Channels - their effects

Ionic channels appear throughout membranes in neurons and are responsible for most of the interesting dynamic behavior. They play a fundamental role in the properties and connections of neurons and neural nets. So far, we have viewed the membrane as a passive cable with inputs and inhomogeneities. However, as we have seen, cables can transmit information only in a very analog fashion and for long distances would require an enormous diameter. (Homework: Given $R_m=100000 \Omega cm^2$ and $R_a=100\Omega cm$ what diameter would you need to get a length constant of a meter? - This is less than the distance traveled to the spinal cord from nerves in the foot. )

This problem can be resolved if there is a means to keep the potential of the cable localized and high. This is exactly the problem that action potentials solve. In a real nerve cell there are many species of ions, calcium, potassium, sodium, chloride, magnesium, to name a few of the most common. The basis for the resting membrane potential is the balance of these ions in the cell and outside of the cell. There are basically two forces at work in absence of ionically selective channels: (i) passive diffusion in which case high concentrations tend to move toward low concentrations and (ii) electric forces which attempt to balance the charges on either side of the membrane.

Hodgkin and Huxley won the Nobel prize for their elegant experimental and theoretical work on the nature of the voltage gated channels in the squid axon. This theory is the basis for all subsequent models of ionic channels in nerve and other membranes. Some of the details may differ, but the basic ideas are the same.

Channels facilitate the passive flow of ions across the membrane. When they are gated by other forces such as calcium or voltage, they can also provide great computational properties to the neuron. Non-gated channels are responsible for the membrane potential. Recall that the equilibrium potential of an ion is given by:

$$E = 2.303\frac{RT}{ZF} \log\frac{\left[C\right]_o}{\left[C\right]_i}$$

where C is the concentration, and at $25^\circ\ C$ we have $2.303RT/ZF=60\ mv$ when Z=+1. Thus, since there are 20 mMoles of potassium inside and 400 outside E_K=-78mV. E_Na=55mV and E_Cl=-60mV. Recall that the membrane potential is found from the Goldman equation:

$$V_m = \frac{RT}{F}\ln \frac{\sum P_j \left[C_j\right]_o} {\sum P_j \left[C_j\right]_i}$$

where P_j are the permeabilities of the ions. At rest

P_K:P_Na:P_Cl=1:.04:.45

during the peak of the action potential

P_K:P_Na:P_Cl=1:20:.45

 

Figure 1: Equivalent circuit for Squid Axon

This is really the correct way to discern the membrane potential, however, in modeling, we will make a much simpler equivalent circuit. We will treat each ion channel as a conductor and a battery. Note that the permeabilities act like conductances and the equilibrium potentials act as batteries. Consider Figure 1 which ignores the pumps for sodium and potassium. Then the equations for the membrane are:  

$$C\frac{dV}{dt} = g_{Na}(E_{Na}-V)+g_{Cl}(E_{Cl}-V)+g_{K}(E_K-V)+I$$ (1)

where I is the applied current. The membrane potential is defined as a steady state of (1) that is the right-hand side must vanish. This enables us to solve forV:

$$V_m = \frac{g_{Na}E_{Na}+g_KE_K+g_{Cl}E_{Cl}+I}{g_{Na}+g_K+g_{Cl}}$$

If $I=0,g_{Cl}=0,g_K=10\times10^{-6}S, g_{Na}=.5\times10^{-6}S$ then V_m=-69mV.

Homework:

1.

Compute the sodium and potassium currents at rest (Hint: The current of an ionic species is I=g(E-V) where E is the reversal potential, g the conductance, and V the resting potential.)

2.

What is the effect of g_Cl on the resting potential. That is if g_Cl is small and positive, will this raise or lower the potential.

3.

Given that E_Ca=150mV suppose that $g_{Ca}=.2\times 10^{-6}S.$ What is V_m?

4.

Suppose that g_Na increases 500 fold as it does during the action potential. What is V_m in this case?

5.

Again ignoring chloride and using the values in the example for the conductances of sodium and potassium, how much current must you inject to increase the potential by 10mV?

6.

Rewrite (1) as

$$CdV/dt = \bar{g}(V_m-V)+I$$

where $\bar{g}$ is the effective conductance. Using the given values for the potassium and sodium conductances and noting that $C=C_m\times A$ where A is the area of the membrane and using $C_m=1\mu F/cm^2$ what is the area of the membrane if the time constant is 1msec. (Hint: The time constant is $C/\bar{g}=C_mA/\bar{g}.$ )