Which revision of the sentence is now properly punctuated?

the peripheral nervous system—which is a network of nerves extending from your brain and spinal cord to all parts of your body carries signals back and forth.
the peripheral nervous system (which is a network of nerves) extending from your brain and spinal cord to all parts of your body carries signals back and forth.
the peripheral nervous system which is a network of nerves extending from your brain and spinal cord to all parts of your body carries signals back and forth.
the peripheral nervous system—which is a network of nerves extending from your brain and spinal cord to all parts of your body—carries signals back and forth.

Punctuation are used to organize a sentence . Moreover, it is used to create clarity and sense or to stress in a sentence. For example quotation marks " ", full stop (.), commas and so on.

A sentence which is punctuated correctly is the peripheral nervous system—which is a network of nerves extending from your brain and spinal cord to all parts of your body—carries signals back and forth.

The above sentence is correct because it makes appropriate use of hyphens.

The hyphen is a punctuation which is used to glue the words together. In other words, the hyphen indicates that the words are linked together. It has certain rules to be used.

To tell the age of things and people, hyphens are often used. It is used with the months, weeks, years and days but except the plural of these.

Also hyphens are used for forming original compound words for humor, vivid writing and special situations.

The answer to your question would be that the sentence that is properly punctuated is the following one: The peripheral nervous system- which is a network of nerves extending from your brain and spinal cord to all parts of your body- carries signals back and forth.

This sentence is an example of the use of the em dash.  The em dash can take the place of commas, parentheses, or colons. In this case, it has been used instead of commas to introduce a nonessential relative clause. As the information provided by the clause is not essential, it should be separated from the rest of the sentence. Most of the times, the clause is offset with commas, but the em dash can also be used in order to enhance readability. Yet, it is more emphatic than the commas.

it creates a visual picture of the man.

it conveys what the scene sounds like.

it describes the actions of the boy.

it reveals the poet’s opinion of the characters.

it encourages readers to care for the characters.

it can make them lose love for themselves