Hodgkin -Huxley neuron model

The standard Hodgkin - Huxley model of an excitatory neuron consists of the equation for the total membrane current, IM, obtained from Ohm's law:

where V denotes the membrane voltage, IK is the potassium current, INa is the sodium current and IL is leakage current carried by other ions that move passively through the membrane. This equation is derived by modeling the potassium, sodium and leakage currents using a simple electrical circuit model of the membrane. We think of a gate in the membrane as having an intrinsic resistance and the cell membrane itself as having an intrincis capacitance as shown in Figure 2.1:

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.

Figure 2.2: The Simple Hodgkin - Huxley Membrane Circuit Model

From circuit theory, we know that the charge q across a capacitator is q = C E, where C is the capacitance and E is the voltage across the capicitor. Hence, if the capacitance C is a constant, we see that the current through the capacitor is given by the time rate of change of the charge

If the voltage E was also space dependent, then we would write E(z,t) to indicate its dependence on both a space variable z and the time t. Then the capacitive current would be
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 as we mentioned previously, the driving force, V - Ec, is the difference between the voltage across the membrane and the equilibrium value for the ion in question, Ec. Note, the ion battery voltage Ec itself might also change in time (for example, extracellular potassium concentration changes over time ). Hence, the driving force is time dependent. The conductance is modeled as the product of a activation, m, and an inactivation, h, term that are essentially sigmoid nonlinearities. The activation and inactivationa are functions of V and t also. The conductance is assumed to have the form

where appropriate powers of p and q are found to match known data for a given ion conductance.We model the leakage current, IL, as
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: