Another simple model for a neuron is the quadratic integrateand-fire model, which takes the form dV dt

= V 2 + I(t), where V denotes the neuron’s membrane potential and I is some input, which we shall here assume to be constant. We’re going to use this example to show how a qualitative analysis can be much easier than solving the equation explicitly, even when the explicit solution appears to be easily found using a computer. However, we’ll also see that the explicit solution can give us critical information that we can’t get from a qualitative analysis.

a. Set I = −1.

i. Separate variables and use a computer to integrate both sides to find the solution.

ii. What is the solution with initial condition V(0) = −2?

How does V(t) behave as t → ∞?

iii. What is the solution with initial condition V(0) = 2?

How does V(t) behave as t increases? How big can t get before you don’t have a solution any more?

iv. Now do a qualitative analysis of the differential equation, and demonstrate the same behaviour that was shown in the previous two parts of this question. What important piece of information can you not obtain from the qualitative approach?

b. Now set I = 1.

i. Separate variables and use a computer to integrate both sides to find the solution.

ii. Show that, no matter what the initial condition is, the solution goes to infinity in finite time.

iii. What will a qualitative approach tell you? What will a qualitative approach not tell you?