A system of four differential equations, such as the Hodgkin-Huxley model, is difficult to analyze, so that normally we are limited to numerical simulations. A mathematical analysis is, however, possible for a system of two differential equations.
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 these cases, more than two variables would have to be eliminated, but the procedure would be completely analogous (258).
We focus on the Hodgkin-Huxley model discussed in Chapter 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 is much faster than that of the variables and . Moreover, the time scale of is fast compared to the membrane time constant of a passive membrane, which characterizes the evolution of the voltage when all channels are closed. The relatively rapid time scale of suggests that we may treat as an instantaneous variable. The variable in the ion current equation (2.5) of the Hodgkin-Huxley model can therefore be replaced by its steady-state value, . This is what we call a quasi steady state approximation which is possible because of the ’separation of time scales’ between fast and slow variables.
Second, we see from Fig. 2.3B that the time constants and have similar dynamics over the voltage . Moreover, the graphs of and in Fig. 2.3A are also similar. This suggests that we may approximate the two variables and by a single effective variable . To keep the formalism slightly more general we use a linear approximation with some constants and set . With , , and , equations (2.4) - (2.5) become
with , and some function . We now turn to the three equations (2.2.2). The equation has disappeared since is treated as instantaneous. Instead of the two equations (2.2.2) for and , we are left with a single effective equation
where is a parameter and a function that interpolates between and (see Section 4.2.2). Eqs. (4.4) and (4.5) define a general two-dimensional neuron model. If we start with the Hodgkin-Huxley model and implement the above reduction steps we arrive at functions and which are illustrated in Figs. 4.3A and 4.4A. The mathematical details of the reduction of the four-dimensional Hodgkin-Huxley model to the two equations (4.4) and (4.5) are given below.
Before we go through the mathematical steps, we present two examples of two-dimensional neuron dynamics which are not directly derived from the Hodgkin-Huxley model, but are attractive because of their mathematical simplicity. We will return to these examples repeatedly throughout this chapter.
The reduction of the Hodgkin-Huxley model to Eqs. (4.4) and (4.5) presented in this paragraph is inspired by the geometrical treatment of Rinzel (439); see also the slightly more general method of Abbott and Kepler (2) and Kepler et al. (258).
The overall aim of the approach is to replace the variables and in the Hodgkin-Huxley model by a single effective variable . At each moment of time, the values can be visualized as points in the two-dimensional plane spanned by and ; cf. Fig. 4.5B. We have argued above that the time course of the scaled variable is expected to be similar to that of . If, at each time, were equal to , then all possible points would lie on the straight line which changes through and . It would be unreasonable to expect that all points that occur during the temporal evolution of the Hodgkin-Huxley model fall exactly on that line. Indeed, during an action potential (Fig. 4.5A), the variables and stay close to a straight line, but are not perfectly on it (Fig. 4.5B). The reduction of the number of variables is achieved by a projection of those points onto the line. The position along the line gives the new variable ; cf. Fig. 4.6. The projection is the essential approximation during the reduction.
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
At rest, we have .
Second, we turn the coordinate system by an angle which is determined as follows. For a given constant voltage , the dynamics of the gating variables and approaches the equilibrium values . The points as a function of define a curve in the two-dimensional plane. The slope of the curve at yields the rotation angle via
Rotating the coordinate system by turns the abscissa of the new coordinate system in a direction tangential to the curve. The coordinates in the new system are
Third, we set and retain only the coordinate along . The inverse transform,
yields and since . Hence, after the projection, the new values of the variables and are
In principle, can directly be used as the new effective variable. From (4.15) we find the differential equation
which is of the form , as desired.
If we introduce and , we find from Eq. (4.17) the variable and from Eq. (4.18) , which are exactly the approximations that we have used in (4.3). The differential equation for the variable is of the desired form and can be found from Eq. (4.20) and (4.21). The resulting function of the two-dimensional model is illustrated in Fig. 4.3A.
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