Ahmadian Yashar, Fumarola Francesco, Miller Kenneth D
Center for Theoretical Neuroscience, Department of Neuroscience, and Swartz Program in Theoretical Neuroscience, and Kavli Institute for Brain Science, College of Physicians and Surgeons, Columbia University, New York, New York 10032, USA.
Center for Theoretical Neuroscience, Department of Neuroscience.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Jan;91(1):012820. doi: 10.1103/PhysRevE.91.012820. Epub 2015 Jan 26.
Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a non-normal matrix. Furthermore, the stochasticity may not be independent and identically distributed (iid) 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 iid elements. M can be non-normal, 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 non-normal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulas 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 formulas 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.
许多学科（包括神经科学和数学生物学）所研究的网络，其连通性可能围绕由非正态矩阵表示的某个潜在平均连通性呈随机分布。此外，连通性矩阵各元素之间的随机性可能并非独立同分布（iid）。更一般地，在许多情况下都会出现理解具有非平凡平均结构和相关性的随机矩阵行为的问题。我们通过对形式为(A = M + LJR)的大型随机(N×N)矩阵进行特征描述来解决这个问题，其中(M)、(L)和(R)是任意确定性矩阵，(J)是一个具有零均值独立同分布元素的随机矩阵。(M)可以是非正态的，(L)和(R)允许具有对行和列索引可分离依赖的相关性。我们首先给出(A)的特征值密度的一般公式。对于非正态的(A)，特征值不足以确定由(A)诱导的动力学，因此我们还给出了一个(N)维线性动力系统中活动幅度和频率功率谱瞬态演化的一般公式，该系统的耦合矩阵由(A)给出。这些量也可以被视为表征非线性网络在固定点附近对小扰动的稳定性和线性响应的幅度。我们推导这些公式，并针对一些受神经生物学模型启发的(M)、(L)和(R)示例进行解析求解。我们还认为，如先前观察到的，在复平面(\Omega)的某些区域中，当(L^{(-1)}(zI - M)R^{(-1)})（对于(z \in \Omega)）的非零奇异值随着(N \to \infty)消失时，(A)的特征值密度支撑之外会存在有限数量的随机分布的离群特征值，并且当(N \to \infty)时这些离群特征值会持续存在。当不存在这样的奇异值且(L)和(R)等于单位矩阵时，对于范数为(\sigma)的(J)，(A)的谱支撑与(M)的(\sigma)伪谱之间在归一化弗罗贝尼乌斯范数（但不是算子范数）上存在对应关系。
