Canard-induced mixed-mode oscillations and bifurcation analysis in a reduced 3D pyramidal cell model

Abstract

This study entails investigation of mixed-mode oscillations (MMOs) with a conductance-based pyramidal cell (PC) model located in the entorhinal cortex layer V. This six-dimensional neuron model was reduced to three dimensions by analysis of the voltage-dependent timescales to illustrate a regime in which the MMOs are generated. Additionally, the mechanism of generation of the MMOs under antiepileptic drug conditions was illustrated in the 3D PC model. Combined with geometric singular perturbation theory (GSPT), this work shows that there is a range of parameters under which the reduced model explains the emergence of MMOs caused by an underlying canard phenomenon. In particular, we theoretically calculate the number of subthreshold oscillations using the relationship with the eigenvalue ratio of the singular perturbation system at the folded singular node, which is consistent with numerical simulations. Furthermore, a slow–fast dynamics analysis of the 3D PC model is performed, where two slow/one fast and one slow/two fast systems with the layer problem and the reduced problem are considered to explain the trajectory on the critical manifold. General one- and two-parameter bifurcation types are also discussed in this work. The first Lyapunov coefficient of the Hopf bifurcation can decide whether the bifurcation is supercritical or subcritical. Bogdanov–Takens (BT) bifurcation was also analyzed in this study and associated with three bifurcation curves near the BT point. Finally, studies on the GSPT and bifurcation analysis are of great importance for further understanding the complex dynamic behaviors and crucial roles of the signal transmission and information processing pathways of the biological nervous system.