What you're referring to (I think) is one model of how synaptic plasticity occurs (it's not the only one, and right now, it's still just a hypothetical model, not a known fact). I haven't looked into this in depth, though we've somewhat recently hired on someone who works in this field, so I've attended his seminars as well as those of other candidates working in that area. Anyway, my understanding is that neurons are spontaneously (? - there'd be some argument if it's spontaneous or directed) producing outgrowths (spines) all the time, but the right signals (again, vague here...complementary transmitters and receptors or surface adhesion molecules) have to be present for that outgrowth to form a bona fide synapse. If there is strong signalling at that synapse, it is reinforced and more likely to become long-lasting, but if it is only weak, or weaker than other nearby synapses, the neuronal outgrowth retracts.
Of course, during development, some of this has to be genetically determined. For example, neural adhesion molecules provide paths for neurons to follow as they migrate into their adult locations. There are developmental stages during which some molecules are turned on or off that help guide different stages of neuronal development.
Much of the "weighting" your talking about, as I've heard it, refers more to after the neurons are in position and there is juxtapositioning of axons and dendrites. Once the major structural components are in place, the neurite outgrowth is where most adult synaptic plasticity takes place.
While my area of research is not in learning/memory, I am interested in neuronal remodeling/synaptic plasticity in a different context. So, I'll likely return to this discussion later with a little better explanations. I'm just posting quickly while gulping down coffee before heading back into the lab, hence the rather vague, incomplete, and not fully thought out explanation above. (Okay, I'm just posting so you'll stick around to continue the discussion later.

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