## Introduction

The Fitzhugh-Nagumo Model is a a two dimensional model that
describes the changes in the membrane potential and ion
conductances at a point in an axon. The Fitzhugh-Nagumo equations
are

x' = c*(y+x-1/3*x^3)

y' = -(x-a+b*y)/c

where x is the membrane potential and y is the recovery variable.
The coefficients, a, b, and c are all positive. In our analysis
of this model, we investigated the fixed points, and their
stability. The model itself is discussed in the background
section, what we did is covered in the analysis, and our findings
are presented in the conclusion.

We found four different behaviors of solutions that occur when
the parameters a, b, and c are varied. The first type is a
periodic solution which represents a deficiency of calcium ions.
The second is that solutions for any initial conditions approach
a fixed resting potential. The third is two stable fixed points
and one unstable saddle point in between them. It represents too
great of a shock being applied to the nerve which causes the
resting potential to rise to an excited state. This is unphysical.
The last type is two spiraling fixed points which represents a
second impulse occuring before the original impulse has been
fully transferred. The second impulse disturbs the original one
and alters its approach to its equilibrium state. If the second
impulse is high enough, the behavior is similar to that in the
third type, wherein the resting potential rises to an excited
state. This is a possible physical solution, unlike the third
type that we found.

Background

Analysis

Conclusions

Andrea Sereny

Karen J. Hirst