Dynamical Systems in Neuroscience:

226 Excitabilityspike?spikerestrestexcitableoscillatoryFigure 7.1: Left: An abstract definition of excitability. There is a spike trajectory thatstarts near a stable equilibrium and returns to it. Right: Excitable systems are nearbifurcations. A modification of the vector field in the small shaded region can resultin a periodic trajectory.that push the state of the neuron to or near the beginning of the large trajectory(small square in Fig. 7.1), thereby initiating the spike. These inputs can be injected byexperimenter via an attached electrode, or they can represent the total synaptic inputfrom the other neurons in the network, or both.7.1.1 BifurcationsThe definition in Fig. 7.1 is quite general, and it does not make any assumptions regardingthe details of the vector field inside or outside of the small shaded neighborhood.Using the theory presented in the previous chapter, we can show that such an excitablesystem is near a bifurcation from resting to oscillatory dynamics.• Bifurcation of a limit cycle. The vector field in the small shaded neighborhoodof the equilibrium can be modified slightly so that the spike trajectory enters thesquare and becomes periodic, as in Fig. 7.1, right. That is, the dynamical systemgoes through a bifurcation resulting in the appearance of a limit cycle.What happens to the stable equilibrium, denoted as “?” in the figure? Depending onthe type of the bifurcation of the limit cycle, the equilibrium may disappear or may losestability. This happens when the limit cycle appears via saddle-node on invariant circleor supercritical Andronov-Hopf bifurcations, respectively. Both cases are depicted inFig. 7.2.