连续时间循环神经网络余维2局部分岔的全局结构
The Global Structure of Codimension-2 Local Bifurcations in Continuous-Time Recurrent Neural Networks
摘要：
如果我们要超越理论神经科学中孤立的特殊情况的研究，我们需要在给定的神经模型上发展更一般的神经回路理论。本文认为这一挑战的背景下，连续时间循环神经网络（CTRNNs），一个简单的，但动态通用的模型，已被广泛用于计算神经科学和神经网络。在这里，我们扩展了以前的工作的参数空间结构的余维-1局部分叉CTRNNs，包括余维-2局部分叉流形。具体来说，我们推导出一般CTRNN的所有通用局部余维2分叉的必要条件，将这些条件专门用于包含一到四个神经元的电路，详细说明这些条件在示例电路中的应用，在可能的情况下推导出这些分叉流形的闭合形式表达式，并演示了这种分析如何使我们能够找到和跟踪几个全球余维1分歧流形，起源于余维2分歧。
英文摘要：
If we are ever to move beyond the study of isolated special cases in theoretical neuroscience, we need to develop more general theories of neural circuits over a given neural model. The present paper considers this challenge in the context of continuous-time recurrent neural networks (CTRNNs), a simple but dynamically-universal model that has been widely utilized in both computational neuroscience and neural networks. Here we extend previous work on the parameter space structure of codimension-1 local bifurcations in CTRNNs to include codimension-2 local bifurcation manifolds. Specifically, we derive the necessary conditions for all generic local codimension-2 bifurcations for general CTRNNs, specialize these conditions to circuits containing from one to four neurons, illustrate in full detail the application of these conditions to example circuits, derive closed-form expressions for these bifurcation manifolds where possible, and demonstrate how this analysis allows us to find and trace several global codimension-1 bifurcation manifolds that originate from the codimension-2 bifurcations.
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