**1. Introduction**

The simplicity of the design of memristor based circuits and the possibility of manufacturing memristors [1–3] using integrated technology make them promising for use in a variety of information storage and processing systems. The construction of neuromorphic systems [4–8] is one of the most important memristor applications where the memristors provide the function of nonvolatile analog memory.

Due to memristor capabilities the wide implementation of memristors is predicted in different circuit application spheres including analog circuits. The properties of memristors [3, 9] open up new possibilities of constructing the memristor based oscillators (MBO) of different types [10–14]. The complex behavior of MBOs is analyzed in some papers (see for instance [15–18]). The inertial property of memristors provides the elimination from oscillator circuits the reactive elements (inductors and capacitors) which are poorly compatible with the requirements of the integrated implementation of neuromorphic systems. By the present time the various types of reactance-less MBO have been proposed [19–28]. This class of oscillators is considered below in the paper.

The neuromorphic systems including artificial neurons (AN) and networks become promising area where the analog memory plays the important role [29–39]. The memory elements are located between neurons and provide restructuring the coupling weight coefficients. Memristors are well suited to the requirements for artificial synapses [9, 40, 41]. The memristor resistance determines the value of the weight coefficients. The change in resistance under the action of current determines the possibility of restructuring the connections.

However, it should be noted that the properties of memristors allow them to be used not only as synaptic elements but also in the artificial neurons themselves. It can be mentioned that the reactance-less MBO consisting of memristor device and an active element, for instance comparator, can be also considered as simple AN model. Such an oscillator element can be inhibited or excited similarly to AN behavior. Its state can be specified by the phase of periodic oscillation.

Advanced AN models [8] that more accurately describe the behavior of biological neurons have high complexity to represent essentially more complex and various dynamical processes. The response of oscillatory AN to the input excitation involves not only changing the state but also changing the character of generation of output pulse train. In this case the number of the pulses and position of the pulses in pulse train depend on input amplitude and transient prehistory.

The complex mathematical model is required to represent such a behavior. This is usually achieved by increasing the order of the model. The complexity of circuits of corresponding oscillatory AN is also must be increased [42, 43] and strict requirements for the precision of circuit parameters must be met.

We present the alternative approach in this paper. We demonstrate that coupled memristor-based reactance-less oscillators have the set of modes with dynamical processes that is enough to provide the desired complex behavior. To support these capabilities at circuit level the approach to MBO construction is presented that based on controlling the comparator threshold. Some advantages of this approach are demonstrated.

Among the advantages of controlling threshold approach in MBO it is essential to point out the opportunity to construct piecewise constant (PWC) oscillators. Recently AN models based on piecewise constant (PWC) oscillators have appeared [44–46]. Such AN models are convenient in practice. PWC oscillators are the oscillators with mathematical models which are systems of ordinary differential equations (ODE) with piecewise constant coefficients. The signals generated by AN in this case are piecewise linear functions of time. PWC oscillators are developed on the base of standard electronic components including amplifiers, logic gates, resistors, capacitors. The transient processes occur in these circuits under constant excitation, for example the charge or discharge of the capacitor at constant current. The analysis of AN behavior of such type and networks based on them is given in papers [47, 48]. The nonlinearity of the memristor characteristics due to the change in its resistance when current flows through device limits the development of PWC memristor based oscillators [49, 50]. Application of the considered approach to control threshold in MBO avoids this restriction because it provides use only changing the sign of the current through the memristor while generation process.

Application in binary oscillator networks is other important capability of the considered coupled reactance-less MBOs. Oscillatory neural networks are promising candidates for solving a number of complex computational problems [51–55]. The most suitable circuit elements for such networks are binary generators with binary output signals [56–58]. In binary oscillator networks (BON) binary signals are exchanged and information is represented by binary streams. The considered coupled reactance-less MBOs can be applied as elementary binary oscillators.

The rest of the paper is organized as follows. Section 2 presents the principle of controlling thresholds in MBO circuits. The circuit version of coupled MBOs with positive couplings and its functionalities are discussed in Section 3. In Section 4 the *Functional Capabilities of Coupled Memristor-Based Reactance-Less Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97808*

functional capabilities of coupled MBOs with inverting connections are given. The main properties of coupled MBO for use in binary generator networks are considered in Section 5. The technique of using phase planes to analyze the behavior of MBOs is widely used in sections.
