Does the standard HTM model have a general solution to the combination of individual columns into a larger representation?
I do understand the general SDR theory and know that it only extends as far as an individual dendrite can reach. For most cells. this is about +/- 8 columns. Most objects present to a much larger field of sensors so how are these individual sensations combined into a unitary object?
A solution? Today? No. But I think the current HTM model is moving in the right direction to solve the binding problem. I think we need to re-think the classic hierarchy to make the HTM model work as an object recognition and learning engine associated with motor activity. If we rethink it correctly, we can start to apply it in a way so multiple hierarchies can be combined at high levels to share learned representations.
Cortical columns higher in the hierarchy should be larger in size. I’m not really sure (I’m not really sure anyone is sure) exactly how big they can be. But the size difference between region 1 and region 6 could be considerable.
But changing the column size doesn’t change the physical reach of an individual dendrite, though, right? I think you need a mechanism like hex grids for allowing an activation to extend beyond the reach of a single dendrite.
But at the higher regions, you might only need to communicate to immediate neighbor columns to efficiently transfer concepts. Someone could work out how thick columns can be before they can only communicate to immediate neighbors. That might be a logical maximum size?
Maybe the long range axons in L1 extend this further since most cells get input from L1.
I’ve read a couple sentences that say similar to orientation columns in V1, high in the hierarchy, similar features are represented in the same column. That means high in the hierarchy, it might not make sense to think about the receptive fields on the sensor, since the receptive fields are on a map of features.
From what I have read, the axons preserve map topology.
This is exactly what grid-forming cells do.
I know your are busy moving and all but you really should look at what my hex-grid thing really is at some point.
I think you’re right, and perhaps we have identified a major overlap here I was missing until this moment. It is hard to say because we have come at this problem from an entirely different angle. These ideas seem related related, but I think we are trying to solve the binding problem through another means. Perhaps we are uncovering the same structures inherent in the system from different avenues, which would honestly be amazing and wonderful. 
I just talked to Jeff about this, and he was skeptical about it. He says he would not say anything about the size of cortical columns, they are not real physical structures in that sense. He says that higher in the hierarchy, a column’s lateral connections spread farther. It is not that they are physically larger.
Are they there books or papers to support this assertion?
(Larger dendrite span in higher levels of a hierarchy)
Will try to get back to you on this, but they are long range layer 3 connections.
I did some quick checking and found that there does seem to be a small increase in dendrite span as you go from V1 to V2 to V3, about 1.5:1 on each step.
This is is not enough to do long range binding.
“Pyramidal neurons in V1 had a mean basal dendritic field area of 1.84 × 104 μm2 (SEM = 2.04 = 2.04 × 103 μm2;n = 21). Layer III pyramidal cells in V2 had a mean basal dendritic field 1.26 times larger (mean = 2.32 × 104 ± 1.78 × 103n = 42) than that of V1 neurons. The mean dendritic field area of layer III pyramidal cells in DL (n = 76) was 1.5 times larger than that in V1 (mean = 2.75 × 104 ± 1.59 × 103 μm2) and that in FST (n = 50) was 2.3 times larger (mean = 4.26 × 104 ± 2.79 × 103 μm2). Our results show that there is a correlation between tangential dendritic field area of basal dendrites of layer III pyramidal neurons and modality of visual processing. The increase in basal dendritic field area of layer III pyramidal cells may allow more extensive sampling of inputs as required by higher-order processing of visual information.”
I don’t know if it’s sufficient, but let’s say binding works by lateral connections allowing columns to vote. Then if a column votes with its neighbor (by which I mean it narrows down the possible objects represented by the union of SDRs in that neighbor), then that neighbor can do the same to its neighbors, extending the reach.
There is considerable overlap of dendrite fields between adjacent columns,
This could be part of the voting of these adjoining columns: “do you see what I see?”
I assume that you place this voting in the proximal dendrites with rising axonal projections.
Yes, I assume it’s proximal, although since it works by turning cells off (maybe), it might be proximal inhibitory synapses. Probably excitatory synapses which cause the voted-for cells to indirectly inhibit other cells in the same column more.
Edit: on second thought, I don’t know because distal connections could elevate firing rates of already active cells. @rhyolight how is voting between columns thought to work?
Will add I heard lateral connections can be up to 6-8mm long in primary visual cortex, reason exists to believe they could potentially be even longer in higher areas. So information from remote columns in a higher area could potentially converge.
Also, if the receptive fields in V1 are extended, then cortical columns in V2 with be extending reach on already extended reach, perhaps. For example, let’s say at each level of the hierarchy each column can only talk to its adjacent neighbor. Indirectly, at level 3, each column can talk to columns up to 3 away.
(Thanks to @subutai for helping me gather the info for this post.)
The long-range lateral connections originating in Layer 2/3 are from axons (not dendrites as has been mentioned). Note that the span of axons does not limit the communication range. @MaxASchwarzer showed last summer that columns can do L2 pooling even if they are indirectly connected – it just takes slightly longer.
These two papers study cortical columns in IT, which is roughly the 4th region up in the visual cortical hierarchy. I have not read them in detail, but they might contain some interesting details.
this neighbour-to-neighbour spreading is part of what Calvin grids are about 
Definitely starting to see some convergence.