Dynamical Systems in Neuroscience:

Excitability 261integrator(saddle-node bifurcation)?resonator(Andronov-Hopf bifurcation)V-nullclinen-nullcline?Figure 7.38: Is there an intermediate mode between integrators and resonators?7.2.11 Transition between integrators and resonatorsConsider the I Na +I K -model or any other minimal model from Chap. 5 that can exhibitsaddle-node or Andronov-Hopf bifurcation, depending on the parameter values. Let usstart with the I Na +I K -model near saddle-node bifurcation, and hence in the integratormode. The intersection of its nullclines at the left knee is similar to the one in Fig. 7.38,left. Now, let us slowly change the parameters toward the values corresponding to theAndronov-Hopf bifurcation with the nullclines intersecting as in Fig. 7.38, right. Atsome point, the behavior of the model must change from integrator to a resonatormode. Is the change sudden, or is it gradual?Any qualitative change of the behavior of the system is a bifurcation. Such abifurcation should somehow combine the saddle-node and the Andronov-Hopf cases;That is, it should have a zero eigenvalue, and a pair of complex-conjugate eigenvalueswith zero real part. Since the I Na +I K -model is two-dimensional, the only way these twoconditions are satisfied is when the model undergoes the Bogdanov-Takens bifurcationconsidered in Sect. 6.3.3. This bifurcation has codimension 2, that is, it can be reliablyobserved when two parameters are changed, in our case E leak and the half-voltage, V 1/2 ,of n ∞ (V ).The top of Fig. 7.39 depicts the phase portrait of the I Na + I K -model at theBogdanov-Takens bifurcation. Notice that the nullclines are tangent near the left knee,but the tangency is degenerate. A small change of the parameter V 1/2 can result eitherin a saddle and a node (middle of the figure) or in a focus equilibrium (bottom of thefigure). The neuron acts as an integrator in the former case and as a resonator in thelatter case.Due to the proximity to a codimension-2 bifurcation, the behavior of the I Na +I K -model is quite degenerate. That is, it could exhibit features that are normallynot observed. For example, the integrator can exhibit post-inhibitory spiking, as inFig. 7.40. This occurs because the shaded region in the figure, bounded by the stablemanifold of the saddle, goes to the left of the resting state. An inhibitory pulse ofcurrent that hyperpolarizes the membrane potential to V < −65 mV and deactivates