Final answer:
A postsynaptic cell can open chemically gated ion channels for fast responses, close channels via G proteins for slow responses, and regulate protein synthesis for metabolic effects.
Step-by-step explanation:
In response to binding a neurotransmitter, a postsynaptic cell can engage in multiple types of responses.
These responses include the fast synaptic potentials caused by the opening of chemically gated ion channels, as well as the slower responses that involve the closing of ion channels through mechanisms involving G proteins and second messenger systems.
Additionally, the postsynaptic cell can regulate protein synthesis and affect its metabolic activities.
Therefore, the correct answer includes a combination of fast synaptic potentials through chemically gated ion channels, modulation of ion channels through secondary messenger systems, and longer-term effects on the cell's metabolic processes.