**3. Memristor**

the cell body. The electrical simulations are transmitted from one neuron to the dendrite of

Another important part of the neuron is the axon. Axon arises from the cell body at a site called axon hillock and extends to over 1 m in length. A neuron can have multiple dendrites, but only one axon. Axon is covered by a layer of dielectric material myelin, known as myelin

The axon terminals of one neuron connect to the dendrites of another neuron through syn‐ apses. Electrochemical signals are transmitted from one neuron to another through synapses. Chemicals known as neurotransmitters are released from the presynaptic neuron, which binds to the receptors located at the dendrites of the postsynaptic neurons. These neurotrans‐ mitters are initially present in small bag‐like structures known as synaptic vesicles that are found at the axonic terminals of the neurons. These synaptic vesicles, when excited, migrate towards the synapse and get attached to the synapse and release the chemical ions through

The major ions that are involved in the process are sodium, potassium, chlorine and calcium. Once released, these ions diffuse through the semipermeable membrane and binds to the receptors which are present on the dendrites of the post‐synaptic neurons. The basic structure

Due to the ion exchange between neurons, a gradient in the ion concentration arises on either side of the semipermeable membrane. Due to this ion concentration difference, a potential will be generated, known as Nernst potential. Changes in the cross‐membrane voltage between the intra‐cellular and extra‐cellular potential will alter the function of the voltage‐dependention

sheath. Before termination, the axon gets divided into a large number of branches.

another neuron at the synaptic terminals.

98 Fourier Transforms - High-tech Application and Current Trends

the semipermeable membrane of the synapses.

of a synapse is shown in **Figure 3**.

**Figure 3.** Structure of synapse.

Memristor is considered to be the fourth fundamental electronic component. The basic state variables in any circuit are voltage (*V*), current (*I*), charge (*q*) and flux (*φ*). The state variables and relations between them are shown in **Figure 4**. Prior to the 1970s, only resistor, capacitor and inductor were known. No component showing the property of memristance was known to the scientific community. It was in 1971 that Leon Chua gave the scientific and logical basis for the existence of a two terminal circuit element called memristor (memristor is the shortened form of 'memory resistor') [3]. He reasoned the existence of the fourth element through symmetry arguments. Although he showed that the memristor has many interesting and valuable circuit properties, he was unable to implement the memristor in the form of a physical device without an internal power supply.

The six different mathematical relations connect the four fundamental circuit variables: volt‐ age (*V*), current (*I*), charge (*q*) and magnetic flux (*φ*). These relations are indicated in **Figure 4**. Since there were no devices that reflected a relation for long between the charge and flux, the memristor was referred to as the missing element, with memristance (*M*), with *dφ* = *Mdq*.

**Figure 4.** Relation between various state variables in an electronic circuit.

One of the most advertised and commercially inclined versions of the memristor was develop by HP Labs that was based on a thin film of titanium dioxide [4, 5]. The main reason that gained attention for this device was the possibility to scale the device beyond the traditional CMOS limits. While, there is debate on the charge transport mechanisms and resistance switching behaviours, the hypothesis is that the hysteresis requires some sort of atomic rear‐ rangement that modulates the electronic current. The HP memristor device consists of a thin film of titanium dioxide (TiO<sup>2</sup> ) sandwiched between two platinum electrodes, with one side of the titanium dioxide doped with oxygen vacancies, TiO2‐x (see **Figure 5**).

The undoped region is insulated and has higher resistance than the doped region. The effec‐ tive resistance within the memristor is determined by the boundary between the doped and

**Figure 5.** Memristor modelled by HP.

undoped regions. Let *D* be the total width of the TiO2 layer and *W* be the width of the doped region. Then the effective resistance of the device is given by *M*eff <sup>=</sup> (*W*/*<sup>D</sup>* ) *<sup>R</sup>*ON + (1 − (*W*/*<sup>D</sup>* ) )*R*OFF, where *R*ON is the resistance of the device if it is completely doped and *R*OFF is the resistance of the device if it is completely undoped, see **Figure 6**.

Under the situation, when a positive voltage at the side of the doped region and negative volt‐ age at the side of the undoped region, the oxygen vacancies move from the doped side to the undoped side, thus, increasing the width (*W*) of the doped region. This results in the overall resistance of the memristor.

If the polarity of applied voltage is reversed, that is, positive potential is applied to the undoped side and negative potential is applied to the doped side, then the width of the undoped region increases, thereby increasing the effective resistance of the device.

When input voltage is withdrawn or when there is no potential difference between the termi‐ nals, the memristor maintains the boundary between the doped and undoped region, since the oxygen ions remain immobile after removal of the input voltage. Thus, the resistance will be maintained at the same value before withdrawing the input voltage.

The resistance of the memristor increases when current flows through it in one direction and the resistance value decreases when the current flows through it in the opposite direction. It can retain the resistance value it had at that point of time, if the current is stopped.

**Figure 6.** Working of a memristor.

Since there were no devices that reflected a relation for long between the charge and flux, the memristor was referred to as the missing element, with memristance (*M*), with *dφ* = *Mdq*.

One of the most advertised and commercially inclined versions of the memristor was develop by HP Labs that was based on a thin film of titanium dioxide [4, 5]. The main reason that gained attention for this device was the possibility to scale the device beyond the traditional CMOS limits. While, there is debate on the charge transport mechanisms and resistance switching behaviours, the hypothesis is that the hysteresis requires some sort of atomic rear‐ rangement that modulates the electronic current. The HP memristor device consists of a thin

The undoped region is insulated and has higher resistance than the doped region. The effec‐ tive resistance within the memristor is determined by the boundary between the doped and

) sandwiched between two platinum electrodes, with one side

(see **Figure 5**).

film of titanium dioxide (TiO<sup>2</sup>

**Figure 5.** Memristor modelled by HP.

of the titanium dioxide doped with oxygen vacancies, TiO2‐x

**Figure 4.** Relation between various state variables in an electronic circuit.

100 Fourier Transforms - High-tech Application and Current Trends

We can see from *i* = *v*/*M*(*q* ) that when there is no voltage difference across the memristor, there is no current through the memristor. When the potential applied is reversed, the width of the undoped region increases resulting in an increase in effective resistance. The high resistance blocks any reverse leakage current and adding more inputs, the collective current does not increase significantly as the effective resistance remains constant.

The *V*‐*I* characteristics of the memristor are shown in **Figure 7**. Generally, indicative of a pinched hysteresis effect [3, 4, 6], the changes in the slope indicate the switching behaviour, with each of the switch having at least two resistance states. With change in operating fre‐ quencies, the resistance values of the state become equal at high frequencies. The frequency dependence of memristor is shown in **Figure 7**.

**Figure 7.** The V‐I characteristics of the memristor device.

Over the years, there have been several efforts to manufacture memristors. The various attempts include polymeric or ionic memristors, resonant tunnelling diode memristors, manganite memristors and spintronic memristors. In addition to the memristor devices, there are circuits that emulate the memristor behaviour, generally referred to as memristive systems.

The crossbar architecture with memristor is used to build ultra‐dense memory cells (RRAM resistive random‐access memory). Another application is the use of memristor for emulating neural circuits that can help develop a range of hardware‐based machine learning methods. The memristors can be also used to implement multilevel memories [7, 8] and analogue mem‐ ories [9]. They also find application in configurable logic arrays [10].
