Dynamical Systems in Neuroscience:

Chapter 6BifurcationsNeuronal models can be excitable for some values of parameters, and fire spikes periodicallyfor other values. These two types of dynamics correspond to a stable equilibriumand a limit cycle attractor, respectively. When the parameters change, e.g., the injecteddc-current in Fig. 6.1 ramps up, the models can exhibit a bifurcation — a transitionfrom one qualitative type of dynamics to another. We consider transitions away fromequilibrium point in Sect. 6.1 and transitions away from a limit cycle in Sect. 6.2.All these transitions can be reliably observed when only one parameter, in our caseI, changes. Mathematicians refer to such as being bifurcations of codimension-1. Inthis chapter we provide definitions and examples of all codimension-1 bifurcations of anequilibrium and a limit cycle that can occur in two-dimensional systems. In Sect. 6.3 wemention some codimension-1 bifurcations in high-dimensional systems, as well as somecodimension-2 bifurcations. In the next chapter we discuss how the type of bifurcationdetermines a cell’s neuro-computational properties.6.1 Equilibrium (Rest State)A neuron is excitable because its resting state is near a bifurcation, i.e., near a transitionfrom quiescence to periodic spiking. Typically, such a bifurcation can be revealed byinjecting a ramp current, as we do in Fig. 6.1. The four bifurcations in the figurehave qualitatively different properties, summarized in Fig. 6.2. In this section weuse analytical and geometrical tools to understand what the differences among thebifurcations are.Recall (see Chap. 4) that an equilibrium of a dynamical system is stable if all theeigenvalues of the Jacobian matrix at the equilibrium have negative real parts. Whena parameter, say I, changes, two events can happen:1. A negative eigenvalue increases and becomes 0. This happens at the saddle-nodebifurcation: the equilibrium disappears.2. Two complex-conjugate eigenvalues with negative real part approach the imaginaryaxis and become purely imaginary. This happens at the Andronov-Hopf167