# Simulating Synaptic Behaviors through Frequency Modulation in a Capacitor–Memristor Circuit

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Comprehensive Physical Model of an Oxide Memristor

_{O}) concentration (n

_{D}) are represented by Equations (1) and (2), respectively. [17,18]. D∇n

_{D}, vn

_{D}and DSn

_{D}∇T are the Fick diffusion flux, drift flux, and Soret diffusion flux, respectively. f is the escape attempt frequency (10

^{12}Hz), a is the hopping distance (0.1 nm), and E

_{a}is the diffusion barrier (0.85 eV).

_{th}are the electrical conductivity and thermal conductivity, respectively [17,18].

_{2}O

_{5}film serving as the resistive switching (RS) layer and the TaO

_{x}film acting as the V

_{O}reservoir. Both the top and bottom electrodes are made of Pd. Figure 1b depicts a sweep voltage curve, clearly showing the set and reset processes. The arrows in the Figure 1b indicate the sweeping direction of voltage.

## 3. The Capacitor–Memristor Circuit with Rectangular Pulses as the Input

## 4. The Theoretical Analysis of the Capacitor–Memristor Circuit

#### 4.1. The Rectangular Pulse as the Input

_{m}in parallel with a parasitic capacitance C

_{m}. The capacitor is considered as a capacitance C

_{c}in parallel with a leakage resistance R

_{c}, as shown in Figure 3b [20,21]. To simplify the analysis, the resistance of the memristor is assumed to be constant. The time constant of this circuit can be determined from Equation (5) [22] as follows:

_{r}, and the falling edge is at t

_{f}. It is assumed that there is no charge accumulation in the capacitor before t

_{r}. The voltage across the memristor is as follows:

_{mf}is the voltage across the memristor at $t\to {t}_{f}^{-}$. In general, the parameters in Equations (5) and (6) satisfy C

_{c}≫ C

_{m}and R

_{c}≫ R

_{m}[21,23]. Thus, the circuit in Figure 3b can be simplified to the circuit in Figure 3c. Equations (5) and (6) can be simplified to

_{cf}is the voltage across the capacitor at $t\to {t}_{f}^{-}$. According to Equation (8), the positive voltage spike across the memristor is close to u regardless of the width of the rectangular pulse. However, the negative voltage spike across the memristor depends on the width of the rectangular pulse and the time constant. A wider pulse and a smaller time constant will result in a higher voltage across the memristor during discharge.

#### 4.2. The Action Pulse as the Input

_{r1}. The voltage across the memristor is

_{cf}is the voltage across the capacitor at $t\to {t}_{f}^{-}$, and u

_{cr2}is the voltage across the capacitor at $t\to {t}_{cr2}^{-}$. The positive voltage spike across the memristor during charging is close to u

^{+}. The negative voltage spike during discharge depends on u

_{cf}and u

^{−}. The existence of the negative rectangular pulse is necessary to increase the voltage across the memristor during discharge of the capacitor. From the equation we can see that the decay rate of the voltage spike is negatively related to the time constant τ. Importantly, if multiple pulses are input into the capacitor–memristor circuit at very short intervals, the charge in the capacitor will remain at the beginning of the subsequent pulse input. The remaining charge will reduce the voltage across the memristor during charging and increase the voltage during discharge, which will eventually affect the change in conductance.

## 5. Imitation of Synaptic Behavior Based on the Capacitor–Memristor Circuit

#### 5.1. Long-Term Depression (LTD)

_{0}is a pre-exponential factor, k is Boltzmann’s constant, and E

_{AC}is the activation energy for conduction [17,24,25]. As illustrated in Figure 5g,h, the conductance exhibits a positive correlation with temperature. The peak voltage drop across the capacitor can be attributed to the decrease in memristor conductance, which subsequently reduces the charging current (Figure 5i).

#### 5.2. Long-Term Potentiation (LTP)

#### 5.3. A Hebbian-Like Learning Mechanism

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Modeling a tantalum oxide memristor. (

**a**) The schematic of the tantalum oxide-based memristor used in the simulation. (

**b**) Calculated DC I-V characteristics of the Pd/TaO

_{x}/Ta

_{2}O

_{5}/Pd device.

**Figure 2.**The simulation results of rectangular pulses as the input. (

**a**) The capacitor–memristor circuit used for the simulation. (

**b**) For the input pulse A (blue) and measurement pulses (green), the measurement pulses are applied directly across the memristor to measure the resistance of the memristor, as shown in subfigure (

**a**). (

**c**) The voltage across the memristor when the rectangular pulse A is used as the input. (

**d**) The voltage across the capacitor when the rectangular pulse A is used as the input. (

**e**) The resistance of the memristor when the rectangular pulse A is used as the input. (

**f**–

**i**) The corresponding results of the input pulse B.

**Figure 3.**The working mechanism of the capacitor–memristor circuit. (

**a**) Rectangular pulse. (

**b**) A schematic diagram of the charging and discharging of the capacitor when a rectangular pulse is applied. (

**c**) Simplified diagram of the charging and discharging of the capacitor when a rectangular pulse is applied.

**Figure 4.**The action pulse is a combination of a positive rectangular pulse and a negative rectangular pulse.

**Figure 5.**Input low-frequency action pulses to the capacitor–memristor circuit to emulate LTD. (

**a**) Low-frequency action pulses with an interval of 2.2 μs. (

**b**) The partial enlargement of subfigure (

**a**). (

**c**) The voltage across the memristor. (

**d**) The partial enlargement of subfigure (

**c**). (

**e**) The conductance of the memristor. (

**f**) The current through the memristor. (

**g**) The conductance of the conducting filament. (

**h**) The temperature of the conducting filament. (

**i**) The voltage across the capacitor.

**Figure 6.**Input high-frequency action pulses to the capacitor–memristor circuit to emulate LTP. (

**a**) High-frequency action pulse with an interval of 0.1 μs. (

**b**) The partial enlargement of subfigure (

**a**). (

**c**) The voltage across the memristor. (

**d**) The voltage across the capacitor. (

**e**) The conductance of the memristor. (

**f**) The current through the memristor.

**Figure 7.**The implementation process of the Hebbian-like mechanism. (

**a**) In the first step, the pulses are only input to memristor M2, while memristor M1 is floating. This step can be regarded as the dog not salivating when there is only the stimulation of the bell ring. (

**b**) In the second step, high- and low-frequency pulses are input to memristors M1 and M2, respectively. During this step, the dog is stimulated by both the ring and the food at the same time. (

**c**) The third step has the same input as the first step. The dog salivates the first few times when only the ring signal is given, but then is indifferent to the stimulus of the ring. These phenomena represent associative memory and forgetting.

**Figure 8.**A Hebbian-like mechanism is implemented in the capacitor–memristors. (

**a**) Input low-frequency action pulse to memristor M2. (

**b**) The partial enlargement of subfigure (

**a**). (

**c**) Input high-frequency action pulse to memristor M1. (

**d**) The partial enlargement of subfigure (

**c**). (

**e**) The voltage across memristor M2. (

**f**) The current through memristor M2. (

**g**) The conductance of memristor M2. (

**h**) The voltage across the capacitor.

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**MDPI and ACS Style**

Yin, K.; Li, J.; Xiong, Y.; Zhu, M.; Tan, Z.; Jin, Z.
Simulating Synaptic Behaviors through Frequency Modulation in a Capacitor–Memristor Circuit. *Micromachines* **2023**, *14*, 2014.
https://doi.org/10.3390/mi14112014

**AMA Style**

Yin K, Li J, Xiong Y, Zhu M, Tan Z, Jin Z.
Simulating Synaptic Behaviors through Frequency Modulation in a Capacitor–Memristor Circuit. *Micromachines*. 2023; 14(11):2014.
https://doi.org/10.3390/mi14112014

**Chicago/Turabian Style**

Yin, Kuibo, Jingcang Li, Yuwei Xiong, Mingyun Zhu, Zhiyuan Tan, and Zhanrui Jin.
2023. "Simulating Synaptic Behaviors through Frequency Modulation in a Capacitor–Memristor Circuit" *Micromachines* 14, no. 11: 2014.
https://doi.org/10.3390/mi14112014