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He states that neither anxiety nor depression is/are playing a permanent role in his life right now.
"is" is correct. The rule is that the noun or pronoun closer to the verb determines what the verb should be. Of course in this sentence both nouns are singular.
But consider the following sentences.
1...Neither the children nor I am interested in doing that.
2...Neither I nor the children are interested in doing that.
neither nor 
Linear Mode
