Problem 46 - Calculus for Biology and Medicine

Question

Use a graphing calculator to study the following example of the Fitzhugh- Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-0.6)(V-1)-w \\ \frac{d w}{d t}=0.03(V-0.6 w) \end{array} $$ Sketch the graph of the solution curve in the \(V-w\) plane when (i) \((V(0), w(0))=(0.8,0)\) and (ii) \((V(0), w(0))=(0.4,0)\).

Step-by-Step Solution

Answer

Graph both scenarios with a tool. Initial conditions move curves differently in the \(V-w\) plane.

Step 1: Set up the Differential Equations

First, understand the given differential equations and identify them for input into a graphing tool. We have two equations: \( \frac{dV}{dt} = -V(V - 0.6)(V - 1) - w \) and \( \frac{dw}{dt} = 0.03(V - 0.6w) \). These equations describe how \(V\) and \(w\) change over time.

Step 2: Configure the Graphing Tool

Use a graphing calculator or software that supports differential equations. Input both differential equations into the calculator. Ensure the tool is set to solve these equations simultaneously as a system, tracking the variables \(V\) and \(w\) over time.

Step 3: Input Initial Conditions for Case (i)

For the first case, set the initial conditions to \((V(0), w(0))=(0.8,0)\). Input these values into your graphing tool. This will determine the starting point on the \(V-w\) plane.

Step 4: Run the Simulation for Case (i)

Allow the graphing tool to compute the solution curve for the initial conditions \((V(0), w(0))=(0.8,0)\). Observe the trajectory on the \(V-w\) plane as the system evolves from the initial conditions. The graph should display a curve showing the relationship between \(V\) and \(w\) over time.

Step 5: Input Initial Conditions for Case (ii)

Adjust the initial conditions to \((V(0), w(0))=(0.4,0)\). Input these new values into your graphing tool to analyze the second scenario.

Step 6: Run the Simulation for Case (ii)

Run the simulation again with the updated initial conditions \((V(0), w(0))=(0.4,0)\). Observe and sketch the solution curve as it appears on the \(V-w\) plane. This will show a potentially different behavior or trajectory compared to the first set of conditions.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that describe how a particular quantity changes over time. They are fundamental in modeling dynamic systems in many fields like physics, engineering, and biology. In the case of the Fitzhugh-Nagumo model, we have two differential equations defined as follows:

\[\frac{dV}{dt} = -V(V - 0.6)(V - 1) - w\]
\[\frac{dw}{dt} = 0.03(V - 0.6w)\]

Each equation outlines how the variables \(V\) and \(w\) change over time. Here:

• \(\frac{dV}{dt}\) describes the rate of change of the variable \(V\).
• \(\frac{dw}{dt}\) represents the rate of change of \(w\).

The behavior of these equations can be complex because they might not have straightforward solutions. This is why utilizing a tool like a graphing calculator can significantly aid in visualizing their evolution over time, thus providing greater intuition into the behavior of the system.

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize mathematical functions and solve complex equations, such as differential equations. In situations where algebraic solutions are challenging or unfeasible, graphing calculators come to the rescue by simulating these equations and drawing graphs that depict the system's behavior over time.

To use a graphing calculator for the Fitzhugh-Nagumo model, perform the following steps:

• Input the system of differential equations into the calculator.
• Configure it to solve the equations simultaneously, ensuring that it tracks both variables, \(V\) and \(w\).

This graphical representation on the calculator helps students and scientists alike to comprehend how variable changes influence the system. It allows users to dynamically observe and interpret the solutions as they appear on the \(V-w\) plane.

Initial Conditions Simulation

Initial conditions are essential when solving differential equations as they establish a starting point for observing how a system evolves. In the Fitzhugh-Nagumo model, the initial conditions determine the initial values for \(V\) and \(w\). By altering these conditions, we can simulate different trajectories and behaviors in the system.

For example, the task requires students to analyze two sets of initial conditions:

• \((V(0), w(0)) = (0.8, 0)\)
• \((V(0), w(0)) = (0.4, 0)\)

By entering these conditions into the graphing calculator, students can run simulations to visualize how the solutions change from their respective starting points. The trajectory observed on the \(V-w\) plane provides insight into whether slight changes in initial conditions can lead to significantly different outcomes, emphasizing the sensitivity and dynamism of such systems. These simulations enhance understanding by demonstrating in real-time how different initial states affect an equation's solution.