Dynamical Systems in Neuroscience:

284 Simple Models1v reset =|b| 1/2bistabilitysaddle-node bifurcationparameter v reset0saddle homoclinic orbit bifurcationv resetv resetv resetsaddle-node homoclinicorbit bifurcationexcitablev resetv resetv resetsaddle-node on invariantcircle bifurcationperiodicv reset-1-1 0 1parameter bFigure 8.3: Bifurcation diagram of the quadratic integrate-and-fire neuron (8.2).current or their combination. Typically, the fast variable has an N-shaped nullcline andthe slower variable has a sigmoid-shaped nullcline. The resting state in such modelsis the intersection of the nullclines near the left knee, as we illustrate in Fig. 8.4a.There, V and u denote the fast and the slow variables, respectively. In Chap. 7 weshowed that many computational properties of biological neurons could be explainedby considering dynamics at the left knee.In Sect. 5.2.4 we derive a simple model that captures the subthreshold behavior ina small neighborhood of the left knee confined to the shaded square in Fig. 8.4 andthe initial segment of the up-stroke of an action potential. In many cases, especiallyinvolving large-scale simulations of spiking models, the shape of the action potentialis less important than the subthreshold dynamics leading to this action potential. So,retaining detailed information about the left knee and its neighborhood and simplifyingthe vector field outside the neighborhood is justified.The simple model˙v = I + v 2 − u if v ≥ 1, then (8.3)˙u = a(bv − u) v ← c, u ← u + d. (8.4)has only four dimensionless parameters. Depending on the values of a and b, it can