Dynamics of the Purkinje cell model (Equations 3a-3e) at several values of J: (a) rapid spiking, at ; (b) AM spiking or headless torus canard, at ; (c) torus canard with head, at ; (d) standard fold/fold cycle bursting at . The upper and middle panel of each frame show, respectively, the time series of the voltage V and the slow gating variable . In the lower panel of each frame, the trajectory of the full system (blue curve) is plotted in projection in the phase space, along with the bifurcation diagram of the fast system at the corresponding value of J. The bifurcation diagrams include branches of fixed points (black curves) and periodic orbits (two red curves, indicating maximal and minimal values of V over the orbit). Solid/dashed curves indicate stable/unstable solutions of the fast system. The labels mark saddle-node bifurcations of fixed points (SNf), saddle-node bifurcations of periodic orbits (SNp), and homoclinic bifurcations (HC). Arrows indicate the direction of drift in for the trajectories.
Burke et al. The Journal of Mathematical Neuroscience 2012 2:3 doi:10.1186/2190-8567-2-3