How is an action potential transmitted across a synapse?

Looking at a cholinergic synapse:The action potential arrives at the presynaptic knob. This causes voltage-gated calcium ion channels to open, and calcium ions to diffuse into the presynaptic knob.The influx of calcium ions causes synaptic vesicles containing the neurotransmitter (acetylcholine) to fuse with the neurone membrane. This releases acetylcholine into the synaptic cleft.Acetylcholine molecules diffuse across the synaptic cleft and bind to receptors on sodium ion channels in the post-synaptic membrane. This causes the sodium ion channels to open, allowing sodium ions to diffuse into the post-synaptic neurone.This causes depolarisation of the axon, generating a new action potential.Acetylcholine is hydrolysed by the enzyme acetylcholinesterase into choline and ethanoic acid. These molecules diffuse back across the synaptic cleft into the presynaptic knob. Here, they are recombined using ATP to reform acetylcholine, ready for the next action potential.