1.5 Cable properties
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1.5 Cable properties


So far, we have learned
that the cell membrane contains ion channels of selective permeability
for specific ions. Ionic currents flowing across
the cell membrane can then drive membrane potential changes that are filtered, in time,
by the membrane capacitance. We’ve considered the cell
to be small and round, and therefore,
isopotential — that is, all parts of the cell membrane have
the same electrical potential. And, indeed, many types
of cells in our body are small, round, and isopotential. However, this is very far
from the situation when it comes to considering neurons. Neurons, in fact, have
very extensive arborisations. These outgrowths from the cell body are filled with cytoplasm
and covered by a plasma membrane, just like the cell body,
but much thinner. Whereas the cell body typically has
a diameter of 10 microns, your neuronal arborisations may have diameters of the order
of one micrometer or less. Typically, these branches of the neuron
extend for hundreds of microns. In the largest arborisations,
the process can extend several meters. Think of neurons in the giraffe brain
that need to communicate along the length of the spinal cord. It turns out that the neuronal arbors
can be considered as leaky electrical cables
with capacitance. These cables transmit and transform
the electrical signals generated in one part of the neuron
to other parts of the neuron. The most important point to realize is
that the membrane potential at different locations
across the neuronal arborisation will be different. Membrane potential can fluctuate strongly
in one part of a neuron and might make little impact
upon other regions of the neuron. To understand why, in this lesson
we’ll study the basic principles of leaky electrical cables
with capacitance. The simplest analogy to think about
in terms of neuronal arborisations and ionic current flow
in these arborisations is to think of a watering hose
as you might use in your garden. Water influx across this leaky cable
will be high at the input end, and if we make small holes
in this garden hose, water will leak out
along the length of this tube. The pressure at different points
in the tube will differ. Here we’ll have a high water pressure, and as the water leaks out,
at the far end of the water hose you’ll have a lower pressure. We can think of the same thing
in terms of the arborisation. We have intracellular cytoplasm, we have axial current flow,
ionic flow of electricity down the length
of the center of the arborisation. This is flanked by
lipid bi-layer membranes, and part of that current flow
will leak out across the ion channels
that are present in the plasma membrane. The current flow that enters here
will therefore decrease in each part of the cable by the amount of current that leaks out
across the plasma membrane. In addition, of course,
we must remember that the lipid bilayer is
an important capacitor, and we therefore need to think
of capacitance of the axial current in part goes to flow
across the ionic channels, in part charges the membrane,
and in part drives the additional current flow
down the length of the arborisation. We can draw the electrical equivalent
of any small length of this cable through thinking about the capacitance
across that unit length provided by the surface area
of the lipid bilayer. There’s the membrane conductance provided by the numbers
and types of ion channels present in that part of the membrane, and of course,
we have the axial current flow that flows down the length
of the arborisation, changing potential depending
upon the axial resistance that’s offered to that current. In general, we can write cable equations that describe this electrical situation. The simplest one is
the steady-state cable equation where we forget about
time-dependent changes and if we forget about
time-dependent changes, we don’t need to consider capacitance because current only flows
into a capacitor during charging, when which is obviously not
at steady state. The membrane potential
at any given point in this cable is therefore given by
the trans-membrane current multiplied by
the trans-membrane resistance. Ohm’s Law: V=IR. We can also see that the drop in potential
between adjacent points in the cable depends upon the axial resistance, and the change in potential across
arbitrarily small areas of the cable depend upon the current
and the resistance, axially. Finally, we can see that
the change in axial current depends upon how much of that current
leaves across the plasma membrane, and so, the change in axial resistance is
equal to minus the membrane current. We can take these three equations
and solve it for the cable equation in one dimension at steady state. We begin by Ohm’s Law. From this equation we can see that Im can also be written as
minus dI Axial over dx. And so, the membrane potential is
therefore equal to: minus Rm, dI axial by dx. The axial current we can rewrite here as being one over R axial, dv by dx. Differentiating this one more time gives us the second derivative here
with respect to space. We can then put this equation in here, and finally we solve for Voltage
as being the ratio of the membrane resistance
to the axial resistance times the second derivative
with respect to space. This second order
differential equation has a solution with an exponential form with respect to space,
length down the cable, and we introduce now the length constant which is equal to the square root
of the membrane resistance divided by the axial resistance. We can plot the exponential drop-off
of voltage across the length of the cable here as the exponential decay
of the membrane potential as the current traverses
this leaky cable at steady state. The distance where
the potential has dropped to 63% — that is, 1/e — is equal to the length constant
of the cell, Lambda. Lambda defined as the square root of membrane resistance
divided by axial resistance has some obvious intuitive properties. If we were to increase
the membrane resistance, there would be less leak of current
across the length of the cable, and the voltage would therefore
drop less across space. Conversely, if we had
a very high axial resistance, then the amount of current
that would leak out would increase and it would be more difficult
for current to flow down here, and we would therefore get
a more sharp drop in the membrane potential. Lambda would be low. The steady-state cable equation
can be rewritten to include the time-dependent charging
of the membrane capacitance. As we saw last week, there’s a time constant involved
with charging the membrane that’s equal to the product
of the membrane resistance and the membrane capacitance, and that can then be included in
this larger partial differential equation that describes the one-dimensional,
time-dependent spread of voltage across space. We have two important constants: we have a length constant that indicates the length scale over which
the membrane potential decreases, and we have another important constant, the time constant, that tells us over what time scale
the membrane potential is filtered as it traverses the neuronal arborisation
in space and time. It’s worth noting that these constants are
actually variable in time and space. The membrane resistance depends upon
how many ion channels are open
at any given time, and in general, the open probability of ion channels is
something that’s highly regulated, and so, the membrane conductance
and membrane resistance varies considerably over time; in fact, that’s what drives
the membrane potential changes in a cell. The axial resistance may not
change over time, but it certainly changes over space, and so, very thick arbors of the neuron
which have large diameters have low axial resistance, whereas the very thin arborisations
that might be far from a neuron might have much higher axial resistances and so the membrane time constants
are not constant across
the neuronal arborisation in space or time, leading to a great deal of complexity. In general, it’s fair to say
that there are no analytical solutions to the cable equation
for real neuronal structures, and therefore,
numerical computer simulations are therefore typically used, and one that’s particularly used often is the one called Neuron, which is freely available for download
at this World Wide Web address, and you can download it
and play with these simulations and see for yourself
how complicated and interesting the spread of neuronal membrane potential
is within individual neurons. So in terms of
membrane potential distributions, neurons are rather complicated. The membrane potential at one point
in a neuron differs from that in a different point of a neuron. The membrane potential is filtered as current flows down
the neuronal arborisations because there’s some leak
across the plasma membrane. Equally, membrane potential changes are
highly filtered in time. A membrane potential change
that occurs rapidly at a distal part
of a neuronal arborisation may have very little impact
upon the cell soma. In order to get a better feel
for how this works, let’s have a look at some example neurons
and how membrane potential varies across the neuronal arborisation. Let’s first think about membrane area. In the distal processes of a neuron, the diameter might be 0.1 micrometers or one micrometer,
or somewhere in between. The largest diameters are
present at the cell body, where the nucleus
and the genetic material is also located, and there we have diameters
of around ten micrometers. So there’s an approximately
hundredfold difference in the diameter of the structures that are located
at different points across a neuron. The surface area of a cable is approximately proportional
to the diameter of that structure. and the surface area,
of course, also for a capacitor, tells us how much capacitance there is, and so, the local capacitance at the soma might be a hundred times higher than the local capacitance
sitting here in a dendrite. A given amount of current or charge
flowing into a small dendrite here with a small amount of capacitance
might then give rise to a very large voltage change, whereas the same charge
flowing into the cell body will give rise to a smaller change
in membrane potential. One might, therefore, imagine
very large potential differences across time and space here
as conductances open here, compared to slower and smaller
membrane potential changes at the cell body, where there’s
considerably larger filtering due to the large amount of surface area
immediately available to this region. Let’s consider a situation where we have
three recording electrodes placed across the arborisation
of a given neuron. At each electrode we could record
the membrane potential at that location, and let’s see what happens
if we inject current into Location One. Let’s take a given point in time
where we begin to inject current. At recording electrode one,
location one on the dendritic arborisation we’re very close
to where the current is being injected. There’s only a small surface area here. The membrane is rapidly charged
and reaches steady state. Current then begins to flow
and charge the capacitors along the way and gradually, we charge location two and location three. And each time, there’s some delay that increases as the current
needs to flow down here, and the rate of charging slows down
because the overall amount of surface area that needs to be charged has increased. The steady-state voltage that’s reached
at these different locations is described by the steady-state
cable equation as we saw before, where we have our length constant Lambda that tells us how much of a drop
in membrane potential we might expect. Now, in real neurons,
the sizes of this dendritic arborisation might be hundreds of micrometers, and that’s exactly the length scale that
also the membrane length constant has. And so if we imagine
that there’s 500 microns between recording electrode one
and recording electrode three, we can then say that this change
in membrane potential will just be 37% of the steady state membrane potential
reached here at this point in the neuronal arborisation. The length constant of neurons, then being on the same scale
as the arborisation of a neuron, we realize that there’s
a considerable impact upon the steady-state membrane potential
across this neuronal arborisation. The situation, however,
becomes even more extreme when we think about brief current pulses. Again, we consider injecting current
here into electrode one, but now just a brief current pulse. We charge the membrane,
and then it relaxes, because of the current spread
rapidly flowing away from that location. What we see at electrode two, however, is a highly filtered version of this
which is much slower, and electrode three,
if we see anything at all, is again slower and smaller. The half-time of these membrane
potential changes is very different comparing these different locations. The amplitude might be forty times larger at distal dendrites compared to what we see at the soma, and equally, the time course
here, which is fast, will be much more extensive at these
small membrane potential fluctuations. Here at the soma,
we may have around ten milliseconds in terms of the half-width
of membrane potential changes, whereas here, at the distal dendrites,
we probably have time constants that are much closer to one millisecond. When currents and conductances
are opening at different times across different locations
of a neuronal arborisation, you can see that there will be
very complex spatiotemporal dynamics where the fast, large signals
taking place here are strongly filtered, giving rise to smaller amplitude,
longer duration potentials at the soma. Membrane potential is, therefore,
clearly very complex in real neurons. So in this lesson we’ve learned
about the complexity of neuronal membrane potential. Neurons have extensive arborisations
that electrically can be considered as leaky cables
with significant capacitance. Membrane potential, therefore, has
complex spatiotemporal dynamics within single neurons,
with significant attenuation and filtering of the signals
across the arbors. In general, this leads
to a complicated situation in terms of information processing, and next week we’ll see
that the nervous system has come across at least one solution
as to how to make reliable signaling across large distances in the form
of the axon potential. That, we’ll discuss next week.

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