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

Abstract

Abstract 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 analyse 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 analyse 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 MMOs and spiking in the singular limit by employing appropriate rescalings and centre manifold reductions. Additionally, we conjecture that the singular canard solution is the limit of a family of canards and provides numerical evidence for the conjecture. Keywords: canardssingular perturbationentorhinal cortex Acknowledgements The authors are deeply indebted to Nancy Kopell for her numerous contributions to this article. The authors would also like to thank John Guckenheimer, Nicola Popović and Martin Wechselberger for fruitful discussions. The research of J. Jalics was supported by a YSU Research Professorship. The research of M. Krupa was supported by NSF grant DMS-0406608. The research of H.G. Rotstein was supported by NSF grant DMS-0817241.