  We consider the long time asymptotic behavior of a large system of $N$ linear
differential equations with random coefficients. We allow for general elliptic
correlation structures among the coefficients, thus we substantially generalize
our previous work [14] that was restricted to the independent case. In
particular, we analyze a recent model in the theory of neural networks [27]
that specifically focused on the effect of the distributional asymmetry in the
random connectivity matrix $X$. We rigorously prove and slightly correct the
explicit formula from [28] on the time decay as a function of the asymmetry
parameter. Our main tool is an asymptotically precise formula for the
normalized trace of $f(X) g(X^*)$, in the large $N$ limit, where $f$ and $g$
are analytic functions.
