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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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