FitzHugh-Nagumo Model - A Two Variable Action Potenital System
Matteo Contino and Christopher Thoroughgood
1. Introduction
Nerve cells communicate with each other via an electrochemical signalling process which can be modelled in a simple circuit. At
the single cell level, the nerve membrane potential, when perturbed, can exhibit a large deviation from its rest state. The nerve
cell “fires” and is referred to as an action potential. A neighbouring nerve cell can sense the voltage change through chemical or
electrical synapses and fires in response. In this way, an action potential can propagate through the tissue.
The squid giant axon is unmyelinated and has a large axon diameter allowing Alan Hodgkin and Andrew Huxley to insert
voltage clamp electrodes inside the lumen of the axon (Hodgkin and Huxley, 1952a). They were able to develop a model which
describes the mechanism of action potentials of a single cell (Hodgkin and Huxley, 1952b). The FitzHugh-Nagumo is a simplified
version of the Hodgkin and Huxley model outlined below.
The characteristics of action potential which is investigated in the FitzHugh-Nagumo model are that a nerve cell is excitable,
may have two stable rest states, and that some nerve cells have a unistable rest state (referred to as the action potential pulse)
(FitzHugh, 1961). The action potential consists of three phases including the resting, depolarisation and repolarisation phases.
If the cell is stimulated in a specific manner, an action potential can be made to oscillate in time. In the oscillating state, there
may be bistability with a stable oscillating state for the same stimulus (Nagumo et al., 1962).
dV
= f (V ) − W + k;
dt
where
1
f (V ) = − V 3
3
(1)
dW
= a(bV − cW )
dt
(2)
Equations (1) and (2) represents a special case of the nondimensionalised FitzHugh-Nagumo model where f (V ) is a polynomial
of the third degree and a, b and c are constant parameters. The FitzHugh-Nagumo model consists of 2 dynamic variables: a
voltage like variable (V ) having cubic nonlinearity allows regenerative self-excitation via a positive feedback, and a recovery
variable (W ) having linear dynamics that provides a slower negative feedback. Numerical analysis of the model is described in
Figure 1 and demonstrates the excitable and oscillatory behaviours of the model for different parameter sets.
2. FitzHugh-Nagumo model and numerical analysis in MATLAB
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 1: Phase portraits (a), (c), (e), (g) with Null clines and evolution plots (b),(d),(f),(h) for the FitzHugh-Nagumo model where four
distinct behaviours can exist. The model consists of four parameters: a, b, c and k. The parameters a and c from the MATLAB code
provided linked in section 3 are fixed to 0.8 and 5 respectivley, whilst for each distinct behaviour parameters b and z vary. In (a) and (b),
b = 0.9 and z = 0 and in (c) and (d), b = 0.9 and z = 2.5. Both scenarios have linearly stable but excitable steady states. In (e) and (f),
b = 0.9 and z = -0.5. The steady state is unstable and periodic limit cycle solutions occur. In (g) and (h), b = 2.5 and z = -0.5. Three
steady states are present, with two stable steady states and one unstable steady state. A perturbation from either of the stable steady
states can effect a switch to the other.
The MATLAB function meshgrid was used and defines the grid onto which the phase portraits are plotted. This grid is
represented by the output coordinate arrays X and Y with the following syntax: [X,Y] = meshgrid(xgv,ygv). The MATLAB code makes use of the function quiver to create the phase portraits. With the syntax quiver(x,y,u,v)plots vectors as
arrows at the coordinates specified in each corresponding pair of elements in x and y. The matrices x, y, u, and v must all be
the same size and contain corresponding position and velocity components.
(a)
(b)
Figure 2: Bifurcation analysis of the FitzHugh-Nagumo. Bifurcation in the forward direction is represented in (a) and the backward
direction in (b). Both show the changes in local stability between the parameters of voltage and current and demonstrates the onset of
sustained oscillations.
When k is increased, the behaviour of the system changes qualitatively from a stable fixed point to a limit cycle. The point
where the transition occurs is called a bifurcation point, and k is the bifurcation parameter. At some point, the fixed point
will loose its stability, which implies that the real part of at least one of the eigenvalues changes from negative to positive. The
bifurcation point was determined with the Eulers method in MATLAB.
(a)
(b)
(c)
Figure 3: pplane analysis of the FitzHugh-Nagumo model using the same parameters as in Figures 1(e) and 1(f) where in comparison, the
output is the same. (a) demonstrates the toolbox and the user friendly nature of entering the differential equations and parameters. (b)
is the resulting phase portrait including null clines, phase paths and vector field of the phase portrait. (c) pplane comes with a number of
ODE solvers.
The programme pplane is an interactive toolbox which can be used within MATLAB to easily produce phase plots. It plots
vector fields for systems of differential equations and allows the user to draw solution curves. Some of its advanced features
include its ability to find equilibrium points and change ODE solvers. This is demonstrated by Figure 3.
3. Links and References
NOTE: The MATLAB code to accompany the handout is linked below
• To download the MATLAB code used in the presentation: http://www2.warwick.ac.uk/fac/sci/moac/people/students/2013/chris_thoroughgood/
ch925/
• To download pplane: http://math.rice.edu/˜dfield/
• Hodgkin-Huxley Experiments:A. L. Hodgkin, A. F. Huxley, Currents carried by sodium and potassium ions through the membrane of giant axon of
Logio, J. P hysio 116, 449-472, 1952a.
• Hodgkin-Huxley Experiments:A. L. Hodgkin, A. F. Huxley, Components of membrane conductance in giant axon of Logio, J. P hysio 116, 449-472,
1952b.
• FitzHugh-Nagumo Experiments: R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophy. J. 1, 445-466,
1961.
• FitzHugh-Nagumo Experiments: J. S. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon IRE 20, 2061-2071,
1962.