Advice on getting a Ph.D with scholarship in Australia (Neuroscience)
Usually, there is no specific timeline for applying for a PhD in Australia, unless you are thinking about particular advertised PhD positions or scholarships which of course have deadlines. After all, because you are an international student, you'd better find a good way to pay your tuition fees unless you are able to finance your project independently. 

Approaching a potential supervisor, you will be asked to provide a research proposal and your CV. You can also include your referees' names and contacts in your CV.  I suggest you introduce yourself and talk briefly about your academic background. Then you can talk about your proposed project and how the professor is suitable for it. Finally, you invite him/her to consider your attached CV and proposal.

You can use below template for emailing potential supervisors:

Dear Professor XX, 

I have graduated (or I will be graduated) from XX with an M.A. degree in XX.

I am interested in carrying out a project, title of your project,  

Based on your research focus on XXX, I am keen to join your research team as a Ph.D. student. 

Please find attached proposal concerning the proposed project and my CV for your perusal. 

I look forward to your reply, 

With Kind regards, 

XXXX