How are action potentials propagated across cholinergic synapses?

In order for action potentials to travel to their intended destination they must cross synapses. These synapses are the spaces between neurones' axon terminals and other neurones' dendrites. In the case of cholinergic synapses, the axon terminal contains vesicles filled with a neurotransmitter called acetylcholine (ACh). When an action potential reaches the terminal the associated depolarisation causes voltage-gated calcium ion (Ca²⁺) channels in the surface membrane to open, allowing and influx of calcium into the axon. This calcium binds to receptors on the ACh vesicle membranes, causing them to migrate to the cell surface membrane and fuse, releasing their contents (ACh) into the synaptic cleft. 

Once ACh is released into the synaptic cleft it diffuses towards the surface membrane of the post-synaptic neurone's dendrites, where it binds to chemically-gated channels. This binding causes these channels to open and allows sodium ions (Na⁺) to enter the cell down its electrochemical gradient. The entrance of positively charged sodium causes depolarisation of the dendrite and, if this depolarisation exceeds the action potential threshold, results in formation and propagation of an action potential down the dendrite. This type of synapse is known as excitatory, however other neurotransmitters and chemicals can affect synapses differently.