Supplementary Figures.

Fig A The ΔV1/2 × Ibias parameter plane, bistability and automatic regimes Bifurcation structure in the ΔV1/2 × Ibias parameter plane Cyan trace: the Hopf bifurcations (HB) as a function of the Ibias and ΔV1/2 model parameters (see also Fig D.A). Dashed-black lines: Existence/creation parameter-range of the 2 additional fixed points (saddle-node in the middle branch, and unstable center/focus in the top branch of the B.D.’s, see Fig 4A and 4C) Red and blue lines: the ΔV1/2 × Ibias range of stable/unstable PO’s (see also Fig G) Notice that outside (below and over) the HB area, there is bistability with a stable fixed point, while for very large Ibias currents (beyond the blue trace) only the stable depolarized fixed point remains (see also Fig C). For very large ΔV1/2 the stable/unstable PO’s between SNC1 and SNC2 detach from the FP’s locus to form an island (see also Fig F and G)
Fig B The Bifurcation Glossary (Codimension 1 and possibly 2) terms illustrated for two ΔV1/2 cases black lines: stable FP’s. They lose (or recover) stability via Hopf Bifurcations red lines: unstable FP’s. In A, unstable FP’s appear (or disappear) via Saddle node (SN) bifurcations at LP1 and LP2. blue lines: Minimum and Maximum voltage of stable periodic orbits. These can appear (or disappear) through SNC bifurcations or supercritical HB’s (e.g. the low amplitude stable cycles in B). cyan lines: Minimum and Maximum voltage of unstable periodic orbits. These can appear (or disappear) through SNC bifurcations or subcritical HB (e.g. HB2 in A), or homoclinic bifurcation (e.g. the lower open ends in A).
Fig C BD’s for a set of ΔV1/2 values—see legend, by stable PO (SPO) color Inset: Zoom in of the ΔV1/2 = 3 mV case for Ibias ∈ [7, 14]μA/cm2
Fig D Combined BD’s and periods of the high-amplitude cycle—in the ΔV1/2 × Ibias parameter space A: Combined BD’s B: max. cycle-periods as a function of ΔV1/2 × Ibias parameters (B) Note that the periods of the low-amplitude cycles (around SNC3 and SNC4) have not been continued.
Fig E BD’s for a set more of ΔV1/2 values (see the panel legends, by SPO color) (see also Fig C)
Fig F Generalized Hopf in codimension-2: Collision of two Hopf Bifurcations (HB) As the two Hopf’s become closer in parameter space, one unstable PO (UPO) is lost. Moreover, the HB type changes from sub-critical to super-critical (see also Fig G)
Fig G The creation of an island in codimension-2 A little before the 2 HB’s collide, the small-amplitude SPO detaches from the UPO. The latter forms an island together with the large-amplitude SPO. (see also Fig F)
Fig H Very complex dynamic organization for ΔV1/2 = 3 mV and Ibias = 9.5 μA/cm2 A: Temporal evolution of membrane voltage V for the SPO (magenta, green trace on Panel C) and the UPO (black, thick cyan trace on Panel C). B: Gate states dynamics for the SPO (green trace on Panel C). Notice the very limited range of the n gate’s variation. C: The unstable FP invariant directions yield transient trajectories which converge mostly to the SPO. This is due to the UPO (thick cyan trace). The transient trajectories starting at the UPO’s single unstable invariant direction converge respectively to the resting FP and to the SPO (thinner dashed cyan traces). See also Fig H.A
Fig I Examples of dynamic objects (phase-space trajectories) corresponding to bifurcations in codimension-2. Ibias variation for ΔV1/2 = 3 mV results in a complex organization (see the B.D. in Fig H—middle row and Inset). The figure presents the rich diversity of dynamic objects (e.g. heteroclinic, homoclinic, stable and unstable periodic orbits) by simulating them for 3 values of the Ibias parameter: 7.44, 7.49 and 9.068 μA/cm2. Heteroclinic (dashed red), homoclinic (blue), stable (green) and unstable (cyan and black, Panel A only) periodic orbits
Fig J An unstable limit cycle as a phase-space separatrix A: Due to the existence of an UPO (cyan), all trajectories (black and red) initiated on the unstable FP’s (red triangle) invariant directions deviate toward the resting FP (blue circle) and not to the SPO (green). Incidently (and like in the B.D.’s on Fig F.C), the UPO’s own invariant direction yields 2 transient trajectories (data not shown) which converge respectively to the resting FP and to the SPO. B: An UPO is absent here unlike Panel A. Hence, the same initial conditions as in Panel A yield transient trajectories half of which (2 out of 4) converge to the SPO.
(PDF)