The theta model, often called the Ermentrout–Kopell model, is one of the simplest models of neuronal bursting oscillations.

The model is Bard Ermentrout and Nancy Kopell are two of the most eminent theoretical neuroscientists alive today.

The variable θ represents the neuron’s membrane potential, but in a complicated way that we don’t explain here.

dθ

dt

= 1 − cos(θ) + I(t)(1 + cos(θ)), for some input function I(t). To make things a lot simpler, in this question we’ll just assume that I(t) is a constant.

For each of the three cases I = −0.2, 0, 0.2 do a qualitative analysis of the theta model, showing all the steady states and whether they are sources, sinks or nodes.