Dynamical Systems in Neuroscience:

174 BifurcationsBoth types of the bifurcation can occur in the I Na,p +I K -model as we show in Fig. 6.7.The difference between top and bottom of the figure is the time constant τ(V ) of theK + current. Since the K + current has high threshold, the time constant does not affectdynamics at rest, but it makes a huge difference when action potential is generated.If the current is fast (top), it activates during the upstroke thereby decreasing theamplitude of action potential, and deactivates during the downstroke thereby resultingin overshoot and another action potential. In contrast, slower K + current (bottom)does not have time to deactivate during the downstroke, thereby resulting in undershoot(short after-hyperpolarization), with V going below the resting state.From the geometrical point of view, the phase portraits in Fig. 6.6b and in Fig. 6.7,bottom, have the same topological structure: there is a homoclinic trajectory (aninvariant circle) that originates at the saddle-node point, leaves its small neighborhood(to fire an action potential), then reenters the neighborhood again, and terminatesat the saddle-node point. This homoclinic trajectory is a limit cycle attractor withinfinite period, which corresponds to firing with zero frequency. This and other neurocomputationalfeatures of saddle-node bifurcations are discussed in the next chapter.Below we only explore how the frequency of oscillation depends on the bifurcationparameter, e.g., on the injected dc-current I.A remarkable fact is that we can estimate the frequency of the large-amplitude limitcycle attractor by considering a small neighborhood of the saddle-node point. Indeed,a trajectory on the limit cycle generates a fast spike from point B to A in Fig. 6.8 andthen slowly moves from A to B (shaded region in the figure) because the vector field(the velocity) in the neighborhood between A and B is very small. The duration of thestereotypical action potentials, denoted here as T 1 , is relatively constant and does notdepend much on the injected current I. In contrast, the time spent in the neighborhood(A, B) depends significantly on I. Since the behavior in the neighborhood is describedby the topological normal form (6.2), we can estimate the time the trajectory spendsthere in terms of the parameters a, b and c (see Ex. 3). This yieldsT 2 =π√ac(b − bsn ) ,where the parameters a, b, and c are those defined in the previous section. So theperiod of one oscillation is T = T 1 + T 2 .In Fig. 6.8, top, we illustrate the accuracy of this estimation using the I Na,p +I K -model, whose topological normal form (6.3) was derived earlier. The duration of theaction potential is T 1 = 4.7 ms, and the duration of time the voltage variable spendsin the shaded neighborhood (A,B) (here −61 ± 11 mV) is approximated byT 2 =π√0.1887(I − 4.51)(ms)The analytical curveω = 1000T 1 + T 2(Hz)