Sure. Although maybe it should be made into another post.
I’ve grown some set of handful figures for having an intuition about the local connectivity requirements of any cortical patch simulation wanting to get realistic axonal and dendritic potentials.
Assuming 15 billion neurons total in cortex [1], and mean unfolded surface of 0.12m² for each hemisphere [2], this is 62500 cells per mm² on average, with some areas maybe twice as dense [3].
Using a regular square lattice, spaced 40µm, we have exactly 25x25 (625) tiles covering one square millimeter, and with that value, about 100 cells per position. This seems to fit nicely into the concept of a developmental minicolumn [3].
The extent of the basal dendritic arbors seems to be a sphere roughly 0.5mm in diameter, quite consistently across cell types, although some segments may try to extend further away if their local neighborhood is arbitrarily starved (cf. experiments with eye occlusions, etc.). Same figure for the 2D diameter of most apical tufts.
But we’re more concerned about the axonal matter, here : the extent of the widest axonal arbors seems to be a fair 3mm in diameter. I’m speaking here about localized lateral projections in same area… either intralayer, or, like, from L4 to L2/3.
Long-range connections among different areas in the hierarchy can of course blow the 1.5mm radius limit away… but long-range aren’t modeled the same way, anyway. And more importantly, if evaluating the diffusion of axonal arbors once they reached the distant area (eg. some arbors from Thalamus to V1), the 3mm diameter figure seems to appear again. So, once a long-range input bit gets sampled by some cells in an area, it can also be sampled among those kinds of potential radius.
So… In the simplest of the topological models I came up with, when you don’t take much care to precisely sort out those “max-range” arbors from more concentrated ones, and each of these axonal arbors is represented by a single point, you’re facing a horizontal sampling range of “potential” afferents, in a 3mm circle around (that’s simply reversing the viewpoint, from axon arbour ‘centers’ to synapses).
If you’re ready to stay in the abstract and clamp that 3mm figure somewhat, you have a blinding-fast computable, relative offset in a 64x64 minicolumns region around your cell, corresponding to a 2.56x2.56mm square.
That takes 12bits. You’ve 4 bits left on your 16b envelope to still chose a particular afferent to that minicolumn (=> 16 distinct afferents per minicolumn). This number is not biologically realistic if it should represent all afferent-centers per minicolumn to the whole sheet, though… and may not be sufficient to represent an HTM sim with deep minicolumns either. But if you’re ready to decompose that problem vertically, in any number of ‘thin-layers’ you need to accurately represent the overall sampling potential you require, then eveything is set:
Each cell or segment can easily be localized in 3D. Sampling 1 thin layer of 16 afferents per such cell or segment (or two of 8 each, allowing more diversity in potential ‘coincidence detection’), you’ve reached the (first) interesting mark of addressing all potential axons to a synapse, in a somewhat-biologically realistic manner, using 16 bits.
The only twist is that you’re sampling from 16 distinct 2D-maps per minicolumnar position, and that each input cell (such as, to take an HTM example, the t-1 from all cells from all minicolumns), may write to several of these, if overall input is spanning more than 16 per minicolumn. Typically 2 minimum for a 32 cells per minicolumn TM implementation (but probably more than this, distributed stochastically, to get a realistic spectrum for distal coincidence detectors).
[edit] As an additional bonus, that “input writing to one or several particular 2D-map(s)” pass may straightforwardly solve the issue of sorting “potentially connectable” from “out-of-potential” inputs for the proximal synapses of HTM during the Spatial Pooling phase. It gets more realistic and straightforward with finer topological schemes than this one, but it’s getting there.
[1] “There are between 14 and 16 billion neurons in the cerebral cortex”
[2] “When unfolded in the human, each hemispheric cortex has a total surface area of about 0.12 square metres”
[3] “Minicolumns comprise perhaps 80–120 neurons, except in the primate primary visual cortex (V1), where there are typically more than twice the number.”