Linear Function Representation of Neural Network
It would seem to depend on the complexity of your neural network. A single perceptron is a linear separator. On the other hand, it is known that multilayer feed-forward neural networks are universal approximators (see here: http://dl.acm.org/citation.cfm?id=70408 ): i.e. they can approximate any function to arbitrary accuracy. Obviously for such a network, you should not expect to be able to express its outputs as a linear function of its inputs.