Analysis of Complex Bursting Patterns in Multiple Timescale Respiratory Neuron Models

Many physical systems feature interacting components that evolve on disparate timescales. Significant insights about the dynamics of such systems have resulted from grouping timescales into two classes and exploiting the timescale separation between classes through the use of geometric singular perturbation theory. It is natural to expect, however, that some dynamic phenomena cannot be captured by a two timescale decomposition. One example is the mixed burst firing mode, observed in both recordings and model pre-B\"{o}tzinger neurons, which appears to involve at least three timescales based on its time course. With this motivation, we construct a model system consisting of a pair of Morris-Lecar systems coupled so that there are three timescales in the full system. We demonstrate that the approach previously developed in the context of geometric singular perturbation theory for the analysis of two timescale systems extends naturally to the three timescale setting. To elucidate which characteristics truly represent three timescale features, we investigate certain reductions to two timescales and the parameter dependence of solution features in the three timescale framework. Furthermore, these analyses and methods are extended and applied to understand multiple timescale bursting dynamics in a realistic single pre-B\"{o}tzinger complex neuron and a heterogeneous population of these neurons, both of which can generate a novel mixed bursting (MB) solution, also observed in pre-B\"{o}tC neuron recordings. Rather surprisingly, we discover that a third timescale is not actually required to generate mixed bursting solution in the single neuron model, whereas at least three timescales should be involved in the latter model to yield a similar mixed bursting pattern. Through our analysis of timescales, we also elucidate how the single pre-B\"{o}tC neuron model can be tuned to improve the robustness of the MB solution