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

The paper presents circuit solutions based on a switch element with S-type I-V characteristic implemented on the classic FitzHugh-Nagumo and FitzHugh-Rinzel models. Proposed simplified electrical circuits allow modeling the integrate-and-fire neuron and burst oscillation mode with emulation of mammalian cold receptor patterns. The circuits are studied using experimental I-V characteristic of an NbO2 switch with a stable section of negative differential resistance (NDR) and a VO2 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. The proposed systems of differential equations with 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.


Introduction
Mathematical modeling of the nerve cell behavior, based on a detailed understanding of the functioning mechanisms of a neuron as a biological object, allows a quantitative prediction of neuron behavior.The parameters of such models reflect a clear biophysical meaning and can be measured, or estimated, in real bioexperiments.However, such modeling does not often allow the possibility of analytical research.To study the behavior of neurodynamic systems, simplified models are used, in particular, the FitzHugh -Nagumo [1][2][3][4][5] and FitzHugh -Rinzel [6] models.Such circuits, on the one hand, should preserve the general principle of generating neural signals, and, on the other hand, should have not very complex circuit solutions in terms of implementing cognitive technologies, for example, spike (SNN) [7] and oscillatory (ONN) neural networks [8][9][10][11][12][13][14].Therefore, the development of neurodynamic models in circuit design is an important task of modern neuroelectronics.
An electrical switching (ES) is an abrupt, significant, and reversible change in the conductivity of an electric element under the applied electric field or a flowing current.Current-voltage characteristic (I-V) of the element contains areas with negative differential resistance (NDR), created by positive feedback of current (Stype I-V) or voltage (N-type I-V) [15][16][17][18].However, this feedback is internal and is not created by external circuit elements, like in radio circuits with operational amplifiers.
A trigger diode is the typical element with an S-type I-V characteristic and the effect of electrical switching.It is formed by three consecutive p-n junctions, where the current instability is caused by the injection of minority carriers into the middle p-n junction [19].Modern silicon technology allows creation of dynistor elements with the S-type I-V characteristic of a wide range of voltages and currents, characterized by stability and low noise level.Among non-silicon materials with ES effect, amorphous semiconductors and transition metal oxides material classes can be highlighted.In amorphous semiconductors material class, phenomena of electrical instability is mainly studied in chalcogenide glasses (CGSs), which contain elements of VI group (Se, S, Te) and elements of the type Si, Ge, Bi, Sb, As, P).In CGSs, the switching can have the stable part of NDR (Figure 1a), the unstable part of NDR (Figure 1b), and the memory effect (Figure 1c) [15].Among transition metal oxides, S-type threshold switching is observed in MOM structures based on Nb2O5, NbO2, TiO2, VO2, Ta2O5, Fe oxide, and some other oxides [16,18].At least in two oxides (NbO2 and VO2), ES is caused by the metalinsulator phase transition (MIT).Recently, the switching effect has been studied in high-temperature superconductivity (HTSC) [20], colossal magnetoresistance (CMR) manganites and heterostructures based on them [21], as well as in various carbon-containing materials, including fullerenes and nanotubes [22,23].
In this paper, we propose circuit solutions based on a generalized switch element with S-type I-V characteristic that implement the classic FitzHugh-Nagumo and FitzHugh-Rinzel models.We study circuits operation using experimental I-V characteristic with a stable (NbO2 switch) and unstable (VO2 switch) NDR sections.In addition, we propose simplified electrical circuits allowing to simulate an integrate-and-fire neuron and burst oscillation mode with emulation of mammalian cold receptor patterns.

S-type switch models controlled by current and voltage
Using an example of a planar VO2 switch, a typical S-type I-V characteristic with an unstable NDR [24] 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 (Uth, Ith) and switching off (Uh, Ih).

Experiment
Model I sw =f(U) If both branches are approximated by straight lines with differential resistances Roff and Ron, the model I-V characteristic has the form Formulas (1-2) characterize the switch element, which state is determined by the voltage on the element (Figure 2b).

Figure 3. Experimental
The I-V characteristic with a stable NDR is an unambiguous and continuous voltage-to-current function that characterizes a current-controlled switch element.Its modeled piecewise linear approximation by straight lines can be written as: with negative resistance in the NDR section In this way, we have identified two types of elements with a stable and unstable NDR on the I-V characteristic, described by a piecewise linear approximation determined by five independent parameters.
Not every circuit can function with elements of both types.For example, when inductors are connected in series with a voltage-controlled element (I-V characteristic with an unstable NDR, see Figure 2a, b), the Therefore, in the diagrams below, we indicate the type of switch with the designation in the form of color resistors (Figure 4), controlled by current (a) or voltage (b).If the circuit can work with both types of switches, the combined designation is used (see Figure 4c).
(a) (b) (c) The circuits, presented in the article, are implemented in the MathCAD and LTSpice software packages, and source files are available in Supplementary Materials.

Relaxation oscillator
An element with an S-shaped NDR can be included in an active RC circuit (Figure 5).If the supply current I0 of the circuit is in the NDR range th 0 h

I <I I 
(5), the circuit generates self-oscillations of the relaxation type, similar to the oscillations of the classical autooscillator on a Pearson-Anson neon lump [26].By modifying the basic circuit, it is possible to compile various neural models.As there are no inductive elements in the illustrated circuit, there is no difference what type of S-element NDR is used to generate relaxation oscillations.

FitzHugh-Nagumo and FitzHugh-Rinzel models
The Hodgkin-Huxley model (HHM) served as the theoretical foundation for the ionic mechanisms involved in the excitation and inhibition in the peripheral and central parts of the nerve cell membrane [27][28][29][30][31][32][33][34][35][36].The FitzHugh-Nagumo (FN) model [27] is the first simplified version of HHM, which only allows modelling of self-excitation and restoration of the membrane potential through positive and negative feedbacks.
The following equations describe the model in a dimensionless form: where G() is a variable describing the dynamics of a membrane potential with a current Iext(), W() is a recovery variable, ,  and 0 are experimentally determined parameters.In this model, with a cubic nonlinearity (Fq(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 [36,37].The FitzHugh-Rinzel (RF) model is the development of the FitzHugh-Nagumo modified generator [38], where additional slowly changing variable Q() is added.The system of equations describe the FR model in dimensionless form as follows [39]: The resulted 3D model with five parameters (0, 1, , , ) helps to study specific reactions of the nerve cell, such as regular and chaotic generation of bursts [40].For the application in ONN and SNN, the model enables frequency and phase coding of information not only with the help of a spikes sequence, but also with bursts synchronization.

FitzHugh-Nagumo model based on a current-controlled switching S-element
To the circuit of the auto-relaxation oscillator (Figure 5a), we add a reactive element -inductance (L), and, in parallel to the supply current I0, we add the resistance (R0), as shown in Figure 6.The inductance introduces an additional degree of freedom to the system (inductance current), which determines the chargingdischarging process of capacity (C0) during switching of the NDR element.In addition, the circuit includes the sources of direct current Ib0 and alternating voltage: where Uin (t) is the input (stimulating) signal relative to the constant bias Ub0.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 3.3), is optional, but they are necessary for a general interpretation of the circuits of FitzHugh-Nagumo (6) and FitzHugh -Rinzel (7) models.The mathematical model of the circuit in Figure 6, expressed in the form of a system of equations based on Kirchhoff's laws, has the following form: where IL and U0 are the inductive current and voltage on the capacitor C0, respectively.The midpoint parameters of the NDR of S-type I-V characteristic are: the system ( 9) is transformed into FitzHugh-Nagumo model ( 6) with the parameters: and a cubic nonlinearity is replaced with a piecewise linear function: In Equation (12), Usw (I) is a function of the argument I=-GImp+Ib0, and is determined from the model I-V characteristic (3).
Figure 7 depicts the functions Fq (G) and Fpw (G) at Ub0 = -Ump and Ib0 = Imp.In this case, the midpoint of the NDR segment for the Fpw (G) function is shifted to the origin, and it corresponds to the best approximation of the Fpw (G) function using cubic nonlinearity Fq (G).
Based on the Equation ( 11), the inductive current IL models the behavior of the membrane potential G, and the voltage on the capacitor U0 reflects the slow recovery potential W. The input signal Uin(t) is the analogous to the input current Iext (t) of the neuron in Equations ( 6) and (7).12) at Ib0 = Imp and Ub0 = -Ump (solid, blue), using the example of I-V characteristic of NbO2 switch (Figure 3).6), using the switching S-element (Figure 3).Electrical circuit parameters: L=0.1 mH, C0=1 nF, R0=1 k, I0= 1 mA, Ib0=0 A, Ub0=0 V and Uin(t)=0 V.
Figure 8 depicts regular oscillations of inductance current IL(t) and voltage U0(t) at capacitance C0 and voltage Usw(t) at the switch, calculated using the example of numerical solution of Equations ( 9) for a switching structure with experimental I-V characteristic of S-type, presented in Figure 3.

FitzHugh-Rinzel model based on a current-controlled switching S-element
Since the FitzHugh-Rinzel model is obtained by adding one more variable to the FitzHugh-Nagumo equations, two options for upgrading the circuit in Figure 6 can be suggested, that is adding an RC filter of either high (Figure 9a) or low frequencies (Figure 9b).
In the first case (Figure 9a), the sequential R1C1 link below the switch generates a voltage signal U1(t) on the capacitance C1, which is modulated by charging-discharging processes.The mathematical model of this circuit (Figure 9a) is represented by a system of equations: ) Using the parameters of S-type I-V characteristic (11), the system of equations ( 13) after the transition to dimensionless variables has the form: The system is converted to the FitzHugh-Rinzel model ( 7) with a piecewise linear function (12) and parameters: Therefore, fluctuations of the induction current model the temporal dependence of the membrane potential of the neuron G, and the oscillations of the U0 and U1 voltages model the recovery potentials Q and W, respectively.In Equation (15), all parameters are positive, except for <0, similar to Rinzel's work [6] with cubic nonlinearity.9a with a switching S-element (Figure 3).Electrical circuit parameters: L=0.025 mH, C0=30 nF, C1=1.75 nF, R0=0.5 k, R1=1 k, I0= 2.7 mA, Ib0=0 A, Ub0=0 V and Uin(t)=0 V.
Figure 10 demonstrates the bursts oscillations of the inductance current IL(t) and voltages U0(t), U1(t), Usw(t) using the example of the numerical solution of equations ( 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: Ib0=0 A, Ub0=0 V and Uin(t)=0 V.
In the second case of the FitzHugh-Rinzel circuit (Figure 9b), the R1C1 integrator (low-pass filter) is included in the oscillating circuit and acts as a voltage modulator U1(t) between capacitance C0 and inductance L due to charging-recharging capacitance C1.Appendix 1 demonstrates how this circuit is converted to the FitzHugh-Rinzel model ( 7) after a linear transformation of variables.

Alternative neural-like circuits based on a switching S-element
In a modified circuit (Figure 3), additional capacity (С1) is connected in parallel with either the S-switch (Figure 11a) or inductance (Figure 11b).The circuits in Figures 11a and 11b are modeled by the following systems of differential equations, respectively: where U0(t) and U1(t) are the voltages on the capacitors С0 and C1, and Isw(U) is a piecewise linear approximation of the I-V characteristic (1) with the condition (2).
In contrast to the system of FitzHugh-Rinzel equations ( 9), equations (16)(17) use the direct Isw(U), and not the inverse I-V characteristic Usw(I), which is a two-valued, discontinuous function.Therefore, mathematical models of these circuits, probably, do not have an analytical solution, cannot be analysed, 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.
Let us demonstrate the operation of the circuit shown in Figure 11b at different temperatures of the switching element.The temperature dependences of the threshold parameters of the I-V characteristic are listed in Table 1, the data correspond to the switching VO2 element described in the previous study [41].12 demonstrates the temperature dependent I-V characteristic using Equations (1-2).Modeled oscillograms of the current circuit (Figure 11b) at different temperatures of the VO2 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 9 to 3 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 [43].
The external influence in the circuits can be implemented similarly to the FitzHugh -Nagumo circuit, by adding sources Ub(t) and Ib(t), and the interaction between the oscillators can be implemented through thermal coupling, described in detail in the studies [10,44,45].
Therefore, we have proposed alternative neural-like schemes based on S-type switches.The VO2 switch, due to its physical properties, can be used as a sensor model object to reproduce the pulse patterns of mammalian cold receptors.

The auto-relaxation oscillator as an integrate-and-fire neuron based on a switching S-element
When applying successive current pulses from various sources, the circuit of the relaxation generator described in Section 2.2 can simulate the operation of an integrate-and-fire neuron.Figure 14    The circuit processing of two successive current pulses is reflected in the oscillogram (Figure 15), where Usw and Isw are the voltage and current on the switch, respectively.When the first current pulse I0 is applied, the capacitor is charged to a voltage U0, which does not reach the switch-on voltage of the switch U0<Uth.Then the discharge process of the capacitor starts through a switch with resistance Roff.When the voltage Usw drops to a certain value U'0, the second current pulse I1 is generated, the switch voltage rises to Usw=Uth, and the switching process and the current pulse Isw 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 ultimately can lead to the generation of a signal at the output of the circuit.

Discussion
The circuit design of the FitzHugh-Nagumo model (Figure 6) can be compared with the well-known circuit on a tunnel diode [1,3], which has an N-type I-V characteristic.The main difference is that the tunnel diode is connected not in series, but in parallel to the inductance.Therefore, the voltage across the diode models the membrane potential G(t), and the inductance current models the recovery potential W(t) of the neuron.
A modification of the FitzHugh-Nagumo circuit, by adding an RC chain, derives the FitzHugh-Rinzel circuit model (Figure 9) based on an element with an S-shaped I-V characteristic.The transformation (G,W,Q, -G, -W, -Q, -) leads the degenerate system (7) to its own form.However, in our FitzHugh-Rinzel model with a piecewise linear function (12) for positive values of , it is necessary to make another replacement Fpw(G) -Fpw(-G) to get the same set of solutions.In the circuit, we change the direction of the supply current I0 and use symmetrical with respect to the inversion (I, U) (-I, -U) branch of the I-V characteristic of the switching S-element.

)(
If, for obtaining relaxation oscillations, the parameters selection of the FitzHugh-Nagumo circuit does not have practical difficulties, the burst mode for the FitzHugh-Rinzel model (Figure 7) is implemented in a narrow range of parameters and its search is not an easy task.The dynamics of the FitzHugh-Rinzel model with a nonlinear piecewise function has not yet been studied, and the values of bifurcations are not yet known.This can be a subject of scientific interest, both from a fundamental and practical point of view.
The mandatory presence of the inductance in the proposed circuits ensures the correct functioning of the circuits only if the S-switch contains a stable NDR section (Figure 3).In practice, the chain elements always I 0 , mA time, ms I sw possess parasitic capacitances, and the switching would not happen instantaneously.Therefore, the FitzHugh-Nagumo and FitzHugh-Rinzel circuits, probably, are capable to function with S-switches with an unstable NDR (Figure 2), however, the possible effect of high-voltage induction on the inductive element may lead to the failure of the switch.In addition, the presence of a stable or unstable NDR is often determined by the manufacturing technology of the switching element.Therefore, the oxides of transition metals, similar to presented here NbO2 and VO2, can be probably used to create elements of both types.
Recent studies highlighted the interest of VO2 switch applications in the SNN, ONN and neural-like circuits [10,44,[46][47][48][49].The study [7] illustrates the implementation of VO2 oscillators with various chaotic and burst modes of spike generation based on two switches.Presented in this paper circuits, containing only one VO2 switch, have obvious advantages.
The alternative circuits, presented in Figure 11, have simpler design and contain fewer elements than the FitzHugh-Rinzel circuits (Figure 7), however, they allow the generation of neural-like burst oscillations.The physical properties of the VO2 switch enable the demonstration of the neuromorphic behavior of the circuit and the reproduction of the functioning of mammalian cold receptors.In the future, assembling a neural network on such elements, using thermal [10,44] or electrical coupling between oscillators, it is possible to create systems with artificial intelligence that have temperature receptors.
Circuit of the auto-relaxation generator that simulates the work of a integrate-and-fire neuron is interesting, as it operates with current pulses.The current pulses at the input are converted into a current pulse of the switch at the output.Such circuits can be connected using current repeaters or directly, through diode bridges and resistors.Creating an effective neural circuit on current pulses can be the subject of future research.
A detailed investigation and operation modes testing of the circuits presented in the current study can be performed using LTspice files (see Supplementary Materials).

Conclusions
In the current study, we proposed circuit solutions that implement the FitzHugh-Nagumo and FitzHugh-Rinzel models with the S-type switching element.Simplified circuits have been developed that allow modeling the integrate-and-fire neuron and the burst oscillation mode with emulation of mammalian cold receptor patterns.
The authors hope the results of this study would spark the interest of researchers in the experimental implementation of the proposed circuits, the mathematical analysis of dynamics of the systems with non-linear S-elements, and the development of oscillator and spike neural networks based on switching S-elements.
Using the parameter Rmp of S-type I-V characteristic (10) and dimensionless time   tRmp/L, the first two equations of system (A1) are re-formulated in matrix form:  Linear transformation of the variables: The coefficients of the FitzHugh-Rinzel model ( 7) are determined from the parameters of the diagram in Figure 9b and from the system of equations (A7) as follows: The characteristic numbers 1 and 2 of the matrix M are always negative, and the modulo of parameter m does not exceed ½.Therefore, all the parameters in Equations (A9) are always positive.For the constructed FitzHugh-Rinzel model with > 0, the piecewise linear function (A8) is the inverse of function (12) (see comments in Discussion section).

Figure 1 .
Figure 1.S-type I-V characteristic with stable NDP segments (a), unstable NDR segments (b) and with bipolar memory effect (c).
exist.During numerical simulation of the circuit, the solution gets unstable with a decrease in the calculation step, since inductance operation does not imply an instantaneous change in current.

Figure 4 .
Figure 4. Designation for a switching element controlled by voltage (a), current (b) and, optionally, current or voltage (c).

Figure 5 .
Figure 5. Relaxation oscillator circuit (a) and the load characteristic with the operating point in the NDR region (b).Current oscillations at I0=1mA, C0=100nF and VO2 switch (I-V characteristic with unstable NDR, see Figure 2a) (c).

Figure 9 .
Figure 9.An oscillator circuit that implements the FitzHugh-Rinzel model with a high-pass R1C1-filter connected in series to the switch (a) and a low-pass R1C1-filter connected between the capacitor С0 and the inductance L (b).

Figure 11 .
Figure 11.Neuro-like circuits based on the S-type switch, generating bursts of oscillation, with a capacitance C1 connected in parallel to the switching element (a) and in parallel to the inductance (b).
captures an electronic circuit with two pulsed current sources I0 and I1.

Figure 14 .
Figure 14. Circuit modeling the integrate-and-fire neuron (a) and an example of the operating point of the circuit in sub-threshold mode.

Figure 15 .
Figure15.Oscillograms of the voltage Usw and the current Is response at the switch when exposed to two successive current pulses from sources I0 and I1.The circuit uses a VO2 switch (Figure2a).
the matrix M of the first two equations of system (A2).As a result, (A2) is reduced to the form: where 1 = -(D(m + ½)+ c) and 2 = -(D(m -½)+ c) are the characteristic numbers of the matrix M. Finally, a dimensionless change of variables is done in (A5): similar to the FitzHugh-Rinzel model, is obtained:

Table 1 .
Temperature dependences of the threshold parameters of the I-V characteristic.