Dynamical Systems in Neuroscience:

Solutions to Exercises, Chap. 9 4330.800.7-10K + gating variable, n0.60.50.40.30.2membrane potential, V (mV)-20-30-40-500.1-600-80 -60 -40 -20 0membrane potential, V (mV)-700 10 20 30 40 50 60 70time (ms)Figure 10.33: Noise-induced bursting a two-dimensional system with co-existence of an equilibriumand a limit cycle attractor; see Ex. 3.0.2w0.1ghost offold limit cycle10.50V(t)0-0.5 0 0.5 1VI(t)-0.50 1000 2000 3000 4000timeFigure 10.34: Rebound bursting in the FitzHugh-Nagumo oscillator; see Ex. 4.3. (Noise-induced bursting) Bursting occurs because noisy perturbations push the trajectory intoand out of the attraction domain of the limit cycle attractor, which coexists with the restingequilibrium; see the phase portrait in Fig. 10.33.4. (Rebound bursting in the FitzHugh-Nagumo oscillator) The oscillator is near fold limit cyclebifurcation. The solution makes a few rotations along the ghost of the cycle before returningto rest; see Fig. 10.34.5. Yes, they can, at the end of a burst. Think of a “fold/Hopf” or “circle/Hopf” burster. Theresting equilibrium is a stable focus right after the termination of a burst, and then it istransformed into a stable node to be ready for the circle or fold bifurcation. Even “circle/circle”bursters could exhibit such oscillations, if the resting equilibrium turns into a focus for a shortperiod of time somewhere in the middle of a quiescent phase. In any case, the oscillationsshould disappear just before the transition to the spiking state.6. (Hopf/Hopf bursting) Even though there is no co-existence of attractors, there is a hysteresisloop due to the slow passage effect through the supercritical Andronov-Hopf bifurcation; seeFig. 10.35. The delayed transition to spiking creates the hysteresis loop and enables bursting.7. (Hopf/Hopf canonical model) First, we restrict the fast subsystem to its center manifold andtransform it to the normal form for supercritical Andronov-Hopf bifurcation, which after appropriatere-scaling, has the formż = (u + iω)z − z|z| 2 .