In 2008 researchers at Hewlett-Packard announced the physical realization of the "memristor" which was theoretically predicted as a fundamental non-linear circuit element by Leon Chua in 1971. Since that time there have been numerous scientific papers applying the concept of memory resistors to a wide range of thin film materials used for a new type of non-volatile memory called ReRAM (resistive random-access memory). It has also been noted that memory resistors may be applicable to explain behavior of biological neurons and some research groups have developed circuit designs exploiting memory resistors as components of neuromorphic electronics. This article reviews the historical background of various forms of memory resistors including the Widrow-Hoff memistor, Chua's memristor model, and a mem-resistor model I developed to correct some of the deficiencies in HP's memristor model. The potential future of memory resistors with respect to artificial intelligence and robotics is briefly discussed.

For information on why the HP's memristor might be considered scientific fraud the reader is referred to

The earliest example of research in memory resistors can be credited to Bernard Widrow of Stanford University and his graduate student Ted Hoff in

So if memistors were physically realized back in 1960 why don't we have any memistor-based electronics today? At least a partial answer to that question can be found in the 1961 report entitled "

Basically what this is describing is the rise of the integrated circuit concept originally developed by

So since electrochemical memistors as used by Widrow in the 1960's conflicted with the emerging trend of solid state integrated circuitry they were eventually discarded. Research in artificial intelligence eventually began relying on integrated circuit designs, the microprocessor, and the software used to run programs on microprocessors. However, the production of a solid-state version of the memistor compatible with the integrated circuit may offer the possibility of continuing the development of ADALINE or similar artificial intelligence circuit architectures.

At the time of Chua's original paper and the later paper by Chua and Kang the memristor was considered a "fourth fundamental circuit element." The other three fundamental circuit elements are the resistor, the capacitor, and the inductor. The resistance of resistors is defined by a functional relationship between current and voltage. The capacitance of capacitors is defined by the functional relationship between charge and voltage. The inductance of an inductor is defined by the functional relationship between current and magnetic flux linkage (the time integral of voltage).The mathematical justification for the existence of the memristor as a fourth fundamental circuit element was based on symmetry with the other three circuit elements in which a fourthfunctional relationship should exist between the electrical quantities of magnetic flux linkage and charge. Thus the mathematical functional relationship defining the four basic elements(according to Chua) is as follows:

Resistor: v = R(i)

Capacitor: q = C(v)

Inductor: ϕ= L(i)

Memristor: ϕ = M(q)

where v = voltage, i = current, q = charge, ϕ = magnetic flux linkage

However, there are a few problems with considering the memristor as a fourth fundamental circuit element. The first problem is that for linear circuit elements in which the memristive function is a linear function there is no effective mathematical difference between the resistor equation and the memristor equation (i.e. differentiating the memristive equation produces the resistor equations). Another problem is that even non-linear memristors would have the same unit as resistors (i.e. ohms) in contrast to capacitors and inductors which each have their own respective unit (i.e. farads, henries).

A more recent reason for memristors to be considered other than a fourth fundamental circuit element results from a 2009 publication entitled "

Memristive system: v = R(x,i) i and dx/dt = m(x,i)

Memcapacitive system: q = C(x,v) v and dx/dt = g(x,v)

Meminductive system: ϕ = L(x,i) i and dx/dt = h(x,i)

where x is a state variable (or, more generally, state vector), dx/dt is the time derivative of the state variable, and m(x,i), g(x,v), and h(x,i) are continuous functions of the respective variables. It is noted that these equations reduce to that of non-linear resistors, capacitors, and inductors provided that dx/dt = 0.

Based on this interpretation it is more reasonable to consider resistors having memory effects as a dynamic generalization of non-linear resistors rather than as a "fourth fundamental circuit element." However, Chua continues to argue that the memristor should be considered a fundamental circuit element since no combination of non-linear resistor, non-linear capacitor, and non-linear inductor can reproduce the dynamic properties of a memristor. This is a fallacious argument since by the same reasoning a diode should be considered a fundamental circuit element because no combination of linear resistor, linear capacitor, and linear inductor can replicate the non-linear properties of a diode. A diode is not considered as a "fundamental" circuit element but rather as one example of a non-linear resistor. In the same fashion Chua's memristor should not be considered as a "fundamental" circuit element but rather as one (and not the only) possible example of a dynamic resistor.

The recent interest in the memristor theory of Chua was sparked by an article in the journal Nature published by researchers at

What the HP researchers attempted to accomplish in their article was to show that the memristor theory could be used to model the resistive switching effects found in a variety of thin film materials. A memristive systems model may be mathematically formulated as:

v = R(x) i and dx/dt = m(i)

where x= state variable, v = voltage, i = current, R(x)=memristance function, m(i) = rate of change of the state variable.

The HP researchers noted that the state parameter x could be equated with the ionic distribution within a thin insulative film. In this case the first equation of the memristive system v=R(x)i may be interpreted as defining the relationship between voltage and electron flow through the thin film at a particular ionic distribution while the second equation dx/dt = m(i) may be interpreted as defining the relationship between the ionic flow and the electron flow. In this respect I believe the HP researchers are correct. In the Nature paper they propose R(x) and m(i) as:

R(x) = R _{ON} (x/D) + R _{OFF} (1-x/D)

m(i) = u _{v} (R _{ON} /D) i

where D = the thickness of the thin film, u _{v} = the ionic mobility, R _{ON} = the on resistance, R _{OFF} = the off resistance.

However, there are several problems with these choice of functions and a direct application of these equations do not match observed measurement. One of the lead HP researchers named Stan Williams recognized this deficiency at the 2008 Memristor and Memristive Systems Symposium and acknowledged that this model is a so-called "freshman model" (see 1:26 at this

1) The models do not take into account memcapacitive effects.

The metal/insulator/metal structures exhibiting memristive effects are by their nature also capacitive. This has not been formally treated and a variety of thin film materials that exhibit the memristive effect (such as perovskites) also exhibit a memcapacitive effect.

2) The models are poorly formulated for electrical engineering applications.

A working engineer is traditionally used to mathematical models based on current as the dependent variable and voltage as the independent variable. For example, this is the way the equations are formulated for the analysis of diodes or the analysis of field effect transistors. However, the formulas presented in the HP paper are based on voltage as the dependent variable making them unwieldy for convenient use.

3) The models do not properly distinguish the internal and external electric field.

If the underlying mechanism is based on ionic drift the model does not properly distinguish between the external field due to an applied voltage and an internal field due to the distribution of ions in the film.

4) The newer models do not actually represent a memristor.

This is something that I am surprised no one has really picked up on yet but the models used by HP in describing Pt-TiO _{2-x} -Pt have diverged from the original Nature paper and no longer describe a memristor but rather the more general class of memristive systems in which dx/dt exhibits an exponential rather than linear relationship with current. As far as I can tell there is STILL no solid state material which can legitimately be called a 2-terminal memristor as originally defined in Leon Chua's 1971 paper.

In 2011, Chua has made an attempt to change the definition of the memristor in the article "

However, in an earlier paper which describes the same type of 2-terminal non-volatile memory devices ("

Clearly there is an inconsistency between the memristor definitions used by Chua from 1976 compared to 2011. It appears that Chua has broadened the definition to save face for HP which has found that their device is not a memristor according to the original definition. This is arguably an example of scientific fraud and at minimum is propaganda to satisfy the business agenda of HP regarding RRAM while elevating the reputation of Chua who is attempting to use his revised memristor definition to take credit for phase change memory, CBRAM, MRAM, and all other forms of 2-terminal non-volatile memories.

The ionic memristor model of HP are deficient for failing to consider the effective mass of ions and repulsive forces between ions and for not properly considering the effects of ion transport on the built-in voltage and depletion width of Schottky junctions or the gap of tunneling junctions. The effective mass may be particularly important in cases where the ions are attached to nanoparticles or are located at the tip of a conducting filament in which case the ionic effective mass can be several orders of magnitude greater than a single ion. Below is an abstract from the article I posted on Cornell ArXiv (March 11, 2011) available at

I posted another article on Mar 24, 2011 extending the harmonic oscillator model to filamentary mem-resistors available at

Another article from May 01, 2011 available at

This article (

It is possible that neural computer designs using materials with similar properties to that of Bernard Widrow's 3-terminal memistor or 2-terminal memresistors will be realized within the next few years. Ironically

One may ask why memory resistive electronics would have any more effect on artificial intelligence than semiconductor-based microprocessors in which neural networks have been simulated in software. The use of software in artificial neural networks typically is based on a

Of course it is unlikely that the development of memory resistive artificial intelligence or robotics achieving the capabilities of human beings will occur overnight. Since it has taken hundred of millions of yearsof evolution to develop human-level intelligence it may take a while for a similar level to occur in autonomous mem-resistive systems which model biological neurons (provided that such modeling was even possible). One way to short-circuit the amount of time necessary may be to use