  The eigendecomposition of the coupling matrix of large biological networks is
central to the study of the dynamics of these networks. For neural networks,
this matrix should reflect the topology of the network and conform with Dale's
law which states that a neuron can have only all excitatory or only all
inhibitory output connections, i.e., coefficients of one column of the coupling
matrix must all have the same sign. The eigenspectrum density has been
determined before for dense matrices $J_{ij}$, when several populations are
considered. However, the expressions were derived under the assumption of dense
connectivity, whereas neural circuits have sparse connections. Here, we
followed mean-field approaches in order to come up with exact self-consistent
expressions for the spectrum density in the limit of sparse matrices for both
symmetric and neural network matrices. Furthermore we introduced approximations
that allow for good numerical evaluation of the density. Finally, we studied
the phenomenology of localization properties of the eigenvectors.
