Synchronous Dynamics of a Hopfield Model with Random Asymmetric Interactions

Abstract

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.