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

TL;DR

This paper develops a comprehensive theoretical framework for understanding the structure of codimension-2 local bifurcations in continuous-time recurrent neural networks, extending previous bifurcation analysis to more complex scenarios.

Contribution

It derives necessary conditions for all generic codimension-2 bifurcations in CTRNNs and applies these to small circuits, advancing the understanding of neural circuit dynamics.

Findings

Derived conditions for codimension-2 bifurcations in CTRNNs

Applied bifurcation analysis to circuits with 1-4 neurons

Identified global bifurcation manifolds originating from codimension-2 points

Abstract

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.