In many regions of the brain, including the mammalian cortex, the strength of synaptic transmission can be bidirectionally regulated by cortical activity (synaptic plasticity). One line of evidence indicates that long-term synaptic potentiation (LTP) and long-term synaptic depression (LTD), correlate with the phosphorylation/dephosphorylation of sites on the [alpha]-Amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor subunit protein GluR1. Bidirectional synaptic plasticity can be induced by different frequencies of presynaptic stimulation, but there is considerable evidence indicating that the key variable is calcium influx through postsynaptic N-methyl-D-aspartate (NMDA) receptors. Here, we present a biophysical model of bidirectional synaptic plasticity based on [Ca[superscript 2+]]-dependent phospho/dephosphorylation of the GluR1 subunit of the AMPA receptor. The primary assumption of the model, for which there is wide experimental support, is that the postsynaptic calcium concentration, and consequent activation of calcium-dependent protein kinases and phosphatases, is the trigger for phosphorylation/dephosphorylation at GluR1 and consequent induction of LTP/LTD. We explore several different mathematical approaches, all of them based on mass-action assumptions. First, we use a first order approach, in which transition rates are functions of an activator, in this case calcium. Second, we adopt the Michaelis-Menten approach with different assumptions about the signal transduction cascades, ranging from abstract to more detailed and biologically plausible models. Despite the different assumptions made in each model, in each case, LTD is induced by a moderate increase in postsynaptic calcium and LTP is induced by high Ca[superscript 2+] concentration. (Contains 1 table and 7 figures.)