I have been conversing with @gmirey for some time on this general topic. We endeavored to get a sense of the scale of the relative dimensional quantities of the temporal layer, inputs and cell/column spacing, number and range of the connections.
You know - the basic facts of making a realistic model. Some of what was discussed was what is known about reach and density of the dendritic arbor in the “input” layer.
We established the key factors would be: what is the spacing of the ascending corticocortical axonal bundles, what is the spacing of the columns/cell bodies, what is the dendritic density, what is the synapse density.
I did a little digging and came up with this:
We are trying to describe a complex 3-dimensional structure composed of messy biological bits.
You indicated that you will be back for layer 2/3 at some future time so let’s focus on lower layers.
Looking at the input corticocortical connections - how apart are they? As indicated earlier - we will skip the massive thalamocortical arbors as I don’t think that you will have to model that as an information path.
These massive thalamocortical arbors are a shotgun to insure that layer IV is induced into resonance with the thalamus
We have to account for cell body spacing, ascending axon bundle spacing, dendrite reach & dendrite arbor spatial density. Fortunately, I am finding papers that offer some figures on all of these items.
Note the massive inter-layer local projections from deep pyramidal axons. Keep in mind that they primarily project on inhibitory inter-neurons, suppressing the losers in local recognition contests.
Numerous papers seem to agree that the spread of the layer 2/3 and lower layers dendrites (radius) is about 300 micrometers (0.3 mm) that gives a diameter of about 0.6 mm at the extreme tips of the dendrites. This is the long tails (in the truest sense of the word) and the average length is somewhat shorter. Also - the dendrites don’t shoot out in a straight path so the 0.5 mm shown in this diagram is a better maximum figure.
See this for more details:
The spatial density of the branching dendrites tends to fill space in a constant density.
so how many synapses are there for a unit of space?
It varies but let’s say 1 per cubic micrometer.
While we are at it - what is the density of the microcolumns and cell bodies?
| Tissue Property or Measurement | Value |
|---|---|
| Y, average interneuron distance. | 20.0 μm (1) (estimated) |
| P, average intercolumn distance. | 26.1 μm (1) |
| ρ, slide neuronal density | 0.0013 neurons/μm2 (1) |
| l | 341 μm (1) |
| s, thickness of the thin slice | 30 μm (1) |
| radius or neurons (average) | 5 μm |
| % interneurons | 20% (2) |
| Model Parameter | Value |
| dn, interneuron distance. | 23.1 μm |
| dc, intercolumn distance. | 29 μm |
| θ | Uniform random [0, 2π] |
| φ | Uniform random, [0, π/3] |
| % omitted neurons | 40% |
| δdn | Gaussian distribution, σ=4.7μm |
| δxn, δzn | Uniform random, [−6 μm, 6 μm] |
| δxc, δzc | Uniform random, [−6 μm, 6μm] |
| N, number of images for average. | 500 |
How handy is that?
The microcolumns are on ~ 26-30 micrometer spacing.
Now we have to match that up with the ascending axonal bundle spacing that pokes through the dendrite arbors. You are looking at area 17 and the Hubel & Weisel paper has a lot to say about what goes on there.
so does this paper:
Organization of pyramidal neurons in area 17 of monkey visual cortex Alan Peters Claire Sethares
Sadly - behind a paywall.
The abstract is very helpful:
Abstract
In sections of area 17 of monkey visual cortex treated with an antibody to MAP2 the disposition of the cell bodies and dendrites of the neurons is readily visible. In such preparations, it is evident that the apical dendrites of the pyramidal cells of layer VI form fascicles that pass into layer IV, where most of them gradually taper and form their terminal tufts. In contrast, the apical dendrites of the smaller layer V pyramidal cells come together in a more regular fashion. They form clusters that pass through layer IV and into layer II/III where the apical dendrites of many of the pyramidal cells in that layer add to the clusters. In horizontal sections taken through the middle of layer IV, these clusters of apical dendrites are found to have an average center‐to‐center spacing of about 30 μm, and it is proposed that each cluster of apical dendrites represents the axis of a module of pyramidal cells that has a diameter of about 30 μm and contains about 142 neurons.
Is this typical for all areas of the cortex? Unfortunately, this useful paper is also behind a paywall.
The Organization of Pyramidal Cells in Area 18 of the Rhesus Monkey - Alan Peters, J. Manuel Cifuentes and Claire Sethares
Maybe you can hunt this down on your own. It has this handy histogram:
So - 20 to 30 micrometer spacing on the axonal bundles too. I suppose that makes sense that it matches up with the microcolumns. That 30-micrometer spacing seems to turn up a lot in these papers.
Like you I have to see pictures to fix this in my mind. 30 um out of maybe 500 um is a little less than 10 percent of the dendrite arbor field. I think it looks something like this:
Yes - before you jump on me - the dendrite arbors should be denser but the picture is too complicated to make out then.
@gmirey responded with this excellent paper:
https://www.ncbi.nlm.nih.gov/pubmed/15260960
and this one:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4370903/
Working with what we have so far:
Since we are visually oriented - a Graphic solution:
30 µm spacing of cell bodies
500 µm / 30 µm = 16.6
Rounding up = 17 = diameter of dendrite reach expressed in terms of cell bodies.
Set up a repeating field of cell bodies … inscribe a 500 µm circle
inspect and remove the cells x 4 corners in the local area…
Any blue cell reaches the center of the inscribed circle.
Since this is a repeating pattern any starting point has the same solution.
The overlap of dendritic reach is:
(17 x 17) – (4 x 13)
289 - 52 = 237 overlapping cells at any given point.
An inscribed area of 500 µm diameter circle gives an area of πr²
3.141 * (250 * 250) = 196350 µm² = input area for any cell body
Assuming that the layer is I µm thick for a first approximation (prolly much bigger!)
Going with the 1 synapse per µm³ and dividing by the 237 overlapping fields
196350 µm² / 237 = 828 synapses per cell per 1 µm of thickness.
It is likely that the dendrites are spaced out through the thickness of the layer. It is very likely that this density is shared with all the layers. (2/3, 4, 6)