  Using the diagrammatic method, we derive a set of self-consistent equations
that describe eigenvalue distributions of large correlated asymmetric random
matrices. The matrix elements can have different variances and be correlated
with each other. The analytical results are confirmed by numerical simulations.
The results have implications for the dynamics of neural and other biological
networks where plasticity induces correlations in the connection strengths
within the network. We find that the presence of correlations can have a major
impact on network stability.
