Throughout this series we’ve started from the same example: I’m 1.85 metres tall. That measurement has a nice property. Measure me with a one-metre ruler or with a millimetre ruler and you get roughly the same answer, because my height is a straight line.
But is everything like that?
The Coastline Paradox
In 1967, the mathematician Benoit Mandelbrot asked a simple-looking question in a famous paper: how long is the coast of Britain?
The answer depends on your ruler. Measure a map with a 100-kilometre ruler and you skip over bays and headlands; you get one length. Use a 10-kilometre ruler and you follow the bays too, so the length grows. Walk the beach with a one-metre ruler and you go around every rock; it grows again. The smaller the ruler, the longer the coast.
Mandelbrot’s idea was that “length” is the wrong question for a shape like this. The right question is how fast the length grows as the ruler shrinks. That rate gives a number describing how rough the shape is: its fractal dimension.
A Dimension Between One And Two
A straight line has dimension 1; a sheet of paper has dimension 2. A rough curve like a coastline sits somewhere in between: not quite a line, not quite a surface. The more twisted and jagged it is, the closer its dimension gets to 2. According to the measurements Mandelbrot cites, the west coast of Britain has a dimension of about 1.25.
This roughness has another interesting property: it often looks similar at every scale. Look at a fern leaf or a romanesco broccoli. A small piece looks like a shrunken copy of the whole. This is called self-similarity.
EEG Is A Coastline Too
Look at an EEG trace on a screen: not a straight line, but a curve that keeps rising and falling, full of twists. Zoom in and new wiggles appear. In other words, EEG looks a lot like a coastline.
One of the most common ways to apply fractal dimension to EEG is the Higuchi method. The idea comes straight from the coastline: we measure the “length” of the signal with different step sizes, that is, different rulers, and look at how the length changes with the step size. The result is a number between 1 and 2. A smooth, slowly oscillating signal is closer to 1; an irregular, fast-changing one is closer to 2.
Why Does It Matter?
Most spectral and temporal features assume the signal comes from a linear system. The brain isn’t a linear system. Nonlinear features such as fractal dimension and entropy capture the signal’s complexity from another angle. They’re used to track changes in complexity in topics such as mental workload, sleep stages, or brain dynamics that change with age.
In our lab, too, we compared linear and nonlinear features in one of our studies on detecting smell stimuli from EEG.
So there are three ways to describe a signal: which rhythms it’s made of (spectral), how it behaves over time (temporal), and how rough it is (fractal). We usually get the best results when we use all three together.
References
- Mandelbrot, B. (1967). How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156(3775), 636–638. DOI: 10.1126/science.156.3775.636
- Higuchi, T. (1988). Approach to an irregular time series on the basis of the fractal theory. Physica D: Nonlinear Phenomena, 31(2), 277–283. DOI: 10.1016/0167-2789(88)90081-4
- Akbuğday, B., Akan, A., Pehlivan, S., Sadıkzade, R. (2022). An Assessment of Linear and Nonlinear Features for Detecting Olfactory Stimulus in EEG. IEEE TIPTEKNO. DOI: 10.1109/tiptekno56568.2022.9960190