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A Computational Study of the Fitzhugh-Nagumo Action Potential System

Anna St. Cyr, Joseph Salomone, and Angela Umstead

Advisors: Dr. Muhammad Usman and Dr. Amit Singh University of Dayton

Abstract

The most celebrated example of mathematical modeling is the Hodgkin-Huxley model of nerve physiology. Their experiments were carried out on a giant axon of a squid, which was large enough for the implantation of

Fitzhugh Nagumo Model

We will use the following form of the Fitzhugh-Nagumo equations by Wilson(1999). 2 v 1 3Fitzhugh Nagumo Mathematical Model v 10 v v r I

t 3 x2

r t

Diffusion Free Fitzhugh-Nagumo Model

Diffusion free Fitzhugh-Nagumo model is given by:

v 1 10 v v 3 r I t 3 r p a 1.25v br t

electrodes. The Hodgkin-Huxley mathematical model for nerve cell action potential is a system of four coupled

ordinary differential equations. The Fitzhugh-Nagumo two-variable action potential system behaves qualitatively like the Hodgkin-Huxley space-clamped system. Being simpler by two variables, action potentials and other properties of the Hodgkin-Huxley model may be visualized as phase-plane plots. We used MATLAB to study

Where v(x,t) and r(x,t) denote the voltage and recovery variables respectively at position x and at time t and I is the injected current. Now we model a pulse traveling along a nerve fiber. We will use MATLAB programs to generate the wave. The following values of parameters are used:

D 1, a 1.5, b 1, and p 0.8 For this set of parameters the steady state values are used as initial conditions : v0 1.5 and r 0 3/8

p a 1.25v br

the numerical solutions as well as the qualitative properties of the model.

The Diffusion Free Fitzhugh-Nagumo Model was solved numerically by using the built in Matlab functions of ODE45, ODE23, and ODE15s.

Diffusion Free Fitzhugh-Nagumo Model

2.5 2 Ode45 Ode23 Ode15s

Fitzhugh-Nagumo Traveling Waves

1.5 1

Voltage

2 1.5

0.5 0 -0.5

Biological Background

The human nervous system controls everything from movement and speech to breathing and digestion. It takes in sensory information, processes it, and then tells the body how to respond. From the central nervous system's control center in the brain and spinal cord, information is constantly moving to and from the vast network of nerve cells in the peripheral nervous system. At the cellular level these messages are passed from neuron to neuron by way of electrical impulses and chemical signaling. In order for a neuron to fire, an electrical impulse known as an action potential must travel along its axon. In order for a message to be passed from one neuron to another it must cross the small space, called the synapse, between the end of one nerve cell's axon and the dendrites of another cell. Because the action potential cannot pass over this gap, when it reaches the end of the axon, chemicals called neurotransmitters are released into the synapse, bind to the dendrites of the next neuron, and cause that neuron to fire. In order for an action potential to begin moving down the axon, ion channels in the cells membrane must open, allowing the voltage between the inside and the outside of the cell to rise above its resting state at -75 millivolts. While at rest only potassium ions can pass through the cell's membrane, yet as the resting potential rises past a certain threshold the action potential begins to travel along the axon opening sodium ion channels along the way. This results in the inside of the cell rapidly becoming more positive than the outside in a step known as depolarization. This depolarizing effect travels along the front of the action potential as it moves down the axon. The cell then repolarizes when potassium ions rush outward across the membrane restoring it to its resting potential. The almost instantaneous change in potential produced by depolarization and repolarization creates a pattern called a spike discharge. When the action potential reaches the end of the axon, it opens calcium ion channels thus increasing the concentration of calcium ions inside the cell. This increase in calcium ions triggers the release of neurotransmitters into the synaptic cleft, and in so doing triggers a nerve impulse in a neighboring neuron. Many models of action potential generation in neurons have been proposed by researchers including the Hodgkin-Huxley, Integrate-and-fire, Morris-Lecar and Fitzhugh Nagumo model.

v o l t a g e

1

-1

0.5

-1.5

0

-2

-0.5 -1 -1.5

0

10

20

30

40

50

60

Time

Phase plans analysis of Fitzhugh-Nagumo Model

v ' = 10 (v - (1/3) v 3 - r) + I r ' = p (a + 1.25 v - b r) a = 1.5 b=1 I = 1.5 p = 0.08

-2 0 20 40 60 80 100 120

2

time

1.5 1

2

Recoovery

0.5

0

1.5

Comparison between Hodgkin-Huxley model and Fitzhugh-Nagumo model In 1950 Hodgkin and Huxley developed a system of non-linear partial differential equations while studying a giant axon of a squid to show the action potential of the nerve axon. This model is very complex containing a system of four coupled differential equations. The Hodgkin-Huxley model is too difficult to solve analytically so in 1961 Fitzhugh and Nagumo created a simplified version. This simplified equation contains two variables opposed to the four variables of the Hodgkin-Huxley model. The Fitzhugh-Nagumo model is obtained from the Hodgkin-Huxley model by combining the variables V and m into a single variable v and combining the variables n and h into a single variable r. The Fitzhugh-Nagumo equations show the qualitative solution to the nerve action impulse model.

v o l t a g e

-0.5

1

-1

0.5

-1.5

0

-2

-0.5 -1 -1.5 -2 0 20 40 60 80 100 120

-4

-3

-2

-1

0 Voltage

1

2

3

4

Conclusion

In this work we have considered a simplified model of action potential generation in neurons known as the

Fitzhugh-Nagumo (FN) model. Unlike the Hodgkin-Huxley, which has four dynamical variables, the FN model has only two

so FN model is easy to explore the dynamical properties. We have solved the system numerically for the solution and traveling waves using MATLAB. Further we have used pplane7, a MATLAB utility developed by Rice University to explore the dynamical properties of the model.

time

References

[1] Excitable Cells. RCN | Digital Cable TV, High-Speed Internet & Phone in Boston, Chicago, New York City, Philadelphia, Washington, D.C. and the Lehigh Valley. 17 July 2003. Web. 17 Mar. 2010. [2] "FitzHugh-Nagumo Model, Mathematical Cell Modeling Simulation Numerical. Web. 09 Apr. 2010. <http://thevirtualheart.org/java/fhn24.html>. Department of Mathematics at Rice University. Web. 09 Apr. 2010. <http://math.rice.edu/~dfield/matlab7>. Knickerbocker, C. J. Nerve Impulse Models. St. Laurence University Department of Mathematics.

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