The "silver dapple" or "silver bay" is caused by the same gene, denoted by the letter "Z". The term silver dapple is somewhat of a misnomer since the gene that produces a black with silver will not always produce the dappled effect.
A black horse with silver is:
E? aa Z?
A bay with silver is:
E? A? Z?
where the ? denotes that the second gene can be either dominant or recessive and make no difference in the resulting horse colour.
The silver gene can be carried by a chestnut/red based horse, but it will not show up externally.
The appearance of the silver horses can also be effected by other genes present. A silver bay with a single cream gene would result in a silver buckskin, for example. With two cream genes present, the silver effect would be diluted right along with the rest of the horse to the point that physically you could not tell that it was present. Similarly, the champagne, and dun genes would change the outward appeanance of the coat, while the tobiano, overo, roan, and appaloosa genes would still display their own effects as normal.