**5.1 Features of binary oscillator networks based on memristor oscillators**

The important properties of coupled MBOs are the simplicity of external managing the conditions of the oscillator injection locking, as well as fast frequency capture under relatively small impact amplitude.

The external excitation can violate the synchronicity of the coupled MBOs. The time to restore synchronization depends on the amplitude of the external impact and coupling strengths between the MBOs. As a result, the coupled MBOs provide the modulation of pulse trains desired for the implementation of oscillatory artificial neurons (AN).

By such a way, it becomes possible to apply the simpler coupling systems of the first order instead of using high-order nonlinear systems with reactive circuit elements and with high requirements for the element parameters.

Note that coupled MBOs belong to the class of binary oscillator and can be exploited in BON on base of integrated technologies. Using binary oscillators with binary output signals [46–48] is one of the promising lines for constructing oscillatory neural networks that are most suitable for integrated technologies. In such binary oscillator networks information is represented by binary streams.

The connection of *N* MBOs is described by system of *N* equations for variables *Ri*ð Þ*t* ð Þ *i* ¼ 1, 2, … , *N* . This system has the view of type Eq. (8):

$$\frac{dR\_i(t)}{dt} = \begin{cases} -\mathbb{1}, \text{if } R\_i(t) > R\_M - r\_i(t), \\\ \mathbb{1}, \text{if } R\_i(t) < R\_m - r\_i(t), \\\ \frac{dR\_i(t - \Delta \tau, \Delta \tau \to 0)}{dt}, \text{if } R\_M - r\_i(t) > R\_i(t) > R\_m - r\_i(t), \end{cases} \tag{15}$$

The outputs of the transmitting MBOs are connected to the inputs of the receiving MBOs directly or via logic gates. Therefore the variables *ri*ð Þ*t* are binary functions of the outputs of the transmitting MBOs:

$$r\_i(t) = k\_i \cdot F\_i \text{ ( $\upsilon\_{out1}, \upsilon\_{out2}, \dots \upsilon\_{outN}$ )}\tag{16}$$

where *Fi*ð Þ … is a logical function of *N* binary variables (0, 1). In this case the relationship between binary variable *vout* and *dR=dt* is unambiguous.

The system of equations Eqs. (15) and (16) describes behavior of BON in autonomous mode. If there are external binary signals, they should be included into the *Fi* functions as additional external variables.

As simple examples of the elements of the binary oscillator networks based on memristor oscillators we can point out ring structure (**Figure 13**) which can be considered as extension of the considered before two coupled identical MBO with positive couplings (**Figure 5**) and also star-like structure (**Figure 14**) with applying the coupled MBO with possible inverting connections.

### **5.2 Some simulation results**

Some simulation results to confirm the features of BON based on MBOs are presented below. To simulate the versions of BON fragments the dimensionless parameters of the variables were used: *RM* ¼ 0*:*8, *Rm* ¼ 0*:*4.

**Figure 13.** *Example of BON ring structure using memristor based oscillators.*

**Figure 14.** *Example of BON star-like structure using memristor based oscillators.*

The circuit element of star-like structure containing two coupled MBOs was selected for simulation. This circuit contains also logical element OR. The coupling strength has the value r = 0.05.

The computed waveforms are given in **Figure 15**. The transient process from starting point to steady state is shown. The oscillators MBO1 and MBO2 have the different initial states: *R*1ð Þ¼ 0 0*:*4, *R*2ð Þ¼ 0 0*:*8. Then the oscillators tend to periodic steady state *R*1ðÞ¼ *t R*2ð Þ*t* and reach it during four periods. The outputs of MBO1 and MBO2 are identical in steady state mode. This state corresponds to logical "0" due to applying the logical function XOR. The change of logical function XOR is shown in the lower curve in **Figure 15**. In this case the pulses of logical "1″ appear under misalignment of the MBO1 and MBO2 outputs.

It can be mentioned that variation of initial state for MBO2 oscillator leads to change in duration of the process of steady state establishing (**Figure 15**). So for *R*2ð Þ¼ 0 0*:*7, 0*:*6, 0*:*5 the transition to periodical steady state is performed for 3, 2 and 1 periods, respectively. In these cases, the difference in the initial states is a multiple of twice the coupling strength factor (here *r* = 0.05). This case corresponds to the complete synchronization of the oscillations (the phase shift is 0).

In connection with this non-multiple case is of interest. Then the initial states *R*1ð Þ¼ 0 0*:*43, *R*2ð Þ¼ 0 0*:*8 are chosen as examples. The corresponding computed waveforms are given in **Figure 16** for different versions.

As can be seen from **Figure 16a** the oscillations in MBO1 and MBO2 are synchronized with the shift at inverting absence. Steady state is reached for time interval of four periods, the periodic pulses are generated at the output, the phase detector XOR generates short mismatch pulses.

#### **Figure 15.**

*The computed waveforms of the transient process from starting point to steady state for coupled MBOs: MBO1 dashed line, MBO2 – dotted line, OR function -solid bold line, function XOR – solid line.*

*Functional Capabilities of Coupled Memristor-Based Reactance-Less Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97808*

**Figure 16.**

*The computed waveforms of the transient process for coupled MBOs with the initial states R*1ð Þ¼ 0 0*:*43, *R*2ð Þ¼ 0 0*:*8*. The graphs correspond to the following versions of coupling types: (a) couplings without inverting; (b) couplings with analog inverting MBO1 input; (c) couplings with inverting of the MBO2 binary output; (d) couplings with joint inverting of both types.*

The timing diagram of MBOs behavior gets more complicated for cases with inverting the coupling signals (**Figure 16b**). The dependences of R (t) can be considered as modulated by triangular oscillations. The output signals of the logical element and the phase detector are converted into complex binary sequences with large period. The similar character of waveforms can be observed in the case of logical inversion (**Figure 16c**) when the logical OR circuit receives at the input the inverted signal from the output of MBO2. By such a way if the difference in the initial states is a non-multiple of twice the coupling strength factor then timing diagram of MBOs behavior is complicated for both types of inversion. It is interesting also that the simultaneous use of both types of inversion leads to another character of waveforms (**Figure 16d**). The full synchronization can be achieved after the transient process.

The presented simulation results illustrate the capabilities of coupled MBOs in its application as elements of Binary Oscillator Networks.
