We noted in Chapter 1 that the EEG signals we are interested in recording are composed of low-amplitude voltage oscillations. While it would be possible to measure each signal relative to electric ground, most of the important biological information is contained in the small fluctuating voltages associated with synaptic events rather than the absolute potential distance from ground. To allow sufficient amplification to see these fluctuating potentials without magnifying the absolute potentials on which they ride, we use a low-frequency filter to eliminate the direct current potentials. How does this work?

When voltage fluctuates over time, current does not flow continuously in the same direction; in fact, it reverses direction at the same frequency as the voltage fluctuation. This is the principle behind the alternating current (AC) we use for household electric power, which in the United States shifts direction 60 times per second or at a frequency of 60 Hertz (Hz). The frequency can be measured by simply counting the number of oscillations that occur in 1 second or alternatively by taking the inverse of the duration of one complete oscillation (the period of the wave, also known as the wavelength), hence

Period (s) = 1/frequency (Hz).

So, for example, an alpha wave that lasts 100 ms (one-tenth of a second) has a frequency of 1/(0.1 seconds) or 10 Hz.

Alternating currents make some electric components work differently than they do under direct current. Most resistors provide the same amount of resistance to current flow in one direction as the other, but some have the property of rectification—they conduct better in one direction than the other. These are known as rectifiers or diodes, and they too have their biological equivalent in certain voltage-gated channels like the potassium delayed rectifier channel that helps terminate some action potentials. Capacitors subjected to direct current simply charge up to the extent of the voltage applied and then conduct no additional current until there is a change in potential. Under AC, they continuously charge and discharge, in one direction then the other, and thus provide very little impediment to current flow. This relationship is called capacitive reactance (X_C), which contributes to the overall impedance of an AC circuit in much the same way that a resistor does for direct current. Capacitive reactance (X_C) is inversely proportional to the oscillation frequency of the circuit and the capacitance:

X_C = 1/(2π × f × C)

So, at higher frequencies, capacitive reactance becomes smaller, and the capacitor (or the cell membrane) becomes less restrictive to current flow. Likewise, as the frequency approaches zero (no current fluctuation, ie, direct current), the reactance approaches infinity, and the capacitor effectively blocks current flow completely.¹

When capacitors and resistors are combined together in AC circuits, some special properties emerge. If the resistor and capacitor are in series (one after the other) between the voltage source and ground, the reactance slows the flow of current through combined “RC” circuit in inverse proportion to the frequency of the oscillation, so low-frequency oscillations are attenuated and high-frequency signals pass
through relatively unaffected. The output of this type of circuit depends on which device comes first, the resistor or the capacitor, and whether the output voltage is measured across the resistor or the capacitor. When the capacitor comes first and the output voltage is measured across the resistor (Fig. 2.1A), the circuit is called a low-frequency filter, since the low frequencies are filtered out. The terminology can be a little confusing, as this type of filter can also be called a high-pass filter, since higher frequencies are passed through without attenuation. The other configuration, with the resistor first and the voltage measured across the capacitor (Fig. 2.1B), is termed high-frequency (or low-pass) filter. In either case, the frequencies attenuated depend on the values of the resistor and capacitor, which determine the time constant (τ) for the filter, according to the simple equation:

τ = R × C.

The time constant results from the time taken for the capacitor to charge and discharge, which occurs at an exponential rate as a voltage change occurs. If the voltage is suddenly stepped to a new level, the capacitor charges according to the exponential equation:

V_out = V_in × e^–t/τ

where e is the base of the natural logarithm (about 2.718). When the time after the voltage step (t) is equal to the time constant (τ), the voltage decays to e^-1 (1/2.718) or about 37% of the input value.

For a low-frequency filter (Fig. 2.1A), the initial fast transient step response will rapidly charge the capacitor, passing the initial voltage through the circuit essentially unchanged, but the connection between the resistor and ground gradually (exponentially) discharges the capacitor and reduces the output voltage to zero. Note that the a DC voltage step produced only a transient response; when the voltage is now stepped back to baseline, the voltage transient that passes quickly through the capacitor is now in the opposite direction, charging the capacitor in the opposite polarity, and the resistor again “bleeds” the voltage on the capacitor down to baseline. As we will see, low-frequency filters can eliminate DC potential shifts and can be quite useful in EEG recordings.

In contrast, if the positions of the resistor and capacitor are reversed (Fig. 2.1B), the high-frequency component of the voltage step is quickly dissipated to ground through the capacitor, but the low-frequency components gradually (exponentially) charge the capacitor so that the full voltage step is eventually seen in the voltage output. This is a high-frequency filter because it removes the fast-frequency components but (depending on the time constant) tends to leave the underlying voltage shifts intact. The step back to baseline is again “rounded off” due to removal of the high-frequency components, but it eventually returns to the initial potential.

Why would we want to filter out high-frequency components of the signal? High-frequency filters are useful for removing unwanted fast components of the signal, which in the case of EEG recordings often means rapid firing motor units of cranial muscles that obscure underlying EEG activity. This activity is often high in amplitude, and a major source of artifact that can obscure the underlying cerebral activity. It also can appear “spiky,” which can be confusing when looking for the epileptiform spikes that are associated with seizures. But as we will see, removing these high-frequency components comes with a “price”—the blunting of faster-frequency cerebral activities such as spike and wave discharges.

A comprehensive understanding of the engineering principles used in the recording of EEG is not required for adequate interpretation. However, the more you understand the technology, the better you can use it. Applying filters can be helpful when used for display purposes after recording but are rarely needed during the acquisition of EEG data in technically adequate recording situations (ie, when electrode impedances are low and extraneous sources of noise/artifact have been minimized). The technologist and electroencephalographer should always remember that filters change both the unwanted artifacts and the physiological data.

An EEG recording is only as good as the equipment it is recorded on, the technologist doing the recording, and the EEGer interpreting the results. Knowing the limitations of your equipment and the software used to run the EEG acquisition and review system is critical for understanding what the signals mean and helps you troubleshoot when something goes wrong.

There are also combination filters that serve specialized functions. In the United States, the 60-Hz power that runs our computers, medical equipment, fluorescent lights, and almost everything else radiates 60 cycles and higher harmonics (multiples) of that frequency as radio-frequency (RF) noise through the air, where any ungrounded wire can act as an antenna to pick it up. In Europe, the AC power frequency is 50 Hz, but the same principles apply. Noise reduction is ideally accomplished by good recording techniques, low impedance attachment of electrodes to the scalp, etc., but particularly in the intensive care unit setting where the ratio of electrical devices to patient may be 10:1, it becomes extremely difficult to eliminate 60-cycle hum. A “notch filter” combines a low-pass (high-frequency) filter that rolls off at just below 60 Hz with a high-pass (low-frequency) filter that rolls off just above 60 Hz. When executed well, this combination filters out the 60-Hz frequency band while leaving both lower and higher frequencies largely intact. Modern digital recording techniques allow such processing to be performed in the digital domain, where the 60-Hz noise can be more precisely targeted with little effect on lower and higher frequencies, but the same caveats apply as for other filter applications.

For both low-frequency and high-frequency filters, a “cutoff frequency” describes the point below or above which frequencies are significantly attenuated. This frequency is determined by the equation:

f_c = 1/(2π × R × C) = 1/(2π × τ)

This is the frequency at which the value of the resistor is equal to the value of the capacitive inductance, and at this point, the voltage attenuation will be (1/√2) or 70.7% of the input voltage. For a low-frequency filter, frequencies below the cutoff frequency are blocked, while for high-frequency filters, the frequencies faster than f_c are reduced. Note that “cutoff” does not mean that all frequencies above or below the value of f_c are eliminated. Rather, there is progressive “roll-off” of amplitude the farther the input frequency is from the “pass band” of frequencies allowed by the filter (we will return to these concepts below). The f_c frequency is really an inflection point or “corner” rather than “cutoff” frequency. The steepness of roll-off has to do with the design of the filter circuit, and additional resistive and capacitive elements (“poles”) can increase the steepness of the amplitude reduction at frequencies beyond the cutoff. Different filter designs can be used depending on the signal processing requirements. Filters commonly carry the names of their designers, such as Bessel, Butterworth, and Chebyshev, each with trade-offs to improve some aspects of performance while damaging others. For example, Chebyshev filters have very steep roll-off of undesired frequencies but cause overshoot “ringing” with a sudden voltage step that adversely affects the response to a rapid shift in voltage. Bessel filters are well-behaved with minimal damage to the phase and wave shape of passed frequencies, which is highly desirable for EEG interpretation.

The reduction in signal amplitude resulting from applying a filter is often considered in terms of its effect on an amplifier’s output, so we will now make a small diversion to talk about EEG amplifiers.

An amplifier does what its name implies—it increases the size of a voltage signal. Amplifiers work by measuring a current or voltage signal and creating a larger version that can drive a recording or display device or be digitized for storage, playback, and analysis later. Fortunately, it is not necessary to be an electrical engineer to interpret

EEG signals; we can treat the amplifier as a “black box” that magnifies cortical potentials so we can see them. But it is important to know some basic principles. EEG signals are recorded using differential amplifiers that magnify the voltage difference between two input terminals (sometimes called grids, because in early tube amplifiers, the input was connected to the grid of the vacuum tube that regulated the passage of amplified current). One input is positive and the other negative, so that the negative input is subtracted from the positive and the resulting
signal is amplified. The amplifier designed for this purpose is known as a comparator and represented as an elongated triangle, with the positive and negative inputs on one side and the output coming out of the opposite vertex (Fig. 2.2). Differential amplifiers are useful for measuring biological signals for several reasons. By subtracting the signal detected at the second input from the signal at the first input and then increasing the size of the resulting signal, they detect and greatly amplify small differences between these signals. The degree of amplification over the original signal is known as the gain and is measured on a log scale in decibels (db) where