Dynamical Systems in Neuroscience:

434 Solutions to Exercises, Chap. 90.5|x|yx1x2Syy0.5x 1Supercritical Andronov-Hopf BifurcationtSlow PassageEffectFigure 10.35: Hopf/Hopf bursting without co-existence of attractors; see Ex. 6 (modified fromHoppensteadt and Izhikevich 1997).Here, u is the deviation from the slow equilibrium u 0 . The slow subsystem˙u = µg(ze iωt + complex-conjugate, u)can be averaged and transformed into the canonical form.8. (Bursting in the I Na,t +I Na,slow -model) First, determine the parameters of the I Na,t -model correspondingto the subcritical Andronov-Hopf bifurcation, and hence the co-existence of theresting and spiking states. Then, add a slow high-threshold persistent Na + current that activatesduring spiking, depolarizes the membrane potential and stops the spiking. During resting,the current deactivates, the membrane potential hyperpolarizes and the neuron starts to fireagain.9. Substitute the slow Na + current in the exercise above with a slow dendritic compartmentwith dendritic resting potential far below the somatic resting potential. As the dendriticcompartment hyperpolarizes the somatic compartment, the soma starts to fire (due to theinhibition-induced firing described in Sect. 7.2.8). As the somatic compartment fires, dendriticcompartment slowly depolarizes, removes the hyperpolarization and stops firing.10. (Bursting in the I Na,p +I K +I Na,slow -model) The time constant τ slow (V ) is relatively small inthe voltage range corresponding to the spike after-hyperpolarization (AHP). Deactivation ofthe Na + current during each AHP is much stronger than its activation during the spike peak.As a result, Na + current deactivates (turns off) during the burst, and then slowly reactivatesto its baseline level during the resting period, as one can see in Fig. 10.36.11. The mechanism of spiking, illustrated in Fig. 10.37, is closely related to the phenomenon ofaccommodation and anodal break excitation. The key feature is that this bursting is notfast-slow.