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

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.

May We Have Your Attention: Analysis of a Selective Attention Task

In this paper we present a deeper analysis than has previously been carried out of a selective attention problem, and the evolution of continuous-time recurrent neural networks to solve it. We show that the task has a rich structure, and agents must solve a variety of subproblems to perform well. We consider the relationship between the complexity of an agent and the ease with which it can evolve behavior that generalizes well across subproblems, and demonstrate a shaping protocol that improves generalization.