Dynamical Systems in Neuroscience:

Two-Dimensional Systems 117rest state to repetitive spiking. When I is small, the phase portrait of the model issimilar to the one depicted in Fig. 4.26 for I = 0. There are two equilibria in thelow-voltage range — a stable node corresponding to the rest state and a saddle. Theequilibria are the intersections of the cubic V -nullcline and the n-nullcline. Increasingthe parameter I changes the shape of the cubic nullcline and shifts it upward, but doesnot change the n-nullcline. As a result, the distance between the equilibria decreases,until they coalesce as in Fig. 4.28 so that the nullclines only touch each other in the lowvoltagerange. Further increase of I results in the disappearance of the saddle and nodeequilibrium, and hence in the disappearance of the rest state. The new phase portraitis depicted in Fig. 4.30; it has only a limit cycle attractor corresponding to repetitivefiring. We see that increasing I past the value I = 4.51 results in transition from restto periodic spiking dynamics. What kind of a bifurcation occurs when I = 4.51?Those readers who did not skip Sect. 3.3.3 in the previous chapter will immediatelyrecognize the saddle-node bifurcation, whose major stages are summarized in Fig. 4.31.As a bifurcation parameter changes, the saddle and the node equilibrium approach eachother, coalesce, and then annihilate each other so there are no equilibria left. Whencoalescent, the joint equilibrium is neither a saddle nor a node, but a saddle-node. Itsmajor feature is that it has precisely one zero eigenvalue, and it is stable on one side ofthe neighborhood and unstable on the other side. In Chap. 6 we will provide an exactdefinition of a saddle-node bifurcation in a multi-dimensional system, and we will showthat there are two important subtypes of this bifurcation, resulting in slightly different0.60.5n-nullcline0.4K + activation variable, n0.30.2V-nullcline0.10heteroclinic orbitheteroclinic orbit-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane voltage, V (mV)Figure 4.26: Two heteroclinic orbits (bold curves connecting stable and unstable equilibria)in the I Na,p +I K -model with high-threshold K + current.