Dynamical Systems in Neuroscience:

224 Bifurcations19. [M.S.] A leaky integrate-and-fire model has the same asymptotic firing rate(1/ln) as a system near saddle homoclinic orbit bifurcation. Explore the possibilitythat integrate-and-fire models describe neurons near such a bifurcation.20. [M.S.] (blue-sky catastrophe) Prove that˙ϕ = ω, ẋ = a + x 2 , if x = +∞, then x ← −∞, and ϕ ← 0 ,is the canonical model (see Sect. 8.1.5) for blue-sky catastrophe. This modelwithout the reset of ϕ is canonical for the fold limit cycle on homoclinic torusbifurcation. The model with the reset x ← b+sin ϕ is canonical for the Lukyanov-Shilnikov bifurcation of a fold limit cycle with non-central homoclinics (Shilnikovand Cymbalyuk 2004, Shilnikov et al. 2005). Here, ϕ is the phase variable onthe unit circle and a and b are bifurcation parameters.21. [M.S.] Define topological equivalence and the notion of a bifurcation for piecewisecontinuous flows.22. [Ph.D.] Use the definition above to classify codimension-1 bifurcations in piecewisecontinuous flows.23. [M.S.] The bifurcation sequence in Fig. 6.40 seems to be typical in 2-dimensionalneuronal models. Develop the theory of Bogdanov-Takens bifurcation with aglobal reentrant orbit.24. [Ph.D.] Develop an automated dynamic clamp protocol (Sharp et al. 1993) thatanalyzes bifurcations in neurons in vitro, similar to what AUTO, XPPAUT, orMATCONT do in models.