Importance of the window function choice for the predictive modelling of memristors
Window functions are widely employed in memristor models to restrict the changes of the internal state variables to specified intervals. Here we show that the actual choice of window function is of significant importance for the predictive modelling of memristors. Using a recently formulated theory of memristor attractors, we demonstrate that whether attractor points exist depends on the type of window function used in the model. Our main findings are formulated in terms of two memristor attractor theorems, which apply to broad classes of memristor models. As an example of our findings, we predict the existence of attractor states in Biolek window function memristors and their absence in memristors described by the Joglekar window function, when such memristors are driven by periodic alternating polarity pulses. It is anticipated that the results of this study will contribute toward the development of more sophisticated models of memristive devices and systems.
During the past decade there have been many publications using window function-based models [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] to describe the response of either discrete memristors or their circuits. While in many cases such models provided reliable predictions, there are situations when the behavior of the circuit depends critically on the choice of window function. It seems that there is little awareness of this fact as, typically, little attention is paid to picking the window function. The purpose of this Letter is to show generally, and illustrate through specific examples, that the choice of window function is of significant importance for the predictive modelling of memristors.
where and are the voltage across and current through the system, respectively, is the memristance (memory resistance), is a vector of internal state variables, and is a vector function 111Voltage-controlled memristive systems are defined similarly .. The window function is a multiplicative factor (normally, ) that enters into the th component of . To ensure a zero drift of across the boundaries, the window functions take zero values at the boundaries, which are often located at and . While it is generally believed that the window functions describe certain physical processes in real memristors, the present authors are not aware of any derivations of window functions either from fundamental physical theories or first-principles modelling. Therefore, at the present level of knowledge, the window functions are rather representations of what we think about how memristors work, and these representations may be close or not so close to the reality.
In what follows we demonstrate that very close memristor models (differing only in their window functions) may exhibit qualitatively different dynamics. Our demonstration uses the recently introduced notion of dynamical memristor attractors . Fig. 1(a) shows the circuit configuration considered in the present study. Here, a single memristor is directly connected to a current source, which drives alternating polarity current pulses through the memristor (the pulse sequence is sketched in Fig. 1(b)). It is shown that the presence or absence of memristor attractors can be directly related to the type of window function used in the memristor model. For instance, we show that when using the Biolek window function, memristors do have attractor dynamics, but not when using the Joglekar window function, even though all other conditions are identical. We thus argue that the predictive modelling of memristors and their circuits requires a further refinement of memristor models in general and window functions in particular. It is important to note that the presence or absence of dynamical attractors can be easily verified experimentally with physical memristors, and conclusions regarding the suitability of the use of a particular memristor model can then be made in particular cases.
The rest of this paper is organised as follows. In Section II (Preliminaries) we introduce the Joglekar and Biolek window functions, as well as dynamical attractors for memristors. The dynamical attractors in two broad classes of memristor models are investigated in Section III, which formulates our main findings in terms of two memristor attractor theorems. In the same section, the theorems are exemplified by Biolek and Joglekar window function memristors. Some concluding remarks are made in Section IV.
Ii-a The window functions used in this paper
This subsection briefly introduces the window functions used in the present paper. As was mentioned above, window functions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] are frequent components of memristor models [6, 15]. Typically, their role is to slow down the change of when the internal state variable approaches a boundary value (such as 0 or 1).
In particular, the Joglekar window function is defined by 
where is a positive integer. In the Biolek approach , the window function is given by
where is the Heaviside step function and is a positive integer. Fig. 2 presents the Joglekar and Biolek window functions plotted for . Note that the Biolek window function (as a function of ) is different for positive and negative currents, and thus has a finite discontinuity at .
Ii-B Dynamical attractors in the context of memristors
Recently, the present authors demonstrated the possibility of attractor dynamics in periodically-driven memristors and memristive networks . In the case of a first-order current-controlled memristive system ( is a scalar and is a scalar function) connected to a current source and subjected to alternating polarity current pulses (see Fig. 1), the conditions for the existence of an attractor point at are given by 
where , , , and are the pulse parameters (defined in Fig. 1).
We also note that such periodically-driven memristors can be described by a memristor potential function 
so that the problem of finding the attractor points (based on Eqs. (5) and (6)) corresponds to the problem of potential function minimisation. To put it differently, every minimum of the potential function represents a dynamical attractor.
Iii Memristor dynamics
Iii-a Theorem 1
We first formulate and prove a theorem concerning the occurrence of attractor dynamics in the memristors described by Eq. (2) of the form
where i) is a continuous function of ; ii) , is monotonically decreasing as a function of with and for ; iii) , is monotonically increasing as a function of with and for . We emphasise that in Eq. (8) represents a quite general class of functions that includes but not limited to the Biolek window function given by Eq. (4).
Theorem 1: There is always a single attractor in the dynamics of single memristors described by Eq. (8) when these are driven by alternating polarity current pulses such that and .
Clearly, there exists a single solution of Eq. (9), since the curves representing the left-hand side and right-hand side of Eq. (9) (as functions of ) must intersect at some point . To verify that is a stable equilibrium point, we substitute Eq. (8) into (6) and divide it by Eq. (9). This leads to
As is a monotonically decreasing function of while is a monotonically increasing function of , the left-hand side of inequality (10) is negative and its right-hand side is positive. Therefore, the inequality (6) is satisfied at the attractor point .
Iii-B Biolek window function memristors
Having established Theorem 1, consider Biolek window function memristors. As , given by Eq. (10), satisfies the conditions listed below Eq. (8), we immediately conclude that there always exists a dynamical attractor in their dynamics (the only required conditions are and ). To find the attractor point we use Eq. (9), which can be written as
In the simplest case of , the suitable solution of Eq. (11) is
where . For , a numerical solution of Eq. (11) can always be found. Fig. 3(a) shows the attractor point corresponding to Eq. (12) as a function of the parameters of the pulse sequence. This plot exhibits a clear symmetry with respect to the parameters of the positive and negative current pulses. The attractor point for larger values of is exhibited in Fig. 3(b). To generate Fig. 3(b), Eq. (11) was solved numerically assuming , which corresponds to the dashed diagonal line in Fig. 3(a). Fig. 3(b) demonstrates an increase in the steepness of the curve with . An important observation is that changes continuously from 0 to 1 in Fig. 3(b). Therefore, a Biolek window function memristor can be reliably set to any desired state by an appropriate choice of the pulse sequence parameters.
It is interesting that Eq. (12), defining the attractor point , is ‘universal’: all the details about the ‘activation’ function are hidden in the constant . Therefore, the same attractor point can be realised with different types of memristors. For instance, one can take (an instantaneous linear drift model) or use if and if (threshold-type model), etc. Here, is a rate constant, and is the current threshold. Figure 4(a) shows an example of attractor dynamics in Biolek window function memristors. The details of the simulation are given in the caption.
It should be mentioned that the memristor potential function given by Eq. (7) can shed some additional light on the attractors in memristor dynamics. Here we just note that in the case of Biolek window function memristors, the potential function can be written as
Iii-C Theorem 2
Next we formulate and prove a theorem concerning the absence of attractors in the dynamics of memristors described by Eq. (2) of the form
where i) is a continuous function; ii) for , and for ; and iii) for .
Theorem 2: There are no attractors in the dynamics of single memristors described by Eq. (14) when they are driven by alternating polarity current pulses such that and . Moreover, for special cases of pulse sequences, neutral equilibrium points are possible.
which, clearly, can not be satisfied simultaneously with Eq. (15) if we assume a zero value for the bracket. One can recognise that the zero bracket in Eq. (16) corresponds to the condition for a neutral equilibrium.
Iii-D Joglekar window function memristors
In the case of Joglekar window function memristors, Eq. (5) takes the form
Moreover, the potential function (Eq. (7)) of the Joglekar window function memristor can be rewritten as
Note that at the neutral equilibrium point, and is a monotonic function of otherwise.
Iv Discussion and conclusions
Historically, window functions were introduced phenomenologically to account for the experimental observation of boundary states of memristors (the ‘on’ and ‘off’ states). To the best of our knowledge, their expressions have never been derived from fundamental physics theories. Irrespective of this fact, several window-function based memristor models have been implemented in SPICE, and their SPICE implementations (see, for instance, Refs. [4, 2, 16, 17, 18]) are considered to be solid tools for predictive circuit simulations. Various prospects for memristor applications have been claimed based on the results of such SPICE modelling.
In this paper we have shown that the choice of the window function is of critical importance for the predictive modelling of memristors. In particular, it has been proven that two broad classes of memristors (exemplified by the Biolek and Joglekar window function memristors) may demonstrate qualitatively different dynamics under the same driving conditions. While in some practical cases this finding may be unimportant, there are situations when it can not be ignored (in particular, in circuits with memristors subjected to periodic, quasi-periodic, and possibly random alternating polarity pulses, etc.). Such situations are relevant to various memristor applications, including spiking neural networks and autonomous oscillating circuits, to name a few.
It should be emphasised that our results are not limited to the Biolek and Joglekar window function memristors. While the window functions described in Refs. [4, 6, 11] satisfy the conditions of Theorem 1, the window functions introduced in Refs. [1, 2, 3, 5, 7, 8, 9, 10, 12] belong to the class covered by Theorem 2. Moreover, experimental studies of dynamical memristor attractors could be used to improve the memristor models. For example, the presence or absence of an attractor in the dynamics of a memristor can serve as an indicator of the model’s validity. We emphasise that our findings are also valid for voltage-controlled memristors driven by voltage pulses, with the appropriate replacement of current by voltage, and can be easily extended to other types of memory circuit elements, such as memcapacitors and meminductors .
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