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Ion Channel Density Regulates Switches between Regular and Fast Spiking in Soma but Not in Axons

Figure 3

Bifurcation diagrams for the hippocampal neuron model.

(A) A saddle node bifurcation in region C1. There are three stationary voltages in the I_stim range of −40 to +50 mA/m². The oscillations occur when the stable stationary potential V_s1 merges with a saddle point voltage V_s2. Type 1 threshold dynamics is generated if the limit cycle involves the merged point, i.e. a saddle-node bifurcation on an invariant circle (SNIC). = 20 µm/s, = 2 µm/s. (B) Subcritical Andronov-Hopf and double-limit cycle bifurcations in region B, = 20 µm/s, = 10 µm/s. The oscillations emerge at I_stim = 84 mA/m², thus when the corresponding stationary point/voltage still is stable. The loss of stability is due to a double-limit cycle bifurcation, characterized in the variable space by the simultaneous appearance of two limit cycles of opposite stability, one yielding stable and persistent oscillations. This bifurcation is not detectable by the Jacobian matrix of the stationary point; instead the bifurcation depends on the global properties of the variable space. The local Andronov-Hopf bifurcation (also named degenerate Andronov-Hopf bifurcation because of the way the limit cycles collapse onto the equilibrium point [21], [29]) occurs at I_stim = 92 mA/m². There is also an additional Andronov-Hopf bifurcation at higher I_stim (524 mA/m², now shown) that terminates the oscillations. (C) For higher values of (region A2) these two Andronov-Hopf points collide and disappear (the non-transversal Andronov-Hopf bifurcation), after which no Andronov-Hopf points are present = 20 µm/s, = 20 µm/s.

doi: https://doi.org/10.1371/journal.pcbi.1000753.g003