SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 165-207, January 2021.
Replica-mean-field models have been proposed to decipher the activity of otherwise analytically intractable neural networks via a multiply-and-conquer approach. In this approach, one considers limit networks made of infinitely many replicas with the same basic neural structure as that of the network of interest, but exchanging spikes in a randomized manner. The key point is that these replica-mean-field networks are tractable versions that retain important features of the finite structure of interest. To date, the replica framework has been discussed for first-order models, whereby elementary replica constituents are single neurons with independent Poisson inputs. Here, we extend this replica framework to allow elementary replica constituents to be composite objects, namely, pairs of neurons. As they include pairwise interactions, these pair-replica models exhibit nontrivial dependencies in their stationary dynamics, which cannot be captured by first-order replica models. Our contributions are two-fold: (i) We analytically characterize the stationary dynamics of a pair of intensity-based neurons with independent Poisson input. This analysis involves the reduction of a boundary-value problem related to a two-dimensional transport equation to a system of Fredholm integral equations---a result of independent interest. (ii) We analyze the set of consistency equations determining the full network dynamics of certain replica limits. These limits are those for which replica constituents, be they single neurons or pairs of neurons, form a partition of the network of interest. Both analyses are numerically validated by computing input/output transfer functions for neuronal pairs and by computing the correlation structure of certain pair-dominated network dynamics.

中文翻译：

---

基于强度的神经网络的配对-重复-平均场极限

SIAM应用动力系统杂志，第20卷，第1期，第165-207页，2021年1月。
已经提出了复制均值场模型，以通过乘胜法解译本来难以分析的神经网络的活动。在这种方法中，人们考虑了由无限多个副本构成的限制网络，这些副本具有与目标网络相同的基本神经结构，但是以随机方式交换峰值。关键是这些复制均值网络是易于处理的版本，保留了感兴趣的有限结构的重要特征。迄今为止，已经针对一阶模型讨论了复制框架，其中基本复制成分是具有独立泊松输入的单个神经元。在这里，我们扩展了此副本框架，以允许基本副本成分成为复合对象，即成对的神经元。由于它们包括成对交互，这些成对复制品模型在其静态动力学中表现出非平凡的依存关系，而一阶复制品模型无法捕获这些依赖关系。我们的贡献有两个方面：（i）我们分析性地描述了具有独立泊松输入的一对基于强度的神经元的平稳动力学。该分析涉及将与二维输运方程有关的边值问题简化为Fredholm积分方程组-这是独立关注的结果。（ii）我们分析一致性方程组，以确定某些副本限制的完整网络动态。这些限制是指复制成分（单个神经元或成对神经元）形成目标网络分区的限制。