Falco main, secondaries include Marth, Falcon, Doc, and working on a Fox. But I play Falco 95% of the time.
As Falco, you're hardly ever playing a true neutral game, unless you count firing lasers outside their range as neutral. One could say that Falco dominates the neutral game with ease, or that he has the strongest neutral game.
But I also disagree slightly with the other posters' definition of "neutral game," I think it also ends when someone shields something, not just when you land a good hit, or any time you enter an advantaged/disadvantaged state compared to the starting state. Also, I think of it a bit differently, and this way applies to all other fighting games, and possibly in any real-time game or even real combat:
In the neutral game, you don't actually cover options, but you can still affect your opponent by threatening options. What's a threat? Well, you threaten options just by making them possible, or more specifically, in range. It's these ranges that provide a sense of zone control, and degree of control over zones is dependent on a few different things:
The range at which it's possible to react to an option is a region of strong control. Falcon's nair has it's own range, but "nair" isn't the option in the neutral game, it's "dash-SH-nair," which covers a maximum distance. Reaction time determines what you can do about moves at various ranges. You can determine whether someone is moving toward you sooner than you can identify the move that's being used, and what you're reacting to decides how much control you have. You might be able to see a fox jumping at you with a nair from a distance, but up closer there's no time to react. Choice-based reaction and knee-jerk reaction is also an important distinction. If you don't have time to react, you won't be able to decide anything in time, and can only do a knee-jerk response or no response, which is difficult to consciously change. Taking note of your opponent's knee-jerk responses and trying to stay conscious of your own is recommended. Knowing how decision time affects behavior is one way to take the guesswork out of reading your opponent.
The strength of the move, the reward for success and risk of failure. Falco may be able to laser at a long distance, but that isn't particularly strong until you're close. However, get too close and powershielding becomes dangerous. A move like Marth F-smash isn't very threatening in the neutral game since using it would be too risky. Standing or dash-dancing around the end of the range of a move adds a chance of whiffing, which increases the risk. Dash-dance also constantly changes the state of neutral game by moving and turning around, but make sure you can handle the mental load at least as much as your opponent.
The tendency to choose options, choice capability and execution capability. It's not quite right to consider all possible options a threat for every opponent. If the Fox opponent can't shinegrab, shielding in response to him jumping at you is good at makes his approach not so threatening. But ability to execute is just as important as willingness to do options. If a Marth never F-smashes in the neutral game, they might as well can't, and the area in front of him becomes less threatening. But showing that you're capable of high-risk choices has the effect of expanding your threat even if you suddenly stop doing them. Those who haven't shown that they're capable of doing certain options appear to not be controlling the ranges that the options control. Interesting mindgames can be played by hiding capability or overplaying it. The metagame can give you a starting point, but that can be used against you (or them).
So, in your mind, trace out the threat ranges from the opponent. Color them by their risk/reward. Darken them by the ability to react to them. Lower their opacity the opponent's capability of choosing the options. Next, draw your own options. Now, you can see the neutral game at a particular moment in time. Next, watch a match and see how it changes over time. Finally, play a match now that you know where you're strong and where they're weak.