Reasoned by its dynamical behavior, the memristor enables a lot of new applications in analog circuit design. Since some realizations have been shown (e.g. 2007, Hewlett Packard), the development of applications with memristors becomes more and more interesting. Besides applications in neural networks and storage devices, analog memristive circuits also promise further applications. Therefore, this article proposes a frequency dependent rectifier memristor bridge for different purposes, for example, using as a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity and describes the circuit theory. In this context it is shown that the Picard Iteration is one possibility to analytically solve the system of nonlinear state equations of memristor circuits. An intuitive picture of how a memristor works in a network in general is given as well and in this context some research on the dynamical behavior of a HP memristor should be done. After all it is suggested to use the memristor bridge as a neuron.

#### Keywords

- Memristor bridge
- rectifier
- neuron
- synaptic circuit
- Hebbian learning
- non-REM sleep

The usage of memristors in analog circuit design enablesnew applications. In [1], an ADC consisting of memristors has been proposed. Other applications are an automatic gain control circuit [2], programmable analog circuits [3], an electrical potentiometer [3] or oscillators [4, 5]. In [6] memristors are used for basic arithmetic operations. In this paper a frequency-dependent rectifier memristor bridge is presented and therefore, a broad description of memristive systems will be given first.

In general, a memristive system is described by the two equations [7]

where

The HP memristor was realized in 2007 by a team of Hewlett Packard researchers led by Stanley Williams [8, 9]. While this memristor is passive, the latest realization of a memristor is that of an active one on the base of niobium oxide [10]. The HP memristor is made of a titanium dioxide layer which is located between two platinum electrodes. This layer is of the dimension of several nanometers and if an oxygen dis-bonding occurs, its conductance will rise instantaneously. However, without doping, the layer behaves as an isolator. The area of oxygen dis-bonding is referred to as space-charge region and changes its dimension if an electrical field is applied. This is done by a drift of the charge carriers. The smaller the insulating layer, the higher the conductance of the memristor. Also, the tunnel effect plays a crucial role. Without an external influence the extension of the space-charge region do not change. The internal state

where

R_{on} is the resistance of the maximum conducting state and R_{off} represents the opposite case. The vector containing the internal states of the HP-memristor is one dimensional. For this reason scalar notation is used. The state equation is

where μ_{v}_{M}

These values are given as an example by the Hewlett Packard Team. The ratio of maximum and minimum value of the memristance can be as large as e.g. 160 [9].

The extension of space-charge region is physically limited. Therefore, a window function was established in order to realize a more realistic model which includes saturation. Several shapes of window functions are possible e.g. [11]. In the following, the window function

is used [12]. The current mathematical description is not optimal for numerical calculations. Excluding external influences, in the case of {_{M}, the voltage over the memristor, for the two cases _{M} > 0 or _{M} < 0, a state change should be possible. The window function results in:

A better stability for numerical simulations is achieved by introducing ε and therefore an extended restriction of definition area. For all simulations in this paper, ε equals 0.02.

The calculations of this part are done by neglecting the window function. The mathematical description using the window function is exceedingly complicated. The aim is to investigate the time behavior of a HP memristor when connected in series with an AC sine current source I_{S}. Using the state equation (5)

dividing of the variables on both sides

and performing integration leads to

whereas x_{0} := _{0}). At (t_{0}+ Δ_{0}+ Δ_{0} (see equation (10)). When using a sine voltage source different results are acquired. In this case the state equation is

and due to _{0} = 0 is

whereas _{0} and −_{0}. In the case of +_{0} the magnitude of Δ

In conclusion, Δ_{0}, if there is a voltage source in series with the memristor. As shown previously, for a supplied current source it does not. Fig. 1(a) illustrates this insight. Since the memristance is determined by the internal state, the change of the memristance behaves similarly. As seen in Fig. 1(b), for a sine voltage source the change of the memristance depends strongly on the internal memristance, however for a sine current source it does not. Note in the case of a sinusoidal signal, the maximum change of the space charge region Δ_{max}_{voltagesource}_{currentsource}

To estimate the dynamical behavior of the HP-memristor in circuits, the frequency f_{cut} for sinusoidal signals is introduced. This is the highest frequency for which a memristor will be able to change from lowest to highest memristance or vice versa. This state change completes exactly at the end of one half period (Δt_{cut} = 0.5 · _{cut}_{cut} depends on the amplitude of the supplied source. Neglecting the window function, the calculations is performed for a single memristor in series with a supplied sine voltage source. This example allows rough estimates for circuits which are more complex. Taking into account that ω equals 2 · π ·

whereas f_{cut} is direct proportional to the amplitude _{0}.

Example: Considering the restricted domain of _{0} = 30_{cut} ≈ 12

If the frequency is higher than f_{cut}, saturation would not be reached. Taking the window function into account, the result of solving the state equation by simulation is _{cut}, if the frequency is increasing the ratio of maximum and minimum value of the memristance will decrease. Note for a supplied sine current source, the calculation for f_{cut} is also possible (Conversion of Eq. (10)). As an example, for an amplitude _{cut} is about 6

In summary, the value of memristance depends on the previous load [13]. If no current is applied, the internal state will be retained. It follows that a memristor acts like a nonvolatile memory, whereas the range of values is continuous [14, 15]. That is an interesting fact comparable to transistor memory technology. The difference between highest and lowest possible memristance is relatively large [9]. Indeed in the true sense, the memristor is not a switch

but it could be used for switching operations. Therefore, depending on the direction and the benchmark the memristor passes higher potentials and blocks for lower ones. Subject to the time shift the behavior of the memristor is frequency dependent. For lim_{f}_{→∞} it behaves like a linear resistor [7] because the change of the internal state is not able to follow the rapid voltage change. For sufficiently small frequencies the nonlinearities are dominating whereas the time shift is directly proportional to the amplitude of the signal.

This circuit (Fig. 2) contains four HP memristors, one AC voltage source and one load resistor. By definition, for a positive voltage the memristors _{1} and _{4} are forward biased, while _{2} and _{3} are reverse biased. In [16] and [17] this circuit has also been presented, but applications differ. While this paper deals with periodic signals and their specifics at different frequencies and initial states, in [16] pulses are used for synaptic weight programming. In [17] the focus lies on generation of nth-order harmonics and the effect of frequency doubling by using this circuit.

At this point the new notation for the memristance

is established. The voltages

are defined from the nodes to ground, whereas the notations

are used for a better understanding. The structure of the circuit implies a nonlinear system of differential equations of the fourth order. The four state equations are

Using these state equations, numerical simulations are possible. For every numerical calculation in this paper the Fehlberg fourth-fifth order Runge-Kutta method is used and _{L}

DC-analysis: Using a DC voltage source, the states change until saturation is reached. The states of _{1} and _{4} will change in the same direction, which is opposite to _{2} and _{3}. For example, for a positive voltage the memristances _{1} and _{4} will decrease while _{2} and _{3} will increase. A DC voltage can be used to set the states of the memristors.

AC-analysis: From this point V_{S} is a sine AC voltage source. Regarding Fig. 3, dependent on the frequency a qualitative difference for the voltage over the load V_{RL} is detectable. For low frequencies (represented by _{0} = 30_{0} = 30

By means of Fig. 4, this frequency selective behavior can be better understood. Note that the asymmetric behavior of the memristances shown in Fig. 4(b) was expected (See equation (12) for a single memristor and its interpretation).

_{cut}: For low frequencies, a complete state change will happen, because the memristors enter saturation. Beginning with an initial value _{1} and _{4} are decreasing, while _{2} and _{3} are increasing until saturation is reached. When the sine voltage source reaches its negative half period the process will be reversed until saturation is reached again and so on. During one period there is a change for the relational operator between both memristances. Thus, every half period, the potential in node one is going to be higher than that in node two, which implies that V_{RL} is always going to be positive.

_{cut}: The memristors do not completely switch from lowest to highest memristance. It follows that they do not reach saturation and the memristors will just reach the initial state after one period. Due to this characteristic, the relational operators between _{1} and _{4} do not change compared to _{2} and _{3} during one period and the relation of the memristances solely depends on the initial conditions (See Fig. 4(b)). For example, if the initial states are equal in all four memristors, the values of _{1} and _{4} would be lower than _{2} and _{3}. Thus, V_{RL} would be positive for the first half period, too. At the second half period, the Memristances _{2} and _{3} keep higher than _{4} and _{1}. Thus, the potential at node “2" is higher than at node “1", which implies that V_{RL} is negative.

There are two reasons why the absolute value for the amplitude of _{RL}_{RL}_{S} changes.

The lower the frequencies, the more the circuit behaves like a Graetz circuit, which is shown in Fig. 5(a). Using diodes instead of memristors in the shown configuration, this circuit is used for rectifying. For very high frequencies the circuit behaves like a Wheatstone bridge (see Fig. 5(b)). This can be proven by using the voltage divider. For lim_{RL→∞} it simply is

and had to be zero for equal resistances, if the conclusion is true. Simulations illustrate that for e.g. _{n}_{RL}_{1}) equals 1.0002.

Using a supply periodic square wave voltage source with

is another interesting example. The results for this are shown in Fig. 6. A frequency dependent behavior is also detectable. For high frequencies the voltage over the load resistor is serrated. For low frequencies this voltage is almost constant with negative peaks. The condition that the initial states of all memristors are equal leads to the behavior presented in Fig. 6. But what happens if the circuit is supplied with a periodic square wave voltage and the initial states are not equal? As shown in Fig. 7(a), for a supplied periodic voltage and high frequencies the variation of memristance depends on the initial conditions. Thus, the voltage curve changes from the saw tooth form. As shown in Fig. 7(b), it is possible to create a curve which is similar to synaptic membrane voltages which are used for Spike-Time-Dependent-Plasticity [18]. The amplitude of the voltage varies for different initial conditions, which can be set by a low frequency or DC voltage.

Depending on the frequency, there are two significant types. Fusing a common Graetz circuit and a Wheatstone circuit in one device is one application possibility. Replacing the independent source with a controlled one leads to further application possibilities. The source is controlled by the load voltage and its frequency is variable, keeping in mind, the amplitude of the output signal decreases by increasing the frequency. Because of that, a frequency controlled regulator is conceivable. If the amplitude exceeds a predefined value, the frequency has to increase to prevent a further amplification. Using the circuit as a saw tooth generator is also possible. As mentioned before, the presented circuit could also be used as a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity.

This chapter will show a connection between the memristor bridge circuit and biological science. In [19] (electro-osmosis in skin) and [20] (cell membranes and the Hodgkin-Huxley potential), examples of the usage of memristive devices in biological systems are given. The Hodgkin-Huxley potential itself has been proposed in [21].

The aim of this paper is to propose using the introduced circuit to model a neuron. For reasons illustrated in Fig. 7, it seems natural to use this circuit to generate output signals which remind of neural impulses. Subsequently in the context of the memristor bridge the terms “neuron" and “cell" are used as synonyms for this memristor bridge circuit. It is assumed that the connection of several memristor bridges models a synaptic circuit (see Fig. 13). In the following, hypotheses on how some biological processes could be applied on the synaptic circuit are given. Periodic square wave signals are not common in the area of synaptic systems, which raises the question: What happens if the signal shown in Fig. 7 is used as input signal instead? Using the voltage waveform shown in Fig. 7 as input voltage of the memristor bridge circuit results in an output waveform as shown in Fig. 8. The observed waveforms are qualitatively similar. Furthermore, it is showed that the amplitude of the output signal is considerably smaller compared to the amplitude of the input signal.

As illustrated in Fig. 7 and 8, it seems, the stronger the initial states of the two memristor pairs differ from each other the larger the amplitude of the output signal becomes. In fact, it strongly depends on whether a pair of memristors (e.g. _{1} and _{4}) reaches a state of low memristance or not (see equations (20) and (21)). For that reason, the output voltage waveform for a set of initial states e.g. _{1}(0), _{4}(0) = 0.73 and _{2}(0), _{3}(0) = 0.27 compared to an other set _{1}(0), _{4}(0) = 0.73 and _{2}(0), _{3}(0) = 0.33 is similar. In order to monitor that the initial states differ by the same value from 0.5, the parameter _{n}_{1}(0_{4}(0_{2}(0_{3}(0_{L} = 1

The state of the whole circuit can be set by applying a DC voltage. If the initial states of all memristors are 0.5, the circuit is not activated. In the other case (

This theory [22] suggests that “when an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part of firing it", the connection between both will improve and signal transmission will become more effective.

Using this theory on the memristor bridge, if cell A and cell B are activated it implies that there is a connection between both. In order to that there is a growth process, the activation has to become larger. A possible circuitry is shown in Fig. 13 and described in the following.

If there are two or more neurons activated at the same time, this could be seen as neural learning process. For example, in Fig. 9 neuron

If two or more neurons are activated and fire at the same phase and frequency, a physical connection could be established. It is conceivable for this connection to increase in strength as the amplitude of the impulses increase. Furthermore, it is essential that several impulses are necessary in order to form a permanent connection. Starting from a non-activated state, by activating a neuron through information

The Non-REM sleep is characterized by high amplitudes and low frequencies [23]. It consists of four stages, which this paper will not further describe. The Electroencephalography (EEG) of a human brain shows among other things that there are different rhythms during one day. Beta rhythms have a spectrum from 13

In the case of the memristor bridge it is assumed that, for example, the operating mode shown in Fig. 7 is the state of awaking. Relative to this, the frequency of the theta rhythm is lower. The frequency of the delta waves is lower than that of the theta rhythm.

As shown in Fig. 11 (a), if the circuit is excited by a lower frequency, which is represented by 7

Interpretation of this results for the Non-REM sleep: One function of the theta rhythm phase could be the alignment of excited neurons. Dependent on the frequency, neurons with a lower state of stimulation or unstimulated neurons do not enter saturation. This could be useful for learning processes.

If the frequency is smaller than that of the theta rhythm, the behavior of the circuit differs. In accordance with delta waves, as an example the frequency is set to 2_{0} = 10_{0} = 10_{n}

Interpretation of this results for the Non-REM sleep: One function of the delta rhythm phase can be seen as the resetting of all neurons to the same not activated state. The possibility of changing between activation and non-activation of all neurons together can be seen as working storage. Thus, the delta rhythm phase can be used to refresh this storage.

The aim of this section is to suggest, how the physical connection could be realized in circuitry. As shown in Fig. 12, the cell _{post}_{L1}. Thus, the cell _{post}_{RL1}, which can be calculated by equations (20) and (21). _{L}_{L1} in parallel with the total memristance of the cell _{post}

_{L}_{RL1} perhaps changes compared to the in Fig. 7 shown waveforms, for example. Simulations show that with a single memristor used as _{L}_{1} = _{2} = _{3} = _{4} = 0.5) and the state of saturation (_{1} = _{4} = 1 and _{2} = _{3} = 0) are tested. Using _{L2} = 1_{L1} equals 100 Ω, the influence of _{Npost}_{1}, _{2}, _{3}, _{4}) is marginal. Note the lower _{L}

Neurons are rather connected with several other neurons than with only one neuron. Regarding Fig. 12 and assuming that _{L1} is split into several resistors with lower resistance, connections with more than one cell are possible. This is shown in Fig. 13. The resistors are the connections for further cells and could be compared to axons. For example, if ten resistors with 100Ω are connected in series, the total resistance is 1_{L2}, whereas the total resistance is 1_{L1} = 100Ω and _{L2} = 1_{in1} in Fig. 13) are given by the voltage divider. In accordance with the Hebbian theory, if the cell _{pre}_{post1}), the amplitude of that output voltage becomes higher compared to the other postsynaptic cells. This implies that the connection between both cells became stronger. The question is, how the activation of the cells can be done automatically by firing of neurons in this circuit itself.

At this point, the Picard Iteration is introduced as a possibility to solve common memristive systems analytically. One advantage related to e.g. the Volterra-series expansion is that the Picard Iteration converges more rapidly. This chapter investigates the possibility of applying this iteration to memristive systems by performing on the memristor bridge circuit. First, a general description of the Picard Iteration is given. A nonlinear, dynamical system with

is given. The iteration is given by

whereas

In the case of the memristor bridge circuit, _{cut}.

For memristive systems using the HP memristor model, the Picard Iteration is performed by

with

Similar to chapter “Memristor bridge circuit" the used notation for the memristance _{n}

1st iteration step: Using the initial states

2nd iteration step:

Separating the variables in this equation leads to

and integration results in

whereas for

Red solid line: analytical solution by Picard Iteration (second iteration step)

_{0} = 30

This circuit (shown in Fig. 15) behaves similar to the memristor bridge, if the applied voltages satisfies three conditions: Amplitude and frequency are the same and the phase is shifted by 180 degrees. The structure of the circuit implies a

nonlinear system of differential equations of the second order whereas the state equations are

with

V_{RL} is given by the equation

For this circuit, the frequency selective behavior is detectable. This circuit can also be used as a programmable generator for synaptic impulses. In general, under the same conditions, the amplitude of the output signal is higher compared to the output signal of the other circuit. If it is possible to grip an input signal and its inversion, the usage of this circuit would reduce the complexity by two memristors in comparison to the preceding circuit. Perhaps this circuit could be used to synchronize two synaptic impulses which are phase shifted by 180 degrees.

In this paper two circuits consisting of HP memristors were presented, for those a frequency selective behavior was detectable. In contrast to high frequencies, operation at low frequencies results in a behavior similar to that of a rectifier. This functional change with frequency is caused by the delay when changing the internal states through an external source. To estimate the dynamical behavior of circuits, the time behavior of a single memristor in series with a periodic source was investigated. The configuration of the circuits leads to a nonlinear system of differential equations which describes the internal states. Using the Picard Iteration is one possibility to solve this system analytically. The frequency selective behavior can be used to realize two modes in one circuit (Graetz and Wheatstone circuit). Other applications are a frequency controlled regulator or a programmable synaptic membrane voltage generator for Spike-Time-Dependent-Plasticity. The behavior of the circuit suggests that the circuit could be used as a neuron. For that reason a synaptic circuit is given as well. In this context it is suggested how a learning process could be applied on this synaptic circuit. Additional, assumptions on the functionality of the Non-REM sleep were made.

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