2.1 Equilibrium potential

Neurons are just as other cells enclosed by a membrane which separates the interior of the cell from the extracellular space. Inside the cell the concentration of ions is different from that in the surrounding liquid. The difference in concentration generates an electrical potential which plays an important role in neuronal dynamics. In this section, we want to provide some background information and give an intuitive explanation of the equilibrium potential.

2.1.1 Nernst potential

From the theory of thermodynamics, it is known that the probability that a
molecule takes a state of energy *E* is proportional to the Boltzmann factor
*p*(*E*) exp(- *E*/*kT*) where *k* is the Boltzmann constant and *T* the
temperature. Let us consider positive ions with charge *q* in a static
electrical field. Their energy at location *x* is
*E*(*x*) = *q* *u*(*x*) where
*u*(*x*) is the potential at *x*. The probability to find an ion in the region
around *x* is therefore proportional to
exp[- *q* *u*(*x*)/*kT*]. Since the
number of ions is huge, we may interpret the probability as a ion density.
For ions with positive charge *q* > 0, the ion density is therefore higher in
regions with low potential *u*. Let us write *n*(*x*) for the ion density at
point *x*. The relation between the density at point *x*_{1} and point *x*_{2} is

A difference in the electrical potential

Since this is a statement about an equilibrium
state, the reverse must also be true. A difference in ion density generates a
difference *u* in the electrical potential. We consider two regions of
ions with concentration *n*_{1} and *n*_{2}, respectively. Solving
(2.1) for *u* we find that, at equilibrium, the
concentration difference generates a voltage

which is called the Nernst potential (Hille, 1992).

2.1.2 Reversal Potential

The cell membrane consists of a thin bilayer of lipids and is a nearly perfect electrical insulator. Embedded in the cell membrane are, however, specific proteins which act as ion gates. A first type of gate are the ion pumps, a second one are ion channels. Ion pumps actively transport ions from one side to the other. As a result, ion concentrations in the intra-cellular liquid differ from that of the surround. For example, the sodium concentration inside the cell ( 60mM/l) is lower than that in the extracellular liquid ( 440 mM/l). On the other hand, the potassium concentration inside is higher ( 400 mM/l) than in the surround ( 20 mM/l).

Let us concentrate for the moment on sodium ions. At equilibrium the
difference in concentration causes a Nernst potential
*E*_{Na} of about
+50 mV. That is, at equilibrium the interior of the cell has a positive
potential with respect to the surround. The interior of the cell and the
surrounding liquid are in contact through ion channels where Na^{+} ions can
pass from one side of the membrane to the other. If the voltage difference
*u* is smaller than the value of the Nernst potential
*E*_{Na},
more Na^{+} ions flow into the cell so as to
decrease the concentration difference. If the voltage is larger than the
Nernst potential ions would flow out the cell. Thus the direction of the
current is reversed when the voltage *u* passes
*E*_{Na}. For this
reason,
*E*_{Na} is called the reversal potential.

As mentioned above, the ion concentration of potassium is higher inside the
cell ( 400 mM/l) than in the extracellular liquid ( 20
mM/l). Potassium ions have a single positive charge
*q* = 1.6×10^{-19} C. Application of the Nernst equation with the Boltzmann constant
*k* = 1.4×10^{-23} J/K yields
*E*_{K} - 77mV at room
temperature. The reversal potential for K^{+} ions is therefore negative.

So far we have considered either sodium or potassium. In real cells, these
and other ion types are simultaneously present and contribute to the voltage
across the membrane. It is found experimentally that the resting potential of
the membrane is about
*u*_{rest} -65 mV. Since
*E*_{K} < *u*_{rest} < *E*_{Na}, potassium ions will, at the resting potential,
flow out of the cell while sodium ions flow into the cell. The active ion
pumps balance this flow and transport just as many ions back as pass through
the channels. The value of
*u*_{rest} is determined by the dynamic
equilibrium between the ion flow through the channels (permeability of the
membrane) and active ion transport (efficiency of the ion pumps).

Cambridge University Press, 2002

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