Dynamical Systems in Neuroscience:

Two-Dimensional Systems 1230-100-10max V(t)membrane voltage, V-20-30-40-50-60-70stableAndronov-Hopf bifurcationis somewhere hereunstableI (V)membrane voltage, V-20-30-40-50-60-70supercriticalAndronov-Hopfbifurcationrestmin V(t)periodic orbitsunstable equilibrium-800 20 40 60 80 100injected dc-current, I-800 20 40 60 80 100injected dc-current, IabFigure 4.35: Andronov-Hopf bifurcation diagram in the I Na,p +I K -model with lowthresholdK + current. a. Equilibria of the model (solution of (4.13)). b. Equilibriaand limit cycles of the model.In Fig. 4.35a we plot the solution of (4.13) as an attempt to determine the bifurcationdiagram for the Andronov-Hopf bifurcation in the I Na,p +I K -model. However,all we can see is that the equilibrium persists as I increases, but there is no informationon its stability or on the existence of a limit cycle attractor. To study the limitcycle attractor, we need to simulate the model with various values of parameter I.For each I, we disregard the transient period and plot min V (t) and max V (t) on the(I, V )-plane, as in Fig. 4.35b. When I is small, the solutions converge to the stableequilibrium, and both min V (t) and max V (t) are equal to the resting voltage. WhenI increases past I = 12, the min V (t) and max V (t) values start to diverge, meaningthat there is a limit cycle attractor whose amplitude increases as I does. This methodis appropriate for analysis of supercritical Andronov-Hopf bifurcations but it fails forsubcritical Andronov-Hopf bifurcations. Why?Figure 4.36 depicts an interesting phenomenon observed in many biological neurons— excitation block. Spiking activity of the layer 5 pyramidal neuron of rat’s visualcortex is blocked by strong excitation, i.e., injection of strong depolarizing current. Thegeometry of this phenomenon is illustrated in Fig. 4.37, bottom. As the magnitude ofthe injected current increases, the unstable equilibrium, which is the intersection pointof the nullclines, moves to right branch of the cubic V -nullcline and becomes stable.The limit cycle shrinks and the spiking activity disappears, typically but not necessarilyvia supercritical Andronov-Hopf type. Thus, the I Na,p +I K -model with low-thresholdK + current can exhibit two such bifurcations in response to ramping up of the injectedcurrent, one leading to the appearance of periodic spiking activity (Fig. 4.34), and thenone leading to its disappearance (Fig. 4.37).