My applications for PhD in Neuroscience got rejected? How to be an extraordinary candidate for PhD? Please take time to read this, I appreciate any advice from any Ph.D. student or mentor, an advice from a neuroscience or health science field person is more useful to me.

I am an international student from India, pursuing my master's degree in Pharmacology at a US university (a private university not classified as a research university). 

Hopefully, I will graduate in May 2018. I have a serious and never-ending interest in doing research in the neuroscience field. I planned to do Ph.D. in neuroscience and took care of all requirements.

My undergrad was in Pharmaceutical Sciences from India, I made an evaluation of my undergrad transcripts and converted into US GPA score by WES, and it is 3.23/4.0. 

My maser's GPA as of now (3 semesters) is 3.64/4.0. 

GRE: 151 Verbal, 153 Quant, 3.0 Analytical writing
TOEFL: Reading - 27, Listening -29, speaking -20, writing -28 (Total - 104)

I don't have any published research papers yet.

I wrote a cohesive Statement of purpose explaining my research experience in my master's program and also my future research interests and mentioned the names of potential advisors that I searched in the respective university faculty websites.

Sent all documents before deadlines. 

I applied to a total of 10 Universities, they are:  

Univ of Alabama Birmingham - rejected
Univ of Texas Austin - awaiting decision
Rockefeller University -awaiting decision
Univ of Illinois Urbana Champaign - rejected
Univ of Iowa - Awaiting decision  

Univ of Kansas - selected for an interview, but got rejected few days after the interview  

Univ of Michigan - Awaiting decision
Icahn School of medicine - rejected
Univ of North Carolina - Awaiting decision
Yale University - rejected  

So, I got rejected from 5 universities, out of which one university interviewed me but rejected later. Now, I am waiting for the decision from 5 more. I feel that my confidence levels are dropping already. 

I am not worried about falling, I have failed many times at many things, and I am sure that I will apply next year or in two years to the same program.

But one thing I noticed was, many of the Universities that rejected me replied: "because of so many extraordinary applications, we could not give admission to an excellent application like your's".

I don't know whether this reply from them is generalized one or specific to me. But I don't know how to be the extraordinary applicant. One bad thing in my application that I am aware of is my GRE. If there are any experienced Ph.D. students here, please suggest me what to do. 

Thank you
You're being rejected because you're applying to top universities where the other candidates are better.  You're getting beat out on both GPA and GRE.

All but two of the schools you listed are R1 universities.  These are the top universities in the country and will be very hard to get into.  You also have a medical school, Icahn, on the list, and that also will be hard to get into, simply because it's a medical school and all of them are always deluged with applicants.  You only have one R2 university, Rockefeller, on the list.

For a top school, you pretty much have to expect that successful PhD candidates will likely have a 3.8 or better GPA and GRE scores that place them in the 80th percentile or better.  Your 3.2 UG and 3.6 master's GPA is low and your GRE scores are very low.

Consider applying to some more R2 schools or even, possibly, some R3 schools.