Brain waves are the time-varying electrical signals measured between two points on the head, for example between the center of the forehead and some other point on the scalp. The resulting oscillations are the summation of voltages due to innumerable sources within the brain, primarily neurons in the cortex. Different populations of neurons fire in synchrony with their neighbors at frequencies ranging from one cycle per second (1 Hertz, written 1 Hz) to over a hundred cycles per second (100 Hz). At any given point, a cacophony of signals can be recorded and at first blush, they make no sense at all. For example, here is the raw EEG signal from location TP10 (near my right ear) during a 7-minute recording as I drank a cup of tea. This particular recording was 21 minutes long, with four 5-minute segments under different conditions. The plot above shows only the first segment, “drinking tea – 1”.

To begin to understand this complex signal, we can decompose it into different frequency bands. What do we mean by that?

Suppose we write a small computer program to draw a sine wave. We plot a sinusoidal oscillation which we shall call D. It has a frequency of 2.5 Hz (we shall pick frequencies that fall within the commonly defined EEG frequency bands):

Then we plot another curve, calling it T. We choose to use a frequency of 5.6 Hz:

Now if we add together these two waves, D and T point by point over the given time interval, we obtain the following curve:

In the same way, if we plot three other waves with frequencies 12 Hz, 19 Hz and 39 Hz, calling them A, B and G, we have:

Now consider what happens when we add together all five sine waves, D, T, A, B and G. Here is the result:

By adding together five simple sine waves of different frequencies we quickly get a complex waveform.

It’s not too much of a stretch to imagine going in the other direction: start with the complex form and decompose it into sinusoidal waves of different frequencies. Essentially, that is what’s being done to extract delta, theta, alpha, beta and gamma waves from raw EEG signals–break them down into components having different frequencies. For the purpose of analyzing EEG recordings, five frequency ranges are defined and given Greek letter names:

EEG Frequency Bands
frequency range name
0.5 – 3 Hz delta
4 – 7 Hz theta
8 – 15 Hz alpha
16 – 31 Hz beta
32 – 100 Hz gamma

The amount that each frequency band contributes to the overall signal is computed using what is known as the Fast Fourier Transform (FFT). The band power in each frequency range can then be expressed as a fraction of the total band power. An FFT algorithm is built into the hardware of the Muse headband device and during a recording, the absolute and relative band power values for five frequency bands at four sensors are streamed wirelessly to a smartphone or computer.

For example, from the 7-minute tea-drinking recording above, the graphs of relative band power for delta, theta, alpha, beta and gamma bands were obtained and are shown below. In each graph, a median value for the indicated time interval is displayed as a horizontal line.

The five graphs above were obtained from only one sensor, TP10 near the right ear. We can summarize the median values for all four sensors using a radar graph as shown below: Radar chart for median relative power at four sensor locations: TP9, FP1, FP2 and TP10

We see that low frequency (0.5 – 3 Hz) delta waves predominate in the right front temple area (rf) and medium frequency (8 – 15 Hz) alpha waves are strongest in the left back (lb) region near the left ear. We now have tools for examining brain waves in more detail.

### One Response to “Understanding brain waves”

1. The simplicity and clarity of your explanation for a layman like myself is an indication that you have a truly well functioning brain; this was a precise answer to a precise google search question (“Are delta brain waves sinusoidal?”).

Thank you

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