Biophysical Basis for Three Distinct Dynamical Mechanisms of Action Potential Initiation
Figure 2
Each class of excitability is derived from a distinct dynamical mechanism of spike initiation.
(A) Phase planes show the fast activation variable V plotted against the slower recovery variable w. Nullclines represent all points in phase space where V or w remain constant. V-nullclines (colored) were calculated at rest (red) and at the onset of stimulation (blue) (Istim is indicated beside each curve); w-nullclines do not change upon stimulation and are plotted only once (gray). Black curves show response of model with direction of trajectory indicated by arrows. Class 1 neuron: Red and gray nullclines intersect at three points (red arrowheads) representing stable (s) or unstable (u) fixed points. Stimulation shifts that V-nullcline upward and destroys two of those points, thereby allowing the system to enter a limit cycle and spike repetitively. The trajectory slows as it passes through constriction between blue and gray nullclines (yellow shading) thereby allowing the neuron to spike slowly, hence the continuous f–I curve. Class 2 neuron: Red and gray curves intersect at a single, stable fixed point. Spiking begins when stimulation destabilizes (rather than destroys) that point. The f–I curve is discontinuous because slow spiking is not possible without the constriction (compare with class 1 neuron). Class 3 neuron: Stimulation displaces but does not destroy or destabilize the fixed point. System variables V,w can follow different paths to the newly positioned fixed point: a single spike is initiated when stimulation instantaneously displaces the quasi-separatrix (dotted curves) so that the system, which existed above the (red) quasi-separatrix prior to stimulation, finds itself below the (blue) quasi-separatrix once stimulation begins; the trajectory must go around the head of the quasi-separatrix (*) to get to the new fixed point – we refer to this mechanism of spike initiation as a quasi-separatrix-crossing or QSC. Dashed black curve shows alternative, subthreshold path that would be followed if trajectory remained above the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as Istim is increased. A bifurcation represents the transition from quiescence to repetitive spiking. Type of bifurcation is indicated on each plot. The range of Istim over which a QSC occurs is indicated in gray and was determined by separate simulations since a QSC is not revealed by bifurcation analysis.