In reflecting on this you may wish to contemplate the difference between the input field and the output or responding units.
Can I suggest that you look at this thread first and see if it helps you with your question:
As I try to understand these things I look to the biology that is being emulated for enlightenment. In this case, as you read this keep in mind that the inhibitory interneurons are spread throughout the cortex and there is a balance between column activations and firing of these inhibitory units. A winning column will activate the inhibitory interneurons which will tend to suppress any additional activity in the nearby neighborhood. This is the basis for the sparsity enforcing inhibition in the HTM theory.
This is emulated (crudely) by the k-winner function in most HTM implementations.
In the biology, the Chandelier cells appear to be in the correct position to provide this regulatory function.
Another likely candidate for this inhibitory-regulatory function is the Basket cell.
HTM theory postulates that some of this inhibition action is local to a single mini-column (the core of temporal memory theory) and some acts on a larger scale encompassing the size of a macro-column.
I am still reading about inhibitory interneurons and it is hard to say that one of these types is the “only” source of sparsity or if it is some sort of collective activity. BTW: there are other types of inhibitory interneurons other than the two I listed above. Suffice it to say that some sort of inhibitory action is being performed and it results in the observed sparse activation of the mini-columns.