When I consider this part of HTM theory it helps me to go back to the biology that inspires the theory.
This part of the theory is modeling the apical dendrite as it snakes through the jungle of other horizontal dendrites and rising projecting axons. It will travel about 500 µm “sideways” (actually, +/- 250 µm centered on the cell body) passing though rising/projecting axons with a mean spacing of about 30 µm.
I say rising/projecting axons because some are from neighboring cells and some are projections from distant maps. To the dendrite they look like the same thing. In the picture below we can see the ascending corticocortical axon from a distant map in red. These axons do make contact at various points in the six layers but we are looking at just L1 for this discussion.
The rising axons individually contact a small local part of the dendrite, in the picture below look at the orange dot to form a mental picture of the relative scale of the cells receptive field in relation to the rising/projecting axons field of possible influence.
Please keep in mind that this picture portrays a vastly reduced number of dendrite arbors - these cells also are also spaced on 30 µm centers but if I put them all in the diagram would be to dense to see anything.
Our dendrite will not necessary have synaptic connections with all of the axons it passes - only the ones that it has learned that it has some meaningful relationship with. You know - the whole fire together/wire together thing ?
So the rectangular field portrayed in the model is actually a circular field emanating from the apex of the pyramidal cell’s soma in the biology. The rectangular shape is a simplification to make the programming easier.
The pool of potential connections corresponds to the axons closest to the path of the dendrite as it snakes through the L1 jungle. There are a much larger pool of rising axons that the dendrite is too far from to form a connection. These rising axons are within the 500 µm diameter circle centered on the cell body but no part of the cells dendrite pass close enough to form a synapse.
Other cells may have a dendrite that does make contact with this rising axon.
So - with this background lets get back to the original questions:
Is it correct to assume that the state of the column is the result of only a limited number of the potential connections in the receptive field (so, in the diagram, only 6 out of the 20 potential connections)?
Yes. These correspond to the axons that the dendrite has formed a synapse with.
Is it also correct that the state of the column depends on both the positive (present, black) state of a number of potential connection and the negative (absent, white) state of a number of other connections?
Not exactly. Yes on the positive (present, black) part and no on the negative (absent, white) state. These negative location may be sampled by some other column but this particular column can’t see as these are they are just too far away from it’s dendrite.
And does this emulate how real synapses and neurons behave?
I hope that this post helps explain that HTM is a simplified version of the neuron/synapse but it does capture the essential features.
More on where the figures I am using come from: