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I'm guessing it may be some combination of the two. If so, which is the more frequent situation?

I'm also guessing it varies depending on the type of memory being formed. Let's take for instance, my memory of opening up this tab and clicking the "Ask Question" button about 30 seconds ago. Surely this can't just be a case of strengthening former synaptic connections, right? It's a completely new and specific memory. So would it be a case of (for the most part) many new synapses being formed to somehow encode this memory? It seems hard to believe that there could be so many new synapses being created constantly in order to encode so many memories. It must happen lightning-fast, no?

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Formation of a synapse does not automatically mean learning. It is the strength of the synapse that makes it a "skill". Hebb's rule for learning states that if a Neuron A repeatedly takes part in firing Neuron B, then the synapse from A to B is strengthened. We should also consider the fact that a neuron can be either excitatory or inhibitory to another neuron. If it is excitatory, the mental habit or skill improves, and otherwise it deteriorates.

Memory formation can be modeled as recurrent neural networks, where outputs continue to occur even after input to network has ceased.

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Hebb's rule, is a simplified view of things. It does not include the multitude of knowledge we have accrued since his first postulate. For example, anti-hebbian is observed in some species and sometimes the elimination of a synapse, spine in this case, is related with memory/learning. "If it is excitatory, the mental habit or skill improves, and otherwise it deteriorates." Sources? Inhibitory neurons have plasticity as well – James Oct 3 '14 at 0:14

Short answer: current evidence is unclear.

Learning As Contrarian mentions, formation of a synapse does not always mean 'learning' as synapses are formed and eliminated all the time, however -- both -- formation and elimination have been found to be linked with learning (interestingly the same spines that are 'learnted' can be 'unlearnted'). So in this sense, the formation of a new synapse or the elimination of an existing one could be 1) 'learning' or it could be 2) a stochastic process unrelated to any learning.

Furthermore, we know that synapses have variable efficacy, which is commonly assumed to underlie learning (evidence on this is less clear). As Contrarian mentions, Hebb's postulate is a nice place to start. It is far from complete however as much evidence suggests that sometimes when 'Neuron A' takes part in the firing of 'Neuron B' a conjoining synapse may be weakened. Hebb's postulate is a very simplified view (and originally didn't even include weakening at all).

Memory When it comes to memory, many people assume that synapses are the substrate for memory, and this may be accurate however it is still an assumption as current evidence is not concrete (for example, here is a paper with evidence that CaMKII, a protein kinase, could be a candidate for the molecular basis of memory, or perhaps a combination of several molecules -- a nice review paper). It is usually assumed that strong synapses represent memories however this may not be the case, it could be that a weak synapse represents a memory.

An important note to this, is the time scale which you elude to. Synapse formation/elimination will take longer than the time for you to remember clicking the 'Ask Question' button. In this case it might be that a group of neurons (commonly called assemblies) are 'remembering' this with their activity, independent of any changes in synapses.

Substantial evidence has been collected to suggest that the basis of a memory is a group of neurons (often termed a cellular assembly) and their connections, simply put: see papers from the Tonegawa Lab.

(Edited to include some references.)

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