Projection methods can be used to reduce the dimensionality of dynamical networks by focussing on a subnetwork -- this is motivated e.g. by models for protein interaction and gene regulatory networks, whose size and complexity requires the application of simplifying coarse-graining techniques in order to derive explanatory insight. In the resulting subnetwork dynamics the influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. It turns out that memory functions can be calculated systematically for such situations; I will describe methods that initially capture the near-steady state dynamics of binary reaction networks that can model protein interactions, progressing to strongly nonlinear gene regulation dynamics and finally to memory functions that are nonlinear themselves.