Dynamical Systems in Neuroscience:

398 Bursting0membrane potential, V (mV)-10-20-30-40-50-60-700 50 100 150 200 250 300 350time (ms)Figure 9.59: Bursting in theI Na,p +I K +I Na,slow -model; see Ex. 10.y21.510.5y-nullcline10x-1-20-0.5-1-3 -2 -1 0 1 2 3xx-nullcline1I(t)0-10 200 400 600timeFigure 9.60: The phase portrait of the system in Ex. 11 shows that there is only onestable equilibrium for any value of I. Yet, the system bursts when I is periodicallymodulated.10. (Bursting in the I Na,p +I K +I Na,slow -model) Explore numerically this model withthe fast subsystem as in Fig. 6.16 and a slow Na + current with parameters:g Na,slow = 0.5, m ∞,slow (V ) with V 1/2 = −50 mV and k = 10 mV, and τ slow (V ) =5 + 100 exp(−(V + 20) 2 /25 2 ). Explain the origin of bursting oscillations whenI = 27 in Fig. 9.59.11. The Bonhoeffer–van der Pol oscillatorẋ = I + x − x 3 /3 − y ,ẏ = 0.2(1.05 + x) ,with nullclines as in Fig. 9.60, is Class 3 excitable. It has a unique stable equilibriumfor any value of I (check). Periodic modulations of I shift the x-nullclineup and down but do not change the stability of the equilibrium. Why does thesystem burst in Fig. 9.60? Explore the phenomenon numerically and explain theexistence of repetitive spikes without a limit cycle.12. Prove that the fast-slow “2+2” systemż = (1 + u + iω)z − z|z| 2 , z ∈ C ,