Three Time-Scales In An Extended Bonhoeffer–Van Der Pol Oscillator

We consider an extended three-dimensional Bonhoeffer–van der Pol oscillator which generalises the planar FitzHugh–Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. (Phys Lett A 374(36):3745–3751, 2010 ) and by Freire and Gallas (Phys Lett A 375:1097–1103, 2011 ). Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. (J Appl Dyn Syst 7(2):361–420, 2008 ) as a canonical form for such systems. Based on results previously obtained in Krupa et al. ( 2008 ), we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer–van der Pol oscillator from the point of view of geometric singular perturbation theory, thus complementing the findings reported in Sekikawa et al. ( 2010 ). In particular, we specify in detail the mixed-mode patterns that are observed upon variation of a bifurcation parameter which is naturally obtained by combining two of the original parameters in the system, and we derive asymptotic estimates for the corresponding parameter intervals. We thereby also disprove a conjecture of Tu (SIAM J Appl Math 49(2): 331–343, 1989 ), where it was postulated that no stable periodic orbits of mixed-mode type can be observed in an equivalent extension of the Bonhoeffer–van der Pol equations.