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I wonder if the following question concerning the axons in the white matter does make sense.

It is common knowledge that white matter is "composed mainly of bundles of myelinated axons" resp. is "mainly made up of myelinated axons, also called tracts" (= "bundle of nerve fibers (axons)".

For the following to make sense the following assumptions are made:

  • All axons making up a bundle are of approximately same length $L$. (Of course not all axons have the same length!)

  • Axons run in parallel as long as possible.

  • Axons normally don't branch inside the white matter. (Or is this a totally unrealistic assumption?)

Assume that it is a sensible definition of a bundle to be the set of axons running densely packed inside a virtual tube of length $l = \alpha L$, $\alpha < 1$. By this definition sub-bundles of a bundle would also be bundles, so we are interested only in maximal bundles.

It's important to notice that the definition of bundles depends on the parameter $\alpha$: smaller $\alpha$ means the axons must run in parallel only for a shorter distance (compared to their common length $L$), larger $\alpha$ means the axons must run in parallel for a longer distance. Smaller $\alpha$ means less but larger bundles, larger $\alpha$ means more, but smaller bundles.

Note that each bundle (independent of $\alpha$) defines two specific cortical regions (the two regions of the source and target neurons of the axons). For the smallest bundles (largest $\alpha$) their diameters correspond well to the size of the source and target regions (see picture below).

See here different partitions into bundles with $\alpha \approx 0.5$ (grey, 1 bundle), $\alpha \approx 0.75$ (blue, 2 bundles) and $\alpha \approx 1$ (red, 4 bundles):

enter image description here

For the sake of simplicity, assume that - for a given $\alpha$ - bundles don't overlap, i.e. each axon is contained in only one bundle. (This might be an oversimplification.)

My question is:

Can a rough estimate be given, how many bundles with a given number $N$ of neurons are there in the white matter (for a given $\alpha$)? (Choose the organism you like or know of.)

I suppose the largest bundle to be found in the brain is the corpus callosum which - as a tube - has a rather short length (small $\alpha$). It contains (in the macaca nemestrina brain) about $10^8$ axons.

For the sake of simplicity, let's consider only powers of ten. Then for a small $\alpha$ which would allow the corpus callosum to be a bundle, we will get as a "null hypothesis" (with purely hypothetical numbers of bundles!):

enter image description here

But maybe the distribution is somehow bimodal, e.g. like this:

enter image description here

Is a reliable answer to this question out of reach today? Don't we know enough about the long-range structure of the white matter? Or did just nobody take the effort to make this statistics? Or can I find it somewhere?

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