Just to clarify, Elavil (amitrptyline) is not an SSRI, it is a TCA (tricyclic antidepressant).
With anxiety disorders, there is not really any proven tried and true therapy that works for everyone. Most literature states that anything that has some sedative properties will be effective, but it is different for everyone. Some may respond to TCA's, others SSRI's, and still others, benzodiazepines (ie xanax, valium, ativan).
As stated above, the SSRI's (and other mood modifiers) attempt to level out the concentrations of certain brain chemicals to avoid the peaks and valleys that occur in some people.
I would check with an actual psychiatrist, rather than a neurologist, to find out what is right for you and your specific situation.