Fig. 7.
Phase-plane explanation for the response regimes in Fig. 4(d). Panels (a1) and (a2) are for the case of quiescence. (a1): Sample time courses of V (solid) and hf (dashed) with I0 = 10 pA and Ihold = −200pA. The filled circle denotes the holding state. (a2): Phase plane analysis for the case in (a1). The dashed line is the V-nullcline with hf =1 at the end of hyperpolarization (Ihold= −200 pA). The shaded areas show where dV/dt > 0 at that time; dV/dt becomes negative as soon as a trajectory crosses the V-nullcline from these areas. The solid lines are all for depolarization during which hf decreases. Similar to the hyperpolarization case, the areas leftward of the left portion of the V-nullcline or under the inverted U-shaped curve (not shown completely) show where dV/dt> 0 corresponding to hf=1 and 0.034, respectively, while dV/dt becomes negative outside these areas. The solid curve with two arrows is the trajectory corresponding to (a1); flow is down and left from the rest state to the holding state during the hyperpolarization and then up and right toward the stable state of depolarization. Similarly, panels (b1) and (b2) show, respectively, the sample time courses (Ihold = −200 pA, I0 = 400 pA) and the corresponding phase plane analysis for the case of regular firing. The curves in (b1, b2) are defined similar to those in (a1, a2). The open square is the resetting point (V= −70 mV, mf = 0.6) which is outside of the loop-like threshold separatrix (curve with double arrows). Panels (c1, c2) and (d1, d2) illustrate the transition mechanism from pauser to buildup with a fixed depolarizing current (I0=130pA) but different levels of pre-hyperpolarization. With strong enough hyperpolarization, Ihold = −150 pA (c1, c2), hf increases to 0.86. Both the holding state (black circle) and the resetting state (d2) are leftward of the threshold separatrix (double arrows), i.e., they are subthreshold, inside the steady state’s attracting domain. The trajectory (single arrow) will converge directly to the steady state upon depolarization. The periodic behavior occurs after the steady state disappears through a saddle-node bifurcation (open circle), i.e., the left and the middle branches of the V-nullcline become tangent with the mf-nullcline (hf=0.31). For (d1, d2), hf is 0.68 just after hyperpolarization (Ihold = −120 pA). The threshold separatrix is more U-shaped and the holding state is superthreshold. The trajectory immediately goes rightward and the system will be reset after an upstroke occurs (hf = 0.57). Because the resetting point (open square) is in the rest state’s attracting domain, the trajectory will converge to the rest state and wait for the resetting state to disappear after which repetitive firing ensues