Repeating a key point: all of this is describing what is happening in the L2/3 layer. At this level all we are working with is patterns and NO predictive memory. The L2/3 is also the layer that talks with other maps though the output axons of the L2/3 cell bodies. Likewise, the sensory inputs and rising axons are projected upwards through all layer terminating in the dense mat of L1.
The projecting axons lateral branches do make connections with inhibitory inter-neurons not shown here.
I am certain that there are connections up and down the minicolumn between the layers but I am not ready to state exactly how these connections work. These connections are key to describing the relationship between pattern recognition and temporal prediction. Working out these rules would be key to understanding the training rules of predictive memory.
Now on to the parts relative to hex-grid theory …
If you look at my entry on the Project : Full-layer V1 using HTM insights post, #34 I put figures on the sizes of the various elements for minicolumn spacing.
Each cell in the mini-column has a few dendrites and each one is at least single SDR and maybe a few.
Referring to this picture from that post - each blue circle is a single mini-column. This is 100 or so cell bodies. Each cell body has it’s own dendrites - say 10 or so for a nice round number.
It turns out that the rising projecting axons are also spaced on this 30 µm spacing so you can assume that each blue circle also has a rising axon bundle.
The large black circle is the reach of the dendrites (+/- 250 µm, or 500 µm total) for the minicolumn in the center of the diagram. That gives the dendrites in each cell in each microcolum access to about 200 or so rising axon clusters. This can be thought of as a “receptive field” of the blue blue mini-columns and rising axons within this black circle for this minicolumn. This is all repeated for the next minicolumn. This is the repeating structure for all minicolumns in the cortex.
In this paper Horizontal Synaptic Connections in Monkey Prefrontal Cortex the lateral connections from the L2/3 cells are given as an average about about 500 µm.
The black beam in this picture is the long distance lateral connection between the two minicolumns so that the two minicolumn “receptive fields” connected by this link of the the hex grid covers the space with very little overlap and very little missed space.
When you factor in the fact that there are several long distance lateral connections emanating in all directions from each cell in the minicolumn you can see that the possibilities to form space covering hex grids is very large. Since these connections from any given cell do not fall on strict angles or length the individual cells can form hex grids with different angle, spacing and phasing.
So recapping, ever minicolumn has 100 cells that each have 10 dendrites, each potentially forming at least one SDR, possibly more. (at least 1000 SDRs per minicolumn)
Each Dendrite, if it went in a straight line from from the cell body passes at least 7 axon projection clusters and with branching probably many more. This means that the area around each minicolum is densely sampled with about 1000 branching dendrites which should end up sampling every rising axon cluster within reach of the dendrites.
The lateral connection link these minicolumns so that if they are responding to a learned pattern, even though it is sampled relatively sparsely, all the space in the resonating hex grid pattern is being sampled and bound together into a single larger unique pattern.
Here is a drawing relating the idealized concept to the messy biological bits.
Also see this post for more on the formation of larger patterns: