与神经网络相关的大型稀疏随机矩阵的谱密度
Spectrum density of large sparse random matrices associated to neural networks
摘要：
大型生物网络的耦合矩阵的特征分解是研究这些网络动力学的核心。对于神经网络，这个矩阵应该反映网络的拓扑结构，并符合戴尔定律，该定律指出神经元只能具有所有兴奋性或所有抑制性输出连接，即，耦合矩阵的一列的系数必须全部具有相同的符号。当考虑多个总体时，稠密矩阵$J_{ij}$的本征谱密度已经确定。然而，这些表达式是在密集连接的假设下推导出来的，而神经回路具有稀疏连接。在这里，我们遵循平均场的方法，以提出精确的自洽表达式的频谱密度的极限稀疏矩阵的对称和神经网络矩阵。此外，我们介绍了近似，允许良好的数值评估的密度。最后，我们研究了特征向量局部化性质的现象学。
英文摘要：
The eigendecomposition of the coupling matrix of large biological networks is central to the study of the dynamics of these networks. For neural networks, this matrix should reflect the topology of the network and conform with Dale's law which states that a neuron can have only all excitatory or only all inhibitory output connections, i.e., coefficients of one column of the coupling matrix must all have the same sign. The eigenspectrum density has been determined before for dense matrices $J_{ij}$, when several populations are considered. However, the expressions were derived under the assumption of dense connectivity, whereas neural circuits have sparse connections. Here, we followed mean-field approaches in order to come up with exact self-consistent expressions for the spectrum density in the limit of sparse matrices for both symmetric and neural network matrices. Furthermore we introduced approximations that allow for good numerical evaluation of the density. Finally, we studied the phenomenology of localization properties of the eigenvectors.
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