As others have said, the correct form is "neither Atkins nor Perkins", however as I said in a comment to @oerkelens answer, the rules at play in using either, or, neither, and nor, aren't as simple as at first sight (well, at my first sight anyway). So, for what it's worth:
A or B
usually means A, or B, or both. Then, by contrast:
Either A or B
usually means one of, but only one of, A or B -- i.e. not both. For those familiar with digital logic, either/or is an "exclusive" or -- i.e. an XOR
By extrapolation then, one might think that the following:
Neither A nor B
is an exclusive nor -- i.e. an XNOR.
But that would in turn imply that the following:
A nor B
could be regarded as a regular (non-exclusive) nor -- i.e. a NOR. But now, returning from the austere purity of digital logic back into the rough and tumble of natural language we see that is not the case. The above "regular (non-exclusive) nor" form of "A nor B" is pretty much non-idiomatic. I can't think of an example where a native English speaker would say "A nor B" without preceding it with "neither".
The overall rule seems to be that when using "or", the use of an initial "either" is optional. Use it to make the "or" exclusive (i.e. one and only one of the choices can be true), and omit it otherwise. However, when using "nor", the use of an initial "neither" is mandatory but it does not imply an exclusive nor. In fact, as far as I can see, there is no such thing as a (simply stated) xnor in English. Hmmm -- need to think about that...
So in summary, by way of examples:
This is idiomatic:
A cow can be regarded as an animal OR a milk machine (and it could be seen as both).
This is also idiomatic:
A blackberry is EITHER a fruit OR a vegetable (but it cannot be both).
And this is idiomatic too:
A cheesecake is NEITHER a fruit NOR a vegetable.
But this is not:
A cow can be regarded as a cheesecake NOR a fruit.
Nor, for good measure, is this:
A cow can be regarded as NEITHER a cheesecake OR a fruit