Order in chaos: Structure of chaotic invariant sets of square-wave neuron models

Bursting phenomena and, in particular, square-wave or fold/hom bursting, are found in a wide variety of mathematical neuron models. These systems have different behavior regimes depending on the parameters, whether spiking, bursting, or chaotic. We study the topological structure of chaotic invariant sets present in square-wave bursting neuron models, first detailed using the Hindmarsh-Rose neuron model and later exemplary in the more realistic model of a leech heart neuron. We show that the unstable periodic orbits that form the skeleton of the chaotic invariant sets are deeply related to the spike-adding phenomena, typical from these models, and how there are specific symbolic sequences and a symbolic grammar that organize how and where the periodic orbits appear. Linking this information with the topological template analysis permits us to understand how the internal structure of the chaotic invariants is modified and how more symbolic sequences are allowed. Furthermore, the results allow us to conjecture that, for these systems, the limit template when the small parameter ε, which controls the slow gating variable, tends to zero is the complete Smale topological template.