![]() ![]() This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.ĭata Availability: Due to patient privacy concerns, data are available upon request by qualified researchers. Received: SeptemAccepted: DecemPublished: February 13, 2020Ĭopyright: © 2020 Varley et al. (2020) Fractal dimension of cortical functional connectivity networks & severity of disorders of consciousness. This supports the hypothesis that level of consciousness and system complexity are positively associated, and is consistent with previous EEG, MEG, and fMRI studies.Ĭitation: Varley TF, Craig M, Adapa R, Finoia P, Williams G, Allanson J, et al. ![]() more complex) networks being associated with higher levels of consciousness. ![]() These results suggest that cortical functional connectivity networks display fractal character and that this is associated with level of consciousness in a clinically relevant population, with higher fractal dimensions (i.e. We also found significant decreases in adjacency matrix fractal dimension and Higuchi temporal fractal dimension, which correlated with decreasing level of consciousness. We found significant decreases in the fractal dimension between healthy volunteers (n = 15), patients in a minimally conscious state (n = 10), and patients in a vegetative state (n = 8), regardless of the mechanism of injury. To test whether brain activity is fractal in time as well as space, we used the Higuchi temporal fractal dimension on BOLD time-series. We used a Compact Box Burning algorithm to compute the fractal dimension of cortical functional connectivity networks as well as computing the fractal dimension of the associated adjacency matrices using a 2D box-counting algorithm. To validate this, we tested several measures of fractal dimension on the brain activity from healthy volunteers and patients with disorders of consciousness of varying severity. We propose fractal shapes as a measure of proximity to this critical point, as fractal dimension encodes information about complexity beyond simple entropy or randomness, and fractal structures are known to emerge in systems nearing a critical point. Next, we'll determine the dimension of the Sierpinski Triangle.Recent evidence suggests that the quantity and quality of conscious experience may be a function of the complexity of activity in the brain and that consciousness emerges in a critical zone between low and high-entropy states. And this describes the Koch Curve - it's wigglier than a straight line, but it doesn't fill up a whole 2-Dimensional plane either.Īs we'll see soon, the more of a plane that a fractal covers the closer its dimensions is to 2. If a line is 1-Dimensional, and a plane is 2-Dimensional, thenĪ fractional dimension of 1.26 falls somewhere in between a line and a plane. We can make some sense out of the dimension, by comparing it to the simple, whole number dimensions. In fact, all fractals have dimensions that are fractions, not whole numbers. What could a fractional dimension mean?įractional dimensions are very useful for describing fractal shapes. We're used to dimensions that are whole numbers, 1,2 or 3. Use a calculator (or Google) to find the value for log(4): So according to the formula D = log(N) / log(r), we can say that D = log(4) / log(3) = 1.26 Order 4 has four times as many pieces as order 3, and each piece is 1/3 the scale. In this case, we can see that the number of pieces in the generator, N, is 4, and the magnification factor is 3, because each section of the generator is 1/3 of the unit length.This same relationship holdsīetween each of the orders of the curve. Remembering that D = log(N) / log(r), we can calculate the dimension D by seeing how the number of units, N, changes with the magnification factor, r. The third order curve follows the same pattern, and it has 64 tiny segments, each of which is 1/27 of the unit length, making a total length of 64/27.Īs the progression continues, the curve gets longer and longer, and eventually becomes infinitely long! Now, it is not very useful to know that a curve is infinitely long,Īnd this is where the concept of Fractal Dimension becomes very useful. Of the unit length, That means the total length of the second order curve is 16/9. The second order of the Koch Curve has had each of the 4 sections of the generator replaced with the same shape, so it has 16 small segments, and each segment is 1/9 The generator (order 1) is made of 4 sections, and each section is 1/3 of the length of the initiator (order 0), which has a unit length of 1. Let's look at the way the length of the curve changes as we iterate the fractal. As we learned in Chapter 2, geometric fractals can be made by starting with a simple generator pattern and replacing every section of the pattern with a smaller copy ![]()
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