Mixed-mode oscillations in a three time scale system of ODEs motivated by a neuronal model

We present a mathematical study of some aspects of mixed-mode oscillation (MMO) dynamics in a three time scale system of ODEs as well as analyze related features of a biophysical model of a neuron from the entorhinal cortex that serves as a motivation for our study. The neuronal model includes standard spiking currents (sodium and potassium) that play a critical role in the analysis of the interspike interval as well as persistent sodium and slow potassium (M) currents. We reduce the dimensionality of the neuronal model from six to three dimensions in order to investigate a regime in which MMOs are generated and to motivate the three time scale model system upon which we focus our study. We further analyze in detail the mechanism of the transition from MMOs to spiking in our model system. In particular, we prove the existence of a special solution, a singular primary canard, that serves as a transition between mixed-mode oscillations and spiking in the singular limit by employing appropriate rescalings and center manifold reductions. Additionally, we conjecture that the singular canard solution is the limit of a family of canards and provide numerical evidence for the conjecture.