Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a nonnormal matrix. Furthermore the stochasticity may not be i.i.d. across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random N × N matrices of the form A = M + LJR, where M, L and R are arbitrary deterministic matrices and J is a random matrix of zero-mean independent and identically distributed elements. M can be nonnormal, and L and R allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of A. For A nonnormal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulae for the transient evolution of the magnitude of activity and frequency power spectrum in an N -dimensional linear dynamical system with a coupling matrix given by A. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulae and work them out analytically for some examples of M, L and R motivated by neurobiological models. We also argue that the persistence as N → ∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A, as previously observed, arises in regions of the complex plane Ω where there are nonzero singular values of L−1(z1 − M)R−1 (for z ∈ Ω) that vanish as N → ∞. When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ-pseudospectrum of M.

We demonstrate the mathematical equivalence of two commonly used forms of firing-rate model equations for neural networks. In addition, we show that what is commonly interpreted as the firing rate in one form of model may be better interpreted as a low-pass-filtered firing rate, and we point out a conductance-based firing rate model.

Identifying the regime in which the cortical microcircuit operates is a prerequisite to determine the mechanisms that mediate its response to stimulus. Classic modeling work has started to characterize this regime through the study of perturbations, but an encompassing perspective that links the full ensemble of the network’s response to appropriate descriptors of the cortical operating regime is still lacking. Here we develop a class of mathematically tractable models that exactly describe the modulation of the distribution of cell-type-specific calcium-imaging activity with the contrast of a visual stimulus. The model’s fit recovers signatures of the connectivity structure found in mouse visual cortex. Analysis of this structure subsequently reveal parameter-independent relations between the responses of different cell types to perturbations and each interneuron’s role in circuit-stabilization. Leveraging recent theoretical approaches, we derive explicit expressions for the distribution of responses to partial perturbations which reveal a novel, counter-intuitive effect in the sign of response functions.

Cortical circuits generate excitatory currents that must be cancelled by strong inhibition to assure stability. The resulting excitatory-inhibitory (E-I) balance can generate spontaneous irregular activity but, in standard balanced E-I models, this requires that an extremely strong feedforward bias current be included along with the recurrent excitation and inhibition. The absence of experimental evidence for such large bias currents inspired us to examine an alternative regime that exhibits asynchronous activity without requiring unrealistically large feedforward input. In these networks, irregular spontaneous activity is supported by a continually changing sparse set of neurons. To support this activity, synaptic strengths must be drawn from high-variance distributions. Unlike standard balanced networks, these sparse balance networks exhibit robust nonlinear responses to uniform inputs and non-Gaussian input statistics. Interestingly, the speed, not the size, of synaptic fluctuations dictates the degree of sparsity in the model. In addition to simulations, we provide a mean-field analysis to illustrate the properties of these networks.