Don’t worry about it.
Your question is very “on topic” for the spatial pooler paper and the fact that you have questions suggests that others may have the same questions. The “sparsing via the inhibition” is not explained very well in the original paper and in the 19 times that the word is mentioned the text does not go very far into explaining how it works in the biology - only that somehow it does. On the plus side - the reference section does point to some good papers.
In the SP paper the inhibition is mentioned in figure 1: "A local inhibition mechanism ensures that a small fraction of the SP mini-columns that receive most of the inputs are active within the local inhibition radius (shaded blue circle).
This is the hand-off from local inhibition to the k-winner-take-all approximation: " The SP models local inhibition among neighboring mini-columns. This inhibition implements a k -winners-take-all computation (Majani et al., 1988; Makhzani and Frey, 2015). At any time, only a small fraction of the mini-columns with the most active inputs become active."
From the forum:
From the linked SP paper:
In the SP paper reference section linked papers discuss the inhibitory inter-neurons in depth.
Porter, J. T., Johnson, C. K., and Agmon, A. (2001). Diverse types of interneurons generate thalamus-evoked feedforward inhibition in the mouse barrel cortex. J. Neurosci. 21, 2699–2710.
PubMed Abstract | Google Scholar
Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., and Wu, C. (2004). Interneurons of the neocortical inhibitory system. Nat. Rev. Neurosci. 5, 793–807. doi: 10.1038/nrn1519
PubMed Abstract | CrossRef Full Text | Google Scholar
Swadlow, H. A. (2002). Thalamocortical control of feed-forward inhibition in awake somatosensory “barrel” cortex. Philos. Trans. R. Soc. Lond. B Biol. Sci. 357, 1717–1727. doi: 10.1098/rstb.2002.1156
PubMed Abstract | CrossRef Full Text
The inter-neuron function is approximated by the k-winner-take-all function:
Majani, E., Erlanson, R., and Abu-Mostafa, Y. (1988). “On the K-winners-take-all network,” in Proceedings of the First International Conference on Neural Information Processing Systems (Lake Tahoe, NV), 634–642.
Google Scholar
http://papers.nips.cc/paper/157-on-the-k-winners-take-all-network.pdf