Dynamical Systems in Neuroscience:

432 Solutions to Exercises, Chap. 90.4u0.200.20.40.60.8x12 1.5 1 0.5 0 0.5 1 1.5 2 2.5 332.521.510.500.511.5x(t)20 50 100 150 200 250 300 350 400 450Figure 10.31: Solution to Ex. 1. Nullclines,hedgehog limit cycle and a burstingsolution of a planar system (modifiedfrom Izhikevich 2000).1K + activation gate, n0.80.60.40.20-80 -60 -40 -20 0membrane potential, V (mV)Figure 10.32: Noise-induced burstingin two-dimensional system; See Ex. 2.Solutions to Chapter 91. (Planar burster) Izhikevich (2000) suggested the systemẋ = x − x 3 /3 − u + 4S(x) cos 40u,˙u = µx ,with S(x) = 1/(1 + e 5(1−x) ) and µ = 0.01. It has a hedgehog limit cycle depicted in Fig. 10.31.2. (Noise-induced bursting) Noise can induce bursting in a two-dimensional system with coexistenceof resting and spiking states. Indeed, noisy perturbations can randomly push thestate of the system into the attraction domain of the resting state or into the attraction domainof the limit cycle attractor, as in Fig. 10.32. The solution meanders between the states,exhibiting a random bursting pattern as in Fig. 9.55,right. neocortical neurons of RS and FStype, as well as stellate neurons of the entorhinal cortex exhibit such bursting; see Chap. 8.