Dynamical Systems in Neuroscience:

256 Excitabilitymembrane potential, V (mV)200-20-40Na+ inactivation gate, h-600.5I=-20h (V)I=10 0.60I=-15-80 mV rest 20 mV0.70 50 100 150-60 -40 -20 0 20 40time (ms)membrane potential, V (mV)00.10.20.30.4I=-10I=-15I=5I=0I=-5I=10h-nullcline1m (V)V-nullclinesFigure 7.32: Inhibition-induced spiking in the I Na,t -model. Parameters are the sameas in Fig. 5.6b, except g leak = 1.5 and m ∞ (V ) has k = 27.hyperpolarizationdeinactivationof I Naexcessof I Nanegativeinjecteddc-currentrestingpotentialinactivationof I Naactivationof I NaspikeFigure 7.33: Mechanism of inhibitioninducedspiking in the I Na,t -model.vation and the Na + conductance, g Na mh, increases. This leads to an imbalance of theinward current and to the generation of the first spike. During the spike, the currentinactivates completely, and the leak and negative injected currents repolarize and thenhyperpolarize the membrane. During the hyperpolarization, clearly seen in the figure,Na + current deinactivates and is ready for the generation of the next spike.To understand the dynamic mechanism of such an inhibition-induced spiking, weneed to consider the geometry of the nullclines of the model, depicted in Fig. 7.32,bottom. Notice how the position of the V -nullcline depends on I. Negative I shifts thenullcline down and leftward so that the vertex of its left knee, marked by a dot, movesto the left. As a result, the equilibrium of the system, which is the intersection of theV - and h-nullclines, moves toward the middle branch of the cubic V -nullcline. WhenI = −2, the equilibrium loses stability via supercritical Andronov-Hopf bifurcation,and the model exhibits periodic activity.Instead of the I Na,t -model, we could have used the I Na + I K -model or any othermodel with a low-threshold resonant gating variable. The key point here is not the ionicbasis of the spike-generation mechanism, but its dynamic attribute — the Andronov-Hopf bifurcation. Even the FitzHugh-Nagumo model (4.11, 4.12) can exhibit thisphenomenon; see Ex. 1.