Dynamical Systems in Neuroscience:

Bifurcations 2051 2b1 2SN23 AH 41BT4a3 AH 4SHOAH2 3SN1SN 1 SN 2SN 1SN 2SHOsupercritical,BTSHOBTs0Figure 6.38: Bogdanov-Takens (BT) bifurcation diagram of the topological normal form(6.11). Abbreviations: AH - Andronov-Hopf bifurcation, SN - saddle-node bifurcation,SHO - saddle homoclinic orbit bifurcation.has two bifurcation parameters, a and b, and the parameter σ = ±1 determines whetherthe bifurcation is subcritical or supercritical. This parameter depends on the combinationof the second-order partial derivatives with respect to the first variable, and it isnon-zero because of the non-degeneracy conditions (Kuznetsov 1995). Bifurcation diagramand representative phase portraits for various a, b and σ are depicted in Fig. 6.38(the case σ > 0 can be reduced to σ < 0 by the substitution t → −t and v → −v).A remarkable fact is that the saddle-node and the Andronov-Hopf bifurcations do notoccur alone. There is also a saddle homoclinic orbit bifurcation appearing near theBogdanov-Takens point.Bogdanov-Takens bifurcation occurs often in neuronal models with nullclines intersectingas in Fig. 6.39a. We show in the next chapter that this bifurcation separatesintegrators from resonators, and it could occur in some layer 5 pyramidal neurons ofrat visual cortex, as we discuss in Sect. 7.2.11 and Sect. 8.2.1. The two equilibria inthe lower (left) knee of the fast nullcline in Fig. 6.39b are not necessarily a saddle anda stable node, but could be a saddle and an (un)stable focus, as in the phase portraitsin Fig. 6.38.