# Switch Elements with S-Shaped Current-Voltage Characteristic in Models of Neural Oscillators

^{*}

## Abstract

**:**

_{2}switch with a stable section of negative differential resistance (NDR) and a VO

_{2}switch with an unstable NDR, considering the temperature dependences of the threshold characteristics. The results are relevant for modern neuroelectronics and have practical significance for the introduction of the neurodynamic models in circuit design and the brain–machine interface. The proposed systems of differential equations with the piecewise linear approximation of the S-type I–V characteristic may be of scientific interest for further analytical and numerical research and development of neural networks with artificial intelligence.

## 1. Introduction

_{2}O

_{5}, NbO

_{2}, TiO

_{2}, VO

_{2}, Ta

_{2}O

_{5}, Fe oxide, and some other oxides [22,24]. In at least two oxides (NbO

_{2}and VO

_{2}), ES is caused by the metal-insulator phase transition (MIT). Recently, the switching effect has been studied in high-temperature superconductivity (HTSC) [26], colossal magnetoresistance (CMR) manganites, and the heterostructures based on them [27], as well as in various carbon-containing materials, including fullerenes and nanotubes [28,29].

_{2}switch) and an unstable (VO

_{2}switch) NDR sections. In addition, we propose simplified electrical circuits, allowing the ability to simulate an integrate-and-fire neuron and burst oscillation mode with the emulation of mammalian cold receptor patterns.

## 2. Materials and Methods

#### 2.1. S-Type Switch Models Controlled by Current and Voltage

_{2}switch, a typical S-type I–V characteristic with an unstable NDR [30] is presented in Figure 2a. Electrical instability causes the presence of a high-resistance (OFF) and low-resistance (ON) branches with threshold voltages (currents) of switching on (U

_{th}, I

_{th}) and switching off (U

_{h}, I

_{h}).

_{off}and R

_{on}, the model I–V characteristic has the form

_{cf}— (cutoff voltage) residual voltage of low-resistance (ON) section. Switching in Equation (1) between OFF and ON states is implemented as follows:

_{2}oscillators [16].

_{2}sandwich switch [31], the S-type I–V characteristic with stable NDR is presented in Figure 3.

#### 2.2. Relaxation Oscillator

_{0}of the circuit is in the NDR range

#### 2.3. FitzHugh–Nagumo and FitzHugh–Rinzel Models

_{ext}(τ), W(τ) is a recovery variable, α, β, and τ

_{0}are experimentally determined parameters. In this model, with a cubic nonlinearity (F

_{q}(G) = G–G

^{3}/3), a bi-stability regime can be observed, when the gravity regions of two stable equilibrium states are separated by a saddle [41,43].

_{0}, τ

_{1}, β, χ, and γ), helps to study specific reactions of the nerve cell, such as the regular and chaotic generation of bursts [45]. For the application in ONN and SNN, this model enables the frequency and phase coding of information, not only with the help of a spikes sequence, but also with bursts synchronization.

## 3. Results

#### 3.1. FitzHugh–Nagumo Model Based on a Current-Controlled Switching S-Element

_{0}, we added the resistance element (R

_{0}), as shown in Figure 6. The inductance introduces an additional degree of freedom to the system (inductance current), which determines the charging–discharging process of capacity (C

_{0}) during switching of the NDR element. In addition, the circuit includes the sources of direct current I

_{b0}and alternating voltage:

_{b}(t) = U

_{b0}+ U

_{in}(t)

_{in}(t) is the input (stimulating) signal relative to the constant bias U

_{b0}. The use of these sources to generate oscillations in the circuit of Figure 6, as well as in the subsequent circuits (Section 3.2 and Section 3.3), is optional, but they are necessary for a general interpretation of the circuits of FitzHugh–Nagumo (Equation (6)) and FitzHugh–Rinzel (Equation (7)) models.

_{L}and U

_{0}are the inductive current and voltage on the capacitor C

_{0}, respectively.

_{sw}(I) is a function of the argument I = –G⋅I

_{mp}+ I

_{b0}, and is determined from the model I–V characteristic in Equation (3).

_{q}(G) and F

_{pw}(G) at U

_{b0}= –U

_{mp}and I

_{b0}= I

_{mp}. In this case, the midpoint of the NDR segment for the F

_{pw}(G) function is shifted to the origin, and it corresponds to the best approximation of the F

_{pw}(G) function using cubic nonlinearity F

_{q}(G).

_{L}models the behavior of the membrane potential G, and the voltage on the capacitor U

_{0}reflects the slow recovery potential W. The input signal U

_{in}(t) is analogous to the input current I

_{ext}(t) of the neuron in Equations (6) and (7).

_{L}(t) and voltage U

_{0}(t) at capacitance C

_{0}and voltage U

_{sw}(t) at the switch, calculated using the example of the numerical solution of Equation (9) for a switching structure with the experimental I–V characteristic of the S-type, presented in Figure 3. Similar oscillograms can be obtained by modeling the FN circuit in LTspice (see Supplementary Materials).

#### 3.2. FitzHugh–Rinzel Model Based on a Current-Controlled Switching S-Element

_{1}C

_{1}link below the switch generates a voltage signal U

_{1}(t) on the capacitance C

_{1}, which is modulated by the charging–discharging processes. The mathematical model of this circuit (Figure 9a) is represented by a system of equations:

_{0}and U

_{1}voltages model the recovery potentials Q and W, respectively. In Equation (15), all parameters are positive, except for γ < 0, similar to Rinzel’s work [8] with cubic nonlinearity.

_{L}(t) and voltages U

_{0}(t), U

_{1}(t), and U

_{sw}(t) using the example of the numerical solution of Equation (13) for a switching structure with the experimental S-type I–V characteristic presented in Figure 3. In the calculations, similarly to the FitzHugh–Nagumo model in Figure 8, the S-type I–V characteristic’s shift to the origin is not used and there is no input (stimulating) signal: I

_{b0}= 0 A, U

_{b0}= 0 V, and U

_{in}(t) = 0 V.

_{1}C

_{1}integrator (low-pass filter) is included in the oscillating circuit and acts as a voltage modulator U

_{1}(t) between capacitance C

_{0}and inductance L due to charging–recharging capacitance C

_{1}. Appendix A demonstrates how this circuit is converted to the FitzHugh–Rinzel model (Equation (7)) after a linear transformation of variables.

#### 3.3. Alternative Neural-Like Circuits Based on a Switching S-Element

_{1}) is connected in parallel with either the S-switch (Figure 11a) or inductance (Figure 11b).

_{0}(t) and U

_{1}(t) are the voltages on the capacitors C

_{0}and C

_{1}, and I

_{sw}(U) is a piecewise linear approximation of the I–V characteristic from Equation (1) with the condition in Equation (2).

_{sw}(U), and not the inverse I–V characteristic U

_{sw}(I), which is a two-valued, discontinuous function. Therefore, mathematical models of these circuits, most likely do not have an analytical solution, cannot be analyzed, and have only a numerical solution. However, these circuits are simpler than the FitzHugh–Rinzel schemes (Figure 7), contain fewer elements, and allow the generation of neural-like bursts.

_{2}element described in the previous study [46].

_{2}element are presented in Figure 13a. A pronounced burst mode can be observed at a temperature of 25 °C. As the temperature rises to 40 °C, a decrease from nine to three in the number of pulses in each pack and an increase in the frequency of repetition of the burst activity are visible. With a further increase in temperature to 50 °C, the bust oscillations become periodic single pulses with an increased frequency. Similar dynamics models the firing patterns of mammalian cold receptors (see Figure 13b) described in the previous study [47].

_{b}(t) and I

_{b}(t), and the interaction between the oscillators can be implemented through the thermal coupling, described in detail in the studies [12,48,49].

_{2}switch, due to its physical properties, can be used as a sensor model object to reproduce the pulse patterns of mammalian cold receptors.

#### 3.4. The Auto-Relaxation Oscillator as an Integrate-And-Fire Neuron Based on a Switching S-Element

_{0}and I

_{1}. Figure 14b demonstrates an example of the operating point of the circuit in sub-threshold mode, when the current and voltage on the switch do not reach the threshold values (U

_{th}, I

_{t}). When the operating point is located on the high-resistance or low-resistance branch of the I–V characteristic, a stable state of the circuit is observed. When the operating point is transferred to the NDR region (for example, by changing the currents I

_{0}and I

_{1}), the generation of spikes begins.

_{sw}and I

_{sw}are the voltage and current on the switch, respectively. When the first current pulse I

_{0}is applied, the capacitor is charged to a voltage U

_{0}, which does not reach the switch-on voltage of the switch U

_{0}< U

_{th}. Then the discharge process of the capacitor starts through a switch with resistance R

_{off}. When the voltage U

_{sw}drops to a certain value U’

_{0}, the second current pulse I

_{1}is generated, the switch voltage rises to U

_{sw}= U

_{th}, and the switching process and the current pulse I

_{sw}are generated on the switch. Therefore, the first impulse sets the switch to the sub-threshold mode, when the operating point of the circuit is near the threshold (see Figure 14b), and the second impulse triggers the switch to turn on. Capacitor C has the role of a current signal integrator, accumulating the charge, which increases the voltage on the capacitor, and it can ultimately lead to the generation of a signal at the output of the circuit.

## 4. Discussion

_{pw}(G) → –F

_{pw}(–G) to get the same set of solutions. In the circuit, we change the direction of the supply current I

_{0}and use symmetrical with respect to the inversion (I, U) → (–I, –U) branch of the I–V characteristic of the switching S-element.

_{0}and C

_{1}in the FR circuits (Figure 9) can be explained as follows. The burst oscillation mode has two time parameters: Low-frequency, which corresponds to the oscillation period of the bursts, and high-frequency, which determines the period of pulses inside the bursts. The two capacitors C

_{0}and C

_{1}differ significantly in nominal value (by more than 15 times, see Figure 10 description). Large capacitance C

_{0}, by gradually charging (from the source) and discharging (through the switch circuit), controls the low-frequency oscillation mode. The voltage U

_{0}(t) (see Figure 10b) changes in a narrow range and sets the operating point of the entire circuit. As a result, the switch either enters the oscillation mode or exits the oscillation mode, forming burst oscillations. The smaller capacitance C

_{1}controls the period of high-frequency pulsed oscillations inside the bursts. Similar considerations can be applied to alternative neural-like circuits (Figure 11).

_{2}and VO

_{2}, can be probably used to create elements of both types.

_{2}switch applications in the SNN, ONN, and neural-like circuits [9,12,48,51,52,53]. The study [9] illustrates the implementation of VO

_{2}oscillators with various chaotic and burst modes of spike generation based on two switches. The circuits presented in this study that contain only one VO

_{2}switch have obvious advantages.

_{2}switch enable the demonstration of the neuromorphic behavior of the circuit and the reproduction of the functioning of mammalian cold receptors.

_{2}film, as this material is used in the manufacturing of bolometric matrices [54] and temperature sensors [55]. A particularly strong dependence exists in the region of the metal to insulator phase transition (MIT), observed near the threshold temperature T

_{t}= 68 °C, in the temperature hysteresis region. Above T

_{t}, the VO

_{2}film operates in the high-conductivity (metal) phase, and, below T

_{t}, the VO

_{2}film exists in the low-conductivity (insulator) phase. The jump in resistance between these phases can reach several orders of magnitude (10–10

^{4}), depending on the structure of the film [55]. Many researchers, including us [56], have demonstrated that the effect of electrical switching occurs due to MIT, when the current passing through the VO

_{2}structure heats it with Joule heat to a temperature T

_{t}. There is a strong dependence of the threshold switching parameters on the ambient temperature T

_{0}[46]. An increase in T

_{0}leads to a decrease in the threshold voltage U

_{th}and, at T

_{0}~ T

_{t}, the switching effect will be suppressed since the VO

_{2}channel will always be in a highly conductive state. This imposes a limitation on the use of a VO

_{2}sensor based on the effect of electrical switching. Most switching elements based on transition metal oxides have strong temperature dependences for U

_{th}, and some structures, for example, based on NbO

_{2}, demonstrate electrical switching up to temperatures of ~300 °C [24]. Therefore, the circuits presented in the current paper, based on elements with the S-shaped I–V characteristic, have the potential for practical application in a wide temperature range. In the future, assembling a neural network on such elements, using thermal [12,48] or electrical coupling between oscillators, it is possible to create systems with artificial intelligence that have temperature receptors.

^{12}. Pairing VO

_{2}switches into simple relaxation circuits (such as Figure 5a) in laboratory conditions is performed using probe stations, or by creating a cell with clamped or soldered contacts.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{mp}of S-type I–V characteristic in Equation (10) and dimensionless time τ ≡ t⋅R

_{mp}/L, the first two equations of system in Equation (A1) are re-formulated in matrix form:

**M**of the first two equations of Equation (A2). As a result, Equation (A2) is reduced to the form:

_{1}= – (D⋅(m + ½)+ c) and λ

_{2}= – (D⋅(m – ½)+ c) are the characteristic numbers of the matrix

**M**. Finally, a dimensionless change of variables is done in Equation (A5):

_{pw}(G) is the piecewise linear function

_{1}and λ

_{2}of the matrix

**M**are always negative, and the modulo of parameter m does not exceed ½. Therefore, all the parameters in Equation (A9) are always positive. For the constructed FitzHugh–Rinzel model with γ > 0, the piecewise linear function in Equation (A8) is the inverse of function in Equation (12) (see comments in the Discussion section).

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**Figure 1.**S-type I–V characteristic with stable NDR segments (

**a**), unstable NDR segments, (

**b**) and with bipolar memory effect (

**c**).

**Figure 2.**Experimental [30] and model I–V characteristic of a planar VO

_{2}switch with an unstable NDR (

**a**) and an electrical key diagram for a model I–V characteristic with U

_{cf}= 0 (

**b**). The VO

_{2}switch parameters for I

_{sw}(U) (1-2) are: U

_{th}= 5.64 V, U

_{h}= 2.12 V, U

_{cf}= 1.754 V, R

_{off}= 10,742 Ω and R

_{on}= 276 Ω.

**Figure 3.**Experimental [31] and the model I–V characteristic of a NbO

_{2}sandwich switch with a stable NDR. Parameters of the NbO

_{2}switch for U

_{sw}(I) (3-4): I

_{th}= 56 μA, I

_{h}= 357 μA, U

_{th}= 0.93 V, U

_{h}= 0.82 V, U

_{cf}= 0.747 V, R

_{NDR}= –365 Ω, R

_{off}= 16.61 kΩ, and R

_{on}= 204.5 Ω.

**Figure 4.**Designation for a switching element controlled by a voltage (

**a**), current (

**b**) and, optionally, current or voltage (

**c**).

**Figure 5.**Relaxation oscillator circuit (

**a**) and the load characteristic with the operating point in the NDR region (

**b**). Current oscillations at I

_{0}= 1 mA, C

_{0}= 100 nF, and VO

_{2}switch (for the I–V characteristic with unstable NDR, see Figure 2a) (

**c**).

**Figure 7.**The functions of potential F (G) for cubic nonlinearity F

_{q}(G) (dash, red) and piecewise linear approximation of F

_{pw}(G) from Equation (12) at I

_{b0}= I

_{mp}and U

_{b0}= –U

_{mp}(solid, blue), using the example of I–V characteristic of NbO

_{2}switch (Figure 3).

**Figure 8.**The time dependences of the induced current I

_{L}(t) (

**a**), the voltage U

_{0}(t) on the capacitance C

_{0}(

**b**) and the voltage U

_{sw}(t) on the switch (

**c**) in the circuit (Figure 6), using the switching S-element (Figure 3). Electrical circuit parameters: L = 0.1 mH, C

_{0}= 1 nF, R

_{0}= 1 kΩ, I

_{0}= 1 mA, I

_{b0}= 0 A, U

_{b0}= 0 V, and U

_{in}(t) = 0 V.

**Figure 9.**An oscillator circuit that implements the FitzHugh–Rinzel model with a high-pass R

_{1}C

_{1}-filter connected in series to the switch (

**a**) and a low-pass R

_{1}C

_{1}-filter connected between the capacitor C

_{0}and the inductance L (

**b**).

**Figure 10.**The time dependences of the induction current I

_{L}(t) (

**a**), voltage U

_{0}(t) at capacitance C

_{0}(

**b**), voltage U

_{1}(t) at capacitance C

_{1}(

**c**), and voltage U

_{sw}(t) at switch (

**d**) in the circuit in Figure 9a with a switching S-element (Figure 3). Electrical circuit parameters: L = 0.025 mH, C

_{0}=30 nF, C

_{1}= 1.75 nF, R

_{0}= 0.5 kΩ, R

_{1}= 1 kΩ, I

_{0}= 2.7 mA, I

_{b0}=0 A, U

_{b0}= 0 V and U

_{in}(t) = 0 V.

**Figure 11.**Neural-like circuits based on the S-type switch, generating bursts of oscillation, with a capacitance C

_{1}connected in parallel to the switching element (

**a**) and in parallel to the inductance (

**b**).

**Figure 12.**Temperature-dependent I–V characteristic, calculated by Equations (1) and (2) and the data in Table 1.

**Figure 14.**Circuit modeling the integrate-and-fire neuron (

**a**) and an example of the operating point of the circuit in sub-threshold mode (

**b**).

**Figure 15.**Oscillograms of the voltage U

_{sw}and the current I

_{s}response at the switch when exposed to two successive current pulses from sources I

_{0}and I

_{1}. The circuit uses a VO

_{2}switch (Figure 2a).

Temperature, °C | U_{th}, V | U_{h}, V | R_{on}, Ω | R_{off}, Ω | U_{cf}, V |
---|---|---|---|---|---|

25 | 5.36 | 1.247 | 53 | 2550 | 0.955 |

40 | 4.052 | 0.93 | 55 | 2216 | 0.758 |

50 | 2.714 | 0.607 | 58 | 1726 | 0.502 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boriskov, P.; Velichko, A. Switch Elements with S-Shaped Current-Voltage Characteristic in Models of Neural Oscillators. *Electronics* **2019**, *8*, 922.
https://doi.org/10.3390/electronics8090922

**AMA Style**

Boriskov P, Velichko A. Switch Elements with S-Shaped Current-Voltage Characteristic in Models of Neural Oscillators. *Electronics*. 2019; 8(9):922.
https://doi.org/10.3390/electronics8090922

**Chicago/Turabian Style**

Boriskov, Petr, and Andrei Velichko. 2019. "Switch Elements with S-Shaped Current-Voltage Characteristic in Models of Neural Oscillators" *Electronics* 8, no. 9: 922.
https://doi.org/10.3390/electronics8090922