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Biophysical Basis for Three Distinct Dynamical Mechanisms of Action Potential Initiation
Figure 9
Competition between kinetically mismatched currents.
(A) Top panels show individual currents in 2D model; bottom panels show how they combine to produce the instantaneous (Iinst) and steady state (Iss) I–V curves. Double-headed arrows highlight effect of βw on the voltage-dependency of Islow. Class 3 neuron: Islow activates at lower V than Ifast, meaning slow negative feedback keeps V from increasing high enough to initiate fast positive feedback at steady state. Fast positive feedback (that results in a spike) can be initiated only if the system is perturbed from steady state. Quasi-separatrix (blue) has a region of negative slope (*) indicating where net positive feedback occurs given the kinetic difference between fast and slow currents: positive feedback that activates rapidly can compete effectively with stronger negative feedback whose full activation is delayed by its slower kinetics. If V is forced rapidly past the blue arrowhead, fast positive feedback initiates a single spike before slow negative feedback catches up and forces the system back to its stable fixed point. Quasi-separatrix is plotted as the sum of all currents but with Islow calculated as a function of w at the quasi-separatrix (see phase plane in Figure 2A) rather than at steady state and is shown here for Istim = 60 µA/cm2. Class 2 neuron: Islow and Ifast activate at roughly the same V. A Hopf bifurcation occurs at the point indicated by the arrow, where 
(see Results). This means that fast positive feedback exceeds slow negative feedback at steady state; as for class 3 neurons, this relies on positive feedback having fast kinetics since the net perithreshold current is still outward (i.e., steady state I–V curve is monotonic). Note that the slope of the steady-state I–V curve is less steep in the class 2 model than in the class 3 model. Class 1 neuron: Islow activates at higher V than Ifast, meaning slow negative feedback does not begin activating until after the spike is initiated. This gives a steady state I–V curve that is non-monotonic with a region of negative slope (*) near the apex of the instantaneous I–V curve. The SNIC bifurcation occurs when ∂Iss/∂t = 0 (arrowhead) because, at this voltage, Ifast counterbalances Ileak and any further depolarization will cause progressive activation of Ifast. (B) Changing ḡfast in the 2D model had equivalent effects on the shape of the steady state I–V curves. Unlike in (A), voltage at the apex of the instantaneous I–V curve (purple arrows) changes as ḡfast is varied; in other words, the net current at perithreshold potentials can be modulated by changing fast currents (which directly impact voltage threshold) rather than by changing the amplitude or voltage-dependency of slow currents. This is consistent with results in Figure 8. (C) Speeding up the kinetics of Islow impacts the onset of class 2 and 3 excitability. Compared with original model (φw = 0.15; black), increasing φw to 0.25 (red) increased Istim required to cause a Hopf bifurcation or a QSC, but did not affect Istim required to cause an SNIC bifurcation; reducing φw to 0.10 (green) had the opposite effect (summarized in right panel). Increasing φw also widened the discontinuity in the class 2 f–I curve and allowed class 2 and 3 neurons to achieve higher spiking rates with strong Istim because of the faster recovery between spikes; reducing φw had the opposite effects.
doi: https://doi.org/10.1371/journal.pcbi.1000198.g009