Although spontaneity can be conditioned by training, it is more an anti-skill than some precise technique. One cannot learn to simulate it, but one can learn to release it, recognize it and channel it. This is not some airy-fairy dogma, but simply a feature of the human nervous system identifiable by modern brain scanning techniques. It is a quite different neurological process, and takes a measurably longer time, to decide to do something rather than simply react to a stimulus. Spontaneity is not simply "thinking quickly."
An actor who is approaching his craft in an analytical way will have little difficulty in simulating a conscious decision made by his character. And in fact there are many modern plays, and especially films, in which complex characters pursue strategic objectives through the strength of their intellects, often after scouring some "inner landscape," other times through the sheer quickness of their wits. Well, it's a choice the writer makes, often as a kind of conspiracy with a particular approach to acting (see below), and certainly a lot of life is like that; not always, but often. So actor, if the glove fits, then go ahead and fake it, simulate it in an intelligent way. Just be careful we don't see the wrong wheels turning behind your eyes and you'll probably get away with it...
But characters who are off-balance, in an "Ah-ha!" moment, or reacting in chaotic or extreme situations, or just plain zany, are in a state that can't be effectively simulated. It is not just the scientist's instruments (FMRI etc.) that can register the difference. Also for the audience it somehow doesn't "ring true." This means that not just the character, but also the actor has to be in an actual state of spontaneity. However, on stage, this spontaneity has to be reliably produced and repeated. The actor's I-myself (also referred to as "system two") has to release her Me-myself ("system one").
An example of "system two" playwriting:
A: Hello. I would love to talk to you. Now. Can you take it? Just say yes or no.
B: You give me a choice?
A: I'll take that as in the affirmative.
An example of "system one" playwriting:
A: Oh my god it's you!
B: Can I help?
A: That would be fantastic!
The first example is much easier to act because it doesn't require spontaneity, and can be acted from the intellect. The second one has to be acted from the body. An actor who is not trained in managing her spontaneity could try to play it from the head, injecting all kinds of objectives and subtext, but the result would be stodgy at best.
There is an interesting exception the rule, and that is the animated cartoon. Spontaneity is often amazingly successfully simulated through the most un-spontaneous of techniques!
An extended flow of spontaneity is often referred to as just that - "flow."