 |pmid= 28297937 |volume=95 |issue= 2 |page= 022140 |journal=Physical Review E|bibcode=2017PhRvE..95b2140C |s2cid=6758362}}</ref><ref name="Caravelli 2021 022140">{{cite journal |last=Caravelli  |display-authors=etal|arxiv=1608.08651 |title=Global minimization via classical tunneling assisted by collective force field formation |year=2021 |doi=10.1126/sciadv.abh1542 |pmid= 28297937 |volume=7 |issue=52 |journal=Science Advances|page=022140 |bibcode=2021SciA....7.1542C |s2cid=231847346 }}</ref>

:<math> \frac{d}{dt} \vec{X} = -\alpha \vec{X}+\frac{1}{\beta} (I-\chi \Omega X)^{-1} \Omega \vec S </math>

as a function of the properties of the physical memristive network and the external sources. The equation is valid for the case of the Williams-Strukov original toy model, as  in the case of ideal memristors, <math>\alpha=0</math>. However, the hypothesis of the existence of an ideal memristor is debatable.<ref>{{Cite journal |last=Abraham |first=Isaac |date=2018-07-20 |title=The case for rejecting the memristor as a fundamental circuit element |journal=Scientific Reports |language=en |volume=8 |issue=1 |page=10972 |doi=10.1038/s41598-018-29394-7 |issn=2045-2322 |pmc=6054652 |pmid=30030498|bibcode=2018NatSR...810972A }}</ref> In the equation above, <math>\alpha</math> is the "forgetting" time scale constant, typically associated to memory volatility, while <math>\chi=\frac{R_\text{off}-R_\text{on}}{R_\text{off}}</math> is the ratio of ''off'' and ''on'' values of the limit resistances of the memristors, <math> \vec S </math> is the vector of the sources of the circuit and <math>\Omega</math> is a projector on the fundamental loops of the circuit. The constant <math>\beta</math> has the dimension of a voltage and is associated to the properties of the memristor; its physical origin is the charge mobility in the conductor. The diagonal matrix and vector <math>X=\operatorname{diag}(\vec X)</math> and <math>\vec X</math> respectively, are instead the internal value of the memristors, with values between 0 and 1. This equation thus requires adding extra constraints on the memory values in order to be reliable.

It has been recently shown that the equation above exhibits tunneling phenomena and used to study [[Lyapunov function]]s.<ref>{{Cite book |last=Sheldon |first=Forrest |title=Collective Phenomena in Memristive Networks: Engineering phase transitions into computation |publisher=UC San Diego Electronic Theses and Dissertations |year=2018}}</ref><ref name="Caravelli 2021 022140" />

===Neuromorphic sensors===
The concept of neuromorphic systems can be extended to [[Sensor|sensors]] (not just to computation). An example of this applied to detecting [[light]] is the [[retinomorphic sensor]] or, when employed in an array, the [[event camera]]. An event camera's pixels all register changes in brightness levels individually, which makes these cameras comparable to human eyesight in their theoretical power consumption.<ref>{{Cite journal |last=Skorka |first=Orit |date=2011-07-01 |title=Toward a digital camera to rival the human eye |url=http://electronicimaging.spiedigitallibrary.org/article.aspx?doi=10.1117/1.3611015 |journal=Journal of Electronic Imaging |language=en |volume=20 |issue=3 |pages=033009–033009–18 |doi=10.1117/1.3611015 |bibcode=2011JEI....20c3009S |issn=1017-9909|url-access=subscription }}</ref> In 2022, researchers from the [[Max Planck Institute for Polymer Research]] reported an organic artificial spiking neuron that exhibits the signal diversity of biological neurons while operating in the biological wetware, thus enabling ''in-situ'' neuromorphic sensing and biointerfacing applications.<ref>{{cite journal |last1=Sarkar |first1=Tanmoy |last2=Lieberth |first2=Katharina |last3=Pavlou |first3=Aristea |last4=Frank |first4=Thomas |last5=Mailaender |first5=Volker |last6=McCulloch |first6=Iain |last7=Blom |first7=Paul W. M. |last8=Torriccelli |first8=Fabrizio |last9=Gkoupidenis |first9=Paschalis |title=An organic artificial spiking neuron for in situ neuromorphic sensing and biointerfacing |journal=Nature Electronics |date=7 November 2022 |volume=5 |issue=11 |pages=774–783 |doi=10.1038/s41928-022-00859-y |s2cid=253413801 |language=en |issn=2520-1131|doi-access=free |hdl=10754/686016 |hdl-access=free }}</ref><ref>{{cite journal |title=Artificial neurons emulate biological counterparts to enable synergetic operation |journal=Nature Electronics |date=10 November 2022 |volume=5 |issue=11 |pages=721–722 |doi=10.1038/s41928-022-00862-3 |s2cid=253469402 |url=https://www.nature.com/articles/s41928-022-00862-3 |language=en |issn=2520-1131|url-access=subscription }}</ref>

=== Military applications ===
{{Unreferenced section|date=January 2023}}
The [[Joint Artificial Intelligence Center]], a branch of the U.S. military, is a center dedicated to the procurement and implementation of AI software and neuromorphic hardware for combat use. Specific applications include smart headsets/goggles and robots. JAIC intends to rely heavily on neuromorphic technology to connect "every sensor (to) every shooter" within a network of neuromorphic-enabled units.