Changes in Repetitive Firing Rate Related to Phase Response Curves for Andronov—Hopf Bifurcations

Abstract

We study specific changes in repetitive firing in the two-dimensional Hindmarsh—Rose (2dHR) oscillatory system that undergoes a bifurcation transition from the supercritical Andronov—Hopf (AH) type to the subcritical Andronov—Hopf (SAH) type. We identify dynamical mechanisms which are responsible for changes of the repetitive firing rate during the AH to SAH bifurcation transitions. These include frequency-shift functions in response to small perturbations of a timescale parameter, its multiplicative parameter, and an external input current in the 2dHR oscillatory system. The frequency-shift functions are explicitly represented as functions relating to the phase response curves (PRCs). Then, we demonstrate that when the timescale is normal and relatively fast, the repetitive firing rate slightly increases and decreases respectively during the AH to SAH bifurcation transition with a change of the intrinsic parameter, whereas it decreases during the SAH to AH bifurcation transition with an increase in the timescale. By analyzing the three different frequency-shift functions, we show that such changes of the repetitive firing rate depend largely on changes of the PRC size. The PRC size for the SAH bifurcation shrinks to the PRC size for the AH bifurcation.