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3.1 Reduction to two dimensions

In this section we perform a systematic reduction of the four-dimensional Hodgkin-Huxley model to two dimensions. To do so, we have to eliminate two of the four variables. The essential ideas of the reduction can also be applied to detailed neuron models that may contain many different ion channels. In this case, more than two variables would have to be eliminated, but the procedure would be completely analogous (Kepler et al., 1992).

3.1.1 General approach

We focus on the Hodgkin-Huxley model discussed in Chapter 2.2 and start with two qualitative observations. First, we see from Fig. 2.3B that the time scale of the dynamics of the gating variable m is much faster than that of the variables n, h, and u. This suggests that we may treat m as an instantaneous variable. The variable m in the ion current equation (2.5) of the Hodgkin-Huxley model can therefore be replaced by its steady-state value, m(t)$ \to$m0[u(t)]. This is what we call a quasi steady state approximation.

Second, we see from Fig. 2.3B that the time constants $ \tau_{n}^{}$(u) and $ \tau_{h}^{}$(u) are roughly the same, whatever the voltage u. Moreover, the graphs of n0(u) and 1 - h0(u) in Fig. 2.3A are rather similar. This suggests that we may approximate the two variables n and (1 - h) by a single effective variable w. To keep the formalism slightly more general we use a linear approximation (b - h) $ \approx$ a n with some constants a, b and set w = b - h = a n. With h = b - w, n = w/a, and m = m0(u), equations (2.4) - (2.5) become

C$\displaystyle {{\text{d}}u\over {\text{d}}t}$ = - gNa[m0(u)]3 (b - w) (u - VNa) - gK $\displaystyle \left(\vphantom{{w\over a}}\right.$$\displaystyle {w\over a}$$\displaystyle \left.\vphantom{{w\over a}}\right)^{4}_{}$ (u - VK) - gL (u - VL) + I , (3.1)


$\displaystyle {{\text{d}}u\over {\text{d}}t}$ = $\displaystyle {1\over \tau}$$\displaystyle \left[\vphantom{F(u,w) + R \, I}\right.$F(u, w) + R I$\displaystyle \left.\vphantom{F(u,w) + R \, I}\right]$ , (3.2)

with R = gL-1, $ \tau$ = R C and some function F. We now turn to the three equations (2.6). The m equation has disappeared since m is treated as instantaneous. Instead of the two equations (2.6) for n and h, we are left with a single effective equation

$\displaystyle {{\text{d}}w \over {\text{d}}t}$ = $\displaystyle {1\over \tau_w}$G(u, w) , (3.3)

where $ \tau_{w}^{}$ is a parameter and G a function that has to be specified. Eqs. (3.2) and (3.3) define a general two-dimensional neuron model. The mathematical details of the reduction of the four-dimensional Hodgkin-Huxley model to the two equations (3.2) and (3.3) are given below. Before we go through the mathematical step, we will present two examples of two-dimensional neuron dynamics. We will return to these examples repeatedly throughout this chapter. Example: Morris-Lecar model

Morris and Lecar (1981) proposed a two-dimensional description of neuronal spike dynamics. A first equation describes the evolution of the membrane potential u, the second equation the evolution of a slow `recovery' variable $ \hat{{w}}$. In dimensionless variables, the Morris-Lecar equations read

$\displaystyle {{\text{d}}u\over {\text{d}}t}$ = - g1 $\displaystyle \hat{{m}}_{0}^{}$(u) (u - 1) - g2 $\displaystyle \hat{{w}}$ (u - V2) - gL (u - VL) + I , (3.4)
$\displaystyle {{\text{d}}\hat{w}\over {\text{d}}t}$ = - $\displaystyle {1\over \tau(u)}$$\displaystyle \left[\vphantom{
\hat{w}- w_0(u)
}\right.$$\displaystyle \hat{{w}}$ - w0(u)$\displaystyle \left.\vphantom{
\hat{w}- w_0(u)
}\right]$ . (3.5)

The voltage has been scaled so that one of the reversal potentials is unity. Time is measured in units of $ \tau$ = RC. If we compare Eq. (3.4) with Eq. (3.1), we note that the first current term on the right-hand side of Eq. (3.1) has a factor (b - w) which closes the channel for high voltage and which is absent in (3.4). Another difference is that neither $ \hat{{m}}_{0}^{}$ nor $ \hat{{w}}$ in Eq. (3.4) have exponents. To clarify the relation between the two models, we could set $ \hat{{m}}_{0}^{}$(u) = [m0(u)]3 and $ \hat{{w}}$ = (w/a)4. In the following we consider Eqs. (3.4) and (3.5) as a model in its own rights and drop the hats over m0 and w.

The equilibrium functions shown in Fig. 2.3A typically have a sigmoidal shape. It is reasonable to approximate the voltage dependence by

m0(u) = $\displaystyle {1\over 2}$ $\displaystyle \left[\vphantom{1+{\rm tanh}\left( {u-u_1\over u_2}\right)}\right.$1 + tanh$\displaystyle \left(\vphantom{ {u-u_1\over u_2}}\right.$$\displaystyle {u-u_1\over u_2}$$\displaystyle \left.\vphantom{ {u-u_1\over u_2}}\right)$$\displaystyle \left.\vphantom{1+{\rm tanh}\left( {u-u_1\over u_2}\right)}\right]$ (3.6)
w0(u) = $\displaystyle {1\over 2}$ $\displaystyle \left[\vphantom{1+{\rm tanh}\left( {u-u_3\over u_4}\right)}\right.$1 + tanh$\displaystyle \left(\vphantom{ {u-u_3\over u_4}}\right.$$\displaystyle {u-u_3\over u_4}$$\displaystyle \left.\vphantom{ {u-u_3\over u_4}}\right)$$\displaystyle \left.\vphantom{1+{\rm tanh}\left( {u-u_3\over u_4}\right)}\right]$ (3.7)

with parameters u1,..., u4, and to approximate the time constant by

$\displaystyle \tau$(u) = $\displaystyle {\tau_w \over {\rm cosh}\left( {u-u_3\over u_4}\right)}$ (3.8)

with a further parameter $ \tau_{w}^{}$.

The Morris-Lecar model (3.4)-(3.8) gives a phenomenological description of action potentials. Action potentials occur, if the current I is sufficiently strong. We will see later on that the firing threshold in the Morris-Lecar model can be discussed by phase plane analysis. Example: FitzHugh-Nagumo model

FitzHugh and Nagumo where probably the first to propose that, for a discussion of action potential generation, the four equations of Hodgkin and Huxley can be replaced by two, i.e., Eqs. (3.2) and (3.3). They obtained sharp pulse-like oscillations reminiscent of trains of action potentials by defining the functions F(u, w) and G(u, w) as

F(u, w) = u - $\displaystyle {1\over 3}$u3 - w  
G(u, w) = b0 + b1 u - w , (3.9)

where u is the membrane voltage and w is a recovery variable (FitzHugh, 1961; Nagumo et al., 1962). Note that both F and G are linear in w; the sole non-linearity is the cubic term in u. The FitzHugh-Nagumo model is one of the simplest model with non-trivial behavior lending itself to a phase plane analysis, which will be discussed below in Sections 3.2 and 3.3.

3.1.2 Mathematical steps (*)

The reduction of the Hodgkin-Huxley model to Eqs. (3.2) and (3.3) presented in this paragraph is inspired by the geometrical treatment of Rinzel (1985); see also the slightly more general method of Abbott and Kepler (1990) and Kepler et al. (1992).

The overall aim of the approach is to replace the variables n and h in the Hodgkin-Huxley model by a single effective variable w. At each moment of time, the values (n(t), h(t)) can be visualized as points in the two-dimensional plane spanned by n and h; cf. Fig. 3.1. We have argued above that the time course of the variable n is expected to be similar to that of 1 - h. If, at each time, n was equal to 1 - h, then all possible points (n, h) would lie on the straight line h = 1 - n passing through the points (0, 1) and (1, 0) of the plane. To keep the model slightly more general we allow for an arbitrary line h = b - a n which passes through (0, b) and (1, b - a). It would be unreasonable to expect that all points (n(t), h(t)) that occur during the temporal evolution of the Hodgkin-Huxley model fall exactly on that line. The reduction of the number of variables is achieved by a projection of those points onto the line. The position along the line h = b - a n gives the new variable w; cf. Fig. 3.1. The projection is the essential approximation during the reduction.

Figure 3.1: Arbitrary points (n, h) are projected onto the line in direction of $ \vec{{e}}_{1}^{}$ and passing through the point (n0(urest), h0(urest)). The dotted line gives the curve (n0(u), h0(u)).

To perform the projection, we will proceed in three steps. A minimal condition for the projection is that the approximation introduces no error while the neuron is at rest. As a first step, we therefore shift the origin of the coordinate system to the rest state and introduce new variables

x = n - n0(urest) (3.10)
y = h - h0(urest) . (3.11)

At rest, we have x = y = 0.

Second, we turn the coordinate system by an angle $ \alpha$ which is determined as follows. For a given constant voltage u, the dynamics of the gating variables n and h approaches the equilibrium values (n0(u), h0(u)). The points (n0(u), h0(u)) as a function of u define a curve in the two-dimensional plane. The slope of the curve at u = urest yields the turning angle $ \alpha$ via

tan$\displaystyle \alpha$ = $\displaystyle {{{\text{d}}h_0\over {\text{d}}u}\vert _{u_{\rm rest} } \over {{\text{d}}n_0\over {\text{d}}u}\vert _{u_{\rm {rest}}}}$ . (3.12)

Turning the coordinate system by $ \alpha$ moves the abscissa $ \vec{{e}}_{1}^{}$ of the new coordinate system in a direction tangential to the curve. The coordinates (z1, z2) in the new system are

$\displaystyle \left(\vphantom{ \begin{array}{c} z_1\\  z_2 \end{array} }\right.$$\displaystyle \begin{array}{c} z_1\\  z_2 \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} z_1\\  z_2 \end{array} }\right)$ = $\displaystyle \left(\vphantom{ \begin{array}{cc} \cos \alpha & \sin \alpha \\  -\sin \alpha & \cos\alpha \end{array} }\right.$$\displaystyle \begin{array}{cc} \cos \alpha & \sin \alpha \\  -\sin \alpha & \cos\alpha \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cc} \cos \alpha & \sin \alpha \\  -\sin \alpha & \cos\alpha \end{array} }\right)$ $\displaystyle \left(\vphantom{ \begin{array}{c} x\\  y \end{array} }\right.$$\displaystyle \begin{array}{c} x\\  y \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} x\\  y \end{array} }\right)$ . (3.13)

Third, we set z2 = 0 and retain only the coordinate z1 along $ \vec{{e}}_{1}^{}$. The inverse transform,

$\displaystyle \left(\vphantom{ \begin{array}{c} x\\  y \end{array} }\right.$$\displaystyle \begin{array}{c} x\\  y \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} x\\  y \end{array} }\right)$ = $\displaystyle \left(\vphantom{ \begin{array}{cc} \cos \alpha & -\sin \alpha \\  \sin \alpha & \cos\alpha \end{array} }\right.$$\displaystyle \begin{array}{cc} \cos \alpha & -\sin \alpha \\  \sin \alpha & \cos\alpha \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cc} \cos \alpha & -\sin \alpha \\  \sin \alpha & \cos\alpha \end{array} }\right)$ $\displaystyle \left(\vphantom{ \begin{array}{c} z_1\\  z_2 \end{array} }\right.$$\displaystyle \begin{array}{c} z_1\\  z_2 \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{c} z_1\\  z_2 \end{array} }\right)$ , (3.14)

yields x = z1 cos$ \alpha$ and y = z1 sin$ \alpha$ since z2 = 0. Hence, after the projection, the new values of the variables n and h are
n' = n0(urest) + z1 cos$\displaystyle \alpha$ , (3.15)
h' = h0(urest) + z1 sin$\displaystyle \alpha$ . (3.16)

In principle, z1 can directly be used as the new effective variable. From (3.13) we find the differential equation

$\displaystyle {{\text{d}}z_1 \over {\text{d}}t}$ = cos$\displaystyle \alpha$$\displaystyle {{\text{d}}n\over {\text{d}}t}$ + sin$\displaystyle \alpha$$\displaystyle {{\text{d}}h\over {\text{d}}t}$ . (3.17)

We use (2.7) and replace, on the right-hand side, n(t) and h(t) by (3.15) and (3.16). The result is

$\displaystyle {{\text{d}}z_1 \over {\text{d}}t}$ = - cos$\displaystyle \alpha$ $\displaystyle {z_1\,\cos\alpha + n_0(u_{\rm rest}) - n_0(u) \over \tau_n(u)}$ - sin$\displaystyle \alpha$$\displaystyle {z_1\,\sin\alpha + h_0(u_{\rm rest}) - h_0(u) \over \tau_h(u)}$ , (3.18)

which is of the form dz1/dt = G(u, z1), as desired.

To see the relation to Eqs. (3.1) and (3.3), it is convenient to rescale z1 and define

w = - tan$\displaystyle \alpha$ n0(urest) - z1 sin$\displaystyle \alpha$ . (3.19)

If we introduce a = - tan$ \alpha$, we find from Eq. (3.15) n' = w/a and from Eq. (3.16) h' = b - w which are the approximations that we have used in (3.1). The differential equation for the variable w is of the desired form dw/dt = G(u, w) and can be found from Eq. (3.18). If we approximate the time constants $ \tau_{n}^{}$ and $ \tau_{h}^{}$ by a common function $ \tau$(u), the dynamics of w is

$\displaystyle {{\text{d}}w \over {\text{d}}t}$ = - $\displaystyle {1\over \tau(u)}$$\displaystyle \left[\vphantom{{w}- w_0(u) }\right.$w - w0(u)$\displaystyle \left.\vphantom{{w}- w_0(u) }\right]$ . (3.20)

with a new equilibrium function w0(u) that is a linear combination of the functions h0 and n0. From Eqs. (3.18) and (3.19) we find

w0(u) = - sin$\displaystyle \alpha$ [cos$\displaystyle \alpha$ n0(u) + sin$\displaystyle \alpha$ h0(u) - c] (3.21)

with a parameter c that is determined by direct calculation. In practice, both w0(u) and $ \tau$(u) are fitted by the expressions (3.7) and (3.8).

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Next: 3.2 Phase plane analysis Up: 3. Two-Dimensional Neuron Models Previous: 3. Two-Dimensional Neuron Models
Gerstner and Kistler
Spiking Neuron Models. Single Neurons, Populations, Plasticity
Cambridge University Press, 2002

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