We have developed a mathematical approach to the study of dynamical biological networks, based on combining large-scale numerical simulation with nonlinear "dimensionality reduction" methods. Our work was motivated by an interest in the complex organization of the signaling cascade centered on the neuronal phosphoprotein DARPP-32 (dopamine- and cAMP-regulated phosphoprotein of molecular weight 32,000). Our approach has allowed us to detect robust features of the system in the presence of noise. In particular, the global network topology serves to stabilize the net state of DARPP-32 phosphorylation in response to variation of the input levels of the neurotransmitters dopamine and glutamate, despite significant perturbation to the concentrations and levels of activity of a number of intermediate chemical species. Further, our results suggest that the entire topology of the network is needed to impart this stability to one portion of the network at the expense of the rest. This could have significant implications for systems biology, in that large, complex pathways may have properties that are not easily replicated with simple modules.