  We derive and study dynamical TAP equations for Ising spin glasses obeying
both synchronous and asynchronous dynamics using a generating functional
approach. The system can have an asymmetric coupling matrix, and the external
fields can be time-dependent. In the synchronously updated model, the TAP
equations take the form of self consistent equations for magnetizations at time
$t+1$, given the magnetizations at time $t$. In the asynchronously updated
model, the TAP equations determine the time derivatives of the magnetizations
at each time, again via self consistent equations, given the current values of
the magnetizations. Numerical simulations suggest that the TAP equations become
exact for large systems.
