Mixed-mode oscillations for slow-fast perturbed systems

Abstract

This article concerns the dynamics of mixed-mode oscillations (MMOs) emerging from the calcium-based inner hair cells (IHCs) model in the auditory cortex. The paper captures the MMOs generation mechanism based on the geometric singular perturbation theory (GSPT) after exploiting the average analysis for reducing the full model. Our analysis also finds that the critical manifold and folded surface are central to the mechanism of the existence of MMOs at the folded saddle for the perturbed system. The system parameters, such like the maximal calcium channels conductance, controls the firing patterns, and many new oscillations occur for the IHCs model. Tentatively, we conduct dynamic analysis combined with dynamic method based on GSPT by giving slow-fast analysis for the singular perturbed models and bifurcation analysis. In particular, we explore the two-slow-two-fast and three-slow-one-fast IHCs perturbed systems with layer and reduced problems so that differential-algebraic equations are obtained. This paper reveals the underlying dynamic properties of perturbed systems under singular perturbation theory.