Dynamical Systems in Neuroscience:

222 Bifurcationsv 2v 11a-11-1Figure 6.53: See Ex. 11.where I ∞ (V ) = I(V, x ∞ (V )) is the steady-state I-V relation of the model. Inparticular, the frequency at the Andronov-Hopf bifurcation iswhere C is the membrane capacitance.10. Determine when the systemundergoes fold limit cycle bifurcation.(frequency) = √ I ′ ∞(V )/(Cτ(V )) ,z ′ = (a + ωi)z + z|z| 2 − z|z| 4 , z ∈ C ,11. Consider a square neighborhood of a saddle equilibrium in Fig. 6.53 (comparewith the inset in Fig. 6.29). Here v 1 and v 2 are eigenvectors with eigenvaluesλ 2 < 0 < λ 1 . Suppose the limit cycle enters the square at the point a = τ(I −I b ),where τ > 0 is some parameter. Determine the amount of time the trajectoryspends in the square as a function of I.12. Determine the bifurcation diagram of the topological normal form (6.13) forsaddle-node homoclinic bifurcation.13. Prove that the systemwith a > 0 undergoes˙v = I + v 2 − u ,˙u = a(bv − u)• saddle-node bifurcation when b 2 = 4I,