Bursting patterns and mixed-mode oscillations in reduced Purkinje model

Abstract

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.