When you stop exercising, muscles quickly wither away. Yet many people claim that the muscles 'remember' some of the previous exercise and it is much easier to regain some of the muscle.

Some story about myonuclei and satelite cells

What is some reliable and scientific source on this phenomenon? How strong is the effect?

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Muscle memory seems to describe two separate concepts.

The one that I've heard about before is entirely based in the brain and is simply that we have long term procedural memory of motor tasks. For instance, I can pick up a piano piece that I haven't seen in ten years and that I barely remember and am able to play it much more easily than I could a new piece. This is the "it's like riding a bike" phenomenon. This is what is described by most of the muscle memory Wikipedia page.

There's also the effect that you seem to be asking about in which muscle strength can be regained more quickly after it is lost than it originally took to gain. This is described in this section of the muscle memory Wikipedia page.

The article about this on Strengtheory has links to sources (not as footnotes, in the text). May be useful to check out.