As we saw in Exercise 19 of Section 1.3, the spiking of a neuron can be modeled by the d...

As we saw in Exercise 19 of Section 1.3, the spiking of a neuron can be modeled by the differential equation dθ/dt = 1 − cos θ + (1 + cos θ)I (t), where I (t) is the input. Assume that I (t) is constantly equal to −0.1. Using Euler’s method with = 0.1, graph the solution that solves the initial value θ(0) = 1.0 over the interval 0 ≤ t ≤ 5. When does the neuron spike?

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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.