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.