The nicotinic acetylcholine receptor (nAChR) and other pentameric ligand-gated ion channels (pLGICs) are native to neuronal membranes with an unusual lipid composition. While it is well-established that these receptors can be significantly modulated by lipids, the underlying mechanisms have been primarily studied in model membranes with only a few lipid species. Here we use coarse-grained molecular dynamics (MD) simulation to probe specific binding of lipids in a complex quasi-neuronal membrane. We ran a total of 50 microseconds of simulations of a single nAChR in a membrane composed of 36 species of lipids. Competition between multiple lipid species produces a complex distribution. We find that overall, cholesterol selects for concave intersubunit sites and PUFAs select for convex M4 sites, while monounsaturated and saturated lipids are unenriched in the nAChR boundary. In order to characterize binding to specific sites, we present a novel approach for calculating a "density-threshold affinity" from continuous density distributions. We find that affinity for M4 weakens with chain rigidity, which suggests flexible chains may help relax packing defects caused by the conical protein shape. For any site, PE headgroups have the strongest affinity of all phospholipid headgroups, but anionic lipids still yield moderately high affinities for the M4 sites as expected. We observe cooperative effects between anionic headgroups and saturated chains at the M4 site in the inner leaflet.We also analyze affinities for individual anionic headgroups. Combined, these insights may reconcile several apparently contradictory experiments on the role of anionic phospholipids in modulating nAChR.