The tympanic membrane was the connective tissue that transmitted sound to the middle ear and separated it from the outer ear.
In Human anatomy, the tympanic membrane was also known as the eardrum. As sound waves entered the external auditory meatus, the tympanic membrane vibrated, which caused the attached ossicular chain in the middle ear to vibrate and ultimately the cochlea in the inner ear to vibrate. Those vibrations stimulated hair cells in the cochlea which sent an electrical signal up the auditory nerve to the cortex.
In Ferengi anatomy, the tympanic membrane was believed to perform the same function as its Human counterpart. When it got infected and was left untreated for more than three weeks, Ferengi could die from it. A massive infection of the tympanic membrane was a chronic condition among Ferengi. When Grand Nagus Zek faked his death in 2369, the reason stated for his death was that he had this chronic infection of the tympanic membrane. (DS9: "The Nagus")
The infection itself was painful for a Ferengi and could cause dizziness. It could even cause a Ferengi to faint when treatment was postponed too long. Some Ferengi had household remedies to treat the infection themselves. Rom tried one of his mother Ishka's remedies, which involved pouring it into his ear, but it did not work. Medical attention was needed to cure it completely. (DS9: "Bar Association")
For Ferengi, the tympanic membrane could also be stimulated with the help of a feather. This was known as a tympanic tickle. When performed by an amateur, the feather could get stuck and medical attention was needed to remove it. Antibiotics were needed to prevent infection of the membrane.