Dynamical Systems in Neuroscience:

82 One-Dimensional Systemsmembrane potential, V (mV)40200-20-40-60-80-100-120excited statesthreshold statessaddle-node(fold) bifurcationrest statessaddle-node(fold) bifurcationI (V)16-890membrane potential, V (mV)40200-20-40-60-80-100-120-1000 -500 0injected dc-current, I (pA)-1000 -500 0injected dc-current, I (pA)Figure 3.32: Bifurcation diagram of the I Na,p -model (3.5).the phase portrait of the system, as we illustrate in Fig. 3.32, right. Each point wherethe branches fold (max or min of I ∞ (V )) corresponds to a saddle-node bifurcation.Since there are two such folds, at I = 16 pA and at I = −890 pA, there are two saddlenodebifurcations in the system. The first one, studied in Fig. 3.25, corresponds to thedisappearance of the rest state. The other one, illustrated in Fig. 3.33, correspondsto the disappearance of the excited state. It occurs because I becomes so negativethat the Na + inward current is no longer strong enough to balance the leak outwardcurrent and the negative injected dc-current to keep the membrane in the depolarized(excited) state.Below the reader can find more examples of bifurcation analysis of the I Na,p - andI Kir -models, which have non-monotonic I-V relations and can exhibit multi-stabilityof states. The I K - and I h -models have monotonic I-V relations and hence only oneequilibrium state. These models cannot have saddle-node bifurcations, as the readeris asked to prove in Ex. 13 and 14.3.3.8 Quadratic integrate-and-fire neuronLet us consider the topological normal form for the saddle-node bifurcation (3.9). From0 = I + V 2 we find that there are two equilibria, V rest = − √ |I| and V thresh = + √ |I|when I < 0. The equilibria approach and annihilate each other via saddle-node bifurcationwhen I = 0, so there are no equilibria when I > 0. In this case, ˙V ≥ I and V (t)increases to infinity. Because of the quadratic term, the rate of increase also increases,resulting in a positive feedback loop corresponding to the regenerative activation ofNa + current. In Ex. 15 we show that V (t) escapes to infinity in a finite time, whichcorresponds to the up-stroke of the action potential. The same up-stroke is generatedwhen I < 0, if the voltage variable is pushed beyond the threshold value V thresh .