Problem 16 - Calculus for Biology and Medicine

Question

Assume the following example of the Fitzhugh-Nagumo model: $$ \begin{array}{l} \frac{d V}{d t}=-V(V-3 / 5)(V-1)-w \\ \frac{d w}{d t}=V-c w \end{array} $$ Find the smallest value of \(c\) for which the model predicts the existence of multiple equilibria.

Step-by-Step Solution

Answer

The smallest value of \(c\) for multiple equilibria is 0.

Step 1: Find the Equilibrium Points

Equilibrium points occur where the derivatives with respect to time are zero. Hence, we have two equations:1. \(-V(V-3/5)(V-1)-w = 0\)2. \(V - cw = 0\)Therefore, the equilibrium points (\(V^*, w^*\)) satisfy:\[-V^*(V^*-3/5)(V^*-1) = w^*\]and\[V^* = cw^*\].

Step 2: Substitute the Second Equation

Substitute \(w^* = V^*/c\) from the second equation into the first equilibrium condition:\[-V^*(V^*-3/5)(V^*-1) = V^*/c\].This simplifies to:\[-cV^*(V^*-3/5)(V^*-1) = V^*\].

Step 3: Simplify the Equation

Rearrange and factor out \(V^*\) from the equation:\[-c(V^*-3/5)(V^*-1) = 1\].Expanding the left-hand side gives:\[-c(V^2 - 8V/5 + 3/5) = 1\].

Step 4: Determine the Conditions for Multiple Equilibria

The quadratic equation \[cV^2 - \frac{8}{5}cV + \frac{3}{5}c = 1\]should have at least two real roots for multiple equilibria, which requires the discriminant \(\Delta\) of this quadratic to be positive.

Step 5: Calculate the Discriminant

The discriminant \(\Delta\) is:\[\Delta = \left(\frac{8}{5}c\right)^2 - 4 \times c \times \left(\frac{3}{5}c - 1\right)\]This simplifies to:\[\Delta = \frac{64}{25}c^2 - 4c \left(\frac{3}{5}c - 1\right)\]Simplifying:\[\Delta = \frac{64}{25}c^2 - \frac{12}{5}c^2 + 4c\]\[\Delta = \frac{64}{25}c^2 - \frac{60}{25}c^2 + 4c\]\[\Delta = \frac{4}{25}c^2 + 4c\]

Step 6: Solve for Positive Discriminant

To have multiple equilibria, solve \(\Delta > 0\):\[\frac{4}{25}c^2 + 4c > 0\]This simplifies to:\[c^2 + 25c > 0\]Factor:\[c(c + 25) > 0\]The roots are \(c = 0\) and \(c = -25\). For \(c(c + 25) > 0\), \(c < -25\) or \(c > 0\). The smallest positive \(c\) is 0.

Key Concepts

Equilibrium Points

In the study of mathematical models, equilibrium points are vital as they represent conditions where the system is in a stable state or unchanging over time. For a system described by differential equations, these are the points where each differential equation is zero. This essentially means there is no change occurring for the variables involved.
For the Fitzhugh-Nagumo model, our goal is to find equilibrium points where both derivatives, \(\frac{dV}{dt}\) and \(\frac{dw}{dt}\), are equal to zero.
Given the equations:

• \(-V(V-\frac{3}{5})(V-1)-w = 0\)
• \(V - cw = 0\)

At equilibrium, the solution \((V^*, w^*)\) must satisfy these conditions. By equating the right-hand sides of both equations to zero, we uncover the equilibrium points.These point to specific values of \(V^*\) and \(w^*\) where the system does not "move," reflecting a steady state. The interplay of these conditions help us understand how equilibrium can be achieved, revealing crucial insights about system stability and the parameter \(c\) in this context.

Quadratic Equation

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). In the context of our model, after some algebraic manipulation, we arrive at a quadratic equation:

• \(cV^2 - \frac{8}{5}cV + \frac{3}{5}c = 1\)

This is vital because the number of real solutions, or roots, to this equation determines the number of equilibrium points the system can have.
Quadratics can have:

• Two real roots (two equilibria)
• One real root (one equilibrium)
• No real roots (no equilibrium)

The challenge lies in manipulating the system equations to achieve a form where quadratics are evident, as these equations simplify the process of determining potential behaviors of the system.
This quadratic form aids in leveraging algebraic techniques to assess system behavior, and in our specific setup, suggests conditions for having multiple equilibria. This insight is what guides our overarching analysis of the Fitzhugh-Nagumo model.

Discriminant

To solve quadratic equations, the discriminant, \(\Delta\), plays a crucial role. It determines the nature and number of the roots of the quadratic equation. The discriminant is calculated as \(b^2 - 4ac\) for any quadratic in the canonical form \(ax^2 + bx + c = 0\).
For our Fitzhugh-Nagumo model, this translates to:

• \[\Delta = \left(\frac{8}{5}c\right)^2 - 4 \times c \times \left(\frac{3}{5}c - 1\right)\]
• Simplified to \[\Delta = \frac{4}{25}c^2 + 4c\]

Analyzing \(\Delta\), it must be greater than zero (\(\Delta > 0\)) to ensure that the quadratic equation has two real roots, leading to multiple equilibria in our model.
The calculations show that multiple equilibria exist when the condition \(c(c + 25) > 0\) is satisfied. This means \(c\) must be less than \(-25\) or greater than \(0\).
Positive conditions reflect realistic, positive feedback mechanisms, and drive our determination of the smallest feasible \(c\) that results in multiple equilibria. Understanding the discriminant's role in this context supports deeper comprehension of solution viability and system dynamics.