Question

Neuron Communications In the FitzHugh-Nagumo model of how neurons communicate, the rate of change of the electric potential v with respect to time is given as a function of v by f(v)=v(a−v)(v−1), where a is a positive constant. Sketch a graph of this function when a=0.25 and 0≤v≤1. Source: Mathematical Biology.

Answer 1

To sketch a graph of the function f(v) = v(a - v)(v - 1) in the given range 0 ≤ v ≤ 1, we need to substitute the value of a = 0.25 into the equation. Let's break it down step-by-step:

1. Plug in the value of a: f(v) = v(0.25 - v)(v - 1).

2. Simplify the equation: f(v) = v(0.25v - v^2)(v - 1).

To sketch the graph, we can analyze the behavior of the function for different values of v:

- As v approaches 0, f(v) also approaches 0.

- As v approaches 1, f(v) also approaches 0.

- The function reaches its maximum value when v = 0.25. At this point, f(v) = 0.25 * (0.25 - 0.25^2)(0.25 - 1).

- The function reaches its minimum value when v = 0.5. At this point, f(v) = 0.5 * (0.25 - 0.5^2)(0.5 - 1).

Based on this information, we can plot the graph of the function f(v) = v(0.25 - v)(v - 1) within the given range 0 ≤ v ≤ 1.