Electricity of Nerves and cells
Biological cells regulate ion movement through their membranes and thus have an electrical
aspect to their operations. The cell membrane is composed of lipids and poses a barrier to
the free flow of water soluble ions. This means it has electrical resistance.
The cell membrane is studded with all sorts of channels through which various chemical
species pass. Some of these channels are passive; opening or closing, but letting the
concentration gradient push the species through like a temperature gradient pushes heat.
Others are active pumps, using energy from ATP to push ions against the gradient.
During normal operation, cells maintain different
concentrations of ions inside and outside the cells.
this leads to an imbalance in electrical charge, the
membrane has an electric potential difference
across it and is acting as a capacitor.
Conduction along nerves is a bit more involved than
Nerves in many higher order organisms are
myelinated, insulated by Schwann cells with gaps
between them about every mm, known as nodes of
Ranvier. At these nodes, the nerve axon’s cell wall
exposed and undergoes the ion exchange which
propagates an electrical impulse down the nerve.
When
this.
is
The ion
exchange
starts
with
+
the cell in a resting state with the Na
concentration inside the cell lower than that
outside and the K+ concentration higher inside
than out. There is less potassium around than
sodium, so the charge does not end up balanced.
Conduction can be sped up if the cycle of potential changes and charge flows has to happen
less frequently, say every mm or so. So the myelin insulating sheath speeds up signal
propagation down the axon. The exposed areas, the nodes of Ranvier, have a length of about
1μm. Signal propagation along the nerve can be modeled with the following circuit:
The electrical properties of the myelin layer are the
same as those for the lipid bilayer but the thickness of
the myelin layer is around 200 times the thickness of
the lipid bilayer. Let’s see what we can find by using
the electrical properties and the signal strength at the
next node.
Experimental measurements show that the amplitude of
the voltage at the next node remains at 99.3% of its
original value.
The start of an
impulse is the opening of the sodium channels, allowing
Na+ ions to flood in, changing the potential across the
membrane from -70mV to +40mV. This is followed by
the closing of the sodium channels and opening of the
potassium channels, dropping the potential below 70mV and then the ion concentrations are returned to
the resting state by the active ion pumps.
In the nerve axon, the ions inside and outside the cell
wall move along the wall from node to node, triggering
the process there.
This propagation of charged ions along the cell
constitutes an electric current and we can calculate a
resistance for the process using the resistivity of the
axoplasm inside the cell.
Getting a feel for the numbers.
For a nerve cell, the thickness of the cell membrane is about 7 nm and the potential
difference between the outer surface and inner surface is around +70 mV in the resting
state.
Electric field: Calculate the electric field in the membrane, both magnitude and
direction.
E=
Resistance of membrane: The resistance can be calculated using Rw = wd/A, with w
=3.6x107 Ωm for lipid membrane and d=thickness. If we use a typical cell radius of
0.05mm and assume it to be cylindrical with an exposed length of 1μm, calculate the
surface area (not including the ends)
Area =
Calculate the resistance of the axon wall at the node:
Rn =
This is an overestimate because it ignores the ion channels.
Capacitance and resistance of membrane: With a potential difference across an insulating
lipid layer, the neuron cell wall is also acting as a capacitor. The dielectric constant for
lipids is about 9. Using the dimensions and electrical properties given, calculate the
resistance and capacitance of the cell wall as a function of length between the nodes.
Cell Dimensions
radius, r
wall thickness, d
Myelinated thickness
Ranvier node length
RA = aL/A = aL/πr2
50 μm
7 nm
200 d
1 μm
Electrical Properties
exoplasm resistivity ρa
2 Ω∙m
wall resistivity ρw
3.6x107 Ω∙m
wall dielectric constant κ
9
wall permeability κεo
8.1x10-11 C/V∙m
C = κεoA/d = κεo2πrL/d
Rw = wd/2πrL
Ra/L =
Cu/L =
Rwu∙L =
myelinated:
Cm/L =
Rwm∙L =
Notice how the internal resistance for a 10μm length compares with the resistance of the membrane
for the same length! So is more charge going to flow out through the membrane or along the axon?
To estimate the distance between nodes, notice that the circuit model has a
voltage divider composed of 2 resistors. The input voltage equals the voltage
across Ra + Rw. The voltage at the next node is the voltage across Rw. Construct
the ratio of Vout/Vin in terms of the resistances and set it equal to 0.93, the
measured ratio of signal strength. Then solve the resulting expression for L for
both myelinated and unmyelinated nerves.
The actual distance between nodes is about 1mm, how did your answer for the myelinated nerve
compare?
For timing and signal speed, we need to calculate the time constant RC. Calculate the RC time
constants for both myelinated and non-myelinated nerves using the distances calculated above.
The correct resistance to use is the one giving the smallest time constant.
The signal velocity is L/RC , calculate this for both myelinated and un-myelinated nerves using the
distance between nodes and the RC time constants calculated above.
myelinated:
un-myelinated:
Typical values for some types of nerves are given below for comparison. Keep in mind that our model
is very simplified and doesn’t include all the ion channels in the walls, so am answer of the same
order of magnitude is as good as we can expect.
Nerve
Typical radius, μm
signal speed (m/s)
brain (un myelinated),
2
20
type C, also some sensory
nerves.
muscle (motor) nerves
10 (4-20)
100
Type A, position, balance,
delicate touch
myelinated
Type B Sensory nerves to
3 (2-4)
30
CNS
Giant Squid Axon,
1 mm = 103
100 (roughly)
unmyelinated, L~5E-4m
Several compromises are made in the construction of nervous systems in organisms.
One is energy vs. signal speed. Larger radii nerves would give faster conduction, as seen for the
giant squid neuron above, but the energy expenditure for repolarization becomes large. Increasing
the spacing between nodes on a myelinated nerve increases the resistance and slows the signal, but
costs less energy for repolarization.
Additionally, there is the size compromise. Only about a third of the nerves in your body are
myelinated. To do so for all of them would take too much space. Your peripheral nerves would be
the size of garden hoses and your spinal cord would be the diameter of a garbage can!
Sources:
Martini, Fredrick H., Fundamentals of Anatomy and PhysiologyU, 7th ed., Pearson, 2006, pp 400-406
Sternheim & Kane, General Physics, 2nd ed., Wiley, 1991, Ch18
Knight, Jones & Field, College Physics, A Strategic Approach, 2007, Ch23, sect8&9.
Sites of interest:
http://www.centenary.edu/attachments/biophysics/bphy304/11a.pdf