Dynamical Systems in Neuroscience:

Bifurcations 209(stable) branches. Due to the relaxation nature of the system, the vector field ishorizontal outside the N-shape fast nullcline, so any transition from Fig. 6.41b toFig. 6.41h must gradually go through the stages in Fig. 6.41c-g. Because the cycle inFig. 6.41f resembles a French duck, at least in the eyes of the French mathematiciansE. Benoit, J.-L. Callot, F. Diener, and M. Diener, who discovered this phenomenon in1977, it is often called a canard cycle.In general, any trajectory that follows the unstable branch is called a canard trajectory.Canard trajectories play an important role in defining thresholds for resonatorneurons, as we discuss in Sect. 7.2.5. It takes of the order of 1/µ units of time toslide along the unstable branch of the fast nullcline. A small perturbation to the leftor to the right could result in an immediate jump to the corresponding stable branchof the nullcline. Hence, the initial conditions should be specified with an unrealisticprecision of the order of e −1/µ to follow the branch, which explains why the canardtrajectories are difficult to catch numerically, let alone experimentally. Consequently,the canard cycles, though stable, also exits in an exponentially small region of valuesof the parameter b. A typical simulation shows an explosion of a stable limit cyclefrom small (Fig. 6.41b) to large (Fig. 6.41h) as the parameter b is slowly varied. Insummary, canard cycles in two-dimensional relaxation oscillators play an importantrole of thresholds, but they are fragile and rather exceptional.In contrast, canard trajectories in three-dimensional relaxation oscillators (one fastand two slow variables) are generic in the sense that they exits in a wide range ofparameter values. A simple way to see this is to treat b as the second slow variable.Then, there is a set of initial conditions corresponding to the canard trajectories.Studying canards in R 3 goes beyond the scope of this book (see bibliography at theend of this chapter).6.3.5 BautinWhat happens when a subcritical Andronov-Hopf bifurcation becomes supercritical,that is, when the parameter a in the topological normal form for Andronov-Hopf bifurcation(6.8,6.9) changes sign? The bifurcation becomes degenerate when a = 0,and the behavior of the system is described by the topological normal form for Bautinbifurcation, which we write here in the complex formż = (c + iω)z + az|z| 2 + a 2 z|z| 4 , (6.12)where z ∈ C is a complex variable, and c, a and a 2 are real parameters. The parametersa and a 2 are called the first and second Liapunov coefficients (often spelled Lyapunovcoefficients). The Bautin bifurcation occurs when a = c = 0 and a 2 ≠ 0, and henceit has codimension 2. It is subcritical when a 2 > 0 and supercritical otherwise. Ifa 2 = 0, then one needs to consider the next term a 3 z|z| 6 in the normal form to get abifurcation of codimension-3, etc.We can easily determine bifurcations of the topological normal form. First of all,(6.12) undergoes Andronov-Hopf bifurcation when c = 0, which is supercritical for