Sequential neural activity has been observed in many parts of the brain and has been proposed as a neural mechanism for memory. The natural world expresses temporal relationships at a wide range of scales. Because we cannot know the relevant scales a priori, it is desirable that memory, and thus the generated sequences, is scale invariant. Although recurrent neural network models have been proposed as a mechanism for generating sequences, the requirements for scale-invariant sequences are not known. This letter reports the constraints that enable a linear recurrent neural network model to generate scale-invariant sequential activity. A straightforward eigendecomposition analysis results in two independent conditions that are required for scale invariance for connectivity matrices with real, distinct eigenvalues. First, the eigenvalues of the network must be geometrically spaced. Second, the eigenvectors must be related to one another via translation. These constraints are easily generalizable for matrices that have complex and distinct eigenvalues. Analogous albeit less compact constraints hold for matrices with degenerate eigenvalues. These constraints, along with considerations on initial conditions, provide a general recipe to build linear recurrent neural networks that support scale-invariant sequential activity.

中文翻译：

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线性循环网络中尺度不变序列活动的生成

已经在大脑的许多部分观察到顺序神经活动，并被提出作为记忆的神经机制。自然世界在广泛的尺度上表达时间关系。因为我们不能先验地知道相关的尺度，所以希望内存和生成的序列是尺度不变的。尽管循环神经网络模型已被提出作为生成序列的机制，但对尺度不变序列的要求尚不清楚。这封信报告了使线性循环神经网络模型能够生成尺度不变序列活动的约束。一个简单的特征分解分析会产生两个独立的条件，这些条件是具有真实、不同特征值的连通性矩阵的尺度不变性所必需的。第一的，网络的特征值必须是几何间隔的。其次，特征向量必须通过翻译相互关联。对于具有复杂且不同的特征值的矩阵，这些约束很容易推广。对于具有退化特征值的矩阵，类似但不那么紧凑的约束条件成立。这些约束以及对初始条件的考虑提供了构建支持尺度不变顺序活动的线性循环神经网络的通用方法。