Abstract
Many physical systems feature interacting components that evolve on disparate time scales. Significant insights about the dynamics of such systems have resulted from grouping time scales into two classes and exploiting the time scale 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-time-scale decomposition. In this work, we are motivated by applications in neural dynamics to focus on a model consisting of a pair of Morris-Lecar systems coupled so that there are three time scales in the full system. We demonstrate that two approaches previously developed in the context of geometric singular perturbation theory for the analysis of two-time-scale systems extend naturally to the three-time-scale setting, where they complement each other nicely. Our analysis explains the dynamic mechanisms underlying solution features in the three-time-scale model. By comparison with certain two-time-scale versions of the same system, we identify some solution properties that truly require three time scales and thus can be viewed as indicators that the presence of three time scales in a system is functionally relevant.