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Chapter 7: Problem 8

The Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron \(v(t)\) obeys the differential equation \(\frac{d v}{d t}=-v\left[v^{2}-(1+a) v+a\right]\) where \(a\) is a constant and \(0< a <1\) (a) For what values of \(v\) is \(v\) unchanging (that is, \(d v / d t=0 ) ?\) (b) For what values of \(v\) is \(v\) increasing? (c) For what values of \(v\) is \(v\) decreasing?

Short Answer

Expert verified
(a) \( v = 0, v = v_1, v = v_2; \) (b) \( v \) between roots; (c) otherwise.

Step by step solution

01

Identify values where dv/dt = 0

To find where the electrical potential is unchanging, set the differential equation equal to zero: \( \frac{dv}{dt} = -v [v^2 - (1+a)v + a] = 0 \). Since the product equals zero, either \( v = 0 \) or \( v^2 - (1+a)v + a = 0 \).
02

Solve quadratic equation

To solve \( v^2 - (1+a)v + a = 0 \), use the quadratic formula: \( v = \frac{ (1+a) \pm \sqrt{(1+a)^2 - 4a} }{2} \). The roots of the equation give values where \( v \) is constant. Calculate \( \sqrt{(1+a)^2 - 4a} \) to find exact roots.
03

Analyze quadratic discriminant

The discriminant \( (1+a)^2 - 4a \) must be evaluated to confirm real roots. Simplify it to \( 1 - 2a + a^2 \), which is always positive since it's a sum of non-negative terms for \( 0 < a < 1 \). Thus, there are two distinct real roots.
04

Determine regions of increasing v

To find where \( v \) is increasing, analyze \( \frac{dv}{dt} > 0 \), or \( -v[v^2 - (1+a)v + a] < 0 \). This inequality suggests evaluating the sign changes in \([-v\)] and \([v^2 - (1+a)v + a]\). Consider intervals from roots found: \([0, v_1) \), \([v_1, v_2] \), and \([v_2, \infty)\).
05

Determine regions of decreasing v

To find where \( v \) is decreasing, use \( \frac{dv}{dt} < 0 \), or \( -v[v^2 - (1+a)v + a] > 0 \). Check the sign in each interval separated by roots and include \( v = 0 \) point evaluation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that involve the rates at which quantities change. They are a potent tool in modeling dynamic systems across many fields. Typically, these equations relate some function with its derivatives.

In the context of the Fitzhugh-Nagumo model, the differential equation \( \frac{d v}{d t} = -v[v^2 - (1+a)v + a] \) represents how the electrical potential in a neuron changes over time. The left side of the equation, \( \frac{d v}{d t} \), stands for the rate of change of the electrical potential \( v \) concerning time \( t \).
  • On the right side, the expression \(-v[v^2 - (1+a)v + a] \) embodies the dynamics influencing the potential.
  • It captures complex interactions where terms have different powers of \( v \), pinpointing non-linear relationships.
By setting \( \frac{d v}{d t} = 0 \), one finds conditions where the potential is constant. Solutions to this provide insight into fixed states or equilibria of the modeled system.
Neuron Electrical Potential
Electrical potential in neurons plays a crucial role in how brain cells communicate. Neurons transmit signals as electrical impulses, and their potential is pivotal in determining how and when these signals are sent.

The Fitzhugh-Nagumo model simplifies complex neuron dynamics to analyze these electrical activities better. By examining the equation associated with \( v(t) \), which indicates electrical potential, one can predict how and when neuron responses occur. Factors like relaxation effects, omitted in this basic form, would further influence this model.
  • The product \(-v[v^2-(1+a) v+a] \) reflects how potential responds to internal states and external inputs.
  • Understanding unchanging, increasing, or decreasing \( v \) informs about neuron excitability and refractory periods.
Quadratic Formula
In solving the mathematical model, the quadratic formula plays a key role. The quadratic equation, \( v^2 - (1+a)v + a = 0 \), emerges from setting the differential equation to zero, offering a path to finding steady states where the potential does not change.

The quadratic formula, \( v = \frac{ (1+a) \pm \sqrt{(1+a)^2 - 4a} }{2} \), provides the roots of this equation. These roots determine the specific values of \( v \), making it constant.
  • The term under the square root, known as the discriminant, is crucial. It reveals whether real solutions exist by determining if this expression is positive.
  • In this model, for \( 0 < a < 1 \), the discriminant \( (1+a)^2 - 4a \) simplifies to a positive value, confirming two distinct real roots.
By calculating these roots, different behaviors of \( v \) in various intervals can be explored easily.
Mathematical Modeling
Mathematical modeling involves creating equations that mirror real-world phenomena. It provides a framework to understand and predict complex systems.

The Fitzhugh-Nagumo model exemplifies this approach, offering a simplified representation of neuron behavior through differential equations. It serves as a foundation for exploring more intricate models and gaining insights into biological processes.
  • Key to effective modeling is identifying essential variables and interactions, simplifying reality while retaining core dynamics.
  • By analyzing this model, researchers can predict responses of neurons to stimuli, making it a vital tool in neuroscience.
Understanding the balance between simplicity and complexity ensures models remain both accessible and insightful.

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Most popular questions from this chapter

A system of differential equations is given. (a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of motion. (b) Obtain an expression for each equilibrium. \(x^{\prime}=x(2-x), \quad y^{\prime}=y(3-y)\)

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