The Global Structure of Codimension-2 Local Bifurcations in Continuous-Time Recurrent Neural Networks
This work addresses the need for more comprehensive theoretical frameworks in computational neuroscience and neural networks, though it is incremental as it builds on prior bifurcation studies.
The paper tackles the challenge of developing general theories for neural circuits by extending the analysis of parameter space structure in continuous-time recurrent neural networks (CTRNNs) from codimension-1 to codimension-2 local bifurcations, deriving necessary conditions for these bifurcations in general CTRNNs and applying them to circuits with up to four neurons to find and trace global bifurcation manifolds.
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.