Memory effects due e.g. to finite propagation times in e.g optical and physiological systems pose a great theoretical and computational challenge. However, by accounting for the spatial propagation of activity, delayed dynamics actually are a simplification over PDE's. We first present recent work on reducing large scale ionic models of "bursting" neural activity to simple two-dimensional dynamical systems with delay; this delay is used to model the backpropagation of firing activity from the cell's body to its dendrites and back. This reduction preserves bifurcations in the full ionic model, enables an analytical study of the bursting dynamics, and provides a drastic computational simplification for real life problems where many such cells are coupled. We then discuss work on delayed bistable systems which arise also in the context of laser and neuron dynamics. We show that bistable dynamics with linear delayed feedback can be understood in terms of the unfolding of a Takens-Bogdanov bifurcation. We provide the two-dimensional normal form for this

infinite-dimensional system in terms of the original parameters; this form again can be used to simplify computations. We also discuss a preliminary computational analysis of the multistability of a cluster of such bistable systems. Finally we describe a memory effect seen in excitable systems, which leads to correlations between firing intervals. Modeled as a dynamic threshold rather than a delay, it leads, in the presence of periodic forcing, to phase locking and chaos. We derive the Lyapunov exponents for this model, and discuss the difficult problems that arise in the analysis of noise-induced crossings when such memory is present.