How does the brain know to start recruiting the muscles?
-look up golgi tendon organ (GTO), muscle spindle, action potential, and motor unit.
What controls the amount of lag between the time you feel the ground and the time you apply the force?
-time from sensory nerve to CNS to motor nerve to motor unit
The force increases gradually, I assume, correct?
-strength can increase gradually (ascending), decrease gradually (descending), or be bell curved, depending on the type of action and range of motion. Actions that allow you to move more weight only at the last half or last quarter of the motion are ascending (ie, squat, bench press). Actions that allow you to move more weight only at the beginning or first half of the motion are descending in strength (ie, upright rowing motion). Actions where you can move more weight during the middle portion are bell-curved in nature (ie, single joint exercises like curling your arm, knee extension). Force changes according to muscle action and velocity of movement- which is different for concentric, eccentric, isometric, etc.
What controls the rate at which that force increases?
-all three answers above, rate and frequency of the action potential, plus the level of training of the individual. For the majority of people (especially in trained marathon runners), at any intensity of exercise, you will recruit first through all the ranges of Type 1 slow-twitch fibers available, then add on more and more of availbe Type 2 fibers only if the stimulus or the intensity is truly high enough (which in most cases will not occur, except through true sprint/power/resistance training). In other words, marathoners and others who are not anaerobically sprint/power trained will first use those fibers that require less activation (type 1) and last the longest. Even through it may be what they may consider an intense running exercise, the level of recruitment of type 2 fibers is still minimal when compared to what would be recruited during an intense power or resistance exercise.
Those who are sprint/power trained can and will practice selective recruitment, meaning they can bypass type 1 fibers and use type 2 fibers initially to perform actions that require high amounts of force in less time. This is why a trained marathoner will not be able to compete with a power trained person in vertical jump tests, sprints, or lifts. The marathoner has to recruit Type 1 through Type 2 to increase the force at a moments notice. The power trained person (when trained well) can jump right into Type 2 to increase the force right then for a that quick motion.