Interplay of Random and Structured Connectivity in the Dynamics of Neural Networks
Synaptic connectivity of neural circuits may range from completely disordered to very structured. Common hypotheses are that high level areas such as the prefrontal cortex are disordered, while early sensory areas like the primary visual cortex are structured. Even at the structured end of the spectrum, however, the connectivity matrix is not completely regular and contains significant randomness. Another general aspect of biological connectivity matrices is nonnormality. Nonnormality gives rise to a hidden feedforward structure between orthogonal activity patterns, and to transient amplification: small perturbations of a stable system from its fixed point leading to large transient responses. ``Balanced amplification" in excitatory-inhibitory neural networks was used  to explain the observed similarity of spontaneous activity patterns with orientation maps in the cat V1 . However,  only considered networks with regular connectivity, without any randomness. The mathematical study of structured, nonnormal random matrices has been underdeveloped. We recently developed tools for studying various phenomena in networks with connectivity matrices of the general form M + J, with M a deterministic matrix representing structure, and J a fully random matrix with zero mean i.i.d. entries. These included general formulae for the distribution of eigenvalues in the complex plane, the magnitude of transient amplification, and the frequency power spectrum of response to external input. Here, using these tools, we study specific examples of M that mimic neural connectivity, have various hidden feedforward structures, and highlight extreme nonnormality. We also extend the results of  which studied the eigenvalue distribution of random matrices of the above type, with an M describing the connectivity of excitatory-inhibitory neurons with all excitatory (inhibitory) synapses having equal average strength.  showed that unlike the M=0 case, whose eigenvalue distribution is the well-known circular law, here the distribution also contains eigenvalues lying significantly outside the traditional circle. We calculate the density of these outliers, and furthermore extend the results to more general M's, like those mimicking the connectivity of orientation tuned simple cells in V1.