All Phase Resetting Curves Are Bimodal, but Some Are More Bimodal Than Others
Figure 4
The firing frequency of Type I excitable cells can be continuously adjusted over a wide range by changing the bias current and can be tuned arbitrarily close to zero (a). The bifurcation diagram for a fixed calcium conductance (b) shows a clear SNIC bifurcation with the fold characterized by the two turning points and , which occur at the SNIC bifurcation point. In all bifurcation diagrams, solid lines mark stable steady states and dashed lines mark unstable ones. Symmetric pairs of solid circles for the same bias current mark peak-to-peak voltages of the stable oscillations, while the open circles mark unstable limit cycle oscillations. On the depolarized steady state branch (continuous line for in panel (b)) there is a subcritical Andronov-Hopf and double-limit cycle bifurcations with stable oscillations emerging at (see the point ) while the corresponding steady state is still stable. The loss of stability is due to a double-limit cycle bifurcation, characterized by the simultaneous appearance of two limit cycles of opposite stability (solid circles—stable and open circles—unstable limit cycles). The local Andronov-Hopf bifurcation, also named degenerated Andronov-Hopf bifurcation [38, 39], occurs at (HP point in panel (b)). The ratio of the amplitudes of the two lobes of the PRC can be continuously tuned over a few orders of magnitude (c) from almost (near the SNIC bifurcation point) up to almost one unit (close to Andronov-Hopf bifurcation marked as HB). The numerically generated PRCs in response to brief square pulses were fitted with a linear combination of the theoretical Type I PRC () and Type II PRC (). The ratio of the coefficient of against shows that very close to the SNIC bifurcation point the Type I PRC “unimodal” character dominates over the Type II “bimodal” character of the PRC (d). When the ratio of the two parameters is close to 1 (see the horizontal dashed line in panel (d)), the two types balance. The combination of a SNIC bifurcation point at low values of the bias current and Andronov-Hopf bifurcation point at large bias currents (SNIC-HB bifurcation diagram) leads to a PRC that is a linear combination of a pure Type I and a pure Type II PRC.
