Dynamical Systems in Neuroscience:

spikingBursting 367foldbifurcationxurestsaddlehomoclinicorbitbifurcationfold bifurcationsaddle homoclinicorbit bifurcationFigure 9.25: “Fold/homoclinic” bursting: The resting state disappears via saddle-node(fold) bifurcation and the spiking limit cycle disappears via saddle homoclinic orbitbifurcation.tential corresponds to the left stable equilibrium, which is the intersection of the leftknee of the fast N-shaped nullcline with the slow nullcline. During resting, the N-shapenullcline slowly moves up, until its knee touches the slow nullcline at a saddle-nodepoint. Right after this moment, the resting state disappears via saddle-node (fold) bifurcation,hence the part “fold” in the name of the burster. After the fold bifurcation,the membrane potential jumps up to the stable limit cycle corresponding to repetitivespiking. During the spiking state, the N-shaped nullcline slowly moves down, and themiddle (saddle) equilibrium moves away from the resting state toward the limit cycle.After a while, the limit cycle becomes a homoclinic trajectory to the saddle, and thenthe cycle disappears via saddle homoclinic orbit bifurcation, hence the part “homoclinic”in the name of the burster. After this bifurcation, the membrane potentialjumps down to the resting state and closes the hysteresis loop.