# Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity

^{1}

^{2}

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## Abstract

**:**

_{2}(Y)/TiN/Ti memristive device. This device is fabricated on the oxidized silicon substrate using magnetron sputtering. The circuit with such nonlinearity is described by a three-dimensional ordinary differential equation system. The effect of the appearance of spontaneous self-oscillations is investigated. A bifurcation scenario based on supercritical Andronov–Hopf bifurcation is found. The dependence of the critical point on the system parameters, particularly on the size of the electrode area, is analyzed. The self-oscillating and excitable modes are experimentally demonstrated.

## 1. Introduction

_{2}(Y)/TiN/Ti memristive device [13,16]. Taking a simplified mathematical description of the voltage-current characteristics of this device, we incorporate it into electronic circuit modeling the FitzHugh–Nagumo (FHN) neuron [27]. Exploring the model and presenting the results in numerical simulations, we analyzed basic bifurcation scenarios of oscillation appearance. A memristive neuron-like generator based on a metal oxide device was also experimentally implemented. This generator can be both in self-oscillating mode and at rest. For the first time, to our knowledge, we are conducting an experimental study of such a memristor-based generator and comparing the results of numerical simulation with experimental data.

_{2}(Y)/TiN/Ti memristive device of this type was studied in our previous work. It was shown that a memristive device induces complex dynamics in a small ensemble of oscillators [31,32,33].

## 2. Model

#### 2.1. Memristive Device

_{m}is the corresponding current. Part (B) consists of linear resistors R

_{6}, R

_{7}, linear inductors L

_{1}, L

_{2}, and linear capacitance C in parallel with them, as well as power supplies E

_{1}, E

_{2}, and a commuted silicon diode D

_{1}. The initial condition V

_{ini}can be loaded into the neuron via an analog switch controlled by a periodic V

_{syn}signal.

_{2}(Y)/TiN/Ti memristor fabricated on the oxidized silicon substrate using magnetron sputtering. The details of technological operations can be found in other works [3,13,16,35,36,37]. To study the structure of the element, we used the high-resolution cross-sectional transmission electron microscopy (XTEM) operating the Jeol JEM-2100F microscope («JEOL» company, Japan) with an acceleration voltage of 200 kV. The cross sections of memristive devices were prepared by conventional technology using the equipment of Gatan Inc. We also developed a custom topology to fabricate the array of paired micro-scale (20 × 20 μm

^{2}) cross-point memristive devices (overall 44 devices) with the described thin-film structure on a silicon chip and mounted the chip in a standard 64-pin package. Electrical measurements and electroforming were carried out at room temperature by using the Agilent B1500A semiconductor device analyzer («ASTANA» company, Russia). Such chips containing several memristive devices are necessary for further studies of complex nonlinear effects in ensembles of oscillators.

_{2}(Y)/TiN/Ti memristor.

_{HRS}/R

_{LRS}≈ 10

^{4}at a reading voltage of U

_{r}= −0.5 V (Figure 3). The voltage values when switching the experimental device were V

_{set}= −5 V and V

_{reset}= 6 V. Positive voltage induced switching from LRS to HRS (RESET), and negative voltage resulted in switching from HRS to LRS (SET). Devices on this basis show a good characteristic for neuromorphic computing, where constant weights are necessary for accurate training and computations [13].

#### 2.2. Mathematical Model of a Memristor

_{m}= −34.8) [30], activated by Joule heating (kT = 4.14 $\times $ 10

^{−21}Joule) and applied electric voltage u. The transition between HRS and LRS was determined by the dynamic contribution to the total current of filaments and, consequently, by the state parameter. In these equations b = 3.2, B = 2.5 $\times $ 10

^{21}are the coefficients determined by quadratic polynomial interpolation from experimental data of a physical device.

_{m}through the memristive device was determined by the transfer of charge carriers through defect states in the oxide material in the region of the filament rupture or the rest of the structure. Such a structure consists of a linear component j

_{lin}, which corresponds to ohmic conductivity ($\rho $ = 10

^{−8}—Coeff for Ohmic current) through filaments (conducting pathways in the memristive structure), and a nonlinear component j

_{nonlin}. The nonlinear transfer of charge carriers (effective barrier E

_{b}= 38.6) is described by the Frenkel–Poole law based on the approximation of current–voltage characteristics in HRS.

_{1}= 30.9, A = 10

^{9}—coefficients corresponding to experimental data. V

_{set}= −3 and V

_{reset}= 3 are the threshold voltages for resistive switching.

_{2}(Y)/TiN/Ti device.

#### 2.3. Memristive Oscillator Model

- The value of the linear and nonlinear components of the current density from (1) is substituted into the general formula of the current density of the memristive device;
- The total current density is substituted into the formula for the current strength of the memristive device ${I}_{\mathrm{m}}\left(u\right)$;
- The resulting current strength is multiplied by the parameters $\gamma $ and d, a nonlinear function $f\left(u\right)$ is obtained, which is then substituted into the differential Equation (4).

_{e}= 4 $\times {10}^{-14}{\mathrm{m}}^{2}$, d = ${10}^{3}\mathsf{\Omega}$.

#### 2.4. Numerical Investigation Methods

^{p}), then it converges on the segment (x

_{0}, X] to the solution of the differential problem with the order O(h

^{p}). The full description of the method is given in [40]. The integration procedure was performed with a step of 0.02 and an error of 10

^{−6}in the application software package for solving MATLAB version R2020b (The MathWorks, USA, purchased from the official website) technical computing problems [41,42,43]. Choosing the initial conditions, we obtained the following results when changing the control parameter.

#### 2.5. Experimental Study of the Generator

_{2}(Y)/TiN/Ti and a resistor representing a load (Figure 5c).

## 3. Results

#### 3.1. Memristive Neuron Model Dynamics

#### 3.2. Bifurcation Analysis

^{2}of the memristive device. To detect possible subcritical modes or other attractors, we ran simulations for three different sets of initial conditions, [u(t = 0), v(t = 0), x(t = 0)], with the following values: y

_{1}= [0.5, 0.25, 0.2], y

_{2}= [−0.5, 0.004, 0.1], y

_{3}= [2, 1, 0.5]. Transient time, e.g., estimation time of the transient process, was varied in the interval 100 ms < t < 50,000 ms with integration step h = 0.02. Next, the amplitude per cycle was calculated as the difference between the maximum and minimum cycle values for one period.

^{2}. Figure 10 illustrates the bifurcation diagram for the smallest device size. It is quite similar to the previous case, but the critical $\mathsf{\epsilon}$ is shifted to a higher voltage.

#### 3.3. Memristive Neural Dynamics Generation

## 4. Conclusions

_{2}(Y)/TiN/Ti memristor. Bipolar resistive switching cycles demonstrated the stable operation and a resistance ratio of R

_{HRS}/R

_{LRS}≈ 10

^{4}at a reading voltage of U

_{r}= −0.5 V. The I–V curve of a memristive device was modeled when ±5 V was applied to the device for 2000 ms. This characteristic qualitatively corresponds to the real structure of Au/Zr/ZrO

_{2}(Y)/TiN/Ti.

_{2}(Y)/TiN/Ti memristor. The other is that the original circuit of the modified FitzHugh–Nagumo neuron used in simulations is known to demonstrate computation by spikes with both integrate–and–fire and resonant–and–fire communication modes [39].

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram of the nonlinear circuit based on memristor. The circuit was modified from [27] by replacing the nonlinear element with the Au/Zr/ZrO

_{2}(Y)/TiN/Ti memristor. Modified with permission from Kazantsev V.B., neural networks; published by 2006 years.

**Figure 2.**The cross-sectional TEM image of the memristive device structure after electroforming. The thickness of the ZrO

_{2}(Y) layer was 40 nm, the thickness of the top Au electrode was 40 nm, and the thickness of the bottom TiN/Ti electrode was 40 nm.

**Figure 3.**Experimental I–V characteristics of the Au/Zr/ZrO

_{2}(Y)/TiN/Ti memristive devices. The arrows indicate the direction of the voltage sweep.

**Figure 4.**I–V characteristics of the memristive device obtained as a result of numerical simulation.

**Figure 5.**(

**a**) Block diagram of a neuron–like memristive generator FHN; (

**b**) device for determining I–V characteristics and memristive chip; (

**c**) analog electric circuit of the FHN neuron containing memristive device. The nonlinearity is set using a memristive device. The capacitor C1 is responsible for the membrane of the neuron, and the power source V1 is associated with a reversible potential (an equilibrium potential).

**Figure 6.**Response of the memristive FHN model on sufficiently strong perturbation current added to the first equation of system (4). The lower panel shows the shape of the current pulse, amplitude—4.42 B, duration—11 ms. Parameter values: $\mathsf{\eta}=0.48,\mathsf{\epsilon}=0.94$.

**Figure 7.**Results of numerical simulation of damped oscillations: (

**A**) time series u and v, (

**B**) phase portraits $\mathsf{\eta}=0.48,\mathsf{\epsilon}=0.24$. Detailed information about the receipt of these figures is contained in Supplementary Materials.

**Figure 8.**Results of numerical simulation of the system in the periodic generation mode: (

**A**) time series u, (

**B**) time series v, (

**C**) phase portraits, $\mathsf{\eta}=0.48,\mathsf{\epsilon}=0.000024$. Detailed information about the receipt of these figures is contained in Supplementary Materials.

**Figure 9.**One–parameter bifurcation diagram for an area of 50 × 50 μm

^{2}. The arrow shows the increased dependency interval. The blue, orange and yellow dependences are determined by y

_{1}, y

_{2}, and y

_{3}initial conditions, respectively.

**Figure 10.**One–parameter bifurcation diagram with a device area of 5 × 5 μm

^{2}. The grey, orange and green dependencies are determined by y

_{1}, y

_{2}, and y

_{3}initial conditions, respectively.

**Table 1.**The parameters obtained in the study of damped oscillations, where $\mathsf{\Delta}$ is the decay decrement,

**λ**is the logarithmic decay decrement,

**T**is the period (second), and

**V**is the frequency (Hz). Also, in the course of the study, the values of

**Q**–factor, cyclic ${\mathsf{\omega}}_{\mathbf{0}}$, and natural $({\mathrm{second}}^{-1}$) frequency were obtained.

Parameter | u | v |
---|---|---|

$\mathsf{\Delta}$ | 2.11 | 2.87 |

λ | 0.74 | 1.06 |

T | 0.0157 | 0.0159 |

V | 63.61 | 62.89 |

Q | 4.24 | 2.97 |

β | 47 | 66 |

${\mathsf{\omega}}_{\mathbf{0}}$ | 399 | 395 |

$\mathsf{\omega}$ | 396 | 389 |

**Table 2.**The parameters obtained during the study of periodic generation, where

**A**is the voltage amplitude (B),

**T**is the oscillation period (second),

**V**is the frequency (Hz), and $\mathsf{\omega}$ is the natural frequency (${\mathrm{second}}^{-1})$.

Parameter | u | v |
---|---|---|

A | 3.97 | 0.017 |

T | 1.753 | 1.725 |

V | 0.57 | 0.58 |

$\mathsf{\omega}$ | 3.58 | 3.64 |

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**MDPI and ACS Style**

Kipelkin, I.; Gerasimova, S.; Guseinov, D.; Pavlov, D.; Vorontsov, V.; Mikhaylov, A.; Kazantsev, V. Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity. *Mathematics* **2023**, *11*, 1268.
https://doi.org/10.3390/math11051268

**AMA Style**

Kipelkin I, Gerasimova S, Guseinov D, Pavlov D, Vorontsov V, Mikhaylov A, Kazantsev V. Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity. *Mathematics*. 2023; 11(5):1268.
https://doi.org/10.3390/math11051268

**Chicago/Turabian Style**

Kipelkin, Ivan, Svetlana Gerasimova, Davud Guseinov, Dmitry Pavlov, Vladislav Vorontsov, Alexey Mikhaylov, and Victor Kazantsev. 2023. "Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity" *Mathematics* 11, no. 5: 1268.
https://doi.org/10.3390/math11051268