Memristor standard cellular neural networks computing in the flux–charge domain
Introduction
Since the discovery at HP laboratories of nanodevices displaying memristive behavior (Strukov et al., 2008), there has been a widespread interest in modeling memristors and studying the nonlinear dynamics and the applicability of nonlinear memristor circuits, see, e.g., Chua (2011), Chua (2015), Chen et al. (2014), Duan and Huang (2014), Mazumder et al. (2012), Mathiyalagan et al. (2016), Nie et al. (2016), Qin et al. (2015), Tetzlaff (2014), Wen et al. (2015), Wu and Zeng (2012) and references therein. Particular attention has been paid to the use of memristors to implement synapses and neurons in neuromorphic architectures, since it is believed, broadly speaking, that memristors are potentially useful and will play a major role in the design of smart computers and future brain-like machines Adamatzky & Chua (2014), Kim et al. (2012a), Sah et al. (2014).
The ideal memristor, originally envisioned by Prof. L. Chua in the seminal paper (Chua, 1971), is a nonlinear device obeying an Ohm’s law but, unlike a resistor, the memristor resistance, also called memristance, depends upon the history of the voltage applied or the current flowing through the memristor. An ideal memristor is then both a nonlinear and a dynamic element. A unique property of an ideal memristor is non-volatility, namely, when current (or voltage) is turned off, the memristor can keep in memory the final value of charge, flux, or memristance (Chua, 2015).
Recently, a new class of neural networks (NNs) has been proposed, where nonlinear dynamic memristors are used within the neurons in place of linear (memoryless) resistors Di Marco et al. (2016a), Di Marco et al. (2016b). Due to the use of memristors during the analog computation, those NNs display peculiar and basically different properties with respect to standard (S) cellular (C) NNs (Chua & Yang, 1988b) and Hopfield (H) NNs (Hopfield, 1984). One salient feature is that the analog processing of a memristor NN takes place in the flux–charge domain, instead of the typical voltage–current domain, as it happens for SCNNs and HNNs. In the case of flux-controlled memristors, the inputs are provided via the initial values of memristor fluxes , the processing is accomplished during the time evolutions of and the result of processing, for convergent memristor NNs, is given by the asymptotic values of fluxes . There are potential advantages in terms of power consumption for NNs operating in the flux–charge domain. Indeed, when a steady state is reached, i.e., the memristor fluxes reach a constant value, the memristor voltages , as well as the capacitor and all other voltages and currents in the memristor NN vanish. Said another way, at a steady state a memristor NN turns off and so the dissipated power is null. Yet, in steady state the memristors act as non-volatile devices keeping in memory the processing result, i.e., the asymptotic values of fluxes . We stress that this is different from SCNNs and HNNs, where voltages, currents and power do not vanish when a steady state is reached, and batteries are needed to hold in memory the processing result.
Let us now compare the equations describing the dynamics of a memristor NN Di Marco et al. (2016a), Di Marco et al. (2016b) with those describing a SCNN (Chua & Yang, 1988b). The memristor NNs Di Marco et al. (2016a), Di Marco et al. (2016b) are obtained from the NN architecture proposed in Kim et al. (2012a), Kim et al. (2012b) by replacing the linear memoryless resistor in a cell with a nonlinear flux-controlled dynamic memristor. In the flux–charge domain the dynamics of a memristor NN satisfy the system of differential equations where is the cell capacitor, is the vector of memristor fluxes, is the interconnection matrix and is a biasing input. Moreover, , where with , is the nonlinear charge–flux characteristic of the flux-controlled memristor, as depicted in Fig. 1 (a) (symbol denotes the transpose).
On the other hand the equations describing in the voltage–currentdomain the dynamics of the SCNN model (Chua & Yang, 1988b) are where and are the cell capacitor and resistor, respectively, is the vector of capacitor voltages, is the interconnection matrix and is a biasing input. Moreover, , where is the typical piecewise linear function with unity gain in the linear region and flat saturation levels shown in Fig. 1 (b).
Note that the memristor NN model (1) is not analogous to model (3) describing a SCNN. The first basic difference concerns the shapes of the nonlinearities in (1) and in (3). Moreover, in (1) the contribution of the neuron interconnections is linear in , while the additive nonlinear term acts as a limiter nonlinearity. Instead, in (3), the contribution of neuron interconnections is given by the nonlinear multiplicative term times the saturation nonlinearity .
On the basis of the previous discussion it is natural to ask whether we can design a NN using memristors whose model in the flux–charge domain is analogous to model (3) describing the dynamics in the voltage–current domain of a SCNN. The crucial advantage would be the possibility to exploit for memristor NNs the bulk of results already available in the literature for studying the dynamics of SCNNs and to design them in order to accomplish a large variety of signal processing tasks (Chua & Roska, 2005). The main result in the paper is that we can give a positive answer to the previous question, i.e., it is indeed possible to design a NN with memristors whose equations in the flux–charge domain are analogous to those of a SCNN (Section 2). This will be accomplished by relying on the newly developed analysis and synthesis method described in Corinto & Forti (2016), Corinto & Forti (2017) and using a realistic memristor model as that proposed by HP (Strukov et al., 2008). The paper then gives a foundation in the flux–charge domain to the dynamics of the model and addresses convergence of solutions in the case of symmetric interconnections between neurons (Section 3). Applications to the solution of image processing tasks in real time are also discussed to confirm that the introduced model has processing capabilities analogous to those of a SCNN (Section 4). The obtained results are compared with existing results in the literature in Section 5. The main conclusions drawn in the paper are collected in Section 6.
Section snippets
Memristor neural network model
Our goal is to design a NN with memristors whose dynamics in the flux–chargedomain is described by a set of differential equations analogous to those in (3) describing a SCNN in the voltage–current domain. The design is based on the method devised in Corinto & Forti (2016), Corinto & Forti (2017). First we find it convenient to recall some basic facts about the method (Section 2.1). Then, we tackle the problem of synthesizing a saturation nonlinearity that closely approximates that of a SCNN in
Convergence results
From standard results on ordinary differential equations it can be easily checked that the -SCNN model (17) enjoys the property of existence, uniqueness with respect to initial conditions, and prolongability of solutions to . However, since the nonlinearity is unbounded, it is not guaranteed a priori that solutions are bounded. The next results show that under a suitable constraint on the norm of we can guarantee boundedness of solutions and also obtain a useful estimate of the norm
Applications
We have seen in Section 2.3 that Eqs. (17) describing the -SCNN model are a good approximation, in the flux–charge domain, of those describing the SCNN model (3) in the voltage–current domain, i.e., the two models are formally analogous. In this section we show by numerical means that the two models indeed display similar processing capabilities when applied to the solution of some processing tasks in real time.
Example In order to illustrate how the dynamics in the flux–charge domain
Discussion
The examples discussed in Section 4 demonstrated that the proposed class of -SCNNs has processing capabilities, in the flux–charge domain, similar to those displayed by SCNNs (Chua & Yang, 1988b) in the typical voltage–current domain. This was confirmed by several other numerical experiments where we applied -SCNNs to other processing tasks. On this basis we may conclude that it is potentially possible to use the bulk of results already available in the literature for SCNNs, see e.g. Chua and
Conclusion
The paper has introduced a class of NNs with memristors, termed -SCNNs, computing and processing signals in the flux–charge domain. The -SCNNs are described in the flux–charge domain by a set of differential equations that are analogous to those describing SCNNs (Chua & Yang, 1988b) in the traditional voltage–current domain. Special care has been devoted to design the neuron nonlinearity of a -SCNN in order that it approximates the nonlinearity of a SCNN. This has been done by using a
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