**2.4 Model equations**

The linear drift model [3] can be applied to describe the behavior of the memristor device. This model involves "instant" voltage–current characteristic for the instantaneous value of resistance

$$
v = R \cdot i \tag{4}$$

and control characteristic given by differential equation:

$$\frac{dR}{dt} = -\mu \frac{(R\_{OFF} - R\_{ON})R\_{ON}}{D^2} \text{ i } = -\gamma \text{i},\tag{5}$$

The model has the following parameters: the high memristor resistor value *ROFF*, the low memristor resistor value *RON*, ion mobility *μ*, the semiconductor film thickness *D*, *γ*-is inertial parameter. These parameters have the following typical values:

*ROFF* = 10 kОhm, *RON* =1kОhm, *μ* = 10�<sup>14</sup> m<sup>2</sup> s �<sup>1</sup> V�<sup>1</sup> , *D* = 10 nm [3], *γ* = 10<sup>9</sup> V A�<sup>2</sup> s �1 .

The switching time of the memristor device under constant current *I* can be estimated as

$$T\_R = \frac{R\_{OFF} - R\_{ON}}{\chi \ I} \approx \frac{R\_{OFF}}{\chi \ I} \,, \tag{6}$$

This time is 100 ms for current value 100 μA.

To describe the behavior of oscillator with comparator it is convenient to exploit the dimensionless parameters and variables. The dimensionless time is also applied. Such dimensionless variables can be obtained by normalizing. The normalization of resistances is performed using division by *ROFF*, respectively for voltages division by ð Þ *I* � *ROFF* is applied and for time - division by *TR*. As a result we have *ROFF* ¼ 1, *RON* ¼ 0*:*1 and *γ* =1.

Taking into account the threshold shift the comparator model with current generator *i* ¼ *i v*ð Þ (**Figure 1a**) is described by the equations

$$i(v) = \begin{cases} \mathbf{1}, \text{if } v > V\_m - kv\_{in} \\\\ -\mathbf{1}, \text{if } -(V\_M - kv\_{in}) < v < V\_m - kv\_{in} \\\\ \mathbf{1}, \text{if } -(V\_M - kv\_{in}) > v \end{cases} \tag{7}$$

Taking into account the binary variables the equations Eqs. (4), (5), and (7) can be transformed to single piecewise constant equation with respect to the variable *R*

$$\frac{dR(t)}{dt} = \begin{cases} -\mathbb{1}, \text{if } R(t) > R\_M - r(t), \\\ \mathbb{1}, \text{if } R(t) < R\_m - r(t), \\\ dR(t - \Delta\tau, \Delta\tau \to 0) \\\ \frac{dR(t - \Delta\tau, \Delta\tau \to 0)}{dt}, \text{if } R\_M - r(t) > R(t) > R\_m - r(t), \end{cases} \tag{8}$$

where *r t*ðÞ¼ *kvin*ð Þ*t =I* –as mentioned above, variable that reflects the change in the threshold under the influence of external signal. To save the oscillation conditions the following restrictions for *r t*ð Þ must be satisfied:

$$R\_{OFF} > R\_M + r(t), R\_M - r(t) > R\_m + r(t), R\_m - r(t) > R\_{ON} \tag{9}$$

The solution of equation Eq. (8) has character of triangular oscillations in the range *Rm* � *r*< *R t*ð Þ<*RM* � *r* .
