# The Global Structure of Codimension-2 Local Bifurcations in Continuous-Time Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2111.04547 · HTML: https://mlsanity.com/paper/2111.04547/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2021.

## Built on by (most cited first, 1 of 1)

- [Seeing double with a multifunctional reservoir computer](https://mlsanity.com/paper/2305.05799/) (2023)

## Abstract (arXiv)

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.