No, I don't think you should give up on your dream entirely, but I think you should change your application list a little. Your original list (Harvard, Columbia, JHU, etc) only includes very difficult to get into schools. I believe you that your GRE score does not accurately reflect your abilities and that you would be able to succeed at these schools, but it is unavoidable that your score will hurt your application. I would drop some of these schools (I think you should drop Harvard, as difficult as that may be to give up on), and add some easier to get into schools in their place.
In addition to lower tier American schools, I think you should also apply to both Master's and PhD programs in Mexico and Canada. I know American schools dominate the research scene, but there are good programs in other countries too, and doing a master's somewhere else is one of the few things you can do to improve your application. I especially recommend Canadian programs because they will give you the experience of living in another country, the opportunity to improve your English, and most of them don't require GRE scores! McGill and the University of Toronto are probably the top two Canadian neuro programs, but there is a longer list here (http://can-acn.org/canadian-graduate-programs-in-neuroscience).
And when you do apply to American schools, make sure to address your low scores in your personal statements, like bsharpe269 said. Your goal is to be so fantastic and so interesting in all other areas of your application that admissions committees want to to talk to you to see if your GRE scores are just a fluke.
Finally, I want to say that it may seem like applying to less prestigious schools is giving up, but really it isn't. There's a lot of good research and great education going on outside of Harvard and MIT, and I think you will be better off if you get in somewhere than nowhere. Last year, many of my friends were applying to graduate schools. One friend, who was applying to math PhD programs, was disappointed in his GRE scores but adjusted his list of schools accordingly. He is now at a lower prestige PhD program, but is having a great time and actually making progress towards his goal of having a career in academia. Another friend applied only to top tier CS grad schools, didn't get into any of them, and a year later is unemployed and miserable. Of course these are just anecdotes, but I really think you will be happier doing the work you want to do at a second choice school than giving up on graduate school entirely.