On the Quantification of Brainwaves
A theoretical perspective on the Welch’s method for processing EEG data
When I started working with EEG data, I was overwhelmed with the breadth of methods available to analyze the measured signals. I quickly stumbled upon the so-called Welch’s method, named after Peter D. Welch, an American mathematician and statistician who worked on signal noise reduction and thereby developed this method.
The idea behind using this is quite subtle: you feed in the data of measured electric potential over time, and you receive back information on the power at different frequencies. While this sounds straightforward, the theory behind it is not as simple.
Measuring electric potential with EEG
EEG, or electroencephalography, is a method to measure the electric potential on our scalps. This measure thereby is an indicator of the brain activity below these electrodes. Thereby, an electric potential signal is recorded at each electrode placed on the participant’s head. This could look like this:
As you can see, at each of the four electrodes on the scalp, we measure an electric potential (in microvolt or µV) over time. These signals that we record do not say anything about the underlying brainwaves yet. To understand what they represent, let’s zoom in in our brain:
The signals that we measure at each electrode result from neural oscillations. While oscillations within a neural cell come from membrane potentials or action potentials that are important in neural firing, these oscillations are too weak to be measurable by EEG. In contrast, neural ensembles, so an assembly of many neural cells in synchrony, produce stronger oscillations. And these we can measure with our instruments.
So let’s imagine each of these neural cells in the figure above resembles a neural ensemble, oscillating in a certain intensity at a specific frequency. Unfortunately, we are not able to measure these different oscillations individually. Rather we measure a combination of these oscillations at each electrode of our EEG apparatus. This combination represents the sum of the individual wave-amplitudes over time:
So, in conclusion, what we measure at each electrode of our EEG apparatus corresponds to the sum of individual oscillations of different frequencies from neural ensembles from all around our brain, which are strong enough to reach our electrode on the scalp.
Electric potential and individual frequencies
Frequencies, amplitudes, and signal power
Let’s come back to our four neural ensembles. Each of these ensembles oscillates with a specific frequency. We can determine this frequency by looking at how much time passes until the oscillation does one full circle.
Additionally, we need to consider the amplitude of the corresponding wave. The amplitude thereby indicates how strong this oscillation is. Therefore, what we get from each neural ensemble is a value for its frequency and one for the signal power, the amplitude.
Plotting the amplitude values for each neural ensemble together over the different oscillation frequencies is exactly what we need to quantify our brainwaves. Considering our four neural ensembles, the corresponding plot would look like this:
As we measure the frequency, the unit for the x-axis is Hertz (Hz). Similarly, we consider the amplitudes for the signal power, which are based on electrical charge or discharge and thus in microvolts (µV) at a specific frequency (Hz).
In reality, it is not as simple with only four neural ensembles. Also, we don’t know how the different ensembles oscillate. Therefore, we need a method to decompose our measured electric potential over time at a specific electrode into the various frequencies and amplitudes.
Signal decomposition and the the Fourier transform
We have seen that the measured electric potential corresponds to many different frequencies with different amplitudes. The idea behind the Fourier transform is to decompose the measured electric potential into individual frequencies. Thereby, we obtain for each frequency value a measure of the signal power at that frequency. The formula of this Fourier transform looks like this:
While the mathematical formula of this Fourier transform looks pretty scary for non-mathematicians, the few things we need to know to understand this formula are the following:
- We integrate our wave function over time since we measure the electric potential in an EEG experiment over a specific period.
- We transform our normal wave frequency to an angular frequency (denoted with 2 π f)
- We obtain a new function that specifies the signal power for each frequency.
Luckily, there are many already existing plugins and functions to conduct this Fourier transform, but understanding how we obtain this signal power — frequency plot is, in my opinion, essential if you do this kind of analysis.
From the Fourier transform to the Welch’s method
Unfortunately, the results we obtain when applying the Fourier transform on EEG data are not as easy to interpret as the one we saw based on our four neural ensembles. Instead, the signal power — frequency plot looks like this:
A major reason for that is temporal dynamics in our brain. When you measure the electric potential on the scalp for a distinct period, it can easily occur that you measure such dynamics that happen once or twice but do not resemble the overall pattern of your measurement.
While the Fourier transform struggles with temporal dynamics and overemphasizes such outliers, the Welch’s method allows smoothening the signal power — frequency function. Instead of taking the whole time span to calculate the signal power at the different frequencies as done in the Fourier transform, the method, suggested by Peter D. Welch, subdivides the whole measuring period into smaller windows on which the Fourier transform is applied. For each of these windows, we then obtain a signal power — frequency function. Averaging then these various functions gives us a more elaborate estimate of the actual brain dynamics:
The windowing of the measurement period can also be further refined by, for example, choosing overlapping windows. However, for the moment, it is sufficient to know that using the Welch’s method, the time span is subdivided into several windows, and the individual results are finally averaged.
From the signal power — frequency plot to the quantification of brainwaves
When people speak about brainwaves, we often hear about delta-waves or theta-waves and their particular functions in the brain. A brainwave is thereby nothing more than a defined frequency range. For example, delta-waves range from 0.5 to 4 Hertz, theta-waves from 4 to 8 Hz, and so on.
If we want to quantify specific brainwaves, we can use the signal power — frequency plot that we generated before. By averaging the signal over the particular frequency range of interest, we get an estimate for this brainwave.
For example, let’s assume we want to measure the signal power of the theta wave at one specific electrode. In this example, we only have two data points, one at 4 Hz and one at 8 Hz. The corresponding signal power values we measured correspond to 20 and 10 µV / Hz, respectively. Averaging these two values, we obtain 15 µV / Hz. This final value ultimately corresponds to a quantification of the theta brainwave at the measurement point.
By repeating this procedure for all electrodes of our EEG apparatus and all brainwave frequencies of interest, we eventually obtain quantifications of the different brainwaves at the specific locations of the electrodes.
Conclusion
The Welch’s method allows to decompose the measured signal of the electric potential at each electrode into its individual frequencies to determine the particular signal power values to ultimately enable us to quantify different brainwaves.
