Prof's neurological health is declining rapidly. What can I (TA) do?

Let the department head know what you've observed. You can write an anonymous letter if you feel uncomfortable or are worried about burning bridges.

(I've been in this situation, as a student. We - the students in the class - wrote a collective letter to the department head. He was able to arrange extra TA support for the professor for the rest of that semester. He also arranged things so that this professor had other duties in the department going forward, but no more teaching.)

If nobody does anything about it, it will eventually blow up, but probably not in the nicest way.

I had a similar experience with a professor that had a more than a small problem with alcohol. We were students, so we didn't want to risk our mark/course by going "against" him/her, so what we did was (as @ff524 suggested) to write some anonymous letters to the department. This lead to him/her going to seek help and everything got fixed in the end.

If nobody would had done this, it would probably have ended in some very unpleasant situation where the school probably would have been forced to fire him/her. Informing the school about the problem was definitely the best option for students, the school and him/herself.

This is just a personal experience in a bit different scenario, but I thought I may be helpful.

It is very wrong to assume that the other members of the department or the administration do not know about this faculty member's condition. It's very likely that they knew about it previously and they know about it now.

However, they are not at liberty to share the details about what they know about their colleague's health with you. This is both an issue of medical privacy as well as a human resources / personnel issue. They may be taking steps themselves to lessen his workload or to arrange a temporary medical leave -- but again, they may not or cannot share those details with you until they are officially public. And they may deeply care for their colleague and not want to publicly embarrass him by removing him from his post before they can arrange a phased retirement.

So when you speak to them, they may only be able to listen and not tell you what they know or what they are going to do. Which seems from the outside like they do not care, but it may be far from that in reality.

Now they might not be aware of some of the impact that his recent decline has had in the classroom. It might be useful to write a private letter or to talk to your chair/dean/provost about how to mitigate the impact on other grad students. But making a big public fuss and public embarrassment might be exactly what the administration may be trying to prevent.