Geometric singular perturbation analysis of a three-timescale coupled reduced Hodgkin–Huxley system

Many physical and biological systems consist of processes that evolve at separate timescales. The mathematical analysis of such systems can be greatly simplified by dividing these systems into subsystems operating on different timescales through the application of geometric singular perturbation theory. In this paper, motivated by applications in neural dynamics, we focus on a four-dimensional model comprising a couple of reduced Hodgkin–Huxley systems, where the variables evolve on three distinct timescales. Through multi-perspective geometric analysis, we obtain a relatively comprehensive view of the oscillatory dynamics of solutions of this inherently three-timescale system. In particular, we reveal that the shapes and relative positions of critical manifold Ms and super-critical manifold Mss are crucial for the emergence and transitions of oscillatory features. We further identify the mechanisms underlying local small-amplitude oscillations. In the pseudo-plateau type bursting, the local oscillations are generated by a supercritical delayed Hopf bifurcation of the fast subsystem and can be converted into a plateauing behavior via parameter changes that alter the geometry of Ms and stability of Mss. In contrast, another type of local oscillation that follows spikes is organized by a special folded saddle singularity and its faux canard, leading to a spike-adding transition. This work yields insights into how multiple timescales interact to produce complex oscillations in a three-timescale coupled system.