Figure 2.1: The Membrane and Gate Circuit Model
Here we show an idealized cell with a small portion of the membrane blown up into an idealized circuit. We see a small piece of the lipid membrane with an inserted gate. We think of the gate as having some intrinsic resistance and capacitance. Now for our simple Hodgkin - Huxley model here, we want to model a sodium and potassium gate as well as the cell capacitance. So we will have a resistance for both the sodium and potassium. In addition, we know that other ions move across the membrane due to pumps, other gates and so forth. We will temporarily model this additional ion current as a leakage current with its own resistance. We also know that each ion has its own equilibrium potential which is determined by applying the Nernst equation. The driving electomotive force or driving emf is the difference between the ion equilibrium potential and the voltage across the membrane itself. Hence, if Ec is the equilibrium potential due to ion c and Vm is the membrane potential, the driving force is Vc - Vm. In Figure 2.2, we see an electric schematic that summarizes what we have just said. We model the membrane as a parellel circuit with a branch for the sodium and potassium ion, a branch for the leakage current and a branch for the membrane capacitance.
From Ohm's law, we know that voltage is current times resistance; hence for each ion c, we can say where we label the voltage, current and resistance due to this ion with the subscript c. This implieswhere gc is the reciprocal resistance or conductance of ion c. Hence, we can model all of our ionic currents using a conductance equation of the form above. Of course, the potassium and sodium conductances are nonlinear functions of the membrane voltage V and time t. This reflects the fact that the amount of current that flows through the membrane for these ions is dependent on the voltage differential across the membrane which in turn is also time dependent. The general functional form for an ion c is thus
where the leakage battery voltage, EL, and the conductance gL are constants that are data driven.Hence, our full model would be
Activation and Inactivation Variables:We assume that the voltage dependence of our activation and inactivation has been fitted from data. Hodgkin and Huxley modeled the time dependence of these variables using first order kinetics. They assumed a typical variable of this type, say m, satisfies for each value of voltage, V: