Do you have some explanation for why you think you would enjoy doing research in neuroscience over the activities you have been doing? Are you thinking of computational neuroscience, or another subfield? As a (former) faculty member in a dept. that included neuroscience, if I evaluated your application, I would have concerns that you would not be happy discarding your lucrative career track to work as an underpaid teaching assistant and researcher in my lab for the next 5 years. On the other hand, if you wanted to work in computational neuroscience, and had some direct experience building applications in this area, and convinced me that you urgently wanted a career in academia rather than the private sector, I might consider your application.
I agree that the most significant gap here is your lack of research experience in neuroscience. Generally, competitive neuroscience PhD applicants have at least 2 years of experience assisting a professor in their lab, or doing neuroscience research in some other capacity (industry, national lab, etc.) You need at least one and preferably two recommendations from professors or PhD-holders who have supervised you in research, as only they can assess your potential to succeed in neuroscientific research at the PhD level. (The other 1-2 should be from professors who had you in class, preferably multiple courses, and in whose courses you earned a high grade.)
You do say that you were elected to membership in Sigma Xi, so if you have research experience, what did that look like?
Your high school achievements (National Merit, Coca-Cola Gold) don’t matter here; also, your non-neuroscience ECs (Princeton business plan competition) won’t matter for your neuroscience application.
Another question is whether your certificate in neuroscience gives you enough coursework to be admissible to a neuroscience program.