A side note on what shawnie has said, sometimes you will need to change the filter but not as often as the manufacturers say you do. They tell you to change your filter every three months (give or take) but in all actuallity the only time you ever swap out your filter is during two events.
1) you had to medicate your tank; after the treatment is done add a new filter with
activated carbon this will react with the meds and remove it from the water entirely this is especially usefull with things like malcryn since it is a broad range antibiotic that if can if used improperly or for too long damage the
nitrogen cycle.
2) the only other time that I can think of to change your filter is when your media bag is falling appart. When I had my 10 gallon going I'd start to find pieces of fiber floating in my tank which meant it was time to change the bag, you may go through a very small mini cycle but usually by that point your cycle is well established.
now onto what everyone else is saying, Unless you have live plants, I have to agree you don't have a cycled tank it was probably new tank syndrome (not neon tetra syndrome) that did your poor fish in also i've never heard of neon tetra sysdrome where did you find that one?
again, you are going to have to cycle the tank, don't worry it happens to everyone who gets a new tank, heck, some of us who have established tanks suddenly have to cycle a tank again as well, anyway, cycle your tank and get a test kit with ammonia, that is the most poisonous ion to fish I don't care what kind of fish they are!! and also by knowing your ammonia levels you can tell what age your cycle is in. Best of luck and let us know if you need any other help,
~Inari