Glia

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https://en.wikipedia.org/wiki/Glia

 

Glia
Illustration of the four different types of glial cells found in the central nervous system: ependymal cells (light pink), astrocytes (green), microglial cells (dark red) and oligodendrocytes (light blue)
Details
Precursor	Neuroectoderm for macroglia, and hematopoietic stem cells for microglia
System	Nervous system
Identifiers
MeSH	D009457

Glia, also called glial cells (gliocytes) or neuroglia, are non-neuronal cells in the central nervous system (brain and spinal cord) and the peripheral nervous system that do not produce electrical impulses. The neuroglia make up more than one half the volume of neural tissue in our body. They maintain homeostasis, form myelin in the peripheral nervous system, and provide support and protection for neurons. In the central nervous system, glial cells include oligodendrocytes, astrocytes, ependymal cells and microglia, and in the peripheral nervous system they include Schwann cells and satellite cells.

Function

They have four main functions:

• to surround neurons and hold them in place
• to supply nutrients and oxygen to neurons
• to insulate one neuron from another
• to destroy pathogens and remove dead neurons.

They also play a role in neurotransmission and synaptic connections, and in physiological processes such as breathing. While glia were thought to outnumber neurons by a ratio of 10:1, recent studies using newer methods and reappraisal of historical quantitative evidence suggests an overall ratio of less than 1:1, with substantial variation between different brain tissues.

Glial cells have far more cellular diversity and functions than neurons, and glial cells can respond to and manipulate neurotransmission in many ways. Additionally, they can affect both the preservation and consolidation of memories. Glia were discovered in 1856, by the pathologist Rudolf Virchow in his search for a "connective tissue" in the brain. The term derives from Greek γλία and γλοία "glue" (English: /ˈɡliːə/ or /ˈɡlaɪə/), and suggests the original impression that they were the glue of the nervous system.

Types

Neuroglia of the brain shown by Golgi's method

Astrocytes can be identified in culture because, unlike other mature glia, they express glial fibrillary acidic protein (GFAP)

Glial cells in a rat brain stained with an antibody against GFAP

Different types of neuroglia

Macroglia

Derived from ectodermal tissue.

Location	Name	Description
CNS	Astrocytes	The most abundant type of macroglial cell in the CNS, astrocytes (also called astroglia) have numerous projections that link neurons to their blood supply while forming the blood–brain barrier. They regulate the external chemical environment of neurons by removing excess potassium ions, and recycling neurotransmitters released during synaptic transmission. Astrocytes may regulate vasoconstriction and vasodilation by producing substances such as arachidonic acid, whose metabolites are vasoactive. Astrocytes signal each other using ATP. The gap junctions (also known as electrical synapses) between astrocytes allow the messenger molecule IP3 to diffuse from one astrocyte to another. IP3 activates calcium channels on cellular organelles, releasing calcium into the cytoplasm. This calcium may stimulate the production of more IP3 and cause release of ATP through channels in the membrane made of pannexins. The net effect is a calcium wave that propagates from cell to cell. Extracellular release of ATP, and consequent activation of purinergic receptors on other astrocytes, may also mediate calcium waves in some cases. In general, there are two types of astrocytes, protoplasmic and fibrous, similar in function but distinct in morphology and distribution. Protoplasmic astrocytes have short, thick, highly branched processes and are typically found in gray matter. Fibrous astrocytes have long, thin, less-branched processes and are more commonly found in white matter. It has recently been shown that astrocyte activity is linked to blood flow in the brain, and that this is what is actually being measured in fMRI. They also have been involved in neuronal circuits playing an inhibitory role after sensing changes in extracellular calcium.
CNS	Oligodendrocytes	Oligodendrocytes are cells that coat axons in the CNS with their cell membrane, forming a specialized membrane differentiation called myelin, producing the myelin sheath. The myelin sheath provides insulation to the axon that allows electrical signals to propagate more efficiently.
CNS	Ependymal cells	Ependymal cells, also named ependymocytes, line the spinal cord and the ventricular system of the brain. These cells are involved in the creation and secretion of cerebrospinal fluid (CSF) and beat their cilia to help circulate the CSF and make up the blood-CSF barrier. They are also thought to act as neural stem cells.
CNS	Radial glia	Radial glia cells arise from neuroepithelial cells after the onset of neurogenesis. Their differentiation abilities are more restricted than those of neuroepithelial cells. In the developing nervous system, radial glia function both as neuronal progenitors and as a scaffold upon which newborn neurons migrate. In the mature brain, the cerebellum and retina retain characteristic radial glial cells. In the cerebellum, these are Bergmann glia, which regulate synaptic plasticity. In the retina, the radial Müller cell is the glial cell that spans the thickness of the retina and, in addition to astroglial cells, participates in a bidirectional communication with neurons.
PNS	Schwann cells	Similar in function to oligodendrocytes, Schwann cells provide myelination to axons in the peripheral nervous system (PNS). They also have phagocytotic activity and clear cellular debris that allows for regrowth of PNS neurons.
PNS	Satellite cells	Satellite glial cells are small cells that surround neurons in sensory, sympathetic, and parasympathetic ganglia. These cells help regulate the external chemical environment. Like astrocytes, they are interconnected by gap junctions and respond to ATP by elevating the intracellular concentration of calcium ions. They are highly sensitive to injury and inflammation and appear to contribute to pathological states, such as chronic pain.
PNS	Enteric glial cells	Are found in the intrinsic ganglia of the digestive system. Glia cells are thought to have many roles in the enteric system, some related to homeostasis and muscular digestive processes.

Microglia

Main article: Microglia

Microglia are specialized macrophages capable of phagocytosis that protect neurons of the central nervous system. They are derived from the earliest wave of mononuclear cells that originate in yolk sac blood islands early in development, and colonize the brain shortly after the neural precursors begin to differentiate.

These cells are found in all regions of the brain and spinal cord. Microglial cells are small relative to macroglial cells, with changing shapes and oblong nuclei. They are mobile within the brain and multiply when the brain is damaged. In the healthy central nervous system, microglia processes constantly sample all aspects of their environment (neurons, macroglia and blood vessels). In a healthy brain, microglia direct the immune response to brain damage and play an important role in the inflammation that accompanies the damage. Many diseases and disorders are associated with deficient microglia, such as Alzheimer's disease, Parkinson's disease and ALS.

Other

Pituicytes from the posterior pituitary are glial cells with characteristics in common to astrocytes. Tanycytes in the median eminence of the hypothalamus are a type of ependymal cell that descend from radial glia and line the base of the third ventricle. Drosophila melanogaster, the fruit fly, contains numerous glial types that are functionally similar to mammalian glia but are nonetheless classified differently.

Total number

In general, neuroglial cells are smaller than neurons. There are approximately 85 billion glia cells in the human brain, about the same number as neurons. Glial cells make up about half the total volume of the brain and spinal cord. The glia to neuron-ratio varies from one part of the brain to another. The glia to neuron-ratio in the cerebral cortex is 3.72 (60.84 billion glia (72%); 16.34 billion neurons), while that of the cerebellum is only 0.23 (16.04 billion glia; 69.03 billion neurons). The ratio in the cerebral cortex gray matter is 1.48, with 3.76 for the gray and white matter combined. The ratio of the basal ganglia, diencephalon and brainstem combined is 11.35.

The total number of glia cells in the human brain is distributed into the different types with oligodendrocytes being the most frequent (45–75%), followed by astrocytes (19–40%) and microglia (about 10% or less).

Development

23-week fetal brain culture astrocyte

Main article: Gliogenesis

Most glia are derived from ectodermal tissue of the developing embryo, in particular the neural tube and crest. The exception is microglia, which are derived from hematopoietic stem cells. In the adult, microglia are largely a self-renewing population and are distinct from macrophages and monocytes, which infiltrate an injured and diseased CNS.

In the central nervous system, glia develop from the ventricular zone of the neural tube. These glia include the oligodendrocytes, ependymal cells, and astrocytes. In the peripheral nervous system, glia derive from the neural crest. These PNS glia include Schwann cells in nerves and satellite glial cells in ganglia.

Capacity to divide

Glia retain the ability to undergo cell divisions in adulthood, whereas most neurons cannot. The view is based on the general inability of the mature nervous system to replace neurons after an injury, such as a stroke or trauma, where very often there is a substantial proliferation of glia, or gliosis, near or at the site of damage. However, detailed studies have found no evidence that 'mature' glia, such as astrocytes or oligodendrocytes, retain mitotic capacity. Only the resident oligodendrocyte precursor cells seem to keep this ability once the nervous system matures.

Glial cells are known to be capable of mitosis. By contrast, scientific understanding of whether neurons are permanently post-mitotic, or capable of mitosis, is still developing. In the past, glia had been considered to lack certain features of neurons. For example, glial cells were not believed to have chemical synapses or to release transmitters. They were considered to be the passive bystanders of neural transmission. However, recent studies have shown this to not be entirely true.

Functions

Some glial cells function primarily as the physical support for neurons. Others provide nutrients to neurons and regulate the extracellular fluid of the brain, especially surrounding neurons and their synapses. During early embryogenesis, glial cells direct the migration of neurons and produce molecules that modify the growth of axons and dendrites. Some glial cells display regional diversity in the CNS and their functions may vary between the CNS regions.

Neuron repair and development

Glia are crucial in the development of the nervous system and in processes such as synaptic plasticity and synaptogenesis. Glia have a role in the regulation of repair of neurons after injury. In the central nervous system (CNS), glia suppress repair. Glial cells known as astrocytes enlarge and proliferate to form a scar and produce inhibitory molecules that inhibit regrowth of a damaged or severed axon. In the peripheral nervous system (PNS), glial cells known as Schwann cells (or also as neuri-lemmocytes) promote repair. After axonal injury, Schwann cells regress to an earlier developmental state to encourage regrowth of the axon. This difference between the CNS and the PNS, raises hopes for the regeneration of nervous tissue in the CNS. For example, a spinal cord may be able to be repaired following injury or severance.

Myelin sheath creation

Oligodendrocytes are found in the CNS and resemble an octopus: they have bulbous cell bodies with up to fifteen arm-like processes. Each process reaches out to an axon and spirals around it, creating a myelin sheath. The myelin sheath insulates the nerve fiber from the extracellular fluid and speeds up signal conduction along the nerve fiber. In the peripheral nervous system, Schwann cells are responsible for myelin production. These cells envelop nerve fibers of the PNS by winding repeatedly around them. This process creates a myelin sheath, which not only aids in conductivity but also assists in the regeneration of damaged fibers.

Neurotransmission

Astrocytes are crucial participants in the tripartite synapse. They have several crucial functions, including clearance of neurotransmitters from within the synaptic cleft, which aids in distinguishing between separate action potentials and prevents toxic build-up of certain neurotransmitters such as glutamate, which would otherwise lead to excitotoxicity. Furthermore, astrocytes release gliotransmitters such as glutamate, ATP, and D-serine in response to stimulation.

Clinical significance

See also: Glioma

Neoplastic glial cells stained with an antibody against GFAP (brown), from a brain biopsy

While glial cells in the PNS frequently assist in regeneration of lost neural functioning, loss of neurons in the CNS does not result in a similar reaction from neuroglia. In the CNS, regrowth will only happen if the trauma was mild, and not severe. When severe trauma presents itself, the survival of the remaining neurons becomes the optimal solution. However, some studies investigating the role of glial cells in Alzheimer's disease are beginning to contradict the usefulness of this feature, and even claim it can "exacerbate" the disease. In addition to affecting the potential repair of neurons in Alzheimer's disease, scarring and inflammation from glial cells have been further implicated in the degeneration of neurons caused by amyotrophic lateral sclerosis.

In addition to neurodegenerative diseases, a wide range of harmful exposure, such as hypoxia, or physical trauma, can lead to the end result of physical damage to the CNS. Generally, when damage occurs to the CNS, glial cells cause apoptosis among the surrounding cellular bodies. Then, there is a large amount of microglial activity, which results in inflammation, and, finally, there is a heavy release of growth inhibiting molecules.

History

Although glial cells and neurons were probably first observed at the same time in the early 19th century, unlike neurons whose morphological and physiological properties were directly observable for the first investigators of the nervous system, glial cells had been considered to be merely "glue" that held neurons together until the mid-20th century.

Glia were first described in 1856 by the pathologist Rudolf Virchow in a comment to his 1846 publication on connective tissue. A more detailed description of glial cells was provided in the 1858 book 'Cellular Pathology' by the same author.

When markers for different types of cells were analyzed, Albert Einstein's brain was discovered to contain significantly more glia than normal brains in the left angular gyrus, an area thought to be responsible for mathematical processing and language. However, out of the total of 28 statistical comparisons between Einstein's brain and the control brains, finding one statistically significant result is not surprising, and the claim that Einstein's brain is different is not scientific (c.f. Multiple comparisons problem).

Not only does the ratio of glia to neurons increase through evolution, but so does the size of the glia. Astroglial cells in human brains have a volume 27 times greater than in mouse brains.

These important scientific findings may begin to shift the neurocentric perspective into a more holistic view of the brain which encompasses the glial cells as well. For the majority of the twentieth century, scientists had disregarded glial cells as mere physical scaffolds for neurons. Recent publications have proposed that the number of glial cells in the brain is correlated with the intelligence of a species. Moreover, evidences are demonstrating the active role of glia, in particular astroglia, in cognitive processes like learning and memory and, for these reasons, it has been proposed the foundation of a specific field to study these functions because investigations in this area are still limited due to the dominance of the neurocentric perspective.

Memristor

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Memristor

Memristor

Memristor developed by the University of Illinois at Urbana-Champaign and National Energy Technology Laboratory
Invented	Leon Chua (1971)
Electronic symbol

A memristor (/ˈmɛmrɪstər/; a portmanteau of memory resistor) is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which also comprises the resistor, capacitor and inductor.

Chua and Kang later generalized the concept to memristive systems. Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably ReRAM.

The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.

As a fundamental electrical component

Conceptual symmetries of resistor, capacitor, inductor, and memristor

Chua in his 1971 paper identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). From this symmetry he inferred the characteristics of a fourth fundamental non-linear circuit element, linking magnetic flux and charge, which he called the memristor. In contrast to a linear (or non-linear) resistor, the memristor has a dynamic relationship between current and voltage, including a memory of past voltages or currents. Other scientists had proposed dynamic memory resistors such as the memistor of Bernard Widrow, but Chua introduced a mathematical generality.

Derivation and characteristics

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

${\displaystyle f({\mathrm{\Phi }}_{\mathrm{m}}(t),q(t))=0}$

The magnetic flux linkage, Φ_m, is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here. Its physical meaning is discussed below. The symbol Φ_m may be regarded as the integral of voltage over time.

In the relationship between Φ_m 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.

${\displaystyle M(q)=\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{m}}}{\mathrm{d}q}}$

Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are;

${\displaystyle M(q(t))=\frac{\mathrm{d}{\mathrm{\Phi }}_{}/\mathrm{d}t}{\mathrm{d}q/\mathrm{d}t}=\frac{V(t)}{I(t)}}$

To relate the memristor to the resistor, capacitor, and inductor, it is helpful to isolate the term M(q), which characterizes the device, and write it as a differential equation.

Device	Characteristic property (units)	Differential equation
Resistor (R)	Resistance (V / A, or ohm, Ω)	R = dV / dI
Capacitor (C)	Capacitance (C / V, or farad)	C = dq / dV
Inductor (L)	Inductance (Wb / A, or henry)	L = dΦm / dI
Memristor (M)	Memristance (Wb / C, or ohm)	M = dΦm / dq

The above table covers all meaningful ratios of differentials of I, q, Φ_m, and V. No device can relate dI to dq, or dΦ_m to dV, because I is the derivative of q and Φ_m is the integral of V.

It can be inferred from this that memristance is charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohm's law R(t) = V(t)/I(t). If M(q(t)) is nontrivial, however, the equation is not equivalent because q(t) and M(q(t)) can vary with time. Solving for voltage as a function of time produces

${\displaystyle V(t)=M(q(t))I(t)}$

This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Nonzero current implies time varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum change in q does not cause much change in M.

Furthermore, the memristor is static if no current is applied. If I(t) = 0, we find V(t) = 0 and M(t) is constant. This is the essence of the memory effect.

Analogously, we can define a ${\displaystyle W(\varphi (t))}$ as menductance.

${\displaystyle i(t)=W(\varphi (t))v(t)}$

The power consumption characteristic recalls that of a resistor, I²R.

${\displaystyle P(t)=I(t)V(t)=I^2(t)M(q(t))}$

As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a constant resistor. If M(q(t)) increases rapidly, however, current and power consumption will quickly stop.

M(q) is physically restricted to be positive for all values of q (assuming the device is passive and does not become superconductive at some q). A negative value would mean that it would perpetually supply energy when operated with alternating current.

Modelling and validation

In order to understand the nature of memristor function, some knowledge of fundamental circuit theoretic concepts is useful, starting with the concept of device modeling.

Engineers and scientists seldom analyze a physical system in its original form. Instead, they construct a model which approximates the behaviour of the system. By analyzing the behaviour of the model, they hope to predict the behaviour of the actual system. The primary reason for constructing models is that physical systems are usually too complex to be amenable to a practical analysis.

In the 20th century, work was done on devices where researchers did not recognize the memristive characteristics. This has raised the suggestion that such devices should be recognised as memristors. Pershin and Di Ventra 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.

The rest of this article primarily addresses memristors as related to ReRAM devices, since the majority of work since 2008 has been concentrated in this area.

Superconducting memristor component

Dr. Paul Penfield, in a 1974 MIT technical report mentions the memristor in connection with Josephson junctions. This was an early use of the word "memristor" in the context of a circuit device.

One of the terms in the current through a Josephson junction is of the form:

${\displaystyle \begin{array}{rl}{i}_M(v)& =ϵ\cos ({\varphi }_0)v\\ & =W({\varphi }_0)v\end{array}}$

where ${\displaystyle ϵ}$ is a constant based on the physical superconducting materials, ${\displaystyle v}$ is the voltage across the junction and ${\displaystyle i_M}$ is the current through the junction.

Through the late 20th century, research regarding this phase-dependent conductance in Josephson junctions was carried out. A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and DiVentra's seminal paper in 2014.

Memristor circuits

Due to the practical difficulty of studying the ideal memristor, we will discuss other electrical devices which can be modelled using memristors. For a mathematical description of a memristive device (systems), see Theory.

A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons ${\displaystyle n_e}$.

${\displaystyle \begin{array}{rl}{v}_M& =R(n_e)i_M\\ \frac{d{n}_e}{dt}& =\beta n+\alpha R(n_e)i_M^2\end{array}}$

${\displaystyle v_M}$ is the voltage across the discharge tube, ${\displaystyle i_M}$ is the current flowing through it and ${\displaystyle n_e}$ is the number of conduction electrons. A simple memristance function is ${\displaystyle R(n_e)=\frac{F}{n_e}}$. ${\displaystyle \alpha ,\beta }$ and ${\displaystyle F}$ are parameters depending on the dimensions of the tube and the gas fillings. An experimental identification of memristive behaviour is the "pinched hysteresis loop" in the ${\displaystyle v-i}$ plane. For an experiment that shows such a characteristic for a common discharge tube, see "A physical memristor Lissajous figure" (YouTube). The video also illustrates how to understand deviations in the pinched hysteresis characteristics of physical memristors.

Thermistors can be modelled as memristive devices.

${\displaystyle \begin{array}{rl}v& =R_0(T_0)\exp \left[\beta \left(\frac{1}{T}-\frac{1}{T_0}\right)\right]i\\ & \equiv R(T)i\\ \frac{dT}{dt}& =\frac{1}{C}\left[-\delta \cdot (T-T_0)+R(T)i^2\right]\end{array}}$

${\displaystyle \beta }$ is a material constant, ${\displaystyle T}$ is the absolute body temperature of the thermistor, ${\displaystyle T_0}$ is the ambient temperature (both temperatures in Kelvin), ${\displaystyle R_0(T_0)}$ denotes the cold temperature resistance at ${\displaystyle T=T_0}$, ${\displaystyle C}$ is the heat capacitance and ${\displaystyle \delta }$ is the dissipation constant for the thermistor.

A fundamental phenomenon that has hardly been studied is memristive behaviour in pn-junctions. 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).

Criticisms

In 2008, a team at HP Labs found experimental evidence for the Chua's memristor based on an analysis of a thin film of titanium dioxide, thus connecting the operation of ReRAM devices to the memristor concept. According to HP Labs, the memristor would operate in the following way: the memristor's electrical resistance is not constant but depends on the current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has previously flowed through it and in what direction; the device remembers its history—the so-called non-volatility property. When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again.

The HP Labs result was published in the scientific journal Nature. 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. Chua also argued that the memristor is the oldest known circuit element, with its effects predating the resistor, capacitor, and inductor. There are, however, some serious doubts as to whether a genuine memristor can actually exist in physical reality. Additionally, some experimental evidence contradicts Chua's generalization since a non-passive nanobattery effect is observable in resistance switching memory. A simple test has been proposed by Pershin and Di Ventra to analyze whether such an ideal or generic memristor does actually exist or is a purely mathematical concept. Up to now, there seems to be no experimental resistance switching device (ReRAM) which can pass the test.

These devices are intended for applications in nanoelectronic memory devices, computer logic, and neuromorphic/neuromemristive computer architectures. In 2013, Hewlett-Packard CTO Martin Fink suggested that memristor memory may become commercially available as early as 2018. In March 2012, a team of researchers from HRL Laboratories and the University of Michigan announced the first functioning memristor array built on a CMOS chip.

An array of 17 purpose-built oxygen-depleted titanium dioxide memristors built at HP Labs, imaged by an atomic force microscope. The wires are about 50 nm, or 150 atoms, wide. Electric current through the memristors shifts the oxygen vacancies, causing a gradual and persistent change in electrical resistance.

According to the original 1971 definition, the memristor is the fourth fundamental circuit element, forming a non-linear relationship between electric charge and magnetic flux linkage. In 2011, Chua argued for a broader definition that includes all two-terminal non-volatile memory devices based on resistance switching. Williams argued that MRAM, phase-change memory and ReRAM are memristor technologies. Some researchers argued that biological structures such as blood and skin fit the definition. Others argued that the memory device under development by HP Labs and other forms of ReRAM are not memristors, but rather part of a broader class of variable-resistance systems, and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents.

In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction. In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors. They indicated inadequacies in the electrochemical modeling presented in the Nature article "The missing memristor found" because the impact of concentration polarization effects on the behavior of metal−TiO_2−x−metal structures under voltage or current stress was not considered. This critique was referred to by Valov et al. in 2013.

In a kind of thought experiment, Meuffels and Soni furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called non-volatility property 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. This critique was finally adopted by Di Ventra and Pershin in 2013.

Within this context, Meuffels and Soni pointed to a fundamental thermodynamic principle: Non-volatile information storage requires the existence of free-energy barriers that separate the distinct internal memory states of a system from each other; otherwise, one would be faced with an "indifferent" situation, and the system would arbitrarily fluctuate from one memory state to another just under the influence of thermal fluctuations. When unprotected against thermal fluctuations, the internal memory states exhibit some diffusive dynamics, which causes state degradation. The free-energy barriers must therefore be high enough to ensure a low bit-error probability of bit operation. Consequently, there is always a lower limit of energy requirement – depending on the required bit-error probability – for intentionally changing a bit value in any memory device.

In the general concept of memristive system the defining equations are (see Theory):

${\displaystyle \begin{array}{rl}y(t)& =g(\mathbf{x},u,t)u(t),\\ \dot{\mathbf{x}}& =f(\mathbf{x},u,t),\end{array}}$

where u(t) is an input signal, and y(t) is an output signal. The vector x represents a set of n state variables describing the different internal memory states of the device. ẋ is the time-dependent rate of change of the state vector x with time.

When one wants to go beyond mere curve fitting and aims at a real physical modeling of non-volatile memory elements, e.g., resistive random-access memory devices, one has to keep an eye on the aforementioned physical correlations. 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 dynamic state equation in its general form then finally reads:

${\displaystyle \dot{\mathbf{x}}=f(\mathbf{x},u(t)+\xi (t),t),}$

where ξ(t) is, e.g., white Gaussian current or voltage noise. On base of an analytical or numerical analysis of the time-dependent response of the system towards noise, a decision on the physical validity of the modeling approach can be made, e.g., would the system be able to retain its memory states in power-off mode?

Such an analysis was performed by Di Ventra and Pershin with regard to the genuine current-controlled memristor. As the proposed dynamic state equation provides no physical mechanism enabling such a memristor to cope with inevitable thermal fluctuations, a current-controlled memristor would erratically change its state in course of time just under the influence of current noise. Di Ventra and Pershin thus concluded that memristors whose resistance (memory) states depend solely on the current or voltage history would be unable to protect their memory states against unavoidable Johnson–Nyquist noise and permanently suffer from information loss, a so-called "stochastic catastrophe". A current-controlled memristor can thus not exist as a solid-state device in physical reality.

The above-mentioned thermodynamic principle furthermore implies that the operation of two-terminal non-volatile memory devices (e.g. "resistance-switching" memory devices (ReRAM)) cannot be associated with the memristor concept, i.e., such devices cannot by itself remember their current or voltage history. Transitions between distinct internal memory or resistance states are of probabilistic nature. The probability for a transition from state {i} to state {j} depends on the height of the free-energy barrier between both states. The transition probability can thus be influenced by suitably driving the memory device, i.e., by "lowering" the free-energy barrier for the transition {i} → {j} by means of, for example, an externally applied bias.

A "resistance switching" event can simply be enforced by setting the external bias to a value above a certain threshold value. This is the trivial case, i.e., the free-energy barrier for the transition {i} → {j} is reduced to zero. In case one applies biases below the threshold value, there is still a finite probability that the device will switch in course of time (triggered by a random thermal fluctuation), but – as one is dealing with probabilistic processes – it is impossible to predict when the switching event will occur. That is the basic reason for the stochastic nature of all observed resistance-switching (ReRAM) processes. If the free-energy barriers are not high enough, the memory device can even switch without having to do anything.

When a two-terminal non-volatile memory device is found to be in a distinct resistance state {j}, there exists therefore no physical one-to-one relationship between its present state and its foregoing voltage history. The switching behavior of individual non-volatile memory devices thus cannot be described within the mathematical framework proposed for memristor/memristive systems.

An extra thermodynamic curiosity arises from the definition that memristors/memristive devices should energetically act like resistors. 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 x_i to another one x_j. Thus, the internal energy of the memristor device in state x_i, U(V, T, x_i), would be the same as in state x_j, U(V, T, x_j), even though these different states would give rise to different device's resistances, which itself must be caused by physical alterations of the device's material.

Other researchers noted that memristor models based on the assumption of linear ionic drift do not account for asymmetry between set time (high-to-low resistance switching) and reset time (low-to-high resistance switching) and do not provide ionic mobility values consistent with experimental data. Non-linear ionic-drift models have been proposed to compensate for this deficiency.

A 2014 article from researchers of ReRAM concluded that Strukov's (HP's) initial/basic memristor modeling equations do not reflect the actual device physics well, whereas subsequent (physics-based) models such as Pickett's model or Menzel's ECM model (Menzel is a co-author of that article) have adequate predictability, but are computationally prohibitive. As of 2014, the search continues for a model that balances these issues; the article identifies Chang's and Yakopcic's models as potentially good compromises.

Martin Reynolds, an electrical engineering analyst with research outfit Gartner, commented that while HP was being sloppy in calling their device a memristor, critics were being pedantic in saying that it was not a memristor.

Experimental tests

Chua suggested experimental tests to determine if a device may properly be categorized as a memristor:

• The Lissajous curve in the voltage–current plane is a pinched hysteresis loop when driven by any bipolar periodic voltage or current without respect to initial conditions.
• The area of each lobe of the pinched hysteresis loop shrinks as the frequency of the forcing signal increases.
• As the frequency tends to infinity, the hysteresis loop degenerates to a straight line through the origin, whose slope depends on the amplitude and shape of the forcing signal.

According to Chua all resistive switching memories including ReRAM, MRAM and phase-change memory meet these criteria and are memristors. However, the lack of data for the Lissajous curves over a range of initial conditions or over a range of frequencies complicates assessments of this claim.

Experimental evidence shows that redox-based resistance memory (ReRAM) includes a nanobattery effect that is contrary to Chua's memristor model. This indicates that the memristor theory needs to be extended or corrected to enable accurate ReRAM modeling.

Theory

In 2008, researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide. For R_ON ≪ R_OFF the memristance function was determined to be

${\displaystyle M(q(t))=R_{\mathrm{O}\mathrm{F}\mathrm{F}}\cdot \left(1-\frac{{\mu }_v{R}_{\mathrm{O}\mathrm{N}}}{D^2}q(t)\right)}$

where R_OFF represents the high resistance state, R_ON represents the low resistance state, μ_v represents the mobility of dopants in the thin film, and D represents the film thickness. 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.

Operation as a switch

For some memristors, applied current or voltage causes substantial change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent to achieve a desired change in resistance. This assumes that the applied voltage remains constant. Solving for energy dissipation during a single switching event reveals that for a memristor to switch from R_on to R_off in time T_on to T_off, the charge must change by ΔQ = Q_on−Q_off.

${\displaystyle E_{\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}}=V^2{\int }_{T_{\mathrm{o}\mathrm{f}\mathrm{f}}}^{T_{\mathrm{o}\mathrm{n}}}\frac{\mathrm{d}t}{M(q(t))}=V^2{\int }_{Q_{\mathrm{o}\mathrm{f}\mathrm{f}}}^{Q_{\mathrm{o}\mathrm{n}}}\frac{\mathrm{d}q}{I(q)M(q)}=V^2{\int }_{Q_{\mathrm{o}\mathrm{f}\mathrm{f}}}^{Q_{\mathrm{o}\mathrm{n}}}\frac{\mathrm{d}q}{V(q)}=V\mathrm{\Delta }Q}$

Substituting V = I(q)M(q), and then ∫dq/V = ∆Q/V for constant VTo produces the final expression. This power characteristic differs fundamentally from that of a metal oxide semiconductor transistor, which is capacitor-based. Unlike the transistor, the final state of the memristor in terms of charge does not depend on bias voltage.

The type of memristor described by Williams ceases to be ideal after switching over its entire resistance range, creating hysteresis, also called the "hard-switching regime". Another kind of switch would have a cyclic M(q) so that each off-on event would be followed by an on-off event under constant bias. Such a device would act as a memristor under all conditions, but would be less practical.

Memristive systems

In the more general concept of an n-th order memristive system the defining equations are

${\displaystyle \begin{array}{rl}y(t)& =g(\mathbf{\text{x}},u,t)u(t),\\ \dot{\mathbf{\text{x}}}& =f(\mathbf{\text{x}},u,t)\end{array}}$

where u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and g and f are continuous functions. For a current-controlled memristive system the signal u(t) represents the current signal i(t) and the signal y(t) represents the voltage signal v(t). For a voltage-controlled memristive system the signal u(t) represents the voltage signal v(t) and the signal y(t) represents the current signal i(t).

The pure memristor is a particular case of these equations, namely when x depends only on charge (x = q) and since the charge is related to the current via the time derivative dq/dt = i(t). Thus for pure memristors f (i.e. the rate of change of the state) must be equal or proportional to the current i(t) .

Pinched hysteresis

Example of pinched hysteresis curve, V versus I

One of the resulting properties of memristors and memristive systems is the existence of a pinched hysteresis effect. For a current-controlled memristive system, the input u(t) is the current i(t), the output y(t) is the voltage v(t), and the slope of the curve represents the electrical resistance. The change in slope of the pinched hysteresis curves demonstrates switching between different resistance states which is a phenomenon central to ReRAM and other forms of two-terminal resistance memory. At high frequencies, memristive theory predicts the pinched hysteresis effect will degenerate, resulting in a straight line representative of a linear resistor. It has been proven that some types of non-crossing pinched hysteresis curves (denoted Type-II) cannot be described by memristors.

Memristive networks and mathematical models of circuit interactions

The concept of memristive networks was first introduced by Leon Chua in his 1965 paper "Memristive Devices and Systems." Chua proposed the use of memristive devices as a means of building artificial neural networks that could simulate the behavior of the human brain. In fact, memristive devices in circuits have complex interactions due to Kirchhoff's laws. A memristive network is a type of artificial neural network that is based on memristive devices, which are electronic components that exhibit the property of memristance. In a memristive network, the memristive devices are used to simulate the behavior of neurons and synapses in the human brain. The network consists of layers of memristive devices, each of which is connected to other layers through a set of weights. These weights are adjusted during the training process, allowing the network to learn and adapt to new input data. 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. They also have the potential to be more energy efficient than traditional artificial neural networks, as they can store and process information using less power. However, the field of memristive networks is still in the early stages of development, and more research is needed to fully understand their capabilities and limitations. For the simplest model with only memristive devices with voltage generators in series, there is an exact and in closed form equation (Caravelli-Traversa-Di Ventra equation, CTD) which describes the evolution of the internal memory of the network for each device. For a simple memristor model (but not realistic) of a switch between two resistance values, given by the Williams-Strukov model ${\displaystyle R(x)=R_{off}(1-x)+R_{on}x}$, with ${\displaystyle dx/dt=I/\beta -\alpha x}$, there is a set of nonlinearly coupled differential equations that takes the form:

${\displaystyle \frac{d\overrightarrow{x}}{dt}=-\alpha \overrightarrow{x}+\frac{1}{\beta }(I-\chi \mathrm{\Omega }X{)}^{-1}\mathrm{\Omega }\overrightarrow{S}}$

where ${\displaystyle X}$ is the diagonal matrix with elements ${\displaystyle x_i}$ on the diagonal, ${\displaystyle \alpha ,\beta ,\chi }$ are based on the memristors physical parameters. The vector ${\displaystyle \overrightarrow{S}}$ is the vector of voltage generators in series to the memristors. The circuit topology enters only in the projector operator ${\displaystyle {\mathrm{\Omega }}^2=\mathrm{\Omega }}$, defined in terms of the cycle matrix of the graph. The equation provides a concise mathematical description of the interactions due to Kirchhoff 's laws. Interestingly, the equation shares many properties in common with a Hopfield network, such as the existence of Lyapunov functions and classical tunnelling phenomena. In the context of memristive networks, the CTD equation may be used to predict the behavior of memristive devices under different operating conditions, or to design and optimize memristive circuits for specific applications.

Extended systems

Some researchers have raised the question of the scientific legitimacy of HP's memristor models in explaining the behavior of ReRAM. and have suggested extended memristive models to remedy perceived deficiencies.

One example attempts to extend the memristive systems framework by including dynamic systems incorporating higher-order derivatives of the input signal u(t) as a series expansion

${\displaystyle \begin{array}{rl}y(t)& =g_0(\mathbf{\text{x}},u)u(t)+g_1(\mathbf{\text{x}},u)\frac{{\mathrm{d}}^{2}u}{\mathrm{d}{t}^2}+g_2(\mathbf{\text{x}},u)\frac{{\mathrm{d}}^{4}u}{\mathrm{d}{t}^4}+\dots +g_m(\mathbf{\text{x}},u)\frac{{\mathrm{d}}^{2m}u}{\mathrm{d}{t}^{2m}},\\ \dot{\mathbf{\text{x}}}& =f(\mathbf{\text{x}},u)\end{array}}$

where m is a positive integer, u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and the functions g and f are continuous functions. This equation produces the same zero-crossing hysteresis curves as memristive systems but with a different frequency response than that predicted by memristive systems.

Another example suggests including an offset value ${\displaystyle a}$ to account for an observed nanobattery effect which violates the predicted zero-crossing pinched hysteresis effect.

${\displaystyle \begin{array}{rl}y(t)& =g_0(\mathbf{\text{x}},u)(u(t)-a),\\ \dot{\mathbf{\text{x}}}& =f(\mathbf{\text{x}},u)\end{array}}$

Implementations

Titanium dioxide memristor

Interest in the memristor revived when an experimental solid-state version was reported by R. Stanley Williams of Hewlett Packard in 2007. The article was the first to demonstrate that a solid-state device could have the characteristics of a memristor based on the behavior of nanoscale thin films. The device neither uses magnetic flux as the theoretical memristor suggested, nor stores charge as a capacitor does, but instead achieves a resistance dependent on the history of current.

Although not cited in HP's initial reports on their TiO₂ memristor, the resistance switching characteristics of titanium dioxide were originally described in the 1960s.

The HP device is composed of a thin (50 nm) titanium dioxide film between two 5 nm thick electrodes, one titanium, the other platinum. Initially, there are two layers to the titanium dioxide film, one of which has a slight depletion of oxygen atoms. The oxygen vacancies act as charge carriers, meaning that the depleted layer has a much lower resistance than the non-depleted layer. When an electric field is applied, the oxygen vacancies drift (see Fast ion conductor), changing the boundary between the high-resistance and low-resistance layers. Thus the resistance of the film as a whole is dependent on how much charge has been passed through it in a particular direction, which is reversible by changing the direction of current. Since the HP device displays fast ion conduction at nanoscale, it is considered a nanoionic device.

Memristance is displayed only when both the doped layer and depleted layer contribute to resistance. When enough charge has passed through the memristor that the ions can no longer move, the device enters hysteresis. It ceases to integrate q=∫I dt, but rather keeps q at an upper bound and M fixed, thus acting as a constant resistor until current is reversed.

Memory applications of thin-film oxides had been an area of active investigation for some time. IBM published an article in 2000 regarding structures similar to that described by Williams. Samsung has a U.S. patent for oxide-vacancy based switches similar to that described by Williams.

In April 2010, HP labs announced that they had practical memristors working at 1 ns (~1 GHz) switching times and 3 nm by 3 nm sizes, which bodes well for the future of the technology. At these densities it could easily rival the current sub-25 nm flash memory technology.

Silicon dioxide memristor

It seems that memristance has been reported in nanoscale thin films of silicon dioxide as early as the 1960s .

However, hysteretic conductance in silicon was associated to memristive effects only in 2009.  More recently, beginning in 2012, Tony Kenyon, Adnan Mehonic and their group clearly demonstrated that the resistive switching in silicon oxide thin films is due to the formation of oxygen vacancy filaments in defect-engineered silicon dioxide, having probed directly the movement of oxygen under electrical bias, and imaged the resultant conductive filaments using conductive atomic force microscopy. 

Polymeric memristor

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. They used a passive layer between electrode and active thin films, which enhanced the extraction of ions from the electrode. It is possible to use fast ion conductor as this passive layer, which allows a significant reduction of the ionic extraction field.

In July 2008, Erokhin and Fontana claimed to have developed a polymeric memristor before the more recently announced titanium dioxide memristor.

In 2010, Alibart, Gamrat, Vuillaume et al. introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor and which exhibits the main behavior of a biological spiking synapse. This device, also called a synapstor (synapse transistor), was used to demonstrate a neuro-inspired circuit (associative memory showing a pavlovian learning).

In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ion-based memristors. The synapse circuit demonstrated long-term potentiation for learning as well as inactivity based forgetting. Using a grid of circuits, a pattern of light was stored and later recalled. This mimics the behavior of the V1 neurons in the primary visual cortex that act as spatiotemporal filters that process visual signals such as edges and moving lines.

In 2012, Erokhin and co-authors have demonstrated a stochastic three-dimensional matrix with capabilities for learning and adapting based on polymeric memristor.

Layered memristor

In 2014, Bessonov et al. reported a flexible memristive device comprising a MoO_x/MoS₂ heterostructure sandwiched between silver electrodes on a plastic foil. The fabrication method is entirely based on printing and solution-processing technologies using two-dimensional layered transition metal dichalcogenides (TMDs). The memristors are mechanically flexible, optically transparent and produced at low cost. The memristive behaviour of switches was found to be accompanied by a prominent memcapacitive effect. 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.

Atomristor

Atomristor is defined as the electrical devices showing memristive behavior in atomically thin nanomaterials or atomic sheets. In 2018, Ge and Wu et al. in the Akinwande group at the University of Texas, first reported a universal memristive effect in single-layer TMD (MX₂, M = Mo, W; and X = S, Se) atomic sheets based on vertical metal-insulator-metal (MIM) device structure. The work was later extended to monolayer hexagonal boron nitride, which is the thinnest memory material of around 0.33 nm. These atomristors offer forming-free switching and both unipolar and bipolar operation. The switching behavior is found in single-crystalline and poly-crystalline films, with various conducting electrodes (gold, silver and graphene). Atomically thin TMD sheets are prepared via CVD/MOCVD, enabling low-cost fabrication. 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. In 2020, atomistic understanding of the conductive virtual point mechanism was elucidated in an article in nature nanotechnology.

Ferroelectric memristor

The ferroelectric memristor is based on a thin ferroelectric barrier sandwiched between two metallic electrodes. 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: R_OFF ≫ R_ON (an effect called Tunnel Electro-Resistance). In general, the polarization does not switch abruptly. The reversal occurs gradually through the nucleation and growth of ferroelectric domains with opposite polarization. During this process, the resistance is neither R_ON or R_OFF, but in between. When the voltage is cycled, the ferroelectric domain configuration evolves, allowing a fine tuning of the resistance value. The ferroelectric memristor's main advantages are that ferroelectric domain dynamics can be tuned, offering a way to engineer the memristor response, and that the resistance variations are due to purely electronic phenomena, aiding device reliability, as no deep change to the material structure is involved.

Carbon nanotube memristor

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

Later it was found that CNT memristive switching is observed when a nanotube has a non-uniform elastic strain ΔL0. It was shown that the memristive switching mechanism of strained СNT is based on the formation and subsequent redistribution of non-uniform elastic strain and piezoelectric field Edef in the nanotube under the influence of an external electric field E(x,t).

Biomolecular memristor

Biomaterials have been evaluated for use in artificial synapses and have shown potential for application in neuromorphic systems. In particular, the feasibility of using a collagen‐based biomemristor as an artificial synaptic device has been investigated, whereas a synaptic device based on lignin demonstrated rising or lowering current with consecutive voltage sweeps depending on the sign of the voltage furthermore a natural silk fibroin demonstrated memristive properties; spin-memristive systems based on biomolecules are also being studied.

In 2012, Sandro Carrara and co-authors have proposed the first biomolecular memristor with aims to realize highly sensitive biosensors. Since then, several memristive sensors have been demonstrated.

Spin memristive systems

Spintronic memristor

Chen and Wang, researchers at disk-drive manufacturer Seagate Technology described three examples of possible magnetic memristors. In one device resistance occurs when the spin of electrons in one section of the device points in a different direction from those in another section, creating a "domain wall", a boundary between the two sections. Electrons flowing into the device have a certain spin, which alters the device's magnetization state. Changing the magnetization, in turn, moves the domain wall and changes the resistance. The work's significance led to an interview by IEEE Spectrum. A first experimental proof of the spintronic memristor based on domain wall motion by spin currents in a magnetic tunnel junction was given in 2011.

Memristance in a magnetic tunnel junction

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).

Extrinsic mechanism

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 (resistive switching). The MTJ consists in a SrTiO3 (STO) tunnel barrier that separates half-metallic oxide LSMO and ferromagnetic metal CoCr electrodes. The MTJ's usual two device resistance states, characterized by a parallel or antiparallel alignment of electrode magnetization, are altered by applying an electric field. When the electric field is applied from the CoCr to the LSMO electrode, the tunnel magnetoresistance (TMR) ratio is positive. When the direction of electric field is reversed, the TMR is negative. In both cases, large amplitudes of TMR on the order of 30% are found. Since a fully spin-polarized current flows from the half-metallic LSMO electrode, within the Julliere model, this sign change suggests a sign change in the effective spin polarization of the STO/CoCr interface. The origin to this multistate effect lies with the observed migration of Cr into the barrier and its state of oxidation. The sign change of TMR can originate from modifications to the STO/CoCr interface density of states, as well as from changes to the tunneling landscape at the STO/CoCr interface induced by CrOx redox reactions.

Reports on MgO-based memristive switching within MgO-based MTJs appeared starting in 2008 and 2009. While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects, another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies and its impact on spintronics. 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 or multiferroicity.

Intrinsic mechanism

The magnetization state of a MTJ can be controlled by Spin-transfer torque, and can thus, through this intrinsic physical mechanism, exhibit memristive behavior. This spin torque is induced by current flowing through the junction, and leads to an efficient means of achieving a MRAM. However, the length of time the current flows through the junction determines the amount of current needed, i.e., charge is the key variable.

The combination of intrinsic (spin-transfer torque) and extrinsic (resistive switching) mechanisms naturally leads to a second-order memristive system described by the state vector x = (x₁,x₂), where x₁ describes the magnetic state of the electrodes and x₂ denotes the resistive state of the MgO barrier. In this case the change of x₁ is current-controlled (spin torque is due to a high current density) whereas the change of x₂ is voltage-controlled (the drift of oxygen vacancies is due to high electric fields). The presence of both effects in a memristive magnetic tunnel junction led to the idea of a nanoscopic synapse-neuron system.

Spin memristive system

A fundamentally different mechanism for memristive behavior has been proposed by Pershin and Di Ventra. The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang. 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. When an external control parameter (such as voltage) is changed, the adjustment of electron spin polarization is delayed because of the diffusion and relaxation processes causing hysteresis. This result was anticipated in the study of spin extraction at semiconductor/ferromagnet interfaces, but was not described in terms of memristive behavior. On a short time scale, these structures behave almost as an ideal memristor. This result broadens the possible range of applications of semiconductor spintronics and makes a step forward in future practical applications.

Self-directed channel memristor

In 2017, Kris Campbell formally introduced the self-directed channel (SDC) memristor. The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide. The SDC device is operational immediately after fabrication. In the Ge₂Se₃ active layer, Ge-Ge homopolar bonds are found and switching occurs. The three layers consisting of Ge₂Se₃/Ag/Ge₂Se₃, directly below the top tungsten electrode, mix together during deposition and jointly form the silver-source layer. A layer of SnSe is between these two layers ensuring that the silver-source layer is not in direct contact with the active layer. Since silver does not migrate into the active layer at high temperatures, and the active layer maintains a high glass transition temperature of about 350 °C (662 °F), the device has significantly higher processing and operating temperatures at 250 °C (482 °F) and at least 150 °C (302 °F), respectively. 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. These factors allow the SDC device to operate over a wide range of temperatures, including long-term continuous operation at 150 °C (302 °F).

Potential applications

Memristors remain a laboratory curiosity, as yet made in insufficient numbers to gain any commercial applications. Despite this lack of mass availability, according to Allied Market Research the memristor market was worth $3.2 million in 2015 and was at the time projected to be worth $79.0 million by 2022. In fact, it was worth $190.0 million in 2022.

A potential application of memristors is in analog memories for superconducting quantum computers.

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. HP prototyped a crossbar latch memory that can fit 100 gigabits in a square centimeter, and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm³). In May 2008 HP reported that its device reaches currently about one-tenth the speed of DRAM. The devices' resistance would be read with alternating current so that the stored value would not be affected. In May 2012, it was reported that the access time had been improved to 90 nanoseconds, which is nearly one hundred times faster than the contemporaneous Flash memory. At the same time, the energy consumption was just one percent of that consumed by Flash memory.

Memristors have applications in programmable logic signal processing, super-resolution imaging physical neural networks, control systems, reconfigurable computing, in-memory computing, brain–computer interfaces and RFID. Memristive devices are potentially used for stateful logic implication, allowing a replacement for CMOS-based logic computation Several early works have been reported in this direction.

In 2009, a simple electronic circuit consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms. It was shown that subjected to a train of periodic pulses, the circuit learns and anticipates the next pulse similar to the behavior of slime molds Physarum polycephalum where the viscosity of channels in the cytoplasm responds to periodic environment changes. Applications of such circuits may include, e.g., pattern recognition. The DARPA SyNAPSE project funded HP Labs, in collaboration with the Boston University Neuromorphics Lab, has been developing neuromorphic architectures which may be based on memristive systems. In 2010, Versace and Chandler described the MoNETA (Modular Neural Exploring Traveling Agent) model. MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware. Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki. In 2011, they showed how memristor crossbars can be combined with fuzzy logic to create an analog memristive neuro-fuzzy computing system with fuzzy input and output terminals. Learning is based on the creation of fuzzy relations inspired from Hebbian learning rule.

In 2013 Leon Chua published a tutorial underlining the broad span of complex phenomena and applications that memristors span and how they can be used as non-volatile analog memories and can mimic classic habituation and learning phenomena.

Derivative devices

Memistor and memtransistor

The memistor and memtransistor are transistor-based devices which include memristor function.

Memcapacitors and meminductors

In 2009, Di Ventra, Pershin, and Chua extended the notion of memristive systems to capacitive and inductive elements in the form of memcapacitors and meminductors, whose properties depend on the state and history of the system, further extended in 2013 by Di Ventra and Pershin.

Memfractance and memfractor, 2nd- and 3rd-order memristor, memcapacitor and meminductor

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.

History

Precursors

Sir Humphry Davy is said by some to have performed the first experiments which can be explained by memristor effects as long ago as 1808. 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. A few years later, in 1968, Argall published an article showing the resistance switching effects of TiO₂ which was later claimed by researchers from Hewlett Packard to be evidence of a memristor.

Theoretical description

Leon Chua postulated his new two-terminal circuit element in 1971. It was characterized by a relationship between charge and flux linkage as a fourth fundamental circuit element. 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.

Twenty-first century

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.

On 23 January 2009, Di Ventra, Pershin, and Chua extended the notion of memristive systems to capacitive and inductive elements, namely capacitors and inductors, whose properties depend on the state and history of the system.

In July 2014, the MeMOSat/LabOSat group (composed of researchers from Universidad Nacional de General San Martín (Argentina), INTI, CNEA, and CONICET) put memory devices into a Low Earth orbit. Since then, seven missions with different devices are performing experiments in low orbits, onboard Satellogic's Ñu-Sat satellites.

On 7 July 2015, Knowm Inc announced Self Directed Channel (SDC) memristors commercially. These devices remain available in small numbers.

On 13 July 2018, MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload.

In 2021, Jennifer Rupp and Martin Bazant of MIT started a "Lithionics" research programme to investigate applications of lithium beyond their use in battery electrodes, including lithium oxide-based memristors in neuromorphic computing.

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, designed to dramatically increase the speed and efficiency of Machine Learning and Artificial Intelligence tasks, optimized for Edge Computing applications.