Mathematical Biology Seminar Paul Bressloff Department of Mathematics, University of Utah Wednesday November 15, 2006 3:05pm in LCB 215 ``Protein receptor trafficking across multiple synapses'' We present a mathematical model of AMPA receptor trafficking between multiple dendritic spines distributed along the surface of a dendrite. Receptors undergo lateral diffusion within the dendritic surface membrane, with each spine acting as a spatially-localized trap where receptors can bind to scaffolding proteins or be internalized through endocytosis. We demonstrate that the lateral diffusion of AMPA receptors within the dendritic membrane generates a form of heterosynaptic competition whose strength depends on spine geometry and membrane diffusivity. We also show how lateral diffusion provides a mechanism for synaptic scaling, which is thought to be important in the homeostatic control of synaptic plasticity. Finally, we explore the consequences of assuming that receptor exocytosis occurs only at the soma, which is a topic of some controversy between experimentalists. In particular, we show that somatic exocytosis can supply distal synapses with receptors only when the properties of the spines/synapses vary with distance from the soma, and that the mean time required for receptors leaving the soma to reach even a few hundred micrometers from the soma is several orders of magnitude longer than the estimated metabolic half-life of an AMPA receptor.