Dynamical Systems in Neuroscience:

Bifurcations 199heteroclinic orbitFigure 6.32: Heteroclinic orbit bifurcation does not change the existence of stability ofany equilibrium or periodic orbit.depending on the saddle quantity, it results in a logarithmic F-I curve, and it impliesthe co-existence of attractors. All methods of analysis of excitable systems near “small”saddle homoclinic orbit bifurcations can also be applied to the case in Fig. 6.31.6.3 Other Interesting CasesSaddle-node and Andronov-Hopf bifurcations of equilibria combined with fold limitcycle, homoclinic orbit bifurcation, and heteroclinic orbit bifurcation (see Fig. 6.32)exhaust all possible bifurcations of codimension-1 on a plane. These bifurcations canalso occur in higher-dimensional systems. Below we discuss additional codimension-1bifurcations in three-dimensional phase space, and then we consider some codimension-2 bifurcations that play an important role in neuronal dynamics. The first-time readermay read only Sect. 6.3.6 and skip the rest.6.3.1 Three-dimensional phase spaceSo far we considered four bifurcations of equilibria and four bifurcations of limit cycleson a phase plane. The same eight bifurcations can appear in multi-dimensional systems.Below we briefly discuss the new kinds of bifurcations that are possible in a threedimensionalphase space but cannot occur on a plane.First, there are no new bifurcations of equilibria in multi-dimensional phase space.Indeed, what could possibly happen with the Jacobian matrix of an equilibrium ofa multi-dimensional dynamical system? A simple zero eigenvalue would result ina saddle-node bifurcation, and a simple pair of purely imaginary complex-conjugateeigenvalues would result in an Andronov-Hopf bifurcation. Both are exactly the sameas in the lower-dimensional systems considered before. Thus, adding dimensions to adynamical system does not create new possibilities for bifurcations of equilibria.In contrast, adding the third dimension to a planar dynamical system creates newpossibilities for bifurcations of limit cycles, some of which are depicted in Fig. 6.33.Below we briefly describe these bifurcations.The saddle-focus homoclinic orbit bifurcation in Fig. 6.33 is similar to the saddlehomoclinic orbit bifurcation considered in Sect. 6.2.4 except that the equilibrium hasa pair of complex-conjugate eigenvalues and a non-zero real eigenvalue. The homoclinicorbit originates in the subspace spanned by the eigenvector corresponding to