Consider the Ermentrout-Kopell model for the spiking of a neuron intr...

Consider the Ermentrout-Kopell model for the spiking of a neuron

introduced in Exercise 19 of Section 1.3. Suppose that the input function I (t) is a constant function, that is, I (t) = I where I is a constant. Describe the bifurcations that occur as the parameter I varies.

Reference

The spiking of a neuron can be modeled∗ by the differential equation

where I (t) is the input. Often the input function I (t) is a constant I. When θ is an odd multiple of π, the neuron spikes.

(a) Using HPGSolver, sketch three slope fields, one for each of the following values of I : I₁ = −0.1, I₂ = 0.0, and I₃ = 0.1.

(b) Calculate the equilbrium solutions for each of these three values.

(c) Using the slope field, describe the long-term behavior of the solutions in each of the three cases.