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According to this source, a neuron can connect to up to 10,000 other neurons. Does this imply that neurons are clustered, that is, not evenly spaced? Argument:

Say axons are length $L$, and dendrites are length $D$. Say a neuron $N$ has 10,000 outgoing connections. Then there must be 10,000 neurons within distance $D$ of $N$'s axon terminal. Then each of those 10,000 neurons are within distance $2D$ of each other by the triangle inequality. Assuming $2D<L$, then all of those 10,000 neurons are less than an axon's distance from each other. But for this to be true, the 10,000 neurons must be closer to each other than they are to $N$. Since this argument holds for any neuron, we get the result that neurons must form clusters of around $10,000$ neurons, or their dendrites must be longer than half their axon length.

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  • $\begingroup$ Why should spacing and number of connections be inversely proportional? Different regions of brain have different types of organisation. Perhaps you should read about it and narrow down your question. In some lower organisms there is no centralized nervous system but a "neural net". You can read about that too. $\endgroup$
    – WYSIWYG
    Commented Oct 21, 2016 at 17:26
  • $\begingroup$ Say each neuron had exactly one connection. Then neurons could be evenly spaced. However here I argue that if neurons have many connections, then they must be clustered. $\endgroup$
    – Jamie
    Commented Oct 21, 2016 at 17:28
  • $\begingroup$ It is not really necessary. Check neural nets in hydra. $\endgroup$
    – WYSIWYG
    Commented Oct 21, 2016 at 17:30
  • $\begingroup$ Hydras have a decentralized nerve network, true, but here I'm referring to the local neural topology. I argue that if a hydra's neurons had thousands of connections, then they would still be locally clustered, even if there is no "central authority" like a brain. $\endgroup$
    – Jamie
    Commented Oct 21, 2016 at 17:36
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    $\begingroup$ Your assumptions about the length of axons and dentrites are incorrect. They can vary tremendously from one neuron to another, and in the same neuron. There could be very short connection lengths between two adjacent neurons, and much longer ones between the same neuron and one on the other side of the brain. $\endgroup$
    – MattDMo
    Commented Oct 21, 2016 at 17:39

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In addition to the comments pointing out other inaccuracies, I think the biggest, most obvious misconception that completely invalidates the whole premise is that axons most certainly do NOT have a single terminal; axons branch and make thousands of individual synapses along their length and at the ends of any branches. For excitatory projection neurons in the cortex, for example, the axons can stretch for many millimeters (or centimeters in larger brains), but also typically branch near their origin to contact the dendrites of neighboring cells.

The number of synapses and number of neurons contacted varies greatly across brain areas, species, cell types, etc.

You might be interested in "small world" organization, such as at link, which I suppose could be considered a form of clustering, but it has absolutely nothing to do with the mathematical distance explanation you are proposing.

The only correct equation you can write here is that the spacing S between neuron A connected to neuron B, must be S<=D+L, where L is the length of A's longest axon, and D is the length of B's longest dendrite; for many pairs, S << D+L. These equations will not lead you directly to any valid conclusions of clustering.

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