Dynamical Systems in Neuroscience:

One-Dimensional Systems 81200I-V relationsteady-state current (pA)0-200-400I=16I=-100I (V)-600-800-100 -80 -60 -40 -20 0 20 40membrane potential, V (mV)Figure 3.31: Equilibria are intersectionsof the steady-state I-V curveI ∞ (V ) and a horizontal line I = const.equilibria coalesce and annihilate each other. As the parameter varies from right toleft, two equilibria — one stable and one unstable — appear from a single point. Thus,depending on the direction of movement of the bifurcation parameter, the saddle-nodebifurcation explains disappearance or appearance of a new stable state. In any case,the qualitative behavior of the systems changes exactly at the bifurcation point.3.3.7 Bifurcations and I-V recordingsIn general, determining saddle-node bifurcation diagrams of neurons may be a dauntingmathematical task. However, it is a trivial exercise when the bifurcation parameteris the injected dc-current I. In this case, the bifurcation diagram, such as the onein Fig. 3.30, is just the steady-state I-V relation I ∞ (V ) plotted on the (I, V )-plane.Indeed, the equationC ˙V = I − I ∞ (V ) = 0states that V is an equilibrium if and only if the net membrane current, I − I ∞ (V ), iszero. For example, equilibria of the I Na,p -model are solutions of the equation0 = I −I ∞ (V ){ }} {(g L (V − E L ) + g Na m ∞ (V )(V − E Na )) ,which follows directly from (3.5). In Fig. 3.31 we illustrate how to find the equilibriageometrically: We plot the steady-state I-V curve I ∞ (V ) and draw a horizontal linewith altitude I. Any intersection satisfies the equation I = I ∞ (V ), and hence is anequilibrium (stable or unstable). Obviously, when I increases past 16, the saddle-nodebifurcation occurs.Notice that the equilibria are points on the curve I ∞ (V ), so flipping and rotating thecurve by 90 ◦ , as we do in Fig. 3.32, left, results in a complete saddle-node bifurcationdiagram. The diagram conveys in a very condensed manner all important informationabout the qualitative behavior of the I Na,p -model. The three branches of the S-shapedcurve, which is the 90 ◦ -rotated and flipped copy of the N-shaped I-V curve, correspondto the rest, threshold, and excited states of the model. Each slice I = const represents