aagopal,
Drexel does not have an official MSTP, if you define MSTP as an NIH-funded program. The schools that do have MSTPs are listed in the link that I provided above.
As far as Drexel's MD/PhD program, a number of my friends have applied and interviewed there in recent years (when it was MCP Hahnemann) and they were not at all impressed. The school has had a lot of financial problems, and funding is an important issue to consider when applying to graduate programs. Drexel's MD/PhD program does NOT provide a stipend during the four medical years of training, where as MSTP programs DO provide a living stipend. This means that you will have to take out a loan to cover living expenses (unless you are independently wealthy) which will accumulate much interest during the 7-8 year program, residency, and fellowships.
Specifically, I am also not too keen on Drexel's neuroscience program, simply because I am not impressed by the work most of the faculty members are doing. Of course, this is my personal opinion which is based on my own research interests in neuroscience.
If you intend to apply MD/PhD this cycle, you need to complete applications very soon. This will involve you writing an essay about your research experience and one about your motivation to pursue the combined degree program (and combining them into one for some schools). Also, you'd have to get research-focused letters of recommendation. There would be no additional fees.
At almost all MD/PhD programs, you certainly can apply during your first year of medical school, and in some cases, even later (although this means your first year of medical school would not be funded). If you are unsure about whether you want to do an MD/PhD program or not, this might be your best option. Do some research over the summer before medical school, and then make up your mind. MSTPs are a huge committment, and are only for those who are truly dedicated to becoming a physician scientist and can't imagine taking any other career route. If a major deciding factor in your case is whether or not you have to fill out a few more applications, perhaps you don't have the necessary drive or motivation at this time.
Again, we are here if you need any help or advice.