Type of non-university, research jobs suitable for a mathematician turned computational neuroscientist, with coding experience I've asked the same question on Math Overflow

Background:

I'm a Ph.D. in pure mathematics (late '13, related to differential geometry, low dimensional topology) turned into computational neuroscientist as postdoc. I've been coding in matlab for the last 9 months, this being my only serious coding experience. I'm also doing some differential geometry and longitudinal statistics and have a minor in statistics. I'm neither an American nor EU citizen, but a research position in the European Union would be ideal (although I'm open to other options.)

My postdoc will not be extended beyond September. So, apart from searching other postdocs, which are the only university positions I can get at this moment, I am therefore also looking for non-university positions (it could be still academic, but the stress here is outside universities) in industry or research labs (e.g. MITRE), preferably permanent. I prefer a research position where I can do pure mathematics and continue coding.

Questions: 


Which types of jobs are there?
Which role do the hired candidates play in them?
How can I find about their job advertisements?
Your questions are rather broad, so let me just point you to some resources.  For the first two questions (and possibly some info about the third) check out some info from the AMS:

http://www.ams.org/profession/career-info/math-work/math-work

http://www.ams.org/profession/career-info/new-phds/new-phds

For 3, some online academic resources to find jobs both in and out of academia are: mathjobs, the AMS's EIMS, the LMS site, the EMS site.  Some universities also have Career Fairs and Career Service Departments which may be able to give you more help in your job search.