Pointless Notes: Pretty Fractals in Your Brain!

Purkinje cells are pretty, neuron cells are pretty, the whole brain is pretty! The spinal cord, however, is ugly. Don't talk about that dumb slimy snake in your back.

All that glitters are Purkinje cells.

So let's measure the fractal dimensions!

It's really straightforward. Just take some pictures of them at various scales and use box-counting.

Fractal patterns for dendrites and axon terminals (1993), Sidiney G.Alves et al.

We find that these three-dimensional biostructures are fractal over at least one decade of length scales, with fractal dimension 1.68 ± 0.08 for the Purkinje cells and 1.28 ± 0.17 for the axon terminals. 

Neural Geometry: Towards a Fractal Model of Neurons (1989), by A.J. Pellionisz:

Fractal model of Purkinje cell vs real Purkinje cell

Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging (2003), Jing Z. Liu

We measured the fractal dimension of human cerebellum white matter skeleton at 2.57 ± 0.01. This indicates that human cerebellum is a highly fractal structure, consistent with conclusions drawn from studies of the cerebellum surface.

Here, "skeleton" just means "morphological skeleton", which roughly means a simplification of the image. Imagine a cake, now image the same cerebellum but with all water sucked out of it, leaving behind a dried corpse. That's like turning the cerebellum into a morphological skeleton.

Is the brain cortex a fractal? (2003), Valerij G. Kiselev

Results obtained in six subjects confirm the fractal nature of the human cerebral cortex down to a spatial scale of 3 mm. The obtained fractal dimension is D = 2.80 0.05, which is in reasonable agreement with previously reported results