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The eigendecomposition of the coupling matrix of large biological networks iscentral 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'slaw which states that a neuron can have only all excitatory or only allinhibitory output connections, i.e., coefficients of one column of the couplingmatrix must all have the same sign. The eigenspectrum density has beendetermined before for dense matrices $J_{ij}$, when several populations areconsidered. However, the expressions were derived under the assumption of denseconnectivity, whereas neural circuits have sparse connections. Here, wefollowed mean-field approaches in order to come up with exact self-consistentexpressions for the spectrum density in the limit of sparse matrices for bothsymmetric and neural network matrices. Furthermore we introduced approximationsthat allow for good numerical evaluation of the density. Finally, we studiedthe phenomenology of localization properties of the eigenvectors.

Spectrum density of large sparse random matrices associated to neural networks