Big-picture, the idea is that different modalities of sensory data (visual, olfactory, etc.) are processed by different minicolumns in the brain, i.e., different subnetworks, each outputting a different firing pattern. These firing patterns propagate across the surface area of the brain, competing with conflicting messages. And then, to quote the OP, "after some period of time a winner is chosen, likely the message that controls the greatest surface area, the greatest number of minicolumns. When this happens, the winning minicolumns are rewarded, likely prompting them to encode a tendency for that firing pattern into their structure." And this happens in multiple layers of the brain.
In other words, there's some kind of iterative mechanism for higher-level layers to find which lower-level subnetworks are most in agreement about the input data, inducing learning.
Capsule-routing algorithms, proposed by Hinton and others, seek to implement precisely this idea, typically with some kind of expectation-maximization (EM) process.
There are quite a few implementations available on github:
https://github.com/topics/capsules