Dynamical Systems in Neuroscience:

Bifurcations 213homoclinic orbit bifurcationsaddle-node bifurcation saddle-node oninvariant circle bifurcationsaddle-nodehomoclinic orbitbifurcationFigure 6.45: Unfolding of saddle-node homoclinic orbit bifurcation.The topological normal form (6.13) is a useful equation, as we will see in the restof the book. It describes quantitative and qualitative features of neuronal dynamicsremarkably well, yet it has only one non-linear term. This makes it suitable for realtimesimulations of huge numbers of neurons. Its bifurcation structure is studied inEx. 12 (see also Fig. 8.3), and the reader should at least look at the solution at theend of the book.6.3.7 Hard and soft loss of stabilityBifurcation is a qualitative change of the phase portrait of a system. Not all changeshowever are equally dramatic. Some are even hardly noticeable. For example, consideran equilibrium undergoing supercritical Andronov-Hopf bifurcation: As a bifurcationparameter changes, the equilibrium loses stability and a small-amplitude stable limitcycle appears, as in Fig. 6.11. The state of the system remains near the equilibrium; itjust exhibits small-amplitude oscillations around it. We can change the parameter inthe opposite direction, an then the limit cycle shrinks to a point and the system returnsto the equilibrium. In neurons, such a bifurcation does not lead to an immediate spike.The neuron remains quiescent; it just exhibits subthreshold small-amplitude sustainedoscillations. Such a loss of stability is called soft: the equilibrium is no longer stable,but its small neighborhood remains attractive. Supercritical pitchfork, cusp and flipbifurcations correspond to soft loss of stability.