Saddle-Node Collisions

Abstract

Model neurons, as well as real neurons, don’t fire when they receive little or no input current, but fire periodically when they receive strong input current. The transition from rest to firing, as the input current is raised, is called a bifurcation. In general, a bifurcation is a sudden qualitative change in the solutions to a differential equation, or a system of differential equations, occurring as a parameter, called the bifurcation parameter in this context, is moved past a threshold value, also called the critical value. (To bifurcate, in general, means to divide or fork into two branches. This suggests that in a bifurcation, one thing turns into two. This is indeed the case in some bifurcations, but not in all.) Because the variation of drive to a neuron is the primary example we have in mind, we will denote the bifurcation parameter by I, and its threshold value by I _c , in this chapter.