The two answers I saw up when I started an answer for you assumed you were talking about a normal power-up state; but your question says, "when your computer falls asleep".
I think you'd have to look at the ACPI setup for your machine to determine this for sure, but I don't think the standard power profiles Windows provides will support any activity like that once the power-saving features are activated.
I'd look at the ACPI specifications, and download a copy of Microsoft Visual C++ Express (for free) and the Windows Driver Kit, and get the MSDN library for .NET, too. There's a programming interface there for ACPI, and if you have the specs, you could conceivably write a driver that would enable you to control the settings for devices individually IF the ACPI BIOS you have on your particular machine supports that degree of configurability for devices. Some devices by default are more configurable; usually the ones which have to be able to 'wake up' if signals arrive at a machine port and are to be sent to them. The disk and display devices are usually the first devices to go down, though, when any change in the power levels for the devices is to be made; and keeping them up preferentially when other devices are going down might be a good trick (and it's the kind of trick you'd have to be able to carry out to run a screensaver when the rest of the system is in a low power state).
That's all a matter of electronics; and although the ACPI configuration can be adjusted (and Microsoft has a free compiler you can download that allows you to compile ASL code to the form OSPM uses) it isn't going to be easy. If you ask, they'd recommend you read the Plug'n Play BIOS Specification, the ACPI Specification, and learn how to write an ACPI Device Function Driver for Windows that will do what you've determined the specs say you can get away with.