A gyrator is a circuit which transforms reactances. Normally gyrators are used to simulate inductors, since inductors are often expensive and bulky.
Gyrators find application in DC holding circuits which are used in most DAAs (Data Access Arrangements). DAAs typically use dry transformers which can not tolerate DC current, and incorporate a DC blocking cap. When the telephone goes off hook, it must pass DC current, and so an inductor can be used, which defeats the purpose of using a dry transformer, or a gyrator can be used.
R1 and R2 are adjusted so that just the right amount of DC current flows. C1 creates a filter which attenuates any high frequencies to the base of Q1. Since the base to emitter current of Q1 is held constant, output of Q2 is held constant and the circuit looks inductive because it doesn't pass high frequencies.
The bridge is to guarantee that the circuit is powered with the correct polarity, regardless of the input polarity. Tip and ring on a telephone circuit can often be reversed.
Gyrator- DC Holding Circuit