Understanding the dynamics of large networks of neurons with heterogeneous
connectivity architectures is a complex physics problem that demands novel
mathematical techniques. Biological neural networks are inherently spatially
heterogeneous, making them difficult to mathematically model. Random recurrent
neural networks capture complex network connectivity structures and enable
mathematically tractability. Our paper generalises previous classical results
to sparse connectivity matrices which have distinct excitatory (E) or
inhibitory (I) neural populations. By investigating sparse networks we
construct our analysis to examine the impacts of all levels of network
sparseness, and discover a novel nonlinear interaction between the connectivity
matrix and resulting network dynamics, in both the balanced and unbalanced
cases. Specifically, we deduce new mathematical dependencies describing the
influence of sparsity and distinct E/I distributions on the distribution of
eigenvalues (eigenspectrum) of the networked Jacobian. Furthermore, we
illustrate that the previous classical results are special cases of the more
general results we have described here. Understanding the impacts of sparse
connectivities on network dynamics is of particular importance for both
theoretical neuroscience and mathematical physics as it pertains to the
structure-function relationship of networked systems and their dynamics. Our
results are an important step towards developing analysis techniques that are
essential to studying the impacts of larger scale network connectivity on
network function, and furthering our understanding of brain function and
dysfunction.Comment: 18 pages, 6 figure