This chapter explores the dynamic behavior of dual flux coupled memristor circuits in order to explore the uncharted territory of the fundamental theory of memristor circuits. Neuromorphic computing anticipates highly dense systems of memristive networks, and with nanoscale devices within such close proximity to one another, it is anticipated that flux and charge coupling between adjacent memristors will have a bearing upon their operation. Using the constitutive relations of memristors, various cases of flux coupling are mathematically modeled. This involves analyzing two memristors connected in composite, both serially and in parallel in various polarity configurations. The new behavior of two coupled memristors is characterized based on memristive state equations, and memductance variation represented in terms of voltage, current, charge and flux. The rigorous mathematical analysis based on the fundamental circuit equations of ideal memristors affirms the memristor closure theorem, where coupled memristor circuits behave as different types of memristors with higher complexity.
In 1969, Leon Chua became the first person to publish non-linear circuit theory against a mathematical foundation . In doing so, it became apparent that there was a hole in the circuit equations at the time. Shortly after, in 1971 he postulated that symmetry implies the existence of a fourth fundamental circuit element to link the missing relationship between charge and flux—that circuit element being the memristor . This research resurfaced and was popularized in 2008, when Hewlett-Packard fabricated the first functional nanoscale memristor . This particular brand of memristor was based on a bi-level titanium dioxide thin film containing dopants which migrate across the width of the memristor when a current is applied to it.
Each fundamental circuit element holds a relationship between any two of either voltage, current, charge, or flux. The memristor thus becomes a fundamental circuit element as it fills the missing gap of the charge-flux relationship. It is important to note that even though
The inherent characteristics of this revolutionary device have enabled its application in a diverse field of areas, including neuromorphic circuits  and non-volatile memory applications . These applications often see arrays of memristors behaving compositely with one another. In addition to the functionality of single discrete memristors, the behaviors of multiple memristors in structures of connectivity have also been analyzed.
Memristors are polarity dependant—while this complicates circuit analysis, it allows for many more configuration permutations than the other fundamental circuits: the resistor, capacitor and inductor. The behavior of two memristor emulators in both serial and parallel connections are experimentally evaluated in Ref. , however, only identical polarity directions are considered. Two charge controlled memristors are connected in series and in parallel in Ref. , with their responses evaluated when polarity is varied. The composite behavior is analyzed by probing the relationships between flux, charge and memristance. The results show novel
Many researchers have sought to use memristors to represent the synapses between neurons in artificial networks, and more recently, a memristive crossbar array has been successfully fabricated which implements a neural network, and is successfully capable of performing limited classifications and simple pattern recognition . By training such networks on sets of known example patterns and tuning the weights of the ‘synaptic’ connections, unknown patterns and images can be recognized. Ultimately, researchers anticipate that networks with a density of 100 billion synapses per square centimeter in each layer should soon be possible by shrinking memristors down to 30 nm across. This indicates highly dense 3D structures with a very large number of memristors within very close proximity of one another will be the norm, and coupling memristor theory is of fundamental significance to this field. The use of memristive crossbar architectures has been gaining much traction in computing large sets of data [11, 12, 13, 14], and the theory behind memristive coupling is absolutely essential in ensuring information is not lost due to undesirable coupling, or by manufacturing more efficient modes of information storage by utilizing coupling theory.
The coupling effects of capacitors and inductors via electric and magnetic fields are well known. The mutual capacitances and inductances of circuits comprised of multiple TiO2 memristors are dependent upon the physical features of each memristor cell , such as size and position. Therefore, coupling is to be expected between adjacent memristors, and must be taken into account when analyzing highly concentrated circuits. In addition to series and parallel connections, coupling has thus been established as a third unique relation in memristive systems .
The behavior of coupled memristors was rigorously analyzed in a systematic manner for the first time in Ref.  with consideration given to all polarity combinations. The theoretical analysis is confirmed in the same paper by use of a separately presented memristor emulator circuit from Ref. . However, the results in the analysis is based on a memristor which exhibits a linear relationship between memductance and flux. This is obviously not the case for many memristors, such as the simplest case of a flux-controlled switching memristor presented in Ref.  where flux is controlled independent of memductance. As such, there is only a very narrow scope of memristors which the research in Ref.  applies to. The results in Ref.  served to broaden this assumption to ideal switching memristors which operate in two states, and obtain new results based on the same constitutive relation equations. This chapter dissects the results in Ref.  and presents them in a more comprehensive format, with the use of fundamental memristor theory to form the basis of the analysis to produce valid results. As such, the findings in this chapter can be applied more broadly and yet maintain the complex behavior which makes the memristor so attractive. The theoretical analysis and analytical solutions provide for novel memductance behavior in terms of flux, charge, voltage and current of ideal memristors. In the process, it is proven that the memristor closure theorem continues to stand for coupled memristors .
2. Coupled memristors
The two types of ideal memristors considered are charge controlled or flux controlled . The relationship between current and voltage of a charge controlled memristor is expressed by
If two flux controlled memristors are considered, the ideal coupled memristive systems can be defined by the following set of equations,
While a general rule cannot be ascertained which would be applicable for all ideal memristors, the most appropriate manner in approaching the task of modeling a pair of coupled memristors is to provide a procedural methodology instead. This is done by way of example with use of a particular type of switching memristor, complete with a known
Instead of assuming a linear relationship between memductance and flux as in Ref. , it is more appropriate to consider the ideal memristor proposed in Ref. , and derive the associated relationship between flux and memductance from a given
For the purposes of this paper, this example of an ideal switching memristor is completely characterized by the following equations:
The memductance can be approximated by
If this specific type of memristor is purely flux coupled with an identical memristor (without any other composite connections), and assuming the simple case of a first order mathematical model of coupling, the individual memductance of each device can be ascertained from Eqs. (5) and (7) as
The coupling strength between these two memristors is reflected by the coupling coefficients
A solvable equation with physical meaning requires assumptions about the physical behavior of the memristors. By considering the special case of identical excitations and voltage history (alternatively, the same initial conditions), and allowing for
When compared to the original hysteresis loop of just one of the two memristors, there are two notable differences: (i) the current spans a larger range of values due to the additive effect of
Despite this being the result of a specific type of switching memristor, it is reasonable to conclude these two changes will occur in all cases of purely coupled switching memristors.
This result can be exploited in neural circuits where synaptic spikes have more complexity than mere ‘ON-OFF states’. On the other hand, it may have an undesirable effect on memristive logic gates where having two states is essential for functionality. Necessary physical precautions must be taken in order to minimize the values of
3. Coupled memristors in serial connections
Two different configurations of serially connected memristors exist according to polarity combinations. The same approximation of the ideal memristors will apply to this section in the same form as in Eq. (7).
3.1. Serial connection with identical polarities
Applying Kirchhoff’s voltage Law (KVL) and equating the current through both memristors, the voltage across and current through
Eq. (11) reflects the complexity of memristive coupling: the derivatives of
This can be analytically solved to give
3.1.1. Serial Case 1: parity at u = 0, u = −2
3.1.2. Serial Case 2: u = 0 → ∞, u = −1 → −∞
3.1.3. Serial Case 3: u = 0 → −1, u = −2 → −1
This case behaves similarly to Case 2, but reversed. As
3.1.4. Serial Case 4: u = −1
Mathematically, there is no solution for
The effect seen here with flux approaching an infinite value is identical to an ideal memristor being connected to a DC source. A constant non-periodic voltage source will also result in flux tending indefinitely towards
This result will not hold true for all ideal memristors . If the memristor was defined by a polynomial
Given a sinusoidal voltage for
For the sake of both attaining a meaningful solution and demonstration, flux is first determined as a function of time where the term from Eq. (15) (2
The assumption used in deriving Eq. (14) was that the initial condition of MR1 was
The memductance (and by extension, current) can therefore be adjusted based on
While a memristor has a variable resistance by its very definition, this variation is limited by the value of
Figure 7 represents memductances derived from Eq. (18) at
3.2. Serial connection with opposite polarities
Following a similar procedure to above where one of two memristors in Figure 8 are flipped such that either terminals
In Figure 10,
Once again, this behavior is mapped against the given charge-flux relationship of the switching memristor characterized by the shape of the curve in Figure 3. The first and final segments of the curve are theoretically non-ending straight lines, and thus, after a voltage pulse is applied for a sufficiently long time interval to increase flux far beyond the upper breakpoint
This behavior is considered comparatively against a single ideal memristor excited by a DC voltage. Suppose a battery with voltage
Ignoring threshold switching effects, the memductance of MR1 reaches a steady state value while MR2 never achieves stability and instead tends towards a perfect conductor. However, these memristors will not display this behavior independently and so it is more practical to consider the two memristors as a single black box device. Equivalent memductance across
4. Coupled memristors in parallel connections
Two different configurations of parallel connected memristors exist according to polarity combinations, just as is the case with serially connected memristors. The same approximation of the ideal memristors will apply to this section in the same form as in Eq. (7). The first case to consider where memristors are configured with identical polarities is depicted in Figure 11.
4.1. Parallel connection with identical polarities
The current passing through
Integration both sides of Eq. (24) yields
Memductance can accordingly be calculated,
In this case the variation between the memductance and flux
4.2. Parallel connection with opposite polarities
A similar procedure can be used in order to ascertain the behavior of anti-parallel connected flux coupled memristors.
In the case shown in Figure 12, the current and flux across terminals
Integrating both sides of Eq. (29) results in a coupled charge-flux relationship as shown below,
For the uncoupled case of
A comprehensive theoretical analysis of flux coupled memristors displays various kinds of new behaviour which are otherwise unattainable from a single memristor. The simplest case of coupling between two switching memristors is shown to have a diverse range of properties when memristors are acting in composite with each other. The results presented only consider bi-state memristors, and as such, we can expect different types of memristors with different charge-flux relationships to expand the types of dynamic behaviors exhibited, with the ability to modify the states attainable by tuning the variables associated with the coupling coefficient (such as physical proximity and device material just to name a couple of examples).
In summary, two serially connected memristors with identical polarities are shown to produce a pair of variable memristors determinable from initial conditions; two serially connected memristors with opposite polarities display behavior often displayed by memristors connected to DC sources, or otherwise resistive behavior. Parallel memristive systems are shown to produce a variation rate in terms of the coupling coefficients. This is a feature that can be determined at the time of fabrication.
Further, what has been considered in this paper is the simplest case of identical memristors with identical initial conditions. The potential application of coupled memristors, in addition to the undoubtedly interesting characteristics of arrays of coupled memristors will serve to open up new avenues of applications, and also provide for guidelines on avoiding undesirable behaviors by having fabrication plants devise methods to reduce the coupling coefficient as low as practicable where design specifications see it fit. In particular, where neural networks will see densely populated circuits which depend on memristors behaving functionally, the effects of coupling must either be mitigated to avoid unexpected and fallible outcomes. The alternative view is that memristive coupling makes it possible to have more than two states between a pair of memristors which would otherwise only be capable of being switched either on or off, and as such, if these intermediary states are quantized, then a large system of many varying states can be produced out of a mere two memristors connected compositely.
This work was supported by the Australia-Korea Foundation—Department of Foreign Affairs and Trade under the AKF00640 grant.
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