NSF GRFP Application for a Rising Undergraduate Senior
In the US in neuroscience, most people apply for grad school without their own funding.

Funding for your research is provided by your research advisor; in your area of interest this will likely come from the NIH and possibly other private organizations specific for AD. You may also be funded for between a semester and a year to rotate with different potential advisors, such as via a T32 training grant from NIH (training programs apply to these grants, not students; students are funded by applying to graduate programs that possess such a grant).

NSF grants for students in neuroscience are very rare (looks like about 60 GRFPs were awarded this year in neuroscience; only 4 don't have an institution listed so I presume applied before being admitted); it's a good goal to shoot for, but not something to rely on. The requirements for the specific fellowship you are looking at specifically exclude disease research because such research is already substantially funded via other means; NSF is trying to fill the other underfunded gaps in very basic work. I would not suggest trying to bypass this by tweaking the language of your application.

NRSA grants (the NIH equivalent) are also somewhat rare, but more achievable. You apply after you are already part of a PhD program, sometime before the end of your second year most often.