Ideally, it would be good to be able to keep up with fields, but practically, I don't see how it's possible. There is so much new work constantly being produced in any one field that it would be quite challenging (though possible) for you to keep up with your two fields of cognitive neuroscience and psycholinguistics, so I think it would be very difficult to keep up with more than that.
Of course, you can skim abstracts in any number of fields that you want, but you have to really decide on how to balance or prioritize depth versus breadth. Yes, breadth is important for interdisciplinary work, but depth is critical if you want to make new contributions to knowledge, as required for a PhD. If you have an insufficiently deep understanding of cognitive neuroscience and psycholinguistics (because you've been trying to keep up with other fields, for which you have only a shallow understanding), I don't see what meaningful contributions you can make to any field. You might creatively try to combine disparate ideas, but you would quickly reveal to experts in the related fields that you don't understand any of the fields sufficiently.
My recommendation is to focus on keeping up with cognitive neuroscience and psycholinguistics and on gaining deep understanding in these two fields. For any other outside field ("such as biology, physics, or computer science"), just be an intelligent, aware person (which you seem to be already), and don't try to "keep up" with them. Rather, enjoy the fact that knowledge is so vast that no one can keep up with everything, but that at least you can meaningfully contribute to expanding knowledge by your deep interdisciplinary knowledge of two fields.