ABSTRACT
In this chapter we make a study of some of the principal properties of the simplified model of nerve impulse transmission (4.4.7) due to FitzHugh and Nagumo. For the space clamped case the model is
du
dt = u(1− u)(u− a)− w + I(t),
dw
dt = bu− γw, (7.1.1)
where 0 < a < 1, b > 0, γ ≥ 0 and I(t) is the total membrane current, which may be an arbitrary function of time. If the space clamp is removed then the model becomes
∂u
∂t =
∂2u
∂x2 + u(1− u)(u− a)− w,
∂w
∂t = bu− γw. (7.1.2)
To begin with we wish to determine whether the model (7.1.1) exhibits the important threshold property mentioned in Section 4.3. In mathematical terms this property is the same as asking whether the system (7.1.1) is excitable. Consider the ordinary differential equation,
dy
dt = y(1− y)(y − a), (7.1.3)
where y = 0, a, 1 are rest states. This equation has stable rest states at y = 0, 1 and an unstable rest state at y = a. These statements are easily checked by observing that if initially y(0) < a then dydt < 0 and so y(t) → 0, whereas if y(0) > a then dydt > 0 and y(t) → 1. The implication of this is that for “small” (i.e., y(0) < a) initial data the solution is attracted to the rest state
y(0) > the is attracted to y = 1. We call the parameter a the threshold and call the equation (7.1.3) excitable. Let us now see if a similar property is present in the system (7.1.1). For
simplicity we only consider the case where I(t) = 0. The case I(t) 6= 0 will be discussed later. With I(t) = 0 the system (7.1.1) can be studied using the techniques developed in Chapter 5. First of all it is an important requirement that (7.1.1) has a unique rest state. That is, on setting
du
dt =
dw
dt = 0,
we require the pair (i.e., the nullclines)
w = u(1− u)(u− a), bu = γw (7.1.4)
to have the unique solution (u,w) = (0, 0). That is, the equation
u(1− u)(u− a) = b γ u
must have the single solution u = 0. For this to be the case the quadratic equation,
u2 − (1 + a)u+ a+ b γ = 0,
can only have complex roots. That is, the parameters a, b, γ must be restricted so that
(1− a)2 < 4 b γ , γ > 0. (7.1.5)
Notice that if γ = 0 then (7.1.1) has the unique rest state (0, 0) without restriction on the parameters a and b. Linearising (7.1.1) about (0, 0) results in the system
du
dt = −au− w,
dw
dt = bu− γw. (7.1.6)
As in Chapter 5 we look for solutions of the form u = α expλt, w = β expλt. Substituting these into equations (7.1.6) leads to the requirement that
(a+ λ)α+ β = 0,
(γ + λ)β − bα = 0, which has nontrivial solutions α and β only if
λ2 + (a+ γ)λ+ b+ aγ = 0. (7.1.7)
FIGURE 7.1.1: Global phase portrait for the system (7.1.8).
