I’m trying to fill some gaps in, and confirm my understanding of HTM’s details, so I have a few very basic questions about the spatial pooler:

The purpose of the spatial pooler is to take in a binary vector with a sparse number of ON bits, and then determine which of the columns of cells “win” (will there always be a set percentage of winning columns, or is the sparsity of the winning columns just a result of the input vector’s sparsity?). The way the winning columns are selected is by computing each column’s overlap score with the input vector, which is done by looking at the bits a column has connected synapses with and taking the sum of all the ON bits; this sum is the column’s overlap score. The reason the spatial pooler does this is to try to make the same (or similar) columns win when similar inputs are fed in. It sort of filters out noise, and forms an invariant representation of variations of a single pattern. Is this a fair description of the spatial pooler’s purpose?

Does the spatial pooler have any role in finding patterns between multiple input fields? For example, if I have
field1
andfield2
, and I encode them each into binary vectors, then concatenate these vectors and feed them into a spatial pooler, does it find spatial patterns between them? E.g. does it learn that whenfield1
's value spikes, thatfield2
's value dips? My current understanding says, yes, it does; but I’m having some difficulty putting into words exactly why that is the case… 
The columns in an HTM layer form synapses with the input vector, but how many, and where at? Is the number of synapses a column can form configured by a parameter, or is there some accepted standard? Can a column form synapses with any bit of the input vector, or is this range restricted? Will there ever be a bit in the input vector that has not formed a synapse with one of the SP’s columns? Also, can two columns form a synapse with the same bit on the input vector?

Once a column determines which bits it will set up synapses with, is it possible for that to change? I know that the permenance of a synapse is variable, and changes constantly, but I’m talking about a column moving a synapse to an entirely different bit on the input vector. My current understanding says, no, you can’t change the position of a synapse during an HTM’s lifetime. The positions of the synapses need to be determined upon the network’s creation and remain the same forever thereafter.

How does one know how many columns a spatial pooler needs to have? Is there a minimum number based on the length of the input vector? Suppose I have an input vector 5 bits long, and a single column in my spatial pooler (I know, completely unrealistic, but it’s just for illustrative purposes…). If I let this single column form 5 synapses, then each bit on the input vector would be covered with a synapse. Or consider a less extreme example, say I have an input vector of 2048 bits, and I let each column in my SP form 2 synapses; then would 1024 columns suffice? Would it even serve a purpose to have any more columns?

This might be a more subjective question? But, what is the most logical way to visualize a spatial pooler? Wouldn’t a 2D grid of cells make the most sense? I know that since HTMs are all currently simulated in computers (there aren’t any hardware implementations of them right now) and there is no actual structure like this in the computer’s memory, but is this the way that one ought to form a mental image of them? That appears to make the most sense to me, because the input is a 1D binary vector, and the SP columns are positioned on top of the vector, and extend synapses toward bits of the vector within a certain range of themselves (or is there no such concept in current HTM implementations of the distance between a column and a particular index of the input vector?). I know nobody is going to make me think of it in a particular fashion  I’m mostly wondering if there is some fundamental concept I might be missing by picturing things this way?
Of course, if this was the best way to picture it, then how do images like these make any sense?:
I understand that in the brain, columns are arranged in three dimensions, but SDRs in the brain aren’t 1D. Hopefully this question makes some sense… I’m having a bit of difficulty pinpointing precisely what I’m trying to ask.
I do believe that is it for now . I think I have a reasonable general, high level understanding of the theory behind the spatial pooler, but there’s just several details I am not quite sure about.