Dynamical Systems in Neuroscience:

Bifurcations 221600K + activation, n0.50.40.3n-nullclineV-nullclinecurrent, I400200non-monotoneI-V relation0.20.1action potential00-80 -60 -40 -20 0membrane voltage, V (mV)-200-80 -60 -40 -20 0membrane voltage, V (mV)Figure 6.52: Ex. 8: This I Na,p +I K -model has a non-monotonic I-V relation, yet theresting state becomes unstable via Andronov-Hopf bifurcation before disappearing viasaddle-node bifurcation. Parameters as in Fig. 4.1a (Chapter 4) except that E leak =−78 mV and n ∞ (V ) has k = 12 mV.7. Determine the stability of the limit cycle near an Andronov-Hopf bifurcation.(Hint: consider the equilibrium r = √ |c/a| in the topological normal form (6.8)).8. The model in Fig. 6.52 has a non-monotonic I-V relation. Nevertheless, therest state loses stability via Andronov-Hopf bifurcation before disappearing viasaddle-node bifurcation. Draw representative phase portraits of the model. Isthe system near Bogdanov-Takens bifurcation?9. Consider a generic two-dimensional conductance-based model˙V = I − I(V, x) , (6.16)ẋ = (x ∞ (V ) − x)/τ(V ) , (6.17)where V and x are the membrane voltage and a gating variable, respectively, Iis the injected dc-current, and I(V, x) is the instantaneous I-V relation, which ofcourse depends on the gating variable x. Here the membrane capacitance C = 1for the sake of simplicity. Show that the eigenvalues at an equilibrium c ± ω aregiven byc = (I V (V, x) + 1/τ(V ))/2andω = √ c 2 − I ′ ∞(V )/τ(V )