Both post-synaptic and postsynaptic are grammatical. So to are pre-synaptic and presynaptic.
The closed version is more common that the hyphenated version. (As shown by another answer here.)
For actual context, here are two paragraphs from the article in question. Note that the second comes some time after the first (the emphasis exists in the referenced text):
At the synapse, the firing of an action potential in one neuron—the presynaptic, or sending, neuron—causes the transmission of a signal to another neuron—the postsynaptic, or receiving, neuron—making the postsynaptic neuron either more or less likely to fire its own action potential.
Chemical transmission involves release of chemical messengers known as neurotransmitters. Neurotransmitters carry information from the pre-synaptic—sending—neuron to the post-synaptic—receiving—cell.
I can personally see no reason for the inconsistent style decision made here. Normally, you pick one way of spelling a word and stick to it throughout—unless there is a particular context that would have it be spelled differently in a certain section.
There might be some reason that the author chose to mix the spelling of the words, but I can't determine what it could have been.
In fact, the use of the hyphenated version seems like a particularly strange choice given that the hyphenated words are used right next to em dashes, which causes a certain inability to parse the sentence quickly.