Memristor Cellular Neural Networks Computing in the Flux-charge Domain

Fernando Corinto; Mauro Forti; Leon O. Chua

doi:10.1007/978-3-030-55651-8_9

Nonlinear Circuits and Systems with Memristors pp 343-372 | Cite as

Memristor Cellular Neural Networks Computing in the Flux-charge Domain

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Fernando Corinto
Mauro Forti
Leon O. Chua

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First Online: 01 November 2020

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Abstract

A memristor is a nonlinear device obeying 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 it. A memristor is then both a nonlinear and a memory element in the (v, i)-domain. Another unique property is nonvolatility, namely, when current (or voltage) is turned off, the memristor can keep in memory the final value of charge, flux, or memristance, thereafter (see Chap. 2).

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Appendix: HP Memristor with Joglekar Window

By integrating (9.5) with Joglekar window (9.6), in the case p = 1, we obtain

$$\displaystyle \begin{aligned}q_M(t)= \frac{1}{\alpha}\int_{x(-\infty)}^{x(t)} \frac{1}{1-(2\sigma-1)^2} d\sigma \end{aligned}$$

where we considered that q_M(−∞) = 0, i.e., the overall charge of the memristor is null at its fabrication. Then

$$\displaystyle \begin{aligned} q_M(t)= F(x(t))= \frac{1}{4 \alpha} \log \left( \frac{x(t)}{1-x(t)} \right) +\tilde C \end{aligned} $$

(9.23)

where $\tilde C$ is a constant depending on the state initial value x(−∞) ∈ (0, 1). Since x(t) ∈ (0, 1) for any t ≥ 0, it turns out that $F:(0,1)\to \mathbb {R}$ is an analytic function. Additionally, F′(x) > 0 for any x ∈ (0, 1), hence F(⋅) is globally invertible, and their inverse function $x=\hat x(q_M): \mathbb {R} \to (-1,1)$ is analytic in $\mathbb {R}$.

We have

$$\displaystyle \begin{aligned}x(t)=\hat x(q_M(t)-\tilde C)= \frac{e^{4 \alpha(q_M(t)-\tilde C)}}{1+e^{4 \alpha(q_M(t)-\tilde C)}} \end{aligned}$$

so that (9.5) can be rewritten as

$$\displaystyle \begin{aligned}v_M(t)=\left[\frac{R_{\mathrm{off}}+R_{\mathrm{on}}e^{4 \alpha(q_M(t)-\tilde C)}}{1+e^{4 \alpha(q_M(t)-\tilde C)}} \right] i_M(t). \end{aligned}$$

Recalling that v_M(t) = dφ_M(t)∕dt and i_M(t) = dq_M(t)∕dt, by integrating between −∞ and t we obtain

$$\displaystyle \begin{aligned}\varphi_M(t)=\int_{q_M(-\infty)}^{q_M(t)} \left[\frac{R_{\mathrm{off}}+R_{\mathrm{on}}e^{4 \alpha(q-\tilde C)}}{1+e^{4 \alpha(q-\tilde C)}} \right] dq \end{aligned}$$

where we considered that φ(−∞) = 0.

Then, we obtain the following flux-charge relation for the HP memristor

$$\displaystyle \begin{aligned} \varphi_M=h(q_M)=\int_{0}^{q_M} \left[\frac{R_{\mathrm{off}}+R_{\mathrm{on}}e^{4 \alpha(q-\tilde C)}}{1+e^{4 \alpha(q-\tilde C)}} \right] dq . \end{aligned} $$

(9.24)

From the previous discussion we have that $h(\cdot ): \mathbb {R} \to \mathbb {R}$ is analytic in $\mathbb {R}$. Moreover, it is easily verified that h(0) = 0,

$$\displaystyle \begin{aligned}h'(q_M)=\left[\frac{R_{\mathrm{off}}+R_{\mathrm{on}}e^{4 \alpha(q_M-\tilde C)}}{1+e^{4 \alpha(q_M-\tilde C)}} \right] >0 \end{aligned}$$

and

$$\displaystyle \begin{aligned}h''(q_M)= - \frac{4 \alpha(R_{\mathrm{off}}-R_{\mathrm{on}}) e^{4 \alpha(q_M-\tilde C)}}{\left (1+e^{4 \alpha(q_M-\tilde C)}\right)^2} <0 \end{aligned}$$

for any $q_M \in \mathbb {R}$.

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Authors and Affiliations

Fernando Corinto
- 1
Mauro Forti
- 2
Leon O. Chua
- 3

1.Department of Electronics & TelecommunicationsPolitecnico di TorinoTorinoItaly
2.Department of Information Engineering and MathematicsUniversity of SienaSienaItaly
3.University of CaliforniaBerkeleyUSA

Memristor Cellular Neural Networks Computing in the Flux-charge Domain

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