Does NMDA-receptor activation depend on neighboring AMPA-receptor activity? This question is coming out of a couple points of confusion after I learned about about NMDA receptors' role in LTP.

I got the impression that after AMPA receptors were activated enough, which depolarizes the cell, that nearby NMDA receptors will allow Na and Ca to come into the cell. 


Is it the local membrane potential, around those receptors that matter for a given voltage-gated NMDAR to open up? Or is there no such thing as 'local membrane potential', and the whole cell body is about the same amount of negative or positive voltage? 
If there's no such thing as 'local membrane potential', does it matter if the AMPA receptors are activated or if other receptors are activated? Does a single neuron usually respond to more than one type of excitatory neurotransmitter? If so, couldn't the activation of any receptor that results in cell depolarization eventually cause NMDAR to open up?
Yes, there most certainly is "local membrane potential".  There are two major levels of compartmentalization in dendritic processing.  For one, most excitatory synapses occur onto "spines", which have a somewhat segregated environment from the rest of the dendrite.  The spine neck has been estimated to have a resistance of up to 1GOhm, although measurements are very difficult, so discussion spans orders of magnitude.  The other level of compartmentalization is that different dendritic branches can function somewhat independently.  

It is actually extremely important that there be "local membrane potential" and that AMPA receptors only activate neighboring NMDA receptors.  That is because Ca2+ coming through NMDA receptors leads (through many steps) to LTP (long-term potentiation) at that synapse.  We want LTP to be synapse-specific (only the activated synapse should be strengthened).  This requires that NMDA currents be specific to the synapse that had the AMPA current.  

Wikipedia has an excellent article on this subject: https://en.wikipedia.org/wiki/Dendritic_spine.  See especially "Modeling spine calcium transients" for discussion of the extent to which NMDA mediated Ca2+ transients are local.