I'm currently a first year undergraduate student in pure math but recently I've become interested in neuroscience and related fields. Is it possible to pursue neuroscience and cognitive sciences without a medical degree?
If yes, do you know any universities in North America or Europe that accepts people with pure math degrees in programs that are related to neuroscience and bio-mathematics?
Sorry if my question is too naive, I don't have much information about academic stuff yet.
At least in the US, a neuroscience or cognitive science PhD program generally admits people with undergrad or masters degrees, which generally can't be medical degrees since many US schools do not offer medical degrees at those levels.
So the answer to your first question is yes.
As for the specifically a math background, I expect that most schools would consider such an application. Even in fields that have corresponding undergrad majors, it's not uncommon for people to enter PhD programs from very different undergrad studies. An application would have to address the question of having an adequate background and understanding of what the field is like.
In one word: absolutely!
For Psychology the top rated public university in the US (UW Madison) states absolutely no specificity for an undergraduate degree for consideration, and in fact no psychology department I could find in my short searching stated any such requirement. Even most engineering schools, for example, don't require an undergraduate degree in engineering for consideration (though they often state a preference) - and this includes what in the US we refer to as "Industrial Engineering", which includes a variety of fields that do work in the cognitive sciences.
For neurology, the picture is a bit different, but the answer is still "yes". For neurology there is "clinical neurology", which are usually programs geared towards medical students (those seeking a medical degree - not necessarily who already have one). However, for pure "neurology" degrees the fact is the same - they don't universally require a similar undergraduate degree.
However, on the other side a word of caution - some degree programs state a preference for a related undergraduate degree. The University of Pittsburgh Neurology department states:
"In general, successful applicants have a BS degree in neuroscience, biology, chemistry, or psychology with a cumulative grade point average of at least 3.4 (on a 4.0 scale), and a cumulative Graduate Record Examination score of at least 160 verbal, 150 quantitative and a 4.5 in analytical writing."
Every department sets its own requirements, and just because one says "in general..." doesn't mean they don't admit people who don't fit their exact stated mold - just that your chances will be higher.
Being an international candidate (I assume), you'll probably have to apply to more programs as the chance for admittance is generally small per application to US schools (it's not uncommon for schools to accept 50% of domestic applications, and but only 1-10% of international applicants), but again - this is not all that uncommon.
If you had the opportunity, it wouldn't hurt to try to direct some independent study or work towards even a single mathematical problem that relates to something like decision or control problems or statistics - which you could then provide as an example of proof that you are interested in cognitive sciences, etc - but you don't have to.
Bottom-line, you will have to figure out what angle of attack you'll want to have on the cognitive sciences. It is a very big field, and you can technically do cognitive science in the field of Business (especially Behavioral Economics), Psychology (where most undergraduates are required to take a course in cognition), Engineering, Multidisciplinary design/art programs (like Human Computer Interaction), Computer Science (especially in Artificial Intelligence), Neuroscience/Neurology, Linguistics, and even Philosophy. None of these require medical degrees, either.
On the downside, if you don't also acquire a medical degree they generally won't let you cut people open and play with their brains or run them through radioactive/high-magnetic machinery by yourself (but you can collaborate with people who can!). If that's OK with you, no medical training is required.
