Eigenvalue spectral properties of sparse random matrices for neural networks

We study randomly connected neural networks to understand the effects of network architectures on network dynamics. Specifically, we examine the impact of sparse connectivity combined with Dale’s law on network stability by analysing the eigenspectrum of the networked Jacobian. Previous work has examined the effects of Dale’s law for fully connected random matrices that are balanced (zero mean) [3] or unbalanced [1] and sparse matrices with constant weights for each population [2]. Our work significantly extends these results by integrating both Dale’s law and sparsity into both balanced and unbalanced matrices with random weights. We find that the stability and density of the eigenspectrum becomes nonlinearly dependent on the sparsity, population means and variances of the weights and give explicit formulae describing this.
Contrary to the balanced, single population case where the sparsity linearly scales the eigenspectrum [5], we find for unbalanced, single populations and (un)balanced two population cases, the sparsity scales the eigenspectrum nonlinearly. This happens for all cases except when both population means are zero, i.e., the balanced single population case. Further, we show the eigenspectral density becomes non-uniform for the above cases which extends previous results. We give an explicit formula for the spectral density when sparsity is introduced with Dale’s law: the spectral radius and density has a nonlinear dependence on the population means, variances and sparsity parameter for both the balanced and unbalanced cases. We deduce new analytical formulas for the eigenspectral radius and outlier, which explicitly describes how these properties nonlinearly change as functions of the sparsity parameter, the individual population means and variances. Sparsity and Dale’s law are fundamental anatomical properties of biological neural networks, and by quantifying the effects of these properties, we gain insight into how network structure influences system stability and neural network behaviour.