Tanaka Hidenori, Nelson David R
Department of Applied Physics, Stanford University, Stanford, California 94305, USA.
School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA.
Phys Rev E. 2019 Jun;99(6-1):062406. doi: 10.1103/PhysRevE.99.062406.
Eigenmodes of a broad class of "sparse" random matrices, with interactions concentrated near the diagonal, exponentially localize in space, as initially discovered in 1957 by Anderson for quantum systems. Anderson localization plays ubiquitous roles in varieties of problems from electrons in solids to mechanical and optical systems. However, its implications in neuroscience (where the connections can be strongly asymmetric) have been largely unexplored, mainly because synaptic connectivity matrices of neural systems are often "dense," which makes the eigenmodes spatially extended. Here we explore roles that Anderson localization could be playing in neural networks by focusing on "spatially structured" disorder in synaptic connectivity matrices. Recently neuroscientists have experimentally confirmed that the local excitation and global inhibition (LEGI) ring attractor model can functionally represent head direction cells in Drosophila melanogaster central brain. We first study a non-Hermitian (i.e., asymmetric) tight-binding model with disorder and then establish a connection to the LEGI ring attractor model. We discover that (1) principal eigenvectors of the LEGI ring attractor networks with structured nearest-neighbor disorder are "quasilocalized," even with fully dense inhibitory connections; and (2) the quasilocalized eigenvectors play dominant roles in the early time neural dynamics, and the location of the principal quasilocalized eigenvectors predicts an initial location of the "bump of activity" representing, for example, a head direction of an insect. Our investigations open up venues for explorations at the intersection between the theory of Anderson localization and neural networks with spatially structured disorder.
一类“稀疏”随机矩阵的本征模，其相互作用集中在对角线附近，在空间中呈指数局域化，这最初是1957年由安德森在量子系统中发现的。安德森局域化在从固体中的电子到机械和光学系统等各种问题中都发挥着普遍作用。然而，其在神经科学中的意义（其中连接可能是高度不对称的）在很大程度上尚未得到探索，主要是因为神经系统的突触连接矩阵通常是“密集的”，这使得本征模在空间上是扩展的。在这里，我们通过关注突触连接矩阵中的“空间结构”无序来探索安德森局域化在神经网络中可能发挥的作用。最近，神经科学家通过实验证实，局部兴奋和全局抑制（LEGI）环形吸引子模型可以在功能上表示果蝇中枢脑中的头部方向细胞。我们首先研究一个具有无序的非厄米（即不对称）紧束缚模型，然后建立与LEGI环形吸引子模型的联系。我们发现：（1）具有结构化最近邻无序的LEGI环形吸引子网络的主特征向量是“准局域化”的，即使具有完全密集的抑制连接；（2）准局域化特征向量在早期神经动力学中起主导作用，并且主准局域化特征向量的位置预测了代表例如昆虫头部方向的“活动峰”的初始位置。我们的研究为在安德森局域化理论与具有空间结构无序的神经网络之间的交叉领域进行探索开辟了道路。
