Properties of networks with partially structured and partially random connectivity

Abstract

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.

Low-dimensional dynamics of structured random networks

“…g ( z i , z j ) = g 1 ( z i ) g 2 ( z j )], the model discussed here overlaps with that studied by Wei and by Ahmadian et al [9, 12], but those works also consider matrix models that are not studied here.…”

Dynamics of random neural networks with bistable units

“…We can evaluate stability using the mean-field solutions x ( η ), rather than the networks values x i that appear in equation 7. In the limit N → ∞, the matrix 7 has an eigenvalue at the point z in the complex plane if [7] $${\int }_{-\infty }^{\infty }{D}_{\sigma }\eta {\left(\frac{gtrue(1-{\text{tanh}}^{2}true(x\left(\eta \right)true)true)}{\mid z+1-s\left(1-{\text{tanh}}^2\left(x\left(\eta \right)\right)\right)\mid }\right)}^2>1,$$ where we use the notation $$D_{\sigma }\eta =\frac{d\eta }{\sqrt{2\pi }\sigma }\exp true(-\frac{{\eta }^2}{2{\sigma }^2}true).$$ If we ask whether there is an eigenvalue at the point z =0, this expression simplifies to $$Q={\int }_{-\infty }^{\infty }{D}_{\sigma }\eta {\left(\frac{g}{{\text{cosh}}^{2}true(x\left(\eta \right)true)-s}\right)}^2>1.$$ We use this latter condition below because the inequality 8 does not support isolated eigenvalues [7], so stability can be assessed by determining whether or not there is an eigenvalue at z =0. Stability requires Q ≤ 1, with the edge of stability defined by Q =1.…”

Coherent chaos in a recurrent neural network with structured connectivity

“…This amplification is related to the way in our network, small fluctuations of the residuals are summed 535 along the output mode and drive coherent fluctuations along the input mode in the regime of passive coherence, but these fluctuations are internally driven by the non-linear dynamics whereas the dynamics in those previous studies were linear. As pointed out in [2], a structured component such as in our model is purely feedforward and can be considered an extreme case of non-normality as it has only zero eigenvalues and therefore all the power in its Schur decomposition is in the off-diagonal. The results here depend on this property and cannot be extended to 540 connectivity with only partial feedforward structure.…”

“…Feedforward structure can be revealed by the Schur decomposition and the extent of feedforwardness can be measured as the fraction of total power (sum-of-squares) that is off the diagonal in the Schur decomposition. As pointed out in [2], a structured component such as in our model is purely feedforward and can be considered an extreme case of non-normality as it has only zero eigenvalues and therefore all its power is in the off-diagonal. The results here depend on this property and cannot be extended to connectivity with partially feed-480 forward structure.…”