Discarding correlation among inputs in a neural network
You could try some feature selection techniques if you want to choose 3.

That said, have you tested the data for multicollinearity?  Maybe you have but I don't think that a1+a2+a3+a4=1 implies a high degree of correlation.

To answer your question though, it really doesn't hurt to try and train a neural net with the four variables.  If one of the inputs is unnecessary, the NN training may well set the weight on one to close to zero without you having to choose one.