We introduce a new formalism for evaluating analytically the cross-correlation structure of a finite-size firing-rate network with recurrent connections. The analysis performs a first-order perturbative expansion of neural activity equations that include three different sources of randomness: the background noise of the membrane potentials, their initial conditions, and the distribution of the recurrent synaptic weights. This allows the analytical quantification of the relationship between anatomical and functional connectivity, i.e. of how the synaptic connections determine the statistical dependencies at any order among different neurons. The technique we develop is general, but for simplicity and clarity we demonstrate its efficacy by applying it to the case of synaptic connections described by regular graphs. The analytical equations so obtained reveal previously unknown behaviors of recurrent firing-rate networks, especially on how correlations are modified by the external input, by the finite size of the network, by the density of the anatomical connections and by correlation in sources of randomness. In particular, we show that a strong input can make the neurons almost independent, suggesting that functional connectivity does not depend only on the static anatomical connectivity, but also on the external inputs. Moreover we prove that in general it is not possible to find a mean-field description à la Sznitman of the network, if the anatomical connections are too sparse or our three sources of variability are correlated. To conclude, we show a very counterintuitive phenomenon, which we call stochastic synchronization, through which neurons become almost perfectly correlated even if the sources of randomness are independent. Due to its ability to quantify how activity of individual neurons and the correlation among them depends upon external inputs, the formalism introduced here can serve as a basis for exploring analytically the computational capability of population codes expressed by recurrent neural networks.

中文翻译：

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用于分析评估点火速率网络模型的互相关结构的形式主义。

我们引入了一种新的形式主义，用于分析性地评估具有递归连接的有限大小点火率网络的互相关结构。该分析执行神经活动方程的一阶微扰展开，该方程包括三个不同的随机性来源：膜电位的背景噪声，它们的初始条件以及突触权重的分布。这允许对解剖学连接和功能连接之间的关系进行分析量化，即，突触连接如何确定不同神经元之间任何顺序的统计依赖性。我们开发的技术是通用的，但为简单起见，我们通过将其应用于规则图所描述的突触连接的情况来证明其功效。这样获得的分析方程式揭示了递归发射率网络以前未知的行为，特别是关于如何通过外部输入，网络的有限大小，解剖连接的密度以及随机源中的相关性来修改相关性的行为。特别是，我们显示出强大的输入可以使神经元几乎独立，这表明功能连接不仅取决于静态解剖连接，还取决于外部输入。此外，我们证明，如果解剖连接太稀疏或我们的三个可变性来源相关，则通常无法找到网络的均值描述àla Sznitman。总而言之，我们展示了一种非常违反直觉的现象，我们称之为随机同步，即使随机来源是独立的，通过它神经元也几乎可以完美地相关。由于它能够量化单个神经元的活动以及它们之间的相关性如何依赖于外部输入，因此这里介绍的形式主义可以作为分析探索循环神经网络表示的人口代码的计算能力的基础。