Some horses due to their sensitive, super alert or spunky nature are prone to bolting which is a sudden and very fast acceleration into gallop at full speed. It is possible to train a horse out of it. However, it will not be a complete absence of any attempts to accelerate. Rather, a horse will stop or slow down immediately upon feeling rider's aids. In time, with the horse in question getting older, the attempts may disappear all together.
Some horses may give an impression that they bolted when in reality they were simply allowed to gain speed over some time until it turned into a very fast and not really controllable gallop. This is not the same as a sudden, very fast and powerful bolt.

Sensitive, very alert horses are also often very forward horses. Running is their way to deal with many problems including spook, exciting/uncomfortable situation, aggression from another horse or human, etc These horses will run first think later and they run fast. Usually you can see that tendency before even the horse is under saddle. These horses will attempt to run on the lunge line. Sometimes you may get a horse who does not appear that spooky, or alert and still they use sudden fast running as means to get away from work, from pressure, or simply because they feel like it.
In reality, it is not so important to know exact reason for each time a horse attempts to bolt, these can be too numerous to count. Once, my horse bolted from a sound of her own poop hitting the ground on the trail :) It is more important to know what to do about it and how to discourage that.

For a horse person/rider it is very important to be observant, calm and confident. Techniques will only work if applied in the right time, with adequate amount of pressure and timely release of it. To be able to do all this person's brain must be very calm, attitude is to observe rather then being busy with his/her own agenda. Another important aspect is to take the horse's responses seriously. This does not mean the handler is paranoid and sees everything as a potential problem. It is rather noticing small changes in the horse's behavior, particularly those which mean lack of focus or obedience. These are important to address rather then ignore.
When a horse is showing lack of focus/obedience of any kind I see two most common mistakes:
1. Avoiding confrontation, pretending that nothing bad is happening, blanking out. The human pretence that nothing bad is happening will not make reality disappear. The horse will see this very well and start putting pressure on the handler/rider to see how far that "blanking out" can go. Unfortunately, some horse handlers do not see their horses pushing them around.
2. Freezing up, absolute absence of any action, waiting. This is usually happens out of fear/concern and is very unproductive behavior which puts a rider in danger on any acting up horse. For a bolting horse it gives him a chance to gain speed.

There are a few techniques I use with any horse but particularly those who show the tendency to run:

• Leading - Start with paying way more attention to how well (attentive and calm) your horse is walking by your side. Honestly, it should be as nice as a well trained dog - not running ahead, not dragging behind, same speed, soft lead rope, respecting your space. Anything less than that requires diligent and relentless schooling. Horses themselves do not know that. If you watch a herd, or even just two horses and they get spooked they run into each other regardless of their rank. They literally bump into each other. This is absolutely must be forbidden toward a human. And this knowledge is taught to a horse, they do not know. Good idea to lead on both sides to accustom a horse to a human presence on both sides of his body. Also, even a well trained horse will trespass once in a while and must be corrected right away.
• Lunging with two lunge lines - This way of lunging gives a handler much more control. I am not a big fan of round pens. I do not have one. I do not even have a fence around my arena except trees. But I am a big fan of double lunging. After initial familiarization with lunging I will start accustoming a horse to two lunge lines. One on inside as normal, the other is on outside from the bit through side surcingle ring or stirrup around the bum into your other hand. Now you have two reins, you can half-halt and you can keep the horse precisely on the line. On a stiff side or with horses that like to pull I will feed the inside line through the bit to the surcingle or saddle ring. This is much more effective way against a potential bolt. I will only start driving after I make sure the horse understood, learned and obeys half-halts. Also, if a horse shows tendency to run this horse will work harder doing transitions, moving around big arena until he settles into quiet way of working.
• Work in Hand - I use work in hand extensively with all young horses. However, with ones who prone to run I will be more attentive to teach them to yield to one rein, to yield to the whip, to do turns on the forehand many times and very well. These aids will be very helpful when I start riding. I will not get on a young horse if they do not know the rein and whip aids taught to them from the ground. It is like getting into a car without brakes and stirring wheel.
• Riding in a small pen, round pen or being lunged - If you have a round pen or a small pen with good footing you can start riding there. I use a helper with a lunge for the introduction of riding to a young horse. Depending on the horse's temperament and learning abilities it can take from a few rides to several months. Again, my gauge is how obedient and responsive my horse is. If the horse attempts to bolt while I am on the lunge line and the only way to stop him is to run to the end of the lunge line we are not ready to ride on our own. I will ride all gaits - walk, trot, canter on the lunge line before going off it. I will push a horse to lengthen the gaits and come back again to test my controls. I will do lots of turns on the forehand big and small. I will welcome potential distractions like wind or horses running in a nearby paddock. If you avoid asking more of your horse in a safe environment you will never experience their potential reactions and they will never experience corrections!
• Riding on your own in large space - When I am positive my horse understands half halts and listens to them I start riding on my own in my normal size arena. I will pay strict attention to my horse's state of mind. If he is tense and looky I will not let my reins long or loose. I will keep him busy with exercises and movements, transitions, etc. After a good piece of work I will let reins long to stretch if the horse feels relaxed but I will keep reins in one hand not in two. This arrangement is much better to retake the reins quickly back. All you have to do is pull up with one hand and take them both much shorter with your free hand.

Sitting through a bolt
If a horse did accelerate very suddenly the most important is to sit the initial moment - a second or two. A rider can fall easily due to unpreparedness and being taken by surprise. However, some may fall later due to fear and stiffness in the body which leads to leaning forward and losing balance making it easy for a horse to shake a rider off without much effort. If you sat the beginning, lean slightly back as you start pulling on the reins. Useing one rein and pulling a horse into a circle is much more effective then pulling on both reins equally. Lifting inside rein up helps you to sit up, prevents a horse from grabbing the bit hard and also makes it easy to start a turn. However, most important is sitting deep into the saddle, slightly leaning back and staying supple enough to sit into the movement rather than being bounced off the horse's back (this can be as little as couple inches or as big as falling off completely).
Several times I used big fence to stop a bolt. I simply held my horse strictly straight into a fence and it stopped her. I cannot advocate this method due to potential danger of a horse trying to jump the fence or a rider not holding him straight bravely enough. This will lead to a last moment quick turn which can unsit the rider.
Fear is the biggest enemy. Even if you are scared do your best not to let it take over and rule your actions. Staying calm when you are in danger on a horse is safer! (easier said then done!!!) :)

Happy riding...