Codimension-2 parameter space structure of continuous-time recurrent neural networks
- 20 June 2022
- journal article
- research article
- Published by Springer Science and Business Media LLC in Biological Cybernetics
- Vol. 116 (4), 501-515
- https://doi.org/10.1007/s00422-022-00938-5
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.Keywords
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