 
Summary: Eigenvalue Spectra of Random Matrices for Neural Networks
Kanaka Rajan and L. F. Abbott
Center for Neurobiology and Behavior, Columbia University,, College of Physicians and Surgeons, New York, New York 10032, USA
(Received 18 July 2006; published 2 November 2006)
The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix
describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can
be chosen randomly from appropriate distributions, making results from random matrix theory highly
relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to
synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or
inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and
inhibitory columns drawn from distributions with different means and equal or different variances.
DOI: 10.1103/PhysRevLett.97.188104 PACS numbers: 87.18.Sn, 02.10.Yn, 05.90.+m, 87.19.La
Knowledge of the statistical properties of eigenvalues of
large random matrices has proven valuable in a wide range
of applications [1,2]. In neuroscience, networks of neurons
are often studied using models in which interconnections
are represented by a synaptic matrix with elements drawn
randomly [3,4]. The distribution of eigenvalues of this
matrix is useful for studying spontaneous activity and
evoked responses in such models [37]. For example, the
