Neurons and other cells are regulated by a great multiplicity of neurotransmitters, modulators, hormones and other chemical messengers, which, through complex networks of extensively diverging and converging pathways, exert a multiplicity of effects. How do we analyze the functioning of such a complex network? If the effects of a transmitter depend on the presence of many other transmitters, how can we predict what they will be? If multiple transmitters act through the same network, how can their actions be specific? If they converge on the same effects, are some of the transmitters redundant? Why are there so many transmitters? Such questions can be addressed using an analytical approach that examines, qualitatively or quantitatively, how the operation of the network globally maps a multidimensional input space of transmitters to a multidimensional output space of effects.