  We study the synchronous dynamics of the Hopfield model when a random
antisymmetric part is added to the otherwise symmetric synaptic matrix. We use
a generating functional technique to derive analytical expressions for the
order parameters at the first time step ($t=1$) and the second time step
($t=2$). We find that the overlap between the target pattern and the state of
the network at $t=1$ is independent of the symmetry of the additional random
part of the synaptic interaction matrix. The result may have bearing on the
theories which use the results at $t=1$ to estimate quantities relevant to the
retrieval performance of the network. The symmetry of the synaptic interaction
matrix becomes effective from the second time step, explicitly through a
correlation function involving spin configurations at different times. This
suggests that the prediction of the long time behavior of the network from the
first few time steps may not be {\it always} correct. This was confirmed by the
numerical simulation which shows that the difference in the long time and the
short time behavior of the network becomes pronounced in presence of asymmetry
in the synaptic interaction matrix. It is also found in simulations that the
size of the basins of attraction of the stored patterns decreases with an
increase in the asymmetry. Moreover, the convergence time for the retrieval
increases. These results are contrary to the expectations from the earlier
studies based on the counting of fixed points. However, the convergence time
for the spurious fixed points increases faster than that for the retrieval
fixed points in the presence of asymmetry in the synapses. This is a positive
feature of the asymmetry so far as the retrieval performance of the network is
concerned.
