The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φm(t) and the amount of electric charge that has flowed, q(t):[1]

[5] In the relationship between Φm and q, the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge:

[6] Pershin and Di Ventra[3] have proposed a test that can help to resolve some of the long-standing controversies about whether an ideal memristor does actually exist or is a purely mathematical concept.

[15] The memristor plays a crucial role in mimicking the charge storage effect in the diode base, and is also responsible for the conductivity modulation phenomenon (that is so important during forward transients).

[17][19] Following this claim, Leon Chua has argued that the memristor definition could be generalized to cover all forms of two-terminal non-volatile memory devices based on resistance switching effects.

[21] They indicated inadequacies in the electrochemical modeling presented in the Nature article "The missing memristor found"[17] because the impact of concentration polarization effects on the behavior of metal−TiO2−x−metal structures under voltage or current stress was not considered.

[25] In a kind of thought experiment, Meuffels and Soni[21] furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called non-volatility property[16] exists in physical reality, its behavior would violate Landauer's principle, which places a limit on the minimum amount of energy required to change "information" states of a system.

To check the adequacy of the proposed model and its resulting state equations, the input signal u(t) can be superposed with a stochastic term ξ(t), which takes into account the existence of inevitable thermal fluctuations.

The instantaneous electrical power entering such a device is completely dissipated as Joule heat to the surrounding, so no extra energy remains in the system after it has been brought from one resistance state xi to another one xj.

The HP Labs group noted that "window functions" were necessary to compensate for differences between experimental measurements and their memristor model due to non-linear ionic drift and boundary effects.

One advantage of memristive networks is that they can be implemented using relatively simple and inexpensive hardware, making them an attractive option for developing low-cost artificial intelligence systems.

[63] In 2004, Krieger and Spitzer described dynamic doping of polymer and inorganic dielectric-like materials that improved the switching characteristics and retention required to create functioning nonvolatile memory cells.

[65] In 2010, Alibart, Gamrat, Vuillaume et al.[66] introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor[67] and which exhibits the main behavior of a biological spiking synapse.

High switching performance, demonstrated synaptic plasticity and sustainability to mechanical deformations promise to emulate the appealing characteristics of biological neural systems in novel computing technologies.

Afterwards, taking advantage of the low "on" resistance and large on/off ratio, a high-performance zero-power RF switch is proved based on MoS2 or h-BN atomristors, indicating a new application of memristors for 5G, 6G and THz communication and connectivity systems.

Switching the polarization of the ferroelectric material by applying a positive or negative voltage across the junction can lead to a two order of magnitude resistance variation: ROFF ≫ RON (an effect called Tunnel Electro-Resistance).

In 2013, Ageev, Blinov et al.[78] reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope.

[89] The magnetic tunnel junction has been proposed to act as a memristor through several potentially complementary mechanisms, both extrinsic (redox reactions, charge trapping/detrapping and electromigration within the barrier) and intrinsic (spin-transfer torque).

Based on research performed between 1999 and 2003, Bowen et al. published experiments in 2006 on a magnetic tunnel junction (MTJ) endowed with bi-stable spin-dependent states[90](resistive switching).

This highlights the importance of understanding what role oxygen vacancies play in the memristive operation of devices that deploy complex oxides with an intrinsic property such as ferroelectricity[95] or multiferroicity.

[2] The mechanism of memristive behavior in such structures is based entirely on the electron spin degree of freedom which allows for a more convenient control than the ionic transport in nanostructures.

These processing and operating temperatures are higher than most ion-conducting chalcogenide device types, including the S-based glasses (e.g. GeS) that need to be photodoped or thermally annealed.

[12] Memristors can potentially be fashioned into non-volatile solid-state memory, which could allow greater data density than hard drives with access times similar to DRAM, replacing both components.

[31] HP prototyped a crossbar latch memory that can fit 100 gigabits in a square centimeter,[104] and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm3).

[22] In September 2014, Mohamed-Salah Abdelouahab, Rene Lozi, and Leon Chua published a general theory of 1st-, 2nd-, 3rd-, and nth-order memristive elements using fractional derivatives.

[20][130] However the first device of a related nature to be constructed was the memistor (i.e. memory resistor), a term coined in 1960 by Bernard Widrow to describe a circuit element of an early artificial neural network called ADALINE.

[1] Five years later he and his student Sung Mo Kang generalized the theory of memristors and memristive systems including a property of zero crossing in the Lissajous curve characterizing current vs. voltage behavior.

[2] On May 1, 2008, Strukov, Snider, Stewart, and Williams published an article in Nature identifying a link between the two-terminal resistance switching behavior found in nanoscale systems and memristors.

[128] In July 2014, the MeMOSat/LabOSat group[131] (composed of researchers from Universidad Nacional de General San Martín (Argentina), INTI, CNEA, and CONICET) put memory devices into a Low Earth orbit.

In the September 2023 issue of Science Magazine, Chinese scientists Wenbin Zhang et al. described the development and testing of a memristor-based integrated circuit.