1) The document describes using the Fitzhugh-Nagumo mathematical model to simulate amyotrophic lateral sclerosis (ALS), also known as Lou Gehrig's disease. 2) MATLAB was used to simulate variations of the Fitzhugh-Nagumo model parameters and visualize the results. 3) The simulations showed that signal propagation along axons can be halted when the diffusion constant ratio decreases below a critical value, mimicking signal blocking seen in ALS. Recovery of the signal was observed when the diffusion constant returned to normal values.

1. Reed R. Woyda
Minnesota State University, Mankato, Department of Mathematics and Statistics
Mathematical Modeling and Computer Simulation of
Amyotrophic Lateral Sclerosis
Introduction
References
1. Fitzhugh R. 1961. Impulses and physiological states in theoretical models of nerve membranes. Biophysical J. 1(6):445-466.
2. G. W. Griffiths, W. W. Schiesser. 2012. Traveling Wave Analysis of Partial Differential Equations., Elsevier, Burlington, MA.
3. A. L. Hodgkin, A. F. Huxley. 1952. Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo.
4. J. Physiol. 116(4):449-472
4. Morris, C., Lecar, H., 1981. Voltage oscillations in the barnacle giant muscle fiber. Biophysical J. 35(1):193-213
Results/Discussion
Method
• All simulations performed in this project were done using the
Fitzhugh-Nagumo (F-N) model, which is comprised of a system
of non-linear partial differential equations which describe a
reaction-diffusion system.
• MATLAB was used to simulate the F-N model, and all variations
of parameters. MATLAB is commonly used for high-level
computing along with the capability of data visualization.
• The Runge-Kutta method, used to approximate solutions of
ordinary differential equations, was used to derive initial
conditions for the reaction-diffusion system.
• Method of Lines technique is used to turn a partial differential
equation into a system of ordinary differential equations and is
used in this study to reduce the F-N model to such a system.
• Amyotrophic Lateral Sclerosis, also known as Lou Gehrig’s
disease, is a progressive neurodegenerative disease that attacks
nerve cells in the brain and spinal cord. ALS has no known cure
and the cause of most cases is unknown. Degeneration of
propagating signals through these neurons in ALS is known as
signal blocking and is thought to occur with inflammation of the
axon.
• Hodgkin and Huxley in 1952 built a mathematical model which
describes the dynamics of nerve cells in the giant squid axon [3].
• In 1961 Richard FitzHugh created a simplified version of the
Hodgkin-Huxley (H-H) model which retains the qualitative but
not quantitative nature [1].
• Then Morris and Lecar in 1981 also produced a simplified and
altered version of the H-H model to model barnacle giant muscle
fibers as voltage-gated potassium and calcium channels [4].
Figure 1 – Solutions of the Fitzhugh-Nagumo equations.
Conditions are a Hopf regime with a = 0.139, ɛ = 0.008, ɤ = 2.54. (a) Perturbations of Iapp = 0.07
produces an action potential. (b) A periodic solution of the equations with Iapp = 0.15. (c) and (d)
depict phase planes for Iapp = 0.07 and 0.15 respectively
Figure 2 Traveling wave solutions of the Fitzhugh-Nagumo reaction-diffusion equations.
Performed with a = 0.139, ɛ = 0.008, ɤ = 2.54. (a) Constant traveling wave with D = 0.03. (b) Signal blocking
at a distance of x1 = 150 with D1 = 0.03 and D2 = D1 *P. (c) Recovery from signal blockage at x2, with D1 =
0.03 and D2 = D1 *P, where P = 0.0372. (d) Cartoon image of axon depicting location of signal blocking, x1,
and location of recovery, x2, along the length of the axon, x.
a b
c
d
x1
x
x2
(1.2)
Method of lines approximation
Derivation of wave speed using the characteristic equation
• Simulations of the non-reaction-diffusion system (figure 1) show that
based on the applied input current, the system exhibits either a single
decaying action potential or a periodic solution.
• As hypothesized, propagation of the action potential, simulated from
the F-N system, was halted when the ratio between D2 and D1 is less
than the critical value P, where P = 0.0372 (figure 2b). Additionally,
when the diffusion constant is returned to D1 at a distance of x2 < xc,
where xc = 12, the action potential is restored and continues to
propagate (figure 2c).
• Deviating lower than the critical ratio xc produces a signal which
decays more rapidly at the point of simulated inflammation, whereas
increasing xc produces a signal which continues to propagate over the
length of the axon.
• Analysis of the wave speed using the characteristic equation gives a
value of c = 0.088 x/t2 (figure 2a) and c = 0.054 for when x > x2
(figure 2b). Thus the wave speed is dependent on the diffusion
constant and when signal blocking phenomena is encountered, the
wave speed decreases to a point such that the voltage goes to zero.
• Future research will now continue to model the Morris-Lecar and
Hodgkin-Huxley systems in order to gain a more quantitative
perspective on the effect of signal blocking.
a b
c d
(1.1)