Rewrite trigger from PostgreSQL to MySQL
You cannot do this in an AFTER INSERT trigger. Why ???

According to the Book



Chapter 19 Page 251 Paragraph 3 under the subheading BEFORE and AFTER TRIGGERS


  The most significant difference between BEFORE and AFTER triggers is that in an AFTER trigger you are not able to modify the values about to be inserted into or updated with the table in question -- the DML has executed, and it is too late change what the DML is going to do.


Just make this as a BEFORE INSERT trigger and update the needed columns:

DELIMITER $$
CREATE TRIGGER connections_INSERT_update_connection_root_trg BEFORE INSERT ON connections 
FOR EACH ROW 
BEGIN
    IF NEW.connection_root IS NULL THEN
        NEW.connection_root = NEW.connection;
    END IF;
END $$
DELIMITER ;


GIVE IT A TRY !!!